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Modified Gravity and Cosmology
Damianos Iosifidis
Supervisor: Jan Pieter Van der Schaar
June 27, 2013
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Contents
1 Introduction 5
2 Conventions/Notations 6
3 General Relativity 83.1 Einstein Equations via the Einstein-Hilbert action . . . . . . . . 8
4 FLRW Universes 114.1 The FLRW-Model . . . . . . . . . . . . . . . . . . . . . . . . . . 114.2 Robertson-Walker metric . . . . . . . . . . . . . . . . . . . . . . 114.3 Matter in FLRW Universes . . . . . . . . . . . . . . . . . . . . . 124.4 The Friedmann equations . . . . . . . . . . . . . . . . . . . . . . 124.5 Derivation of Friedmann equations by varying with respect to a(t) 154.6 Critical density and density parameter . . . . . . . . . . . . . . . 174.7 Cosmological length-Scales . . . . . . . . . . . . . . . . . . . . . . 19
4.7.1 Particle Horizon . . . . . . . . . . . . . . . . . . . . . . . 194.7.2 Curvature Scale . . . . . . . . . . . . . . . . . . . . . . . . 204.7.3 Hubble Horizon and Wavelengths . . . . . . . . . . . . . . 21
5 The Standard Cosmological Model 215.1 Problems of the Standard Cosmological Model . . . . . . . . . . 235.2 The Horizon problem . . . . . . . . . . . . . . . . . . . . . . . . . 23
5.2.1 The flatness problem . . . . . . . . . . . . . . . . . . . . . 245.2.2 The Magnetic Monopole problem . . . . . . . . . . . . . . 25
5.3 Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.3.1 Inflationary dynamics . . . . . . . . . . . . . . . . . . . . 255.3.2 Slow-roll regime . . . . . . . . . . . . . . . . . . . . . . . 275.3.3 Slow-roll parameters . . . . . . . . . . . . . . . . . . . . . 295.3.4 Number of e− folds . . . . . . . . . . . . . . . . . . . . . 31
5.4 Resolving the problems . . . . . . . . . . . . . . . . . . . . . . . 325.4.1 Resolving horizon problem . . . . . . . . . . . . . . . . . . 325.4.2 Resolving Flatness . . . . . . . . . . . . . . . . . . . . . . 335.4.3 Resolving magnetic monopoles abundance . . . . . . . . . 33
6 Modified Gravity: Why and how? 346.1 An Aside: The 3 formulations of Gravity . . . . . . . . . . . . . . 35
6.1.1 Metric− Theories of Gravity . . . . . . . . . . . . . . . . 366.1.2 Palatini Gravity . . . . . . . . . . . . . . . . . . . . . . . 366.1.3 Metric−Affine Gravity . . . . . . . . . . . . . . . . . . 37
7 f(R)-Theories of Gravity 387.1 Field Equations of f(R)-theories . . . . . . . . . . . . . . . . . . . 387.2 The ’Scalaron’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407.3 Equivalence of Metric f(R) and Brans-Dicke theory for ωBD = 0 . 417.4 Conformal Transformations-Einstein and Jordan frames . . . . . 42
7.4.1 Conformal transformations . . . . . . . . . . . . . . . . . 427.5 Einstein and Jordan frames . . . . . . . . . . . . . . . . . . . . . 437.6 Conformal transformations in FLRW universes . . . . . . . . . . 457.7 Maximally Symmetric Spaces in f(R)-theories . . . . . . . . . . . 46
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7.8 Scalaron degree of freedom and stability . . . . . . . . . . . . . . 477.9 Maximally Symmetric Spaces in the Einstein frame . . . . . . . . 497.10 Extension of the variational technique δa(t), δN(t) to f(R)-theories 507.11 Starobinsky’s model of inflation . . . . . . . . . . . . . . . . . . . 527.12 Dynamics in the Einstein frame . . . . . . . . . . . . . . . . . . . 577.13 Connection with observations . . . . . . . . . . . . . . . . . . . . 627.14 Starobinsky’s model with a Cosmological constant . . . . . . . . 637.15 Theory in the Einstein frame . . . . . . . . . . . . . . . . . . . . 69
8 More General Theories of Modified Gravity 708.1 Field Equations for a general f(R,φ) theory . . . . . . . . . . . . 708.2 Gauss-Bonnet Gravity . . . . . . . . . . . . . . . . . . . . . . . . 728.3 Field Equations for a General f(R,G)-Theory . . . . . . . . . . . 72
8.3.1 Gauss-Bonnet term for a K = 0 FLRW Cosmology . . . . 758.3.2 Modified Friedmann equations for f(R,G) . . . . . . . . . 77
8.4 Field equations of a more general theory . . . . . . . . . . . . . . 78
9 Primordial Black hole Formation 819.1 Pair creation of black holes in Einstein Gravity . . . . . . . . . . 819.2 Pair creation for the theory f(R) = R+ αRn, n 6= 2 . . . . . . . 829.3 Pair creation for the theory f(R) = R+ αR2 . . . . . . . . . . . 83
10 An interesting possibility 85
11 Conclusions/Results 90
12 Appendix 9112.1 Metric Compatibility and the Levi-Civita Connection . . . . . . 9112.2 Definitions and useful identities . . . . . . . . . . . . . . . . . . . 9312.3 Relations between conformally transformed quantities . . . . . . 95
13 Variations 9613.1 Some basic Variations . . . . . . . . . . . . . . . . . . . . . . . . 9613.2 Variation of the ′squared′ Riemann and Ricci tensors . . . . . . 9913.3 General Variations . . . . . . . . . . . . . . . . . . . . . . . . . . 100
14 Calculus of Variations 104
15 Principle of Least Action 10715.1 With dependence on higher derivatives . . . . . . . . . . . . . . . 10815.2 Classical Particle Mechanics . . . . . . . . . . . . . . . . . . . . . 109
16 Definitions and Some Proofs 11016.1 The Weyl/Conformal Tensor . . . . . . . . . . . . . . . . . . . . 11016.2 The generalized Kronecker delta . . . . . . . . . . . . . . . . . . 11216.3 Lovelock Scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . 11216.4 Energy-momentum tensor for a perfect fluid . . . . . . . . . . . . 11316.5 Operators in flat(K = 0) FLRW spacetime . . . . . . . . . . . . . 11416.6 An identity coming from Gauss-Bonnet term . . . . . . . . . . . 11516.7 Derivation of the continuity equation for FLRW universes . . . . 116
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Abstract
The present thesis is a study of Modified Theories of Metric Gravityand their implications to cosmology. We first present the Standard Cosmo-logical Model based on Einstein’s Gravity and then discuss its difficultiesin explaining the late-time accelerated expansion as well as the naturalinclusion of an inflationary era. We then introduce and review some of thebasic theories of modified gravity such as f(R), scalar-tensor-f(R, φ) andGauss-Bonnet-f(R,G) Gravity. We explicitly derive general variations oftensors related to Gravity and subsequently apply these variations to de-rive the field equations for the aforementioned theories. In the case ofFLRW universes, we also have extended to f(R), f(R,G) and f(R, φ) amethod to derive the field equations which evolves variations with respectto the scale factor. Throughout the thesis, special attention is given onf(R)-theories of Gravity and their cosmological consequences. In par-ticular, we review Starobinsky’s model of inflation in both Einstein andJordan frames, and we go on to study this model including a cosmologicalconstant as well. Parametric solutions of the scale factor and cosmic timeare found in this case and the total number of e− folds is also obtainedand compared with the case of a zero cosmological constant. The limitof a small cosmological constant compared to the parameter of Starobin-sky’s model is also studied. In addition, for the latter model we inves-tigate how does the probability of primordial Nariai black holes changecompared to GR(with Λ). Finally, we study a model in which a termproportional to the trace of the energy-momentum tensor has been addedto the Einstein-Hilbert action. In particular, for the latter we derive themodified Friedmann equations along with the new continuity equationand study the possibility of accelerated expansion without introducingnew forms of matter-energy or additional fields.
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1 Introduction
The past century has marked as the golden age of Cosmology. Ground-brakingobservations have made Cosmology a scientific area and not a philosophicalstudy as it used to be. Einstein’s theory has the remarkable property of beingable to be applied on large (cosmological) scales. It is worth noting that GRcould succesfully explain a great part of observations. However, General rela-tivity alone could not fully explain all the observational data and there weremany questions that remained unanswered. Thus, one can imagine that GRmight not be the end of the story. For instance, the late-time accelerated ex-pansion cannot be predicted by Einstein’s theory. In particular, as can be seenby the Friedmann equations, the expansion is always deccelerated for conven-tional matter. Therefore, the modification of Einstein’s theory comes into thegame. In addition, more foundamental theories such as String Theory, predicthigher order terms contributing to the Gravity action when the curvature ishigh. Thus, both observations and theory seem to lead to the conclusion thatGravity should be modified somehow. In the present thesis we study modifiedtheories of Gravity and subsequently arrive at conclusions relevant to Cosmol-ogy. A general formulation of the Variational Principle applied to Gravity isalso given in the thesis. Having the general variations of tensors related toGravity, we apply our results to derive the field equations for a variety of Modi-fied Gravities such as f(R), f(R,G), f(R,φ), etc. Then, we derive the modifiedFriedmann equations for each theory by using an alternative variational methodwhich includes variations with respect to the scale factor. Special attention isgiven on f(R)-theories of Gravity as these are the simplest extensions of Ein-stein’s Gravity, easy to study and provide additional insight to our knowledgeof Gravity. As a particular model of f(R)-Gravity we review and analyticallystudy Starobinsky’s model of inflation. In this model a term quadratic in theRicci scalar is added to the Einstein Hilbert-term. It is rather interesting thatsuch a simply model has such a remarkable agreement with observations. Inaddition, is very economical since it only has one free parameter and makescertain predictions that distinguish it from the other inflationary models. Wethen, go on to include a cosmological constant to the latter model and study itsdynamics. Parametric solutions for the scale factor are found for such a case.In the limit where the cosmological constant is sufficiently small compared tothe Starobinsky parameter we recover Starobinsky’s model as expected. Thelatter model can be seen as a unification model of the early inflationary era(the term ∼ R2 dominates) and the late-time accelerated expansion (the Λ > 0term dominates). For this model we also compute how does the probabilityof Nariai primordial black holes change compared to GR (with a cosmologicalconstant).After doing so, we change subject but still in modified Gravity andderive the field equations for a model that a term proportional to the trace ofthe energy-momentum tensor has been added to the Einstein-Hilbert term. Wethen derive the modified Friedmann equations for such a model and realize thepossibility, under certain conditions, of having accelerated expansion withoutviolating the Strong Energy Condition(ρ+ 3p > 0).
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2 Conventions/Notations
A list of conventions/notations that are used throughout the thesis is givenbelow. We try to be as standard and self-consistent as possible with the con-ventions and there is an indication whenever a different notation is used.
Conventions/Notationsδνµ Kronecker’s deltaεµνρσ Totally antisymmetric Levi-Civita symbolεµνρσ =
√−gεµνρσ Levi-Civita tensor
δµ1µ2...µkν1ν2...νk
Generalized Kronecker deltagµν = gµν(xα) Metric tensor of general spacetimesg = det(gµν) Determinant of the metric tensorTr(Aµν) = gµνAµν Generalization of the trace of a matrix Aµν in curved spacetimesηµν = diag(−1, 1, 1, 1) Minkowski metric tensorΓα
µν Levi-Civita connection (or Christoffel symbols)∇µ Covariant derivative with respect to the Levi-Civita connectionRµ
νρσ Riemann (or curvature) tensorRµν = Rα
µαν = gαβRαµβν Ricci tensorR = gµνRµν Ricci scalar (or curvature scalar)Gµν ≡ Rµν − R
2 gµν Einstein tensorCµνρσ Weyl (or conformal) tensor2 ≡ gµν∇µ∇ν = ∇µ∇µ Box (or D’Alambertian) operator in curved spacetimesDµν ≡ gµν2−∇µ∇ν
uµ 4-velocity fieldS Total actionSM = SM [gµν ,Ψ] Matter part of the actionΨ Collectively denotes matter fieldsL Lagrangian densityLM Lagrangian density of the matterTµν ≡ − 2√
−g
δSM [gµν ,Ψ]δgµν Energy-momentum (or stress-energy) tensor
G Gauss-Bonnet termt Cosmic timeη =
∫dt
a(t) Conformal time˙≡ d
dt Differentiation with respect to cosmic timea(t) Scale factorH(t) ≡ a
a Hubble parameterp(t) Pressure of a perfect fluidρ(t) Density of a perfect fluidρc Critical densityΩ(t) ≡ ρ
ρcDensity parameter
ψ, φ Scalar fieldsN Number of e− foldsε = ε1 First slow-roll parameterη = ε2 Second slow-roll parameterκ = 8πG Gravitational constantG Newton’s Gravitational constantm2
pl = κ−1 Squared Plank mass
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We furthermore adopt natural units, that is, c = 1 = ~. Notice also, thatwe have chosen the convention ηµν = diag(−1, 1, 1, 1) for the Minkowski metrictensor as this is the most common choice made in GR textbooks. As a result,for the Robertson-Walker metric we have1
ds2 = −N2(t)dt2 + a2(t)(
dr2
1−Kr2+ r2(dθ2 + sin2 θdφ2)
)(1)
We denote the symmetric and antisymmetric parts of a tensor with round andsquare brackets respectively. For instance, given a rank-2 tensor Tµν , its sym-metric part is expressed as
T(µν) =12(Tµν + Tνµ) (2)
while the antisymmetric reads
T[µν] =12(Tµν − Tνµ) (3)
As a final remark, we use the standard notation for the indices. Namely, theGreek indices µ, ν, ρ, ... etc. run from 0 to 3 while Latin indices run over thespatial part of spacetime, that is i, j, k, ... = 1, 2, 3.
1One usually adopts the gauge N(t) = 1.
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3 General Relativity
The 20− th century has been one of the most productive centuries in terms ofthe development of fundamental physics. As we know, the two ground-brakingtheories had been developed throughtout that century. These are, of course,General Relativity and Quantum-mechanics. The former describing gravity onscales well greater than the Planck scale (∼ 10−35m) and the latter being ableto explain what happens on atomic scales and high energies. Both of themseem to remarkably describe physics to high accuracy, each of them in its ownscale. However, any attempt of combining the two has failed till now. Quan-tum theories of gravity may give a solution but this of course remains to beseen. Throughout the thesis we concentrate on General Relativity2 and, in par-ticular, modifications/extensions of it, since this theory is applicable on large(cosmological) scales. We review the cosmology arising from GR, explain itssuccesses and discuss its shortcomings. We particularly discuss th inability ofGR to naturally include an inflationary era (without additional fields) alongwith its problems to explain the late-time accelerated expansion. Here we de-rive Einstein equations of General Relativity by using the variational Principleand in subsequent chapter we use this method to derive the field equations fora variety of Modified Gravities.
3.1 Einstein Equations via the Einstein-Hilbert action
As it is known in the scientific history, Hilbert was able to derive the Einsteinequations well before the formulation of General relativity and therefore withoutknowing their great significance. The method used by Hilbert was that of leastaction. More specifically, the variation of the so-called Einstein-Hilbert actionwith respect to the metric tensor yields the Einstein field equations in vacuum.Adding a matter action as well, and varying the total action with respect tothe metric tensor, leads then to the full Einstein equations under the presenceof matter. We outline this method here, for the most straightforward casewhich is the Einstein-Hilbert action, as we are going to be using it further onwhen studying modified theories of Gravity. The Einstein-Hilbert action is thefollowing
SEH =12κ
∫d4x
√−gR (4)
where the factor 1/2κ has been chosen as such, in order to give us the exactEinstein equations after the variation is done. Adding a matter action to thelatter, we have
SGR = SEH + SMatter[gµν ,Ψ] =∫d4x
√−g[ 12κR+ LM
](5)
where ψ collectively denotes the matter fields. By varying the above action withrespect to the metric tensor and using the principle of least action, it followsthat
δSGR = 0 (6)
0 = δSEH + SMatter =∫d4x[ 12κδ(√−gR) +
δ
δgµν(√−gLM )δgµν
](7)
2A nice introduction to GR is given in Carroll’s notes [1].
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Defining now, the energy momentum tensor-Tµν via
Tµν ≡ −2√−g
δ(√−gLM )δgµν
(8)
and using the relations ( which we analytically prove in the appendix)
δ√−g = −1
2√−ggµνδg
µν (9)
δR = (Rµν +Dµν)δgµν (10)
whereDµν ≡ gµν2−∇µ∇ν (11)
and the fact that∫d4x
√−gDµνδg
µν =∫d4x
√−gδgµνDµν1 =
∫d4x
√−gδgµν(gµν2−∇µ∇ν)1 = 0 (12)
we arrive at0 =
∫ √−gδgµν
[ 12κ
(Rµν −
gµν
2R)− 1
2Tµν
](13)
and since in the last one, the variation δgµν is arbitrary, it must hold that
Rµν −gµν
2R = κTµν ⇒ (14)
Gµν = κTµν (15)
where we have defined the Einstein tensor Gµν ≡ Rµν − gµν
2 R. The latter are ofcourse the Einstein Field Equations of Gravity. Notice that we have not includeda cosmological constant. However, this can be done in a straightforward manner.Indeed, adding a term −2Λ into the Gravitational action, the latter becomes
SGR =12κ
∫d4x
√−g[R− 2Λ
]+ SM [gµν ,ΨM ] (16)
And the only additional variation we employ is
δ
∫d4x
√−g(−2Λ) =
∫d4x(−2Λ)δ
√−g =
∫d4x
√−gΛgµνδg
µν (17)
By using again the variational principle we arrive at
Rµν −gµν
2R+ Λgµν = κTµν (18)
which are the Einstein field equation with a cosmological constant. It is worthnoting that depending on which side of the equation we place Λ, we have differentinterpretations of it. When it is placed on the left hand side (as above) it isseen as a geometrical effect. On the other hand, when it is written as
Rµν −gµν
2R = κTµν − Λgµν (19)
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it is interpreted as an energy-matter contribution, oftentimes thought of asvacuum energy. Having Einstein’s equations we can apply them on large scalesand study the cosmology that follows out of them. It is rather remarkable thefact we do have equations that can describe our (observable) universe! In thenext chapter we define the appropriate cosmological principles and quantitiesand then apply them to Einstein’s theory. In subsequent chapters we study thecosmology arising in modified theories of Gravity.
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4 FLRW Universes
4.1 The FLRW-Model
When we observe the universe on large scales3 it seems to be both isotropic andhomogeneous. Isotropy means that it is the same in every direction we maylook at. Homogeneity reflects the fact that the density is the same everywhere.The statement that the universe is both isotropic and homogeneous (on largescales) is called the Cosmological Principle. In a more informal way the latterprinciple expresses the belief that our position in the universe has nothing spe-cial. Any other observer at a different galaxy would observe the same large scalestructure as we do. Thus, our (observable) universe seems to be well describedby the simplest solutions to Einstein’s equations, the isotropic and homogeneousFriedmann− Lemaitre−Robertson−Walker models4.
4.2 Robertson-Walker metric
A mathematical description of an isotropic and homogeneous universe is achievedby the use of the so-called Robertson-Walker metric. Let us now present thismetric here. Firstly, because of the isotropy of the spatial part of spacetime(any direction is alike), no 3 − vector components of the metric are allowed.Therefore, we have
gi0 = g0i = 0 (20)
where i = 1, 2, 3. This is so because any non-zero component gi0 would definea specific direction and therefore would spoil isotropy. Using this, we have forthe line element
ds2 = gµνdxµdxν = g00dt
2 + gijdxidxj + 2g0i︸︷︷︸
=0
dtdxi =
= g00dt2 + gijdx
idxj = −N2(t)dt2 + a2(t)γijdxidxj (21)
where N2(t) ≡ −g00 and a(t) is a time dependent function, the so-called scalefactor, which measures the separation of two points of spacetime at any instantof cosmic time-t. Now, one usually takes the gauge N(t) = 1, which we alsodo here5. In addition it can be shown that the most general spatial part of themetric is given by
γij =δij(
1 + K4 x
kxlδkl
)2 (22)
where K can always be normalized in such a way to take the values −1, 1 or 0.The line element is therefore given by
ds2 = −dt2 + a2(t)δijdx
idxj(1 + K
4 (x2 + y2 + z2))2 (23)
3We consider large scales to be those which are greater than 108-light years. Recall that1-light year is approximately equal to 1016meters.
4See [2] for a mathematical formalism of cosmology.5We should mention though, that when variations with respect to the scale factor are
performed, one should also vary with respect to N(t) and take the gauge N(t) = 1 at the endof the day.
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or, in spherical coordinates
ds2 = −dt2 + a2(t)( dr2
1−Kr2+ r2(dθ2 + sin2 θdφ2)
)(24)
and the metric tensor can be read off to be
gµν = diag
(−1,
a2(t)1−Kr2
, a2(t)r2, a2(t)r2 sin2 θ
)(25)
4.3 Matter in FLRW Universes
The isotropy together with the homogeneity of the FLRW universes restrictshighly the contain of matter that can appear in them. Indeed, the only formof matter that can exist in such universes is that of a perfect fluid. This is sobecause any non-vanishing viscosity would spoil isotropy.. Thus, any fluid livingin FLRW universes has the perfect fluid form
Tµν = (ρ+ p)uµuν + pgµν (26)
where ρ(t) and p(t) are the density and pressure of the isotropic fluid respectivelyand uµ is the 4−velocity of an observer the co-moving with the fluid. Note alsothat ρ and p are functions only of t. This is so because any spatial dependencewould spoil the homogeneity that we demand to have. In the case at hand gµν
is the one given at (25). Note also that uµ is normalized according to
uµuµ = −1 (27)
as can be readily seen by the definition
uµ ≡ dxµ
dτ(28)
wheredτ2 = −ds2 = −gµνdx
µdxν (29)
is the so-called proper time. It is worth noting that in Friedmann universesthe only allowed quantities are scalars with time-dependence (at most). In-deed, this is so because isotropy does not allow any vector-like dependence (thisdefines a specific direction) and only allows the presence of scalars. On theother hand, homogeneity kills any spatial dependence that the given quantitymight had and only leaves the choice of time dependence. In other words, themathematical quantities that live in an isotropic and homogeneous universe aretime-dependent scalars → f(t).
4.4 The Friedmann equations
Up to now, our approach to cosmology was fully theory-independent namelyit holds not only in GR, but also in modified theories of Gravity. Here wefind the Cosmological solutions for Einstein’s theory of Gravity. Let us nowderive the equations that govern the evolution of our universe, the so-calledFriedmann equations. The Friedmann equations are two differential equationsin time, that include the scale factor a(t), the density ρ(t) and the pressure p(t)
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of the fluid. The latter two equations together with a third one, the so-called thecontinuity equation form a set of 3 equations (two of them are independent) with6 3 unknown dynamical variables (a(t), ρ(t) and p(t)) which when found fullydescribe our system. To find the Friedmann equations we firstly use the form ofthe Ricci tensor computed for the Robertson-Walker metric, the non-vanishingcomponents of which are
Rij =
[a
a+ 2
(a
a
)2
+ 2K
a2
]gij , R00 = −3
a
a(30)
We then find the Ricci tensor to be
R = gµνRµν =6a2
[aa+ a2 +K
](31)
As long as the matter part is concerned, taking into account that for a co-movingobserver it holds that
uµ = δµ0 = (1, 0, 0, 0) (32)
we have, for the non-vanishing components of the energy momentum tensor,
Tij = pgij , T00 = ρ (33)
Then, from the Einstein field equations (with a cosmological constant)
Rµν −R
2gµν = κTµν − Λgµν (34)
taking the 00-component and using all the above along with g00 = −1, we have
−3a
a+ 3
a
a+ 3
[(a
a
)2
+K
a2
]= κρ+ Λ ⇒
(a
a
)2
=κ
3ρ+
Λ3− K
a2(35)
This is the first of the Friedmann equations. The second one follows by takingthe ij-components of the Einstein equations, leading to[
−2a
a−(a
a
)2
− K
a2
]gij = (κp− Λ)gij ⇒
2a
a+(a
a
)2
= −κp+ Λ− K
a2(36)
The latter is the second Friedmann equation. When Λ = 0, depending on thevalue of Kone distinguishes three case. The case K = 0 corresponds to a flatopen universe. When K = +1 one has a closed positively curved universe,while for K = −1 we get a negatively curved open universe. We will see how do
6Notice, however, that one usually imposes a barotropic fluid relation-p = wρ betweenpressure and density, where w is, in general, a time-dependent quantity. The term barotropicin fluid dynamics, refers to that fluid for which the pressure is only a function of the energydensity→ p = p(ρ). This equation is found by thermodynamical considerations and is used toclose the system of equations.
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these cases arise in a more formal way when we introduce the so-called densityparameter-Ω(t). Now, subtracting the two Friedmann equations, one can formwhat is known as Raychaudhuri or acceleration (due to a) equation, which isthe following
a
a= −κ
6(ρ+ 3p) +
Λ3
(37)
From the above we see that the acceleration/deceleration of the universe isindependent of the curvature (the term that contains K has dropped out) andwhen Λ = 0 depends only on the matter constituents (ρ and p) of the universe.It is also apparent from the latter that for conventional matter it holds that(ρ+ 3p) > 0 and one always has
a
a∝ −(ρ+ 3p) < 0 ⇒ a < 0 (38)
Thus, accelerated expansion cannot be realized with ordinary matter and oneneeds to introduce new forms of matter-energy in order to reproduce such anexpansion. On the other hand, as can be seen from (37) when Λ > 0 acceleratedexpansion can occur if
Λ >κ
2(ρ+ 3p) (39)
One needs one more equation to close the system.7 This comes from the energy-momentum conservation
∇µTµν = 0 (40)
which is a direct consequence of the Bianchi identities. Note that this can alsobe proved by using the diffeomorphism invariance of the matter action. As weprove in the appendix, for Friedmann universes one has
∇µTµν = −δ0ν[ρ+ 3H(ρ+ p)
](41)
and thus, combining the above two, for ν = 0
ρ+ 3H(ρ+ p) = 0 (42)
This is known as the continuity equation. Note also that we have defined thequantity H(t), through
H(t) ≡ a
a=d ln adt
(43)
which is called the Hubble parameter and gives the expansion rate of the uni-verse. This is a generalization (time-dependent) of the Hubble parameter-H0
appearing in Hubble’s velocity law for distant galaxies
vi = H0ri (44)
where ri is the distance of the observed galaxy from us and v its velocity withrespect to our galaxy. The value of the Hubble parameter H0 is estimated , by
7However, from the two Friedmann equations and the continuity equation only two of themare independent. For instance, by taking the time derivative of the 1st Friedmann equationand upon using the continuity equation we arrive at the 2nd Friedmann equation. In fact,the system closes upon imposing a relation between ρ and p, following by thermodynamicalconsiderations, as we have already mentioned.
14
observations, to be H0 ∼ 2 · 10−18sec. From the latter one can estimate the ageof the universe
t0 ∼1H0
∼ 4 · 1017sec ∼ 13.8 billion years! (45)
This gives a lower bound on universe’s age since in arriving at the above pro-portionality we assumed conventional matter (for instance dust) dominancethroughout all eras. A more complicated analysis can be used in order to ar-rive at a more accurate resulta, however the order of magnitude is in perfectagreement with observations which indictate a an age of 13, 9−14-billion years.More information on Friedmann models and cosmology can be found in the nicereview [3]. In what follows we give another method on deriving the Friedmannequations and then discuss the current model of Cosmology.
4.5 Derivation of Friedmann equations by varying withrespect to a(t)
As it is known, for the case of FLRW universes one can obtain the Friedmannequations by writing the total action as a functional of the scale factor andthen vary with respect to it. A second variation with respect to N(t) must alsobe done in order to arrive the full equations. The take N(t) = 1 can then betaken in the end of the day. We prove this here for a flat (K = 0) Friedmanncosmology and in subsequent chapter we extend this method and derive themodified Friedmann equations for a general f(R)-Theory. Now, using
R =6a2
[aa+ a2
],√−g =
√−detgµν = a3 (46)
andTij = pa2δij , T00 = ρ (47)
the action of General Relativity becomes
S =∫d4x
√−g[ 12κR+ LM
]=
=∫d4x[ 3κ
(a2a+ aa2) +√−gLM
](48)
By varying the above with respect to the scale factor we obtain
δaS = 0 ⇒
0 =∫d4x[ 3κ
(2aaδa− 2a2aδa+ a2δa+ 2a2aδa) +δ(√−gLM )δa
δa]
=
=∫d4x[ 3κ
(2aaδa+ a2δa) +δ(√−gLM )δa
δa]
(49)
where we have integrated by parts and dropped the surface term that appeared.Now, using the chain rule
δ(√−gLM )δa
=δ(√−gLM )δgµν
δgµν
δa(50)
15
and alsoδg00
δa= 0 ,
δgij
δa= − 2
a3δij (51)
we arrive at0 =
∫d4x[ 3κ
(2aa+ a2) + Tijδij]δa (52)
and since the last must hold for any variation δa we conclude that
3κ
(2aa+ a2) + Tijδij = 0 (53)
butTijδ
ij = pa2δijδij = 3pa2 (54)
and therefore2a
a+( aa
)2
= −κp = −8πGp (55)
which is one of the Friedmann equations as stated. Now, in order to obtain the2nd Friedmann equation we start by the original form of a flat Robertson-Walkermetric
ds2 = −N2(t)dt2 + a2(t)(dx2 + dy2 + dz2) (56)
and we only take the gauge N(t) = 1 after the variation with respect to N(t)has been performed. In this case, one has
√−g = a3N (57)
as well asδg00
δN=
2N3
(58)
δgij
δN= 0 (59)
The Ricci scalar will then be
R[N, a] = gµνRµν = g00R00 + gijRij = − 1N2(t)
R00 + gijRij (60)
such that
δR
δN=
δ
δN
(− 1N2(t)
R00 + gijRij
)=
2N3
R00 + 0 ⇒
δR
δN=
2N3
R00 (61)
In addition, the variation of the matter part of the action reads
δ(√−gLM )δN
=(− 2√
−gδ(√−gLM )δgµν
)︸ ︷︷ ︸
≡Tµν
δgµν
δN
(− a3N
2
)=
= −a3N
22N3
T00 = −a3ρ
N2⇒
δ(√−gLM )δN
= −a3ρ
N2(62)
16
where we have also employed the chain rule. Varying the total action it followsthat
0 = δNS[a,N ] =
= δN
∫d4x
(a3N
2κR[a,N ] +
√−gLM
)=
=∫d4x
((δN)a3R+ a3N
δR
δNδN +
(√−gLM )δN
δN
)=
=∫d4xa3(δN)
(R[N, a] +
2N2
R00 −2κN2
ρ
)(63)
and since the last one must hold true for any variation δN , we conclude that
R[N, a] +2N2
R00 −2κN2
ρ = 0 (64)
We can now take the gauge N(t) = 1 to arrive at
R+ 2R00 = 2κρ⇒
6
[a
a+(a
a
)2]− 6
a
a= 2κρ⇒
6(a
a
)2
= 2κρ⇒(a
a
)2
=κ
3ρ (65)
which is the 2nd Friedmann equation. This is a nice alternative way of derivingthe field equations in the case of an FLRW background. In a later chapter, weextend this method to obtain the modified Friedmann equations in the case off(R) as well as f(R,G) -Gravities.
4.6 Critical density and density parameter
As we have already stated, for Λ = 0 a spatially flat geometry corresponds toK = 0. In this case the first Friedmann equation reads
H2 =κ
3ρc (66)
This means that a universe is flat exactly then when its density satisfies
ρc(t) =3H2(t)κ
(67)
where ρc is called critical density. For a given universe we then define thedensity parameter-Ω(t) to be the ratio of the density of the universe over thecritical density, namely the one for which the universe is flat. In mathematics
Ω ≡ ρ
ρc(68)
17
where ρ is the density of the given universe. Note that when ρ = ρc one hasΩ = 1, while for ρ > ρc and ρ < ρc we have Ω > 1 and Ω < 1 respectively.Using the above definition we may write the first Friedmann equation as
H2 =κ
3ρ− K
a2⇒
1 = Ω− K
a2H2⇒
Ω− 1 =K
(aH)2(69)
From which is clear that K = 0 corresponds to Ω = 1, while for K = 1 one hasΩ > 1 and for K = −1 → Ω < 1. Thus, we can distinguish the following threecases.Case 1: Ω = 1
In this case the total density of the universe exactly equalizes the criticaldensity and a flat universe follows. For such a configuration the density param-eter remains constant throughtout the whole evolution of the universe. Indeed,taking the time derivative of Ω and upon using Friedmann equations one finds
Ω = −(1 + 3w)(1− Ω)ΩH (70)
where we have assumed the presence of a barotropic fluid, obeying p = wρ. Thelatter differential equation has two fixed points8
Ω∗ = 0
andΩ∗ = 1
The former describes an empty universe and the latter is the relevant one toour discusion. We see that since Ω = 1 is a fixed point, a universe that startswith that value of the density parameter continues to retain it during its wholeevolution. Therefore, a flat universe Ω = 1 (or equivalently K = 0) remains flatthroughout cosmic evolution. In this case on has an ever-expanding universewith decreasing expansion rate (for conventional matter).Case 2: Ω > 1
Let us suppose that we start off we a density parameter which is greater thanone (Ω0 > 1), that is, the density in such a universe exceeds ρc. For conventionalmatter (1+3w) > 0 and for the above initial condition (1−Ω0 < 0) we see thatone has
Ω > 0
at all times. This indicates that the effects of a closed geometry become moreimportant as timeCase 3: Ω < 1
In exactly the same way as before, but now assuming an initial condition1− Ω0 > 0 we now arrive at
Ω < 08Recall that for a system of first order differential equations ~x = ~F (~x), the fixed points-~x∗
are determined by solving ~x∗ = ~0 ⇒ ~F (~x∗) = ~0.
18
at all times. Thus, we see that a negatively curved geometry becomes more andmore important with the evolution of time. As a result we see that startingwith a geometry that is not flat (either positively or negatively curved) thevalue of Ω goes away from one as time goes on. We can therefore conclude thatthe effects of non-flat geometries become stronger and stronger at late times.In our universe, that was of course a problem that needed explanation. Moreaccurately, since today the universe appears to be flat to a good approximation,this means that at earlier times (for conventional matter) was even more flat (Ωwas closer to unity), which implies immediately some fine tuning. As we willsee in what follows inflation can give an explanation to the above problem. Letus continue now by giving some useful quantities which are used when studyingcosmology.
4.7 Cosmological length-Scales
There are a number of interesting cosmological length-scales of physical signif-icance, that are used in order to extract useful results concerning cosmology.Here we give the definitions of the most important among them and discusstheir physical meanings in each case. Note that since we have set the speed oflight c = 1, the length scales we discuss below are also expressed as time scaleswhenever this is convenient.
4.7.1 Particle Horizon
The belief that our universe has a finite age, together with fact that the lightpropagates with a finite and constant speed, lead to the fact that there exists aradius between which the regions can affect each other and out of which cannot.We call these, regions of casual connection and the latter radius is known asparticle horizon. Given a spacetime event in an FLRW universe, events thattake place at distances greater than that of particle horizon cannot affect andbe affected by this given event. Let us give now the definition of the particlehorizon in mathematical terms. We start by noting that light travels along nullgeodesics (because of its zero mass), namely
ds2 = 0 (71)
One then, can always fix the angles θ and φ to constant values (so that dθ =0 = dφ),and the Robertson-Walker metric is written as
0 = ds2 = −dt2 + a2(t)dr2
1−Kr2⇒
∫ r(t)
0
dr√1−Kr2
=∫ t
0
dt′
a(t′)(72)
Now, we state the left hand side of the above is the co-moving distance traveledby the photon till cosmic time t. Indeed, from the form of the Robertson-Walkermetric
ds2 = −dt2 + a2(t)dσ2 (73)
and focusing on the co-moving spatial part of the above metric we have
dσ2 =dr2
1−Kr2+ r2(dθ2 + sin2 θdφ2) (74)
19
and by taking a radial trajectory (θ = const.,φ = const.)
dσ2 =dr2
1−Kr2(75)
which when integrated gives us the co-moving distance
σ(t) =∫ r(t)
0
dr√1−Kr2
(76)
Now, in order to get the so-called physical (or proper) distance we multiplythe result with the scale factor at cosmic time t. Taking all into account, theparticle horizon will be
dp(t) ≡ a(t)∫ r(t)
0
dr√1−Kr2
= a(t)∫ t
0
dt′
a(t′)(77)
Using the fact that
H =dH
dt=dH
da
da
dt=dH
daa =
dH
daaH ⇒
dt =da
aH(78)
,where the chain rule has been used, the particle horizon is oftentimes given inthe form
dp(t) = a(t)∫ a(t)
0
da
a2H(a)(79)
where H is seen here as a function of the scale factor. To recap, the particlehorizon defines the size of the largest casually connected region, namely thedistance that a photon has traveled after the big bang. Since nothing can travelfaster than the speed of light, regions that are separated at a distance greaterthan the particle horizon are not in casual connection and cannot affect eachother.
4.7.2 Curvature Scale
As it is known from basic manifold analysis, any curved space can locally beapproximated by flat space. In mathematical terms, a small region around apoint on a smooth surface can be approximated by the tangent plane of themanifold at this point. We can also see this in our everyday life. Even thoughthe surface of the earth is, to a good approximation, spherical when it comes tomeasure distances we use straight rulers which reflects the fact that we live ina locally Euclidean space. Indeed, this is true because compared to the radiusof the earth (around 6.500 km ) our everyday distances (which vary from somemeters to at most kilometers) can be considered negligible. The curvature radiusof the earth is much bigger than our everyday scales. However, when travelingwith a plane from ,say, Australia to Netherlands one must take into accountthe curvature of the earth. The same argument holds for curved spacetimes.In particular, for FLRW universes the distance beyond which the curvature of
20
space can no longer be ignored is set by the scale factor itself. To be moreprecise, the curvature scale is given by
λK(t) =a(t)|K|
(80)
from which we see that, for positively and negatively curved universes (|K| =1) the distance beyond which phenomena associated with curvature becomeimportant is a(t). On the other hand, for flat (K = 0) spacetimes the curvaturephenomena appear at spatial infinity, which is another way of saying, there isno curvature.
4.7.3 Hubble Horizon and Wavelengths
Another quantity of interest is the so-called Hubble-horizon or Hubble-radius.The Hubble-horizon gives an estimation for the time9 required for the variousphysical phenomena to be developed. Its definition is the following
λH(t) =1
H(t)(81)
Now, the evolution of the universe influences all physical quantities that havelength scales. In particular, because of the expansion/contraction of the uni-verse, all the wavelengths must evolve in a way that is proportional to thescale factor. Given a wavelength λk , its evolution ( stretching/contraction) isdescribed by
λk(t) =a(t)k
(82)
where k is the wavenumber of the given wavelength.
5 The Standard Cosmological Model
It is worth noting that our observable universe seems to be well, to some ex-tend, described by the simplest solutions to Einstein equations, the isotropicand homogeneous Friedmann − Lemaitre − Robertson − Walker (FLRW)-Models. The logical belief that the Cosmological Principle holds true, namelythe fact that on large scales the universe appears to be both homogeneous andisotropic leads to the adoption of a metric that meets these demands, the so-called Robertson −Walker metric. The form of this metric also restricts thematter content that can appear in such universes. The latter has the form ofa perfect fluid. Taking all the above into account, and by using Einstein equa-tions one arrives to the 2 differential equations that describe the behaviour ofour universe; the Friedmann equations. A space that is both homogeneous andisotropic has only one choice, to expand in dimensions. This fact is capturedin the scale factor-a(t) which in fact gives the evolution of the universe whenis found by solving the Friedmann equations. The Standard Big-Bang Cos-mological Model consists of a sequence of eras, during which the universe wasdominated by different forms of matter/energy.
9Remember that we have set c = 1 and therefore times scales can be seen as length scales.
21
A trip into universe’s life from the present all the way down to the begin-ning would be as follows. Today, almost 13.8-billion years (∼ 1017sec) afterthe Big-Bang, there is a mysterious fluid (or field) driving our universe into anaccelerated expansion[4]. As it can be seen from the Friedmann equations, forconventional matter (ρ + 3p > 0) accelerated expansion is not a choice10. Auniverse that is dominated by conventional matter always experiences deceler-ated expansion. Thus, one attributes this behaviour (accelerated expansion)due to some unknown form of energy, yet to be discovered. The latter is, nowa-days, called dark energy. The term dark here is used in order to indicate ourignorance about it.
Going back nearly 2 billion years from today, the dominant componentof the universe is now pressure-less (p = 0) matter referred to as dust. Thisconsists of particles (or even galaxies) that have velocities much smaller than thespeed of light11. This phase of the universe is known as matter-era, and for aflat spatial geometry, during this era the scale factor and the Hubble parameterevolve as
a ∝ t2/3 (83)
H =23t
(84)
with the matter density obeying
ρd =4
3t2(85)
where the subscript d stands for dust12. It is remarkable the fact that theend of the radiation era leaves a blueprint that can actually be observed today.This is the so-called Cosmic-Microwave-Background (CMB) radiation. It is themoment when the first atoms were formed and the radiation observed is the onecoming from the quantum jumps (jumps of electrons from exited states to theground state in atoms) of the electrons when those atoms where formed.
As we go even more backwards in time, the universe gets hotter and hotterand there is a time (around teq ∼ 104 years) when the dust becomes comparablewith what is known as radiation. Beyond that time, the dominant constituentsof the universe are photons along with high-energy particles (kBT mc2)which are collectively referred to as radiation13. For a flat (K = 0) geometry,a radiation dominated universe behaves as
a ∝ t1/2 (86)
H =12t
(87)
as well asρr =
34κ
1t2
(88)
where r indicates radiation. It is only then, earlier than the radiation era inthe expansion history of the universe, when inflation is assumed to occur. The
10However, by modifying Gravity accelerated expansion can be realized for conventionalmatter [5],[6], [7]
11More accurately, it holds that kBT mc212Note that the notation ρm is also frequently used in the bibliography, where the subscript
m here stands for matter.13The barotropic equation for radiation is p = ρ/3.
22
time that inflation starts is believed to be ∼ 10−36sec and lasts till 10−34secafter the Big-Bang. We should mention here that inflation is far from being afundamental and concrete theory. It is merely a paradigm of how the thingsin the early universe had might be evolved. This is so because one can alwaysadjust the free parameters of inflation so to solve the problems of the StandardCosmological Model we want it to solve. Furthermore, inflation itself brings itsown problems and rises new questions that need to be answered. Nevertheless,the development of physics allows one to make conjectures about how doesthe universe behaves after ∼ 10−36sec14. However, inflation does provide wellformulated and simple mechanisms that can address many of the puzzles ofStandard Cosmology.
If one wishes to go back beyond ∼ 10−43sec then all the theories thathave been constructed so-far, brake down. This period is called the Plank era.As it is known, given a length scale l and a mass scale m the quantum effectsbecome important when the size of the former is comparable with the Compton-wavelength. More accurately, quantum effects cannot be ignored as long as
l < λQ (89)
whereλQ =
~mc
(90)
Hence, in the early stage of universe’s life there exists a time when the latterhad a size comparable to the Compton wavelength and quantum effects wereessential. There is no viable theory that can be used in order to explore thePlank era. Nevertheless, Quantum theories of Gravity are developing and theremight be a significant progress in the near future.
5.1 Problems of the Standard Cosmological Model
The Standard Big-Bang model of cosmology, even though it could successfullyexplain many of the observational data, was plagued by three basic problems.We discuss these problems in detail below.
5.2 The Horizon problem
Because of universe’s expansion/contraction any physical length-scale l evolvesin a way proportional to the scale factor (stretching of spacetime)
l ∝ a(t) (91)
For a flat FLRW universe filled with conventional barotropic matter (ρ+3p > 0,p = wρ) the scale factor goes as
a(t) ∝ tn (92)
where n = 23(1+w) < 1. Therefore, for a given scale l one has
l ∝ tn (93)
14These are merely representative, the exact starting and ending of inflation are theory-dependent.
23
Now, for such a model the particle (or cosmic) horizon is almost identified withthe Hubble horizon
dp ≈ λH ∝ t (94)
where
dp = a(t)∫ t
0
dt′
a(t′)=∝ tn
∫ t
0
dt′
(t′)n∝ t (95)
andλH =
1H
=t
n∝ t (96)
the particle and Hubble horizons respectively. Thus, it follows that
l
dp∝ tn−1 (97)
and since n − 1 < 0 we see that the ratio l/dp decreases with time. This inturn means that regions that are in casual connection today were out of eachother’s particle horizon at earlier times. As a result, antipodal points on thesky that are in casual connection today were not in the far past. Thus, theseregions were separate, could not affect each other and were not in equilibrium.This contradicts the observations of the CMB where the isotropy is apparent toa large degree. The latter is known as the Horizon Problem.
5.2.1 The flatness problem
It is rather astonishing the fact that the density parameter observed today isvery close to unity (Ω0 ≈ 1). The latter implies a universe with spatial geometrythat is almost Euclidean. This observation along with the fact that in Friedmannmodels the impact of geometry increases with time, lead to the conclusion thatin the past the density parameter was even closer to unity. To be more specific,using appropriate combinations of the Friedmann equations, it follows that
dΩ(1− Ω)Ω
= −(1 + 3w)da
a(98)
which can be directly integrated, to give
|1− Ω|Ω
∝ a1+3w (99)
In addition, using the fact that the dominant contribution to the temperaturecomes from radiation (w = 1/3), for which we know that
ρr ∝ T 4 (100)
but alsoρr ∝ a−4 (101)
such thata ∝ 1
T(102)
Substituting the above into (699) we arrive at (upon setting w = 1/3)
Ω0
|1− Ω0||1− Ω|
Ω=(T0
T
)1+3w
=(T0
T
)2
(103)
24
where the subscript 0 indicates present values. Recent observations reveal that
|1− Ω0| ∼ 10−2 (104)
From the CMB we also know that T0 ∼ 2.7K ∼ 10−13GeV , and applying theabove equation at Plank’s time (∼ 10−44sec) we conclude that
|1− Ωpl| ∼ 10−66 (105)
where we have used that Tpl ∼ 1019GeV . Given the fact that our universe isdescribed by an FLRW model it then follows that in the end of Plank era thelatter appears to have a geometry that is to a high accuracy (10−66) Euclidean.Such a small curvature immediately rises the question of fine -tuned initialconditions. Why did the universe have such a small curvature to begin with?
5.2.2 The Magnetic Monopole problem
The Standard Model of Particle Physics predicts the existence of magneticmonopoles. These are isolated magnetic charges and can be created in abun-dance when the temperature of the universe exceeds the value of ∼ 1028K.According to the theory, magnetic monopoles appear to be stable and becauseof their big mass (∼ 10−9gr) they should be dominant. The problem is thatmagnetic monopoles have never been observed.
5.3 Inflation
The Standard Cosmological Model, when constructed, was able to successfullyexplain most of the observational data. However, there are a number observa-tions that the latter cannot address. The most important among these are theflatness,the horizon and the magnetic monopole problems. In order for theseproblems to be solved, an idea called inflation had been proposed in 80′s byGuth [8], and later on revisited by Linde [9]. Similar ideas also appeared fewyears ago in a paper of Kazanas [10]15. The inflationary paradigm goes as fol-lows. There exists a time in the early universe, around 10−36 − 10−34 sec afterthe Big-Bang, when the universe experienced accelerated expansion, in partic-ular quasi-exponential. This expansion occurred because of the existence (anddominance over the other forms of matter) of a scalar field-φ dubbed inflaton.Inflation is driven by that field. Such an expansion can, in principle, resolve theaforementioned problems and in addition can generate perturbations, in a log-ical manner, that afterwards develop and give the present large scale structureof the universe. Introduction to inflation can be found for instance in [12], [1]and [9]. In what follows we study the dynamics of inflation and then see howdoes inflation intend to solve the basic problems of the Standard CosmologicalModel.
5.3.1 Inflationary dynamics
Let us study now the dynamics of Inflation. The first assumption of the Infla-tionary paradigm is that there exists a scalar field-φ, called inflaton, which is
15See also [11] for a recent personal perspective on inflation by the same author author.
25
minimally coupled to gravity16. The second assumption is that for some reasonthis field was dominating over the other forms of matter-energy in the earlyuniverse and exactly at that period was slowly-rolling17. To study the dynam-ics now, we consider a pseudo-Riemannian spacetime filled with this inflatonfield-φ, which is minimally coupled to Gravity. Then, the Lagrangian densityof the field will be
Lφ = −[12∇αφ∇αφ+ V (φ)
](106)
Then, in the context of Einstein’s Gravity, the total action reads18
S =12κ
∫d4x
√−gR+ SM [gµν , φ] =
∫d4x
√−g[
12κR+ Lφ
]⇒
S =∫d4x
√−g[
12κR− 1
2∇αφ∇αφ− V (φ)
]=
=∫d4x
√−g[
12κR− 1
2gµν∇µφ∇νφ− V (φ)
](107)
The latter needs to be varied independently with respect to the metric tensoras well as the scalar field. Varying with respect to the metric tensor, we have
δgS = 0 =
=∫d4x
[12κ
δ(√−gR)δgµν
δgµν − (δgµν)∇µφ∇νφ
2√−g − 1
2gµν(δgµν)
√−gLφ
](108)
but,as we prove in the appendix
δ(√−gR)δgµν
δgµν =(Rµν −
R
2gµν
)√−gδgµν (109)
Using this, the above variation reads
0 =∫d4x
√−gδgµν
[(Rµν −
R
2gµν
)− κ(∇µφ∇νφ+ gµνLφ
)](110)
and we see that by identifying
T (φ)µν ≡ ∇µφ∇νφ+ gµνLφ = ∇µφ∇νφ− gµν
(12∇αφ∇αφ+ V (φ)
)(111)
we arrive at the field equations
Rµν −R
2gµν = κT (φ)
µν (112)
Note also that as a direct consequence of the Bianchi identities, one has
∇µ
(Rµν −
R
2gµν
)= 0 (113)
16Minimally coupled means that the only coupling of the field with Gravity is through the√−g (because it is contained in the volume element) term. Namely, there are no terms such
as Rφ, etc., contained in the action.17We define what we mean by slow-roll in subsequent chapter.18Since the dominant component is the scalar field, any other form of matter that could
appear has been neglected.
26
which results in∇µT (φ)
µν = 0 (114)
and therefore we conclude that the energy-momentum tensor of the scalar fieldis covariantly conserved. Now, in order to close the system of the field equations,we vary the action with respect to the inflaton field-φ, to get
δφS = 0 ⇒
0 = δφ
∫d4x
√−g[
12κR− 1
2∇µφ∇µφ− V (φ)
]⇒
0 = δφ
∫d4x
√−g[
12κR− 1
2∂µφ∂µφ− V (φ)
]⇒
0 =∫d4x
√−g[−1
22∂µφδ(∂µφ)− V,φδφ
]⇒
0 =∫d4x
√−g[− ∂µφ∂µ(δφ)− V,φδφ
]⇒
0 =∫d4x
√−g[−∇µφ∇µ(δφ)− V,φδφ
]⇒
0 = −∫d4x
√−g∇µ(δφ∇µφ)︸ ︷︷ ︸
=surface term
+∫d4x
√−gδφ
[∇µ∇µφ− V,φ
]⇒
0 =∫d4x
√−g(δφ)
[−2φ+ V,φ
](115)
where we have used the fact that the covariant derivative is reduced to thepartial one when acting on scalars, along with the commutativity property of thevariation δ with the partial derivatives ∂µ. Note also, that a surface term thatappeared has been dropped since it does not modify the equations of motion.Now, since the latter must hold true for any arbitrary variation δφ, we concludethat
2φ− V,φ = 0 (116)
which is the propagation equation for the scalar field φ. To recap, the full fieldequations are
Rµν −R
2gµν = κT (φ)
µν (117)
2φ− V,φ = 0 (118)
whereT (φ)
µν ≡ ∇µφ∇νφ− gµν
(12∇αφ∇αφ+ V (φ)
)(119)
We should mention here that all the above hold true for any spacetime andscalar field φ. To describe the inflationary era we should also demand an FLRWbackground together with the slow-roll conditions on φ. This is done in thefollowing subsection.
5.3.2 Slow-roll regime
In the previous subsection we derived the field equations for Einstein Grav-ity, where a scalar field was also present in the spacetime (general) and which
27
was minimally coupled to Gravity. However, no assumptions (apart from itsdominance) about our scalar field have been made and, in addition, the space-time had left unspecified. Here, we assume an FLRW background, namely aspacetime described by the Robertson-Walker metric
ds2 = −dt2 + a2(t)( dr2
1−Kr2+ r2(dθ2 + sin2 θdφ2)
)(120)
where K is normalized to take the values K = 0,±1. This in turn, as wehave already seen, restricts the form of the matter that can be present in thisspacetime. The latter can only have the form of a perfect fluid, with the energy-momentum tensor given by
T (φ)µν = (ρ+ p)uµuν + pgµν (121)
Additionally, the homogeneity and isotropy of our spacetime restrict the coor-dinate dependence of the various quantities that live in it. To be more specific,every quantity in the FLRW spacetime can only have time dependence (at most).Therefore, our scalar field φ does not have much of a choice, it can only dependon time φ(xα) → φ(t). Then, one has
∇µφ = ∂µφ = δ0µφ = −uµφ (122)
where uµ = (1, 0, 0, 0) = δµ0 is the 4− velocity of a comoving observer, which is
normalized according touµu
µ = −1 (123)
from which it also follows that
uµ = (−1, 0, 0, 0) = −δ0µ (124)
Upon substituting the above results into (111), we obtain
T (φ)µν = (−uµφ)(−uν φ)−
12uαu
α︸ ︷︷ ︸=−1
+V (φ)
gµν ⇒
T (φ)µν = φ2uµuν +
(12φ2 − V (φ)
)(125)
The comparison of the last with the perfect fluid energy momentum tensor of(121) reveals
ρ+ p = φ2 (126)
p =12φ2 − V (φ) (127)
From which we can express the density and the pressure in terms of the inflatonfield as
ρ =φ2
2+ V (φ) (128)
p =12φ2 − V (φ) (129)
As it can be seen from the latter one, when the potential term exceeds thekinetic term (1/2)φ2 one has negative pressure. Thus, one can in principle adjust
28
the field so to achieve (ρ + 3p) < 0 for some negative pressure configuration.This will yield an accelerated expansion a > 0 as can be readily seen from theRaychaudhuri equation
a
a= −κ
6(ρ+ 3p) > 0 (130)
for (ρ + 3p) < 0. Here comes the slow-roll assumption. Suppose there exists aregime in which the field is slowly varying and the potential is nearly constant,say V0. Then, the following will hold
12φ2 ± V (φ) ≈ ±V (φ) ≈ ±V0 = const. (131)
and as a result
ρ =φ2
2+ V (φ) ≈ V (φ) ≈ V0 (132)
p =12φ2 − V (φ) ≈ −V (φ) ≈ −V0 (133)
so thatp ≈ −ρ (134)
which will giveρ+ 3p ≈ −2ρ < 0 (135)
and therefore accelerated expansion (a > 0). There are various inflationarymodels each one with a specific potential and therefore different predictions,in general. However, in all inflationary models the scale factor evolves quasi-exponential in the end of inflation. This is the mechanism of inflation. Let usrecap by mentioning again the crucial assumptions of the inflationary paradigm.Firstly, it is assumed that there exists a scalar field φ (inflaton) that is minimallycoupled to gravity and dominates over the other forms of matter-energy in theearly universe. Secondly, during its dominance the field is slowly varying and hasa (nearly) flat potential. Then, the scalar field can be seen as a barotropic fluid(p = wρ) corresponding to w ≈ −1, which has negative pressure. Such a fluidviolates the strong energy condition, ρ + 3p > 0 and results in an acceleratedexpansion a > 0. Finally, the scalar field exits the slow-roll regime and oneenters the radiation era19 of the Standard Cosmological Model.
5.3.3 Slow-roll parameters
As we have seen, a scalar field that is minimally coupled to Gravity , for FLRWuniverses can be thought of as a perfect fluid with density and pressure that aregiven by
ρ =12φ2 + V (φ) (136)
p =12φ2 − V (φ) (137)
19Actually, after the slow-roll regime the field is assumed to reach the bottom of the potentialand starts oscillations around it. This is a mechanism for particle production and is calledreheating. The whole procedure of reheating is not yet fully understood. In addition theduration of reheating is small enough and is not considered to be an extra era of universe’slife.
29
respectively. Then, the first of the Friedmann equation’s is written as20
H2 =κ
3ρ =
κ
3
[12φ2 + V (φ)
](138)
In addition, using the fact that for FLRW universes the action of the box-operator on scalars is given by
2f = −[ d2
dt2+ 3H
d
dt
]f (139)
The propagation equation (118) of the inflaton is written as
φ+ 3Hφ+ V,φ = 0 (140)
Now, to be more accurate, there are two slow-roll conditions. The slow rollcondition we saw before
12φ2 ± V (φ) ≈ ±V (φ) ≈ ±V0 = const. (141)
is the first one among these. The second one is to neglect the term φ , comparedto the other two terms, in equation (140) which gives
3Hφ+ V,φ ≈ 0 ⇒
φ ≈ −V,φ
3H(142)
Having the two aforementioned slow-roll conditions, we define the two so-calledslow-roll parameters. We start by writing the first slow-roll condition condition
12φ2 V (φ) (143)
and substitute φ from the second one, to get
12
(V,φ
3H
)2
V ⇒
16V 2
,φ 3H2︸︷︷︸≈κV
V = κV 2 ⇒
16κ
(V,φ
V
)2
1 (144)
where, in the second line we used the first slow-roll condition to write 3H2 ≈ κV .Now, we define the first slow-roll parameter ε , via
ε ≡ 12κ
(V,φ
V
)2
(145)
Then, condition (143) translates to (up to some irrelevant numerical factor)
ε 1 (146)20We should also include the curvature term −K/a2 here. However, this term becomes
quickly negligible once inflation starts. That is, all curvature goes quickly away due to infla-tion.
30
Note that by definition one always has ε > 0. This requirement ensures thatthere is a slow-roll regime. One should also demand that inflation lasts for aprolonged period of time. Then, one defines the second slow-roll parameter
η ≡ 1κ
V,φφ
V(147)
and inflation lasts long enough, provided that
|η| 1 (148)
So long as both ε and η are small compared to unity, inflation occurs. Inflationends when these requirements are no longer met, namely when
εf ∼ 1 , |ηf | ∼ 1 (149)
We should mention here, that the above conditions on the slow-roll parameterseven though sufficient they are not necessary. Indeed, the slow-roll conditionsimposed by the smallness of the slow-roll parameters, only restrict the form ofthe potential and not the value of φ. That is, the kinetic term could be largeregardless of the slow-roll assumptions. However, a sufficient as well as necessarycondition does exist and can be found by using the so-called Hamilton-Jacobiformalism [13].
5.3.4 Number of e− folds
In order for inflation to solve the problems that is supposed to solve, it hasto last a certain amount of time. Now, since during inflation the scale factorexpands nearly exponentially, by taking the logarithm of the latter we get justa number. We define the number of e− folds to be
N (t) ≡ ln(a(tf )a(t)
)=∫ tf
t
Hdt (150)
where a(tf ) is the value of the scale factor at the end of inflation. Then, thetotal amount of e− folds, which gives the duration of inflation, is given by
N (tf ) ≡ ln(a(tf )a(ti)
)=∫ tf
ti
Hdt (151)
Now, since inflation occurs in the slow-roll region, one has
H2 ≈ κ
3V (152)
as well as3Hφ ≈ −V,φ (153)
so that, dividing the latter two we arrive at
H ≈ −κ(V
V,φ
)φ = −κ
(V
Vφ
)dφ
dt⇒
Hdt ≈ −κ(V
V,φ
)dφ (154)
31
Thus, the above number of e− folds may also be expressed as
N (t) =∫ tf
t
Hdt ≈ −κ∫ φf
φ(t)
(V
V,φ
)dφ⇒
N (t) ≈ −κ∫ φf
φ(t)
(V
V,φ
)dφ (155)
We see that given a potential for inflation, the number of e− folds can alwaysbe determined upon using the above relation. We should mention here thatfor a typical inflationary model one demands N ∼ 60 e − folds in order forthe Problems of the Standard Cosmological Model to be resolved. Let us con-tinue now and see how does inflation intend to solve the problems we discussedpreviously.
5.4 Resolving the problems
Let us now psesent the way in which inflation is used to tackle the problemsdiscussed. Even though some equations are used, our approach here is morequalitatively and gives the general mechanisms through which inflation providesa solution to the three most important cosmological problems.
5.4.1 Resolving horizon problem
Let us see now how does inflation resolves the horizon problem. As it is knownthe particle or cosmic horizon (dp ≈ t)21 defines a region the parts of which arein casual connection with one another, namely they exchange energy and mo-mentum and therefore can be in an equilibrium state. On the onset of inflationin casual connection are length scales-l which satisfy
l ≤ (dp)i ≈ cti ∼ 10−28m (156)
In the end of inflation all length scales have been inflated by a factor
af
ai= eH0(tf−ti) ≈ e(tf−ti)/ti ≈ 1043 (157)
Thus, in the end of inflation the above length-scale has increased to
l′=af
ail (158)
As a result the casual connected region now obeys
l′≤ 1043(dp)i ∼ 1015m (159)
and is extremely bigger than the corresponding particle horizon-(dp)f ≈ ctf ∼10−26m. Even better is the fact that the casual region after inflation is muchbigger than our observable universe today. Note that the size of our observableuniverse is given by the value particle horizon today, which is
(dp)0 ≈ ct0 ∼ 104Mpc ∼ 3 · 1026 (160)21We have adopted the natural system of units for which it holds that ~ = 1 = c = kB .
However, when we want to arrive at certain results with appropriate units we substitute thelatter by dimensional analysis. For instance, here we have dp ≈ t = ct.
32
Now, assuming that the inflationary era is followed by the usual radiation era(for which we know that a ∝ T−1), we have
l0 =(Tf
T0
)lf (161)
where f denotes values at the end of inflation while the subscript 0 denotespresent values. According to the above , a region with dimensions ∼ 1015m inthe end of inflation, today corresponds to a length scale
l0 ∼ 1020Mpc ∼ 3 · 1042m (162)
where we have used that Tf ∼ 1027K and T0 ∼ 2, 7K. Note that l0 is manyorders of magnitude greater than the dimensions of the observable universetoday. Thus, we conclude that the existence of an inflationary era brings theentire observable universe today, into casual connection. This can explain theisotropy and homogeneity of the universe observed today avoiding, in addition,fine-tuned initial conditions. Given any initial curvature Ωi, inflation brings itto a value Ωf which is very close to unity. Thus, in the end of inflation thespace has become flat.
5.4.2 Resolving Flatness
The degree of curvature of the spatial part of spacetime is determined by thedifference |1− Ω|. Now, during inflation
a ∼ eH0t (163)
with H0 ≈ Hi ≈ 1/ti. Then, using
1 = Ω− K
(aH)2= Ω− K
a2(164)
it follows that|1− Ωf ||1− Ωi|
= e−2H0(tf−ti) ≈ e−2(tf−ti)/ti (165)
and setting ti ∼ 10−36sec together with tf ∼ 10−34sec we have that
|1− Ωf | ∼ 10−86|1− Ωi| (166)
From the last one we conclude that any initial curvature that might existed atthe beginning, has been diminished in the end of inflation. This is how inflationsolves the flatness problem.
5.4.3 Resolving magnetic monopoles abundance
The exponential expansion of the dimensions of the universe during inflationdecreases exponentially the density of the magnetic monopoles. For instance,requiring 60 e− folds it follows that
af
ai= eN = e60 ≈ 1026 (167)
The latter implies that today we can expect one monopole in each galaxy. If weassume N ∼ 68 then the density drops at one monopole in the entire observableuniverse! Therefore, inflation solves the monopole problem by decreasing thedensity of the latter dramatically.
33
6 Modified Gravity: Why and how?
Soon after the development of General Relativity by Einstein, many alternativesto GR had been proposed. Most of them however, were lacking simplicity aswell as observational fitting. Nevertheless, modification proposals have beenpublished by researchers ever-since. There are a number of reasons that allowone to modify gravity. Firstly, from a mathematical perspective there is noa-priori demand that forces one to consider the Einstein-Hilbert action (linearin R) as the fundamental action of gravity. However, if we insist obtainingfield equations that are at most second order partial differential equations withrespect to the metric tensor, then the only ones that one can arrive at areEinstein’s equations22. We should mention though, that the map from theaction to the Einstein equations is not one to one, namely the Einstein Hilbertaction is not the only one that gives Einstein equations when is varied. To bemore specific we give the following theorem.Lovelocks’s Theorem: Let S gravitational action that depends only on themetric tensor and its first and second derivatives
S =∫d4x
√−gL(gµν , ∂αgµν , ∂β∂αgµν) (168)
Then, by extremising with respect to the metric tensor
δS =∫d4xδgµνEµν [L] (169)
where
Eµν [L] =d
dxα
[∂L
∂(∂αgµν)− d
dxβ
(∂L
∂(∂β∂αgµν)
)]− ∂L∂(gµν)
(170)
The Euler-Lagrange equations are given by
Eµν [L] = 0 (171)
Lovelock’s theorem states that the only field equations that can be obtained forsuch an action (in 4 dimensions) that contain no higher than 2nd-order deriva-tives of the metric tensor are Einstein equations (in vacuum) with a cosmologicalconstant
Eµν =√−g[α
(Rµν − R
2gµν
)+ Λgµν
]= 0 (172)
where α and Λ are arbitrary constants. It is worth noting that the abovetheorem is very restrictive. It says that the only field equations that one canget (in 4−dim) that are second order partial differential equations with respectto the metric tensor are actually Einstein equations. The action that is used toderive Einstein equations is the so-called Einstein-Hilbert action
SEH =12κ
∫d4x
√−gR (173)
22This is true only for metric theories, namely when the only independent field in the actionis the metric tensor. Of course this is not true in Palatini and Metric-Affine theories of Gravitywhere the connection is independent of the metric tensor.
34
but, in general, any action of the form
S =∫d4x
√−gL (174)
withL = αR− 2Λ + βG + γI (175)
also yields Einstein equations. Here β and γ are also arbitrary constants and
G ≡ R2 − 4RµνRµν +RµνρσR
µνρσ (176)
andI ≡ εµνρσRαβ
µνRαβρσ (177)
the former is called the Gauss− Bonnet term and the latter (when integratedover spacetime) is referred to as signature or index. For both of them it holdsthat
Eµν [G] = 0 (178)Eµν [I] = 0 (179)
and therefore they do not contribute to the field equations. The former is trueonly in 4-dimensions (or lower) and the latter for any dimensionality. Bothof them when integrated over spacetime and multiplied with the appropriatenumerical factors give topological invariants for the given manifold (spacetime).The restriction imposed by the Lovelock theorem indicates that in order to getfield equations that differ from Einstein’s one should do one (or more) of thefollowing
• Allow for field equations that contain partial derivatives of the metric tensorof degree higher than two
• Consider additional fields (apart from the metric tensor)
• Do not assume either metric compatibility or torsion-free connection, or both
• Work in spaces with dimensionality greater than 4
• Give up locality
In the present thesis we study theories that admit higher order partial differentialequations with respect to the metric tensor. We also consider modifications thatarise due to the addition of other fields apart from the metric tensor Namely, weconcentrate to the first two possibilities of the aforementioned cases. The thirdcase arises in the so-called Palatini and Metric− Affine theories of Gravity.We define below what we exactly mean by Metric, Palatini and Metric-AffineGravity.
6.1 An Aside: The 3 formulations of Gravity
There exist 3 variational approaches to Gravity. These are the Metric, thePalatini and theMetric−AffineGravities. Each one of them predicts differentdynamics, in general, but in some particular cases they might coincide, as wediscuss in what follows. Before introducing any of them, we should point outthat throughtout the current thesis we are solely use the Metric approach. Letus now explore the aforementioned approaches.
35
6.1.1 Metric− Theories of Gravity
In Metric− Theories of Gravity one makes two assumptions. Firstly, that theconnection is metric compatible, that is
∇αgµν = 0 (180)
along with the assumption of a torsion-free connection
Γα[µν] = 0 ⇒ Γα
µν = Γανµ (181)
The above two conditions completely fix the connection to be the Levi−Civitaconnection
Γαµν =
12gαβ(∂µgβν + ∂νgβµ − ∂βgµν) (182)
Then, the only independent quantity on the manifold is the metric tensor. Thisdefines a Riemannian geometry (pseudo-Riemannian in our case) and the spaceis fully described by the metric tensor. We should point out that this need notbe the case in general. Indeed, the metric tensor and the connection define, ingeneral, different properties on the manifold. For the former defines distancesand angles between vectors, while the latter defines parallel transfer of vectorand tensor fields on the manifold. Thus, in the case of Metric Gravity there isnot much of a choice, if one were to write a Gravity action this can only dependon the metric tensor. The mathematical expression of such an action is
S = SG[gµν ] + SM [gµν ,Ψ] (183)
with both gravity and matter actions metric-dependent only. Here SG and SM
stand for the Gravity and matter parts of the action respectively. The presentthesis is solely formulated along the lines of the above approach. We do ,however,present the other two formulations as well, just for completeness.
6.1.2 Palatini Gravity
In Palatini Gravity no a-priori assumptions about the compatibility of the metricor the torsionlessness of the connection are made. Thus, metric and connectionare independent fields, both fundamental with their own geometrical signifi-cance. However, one assumption is made; that the matter part of the actiondoes not depend on the connection. A general action in the Palatini formulationreads
S = SG[gµν ,Γκαβ ] + SM [gµν ,Ψ] (184)
Then, in order to derive the field equations one has to independently vary withrespect to the metric tensor-δg as well as with respect to the connection-δΓ. Notethat the connection here is not symmetric in general and also the metric com-patibility does not hold true. As a result the connection-Γκ
αβ is not (in general)the Levi-Civita connection. It is said in the literature that if one chooses theEinstein-Hilbert Lagrangian density, that is LG = R , then the Palatini proce-dure coincides with Einstein’s theory formulated in Metric-approach. However,this is not true in general. That is, additional assumptions must also be made forthis coincidence to occur. This can be accomplished by either imposing metriccompatibility (∇αgµν = 0) or considering a torsion-less connection (Γα
[µν] = 0)as shown in [14]. A thoroughly discussion on this subject and related topics canalso be found there as well as in [15], [16].
36
6.1.3 Metric−Affine Gravity
A generalization of the Palatini-Gravity is the so-called Metric−Affine Grav-ity in which the matter action does depend on the independent connection aswell. The general action is then written as
S = SG[gµν ,Γκαβ ] + SM [gµν ,Γκ
αβ ,Ψ] (185)
Exactly this dependence of the matter action on the connection, defines a newtensor
∆ µνα ≡ − 2√
−gδSM [gµν ,Γα
µν ]δΓα
µν
(186)
which is called the Hypermomentum− tensor. Note that the above quantity isindeed a tensor. This is so because even though Γµ
αβ is not a tensor, the variationδΓµ
αβ is a tensor as we show in the appendix. It is worth noting that there seemsto exist a relation between the spin of a particle (intrinsic property) and thenon-vanishing of the Hypermomentum-tensor which results in different gravita-tional effects. In particular it can be shown that the Hypermomentum tensoridentically vanishes for spinless particles but has a non-zero value for particlesthat have spin. Under certain assumptions, Metric-Affine Gravity can repro-duce Einstein’s theory and therefore be compatible with observational analysis.We should also point out that among these three approaches, Metric−AffineGravity is the least studied, mostly because of its complexity. The dynamicsof the latter has been studied to some extend in [17], however no cosmologicalsolutions have ever been found. Certainly, in the near future, more things aboutMetric−Affine Gravity and its cosmological implications will be known.
37
7 f(R)-Theories of Gravity
The most straightforward generalizations of General Relativity are the so-calledF(R)-theories, in which the Ricci scalar appearing in the Einstein-Hilbert actionis replaced by a general function of the latter, R → F (R). Nice discussions andreviews on f(R)-Gravity can be found in [18],[16],[19], [20],[21] and [22] forinstance. Cosmological implications of those theories are given in [23], [24],[14],[6]. In this section we shall analytically derive the field equations for f(R)-theories and then discuss cosmological implications together with some specificexamples. In particular we are going to review and study Starobinsky’s modelof inflation and then go on to study the implications of adding a cosmologicalconstant to it.The dynamics of the latter is also studied, to some degree, in bothJordan and Einstein frames. Let us start now with the derivation of the fieldequations of f(R)-Gravity.
7.1 Field Equations of f(R)-theories
The action of an f(R)-theory is given by
S =∫d4x
√−g 1
2κf(R) + Smatter[gµν ,Ψ] (187)
where Ψ collectively denotes any matter fields. Varying the above action, withrespect to the metric tensor, we arrive at
δS = 0 ⇒
0 = δ
∫d4x
√−g 1
2κf(R) + δSmatter[gµν ,Ψ] (188)
0 =∫d4x
√−g2κ
[−gµν
2f(R)δgµν + f,R δR
]+∫d4x
δ(√−gLM )δgµν
δgµν (189)
and upon usingFδR = F (Rµν +Dµν)δgµν (190)
23
which holds for any scalar F, we can furthermore, using twice the Leibnizrule, deduce ∫
d4x√−gFDµνδg
µν =∫d4x
√−gδgµν(DµνF ) (191)
where we have dropped any surface terms that appeared. Such that∫d4x
√−gFδR =
∫d4x
√−gF (Rµν +Dµν)δgµν =
=∫d4x
√−gδgµν [(Rµν +Dµν)F ] (192)
23Recall that we have defined the differential tensorial operator Dµν = gµν2−∇µ∇ν
38
Therefore, setting F = f,R to the equation right above and substituting backto the variation we obtain
0 =∫d4x
√−g2κ
[−gµν
2f(R)δgµν + δgµν(Rµν +Dµν)F
]+∫d4x
δ(√−gLM )δgµν
δgµν
(193)and defining, as usual, the energy-momentum tensor of the matter fields24 via
TMµν = Tµν ≡ −
2√−g
δSmatter
δgµν(194)
or equivalently
Tµν ≡ −2√−g
δ(√−gLm)δgµν
(195)
we finally arrive at
0 =∫d4x
√−gδgµν
[(−gµν
2f(R) + (Rµν +Dµν)f,R
) 12κ− Tµν
2
](196)
and since this must hold for any variation δgµν we conclude that
0 =(−gµν
2f + (Rµν +Dµν)f,R
) 12κ− Tµν
2⇒
−gµν
2f + (Rµν +Dµν)f,R = κTµν (197)
where f,R = ∂f/∂R and Dµν = gµν2 − ∇µ∇ν . The latter are the fieldequations for a general f(R)-theory. Firstly, we mention that in f(R)-theoriesthe energy momentum tensor continues to be covariantly conserved. Let usprove this statement. Acting with ∇µ on the above equation it follows that
κ∇µTµν = −12∇νf +∇µ(Rµνf,R ) +∇µDµνf,R (198)
and using the chain rule, we have
∇νf = (∇νR)∂f
∂R= (∇νR)f,R (199)
Furthermore, for any scalar ψ it holds that25
∇µDµνψ = −(2∇ν −∇ν2)ψ = [∇ν ,2]ψ = −Rµν∇µψ (200)
and therefore by setting ψ = f,R we have
∇µDµνf,R = −Rµν∇µf,R (201)
Substituting these results into (608) we arrive at
κ∇µTµν = −12(∇νR)f,R +(∇µRµν)f,R +Rµν(∇µf,R )−Rµν∇µf,R⇒
24We will be abbreviating T Mµν with Tµν for the energy-momentum of the matter fields and
only explicitly include the index M when additional energy contributions of a different natureare present.
25This is proven in the Appendix.
39
κ∇µTµν = −12(∇νR)f,R +(∇µRµν)f,R
and using the identity
∇µRµν =12∇νR (202)
we complete the proof∇µTµν = 0 (203)
Secondly, not only the energy-momentum tensor of the matter field is con-served (proved above), but also the effective energy-momentum tensor comingfrom the additional terms with respect to GR. Indeed, the field equations off(R), after a trivial rearrangement, can be recast to our familiar Einstein equa-tion’s form
Gµν = κT effµν + κTM
µν (204)
whereκT eff
µν ≡ f −R2
gµν + (1− f,R)Rµν −Dµνf,R (205)
Note that in the case of GR ⇒ f(R) = R and the above tensor identicallyvanishes. Now, acting with ∇µ on (204) we obtain
∇µGµν = κ∇µT effµν + κ∇µTM
µν (206)
but ∇µGµν = 026 and as we proved before ∇µTMµν , therefore
∇µT effµν = 0 (207)
as well. Thus, effective energy-momentum tensor is also conserved, as stated.Thirdly, there is a drastically different feature that f(R) ( for f(R) 6= 0) theorieshave in comparison to GR. This is the additional scalar propagating degree offreedom which we exploit in the following subsection.
7.2 The ’Scalaron’
As mentioned previously, in f(R)- theories there is an additional scalar propa-gating degree of freedom. To see this, we take the trace (contraction with gµν)of the field equations (197) in vacuum (Tµν = 0) to arrive at
−2f +Rf,R + 32f,R = 0 (208)
which can be written as
2f,R =13
(2f −Rf,R
)(209)
and defining
−dVeff
dR≡ 1
3
(2f −Rf,R
)(210)
it follows that
2f,R = −dVeff
dR(211)
26See appendix.
40
from which we can now recognize a scalar propagating degree of freedom f,R thatgoes with the name ′scalaron′. This propagating degree of freedom becomesmore clear when one goes to the so-called Einstein frame. In this frame we haveEinstein’s theory plus a scalar field with potential which is minimally coupledto Gravity, however non-minimally coupled to matter. We discuss this in moredetail in subsequent chapter where we also define Jordan and Einstein frames.
7.3 Equivalence of Metric f(R) and Brans-Dicke theoryfor ωBD = 0
As we have already stated, in f(R)-theories there is an additional propagatingscalar degree of freedom in comparison to GR. As a result, these theories canbe seen as Brans-Dicke theories with the appropriate choice of the Brans-Dickeparameter ωBD. More precisely, Metric f(R)-theories are equivalent to Brans-Dicke theories with ωBD = 027. To see this we introduce an auxiliary field28 χand consider the following action
S =12κ
∫d4x
√−g[f(χ) + f,χ(R− χ)
]+ SM (gµν ,ΨM ) (212)
Under the assumption f,χχ 6= 0, variation of the above with respect to χ yields
0 = δχ
∫d4x
√−g[f(χ) + f,χ(R− χ)
](213)
and upon usingδχf = f,χδχ (214)
along withδχf,χ = f,χχδχ (215)
we arrive at
0 =∫d4x
√−g[f,χδχ+ δχf,χχ(R− χ)− f,χδχ
]=
=∫d4x
√−gδχ[f,χχ(R− χ)] = 0 (216)
Since the latter must vanish for any variation δχ and by assumption f,χχ 6= 0we conclude that
χ = R (217)
this is an algebraic equation in χ and can be replaced back to our action to giveexactly an f(R) theory as stated. Furthermore, defining
φ ≡ f,χ (218)
one can solve the latter with respect to χ and find χ = χ(φ). The action canthen be written as
S =12κ
∫d4x
√−g[φR− V (φ)
]+ SM (gµν , ψM ) (219)
27For Palatini f(R)gravity this equivalence occurs for ωBD = −3/2.28Auxiliary means that it contains no kinetic term in the Lagrangian and therefore has no
dynamics. As the name suggests its introduction is made for convenience reasons and in theend it must be absent from the theory.
41
Recall now the action for the Brans-Dicke theory
SBD =12κ
∫d4x
√−g[φR− ωBD
φ(∇µφ∇µφ)− V (φ)
](220)
Thus, from the latter two we see that f(R) is recovered from Brans-Dicke whenthe second term in (220) is absent, namely when ωBD = 0. Nevertheless, weshould mention again that this result holds for Metric-f(R) theories.
7.4 Conformal Transformations-Einstein and Jordan frames
There exists also a relation between f(R)-theories and specific forms of Scalar-Tensor theories of Gravity by means of a conformal transformation. Firstly, wedefine conformal transformations and then apply them in order to derive theequivalence.
7.4.1 Conformal transformations
Consider a Riemannian manifold (M, g) of dimension n. The map
gab → gab = Ω2(x)gab (221)
where Ω(x) is an arbitrary smooth spacetime function, leaves the angle ofany two vectors of the space, invariant and therefore is called a conformaltransformation. Indeed, let xa and yb, where a, b = 1, 2, ..., n be two arbitraryvectors on the manifold. By definition, the angle between the two is given by
cos θ ≡ (x · y)√(x · x)(y · y)
=xaybgab√
(xcxdgcd)(yeyfgef )(222)
which under the conformal transformation (221) becomes
cos θ → xaybgab√(xcxdgcd)(yeyf gef )
=Ω2xaybgab
Ω2√
(xcxdgcd)(yeyfgef )= cos θ (223)
as stated. Thus, under conformal transformations the angles between vector donot change. However, the distance of points on the manifold does change. Thisis already obvious from the squared line element ds2 = gabdx
adxb which undera conformal transformation changes to
ds2 → ds2 = gabdxadxb = Ω2ds2 (224)
and therefore, given any two points A and B on the manifold their distance-lAB
after the conformal transformation has been performed is given by
lAB =∫ B
A
ds =∫ B
A
Ωds 6= lAB (225)
where lAB =∫ B
Ads is their original distance. Conformal transformations are
used to define equivalences among Gravity theories. In the following subsectionswe present such an equivalence through a conformal transformation that relatesf(R) with Scalar-Tensor theories of Gravity.
42
7.5 Einstein and Jordan frames
Consider an f(R) theory, then the action is given by
S =∫d4x
√−g 1
2κf(R) + Smatter[gµν ,Ψ] (226)
As we have proved before when the action is given in this form, the energy-momentum tensor of the matter fields is covariantly conserved.Definition: We define the Jordan−frame to be exactly this frame at which theenergy momentum-tensor of the matter fields is covariantly conserved, namely
∇µT (J)µν = 0 (227)
Furthermore, in the Jordan frame, test particles follow geodesics of the metric.We now consider the conformal transformation29
gµν = Ω2gµν (228)
As an immediate consequence of the latter we have√−g = Ω−4
√−g (229)
In addition, under the conformal transformation the Ricci scalar transformsaccording to
R = Ω2(R+ 6∼2ω − 6gµν∂µω∂νω) (230)
where we have definedω ≡ lnΩ (231)
and the box operator associated with the transformed metric ˜gµν ,
∼2φ =
1√−g
∂µ(√−ggµν∂νφ) (232)
for any scalar φ. Now, as we have seen we can recast our action into
S =∫d4x
√−g[ 12κRf,R − U
]+ SM [gµν ,Ψ] (233)
whereU =
12κ
(Rf,R − f) =12κ
(RF − f), F ≡ f,R (234)
The latter action transforms, under the conformal transformation, to
S =∫d4x√−g[
12κFΩ−2R− 1
2κFΩ−2(6gµν∂µω∂νω)− Ω−4U
]+
3κ
∫d4x√−gFΩ−2∼2ω + SM [Ω−2gµν ,Ψ] (235)
From the latter we see that by choosing the parameter of the transformation-Ωappropriately, one can isolate an Einstein-Hilbert term in the action30 and write
29Note that under a conformal transformation of the metric the spacetime points xµ do notchange (fixed on the manifold) relative to the manifold, only the structure of the manifold(through gµν) does so.
30Namely a term that is linear in the Ricci scalar.
43
the remaining terms as an effective scalar filed propagating in the spacetime.Indeed, if we choose
FΩ−2 = 1 ⇒ Ω2 = F = f,R (236)
under the assumption that f,R > 0, the above action is written as follows
S =∫d4x√−g[
12κR− 3
κgµν∂µω∂νω −
1F 2
U
]+
3κ
∫d4x√−g∼2ω︸ ︷︷ ︸
=surface term
+SM [1Fgµν ,Ψ] (237)
As can be easily seen now, the second term in the action corresponds to thekinetic term of the scalar field up to a trivial redefinition of ω. Indeed, toobtain the canonical factor 1/2 in front of the kinetic term, we simply define anew field φ (or redefine ω if you wish), such that√
3κω =
1√2φ (238)
and since ω = ln Ω = 12 lnF we also have
F = e√
2κ3 φ (239)
In addition, the surface term does not affect the field equations and thereforecan be dropped. Taking all the above into account, the action reads
S =∫d4x√−g[
12κR− 1
2gµν∂µφ∂νφ− V (φ)
]+ SM [e−
√2κ3 φgµν ,Ψ] (240)
whereV (φ) ≡ U
F 2=
12κ
RF − fF 2
(241)
This action now is simply the action of GR with an additional field-φ propa-gating in the spacetime, which is minimally coupled to gravity, however is nonminimally coupled to matter due to the coupling of the field with the metricin the matter action-SM [e−
√2κ3 φgµν ,Ψ]. The action has now been expressed in
what is called the Einstein−frame. To be more accurate we give the followingdefinition.Definition: The Einstein−frame is defined to be exactly that frame for whichthe Ricci scalar enters the action only linearly (as in Einstein-Hilbert action).For this frame however the energy-momentum tensor associated with the matterfields is no longer covariantly conserved due to the coupling of the scalar fieldwith the matter fields as can be seen from the expression of the matter action
SM [e−√
2κ3 φgµν ,Ψ] ⇒ ∇µTµν 6= 0 (242)
From now on we will denote the quantities that are expressed in the Einstein−frame with a tilde (as we already did so far) in order to have a clear distinctionbetween Einstein and Jordan frames. Let us now relate the energy-momentumtensors of the two frames. By definition
Tµν = − 2√−g
δSM [gκλ,Ψ]δgµν
= − 2√−g
δSM [Ω2gκλ,Ψ]δgµν
=
44
Ω4
(− 2√
−gδSM [Ω2gκλ,Ψ]
δgαβ
)︸ ︷︷ ︸
≡Tαβ
δgαβ
δgµν(243)
and sincegαβ = Ω−2gαβ (244)
one hasδgαβ
δgµν= Ω−2 δg
αβ
δgµν= Ω−2δα
µδβν (245)
Using this, we deriveTµν = Ω4Ω−2Tαβδ
αµδ
βν ⇒
Tµν = Ω2Tµν (246)
As we have seen the Einstein frame is obtained for the special case
Ω2 = F = f,R (247)
Then,
Tµν =1FTµν (248)
7.6 Conformal transformations in FLRW universes
Let us now find relations between cosmological quantities expressed in the Ein-stein frame in terms of those given in the Jordan frame for f(R)-theories. Westart with the Robertson-Walker line element for a spatially flat universe
ds2 = −N2(t)dt2 + a2(t)(dx2 + dy2 + dz2) (249)
where one usually takes the gauge N(t) = 1. The relation with the Einsteinframe is found by using
ds2 = Fds2 , F = f,R (250)
Expanding the latter, we obtain
−N2(t)dt2 + a(t)(dx2 + dy2 + dz2
)= −FN2(t)dt2 + Fa2(t)
(dx2 + dy2 + dz2
)(251)
From which we read offN2(t) = FN2(t) (252)
as well asa2(t) = Fa2(t) (253)
From the former we see that even when one chooses the gauge N(t) = 1 →N(t) =
√F 6= 1. In order to have the usual term −dt2 in the expression for the
transformed line element we define a new cosmic time t related to the usual t,through
dt = N(t)dt =√Fdt (254)
in the gauge N(t) = 1. Then, the line element in the Einstein frame is given by
ds2 = −dt2 + a(t)(dx2 + dy2 + dz2) (255)
45
wherea(t) =
√Fa(t) (256)
Now we want to find a relation between the Hubble parameters in the twoframes. Firstly, we compute
d
dta(t) =
d
dt(√Fa(t)) =
dt
dt
d
dt(√Fa(t)) =
=1√F
(F
2√Fa(t) + a(t)
)(257)
Such that
H(t) =ddta(t)
a(t)=
1√F
(F
2√
Fa(t) + a(t)
)√Fa(t)
=
=1F
(F
2√F
+H(t)
)⇒
H(t) =1F
(F
2√F
+H(t)
)(258)
From this we can also conclude that
H(t)dt =
(1 +
F
2HF
)Hdt (259)
Now, let N represent the number of e − folds in the Jordan frame and N bethe corresponding number of e− folds in the Einstein frame. Then,upon using(259) we arrive at the following relation between the two
N ≡∫ tf
ti
Hdt =∫ tf
ti
Hdt
[1 +
F
2HF
]=
=∫ tf
ti
Hdt︸ ︷︷ ︸≡N
+12
∫ tf
ti
d(lnF ) ⇒
N = N +12
ln(F (Rf )F (Ri)
)(260)
where Ri = R(ti) and Rf = R(tf ).
7.7 Maximally Symmetric Spaces in f(R)-theories
The field equations for a general f(R)-theory with a cosmological constant, are
−f2gµν + (Rµν +Dµν)f,R = −Λgµν + κTµν (261)
where Dµν = ∇µ∇ν − gµν2. Contracting the latter with gµν we arrive at
−2f +Rf,R + 32f,R = −4Λ + κT (262)
46
From this we conclude that, contrary to GR, even in the case of an emptyspace (Tµν = 0) and the absence of a cosmological constant (Λ = 0), onedoes not necessarily have R = 0. It is a general aspect of modified theoriesof gravity the existence of additional solutions compared to Einstein gravity.For instance, Birkhoff’s theorem on uniqueness of the Schwarzschild solutionas the only spherically symmetric solution in vacuum, no longer holds true. Inaddition, f(R)-theories admit a larger variety of maximally symmetric solutions.Indeed, setting Tµν = 0 ⇒ T = 0 to the above, for maximally symmetric31
(R = constant) one hasRf,R − 2f = −4Λ (263)
This is an algebraic equation32 in R. Solutions with R = R0 = 0 will give theMinkowski spacetime, while those with R = R0 > 0 or R = R0 < 0 will give de-Sitter and anti-de-Sitter spacetimes respectively. We should mention here thatfor f(R)-theories we can have de-Sitter/anti-de-Sitter spacetimes even whenΛ = 0 as (263) possesses solutions even when the cosmological constant is zero.The latter can be found by solving the algebraic
Rf,R − 2f = 0 (264)
and again, by finding the solutions we specify the nature of the maximallysymmetric space.
7.8 Scalaron degree of freedom and stability
Now, taking the trace of the field equations (197) we arrive at
−2f +Rf,R + 32f,R = κT (265)
Perturbing the last one we can study the stability of given spacetimes. Let usstudy now under what conditions does a maximally symmetric space (R = R0 =const.) 33 retain its stability. Starting with a maximally symmetric space withR = R0 we perturb the Ricci scalar as follows
R0 → R0 + δR (266)
and subsequently we study the behaviour of R by using (265). Define
K ≡ −2f +Rf,R + 32f,R (267)
Then, for vacuum (T = 0) perturbations we have
δK = K[R0 + δR]−K[R0] = 0 (268)
We expand to first order
f(R0 + δR) = f(R0) + f,R
∣∣∣R=R0
δR+O((δR)2
)≈
31Maximally symmetric spaces are those which possess a constant scalar curvature (R =
constant = R0). This immediately implies that Rµν = R04
gµν . Spaces with the last propertyare oftentimes called Einstein spaces in the literature.
32Note that this is not a differential equation in R, it only holds for specific values of R.33In the case where R = R0 = 0 our maximally symmetric space is that of Minkowski and
for R = R0 > 0 and R = R0 < 0 one has a de-Sitter or an anti-de-Sitter space respectively.
47
≈ f0 +A0δR (269)
where we have set f0 ≡ f(R0) and A0 ≡ f,R
∣∣∣R=R0
. Next we compute
f,R(R0 + δR) = f,R
∣∣∣R=R0
+ f,RR
∣∣∣R=R0
δR+O((δR)2
)≈
A0 +B0δR (270)
Note that we have also defined B0 ≡ f,RR
∣∣∣R=R0
. Finally, we calculate the
behaviour of the product term Rf,R under the variation. We have
(R0 + δR)f,R(R0 + δR) ≈ (R0 + δR)(A0 +B0δR) ≈
≈ R0A0 + (R0B0 +A0)δR+O((δR)2
)≈
≈ R0A0 + (R0B0 +A0)δR (271)
where again we have neglected any higher order terms that appeared. Thus,substituting the above results into (268), one has
0 = δK = K[R0 + δR]−K[R0] == −2f0 − 2A0δR+R0A0 + (R0B0 +A0)δR
+3B02δR+ 2f0 −R0A0 == (−A0 +R0B0)δR+ 3B02δR = 0 ⇒[
2 +13
(R0 −
A0
B0
)]δR = 0 (272)
from which we see that if we identify
m2 ≡ −13
(R0 −
A0
B0
)(273)
a propagating degree of freedom appears
(2−m2)δR = 0 (274)
which is dubbed scalaron. We state now, that the mass of the scalaror must bereal (m2 > 0) in order to avoid instabilities. Indeed, for the FLRW metric thebox operator assumes the form
2 = −(d2
dt2+ 3H0
d
dt
)(275)
where we also took into account that H = H0 = const. since we are consideringdS/AdS spaces. Thus, the above propagation equation reaches
¨(δR) + 3H0˙(δR) +m2(δR) = 0 (276)
The last one is a 2nd-order linear differential equation with constant coefficientsand is reducible to a 2nd-order algebraic equation. Indeed, by making the ansatz
δR ∝ eλt (277)
48
the latter givesλ2 + 3H0λ+m2 = 0 (278)
which we can now immediately solve to find λ. The discriminant is
∆ = (3H0)2 − 4m2 (279)
and we see that when m2 < 0 we have that ∆ > 0 always and
λ+,− =−3H0 ±
√∆
2(280)
but for the solution λ+ we have that
−4m2 > 0
(3H0)2 − 4m2 > (3H0)2√
∆ > 3H0
−3H0 +√
∆ > 0λ+ > 0 (281)
which will in turn give a term that goes as
∝ eλ+t (282)
which exponentially diverges for large t and as a result even for small pertur-bations, δR grows exponentially large which results in an unstable spacetime.However, in the case m2 > 0 the space is stable. Indeed, if ∆ < 0 this is trivialsince the solutions contain an exponentially decaying term multiplied by sinesand cosines (coming from the imaginary parts of λ’s). Now, when ∆ < 0 andm2 > 0 one has
λ+,− =−3H0 ±
√∆
2=
3H0
2
[− 1±
√1−
( 2m3H0
)2](283)
but since √1−
( 2m3H0
)2
< 1 (284)
for m2 > 0, both solutions are negative and thus decay exponentially. Asa result, initial perturbations become smaller and smaller as time increases at-tributing, therefore, stability to the space. Stability analysis for general modifiedGravities is given in [25]
7.9 Maximally Symmetric Spaces in the Einstein frame
As we have seen in the previous section, maximally symmetric solutions (R =R0 = const.) of f(R)-theories with a cosmological constant, obey
Rf,R − 2f = −4Λ (285)
which then has to be solved with respect to R. Note that this result holdswhen the action is expressed in the Jordan frame. One would like to knowhow does the form of the Ricci scalar change when transforming to the Einstein
49
frame. To see this, suppose we have found a maximally symmetric solutionR = R0 = const. in the Jordan frame. Then, as we know for a general R thetransformed Ricci scalar R (for general R) to the Einstein frame is given by
R = Ω−2R− 6∼2ω + 6gµν∂µω∂νω (286)
where ω = ln Ω and in order to obtain the Einstein frame the choice Ω2 = F =f,R is made. Then, for R = R0 = const. we also have ω = lnF (R0) = const. sothat both second and third term of the right hand side of the above equationvanish, yielding
R0 =R0
F (R0)(287)
7.10 Extension of the variational technique δa(t), δN(t) tof(R)-theories
Let us extend now, the trick to derive the Friedmann equations by varying withrespect to the scale factor, to also include f(R)-theories. We do so here for theflat case (K = 0). We have
S =∫d4x
√−g[ 12κf(R) + LM
]=∫d4x[ 12κa3f(R(a)) +
√−gLM
](288)
and varying with respect to a(t), it follows that
δaS = 0 =∫d4x[3a2fδa+ a3f,RδR− 2κ
a3
2Tµν
δgµν
δaδa]
(289)
where we have used the chain rule to write
δ(√−gLM )δa
=δ(√−gLM )δgµν
δgµν
δa= − 2√
−gδ(√−gLM )δgµν︸ ︷︷ ︸
≡Tµν
(− a3
2
)δgµν
δa=
= −a3
2Tµν
δgµν
δa(290)
But sinceg00 = −1, gij =
1a2δij (291)
it holds thatδg00
δa= 0,
δgij
δa= − 2
a3δij (292)
and thereforeδ(√−gLM )δa
= δijTijδa = 3a2pδa (293)
Now, as we have proved, for any scalar F it holds that∫d4x
√−gFδR =
∫d4x
√−gδgµν(Rµν +Dµν)F (294)
so that, in our case we have (using√−g = a3, F = f,R)∫
d4x√−gFδR =
∫d4xa3f,R(δa)
δR
δa=∫d4xa3 δg
µν
δa(Rµν +Dµν)f,R
50
=∫d4xa3δa
(− 2a3
)δij(Rij +Dij)f,R =
−6∫d4xa2δa
[a
a+ 2
(a
a
)2]f,R + 6
∫d4xa2δa
(d2
dt2+ 2H
d
dt
)f,R (295)
Substituting all the above into our expression for the variation, we arrive at
0 =∫d4xa2δa
[3f − 6
[ aa
+ 2(a
a
)2 ]f,R
]
+∫d4xa2δa
[6( d2
dt2+ 2H
d
dt
)f,R − 6κp
](296)
From which it follows the 1st modified Friedmann equation for f(R)-Gravity,
f
2− 2
[a
a+ 2
(a
a
)2]f,R +
(d2
dt2+ 2H
d
dt
)f,R = κp (297)
Now,as in the case of GR in order to obtain the second Friedmann equation westart by the flat Robertson-Walker metric
ds2 = −N2(t)dt2 + a2(t)(dx2 + dy2 + dz2) (298)
without taking the gauge N(t) = 1 yet. Then, we have
S = S[a,N ] =∫d4x
[12κa3Nf
(R[a,N ]
)+√−gLM
](299)
and varying with respect to N(t) it follows that
0 = δN
∫d4x
[a3Nf
(R[a,N ]
)+ 2κ
√−gLM
]=
=∫d4x
[(δN)a3f +Na3f,R
δR
δNδN + 2κ
δ(√−gLM )δN
δN
](300)
As we have already proved∫d4x
√−gFδR =
∫d4x
√−gF (Rµν+Dµν)δgµν =
∫d4x
√−gδgµν(Rµν+Dµν)F
for any scalar F. In our case√−g = a3N and F = f,R so that∫
d4x√−gFδR =
∫d4xa3NF (Rµν +Dµν)
δgµν
δNδN =
=∫d4xa3N(δN)(R00 +D00)f,R
2N3
(301)
We also have proved in previous chapter
δ(√−gLM )δN
= −a3ρ
N2(302)
51
Substituting all the above into our expression for the variation, we arrive at
0 =∫d4xa3(δN)
[f +
2N3
(R00 +D00)f,R − 2κ1N3
ρ
](303)
From which we conclude that
f +2N2
(R00 +D00)f,R − 2κ1N2
ρ = 0 (304)
Finally, using
R00 = −3a
a(305)
andD00φ = 3H
d
dtφ (306)
where the last one only holds for FLRW backgrounds, and taking the gaugeN(t) = 1, the above equation assumes the form
f
2+ 3
(− aa
+Hd
dt
)f,R = κρ (307)
which is the 2nd modified Friedmann equation for f(R)-Gravity. Note that whenf(R) = R one recovers (
a
a
)2
=κ
3ρ (308)
as expected. We continue now by studying a particular choice of f(R), in whicha term quadratic in the Ricci scalar is added to the Einstein-Hilbert action.This model which is called the Starobinsky model is in a remarkable agreementwith the recent observational cosmological data and provides a natural way toinclude an inflationary era. It is also worth noting that this particular modelwas the first model proposed for inflation.
7.11 Starobinsky’s model of inflation
An alternative model to scalar field inflation was proposed by Starobinsky [26].This model simply includes a quadratic term in the Lagrangian of gravity. Tobe more precise, the action reads
S =12κ
∫d4x[R+ αR2
]+ SM [gµν ,ΨM ] (309)
which is, of course, a specific choice of f(R) given by
f(R) = R+ αR2 (310)
where α is a parameter of the model which has dimensions of inverse masssquared as can be easily seen from dimensional analysis. Inflation occurs whenthe quadratic term dominates, moreover there is a graceful exit from the infla-tionary era when the linear term becomes comparable with the former one. Letus motivate this choice here. Firstly, assuming that GR comes as a low curva-ture limit a more fundamental theory, then at early times when the curvature is
52
strong one should include higher order contributions. To second order, the onlyindependent scalar quantities one can form are R2, RµνR
µν and RµνρσRµνρσ.
Any other second order combination depends on these. For instance let us takethe contraction RµνρσR
µρgνσ, then
RµνρσRµρgνσ = RνµσρR
µρgνσ = RµρRµρ (311)
which is the second of the combinations we gave above. Thus, adding all thesecond order combinations (above three) to the Einstein Hilbert action we arriveat
S =12κ
∫d4x
√−g[R+ βR2 + γRµνR
µν + λRµνρσRµνρσ
](312)
where β, γ and λ are parameters with the appropriate units. Now, in 4 − dimthe combination
G ≡ R2 − 4RµνRµν +RµνρσR
µνρσ (313)
which is called Gauss-Bonnet term, is a total derivative. In particular it can beshown that √
−gG = ∂αVα (314)
withV α =
√−gεαβγδε µν
ρσ Γρµβ
[12Rσ
νγδ +13Γσ
λγΓλνσ
](315)
which forces ∫d4x
√−gG =
∫d4x∂αV
α (316)
to be a surface term and therefore it gives no contribution to the field equations,namely ∫
d4x√−gG =
∫d4x
√−g[R2 − 4RµνR
µν +RµνρσRµνρσ
]=
=∫d4x∂αV
α = surface term
As a result we can solve with respect to either one of the three combinations interms of the other two. We have∫
d4x√−gRµνρσR
µνρσ =∫d4x
√−g[−R2 + 4RµνR
µν]
+ surface term
Substituting this back to the action the latter recasts to
S =12κ
∫d4x
√−g[R+ (β − λ)R2 + (γ + 4λ)RµνR
µν]
+ surface term
and since the surface term does not contribute to the field equations we maydrop it and write
S =12κ
∫d4x
√−g[R+ (β − λ)R2 + (γ + 4λ)RµνR
µν]
(317)
Moreover, for the Robertson-Walker metric the Weyl (or conformal) tensor iden-tically vanishes (see [27])
Cµνρσ = 0 (318)
53
but it holds that
CµνρσCµνρσ =
13R2 − 2RµνR
µν +RµνρσRµνρσ (319)
which can be written as
CµνρσCµνρσ = 2
(RµνR
µν − R2
3
)+ G (320)
where G is the Gauss-Bonnet term. Integrating the latter one over spacetimeand using the fact that the left hand side is zero, along with the fact that theintegration of G will give a surface term, we have
0 =∫d4x
√−g2
(RµνR
µν − R2
3
)+ surface term⇒
∫d4x
√−gRµνR
µν =∫d4x
√−gR
2
3+ surface term (321)
Upon further substitution of the above into our action (317), we finally arriveat
S =12κ
∫d4x
√−g[R+
(β +
λ+ γ
3
)R2
](322)
orS =
12κ
∫d4x
√−g[R+ αR2
](323)
where we have setα = β +
λ+ γ
3(324)
and we see that our original theory that includes all the independent quadraticscalar combinations of the Riemann tensor is indeed reduced to Starobinskytheory.
In order to have now a viable model, it must hold that α > 0. Indeed, fromthe stability analysis that we performed in previous chapter, we proved that thespace under investigation is stable if and only if
m2 > 0 (325)
namely, the mass of the scalaron has to be real. Now, from the trace equation(in vacuum) of f(R)
−2f +Rf,R + 32f,R = 0 (326)
which, for f(R) = R+ αR2 reduces to
−R+ 6α2R = 0 ⇒
(2− 16α
)R = 0 (327)
and comparing with the usual form of the Klein-Gordon equation34 we concludethat if we associate a mass M2 > 0 for the scalaron, we have the identification
16α
≡M2 > 0 (328)
34Note that with our conventions for the sign of the metric tensor, the Klein-Gordon equa-tion for a scalar field φ reads (2−M2)φ = 0.
54
therefore, α > 0 as stated. From here on we are going to use M2 as ourparameter. Now, from the modified Friedmann equations (for K = 0)
3(f,RH2 +H ˙f,R) +
12(f −Rf,R) = κρ (329)
(− d2
dt2+H
d
dt− 2H
)f,R = κ(ρ+ p) (330)
for the choice f(R) = R+ 1M2R
2 in the absence of matter (ρ = 0 = p) the aboveread
H − H2
2H+M2
2H + 3HH = 0 (331)
(2−M2)R = 0 (332)
or if we use the form of the box operator on scalars (in a flat FLRW background)
2φ = −( d2
dt2+ 3H
d
dt
)φ (333)
for any scalar φ, we arrive at
R+ 3HR+M2R = 0 (334)
Now, in order to have an inflationary era the following requirement must be met∣∣∣ HH2
∣∣∣ 1 (335)
from which it also follows that
H H3 (336)
Using this, the first two terms on the left hand side of (331) can be droppedand one has
M2
2H + 3HH ≈ 0 (337)
but H 6= 0, and thus we get
H = −M2
6(338)
which, assuming that for t = ti (onset of inflation) → H = Hi, the above canimmediately be integrated to give
H(t) = Hi −M2
6(t− ti) (339)
Therefore, the scale factor will be given by
d(ln a)dt
= H = Hi −M2
6(t− ti) ⇒
a(t) = AeHit−M212 (t−ti)
2(340)
where A is an integration constant. Assuming that for t = ti → a = ai we getA = aie
−Hiti , such that
a(t) = aieHi(t−ti)−M2
12 (t−ti)2
(341)
55
Thus, we see that we have a quasi-exponential expansion so long as
Hi M (342)
which we demand to hold from now on. A general requirement is for the massM to be smaller than the Hubble parameter at any given time during inflation.The Ricci scalar will then be
R = 6(2H2 + H) = 12H2 −M2 (343)
Recall at this point that inflation ends when the slow-roll parameter 35
ε1(t) ≡ −H
H2(344)
becomes comparable to the unity. Namely, the end of inflation is characterizedby the condition
ε1(tf ) ∼ 1 (345)
In our case
ε1(t) ≡ −H
H2≈ M2
61H2
(346)
and thus, inflation lasts so long as
H >M√
6(347)
while, ends at
Hf ∼M√
6(348)
The latter can now be used in order to estimate the duration of inflation (tf−ti).Indeed, for t = tf equation (339) gives
Hf = Hi −M2
6(tf − ti) ⇒
(tf − ti) =6M2
(Hi −
M√6
)(349)
What is of interest now, is the number N of e−folds, which gives a measureof how long does inflation have to last in order to solve the problems for whichit has been introduced to. By definition
N ≡∫ tf
ti
Hdt = ln(af
ai
)= Hi(tf − ti)−
M2
12(tf − ti)2 (350)
Our task is to now relate the above number of e − folds with the value of theslow-roll parameter on the onset of inflation ε1(ti). To do so, set t = ti into(346) to get
ε1(ti) =M2
6H2i
(351)
35Of course, inflation lasts so long as ε1(t) < 1.
56
Having this, we compute
N = Hi(tf − ti)−M2
12(tf − ti)2 =
= −M2
12
[(tf − ti)2 − 2
6M2
Hi(tf − ti) +(
6Hi
M2
)2
−(
6Hi
M2
)2]
=
= −M2
12
[(tf − ti −
6Hi
M2
)2]
︸ ︷︷ ︸≡ 6
M2
+12
6H2i
M2︸ ︷︷ ︸≡1/ε1(ti)
=
= −M2
126M2
+12
1ε1(ti)
=12
(−1 +
1ε1(ti)
)⇒
N =12
(−1 +
1ε1(ti)
)(352)
where in the second line we added and subtracted the same term in order tocomplete the square. Thus, the value ε1(ti) in terms of the number of e− foldsis given by
ε1(ti) =1
2N + 1(353)
A typical value of N that is required for the flatness and horizon problems tobe solved is of the order
N ∼ 70 (354)
which results inε1(ti) ∼ 0, 00709 ∼ 7, 1× 10−3 (355)
Note also that at the end of inflation the Ricci scalar has decreased to
Rf ≈ 12H2f −M2 = M2 (356)
One can see, therefore, that indeed inflation ends when
16M2
R2 ∼ R ∼M2 (357)
For reviews of the Starobinsky model, the reader is also referred to take a lookat [20], [28] as well as [29].
7.12 Dynamics in the Einstein frame
Let us now study Starobinsky’s model in the Einstein frame. Note that in thisframe, in the absence of matter fields, the comparison with usual scalar fieldinflation is immediate since no matter fields are present. This can be seen fromthe action in the Einstein frame (with no matter fields), which is
S =∫d4x√−g[
12κR− 1
2gµν∂µφ∂νφ− V (φ)
](358)
whereV (φ) =
U
F 2
12κ
RF − fF 2
(359)
57
andF = e
√2κ3 φ (360)
This can be seen as Einstein Gravity plus a minimally coupled (in the absenceof matter)36 scalar field which field can be identified with the inflaton. Notethat in our case
f = R+1
6M2R2 (361)
andF = f,R = 1 +
13M2
R (362)
Thus, from the above one can solve with respect to R, to get
R = 3M2(e√
2κ3 φ − 1
)(363)
Having these we can now compute our potential which is given by
V (φ) =3M2
4κ
(1− e−
√2κ3 φ)2
(364)
From this we see that in the limit√κφ 1 the exponential term is highly
suppressed, such that
V (φ) ≈ 3M2
4κ= const. (365)
and one arrives at slow-roll inflation. On the other hand, in the limit√κφ 1
one has
V (φ) ≈ 3M2
4κ
(1− 1 +
√2κ3φ
)2
=12M2φ2 (366)
which tells us that in this region the field oscillates around the origin and we havereheating. To find the relation between the cosmic time in he Einstein framewith the corresponding in Jordan frame, we first note that during inflation
R ≈ 12H2 1 (367)
so that
F = 1 +1
3M2R ≈ 1
3M2R ≈ 4H2
M2(368)
and upon using the definition
dt =√Fdt (369)
we integrate and use the above approximate relation, to get
t =∫ t
ti
√Fdt ≈ 2
M
∫ t
ti
Hdt =
=2M
(Hi(t− ti)−
M2
12(t− ti)2
)(370)
36As has been pointed out before, when matter fields are present, the scalar φ couples tothem.
58
where we have rescaled ti in order to have ti = 0 for t = ti. The end of inflationin Jordan frame
tf ≈ ti +6Hi
M2(371)
now translates to
tf ≈2M
∫ tf
ti
Hdt =2MN (372)
in the Einstein frame. In order now to get a relation between the scale factors,we start by again completing the square, this time in (370) to get
t ≈ 1M
6M2
H2i −
(Hi −
M2
6(t− ti)2
)︸ ︷︷ ︸
≡H(t)
=
=1M
6M2
(H2i −H2) (373)
from which it follows that
H(t)Hi
=
√1− M3
6H2i
t ≈ 1− M2
12H2i
Mt (374)
where we have used H2i /M
2 1 to approximate the square root. In addition,combining (341) and (372) we obtain
a(t)ai
= eM2 t (375)
Substituting all the above into the definition
a(t) =√Fa(t) (376)
we finally arrive at
a(t) ≈ 2MH(t)a(t) ≈
(1− M2
12H2i
Mt
)2Hiai
Me
M2 t (377)
and if we define ai = 2Hiai/M we have
a(t) ≈(
1− M2
12H2i
Mt
)aie
M2 t (378)
and we observe that in the Einstein frame the scale factor grows quasi-exponentially.Consequently, the Hubble parameter in the Einstein frame will be
H(t) =da(t)
dt
a(t)≈ M
2
[1− M2
6H2i
(1− M2
12H2i
Mt
)−2]
(379)
What is good report now, is that in the Einstein frame we can apply the basicinflation theory that we had presented in a previous chapter. The slow-rollparameters can be computed by37
ε(φ) ≡m2
pl
2
(V,φ
V
)2
(380)
37Note that in the Jordan frame we use ε1 and ε2 for the first and second slow-roll parameterrespectively, while in the Einstein frame we will just denote them by ε and η.
59
η(φ) ≡ m2pl
Vφφ
V(381)
where κ has been replaced by the Planck mass through their relation κ ∼ m−2pl .
The number of e− folds (or amount of inflation if you wish) can also be easilyfound by using
N ≈ 1m2
pl
∫ φ
φf
V
V,φdφ (382)
Note that in the slow-roll limit N ≈ N . To find φf now we use the the factthat in the end of inflation
εf ∼ 1 (383)
which when applied for our potential
V (φ) =3M2
4κ
(1− e−
√2κ3 φ)2
(384)
gives
φf =
√32
ln [2√
3− 3] ≈ −0.94mpl (385)
It can be shown that ( see [30])
ε =34
1N 2
+O(
ln2NN 3
)(386)
η = − 1N
+3 lnN4N 3
+54
1N 2
+O(
ln2NN 3
)(387)
For the spectral index38 and scalar to tensor ratio one finds ([30])
ns = 1 + 2η − 6ε (388)
andr = 16ε (389)
respectively. While for the slope of the tensor primordial spectrum (associatedwith the gravitational waves) one has
nt = −2ε (390)
Thus, for the spectral index ns, we have to leading order
ns ≈ 1− 2N
+3 lnN2N 3
− 2N 2
(391)
From observations we know that
r < 0.22 (392)
38This is the slope of the scalar power spectrum which is associated with density perturba-tion. It measures the deviation from scale invariance. The departure from scale invariance isspecified by how far from unity this index lies. That is, for a scale invariant spectrum one hasns = 1.
60
Note that Starobinsky’s model, in particular, predicts a very low scalar to tensorratio r which distinguishes it from the other cosmological models. Observationsof the spectral index also give
ns = 0.960± 0.013 (393)
and using the amplitude of initial perturbations
∆2R =
m4plV
24π2ε(394)
whose value is an observable, we find(V
ε
)1/4
= 0.027mpl = 6.6 · 1016GeV (395)
which fixes the parameter (only one!) of the model (M), to be
M
mpl=
√323
(2.7)2 · 10−4 e
√23
φimpl
(1− e√
23
φimpl )2
= (3.5± 1.2)10−5 (396)
so that we haveM ∼ 10−5mpl ∼ 1014GeV (397)
Now, a typical value for the number of e − folds is 55 for which number weobtain the theoretical values
r = 3.5 · 10−3 (398)
andns = 0.965 (399)
which is in a perfect agreement with the observational value (393). It is ratherremarkable that Starobinsky’s model of inflation which is the 1st inflationarymodel, comes naturally from higher order corrections of the Gravitational ac-tion, and is very economical (only one free parameter), has such a perfect agree-ment with observations! The recent Planck data [31],[32] seems to favour thismodel even more as we discuss below.
61
7.13 Connection with observations
Figure 1: Recent Planck data
In the above plot we have the contraints on inflationary models from therecent39 Planck data [31],[32]. More accurately, above we have the plot oftensor to scalar ratio (r) against the Primordial tilt (ns) and the observationalcontraints in these parameters. In the same plot we also have the theoreticallycomputed range of those parameters for the various inflationary models. Itis worth noting that, as it is obvious from the plot, Starobinsky’s model ofinflation is in a perfect agreement with observations. It sits exactly in the center(orange line) of the region defined by the observational contraints (orange line).In addition, the prediction of a very small tensor to scalar ratio , which is afeature of Starobinsky’s inflation and distinguishes it from other inflationarymodels, seems to be also favoured by data. We see also that the inflationarymodel with potential ∝ φ3 has been ruled out by observations (green line). Torecap, Starobinsky’s model of inflation which has a natural explanation as a highcurvature correction to Gravity and only one free parameter (very economic!)has currently the best possible fitting to data! Interstingly enough, that wasthe first model of inflation ever proposed.
39The present thesis was written in 2013. The same year the results from the Planck satelitewere given.
62
7.14 Starobinsky’s model with a Cosmological constant
Let us now study some features of the Starobinsky model when one adds apositive cosmological constant Λ > 0. Firstly, we recognize dS spacetime as ourmaximally symmetric space, exactly like in GR. Indeed, for f(R) = R + αR2,Λ 6= 0 equation (604) gives
R0 = 4Λ (400)
exactly as in GR. However, we should mention here that this is merely a coinci-dence, due to the fact that the contributions of the quadratic term R2 exactlycancel one another in (604). For a general f(R) = R+αRn, n 6= 2 the situationis different as we will explore in later chapter. We state now, that stabilityrequires that α > 0. Recall that in order to have a stable maximally symmetricspace, the mass of the scalaron has to be real (m2 > 0) as we proved in previoussubsection. There, we had shown that the scalaron mass is given by
m2 ≡ 13
(R0 −
A0
B0
)(401)
where, A0 ≡ f,R
∣∣∣R=R0
and B0 ≡ f,RR
∣∣∣R=R0
. For the case at hand, we have
m2 = −13
(R0 −
1 + 2αR0
2α
)=
16α
(402)
thus, we see that we must have α > 0 in order to be able to exit from theinflationary era, as we have already proven in previous chapter. Now, in thiscase the 00-component of the field equations for a flat FLRW background gives
H − 12H2
H+
12M2H + 3HH =
ΛM2
61H
(403)
while the trace of the field equations, now yields
R+ 3HR+M2R = 4Λ (404)
Again, since during inflation∣∣∣H/H2
∣∣∣ << 1 the first two terms on the left handside of equation (403) can be neglected, and one is left with
12M2H + 3HH ≈ M2Λ
61H⇒
M2H2 + 6H2 dH
dt=M2Λ
3⇒
6M2
H2dH(Λ3 −H2
) = dt (405)
Now, writing H2 = −(
Λ3 −H
2)
+ Λ3 and integrating over dt the latter becomes
∫dt =
6M2
∫ −(
Λ3 −H
2)
+ Λ3
Λ3 −H2
dH =
63
=6M2
[−∫dH +
Λ3
∫1
Λ3 −H2
dH︸ ︷︷ ︸=I
]⇒
M2
6t+ C0 = −H + I (406)
where C0 a constant to be determined in what follows. We compute
I =Λ3
∫1
Λ3 −H2
dH =
√Λ3
∫ d(√
3ΛH)
1−(√
3ΛH)2 (407)
Setting u =√
3ΛH we have
I =
√Λ3
∫du
1− u2=
√Λ3
12
∫(1 + u) + (1− u)(1− u)(1 + u)
du =
=
√Λ3
∫ [1
1 + u+
11− u
]du =
√Λ3
ln∣∣∣∣1 + u
1− u
∣∣∣∣ ==
√Λ3
ln
∣∣∣∣∣∣1 +
√3ΛH
1−√
3ΛH
∣∣∣∣∣∣ (408)
Thus, we finally arrive at
H −√
Λ3
ln
∣∣∣∣∣∣1 +
√3ΛH
1−√
3ΛH
∣∣∣∣∣∣ = −M2
6t− C0 (409)
Some remarks can immediately be drawn from the latter. Firstly, note that inthe Λ → 0 limit one recovers Starobinsky’s inflation
H ≈ −M2
6t− C0 (410)
Indeed, it holds
limΛ→0
√Λ3
ln
∣∣∣∣∣∣1 +
√3ΛH
1−√
3ΛH
∣∣∣∣∣∣ = limΛ→0
√Λ3
ln
∣∣∣∣∣∣√
3ΛH
−√
3ΛH
∣∣∣∣∣∣ == 0 · 0 = 0 (411)
such that
limΛ→0
[H −
√Λ3
ln
∣∣∣∣∣∣1 +
√3ΛH
1−√
3ΛH
∣∣∣∣∣∣]
= limΛ→0
[− M2
6t− C0
]=
= limΛ→0
H − limΛ→0
√Λ3
ln
∣∣∣∣∣∣1 +
√3ΛH
1−√
3ΛH
∣∣∣∣∣∣︸ ︷︷ ︸=0
= limΛ→0
[− M2
6t− C0
](412)
64
and thus
H ≈ −M2
6t− C0 (413)
in the limit Λ → 0. Thus, we see that we recover Starobinsky’s inflation forC0 = −
(Hi + M2
6 ti
). One would like to solve (419) with re spect to H = H(t)
and then integrate again, to find the behaviour of the scale factor a(t). However,from the implicit form of (419) this seems highly unlikely. Nevertheless, thereis something we can do. A game with the derivatives reveals
H =dH
dt=dH
da
da
dt=dH
daa =
dH
daaH (414)
Substituting this into
12M2H + 3HH ≈ M2Λ
61H⇒
we arrive at
M2
6=
H2
Λ3 −H2
H →H=aH dHda
M2
6=
H2
Λ3 −H2
aHdH
da⇒
M2
3da
a=
H2
Λ3 −H2
d(H2) ⇒
M2
3
∫d(ln a) =
∫H2
Λ3 −H2
d(H2)︸ ︷︷ ︸call it I
+C (415)
where C is a constant of integration which, clearly does not affect the number ofe− folds, and thus our final results will be independent of it. Now, to calculateI set z = H2 and A = Λ
3 to get
I =∫
H2
Λ3 −H2
d(H2) =∫
z
A− zdz = −
∫(A− z)−A
A− zdz =
= −∫ [
1− A
A− z
]dz = −
[z +A ln |A− z|
]= −
[H2 +
Λ3
ln∣∣∣Λ3−H2
∣∣∣](416)
such that
M2
3ln a = −
[H2 +
Λ3
ln∣∣∣Λ3−H2
∣∣∣]+ C ⇒
ln a = − 3M2
H2 − ln∣∣∣Λ3−H2
∣∣∣ ΛM2
+ C ⇒
ln
[a
(Λ3−H2
) ΛM2]
= − 3M2
H2 + C (417)
65
Exponentiating the latter one, we finally obtain
a = C(
Λ3−H2
)− ΛM2
e−3
M2 H2(418)
where we have redefined the integration constant eC → C. In total we have thedoublet
H −√
Λ3
ln
∣∣∣∣∣∣1 +
√3ΛH
1−√
3ΛH
∣∣∣∣∣∣ = −M2
6t− C0 (419)
along with
a = C(
Λ3−H2
)− 3ΛM2
e−3
M2 H2(420)
or if we also exponentiate (419)
e
1−√
3ΛH
1 +√
3ΛH
q
Λ3H2
= e−M26H t−C0
H (421)
a = C(
Λ3−H2
)− ΛM2
e−3
M2 H2(422)
More accurately one should have the parametric form a = a(H), t = t(H)which we also give here
a(H) = C(
Λ3−H2
)− ΛM2
e−3
M2 H2(423)
t(H) =6M2
−H +
√Λ3
ln
∣∣∣∣∣∣1 +
√3ΛH
1−√
3ΛH
∣∣∣∣∣∣− C0 (424)
where we have redefined C0 to absorb the multiplicative factor M2/6 in it.Note that even though we do not have an explicit function a(t), we have
found a parametric solution a = a(H), t = t(H) . Having the doublet a =a(H), t = t(H) we can compute the sign of the derivatives that are of specialinterest (e.g. a, a) and find qualitatively features of the behaviour of the system.In addition, the quantity we are really interested in is H2 as this is all one needsin order to relate the number of e−folds N with the slow-roll parameter on theonset of inflation. This will become clear in what follows. Again, one would liketo find a relation ε1(ti) = f(N ) to constrain the value of the slow-roll parameterat the onset of inflation. We start by dividing (405) with 1/H3, to arrive at
12M2 1
H2+ 3
H
H2≈ M2Λ
61H4
but since ε1(t) = −H/H2, it follows that
ε1(t) = −M2
6Λ3
[1H4
− 3Λ
1H2
](425)
66
Multiplying through by H4/ε1 and setting x = H2 it follows that
x2 − M2
6ε1x+
M2Λ18ε1
= 0 (426)
which has solutions
x+,− =M2
61
2ε1
[1±
√1− 8Λε1
M2
](427)
However, demanding to recover the solution
H2 → M2
6ε1(428)
in the limit where the cosmological constant is zero Λ → 0, the root with theminus sign (x−) is not allowed and we have the unique solution
H2 = x+ =M2
61
2ε1
[1 +
√1− 8Λε1
M2
](429)
Now, since ε1 1 for a slow roll inflation, we Taylor expand for small ε1, to get
H2 =M2
61
2ε1
[1 +
√1− 8Λε1
M2
]≈ M2
61
2ε1
[1 + 1− 4Λ
M2ε1
]=
=M2
61ε1
[1− 2Λ
M2ε1
]⇒
H2 =M2
6
[1
ε1(t)− 2ΛM2
](430)
We should mention here, that this relation H2 = f(ε1) is all we need in orderto compute the number of e − folds. Indeed, since the parametric solutiona = a(H) has a dependence only on H2, expressing H2
i ,H2f in terms of ε1(ti)
and ε1(ti) respectively, we can find ai(εi) and af (εf ). Thus, the number ofe − folds can be determined in terms of εi40. Setting t = ti and subsequentlyt = tf to (430) we have
H2i =
M2
6
[1
ε1(ti)− 2ΛM2
](431)
as well as
H2f =
M2
6
[1− 2Λ
M2
](432)
where we also took into account that εf ∼ 1. As stated before, the integrationconstant C in the expression for the scale factor drops out of the calculation sincewe are interested in the ratio af/ai which enters in the number of e − folds.The number of e− folds is given by the expression
N ≡∫ tf
ti
Hdt = ln(af
ai
)(433)
40Note that since at the end of inflation εf ∼ 1 (fixed) the final result will depend only onεi.
67
we compute
af
ai=
(Λ3 −H
2f
Λ3 −H
2i
)− ΛM2
e−3
M2 (H2f−H2
i ) (434)
Using the above relations we obtain
N = ln(af
ai
)= − Λ
M2ln
(Λ3 −H
2f
Λ3 −H
2i
)− 3M2
(H2f −H2
i ) =
= − ΛM2
ln
(2Λ3 − M2
62Λ3 − M2
61
ε1(ti)
)− 1
2
(1− 1
ε1(ti)
)(435)
Now, since
ε1(ti) 1 ⇒ 1ε1(ti)
1 (436)
we can approximate
2Λ3− M2
61
ε1(ti)≈ −M
2
61
ε1(ti)(437)
such that
ln
(2Λ3 − M2
62Λ3 − M2
61
ε1(ti)
)≈ ln
[ε1(ti)
(1− 4Λ
M2
)](438)
In addition, since ε1(ti) is small, the term ln ε1(ti) can be neglected comparedto 1/ε1(ti). Then, one is left with
N ≈ − ΛM2
ln(
1− 4ΛM2
)+
12
(−1 +
1ε1(ti)
)⇒
ε1(ti) =1
(2N + 1) + ΛM2 ln
(1− 4Λ
M2
)2 (439)
Let us now comment on our result. Firstly, note that in the limit Λ → 0 werecover our result from the previous chapter
ε1(ti)Λ→0−→ 1
(2N + 1)(440)
as we should. Thus, we see that if our model is seen as a unification model(of inflation and late-time accelerated expansion) which means Λ/M2 1, werecover Starobinsky’s model immediatelly. Secondly, we notice that the secondterm on the denominator of the above expression is always negative, for eithersign of Λ, resulting in the increase of the slow-roll parameter. Considering anarbitrary ti → t in the latter, we can can the second slow-roll parameter η, byits definition
η ≡ ε
εH=d ln εdN
= −2ε (441)
68
7.15 Theory in the Einstein frame
Again, one is interested in studying the dynamics of the model in the Einsteinframe and spot any non-negligible changes that might exist. Note that now,since there is also a term −2Λ in the action, after the conformal transformationthe potential will acquire an extra contribution with respect to the model Λ = 0.Indeed, on going to the Einstein frame, the following will hold∫
d4x√−g(−2Λ) =
∫d4x√−gF−2(−2Λ) (442)
but in our caseF = e
√2κ3 φ (443)
so that ∫d4x
√−g(−2Λ) =
∫d4x√−g(−2Λ)e−2
√2κ3 φ (444)
Therefore, the action in the Einstein frame will be
S =∫d4x√−g[
12κR− 1
2gµν∂µφ∂νφ− V (φ)− 1
2κ2Λe−2
√2κ3 φ
](445)
where
V (φ) =3M2
4κ
(1− e−
√2κ3 φ)2
(446)
Thus, we have an effective potential
Veff (φ) ≡ V (φ) +Λκe−2√
2κ3 φ =
3M2
4κ
(1− e−
√2κ3 φ)2
+Λκe−2√
2κ3 φ (447)
and the action (in the Einstein frame) is written as
S =∫d4x√−g[
12κR− 1
2gµν∂µφ∂νφ− Veff (φ)
](448)
Note however, that by taking the limit√κφ 1, we reproduce the same slow-
roll regime as in the case with Λ = 0. Indeed, the condition√κφ 1 implies
that the exponential terms in the potential are highly suppressed, namely
e−√
κ3 φ → 0, e−
√2κ3 φ → 0 (449)
and one has
Veff (φ) ≈ 3M2
4κ= const. (450)
which is identical to (365). As a result, the addition of a cosmological constantdoes not spoil the slow-roll regime of the Starobinsky model. On the otherhand, the reheating regime does acquire a modification which, however is highlysuppressed when the cosmological constant is of the order of energy scales ofthe vacuum today, namely when Λ/M2 1. Indeed, in the limit
√κφ 1 to
second order, we have
e−√
κ3 φ ≈ 1−
√κ
3φ+
(√κ
3φ
)2
(451)
69
and the effective potential goes as
Veff (φ) = V (φ) +Λκe−2√
2κ3 φ =
3M2
4κ
(1− e−
√2κ3 φ)2
+Λκe−2√
2κ3 φ ≈
≈ 3M2
4κ
1− 1 +
√2κ3φ−
(√2κ3
)2
φ2
+Λκ
1− 2
√2κ3φ+
(2
√2κ3
)2
φ2
≈ M2
2φ2
(1−
√2κ3φ
)2
+Λκ
(1− 2
√2κ3φ+
8κ3φ2
)≈
≈(M2
2+
8Λ3
)φ2 − 2Λ
√23κφ+
Λκ
+O(√κφ)3
)⇒
Veff (φ) ≈(M2
2+
8Λ3
)φ2 − 2Λ
√23κφ+
Λκ
(452)
Upon completing the square, the last reaches
Veff (φ) ≈ b(φ− φ0)2 +Λκb
(1− 2
3Λb
)(453)
where
b =M2
2+
8Λ3
(454)
and
φ0 =Λb
√23κ
(455)
from which we observe that compared to the case Λ = 0 we have a shift φ0 ofthe center of oscillations together with a constant shifting
V0 ≡Λκb
(1− 2
3Λb
)(456)
on the potential.
8 More General Theories of Modified Gravity
In this section we shall explicitly derive the field equations for more generalmodifications of gravity. To be more specific, the field equations of scalar-tensor f(R,φ), Gauss-Bonnet f(R,G) and of a general theory f(R,RµνR
µν ,RµνρσR
µνρσ, RµνρσRµρRνσ) will be obtained. In the case of Gauss-Bonnet we
also give the field equations for an FLRW background, i.e. the modified Fried-mann equations for such theories. Such theories can give an explanation for thelate-time accelerated expansion ([7], [5]). They can also be used as unificationmodels [33]. We start now with the scalar-tensor f(R,φ) case.
8.1 Field Equations for a general f(R, φ) theory
For this theory, our starting action is
S = Sf(R,φ) + SM [gµν ,ΨM ] =∫d4x
√−g[ 12κLf + LM
]=
70
=12κ
∫d4x
√−g[f(R,φ)− ω(φ)
2gµν∇µφ∇νφ− V (φ)
]+∫d4x
√−gLM
(457)
where the independent fields are the metric tensor gµν as well as the scalar field-φ and we have to vary with respect to both independently in order to obtainthe full field equations. Firstly, let us vary with respect to the metric. We have
δgS = 0 ⇒
0 =∫d4x
√−gδgµν
[− gµν
2Lf +(Rµν+Dµν)f,R−
ω
2∇µφ∇νφ−2κ
12T (M)
µν
](458)
Therefore−gµν
2Lf + (Rµν +Dµν)f,R −
ω
2∇µφ∇νφ = κT (M)
µν (459)
where
Lf = f(R,φ)− ω(φ)2
gµν∇µφ∇νφ− V (φ) (460)
andT (M)
µν = − 2√−g
δSM
δgµν(461)
Varying now with respect to φ we obtain
δφS = 0 ⇒
0 =∫d4x
√−g[f,φδφ− (δφ)
ω,φ
2∇µφ∇µφ− ω∇µφδ(∇µφ)− V,φδφ
](462)
Now, since δ and ∂µ commute and the covariant derivative (∇µ) is reducedto partial one (∂µ) when acting on scalars we can write the third term in theintegral as follows
ω∇µφδ(∇µφ) = ω∇µφδ(∂µφ) = ω∇µφ∂µ(δφ) =
= ω∇µφ∇µ(δφ) = ∇µ
[ω(∇µφ)δφ
]− (δφ)∇µ
(ω∇µφ
)=
= ∇µ
[ω(∇µφ)δφ
]− (δφ)
[ω,φ∇µφ∇µφ+ ω∇µ∇µφ
]=
= ∇µ
[ω(∇µφ)δφ
]− (δφ)
[ω,φ∇µφ∇µφ+ ω2φ
](463)
where in the second line we have used Leibniz rule and in the third one we haveapplied the chain rule to write ∇µω = ∂µω = ω,φ∂µφ = ω,φ∇µφ. Substitutingthe latter result into the variation of the action, we obtain
0 =∫d4x
√−g(δφ)
[f,φ −
12ω,φ∇µφ∇µφ+ ω,φ∇µφ∇µφ+ ω2φ− V,φ
]−∫d4x
√−g∇µ
[ω(∇µφ)δφ
]︸ ︷︷ ︸
surface term
⇒
0 =∫d4x
√−g(δφ)
[f,φ +
12ω,φ∇µφ∇µφ+ +ω2φ− V,φ
](464)
71
where we have dropped the surface term that appeared. Note that the last onemust be equal to zero for any variation δφ, therefore we conclude that
f,φ +12ω,φ∇µφ∇µφ+ +ω2φ− V,φ = 0 (465)
Thus, an f(R,φ)-theory is described by the set of field equations
−gµν
2
(f − ω(φ)
2gαβ∇αφ∇βφ− V (φ)
)+ (Rµν +Dµν)f,R
−ω2∇µφ∇νφ = κT (M)
µν (466)
f,φ +12ω,φ∇µφ∇µφ+ +ω2φ− V,φ = 0 (467)
Note that for φ = 0, that is no scalar field is present, the theory reduces tof(R) as expected. We continue now by giving the field equations for the Gauss-Bonnet theory.
8.2 Gauss-Bonnet Gravity
In this chapter we study f(R,G)-theories of Gravity, the so-called Gauss-BonnetGravities. On the first part of this section we will derive the field equations fora general f(R,G)-theory. On the second part we are going to concentrate onFLRW universes, namely will derive the modified Friedmann equations arisingin the Gauss-Bonnet Gravity. The motivation of including the Gauss-Bonnetterm
G ≡ R2 − 4RµνRµν +RµνρσR
µνρσ (468)
is that this is the second Lovelock scalar. The first one is the Ricci scalar41.Lovelock scalar are specific scalars which when linearly enter the action, the fieldequations that one obtains include no higher than 2nd-order partial derivativesof the metric tensor. In general, the so-called Lovelock Gravity is a generaliza-tion of Einstein’s theory. In the case of a 4− dim spacetime Lovelock’s theorycoincides with GR because of the fact that the Gauss-Bonnet term is a topo-logical invariant and therefore does not modify the field equations. For higherdimensions Lovelock’s Gravity differs from GR but the field equations continueto have only up to 2nd-order partial derivatives of the metric tensor. Let usproceed now to obtain the field equations.
8.3 Field Equations for a General f(R,G)-Theory
A generalization of the f(R)-Gravity is the so-called f(R,G)-Gravity in whichnow f is a general function of both the Ricci scalar and the Gauss-Bonnet term.We should mention here, that the Gauss-Bonnet term alone is a total deriva-tive42. However, when coupled to some scalar it no longer reduces to a surfaceterm. Therefore, it is interesting to study models where the Gauss-Bonnet term
41See appendix42Therefore does not contribute to the equations of motion.
72
is coupled to some scalar43. Let us now derive the field equations for f(R,G)-Gravity. A general f(R,G)-Theory is given by the following action,
S = SG−B + SM =∫d4x
√−g[ 12κf(R,G) + LM
](469)
where G is the so-called Gauss-Bonnet term and is given by
G ≡ R2 − 4RµνRµν +RµνρσR
µνρσ (470)
Varying the above action with respect to the metric tensor we obtain
0 = δS =∫d4x[ 12κf(R,G)δ
√−g+ 1
2κ√−gδf(R,G)+
δ(√−gLM )δgµν
δgµν]
(471)
Now, using
δ√−g = −1
2√−ggµνδg
µν (472)
δf(R,G) = f,RδR+ f,GδG (473)
and the definition
Tµν ≡ −2√−g
δ(√−gLM )δgµν
(474)
we arrive at
0 =∫d4x
√−g[− gµν
2fδgµν + f,RδR+ f,GδG − κTµνδg
µν]
(475)
Furthermore, as we have proved, it holds that
δR = (Rµν +Dµν)δgµν (476)
and, in addition
δG = 2RδR− 4δ(RµνRµν) + δ(RµνρσR
µνρσ) (477)
Upon using these, the latter equation becomes
0 =∫d4x
√−g[− gµν
2fδgµν + (f,R + 2Rf,G)δR
−4f,Gδ(RµνRµν) + f,Gδ(RµνρσR
µνρσ)− κTµνδgµν]
(478)
Now, setting F = f,G to equation (683) and integrating over spacetime, wearrive at ∫
d4x√−gf,Gδ(RµνρσR
µνρσ) =∫d4x
√−gδgµν2f,GRµ
αβγRναβγ + 2∫d4x
√−gf,GRα
νρσδRανρσ (479)
43A stability analysis for viability of Gauss-Bonnet Gravity has been carried out in [34],[25].For more information on cosmological perurbations in modified Gravity, see [35].
73
As we prove in the chapter of general variations in the appendix, for a generaltensorial field Tα
νρσ(x) it holds that∫d4x
√−g Tα
νρσδRανρσ =
=∫d4x
√−gδgµν
[∇[ρ∇σ]Tµνρσ +∇ρ∇σTµρ[νσ] +∇ρ∇σTσµ[ρν]
](480)
and by setting Tµνρσ = f,GRµνρσ and employing the symmetries of the Riemanntensor, it follows that ∫
d4x√−gf,GRα
νρσδRανρσ =
=∫d4x
√−gδgµν
[2∇(ρ∇σ)(f,GRµρνσ)
](481)
Collecting the various terms, we finally obtain∫d4x
√−gf,Gδ(RµνρσR
µνρσ) =∫d4x
√−gδgµν
[2f,GRµ
αβγRναβγ + 4∇(ρ∇σ)(f,GRµρνσ)]
(482)
As far as the term f,Gδ(RµνRµν) is concerned, we use∫d4x
√−gT νσδRνσ =
=∫d4x
√−gδgµν
[− 1
2∇λ∇µ(Tλ
ν + Tνλ) +
12gµν∇α∇βTαβ +
122Tµν
](483)
and set Tµν = f,GRµν , to get∫d4x
√−gf,GR
νσδRνσ =∫d4x
√−gδgµν
[−∇λ∇(µ(f,GRν)λ) +
12gµν∇α∇β(f,GRαβ) +
122(f,GRµν)
](484)
Also, using equation (684) for F = f,G and integrating over spacetime, we obtain∫d4x
√−gf,Gδ(RµνR
µν) =∫d4x
√−gf,G
[2RµνδRµν + 2RµαR
ανδg
µν]
=
=∫d4x
√−gδgµν
[− 2∇λ∇(µ(f,GRν)λ) + gµν∇α∇β(f,GRαβ)+
2(f,GRµν) + 2f,GRµαRα
ν
](485)
Substituting all the above into equation (478) we finally arrive at
0 =∫d4x
√−g[− gµν
2fδgµν + (f,R + 2Rf,G)δR
−4f,Gδ(RµνRµν) + f,Gδ(RµνρσR
µνρσ)− κTµνδgµν]
=
74
=∫d4x
√−gδgµν
[− gµν
2f + (Rµν +Dµν)(f,R + 2Rf,G)
+8∇λ∇(µ(f,GRν)λ)− 4gµν∇α∇β(f,GRαβ)−42(f,GRµν) + 2f,GRµαR
αν
+2f,GRµαβγRναβγ + 4∇(ρ∇σ)(f,GRµρνσ)− κTµν
](486)
and since the last is true for any variation δgµν , we conclude that
−gµν
2f + (Rµν +Dµν)(f,R + 2Rf,G) + 8∇λ∇(µ(f,GRν)λ)
−4gµν∇α∇β(f,GRαβ)− 42(f,GRµν) + 2f,GRµαRα
ν
+2f,GRµαβγRναβγ + 4∇(ρ∇σ)(f,GRµρνσ) = κTµν (487)
The latter are the field equations for a general f(R,G)-Theory. Recall thatf,R ≡ ∂f
∂R and f,G ≡ ∂f∂G and the differential tensorial operator Dµν has defined
asDµν ≡ gµν2−∇µ∇ν (488)
8.3.1 Gauss-Bonnet term for a K = 0 FLRW Cosmology
In this section we compute the Gauss Bonnet term for a flat (K = 0) FLRWCosmology and we confirm that it is indeed a total derivative. Of course, sincethere is no preferred spatial direction in FLRW universes, the latter is a timederivative. In order to compute the Gauss-Bonnet term we need to find theinteresting contractions, R2, RµνR
µν and RµνρσRµνρσ which , themselves, will
be proven to be very useful in the models we are going to study later on. ForK = 0 we have
g00 = −1, gij = a2(t)δij (489)
gijgij = 3,
√−g = a3 (490)
as well asR00 = −3
a
a= −3(H +H2) (491)
Rij =
[a
a+(a
a
)2]gij = (H + 3H2)gij (492)
such thatR = R00g
00 +Rijgij = 6(H + 2H2) (493)
Therefore, one hasR2 = 36(H2 + 4H2H + 4H4) (494)
Now, regarding the RµνRµν term, we have
RµνRµν = (R00)2(g00)2 +RijR
ij = 9(H +H2)2 + 3(H + 3H2)2 =
= 3[4H2 + 12HH2 + 12H4
]⇒
RµνRµν = 12
[H2 + 3HH2 + 3H4
](495)
75
From the above results we can see that in FLRW the combination RµνRµν− 1
3R2
is also a total derivative. Indeed, using the latter results it follows that
RµνRµν − 1
3R2 = −H2(H +H2) = −H2 a
a(496)
so that ∫d4x
√−g[RµνR
µν − 13R2
]= −
∫d4xa3H2 a
a=
= −∫d4xa2a = −1
3
∫d4x
d
dt(a3) = surface term (497)
a surface term as stated. Let us now compute the non-vanishing componentsof the Riemann tensor. Due to the symmetries of the FLRW spacetime44, theonly non-vanishing Christoffel symbols are
Γij0 =
a
aδij , Γ0
ij = aaδij (498)
Of course, the form of the non-vanishing Christoffel symbols does not come asa surprise. Since there is not preferred direction in Friedmann Cosmologies,vectorlike components such as Γ0
i0, Γi00 and so on, must identically vanish. In
addition, since the only available tensor is the metric tensor with the form
g00 = −1, gij = a2(t)δij (499)
we expect that the non-vanishing Christoffel symbols will be given in terms ofthe δij , the scale factor and its derivatives, as is indeed the case. Additionally,the fact that the metric tensor is only time dependent forces the Christoffelsymbols also to depend only on time, as can be easily seen from their formabove. Therefore, the following is true
Γρµν = Γρ
µν(t) ⇒
∂iΓρµν = 0 (500)
for every ρ, µ, ν ∈ 0, 1, 2, 3 and i ∈ 1, 2, 3.Being equipped with these rela-tions, we are now in a position to compute the non-vanishing components of theRiemann tensor. We start by computing the spatial part Ri
jkl firstly. We have
Rijkl = 2 ∂[kΓi
l]j︸ ︷︷ ︸=0
+2Γie[kΓe
l]j =
= Γi0kΓ0
lj − Γi0lΓ
0kj + Γi
mkΓmlj︸ ︷︷ ︸
=0
−ΓimlΓ
mkj︸ ︷︷ ︸
=0
= Hδikaaδlj −Hδi
l aaδkj (501)
and using that δij = gij/a2 the latter assumes the form45
Rijkl = H2(δi
kglj − δilgkj) = 2H2δi
[kgl]j (502)
44Note that the computations here are done for a flat FLRW geometry, namely K = 045One can anticipate this form for Ri
jkl as the 3 − dim spatial space is maximally sym-
metric. Recall that the Riemann tensor for a maximally symmetric space is given byRabcd = K(gacgbd − gadgbc), for some constant K, where a,b,c,d run over the dimension-ality of the manifold. Note also, that for a maximally symmetric space the Ricci scalar is aconstant.
76
such thatRi
jklRjkl
i = 9H4 (503)
Again, we note that vectorlike components such as, Ri000, R
00i0 and so forth,
vanish identically since there is no special spatial direction. We now proceed tocompute the components Ri
0j0. For these, one has
Ri0j0 = 2∂jΓi
0]0 + 2Γie[jΓ
e0]0 =
= −∂0Γij0 + Γi
0j Γ000︸︷︷︸
=0
− Γi00︸︷︷︸
=0
Γ0j0 + Γi
jk Γk00︸︷︷︸
=0
−Γik0Γ
kj0 =
= −
[a
a−(a
a
)2]δij −
(a
a
)2
δikδ
kj︸︷︷︸
=δij
=
=[− aa
+H2 −H2
]δij = − a
aδij ⇒
Ri0j0 = − a
aδij (504)
Taking all the above into account we arrive at
RµνρσRµνρσ = 12
[H2 + 2HH2 + 2H4
](505)
Thus, we have for the Gauss-Bonnet term
G = R2 − 4RµνRµν +RµνρσR
µνρσ =
= 12 · 2(HH2 +H4) = 24H2(H +H2) ⇒
G = 24H2(H +H2) (506)
so that√−gG = 24a3 a
2
a2
a
a= 24a2a =
d
dt(8a3) ⇒
√−gG =
d
dt(8a3) (507)
Indeed a total derivative.
8.3.2 Modified Friedmann equations for f(R,G)
Using the same method (variation with respect to a(t)) as we did when arrivingat the modified Friedmann equations in the case of f(R), we have here
0 = δa
∫d4x
(12κf(R,G) +
√−gLM
)=
=∫d4x
(3a2fδa+ a3(f,RδaR+ fGδaG)δa+
δ(√−gLM )δa
δa
)(508)
from which it follows that
f
2− 1
2(Rf,R + Gf,G) + AfR + 4H2AfG
77
+(2H + 3H2)f,R = −κp (509)
where A ≡ d2/dt2 + 2Hd/dt. This is the 1st modified Friedmann equationfor Gauss-Bonnet Gravity. The second one is obtained by varying again withrespect to N(t) and taking the gauge N(t) = 1 in the end of the day, as we didmany times till now. After doing so, we arrive at
f − (Rf,R + Gf,G) + 6Hf,R + 24H3f,G + 6H2f,R = 2κρ⇒ (510)f
2− 1
2(Rf,R + Gf,G) + 3Hf,R + 12H3f,G + 3H2f,R = κρ (511)
which is the 2nd modified Friedmann equation for f(R,G).
8.4 Field equations of a more general theory
We derive here the field equations for a general
f(R,RµνRµν , RµνρσR
µνρσ, RµνρσRµρRνσ)− theory (512)
We should mention here that this type of theory has not been fully stud-ied in literature due to its complexity. The scalar combination that is re-sponsible for such a complexity is RµνρσR
µρRνσ which gives a 3rd-order con-tribution of the Riemann tensor. Indeed, the field equations of the theoryf(R,RµνR
µν , RµνρσRµνρσ) have already been obtained and the theory has been
studied in the literature to some small degree. Both of these theories includedifferential equations for the metric tensor that are of 4− th-order. This is easyto conclude from the functional form of ′f ′ which includes scalar combinationsof the Riemann tensor, however no derivatives of it. For instance, non-local46
theories of the type f(R,2R) possess field equations that contain derivatives ofthe metric tensor that are up to 6 − th order. Let us demonstrate here a wayto find what the degree of the field equations will be before obtaining the fieldequations for any given theory. We start by noting that the Riemann tensor
Rµνρσ = 2∂[ρΓ
µσ]ν + 2Γµ
α[ρΓασ]ν (513)
contains terms of derivatives of the metric tensor up to second order. Thisfollows immediately from the fact that the Christoffel symbols
Γαµν =
12gαλ(∂µgλν + ∂νgλµ − ∂λgµν) (514)
include first order derivatives of the metric tensor. We symbolically write
Γαµν ∼ Γ ∼ (∂g) (515)
Then,Rµ
νρσ ∼ R(1) ∼ ∂Γ ∼ (∂2g) (516)
where we have defined the index-free shorthand notation R(1) for the Riemanntensor, Γ for the Christoffel symbols, and g for the metric tensor. Being con-tractions of the Riemann tensor, the Ricci tensor and scalar will also behaveas
R(2) ∼ (∂2g) (517)
46The non-locality arises due to 2, since the box operator contains partial derivatives which,of course, express non-locality.
78
R(3) ∼ (∂2g) (518)
where R(2) and R(3) are shorthand notations for the Ricci tensor and scalarrespectively. It then, immediately follows that any given function
f(R(1), R(2), R(3), g) = f(∂2g, ∂g, g) (519)
will give field equations that are of 4−th order with respect to the metric tensor.Indeed, the variation of the action
S =∫d4x
√−gf(R(1), R(2), R(3), g) + SM [gµν ,Ψ] =
=∫d4x
√−gf(∂2g, ∂g, g) + SM [gµν ,Ψ] (520)
will yield a contribution
δf(∂2g, ∂g, g) ∼ f,XδX + f,Y δY + fgδg (521)
where we have set X ≡ (∂2g) and Y ≡ (∂g). Focusing only on the term thatcontains X, and suppressing any other dependence, we see that in order toobtain a term that is proportional to (δg) in the action, one needs to partiallyintegrate the term
f,XδX = F (∂2g)δ(∂2g) (522)
twice in order to isolate the term (δg) and get a contribution that goes like
∼ (δg)∂2F (∂2g) ∼ (δg)(∂4g) (523)
where we used the fact that the derivative of the function f(∂2g) with respectto ∂2g will give a function F which also depends on ∂2g. Thus, from the latterwe see that the equations of motion will contain derivatives of the metric tensorwhich are up to 4 − th order. We state now that the theory f(R,2R) willgive field equations that are of 6 − th order. Indeed, since the box operator 2
includes in it second order derivatives, the term 2R goes as
2R ∼ ∂2R (524)
but sinceR ∼ ∂2g (525)
one has2R ∼ ∂4g (526)
As a result, setting 2R = A we will have
δf(R,2R) ∼ (δA)fA ∼ F (A)(δA) ∼ F (∂4g)(δ∂2) (527)
where again we have used the fact that the derivative of f(A) with respect to Awill in general yield a function F of A and we have only focused on the termsof highest derivatives. Once again one must partially integrate twice the latterin order to remove the two partial derivatives of the term (δ∂2g) and thereforethe above will give a contribution
δf(R,2R) ∼ F (∂4g)(δ∂2g) ∼ (δg)∂2F (∂4g) ∼ (δg)(∂6g) (528)
79
to the field equations, which has in it 6 − th order derivatives of the metrictensor. Thus, we see that indeed the theory f(R,2R) is a 6 − th order theoryof Gravity. Such theories are equivalent to Einstein theory plus two additionalscalar fields propagating in spacetime coupled with matter fields. Recall that aswe have already shown, specific 4−th order theories of Gravity, such as f(R) areequivalent to GR with one additional scalar field coupled to matter. Therefore,one can easily by induction arrive to the following statement. Statement: Ageneral theory of Gravity given by
S =∫d4x
√−gf(R,2R, ...,2nR) + SM [gµν ,Ψ] (529)
where R is the Ricci scalar, n ∈ N, possesses field equations that are of (2n+4)-order and the theory is equivalent to GR plus (n + 1) scalar fields. Let usnow come back to the derivation of the field equations. We proceed as follows.Firstly, we write down our action
S =12κ
∫d4x
√−gf(R,RµνR
µν , RµνρσRµνρσ, RµνρσR
µρRνσ) + SM [gµν ,ΨM ]
Then, we setX = RµνR
µν (530)
Y = RµνρσRµνρσ (531)
Z = RµνρσRµρRνσ (532)
such thatS =
12κ
∫d4x
√−gf(R,X, Y, Z) + SM [gµν ,ΨM ] (533)
Varying with respect to the metric tensor it follows that
0 = δgS =
=∫d4x
√−g[−gµν
2fδgµν + f,RδR+ f,XδX + f,Y δY + f,ZδZ − κTµνδg
µν]
(534)
and by using the results we have derived for variations of the Riemann and Riccitensor as well as the Ricci scalar, we finally arrive at
(Rµν +Dµν)f,R + 2f,XRµαRαν − 2∇λ∇µ(f,XR
λν)
+gµν∇α∇β(f,XRαβ) + 2(f,XRµν) + 4∇ρ∇σ(f,Y Rµρνσ) + f,Y Rαβγ
µ Rναβγ
+∇[ρ∇σ](f,ZRµρRνσ) +∇ρ∇σ[f,Z(RµνRρσ −Rσ(µRν)ρ)
]+∇λ∇µ
[f,ZRα(λν)βR
αβ]
+ gµν∇κ∇λ[f,ZRακβλR
αβ]
+ 2(f,ZRαµβνRαβ)
+RαβµσR
βσRανf,Z + 2RανρσR
σµ R
ρα f,Z = κTµν (535)
where f,N ≡ ∂f∂N with N = R,X, Y, Z. Recall also that parentheses denote
symmetric parts of tensors and that Dµν = gµν2 − ∇µ∇ν . The latter are thefield equations for an f(R,RµνR
µν , RµνρσRµνρσ, RµνρσR
µρRνσ)-theory. We seethe complexity arising here since only the field equations took (almost!) half apage to be written down.
80
9 Primordial Black hole Formation
As it was explained in a former section, it is believed that there exists an earlystage of the universe where the last is dominated by a scalar field. In theslow-roll regime of this scalar field φ (inflanton) the density associated withthe latter is approximately constant and results in an approximate effectivecosmological constant. As we have mentioned, an empty space with a positivecosmological constant is called a de-Sitter space. Even though de-Sitter spaceis stable under classical perturbations, the story changes when one includesQuantum-mechanics. Then, Quantum-mechanical effects lead to black hole-pair production which in turn can produce a non-perturbative instability. Asit is known, in pure Einstein Gravity (plus a cosmological constant Λ) the rateper unit volume of creating a black hole-pair (during Inflation) goes as
Γ ∝ e−π/ΛG (536)
and from that we can infer that the probability of pair creation is relatively smalland starts becoming considerable for energies close to Plank scale. However,including a Gauss-Bonnet term with proportionality constant α > 0 one obtains
Γ ∝ e−π/ΛG+4πα/G (537)
as has been shown in [36]. Thus, one sees that even though the field equations(Einstein’s here) assume no modifications, the pair creation is enhanced in aconsiderable way when the topological Gauss-Bonnet term is included. Whendealing with modified theories of Gravity, therefore, one may ask how do theadditional terms of these theories modify the pair creation rate? This will beconsidered in what follows. In particular, we study how does the creation ratechange compared to GR, for the theory
S =12κ
∫d4x
√−g[R+ αR2 − 2Λ
](538)
which is Starobinsky’s theory plus a cosmological constant , the dynamics ofwhich studied in former chapter.
9.1 Pair creation of black holes in Einstein Gravity
Using the semi-classical treatment of pair creation (following from the No −boundary Proposal. See [37] for an exact definition and subsequent analysis)one finds that the pair creation rate is given by
Γ ∝ e−(Iobj−Ibg) (539)
where Iobj and Ibg are, the Euclidean action of the object under considerationand the Euclidean action of the background spacetime. Thus, the pair creationof an object in a given background spacetime is given the ratio of the probabil-ity47 to have the universe with this object contained in it over the probabilityto have it empty48
47From the No− boundary Proposal follows that the eigenfunction of a universe X is givenby Ψ ∝ e−IX , with IX Euclidean. Therefore, the probability of ′measuring′ a universe X isP ∝ |Ψ|2.
48The term empty here refers to the absence of objects like black holes, etc.
81
9.2 Pair creation for the theory f(R) = R + αRn, n 6= 2
Let us first seek maximally symmetric solutions for the case
f(R) = R+ αRn , n 6= 1 (540)
Thenf,R = 1 + nαRn−1 (541)
and (604) givesR+ nαRn − 2R− 2αRn = −4Λ ⇒
R+ (2− n)αRn = 4Λ (542)
Let us now consider the case n 6= 2, as the case n = 2 is of special interest andwill be dealt in the next subsection. Furthermore let us set Λ = 0. Then, aswe have already stressed out, even in the absence of a cosmological constant,our space possesses de-Sitter/anti-de-Sitter solutions. Indeed, from the latterwe have
R[1− (n− 2)αRn−1
]= 0 (543)
from which it follows that one maximally symmetric solution is Minkowski space-time (R = 0 case) and for n 6= 2 there are also maximally symmetric (meaningconstant curvature) solutions satisfying
Rn−1 =1
(n− 2)α(544)
Now, the cases n = 2k + 1 with k ∈ N, and n 6= 2k + 1 should be consideredseparately.Case 1: n = 2k + 1 , k ∈ N . In this case
R2k =1
(n− 2)α(545)
and solutions exist only when
(n− 2)α > 0 (546)
Then
R = ±[
1(n− 2)α
] 1n−1
= ±[(n− 2)α
] 11−n
, n = 2k + 1, k ∈ N (547)
and we see that both de-Stter and anti-de-Sitter solutions exist, correspondingto plus and minus signs of the above equation respectively. The Ricci tensor isthen given by
Rµν = ±14
[(n− 2)α
] 11−n
gµν (548)
Case 2: n 6= 2k + 1 , k ∈ N . Here, solutions exist for either sign of thequantity (n − 2)α. Depending on the latter sign one can have an either de-Sitter or anti-de-Sitter solution but not both. More precisely,
R = + [(n− 2)α]1
1−n (549)
82
9.3 Pair creation for the theory f(R) = R + αR2
We now consider the case n = 2 and note that for α = 16M2 > 0 we get the
Starobinsky model . In this case (n = 2) the second term in equation (542)exactly vanishes and one is left with
R = 4Λ (550)
and consequentlyRµν = Λgµν (551)
so we observe that in this case the maximally symmetric solution coincides withthat of GR and is the de-Sitter one49. Writing down our action
I =12κ
∫d4x
√−g[R+ αR2 − 2Λ
](552)
for the case above (maximally symmetric spaces) the Euclidean action for acompact manifold, reaches
IE = −ΛκV4 −
8ακ
Λ2V4 (553)
Thus, for empty dS space one has
IE [dS] = −ΛκV4[dS]− αΛ2V4[dS] (554)
where V4[dS] = 83π
2L4. And for dS space with the presence of a Nariai blackhole in it
IE [Nariai] = −ΛκV4[Nariai]−
8ακ
Λ2V4[Nariai] (555)
where V4[Nariai] = 169 π
2L4. So that
IE ≡ IE [dS]− IE [Nariai] =(Λκ− 8α
κΛ2)V (556)
whereV ≡ V4[dS]− V4[Nariai] =
89π2L4 (557)
Following the same prescription with the preceding section, we have for the paircreation rate
Γ ∝ e−IE [Nariai]
e−IE [dS]= eIE = e
(Λκ−
8ακ Λ2
)89 π2L4
(558)
but
L =
√3Λ
(559)
and thereforeΓ ∝ e−
πL23G +8 π
G α (560)
From the last one we see that compared to Einstein Gravity, here we havean additional contribution coming from the quadratic term of the Ricci scalar
49Assuming that Λ > 0.
83
that we added to the action. Moreover, as we have shown in previous chapterα > 0 and thus the probability is enhanced compared to GR. The study of paircreation rates is interesting because a very high creation rate contradicts largescale observations, since the evolution of our universe changes drammaticallywhen may primordial black holes are formed. This allows one to rule out modelsthat predict very high creation rates.
84
10 An interesting possibility
There exist, in the literature, the so-called f(R, T ) where now f is a function notonly of the Ricci scalar but also of the trace of the energy-momentum tensorT = gµνTµν (see [38], [39]). Using our results for the general variations it isquite straightforward to obtain the field equations for a general f(R, T )-theory.However, here we will derive the field equations and study the cosmologicalimplications of the linear model
S =12κ
∫d4x
√−g[R+ βT
]+ SM [gµν ,ΨM ] (561)
where we have added a linear term in T to the Einstein-Hilbert action. Letus motivate this choice. Our argument goes as follows. Since in the Einstein-Hilbert action the Gravity information is carried one could, in principle, also adda matter description to it. Since the corresponding scalar to R that expressesthe matter dependence is T we simply add this to the action along with aparameter β which is the parameter of our model. In addition, as it is clearfrom the latter action, our model does not modify Einstein’s theory in vacuum,namely the vacuum solutions of Einstein theory continue to hold true here aswell. Let us now derive the field equations of our model. As we have alreadyproved the variation of the Einstein-Hilbert part of the action will simply be
δ
∫d4x
√−gR =
∫d4xδ
(√−gR
)=∫d4x
√−gδgµν
(Rµν −
R
2gµν
)(562)
The variation of the matter action will as usual give∫d4x
δ(√−gLM )δgµν
δgµν =∫d4x
√−g(−Tµν
2
)δgµν (563)
As long as the new T -term is concerned, we have for its variation
δgT = δ(Tµνgµν) = δgµνTµν + gµν δTµν
δgαβδgαβ = δgµνTµν + gαβ δTαβ
δgµνδgµν =
= δgµν
(Tµν + gαβ δTαβ
δgµν
)(564)
Therefore
δ(√−gT ) =
√−gδgµν
(−gµν
2T + Tµν + gαβ δTαβ
δgµν
)(565)
Thus, the variation of our action, yields
δgS = 0 ⇒
0 =∫d4x
√−gδgµν
[12κ
(Rµν −
R
2gµν
)− 1
2Tµν +
β
2κ
(−gµν
2T + gαβ δTαβ
δgµν+ Tµν
)](566)
Since the latter must hold true for any variation δgµν , we conclude that
Rµν −R
2gµν = (κ− β)Tµν − βgαβ δTαβ
δgµν+ β
gµν
2T (567)
85
These are the field equations of the theory given by (561). Now, since we areinterested in the cosmological implications of the theory we will apply the fieldequations for the case of a perfect fluid, which is the only form of matter-energythat can live in an FLRW universe. Recall that for a perfect fluid, one has
Tµν = (ρ+ p)uµν + pgµν (568)
or in the corresponding contravariant form
Tµν = (ρ+ p)uµν + pgµν (569)
Having this, we can calculate
gαβ δTαβ
δgµν= p gαβ δgαβ
δgµν︸ ︷︷ ︸=−gµν
= −pgµν (570)
As a result, for a perfect fluid the field equations assume the form
Rµν −R
2gµν = (κ+ β)Tµν + βpgµν + β
gµν
2T (571)
orRµν −
R
2gµν = (κ+ β)Tµν +
β
2gµν(2p+ T ) (572)
We now consider an FLRW background. Then, the 00-component of the aboveequations gives
3
[(a
a
)2
+K
a2
]= (κ+ β)ρ− β
2(2p+ T ) =
= (κ+ β)ρ− β
2(−ρ+ 5p) ⇒(
a
a
)2
=(κ
3+β
2
)ρ− 5
6βp− K
a2(573)
while the ij-component yields
−
[2a
a+(a
a
)2
+K
a2
]gij = (κ+ β)pgij +
β
2(2p+ T )gij
(574)
so that
2a
a+(a
a
)2
=β
2ρ−
(κ+
7β2
)p− K
a2(575)
Thus, combining the above two modified Friedmann equations we get an accel-eration equation that goes as
a
a= −κ
6
[ρ+
(3 +
8βκ
)p
](576)
Interestingly enough, if the parameter β is appropriately chosen one can achieveaccelerated expansion (a > 0) without violating the strong energy condition (ρ+
86
3p > 0). Clearly, the parameter β has to be negative in order to get acceleratedexpansion as can be seen from the above equation. Thus, an inflationary era ora late-time accelerated expansion could both be realized in this model. Now,to close the system of equation one would like to have a continuity equation asin the case of Einstein Gravity. There, the continuity equation follows from thelocal energy-momentum conservation
∇µTGRµν = 0 (577)
but in our case ∇µTµν 6= 0. However, defining the effective energy-momentumtensor
κT effµν ≡ (κ+ β)Tµν +
β
2gµν(2p+ T ) (578)
this is ,in fact, conserved. Indeed, from the field equations
∇µ
(Rµν −
R
2gµν
)= κ∇µT eff
µν (579)
but from Bianchi identities, it follows that
∇µ
(Rµν −
R
2gµν
)= 0 (580)
and as a result∇µT eff
µν = 0 (581)
Now, in order to find the modified continuity equation, we expand the last one
0 = ∇µT effµν = (β + κ)∇µTµν +
β
2∇ν(−ρ+ 5p) (582)
As we prove in the appendix, for FLRW universes,
∇µTµν = −δ0ν[ρ+ 3H(ρ+ p)
](583)
and, of course, it holds that
∇ν(−ρ+ 5p) = ∂ν(−ρ+ 5p) = δ0ν(−ρ+ 5p) (584)
Substituting these, into the conservation equation (582), and taking ν = 0, wearrive at
−(β + κ)[ρ+ 3H(ρ+ p)
]+β
2(−ρ+ 5p) = 0 ⇒
ρ+ 3H(ρ+ p)− β
2(β + κ)(−ρ+ 5p) = 0 (585)
In this form we can recognize the last term as a correction to the usual continuityequation. It can also be written as(
1 +β
2(β + κ)
)ρ− 5β
2(β + κ)p+ 3H(ρ+ p) = 0 (586)
Assuming a barotropic fluid satisfying
p = wρ (587)
87
as usual, the latter recasts to(1 +
β
2(β + κ)(1− 5w)
)ρ+ 3Hρ(1 + w) = 0 (588)
orρ+ 3Hρ(1 + weff ) = 0 (589)
where we have defined
1 + weff ≡1 + w
1 + β2(β+κ) (1− 5w)
(590)
Now as can be easily shown (see [39] for instance) in such a theory, testingparticles do not follow geodesics in general. More precisely, one has
uν∇νuµ =
d2xµ
ds2+ Γµ
ναuνuα = fµ ⇒
d2xµ
ds2+ Γµ
ναuνuα = fµ
(591)
wherefµ = κ
∇νp
(ρ+ p)[κ+ f,T
] (gµν + uµuν) (592)
In the latter f,T ≡ ∂f∂T . Note that in our case
f(R, T ) = R+ βT (593)
so thatf,T = β (594)
We state now that in FLRW universes, particles do follow geodesics. Indeed,for Friedmann universes we have
gµ0 = −δµ0 (595)
as well as∇µφ = ∂µφ = δ0µφ (596)
for any scalar φ. Assuming a comoving observer as usual
uµ = −δ0µ (597)
anduµ = +δµ
0 (598)
Using the above, equation (592) becomes
fµ =κ
(κ+ β)p
(ρ+ p)
(δ0νg
µν + δ0νuµuν
)=
=κ
(κ+ β)p
(ρ+ p)
(gµ0 + uµ
)=
κ
(κ+ β)p
(ρ+ p)
(− δµ
0 + δµ0
)= 0 (599)
88
This can also be seen from the fact that
uµ∇µuν = 0 (600)
for the Robertson-Walker metric, as we prove in the appendix. Thus, in anFLRW universe testing particles will follow geodesics. In this model we studiedthe possibility of having an accelerated expansion without violating the strongenergy condition (ρ+ 3p > 0). Our initial motivation was to add in the actionthe trace of the energy-momentum tensor since the latter contains in it thedynamical characteristics of our fluid. In addition in vacuum (T = 0), ourmodel coincides with GR. However, we merely pointed out a possibility andadditional observational contraints have to be taken into account for this modelto be viable.
89
11 Conclusions/Results
In this thesis we studied Modified Theories of Gravity and implications to Cos-mology. In order to do so, we analytically derived general variations of ten-sors related to Gravity and then applied them to derive the field equations forf(R), f(R,φ), f(R,G) as well as more general theories. An alternative way toderive the Friedmann equations (which includes variations with respect to thescale factor) was also extended to f(R), f(R,φ), etc. in the thesis. In addition,throughtout the thesis we payed special attention on f(R)-theories and theirconsequencies. We then, went on to review and study in detail, Starobinsky’smodel of inflation which is a special case of f(R)-theories and saw how is thismodel favoured by recent observations. Then, we continued by adding a cosmo-logical constant to it, which can be seen as a unification model, and we were ableto find parametric solutions of the scale factor during inflation. In the literaturethere are also other unification models [40],[33], [41] We also confirmed thatwhen the energy scale of the cosmological constant is much smaller than thatof the Starobinsky’s parameter we recover Starobinsky’s inflation. Having donethis, we studied how does the probability of primordial black hole formationchanges in this context. Finally, we considered a model in which a term linearin the trace of the energy-momentum tensor is added to the Einstein-Hilbertaction. For this model we analytically derived the general field equations andthen in the case of an FLRW background we obtained the modified Friedmannequations. An investigation of the modified acceleration equation revealed to usthat accelerated expansion can be realized, under certain conditions, withoutviolating the Strong Energy Condition. A further research could include thecomputation of observables for the latter theory and comparison with data, tofind out whether is viable or not.
90
12 Appendix
12.1 Metric Compatibility and the Levi-Civita Connec-tion
The formulation of General Relativity is based upon the crucial assumption thatthe connection is metric compatible as well as symmetric. These requirementsforce the latter to acquire the form
Γρµν =
12gρα(∂µgνα − ∂νgµα − ∂αgµν) (601)
which is often called the Levi-Civita connection. We should mention here that,in general, the metric tensor gµν and the connection Γρ
µν , need not be relatedto each other. The former defines angles between vectors and measures dis-tances while the latter defines parallel transport of vector and tensor fields onthe manifold. In the so-called Metric Theories of Gravity the aforementionedassumptions for the connection are made and therefore the geometry of thespace is uniquely determined by the form of the metric tensor. The form ofthe Levi-Civita connection lies upon the following theorem which we state andprove here.Theorem: Let (M, g) be an n-dimensional Riemannian space. Then, thereexists a unique, symmetric connection that is compatible with the metric. Thelatter is called the Riemann, or the Levi-Civita50 connection and is given by
Γlcd =
12gbl(∂cgdb + ∂dgbc − ∂bgcd) (602)
Proof: In order to prove the theorem we are going to use the following defini-tion. Definition : In a Riemannian space a connection is said to be compatiblewith the metric iff for any pair of vectors ua, vb on the manifold, the innerproduct (uv) does not change upon the parallel transportation of the vectorsalong a given curve. Namely, let C : xa = xa(λ), where a = 1, 2, ..., n , be agiven curve in the space, parametrized by a parameter λ. Then, it holds that
d
dλ(uv)|along C = 0 (603)
Expanding the last, we obtain
d
dλ(uv) =
d
dλ(uava) =
d
dλ(uavbgab) =
=dua
dλvbgab +
dvb
dλuagab + (∂cgab)
dxc
dλuavb =
=dua
dλvbgab +
dva
dλudgda + (∂cgab)
dxc
dλuavb
where we have used the chain rule to write ddλgab = (∂cgab)dxc
dλ . Now, since bothu and v are parallel transported along C, they satisfy
dua
dλ+ Γa
dcud dx
c
dλ= 0 (604)
50Mathematicians use the term Riemann connection, while, physicists tend to use the termLevi-Civita connection. However, frequently they both agree to refer to them as Christoffelsymbols
91
as well asdva
dλ+ Γa
bcvb dx
c
dλ= 0 (605)
and by substituting these into equation (604) we arrive at
d
dλ(uv) = −Γa
dcgabudvb dx
c
dλ− Γa
bcvb dx
c
dλgdau
dvb dxc
dλ+ (∂cgab)
dxc
dλuavb =
= −Γadcgabu
dvb dxc
dλ− Γa
bcgdaudvb dx
c
dλ+ (∂cgdb)
dxc
dλudvb =
= (−Γadcgab − Γa
bcgda + ∂cgdb)udvb dxc
dλ⇒
d
dλ(uv) = (−Γa
dcgab − Γabcgda + ∂cgdb)udvb dx
c
dλ(606)
Therefore, metric compatibility is assured iff
d
dλ(uv) = (−Γa
dcgab − Γabcgda + ∂cgdb)udvb dx
c
dλ= 0 (607)
and since the vectors ud and vb are arbitrary and for a smooth curve we havedxc
dλ 6= 0, we conclude that it must hold
Γadcgab + Γa
bcgda − ∂cgdb = 0 (608)
In addition, cyclically permuting the indices c,d and b we get
Γabdgac + Γa
cdgba − ∂dgbc = 0 (609)
permuting once more
Γacbgad + Γa
dbgca − ∂bgcd = 0 (610)
Now, imposing the crucial assumption that the connection is symmetric51 inits lower indices, namely Γa
bc = Γacb we can add equations (608) and (609) and
subtract (610) from them, to get
2Γacdgab − ∂cgdb − ∂dgbc + ∂bgcd = 0 (611)
furthermore, multiplying (and contracting at the same time) by gbl and usingthe identity gabg
bl = δla it follows that
Γlcd =
12gbl(∂cgdb + ∂dgbc − ∂bgcd) (612)
The latter is indeed the Levi-Civita (or Riemann) connection as we stated.The uniqueness is, of course, apparent.We should mention once again here theassumptions that were made in order to arrive at the Levi-Civita connection.Firstly, the metric compatibility condition of the connection
d
dλ(uv)|along C = 0 (613)
51The terms torsion-free or torsionless are also been, frequently, used to denote a symmetricconnection.
92
followed by torsionlessnessΓa
[bc] = 0 (614)
Notice that General Relativity meets both the above requirements. The gen-eral class of theories satisfying the aforementioned assumptions is called MetricGravity. One should also comment at this point that equation (608) is identicalto
∇cgdb = 0 (615)
namely, the metric is covariantly constant. In physics textbooks of GR the lastequation is commonly the one that is referred to as the compatibility condition.Of course the latter is identical with our definition of metric compatibility givenby (603). Finally, we notice that setting u = v the compatibility conditionreaches
d
dλ(||u||2)|along C = 0 (616)
which reflects the fact that in such spaces a known Euclidean.property is as-sumed to continue to hold true, that of a vector preserving its norm when isparallel transported along a given curve.
12.2 Definitions and useful identities
Riemann Tensor Rµναβ
(∇α∇β −∇β∇α)uµ = Rµναβu
ν (617)
where uµ is an arbitrary vector field. Alternatively, in terms of the connection
Rµνρσ = 2∂[ρΓ
µσ]ν + 2Γµ
α[ρΓασ]ν (618)
Covariant derivative on scalars
∇µφ ≡ ∂µφ (619)
Covariant derivative on contravariant and covariant vector fields respectively
∇αuµ = ∂αu
µ + Γµαρu
ρ (620)
and∇αuµ = ∂αuµ − Γρ
αµuρ (621)
and since Γρµν is symmetric in µ↔ ν it follows that
∇[µuν] = ∂[µuν] (622)
such that if we set uµ = ∇µψ = ∂µψ we get
∇α∇βψ = ∇α∇βψ (623)
and therefore by setting uµ = ∇µψ to the very first definition of the Riemanntensor we have
(∇α∇β −∇β∇α)∇µψ = Rµναβ∇νψ (624)
and upon contacting µ and α we finally arrive at
∇µ∇β∇µψ −∇β2ψ = Rνβ∇νψ ⇒
93
2∇βψ −∇β2ψ = Rνβ∇νψ ⇒
(2∇β −∇β2)ψ = Rνβ∇νψ (625)
where ψ is an arbitrary scalar field. Let us now prove some extremely usefulrelations between covariant derivatives, Riemann tensor, Ricci tensor and Ricciscalar. We start from the Bianchi identities, namely
∇ρRµνκλ +∇κRµνλρ +∇λRµνρκ = 0 (626)
and upon contraction with gµκ we obtain
∇ρRνλ +∇µRµνλρ −∇λRνρ = 0 ⇒
∇µRµνλρ = ∇λRνρ −∇ρRνλ ⇒
∇µRµνλρ = 2∇[λRρ]ν (627)
contracting, furthermore, with gνρ it follows that
∇µRµλ = ∇λR−∇νRνλ ⇒
∇µRµλ =12∇λR (628)
In addition, acting with the operator ∇λ on the above equation, we take
∇µ∇νRµν =122R (629)
Let us also prove a very nice identity. We start from the equation proved before
∇ρRµναρ = ∇νRµα −∇µRνα (630)
and act on it with ∇ν to get
∇ν∇ρRµναρ = 2Rµα −∇ν∇µRνα = 2Rµα −∇ν∇µRνα (631)
Now, for a general tensorial field of rank -(1,1), namely Tµν(x) the definition
of the Riemann tensor gives
2∇[ρ∇µ]Tνα = Rν
βρµTβα −Rβ
αρµTνβ (632)
and by setting Tµν = Rµ
ν we obtain
2∇[ρ∇µ]Rνα = Rν
βρµRβ
α −RβαρµR
νβ
and contracting in ρ and ν
∇ν∇µRνα = ∇µ∇νR
να +RβµR
βα −Rβ
ανµRνβ ⇒
∇ν∇µRνα = ∇µ∇νR
να +RρµR
ρα −RµναρR
νρ (633)
In addition, using
∇νRνα = ∇νRνα =
12∇αR (634)
94
the last assumes the form
∇ν∇µRνα =
12∇µ∇αR+RρµR
ρα −RµναρR
νρ (635)
and substituting the above back to (631) we arrive at
∇ν∇ρRµναρ = 2Rµα −12∇µ∇αR−RρµR
ρα +RµναρR
νρ (636)
Let us elaborate again with the definition of the Riemann tensor, this time fora rank-(2, 0) tensor. We have
2∇[ρ∇λ]Rνβ = Rν
κρλRκβ +Rβ
κρλRνκ (637)
and contracting in β and ρ we obtain
∇ρ∇λRνρ −∇λ∇ρR
νρ = RνκρλR
κρ +RκλRνκ (638)
Furthermore, multiplying the latter by52 gλµ we get
∇ρ∇µRνρ −∇µ∇ρRνρ = Rν µ
κρ Rκρ +RκµRνκ ⇒
∇ρ∇µRνρ −∇µ∇ρRνρ = RνκρµRκρ +Rκ
µRνκ (639)
and exchanging µ↔ ν,
∇ρ∇νRµρ −∇ν∇ρRµρ = RµκρνRκρ +Rκ
νRµκ (640)
Now, adding the last two equations and using the symmetries of Riemann andRicci tensors along with the relation
∇αRαβ =12∇βR (641)
it follows that2∇ρ∇(µRν)ρ = 2RµκρνRκρ + 2RµκRν
κ ⇒
∇ρ∇(µRν)ρ =12∇(µ∇ν)R+RµκρνRκρ +RµκRν
κ (642)
12.3 Relations between conformally transformed quanti-ties
As we have already seen a conformal transformation (M, g) 7→ (M, g) is givenby
gab = Ω2(x)gab (643)
Here we consider general n − dim spaces, and therefore a,b=1,2,3,...,n. Now,since
gabgab = n = gabg
ab (644)
we deduce that the inverse metric tensor transforms according to
gab = Ω−2gab (645)52In fact, this is multiplication together with contraction but we simply refer to it as mul-
tiplication.
95
The square root of the determinant will then be transformed as√|g| = Ωn
√|g| (646)
Notice that here we take the absolute value of the determinant in order to beable to easily recover the results of GR (−g = |g|) where the space is pseudo-Lorentzian (det(gµν) = g < 0) and one needs to take
√−g.
13 Variations
13.1 Some basic Variations
We present and prove here some of the basic variations that have ,extensively,beenused throughout the derivations. Let us start by the definition of the inversemetric tensor
gµνgνα = δα
µ (647)
and vary the last by noting that δαµ is constant, to get
0 = (δgµν)gνα + (δgνα)gµν (648)
Now, contracting with gαβ we arrive at
0 = (δgµν)δνβ + (δgνα)gµνgαβ ⇒
δgµβ = −gµνgαβ(δgνα) (649)
An alternative derivation of the same relation goes as follows,
δgµν = δ(gµαgνβgαβ) =
= (δgµα)gνβgαβ + gµα(δgνβ)gαβ + gµα)gνβ(δgαβ) =
= (δgµα)gνβgαβ + δgµν + δgνµ ⇒
δgµν = (δgµα)gνβgαβ + δgµν + δgνµ ⇒
δgµν = −gµαgνβδgαβ (650)
Now varyinggµνg
µν = 4 (651)
we getgµνδgµν = −gµνδg
µν (652)
Of course, the above results hold true for an general n−dimensional Riemannian(or pseudo-Remannian) space namely
gabgbc = δc
a (653)
andgabg
ab = n (654)
so thatδgab = −gacgbdδg
cd (655)
96
as well asgabδgab = −gabδg
ab (656)
where the indices a, b, c, d, ... run over the dimensionality of the space (namelyfrom 1 to n). Let us now compute the variation of the square root of thedeterminant of the metric tensor. We do the calculation fully general for n−dimRiemannian spaces and then apply it for the pseudo-Riemannian 4− dim spaceof General relativity . Firstly, we write the determinant simply as
g ≡ det(gab) (657)
We have thatδ(√g) =
12√gδg (658)
Now, for any square n× n- matrix A it holds that
det(A) = eTr(A) (659)
Setting A → gab in the above, we arrive at
g = det(gab) = eTr(gab) (660)
and under the variation gab → gab + δgab it follows that
det(gab + δgab) = eTr(gab+δgab) = eTr(gab)+Tr(δgab) = eTr(gab)︸ ︷︷ ︸≡g
eTr(δgab) (661)
where on going from the second to the third equality we employed the linearityof the trace. Since the variations δg are small, in the expansion of eTr(δgab) wecan neglect second and higher order terms
((δg)2 ≈ 0
)and we shall have
eTr(δgab) ≈ 1 + Tr(δgab) (662)
and thereforedet(gab + δgab) ≈ g
(1 + Tr(δgab)
)(663)
but, by the definition of the trace
Tr(δgab) = gabδgab (664)
so thatdet(gab + δgab) ≈ g
(1 + gabδgab
)(665)
Using the latter to the definition of the variation, we arrive at
δg = δ(det(gab)
)= det(gab + δgab)− det(gab) ≈
≈ g(1 + gabδgab
)− g = ggabδgab (666)
It has been proven before that
gabδgab = −gabδgab (667)
97
thus we finally obtainδg = −ggabδg
ab (668)
It also follows that
δ(√g) =
12√gδg = −1
2g√ggabδg
ab = −12√ggabδg
ab ⇒
δ(√g) = −1
2√ggabδg
ab (669)
Now, in order to get the expression for the 4 − dim pseudo-Riemannian spaceof General relativity, we simply replace g → −g and let the indices run from 0to 3 (The usual Greek ones). We then have
δ(√−g) = −1
2√−ggµνδg
µν (670)
where µ, ν = 0, 1, 2, 3. In addition eq. (668) provides a nice formula that allowsone to compute in compact form, variations of any power of the determinant.Indeed, for a general Riemannian space, the variation of the n− th53 power ofg will be
δ(gn) = ngn−1δg = −ngn−1ggabδgab = −ngngabδg
ab ⇒
δ(gn) = −ngngabδgab (671)
If we now sum over the body of all natural numbers (n ∈) we obtain
∞∑n=0
δ(gn) = −( ∞∑
n=0
ngn)gabδg
ab (672)
Assuming now that g < 1, in order to calculate the sum appearing on the righthand side we start by
11− x
=∞∑
n=0
xn , |x| < 1 (673)
and differentiate with respect to x, to get
1(1− x)2
=∞∑
n=0
nxn−1 , |x| < 1 (674)
multiplying through by x
x
(1− x)2=
∞∑n=0
nxn , |x| < 1 (675)
Thus, setting x = g to the last one, we obtain
g
(1− g)2=
∞∑n=0
ngn , |g| < 1 (676)
53This n here has nothing to do with the dimension of spacetime, it is merely an arbitraryreal number
98
so that equation (672) assumes the nice form∞∑
n=0
δ(gn) = − g
(1− g)2gabδg
ab , |g| < 1 (677)
We should mention here that we arrived to this compact form because of ourassumption that |g| < 1. This is a necessary condition we must impose in orderfor the sum to converge. When |g| ≥ 1 the sum diverges and such a compactformula does not exist.
13.2 Variation of the ′squared′ Riemann and Ricci tensors
In this section the variations of RµνρσRµνρσ and RµνR
µν of gravity are derived.We compute for the former
δ(RµνρσRµνρσ) = RµνρσδR
µνρσ +RµνρσδRµνρσ (678)
Now,RµνρσδR
µνρσ = Rµνρσδ(gµαgνβgργgσδRαβγδ) =
= RµνρσδRµνρσ + 4RµγβδRαβγδδg
µα = RµνρσδRµνρσ + 4RµαβγRναβγδg
µν
(679)so that
δ(RµνρσRµνρσ) = 2RµνρσδRµνρσ + 4Rµ
αβγRναβγδgµν (680)
Note now that the second term in the above equation is of the desired form. We, therefore, only need to manipulate the first term. We have
2RµνρσδRµνρσ = 2Rµνρσδ(gµαRα
νρσ) == 2RµνρσgµαδR
ανρσ + 2RµνρσRα
νρσδgµα =
= 2RανρσδRα
νρσ − δgαβgµαRβνρσ2Rµνρσ (681)
and therefore
δ(RµνρσRµνρσ) = 2Rα
νρσδRανρσ + 2Rµ
αβγRναβγδgµν (682)
If we now couple a scalar-F to the above variation we obtain
Fδ(RµνρσRµνρσ) = 2FRα
νρσδRανρσ + 2FRµ
αβγRναβγδgµν (683)
This relation will be proven to be very useful in the variations we will be con-sidering later on. For the variation of the form Fδ(RµνR
µν), where F is anarbitrary scalar, one has
Fδ(RµνRµν) = F
[RµνδRµν +RµνδR
µν]
=
= F[RµνδRµν +Rµνδ(gµαgνβRαβ)
]=
= F[RµνδRµν +RαβδRαβ +RµνRαβ(gνβδgµα + gµαδgνβ)
]=
= F[2RµνδRµν + 2RµαR
ανδg
µν]⇒
Fδ(RµνRµν) = F
[2RµνδRµν + 2RµαR
ανδg
µν]
(684)
99
13.3 General Variations
In what follows we are going to compute variations of several useful quantitiesfor general n− dim spaces which ,of course, include the spacetime of GR as thespecial case n = 4. We proceed now by firstly computing the variation of theLevi-Civita connection, δΓa
bc under the variation of the metric tensor
gab → gab + δgab (685)
where a, b = 1, 2, ..., n By definition
δΓabc = Γa
bc[gde + δged]− Γa
bc[gde] (686)
if we now expand Γabc[g
de +δged] keeping only first order terms54 in δgab we have
Γabc[g
de + δged] =12(gal + δgal) [∂b(gcl + δgcl) + ∂c(gbl + δgbl)− ∂l(gbc + δgbc)]
≈ Γabc[g
de] +12gal [∂b(δgcl) + ∂c(δgbl)− ∂l(δgbc)] +
+12(δgal) [∂bgcl + ∂cgbl − ∂lgbc] (687)
Now, since δgab is a tensor (being the difference of two tensors) we may write
∇bδgcl = ∂bδgcl − Γebcδgel − Γe
blδgec (688)
∇cδgbl = ∂cδgbl − Γebcδgel − Γe
clδgeb (689)
∇lδgcb = ∂lδgcb − Γeclδgeb − Γe
blδgec (690)
adding the former two and subtracting from them the last one k that you havewrite access to this file or that enough disk space is available.we arrive at
∂b(δgcl) + ∂c(δgbl)− ∂l(δgbc) = ∇bδgcl +∇cδgbl −∇lδgcb + 2Γebcδgel (691)
In addition∇bgcl = ∂bgcl − Γe
bcgel − Γeblgec (692)
∇cgbl = ∂cgbl − Γebcgel − Γe
clgeb (693)
∇lgcb = ∂lgcb − Γeclgeb − Γe
blgec (694)
and ,again, adding the first two and subtracting the third one, also using thefact that ∇agbc = 0, we get
0 = ∂bgcl + ∂cgbl − ∂lgcb − 2Γebcgel ⇒
∂bgcl + ∂cgbl − ∂lgcb = 2Γebcgel (695)
and upon substitution of (696) and (708) into (687), it follows that
Γabc[g
de + δged] ≈ Γabc[g
de]+
+12gal [∇b(δgcl) +∇c(δgbl)−∇l(δgbc) + 2Γe
bcδgel] +12δgalΓe
bcgel
54This means we drop terms such as δg∂δg,(δg)2 and so on.
100
and finally, by using(δgal)gel = −(δgel)gal (696)
the last becomesΓa
bc[gde + δged] ≈ Γa
bc[gde]+
+12gal [∇b(δgcl) +∇c(δgbl)−∇l(δgbc)] + galΓe
bcδgel − (δgel)Γebcg
al =
12gal [∇b(δgcl) +∇c(δgbl)−∇l(δgbc)] (697)
Such thatδΓa
bc = Γabc[g
de + δged]− Γabc[g
de] =
12gal [∇b(δgcl) +∇c(δgbl)−∇l(δgbc)] ⇒ (698)
and therefore
δΓabc =
12gal [∇b(δgcl) +∇c(δgbl)−∇l(δgbc)] (699)
Let us comment on our result. We see that even though Γabc is not a tensor,
δΓabc is a tensor. The fact that Γa
bc is not a tensor can be easily seen from thefact that there are terms in it such as ∂bglc which as we know do not transformas tensors under a coordinate transformation. On the other hand, as we can seein (699) when varying the connection this term is replaced by ∇bδglc which bydefinition transforms as a (0, 3)-rank tensor. Therefore δΓa
bc is, indeed, a tensor.Now let us compute the variation of the Riemann tensor under the variation
Γabc → Γa
bc + δΓabc (700)
which in the case that the connection is related with the metric tensor, such asthe Levi-Civita connection, is equivalent to the variation gab → gab + δgab. Weshould point out, though, that for Palatini and Metric affine theories of gravitythe above statement is no longer true because in these theories metric andconnection are not related. On the other hand, in the case of General Relativity,which is a Metric-Theory, this is true as the metric tensor and connection arerelated, the latter being the Levi-Civita connection. Under the variation Γa
bc →Γa
bc + δΓabc the Riemann tensor changes as follows
Rabcd[Γ
ekl + δΓe
kl] == 2∂[cΓa
d]b + 2∂[cδΓad]b + (Γa
ec + δΓaec)(Γ
edb + δΓe
db)− (Γaed + δΓa
ed)(Γecb + δΓe
cb) =
= 2∂[cΓad]b + Γa
ecΓedb + Γa
ecδΓedb + Γe
dbδΓaec − Γa
edΓecb − Γa
edδΓecb − Γe
cbδΓaed +O(δΓ2)
= 2∂[cΓad]b + 2Γa
e[cΓed]b︸ ︷︷ ︸
≡Rabcd[Γe
kl]
+2∂[cδΓad]b + 2Γa
e[cδΓed]b − 2Γe
b[cΓad]e +O(δΓ2)
= Rabcd[Γ
ekl] + 2∂[cδΓa
d]b + 2Γae[cδΓ
ed]b − 2Γe
b[cΓad]e +O(δΓ2) (701)
such that to first order in δΓ we have
Rabcd[Γ
ekl + δΓe
kl] ≈ Rabcd[Γ
ekl] + 2∂[cδΓa
d]b + 2Γae[cδΓ
ed]b − 2Γe
b[cΓad]e (702)
101
and therefore the variation of the Riemann tensor will be
δRabcd ≡ Ra
bcd[Γekl + δΓe
kl]−Rabcd[Γ
ekl] =
= 2∂[cδΓad]b + 2Γa
e[cδΓed]b − 2Γe
b[cΓad]e (703)
Now, since δΓabc is a tensor, we have
∇cδΓadb = ∂cδΓa
db + ΓaceδΓ
edb − Γe
cdδΓaeb − Γe
cbδΓaed (704)
as well as∇dδΓa
cb = ∂dδΓacb + Γa
deδΓecb − Γe
cdδΓaeb − Γe
dbδΓaec (705)
and the subtraction of them leads to
2∇[cδΓad]b = ∂[dδΓa
c]b + 2Γae[cδΓ
ed]b − 2Γe
b[cΓad]e (706)
And upon using the above relation, the variation of the Riemann tensor reaches
δRabcd = 2∇[cδΓa
d]b = ∇cδΓadb −∇dδΓa
cb (707)
Now let us use the above derivations for our physical 4 − dim spacetime andobtain general variations containing the Riemann and Ricci tensor. Firstly,using the relation
gαλδgνλ = −gνλδgαλ (708)
we have
δΓασν =
gαλ
2[∇σδgνλ +∇νδgσλ −∇λδgσν ] =
= −12[gνλ∇σδg
αλ + gσλ∇νδgαλ +∇αδgσν ] (709)
and therefore the variation of Riemann tensor becomes
δRανρσ = ∇ρ(δΓα
σν)−∇σ(δΓαρν) =
= −∇ρ
2[gνλ∇σδg
αλ + gσλ∇νδgαλ +∇αδgσν ]
+∇σ
2[gνλ∇ρδg
αλ + gρλ∇νδgαλ +∇αδgρν ] =
= gνλ∇[σ∇ρ]δgαλ + gλ[ρ∇σ]∇νδg
αλ +∇[σ∇αδgρ]ν ⇒
δRανρσ =
(gνλ∇[σ∇ρ] + gλ[ρ∇σ]∇ν
)δgαλ +∇[σ∇αδgρ]ν (710)
Thus, for any (1, 3) tensorial field T νρσα (x) it holds that∫
d4x√−g Tα
νρσδRανρσ =
∫d4x
√−g[Tαλ
ρσ∇[σ∇ρ]δgαλ+
12(Tα
νλ
σ∇σ∇ν − Tανρ
λ∇ρ∇ν
)+
12Tα
νρσ(∇σ∇αδgρν −∇ρ∇αδgσν
)]=
=∫d4x
√−gδgµν
[∇[ρ∇σ]T
ρσµν +
12∇λ∇σTµλνσ −
12∇ρ∇λTµ
λρν
]+∫d4x
√−g[12δgρν∇σ∇αT
ανρσ − 12δgσν∇ρ∇αT
ανρσ
](711)
102
Note that we have used the Leibniz rule twice and dropped any surface terms,on going from the second to the third line. Using, in addition, the fact that
(δgρν)∇σ∇αTανρσ = (δgρν)gνβ∇σ∇αT
α ρσβ =
= −(δgνβ)∇σ∇αTα σ
βν = −(δgµν)∇σ∇ρTρµνσ (712)
we finally arrive at∫d4x
√−g Tα
νρσδRανρσ =
∫d4x
√−gδgµν
[∇[ρ∇σ]Tµνρσ +∇ρ∇σTµρ[νσ]
+12∇σ∇ρTρµσν −
12∇σ∇ρTρµνσ
]⇒
∫d4x
√−g Tα
νρσδRανρσ =
=∫d4x
√−gδgµν
[∇[ρ∇σ]Tµνρσ +∇ρ∇σTµρ[νσ] +∇ρ∇σTσµ[ρν]
](713)
Now from
δRανρσ =
(gνλ∇[σ∇ρ] + gλ[ρ∇σ]∇ν
)δgαλ +∇[σ∇αδgρ]ν (714)
contracting in α and ρ we take the variation of the Ricci tensor under gµν →gµν + δgµν ,
δRνσ = δRανασ =(
gνλ∇[σ∇α] + gλ[α∇σ]∇ν
)δgαλ +∇[σ∇αδgα]ν
=12
[gλν∇σ∇αδg
αλ − gλν∇α∇σδgαλ + gλα∇σ∇νδg
αλ − gλσ∇α∇νδgαλ
+∇σ∇ρgραδgαν −∇α∇αδgσν
](715)
and upon using (708), and relabeling the dummy indices ρ → α, α → λ in thefifth term, this term cancels against the first one and, after some gathering ofthe various terms, we obtain
δRνσ = −gλ(ν∇α∇σ)δgαλ +
12gλα∇σ∇νδg
αλ − 122δgσν (716)
or
δRνσ = −gλ(ν∇α∇σ)δgαλ +
12gλα∇σ∇νδg
αλ +12gσαgνβ2δgαβ (717)
where we have used the identity
δgσν = −gσαgνβ(δgαβ) (718)
Such that, for a general tensorial field- Tµν(x) it will hold∫d4x
√−gT νσδRνσ =
103
=∫d4x
√−g[− 1
2δgµν(∇λ∇µT
λν +∇λ∇µTν
λ)+
δgµν(12gµν∇α∇βT
αβ) + δgµν 122Tµν
]⇒
∫d4x
√−gT νσδRνσ =
=∫d4x
√−gδgµν
[− 1
2∇λ∇µ(Tλ
ν + Tνλ) +
12gµν∇α∇βTαβ +
122Tµν
](719)
Setting Tµν = gµν in the latter equation and using the compatibility of themetric (only holds true for metric-theories)
∇αgµν = 0 (720)
we conclude that ∫d4x
√−ggµνδRµν = 0 (721)
using this, we have∫d4x
√−gδR =
∫d4x
√−gδ(gµνRµν) =
∫d4x
√−g[(δgµν)Rµν + gµνδRµν
]=
= 0 +∫d4x
√−g(δgµν)Rµν =
∫d4x
√−g(δgµν)Rµν (722)
In addition, as we show in the appendix
δ√−g = −1
2√−ggµνδg
µν (723)
Therefore,
δ
∫d4x
√−gR =
∫d4xδ(
√−gR) =
∫d4x[(δ√−g)R+
√−gδR
]=
=∫d4x
√−gδgµν
(Rµν −
R
2gµν
)⇒
δ
∫d4x
√−gR =
∫d4x
√−gδgµν
(Rµν −
R
2gµν
)(724)
This result will be used when deriving the Einstein field equations.
14 Calculus of Variations
Let us see how the Calculus of Variations arises out of pure mathematical cu-riosity. Consider the integral55
I[y(x)] =∫ x2
x1
dxf(y(x), y′(x), x) (725)
55We call I[y(x)] a functional. A functional is a map Cn → R , where Cn is the set ofn − times differentiable continuous functions. It should be differentiated from a functionwhich is a map R → R.
104
where y(x) is a function of x, y′(x) = dydx and f an arbitrary function of the
given arguments. We ask now, for which curve y(x) does the above integral getits extreme value (maximum or minimum)? Note that if we had a function I(ε)we would know how to proceed, the extremum (or extrema if there are morethan one) occurs exactly there where
dI
dε= 0 (726)
However, in our case, after the integration is performed we are left with a number(which of course cannot be varied). Here is where the magic of mathematicscomes in. Suppose we have found that function which extremizes the integral,call it y(x). We then consider a family of curves parametrized by ε , deviatingby this solution. We denote them by
Y (x, ε) (727)
and we demand that they continuously depend on ε and for ε = 0 we recoverour extreme curve y(x), namely
Y (x, 0) = y(x) (728)
In addition all the curves should end up to the same points on the plane, namely
Y (x1, ε) = y(x1) = y1, Y (x2, ε) = y(x2) = y2 (729)
Now, we have constructed a function-I(ε) which we are allowed to vary withrespect to ε. One then has
dI
dε=
d
dε
∫ x2
x1
dxf(Y (x, ε), Y ′(x, ε), x) =∫ x2
x1
dx∂
∂εf(Y (x, ε), Y ′(x, ε), x) =
and using the chain rule
dI
dε=∫ x2
x1
dx[ ∂f∂Y
∂Y
∂ε+
∂f
∂Y ′∂Y ′
∂ε
](730)
but since partial derivatives commute, we may write
∂Y ′
∂ε=
∂
∂x
(∂Y∂ε
)(731)
and therefore
dI
dε=∫ x2
x1
dx[ ∂f∂Y
∂Y
∂ε+
∂f
∂Y ′∂
∂x
(∂Y∂ε
)]=
=∫ x2
x1
dx[ ∂f∂Y
∂Y
∂ε+
∂
∂x
( ∂f∂Y ′
∂Y
∂ε
)− ∂Y
∂ε
∂
∂x
( ∂f∂Y ′
)](732)
Now we remove the parameter ε by setting ε = 0 since the introduction of itwas made for auxiliary reasons. Then, partial derivatives with respect to x arereduced to total ones
∂
∂x→ d
dx(733)
105
and by definitionY (x, ε)
∣∣∣ε=0
= Y (x, 0) = y(x) (734)
Note also that since the parametrization is regular, one has
∂Y
∂ε6= 0, ∀ ε (735)
The extremum then occurs exactly there where
dI
dε
∣∣∣ε=0
= 0 (736)
Taking all the above into account, one arrives at
0 =∫ x2
x1
dx[ ∂f∂Y
∣∣∣ε=0
∂Y
∂ε
∣∣∣ε=0
+d
dx
( ∂f∂Y ′
∂Y
∂ε
)∣∣∣ε=0
− ∂Y
∂ε
∣∣∣ε=0
d
dx
( ∂f∂Y ′
)∣∣∣ε=0
]=
=[( ∂f∂Y ′
∂Y
∂ε
)∣∣∣ε=0
]∣∣∣x2
x1
+∫ x2
x1
dx(∂Y∂ε
)∣∣∣ε=0
[∂f∂y− d
dx
( ∂f∂y′
)](737)
Now as an immediate consequence of (729) we have that
∂Y
∂ε
∣∣∣ε=0, x=xi
= 0, i = 1, 2 (738)
at the end points x1, x2. Indeed, Taylor expanding Y (x, ε) in ε around thesolution y(x) it follows that
Y (x, ε) ≈ y(x) +∂Y
∂ε
∣∣∣ε=0
ε+O(ε2) (739)
and by evaluating the latter at xi (i = 1, 2) we obtain
yi ≈ yi +∂Y
∂ε
∣∣∣ε=0,x=xi
ε⇒
∂Y
∂ε
∣∣∣ε=0,x=xi
= 0 (740)
to first order in ε. Using this we see that the first term in the second line of(737) vanishes and we are left with∫ x2
x1
dx(∂Y∂ε
)∣∣∣ε=0
[∂f∂y− d
dx
( ∂f∂y′
)](741)
and since this must be true for any(
∂Y∂ε
)∣∣∣ε=0
we conclude that
∂f
∂y− d
dx
( ∂f∂y′
)= 0 (742)
We are now in a position to answer the question we raised at the beginning ofthis section. The function y(x) that extremizes the integral
I[y(x)] =∫ x2
x1
dxf(y(x), y′(x), x) (743)
106
can be found by solving the differential equation (742). It is, in fact, an as-tonishing result. However, the whole derivation was somewhat involved. Thereexists an equivalent method that can be used in a more straightforward mannerand appears to be more practical in applications. Let us unfold it here. To firstorder in ε we can write
Y (x, ε) ≈ y(x) + εg(x) (744)
where g(x) = (∂εY )|ε=0. We then define the deviation from the extreme pathy(x) via
δy := Y (x, ε)− y(x) = εg(x) (745)
so that Y (x, ε) = y(x) + δy. We also define the variation of the functionalthrough
δI := I[Y ]− I[y] = I[y + δy]− I[y] (746)
Then, for small deviations, expanding in ε one has
δI = I[y + δy]− I[y] = I[y + εg(x)]− I[y] ≈
≈ I[y] +dI
dε
∣∣∣ε=0
εg(x)︸ ︷︷ ︸=δy
−I[y] =dI
dε
∣∣∣ε=0
δy
and thereforeδI = I[y + δy]− I[y] ≈ dI
dε
∣∣∣ε=0
δy (747)
to first order in ε. The latter implies that the condition-dIdε
∣∣∣ε=0
= 0 for the
extreme curve y(x) is translated into the equivalent restriction
δI = I[y + δy]− I[y] = 0 (748)
and due to the fact that the derivation was made for small ε to first order,the above equation is called the first variation. This method is indeed morestraightforward to use as the following examples demonstrate.
15 Principle of Least Action
The principle of least action seems to be the most profound and useful tool intheoretical physics. Simply put, it is the application of Calculus of Variationsfor physical problems. There are several reasons contributing to this privilegedposition of this principle. First of all, its apparent simplicity and elegancelead to straightforward examinations of the systems under consideration . Inaddition, it provides a new, powerful way to look at known problems, givesadditional intuition, and the symmetries of the system appear in an apparentway. Another reason is its Universality. Namely, it is used in many differentareas of physics with great success in all cases. For instance, it is used inClassical mechanics, Electrodynamics, Particle Physics, and in Gravity as wehave solely used it throughout this thesis. These are but few reasons telling uswhy the Principle of Least Action is considered to be the most profound conceptof modern theoretical physics. In the following subsections we present some ofits many applications and examine the physical significance in each case.
107
15.1 With dependence on higher derivatives
Let us now allow for a Lagrangian which depends on higher order (greater thanthe first) derivatives. The form of such a Lagrangian is
L = L(y(x), y′(x), ..., y(n)(x), x
)(749)
where, with y(n)(x) we denote the n − th derivative of y(x) with respect to x,namely y(n)(x) := dn
dxn y(x). Now we also assume that all the derivatives up to(n− 1)-order of the variation vanish at the boundaries, along with δy. In words
δy|ti= 0 = δy′|ti = ... = δy(n−1)|ti (750)
where i = 1, 2. By definition we have
δL = L(y + δy, y′ + δy′, ..., y(n) + δy(n), x)− L(y, y′, ..., y(n), x) (751)
and Taylor expanding L(y + δy, y′ + δy′, ..., y(n), x) we arrive at (keeping onlylinear terms56)
L(y + δy, y′ + δy′, ..., y(n), x) ≈ L(y, y′, ..., y(n), x)+ (752)∂L
∂yδy +
∂L
∂y′δy′ + ...+
∂L
∂y(n)δy(n)
and thereforeδL =
∂L
∂yδy +
∂L
∂y′δy′ + ...+
∂L
∂y(n)δy(n) (753)
The variation of the action will then be
δS = δ
∫dxL(y, y′′, ..., y(n), x) =
∫dxδL =
=∫ x2
x1
dx(∂L∂y
δy +∂L
∂y′δy′ + ...+
∂L
∂y(n)δy(n)
)(754)
Now, to manipulate the terms involving derivatives of δy we first compute∫ x2
x1
dx∂L
∂y′δy′ =
∫ x2
x1
dx[ ddx
( ∂L∂y′
δy)− δy d
dx
( ∂L∂y′
)]=
=( ∂L∂y′
δy)∣∣∣x2
x1︸ ︷︷ ︸=0
−∫ x2
x1
dxδy[ ddx
( ∂L∂y′
)]= −
∫ x2
x1
dxδy[ ddx
( ∂L∂y′
)]⇒
∫ x2
x1
dx∂L
∂y′δy′ = −
∫ x2
x1
dxδy[ ddx
( ∂L∂y′
)](755)
continue with∫ x2
x1
dx∂L
∂y′′δy′′ =
∫ x2
x1
dx[ ddx
( ∂L∂y′′
δy′)− δy′ d
dx
( ∂L∂y′′
)]=
=( ∂L∂y′′
δy′)∣∣∣x2
x1︸ ︷︷ ︸=0
−∫ x2
x1
dx[ ddx
(δy
d
dx
∂L
∂y′′
)− δy d
2
dx2
( ∂L∂y′′
)]=
56Since the variations δy,... etc are small we can drop quadratic and higher order terms.
108
=[δy
d
dx
( ∂L∂y′′
)]∣∣∣x2
x1︸ ︷︷ ︸=0
+∫ x2
x1
dxδy[ d2
dx2
( ∂L∂y′′
)]=
=∫ x2
x1
dxδy[ d2
dx2
( ∂L∂y′′
)]⇒
∫ x2
x1
dx∂L
∂y′′δy′′ =
∫ x2
x1
dxδy[ d2
dx2
( ∂L∂y′′
)](756)
now we see the pattern, all the derivatives on δy are now acting on the termsthat contain partial derivatives of L with respect to y, and for an odd numberof partial integrations we pick up a factor −1 whilst for even number a factorof +1. Indeed, for any term ∂L
∂y(k) , with k = 0, 1, 2, ..., n the k− th derivative itwill hold∫ x2
x1
dx∂L
∂y(k)δy(k) =
∫ x2
x1
dx[ ddx
(δy(k−1) ∂L
∂y(k)
)− δy(k−1) d
dx
( ∂L
∂y(k)
)]=
=(δy(k−1) ∂L
∂y(k)
)∣∣∣x2
x1︸ ︷︷ ︸=0
−∫ x2
x1
dxδy(k−1)[ ddx
( ∂L
∂y(k)
)]= ...︸︷︷︸
k−times
=
= (−1)k
∫ x2
x1
dxδy[ dk
dxk
( ∂L
∂y(k)
)]⇒
and so, indeed∫ x2
x1
dx∂L
∂y(k)δy(k) = (−1)k
∫ x2
x1
dxδy[ dk
dxk
( ∂L
∂y(k)
)](757)
Therefore, the variation yields
δS = δ
∫ x2
x1
dxL(y, y′′, ..., y(n), x) =∫ x2
x1
dxδL =
=∫ x2
x1
dx(∂L∂y
δy +∂L
∂y′δy′ + ...+
∂L
∂y(n)δy(n)
)=
=∫ x2
x1
dxδy[∂L∂y
− d
dx
( ∂L∂y′
)+ ...+ (−1)n dn
dxn
( ∂L
∂y(n)
)]=
=∫ x2
x1
dxδy[ n∑
k=0
(−1)k dk
dxk
( ∂L
∂y(k)
)]= 0 (758)
and for the last one to hold true we must haven∑
k=0
(−1)k dk
dxk
( ∂L
∂y(k)
)= 0 (759)
15.2 Classical Particle Mechanics
Consider now a classical particle moving in an 1− dim potential V (q), where qis a canonical coordinate. The Lagrangian for this system is simply
L(q, q) = T − V =12mq2 − V (q) (760)
109
The Principle of Least Action gives
δ
∫ t2
t1
dtL(q, q) = 0 ⇒ (761)
∫ t2
t1
dt δL(q, q) = 0 (762)
and as has been proven before for the general case
δL =∂L
∂qδq +
∂L
∂qδq (763)
with δq = 0 at the endpoints t1, t2. Partially integrating the second term, itfollows that
δL =∂L
∂qδq +
d
dt
(∂L∂qδq)− δq d
dt
(∂L∂q
)(764)
so that, equation (762) gives
0 =∫ t2
t1
dt[∂L∂qδq +
d
dt
(∂L∂qδq)− δq d
dt
(∂L∂q
)]=
=(∂L∂qδq)∣∣∣t2
t1+∫ t2
t1
dtδq[∂L∂q
− d
dt
(∂L∂q
)]=
=∫ t2
t1
dtδq[∂L∂q
− d
dt
(∂L∂q
)](765)
Now since t1 and t2 are arbitrary and the latter should vanish for any variationδg we conclude that
∂L
∂q− d
dt
(∂L∂q
)= 0 (766)
this is the famous Euler-Lagrange equation.
16 Definitions and Some Proofs
16.1 The Weyl/Conformal Tensor
In gravity there exists a unique quantity that is invariant under conformal trans-formations
gµν = Ω2(x)gµν (767)
This quantity is the Weyl tensor which, in terms of the Riemann and metrictensor, by
Cµνρσ = Rµνρσ−12(gµρRνσ +gνσRµρ−gνρRµσ−gµσRνρ)+
R
6(gµρRνσ−gµσRνρ)
(768)in 4 dimensions. We should mention here that unlike the Ricci tensor and scalarwhich desribe the curvature around a spacetime point due to the presence ofmatter (or energy) at this point, the Weyl tensor describes the curvature around
110
this point due to the presence of matter in another point. For a general n−dimmanifold its form is
Cabcd = Rabcd −2
n− 2(ga[cRd]b − gb[cRd]a) +
2(n− 1)(n− 2)
Rga[cgd]b (769)
By construction, it is trace-free
Cabac = 0 = Ca
abc = Cabca (770)
in all its indices, as can be easily checked. In 4 − dim gravity its square (con-traction with itself) is given by
CµνρσCµνρσ =
13R2 − 2RµνR
µν +RµνρσRµνρσ (771)
and as is obvious from the latter, it is related with the Gauss-Bonnet termthrough
CµνρσCµνρσ = 2
(RµνR
µν − R2
3
)+ G (772)
where G is the Gauss-Bonnet term. There have been proposed theories of Grav-ity which have the conformally invariant Gravity action
S =∫d4x
√−gCµνρσC
µνρσ (773)
Now, since the Gauss-Bonnet term does not affect the field equations, (surfaceterm) using the above equation the action becomes
S =∫d4x
√−gCµνρσC
µνρσ =
= 2∫d4x
√−g(RµνR
µν − R2
3
)+ surface term (774)
We point out here, that this theory cannot be a theory of Gravity because itlacks cosmological solutions. Indeed, as we have proved in previous chapter forflat (K = 0) FLRW universes
RµνRµν − 1
3R2 = −H2(H +H2) = −H2 a
a(775)
so that ∫d4x
√−g[RµνR
µν − 13R2
]= −
∫d4xa3H2 a
a=
= −∫d4xa2a = −1
3
∫d4x
d
dt(a3) = surface term (776)
and the variation of (774) with respect to the scale factor is trivial, yielding nocosmological equations of motion for the scale factor. As a result, the theorygiven by (774) cannot describe a valid theory of Gravity. The above result canalso be seen, simply by the fact that the FLRW spacetime is conformally flat,namely
Cµνρσ = 0 (777)
for an FLRW background. Therefore, any theory containing the square of theWeyl tensor alone in the action does not have cosmological solutions.
111
16.2 The generalized Kronecker delta
The generalized Kronecker delta is defined by
δµ1µ2...µkν1ν2...νk
≡ det
δµ1ν1... δµ1
νk
. .
. .δµkν1... δµk
νk
(778)
which can also be written in the more compact form
δµ1µ2...µkν1ν2...νk
≡ 1k!δµ1[ν1δµ2ν2...δµk−1
νk−1δµk
νk] (779)
where the square brackets denote antisymmetrization. For instance, we explic-itly compute
δµνρσ = det
(δµρ δµ
σ
δνρ δν
σ
)= δµ
ρ δνσ − δµ
σδνρ (780)
16.3 Lovelock Scalars
The Lovelock scalars are scalar combinations of the Riemann and metric tensorwhich when they appear in the gravitational action give rise to field equationsthat are of second order with respect to the metric tensor. The so-called Love-lock Gravity exist of a Lagrangian density formed by linear combinations ofsuch scalars. The Lovelock scalars are, in fact, dimensionally extended Eulerdensities. The n− th.....
Ln =12nδµ1ν1...µnνn
α1β1...αnβnRα1β1
µ1ν1...Rαnβn
µnνn=
12nδµ1ν1...µnνn
α1β1...αnβn
n∏k=1
Rαkβkµkνk
(781)where Rµν
ρσ is the Riemann tensor. Lovelock Gravity consists of the sum of suchLagrangian densities. To be more precise, a Lovelock Gravity in d-dimensionsis given by
S =∫ddx
√−gL (782)
withL =
∑n
αnLn (783)
Note that the only terms that contribute to the field equations are those whichsatisfy n < d
2 and therefore in 4− dim the only contributing term is
L1 =12δµ1ν1α1β1
Rα1β1µ1ν1
=12(δµ1
α1δν1β1− δν1
α1δµ1β1
)Rα1β1µ1ν1
=
=12(Rµ1ν1
µ1ν1−Rν1µ1
µ1ν1) = Rµ1ν1
µ1ν1= gαµ1gβν1Rαβµ1ν1 =
= gβν1Rαβαν1
= gβν1Rβν1 = R⇒L1 = R (784)
and therefore, Lovelock’s theory in 4 − dim coincides with Einstein Gravity.This is true for lower dimensions, however does not hold true for spaces of
112
dimensionality greater than 4 due to the fat that additional scalar combinations(apart from R) appear. For n− 2 a similar computation to the above, reveals
L2 =122δµ1ν1µ2ν2α1β1α1β1
Rα1β1µ1ν1
Rα2β2µ2ν2
= R2 − 4Rµν +RµνρσRµνρσ (785)
which is of course the Gauss-Bonnet term. We mention again that in 4−dim thiscombination does not affect the field equations when linearly added, however itdoes give modifications for higher dimensional theories.
η ≡ ε
εH(786)
16.4 Energy-momentum tensor for a perfect fluid
Following a fully covariant way, we prove here that the energy-momentum tensorfor a perfect fluid in curved spacetimes is given by
Tµν = pgµν + (ρ+ p)uµuν (787)
where p and ρ are the pressure and the density of the isotropic fluid respec-tively, gµν the metric tensor of the spacetime, and uµ the velocity field of anobserver following the motion of the fluid. For Minkowski spacetime (ηµν =diag(−1, 1, 1, 1)) the non-vanishing components of Tµν are known to be
T00 = T 00 = ρ, Tij = pδij (788)
Let us see now, how one may approach the problem and find Tµν for a perfectfluid in curved spacetimes. Since Tµν is a rank − 2 tensor, we should be ableto express it in terms of fundamental rank − 2 tensors of the given spacetime.Let us now search for such tensors. One choice is obvious, it is of course themetric tensor gµν . This contribution comes from geometry. We should also havesome kinematic characteristics contained in the form of the energy-momentumtensor. Such a kinematic quantity is the 4− velocity uµ of an observer movingin the spacetime. But this is a rank − 1 tensor, namely a vector. Hopefully,we can form a symmetric rank − 2 tensor out of uµ by forming what is calledthe tensor product uµ ⊗ uν . Thus, Tµν will be a linear combination of gµν anduµuν . However, this is not the end of the story, we also need to attribute somefluid characteristics. These will be the pressure p together with density ρ. Sincethe latter are scalars, the way they come into the game is as coefficients of thetwo tensors mentioned before. More accurately, one should consider functionsof p and ρ as coefficients. Therefore, the general form of the energy-momentumtensor is
Tµν = A(p, ρ)gµν +B(p, ρ)uµuν (789)
where A(p, ρ) and B(p, ρ) are continuous, differentiable functions (of p and ρ ofcourse!) to be determined. We proceed as follows. The latter equation shouldalso hold true locally, namely when Gravity can be ignored. Then, Tµν willbe that of special relativity as given by (788) and gµν will be reduced to theMinkowski tensor ηµν . In addition, considering a co-moving observer, that isone that follows the smoothly expansion/contraction of the fluid, we have
uµ = (1, 0, 0, 0) = δµ0 (790)
113
such thatuµ = −δ0µ (791)
Taking all the above into account, one has locally
TSRµν = A(p, ρ)ηµν +B(p, ρ)δ0µδ
0ν (792)
where SR stands for Special Relativity. Contracting the last with ηµν it followsthat
−ρ+ 3p = 4A−B (793)
On the other hand, contraction with δµ0 δ
ν0 yields
ρ = −A+B (794)
and by adding the latter two we obtain
A = p (795)
and substituting this to either of the above ones, we get
B = ρ+ p (796)
Thus, the energy-momentum tensor for a perfect fluid in curved spacetimes isgiven by
Tµν = pgµν + (ρ+ p)uµuν (797)
16.5 Operators in flat(K = 0) FLRW spacetime
The symmetries of FLRW universes exclude any spatial dependence of scalarsliving in these spacetimes.57 Therefore, any scalar quantity-φ is time dependentonly, namely φ→ φ(t). As an immediate result we have that
∇µφ = ∂µφ = ∂µφ(t) = φδ0µ (798)
such that∇µ∇νφ = ∇µ (φδ0ν)︸ ︷︷ ︸
call it uν
= ∇µuν =
= ∂µuν − Γαµνuα = ∂µ(φδ0ν)− Γα
µνδ0αφ =
= δ0µδ0ν φ− Γ0
µν φ⇒
∇µ∇νφ = δ0µδ0ν φ− Γ0
µν φ (799)
Now, for the Christoffel symbols we have
Γ000 = 0, Γ0
ij = aaδij = Hgij (800)
and therefore∇0∇0φ = φ (801)
and∇i∇jφ = −aaφδij = −Hφgij (802)
57Remember that there is no preferred direction or any special points. Everywhere is alike.
114
We can now compute the action of D’Alambertian (or just ”Box”) on a scalar,
2φ ≡ gµν∇µ∇νφ = −∇0∇0φ+ gij∇i∇jφ =
= −φ−Hφgijgij = −(φ+ 3Hφ) ⇒
2φ = −(φ+ 3Hφ) = −( d2
dt2+ 3H
d
dt
)φ (803)
In section (?) we had defined the tensorial-differential operator Dµν via
Dµν ≡ gµν2−∇µ∇ν (804)
In a flat FLRW background, the action of the latter on scalars, yields58
Dµνφ = (gµν2−∇µ∇ν)φ = −(gµν + δ0µδ0ν)φ+ (Γ0
µν − 3Hgµν)φ (805)
The only non-vanishing components of which, are
D00φ = 3Hφ (806)
Dijφ = −(φ+ 2Hφ)gij (807)
It is aslo true that
Dφ = gµνDµνφ = (42−2)φ = 32φ (808)
So that a new tensorial operator can be defined via
Dµν = Dµν −34gµν2 (809)
which is traceless (D = gµνDµν = 0) by construction.
16.6 An identity coming from Gauss-Bonnet term
Interestingly enough, if one expresses all the quantities in the Gauss-Bonnetterm as rank-(2, 0) tensor contractions with the metric tensor, one can deducean additional identity. We have
G = R2 − 4RαβRαβ +RαβγδR
αβγδ (810)
and our goal is to express the latter as
G −R2 + 4RαβRαβ −RαβγδR
αβγδ = 0 = Aµνgµν (811)
for some tensor Aµν and then conclude that
Aµν = 0 (812)
We first start with G which can straightforwardly be written as
G = 414G = gµνgµν
14G =
(gµν
4G)gµν (813)
58Note that the outcome of the action of Dµν on a scalar is a symmetric rank-2 tensor.This symmetry is only present for scalars and is an immidiate consequence of the fact that∇µ∇νφ = ∇ν∇µφ for any scalar φ.
115
Now we continue we the term R2 which we express as
R2 = RR = RRµνgµν = (RRµν)gµν (814)
In addition, sinceRαβR
αβ = (RµρRνρ)gµν (815)
but alsoRµρσνgµν = −Rρσ ⇒
RρσRρσ = −RµρσνgµνRρσ = − (RµρσνRρσ) gµν (816)
we may write
RαβRαβ =
12RαβR
αβ +12RαβR
αβ =
=12(RµρRν
ρ)gµν −(
12RµρσνRρσ
)gµν ⇒
RαβRαβ =
[12(RµρRν
ρ −RµρσνRρσ)]gµν (817)
As for the contraction RαβγδRαβγδ one has
RαβγδRαβγδ = (RµαβγRν
αβγ)gµν (818)
Therefore, substituting the above results to equation (810) we obtain
0 = G −R2 + 4RαβRαβ −RαβγδR
αβγδ =
=[gµν
4G −RRµν + 4
12(RµρRν
ρ −RµρσνRρσ)−RµαβγµRναβγ
]gµν (819)
and since the last must always be true for any gµν we conclude that
gµν
4G −RRµν + 4
12(RµρRν
ρ −RµρσνRρσ)−RµαβγRναβγ = 0 ⇒
gµνG = 4RRµν + 8(RµρσνRρσ −RµρRνρ) + 4RµαβγRν
αβγ (820)
16.7 Derivation of the continuity equation for FLRW uni-verses
We prove here that the (covariant) conservation of the energy-momentum tensor
∇µTµν = 0 (821)
implies the continuity equation
ρ+ 3H(ρ+ p) = 0 (822)
for FLRW universes. We start from the fact that the energy-momentum tensorin a homogeneous and isotropic universe, has to have the perfect fluid form
Tµν = (ρ+ p)uµuν + pgµν (823)
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and we take a comoving observer, namely one for which the 4 − velocity fieldsatisfies
uµ = (1, 0, 0, 0) = δµ0 (824)
it also follows thatuµ = −δ0µ (825)
as a direct consequence of the fact that
uµuµ = −1 (826)
Recall also that since any scalar quantity f in such universes cannot have anyspatial dependence, it follows that f(~x, t) → f(t) and thus
∇µf = ∂µf = δ0µf (827)
Then, acting with ∇µ on the above energy-momentum tensor, we have
∇µTµν = uνuµ∇µ(ρ+ p) + (ρ+ p)[uν∇µuµ + uµ∇µuν
]+ gµν∇µp (828)
Computing each term separately, we have
uνuµ∇µ(ρ+ p) = uνuµ∇µ(ρ+ p) = −δ0νδ
µ0 ∂µ(ρ+ p) = −δ0ν(ρ+ p) (829)
uµ∇µuν = uµ∇µuν = uµ[∂µuν︸ ︷︷ ︸
=0
−Γαµνuα
]=
= −Γαµνuαu
µ = +Γαµνδ
0αδ
µ0 = Γ0
0ν = 0 (830)
uν∇µuµ = uν∇µuµ = −δ0ν(∂µu
µ︸ ︷︷ ︸=0
+Γµµαu
α) =
= −δ0νΓµµαδ
α0 = −δ0νΓµ
µ0 = −δ0ν(Γ110 + Γ2
20 + Γ330) = −δ0ν3H (831)
and, of coursegµν∇µp = ∇νp = ∂νp = δ0ν p (832)
Upon substituting all the above into (828) it follows that
∇µTµν = −δ0ν(ρ+ p)− δ0ν3H(ρ+ p) + δ0ν p =
= −δ0ν[ρ+ 3H(ρ+ p)
](833)
and thus, the requirement
∇µTµν = 0 (834)
is translated into−δ0ν
[ρ+ 3H(ρ+ p)
]= 0 (835)
for ν 6= 0 the latter is trivially satisfied, and for ν = 0 we obtain
ρ+ 3H(ρ+ p) (836)
as stated.
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