University of Perugia

Department of Physics and Geology

Degree course in Physics

Final Thesis

Inflation and the Theory ofCosmological Perturbations

Supervisor:

Prof. Gianluca Grignani

Graduant:

Marco Quaglia

Academic Year 2013/2014

Abstract

This thesis is about inflation. The understanding of initial conditionsthat our universe may have experienced is not only a formal description ofa good scientific theory, but also a crucial answer to some of the deepestquestions of mankind: "What is the universe? How did it come to be?". Sucha theory is currently a hot topic because of experimental observations, so itis worth spending some time trying to understand its basic principles andtheoretical predictions. Inflation provides a consistent description of whatwe see today all around us and improves the Standard Big Bang Theory bysolving elegantly some of its problems. However, a theory needs not onlyto be elegant, but also to be verified through the sperimental method, thatis by observations. Nowadays a big effort from the scientific community isbeing made to test predictions of this theory, namely find a trace in theCosmic Microwave Background of primordial gravitational waves, which arecharacteristic of a model of inflation which is subject of this thesis.

Questa tesi tratta l’Inflazione. La comprensione delle condizioni inizialiche il nostro universo può aver attraversato non soltanto è una descrizioneformale di una buona teoria scientifica, ma anche una risposta cruciale allepiù profonde domande del genere umano: "Che cos’è l’universo? Come ènato ed evoluto?". Questa teoria è attualmente un tema caldo per via diosservazioni sperimentali, perciò vale la pena impegnare del tempo cercandodi capire i suoi principi di base e le sue previsioni teoriche. L’inflazionefornisce una descrizione coerente di ciò che vediamo oggi intorno a noi emigliora la Teoria Standard del Big Bang risolvendo con eleganza alcuni deisuoi problemi. Tuttavia, una teoria non solo deve essere elegante, ma ancheverificata attraverso il metodo scientifico, cioè da osservazioni. Ad oggi èstato fatto un grande sforzo da parte della comunità scientifica per testare leprevisioni di questa teoria, e cioè trovare una traccia nella radiazione cosmicadi fondo (CMB) di onde gravitazionali primordiali, che sono caratteristichedi un modello di inflazione che è oggetto di questa tesi.

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Contents

1 Introduction 51.1 Cosmological Principle . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 The inevitable Singularity . . . . . . . . . . . . . . . . . . . . . . . 6

2 Standard Comsmology 72.1 Friedmann-Lemaitre-Robertson-Walker metric . . . . . . . . . . . . 72.2 Shortcomings of standard Big-Bang Theory . . . . . . . . . . . . . 19

3 Inflationary Cosmology 233.1 Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2 Inflaton field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4 Cosmological Perturbations 314.1 Gravitational Instability . . . . . . . . . . . . . . . . . . . . . . . . 314.2 Quantum fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . 384.3 The consistency relation . . . . . . . . . . . . . . . . . . . . . . . . 42

Bibliography 47

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CHAPTER 1Introduction

1.1 Cosmological PrincipleEvery theory starts from principles. For most of the twentieth century, the homo-geneity and isotropy of the universe had to be taken as an assumption, known asthe “Cosmological Principle”.

Homogeneity and IsotropyWe are immersed in inhomogeneity. The everyday life naively proves that it so. Forinstance, human beings are inhomegeneities. And luckily for us, it is so. Withoutinhomogeneities we would not exist. But there is more than what meets the eye.It is indeed interesting to jump from home scale to cosmic distances and withthe right instruments unravel the composition of the universe at unimaginabledistances. We will then actually find that the latter is homogeneous.Homogeneity means that the physical conditions are the same at every point ofany given hypersurface.Isotropy means that the physical conditions are identical in all directions whenviewed from a given point on the hypersurface. Isotropy at every point automaticallyenforces homogeneity. The observable patch of the universe is of order 3000 Mpc1.Redshift surveys suggest that the universe is homogeneous and isotropic only whencoarse grained on 100 Mpc scales.

Why is the universe homogeneous in the scale 100-3000 Mpc? Why is it nothomogeneous under that scale? Finding an answer to these two questions is whatCosmology is all about. To do so, one needs to taste all fields of physics. Indeed,

11 parsec = 1 pc = 3.26 lightyears = 3.086× 1018cm 1 Mpc = 106 pc

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Cosmology uses methods from nearly all fields of theoretical physics, among whichare General Relativity, thermodynamics and statistical physics, nuclear physics,atomic physics, kinetic theory, particle physics and field theory. Although theCosmological Principle has been proved by experimental facts and its validity isrestricted to a limited range of scales, it keeps the rank of principle rather thanthe rank of law. Nevertheless, the most important feature of our universe is itslarge scale homogeneity and isotropy. This feature ensures that observations madefrom our single vantage point are representative of the universe as a whole and cantherefore be legitimately used to test cosmological models.

1.2 The inevitable SingularityIf we assume the Cosmological Principle and accept the experimental fact thatthe universe is expanding, then what we realize is that, reversing the time andso letting the universe contract, at some point in the past the scale factor of theuniverse, which states the dimension of the space, must vanish. in other wordseverything, including matter, dark matter, dark energy and radiation, must squeezeinto a tremendously small volume.2 Physicists call this moment in time singularity.It is very rare that a cosmological theory does not predict the singularity, thoughthere exist some cases (involving certain values of a cosmological constant) in whichit is possible to avoid it.

2Saying that space is contracting means that the space is truely becoming smaller. Then,simply, everything within it must follow the contraction. Such contraction reaches the scaleof Planck lenght, which corresponds to a volume V = l3p ∼ 10−99cm3. Just as in the case of aballoon that is deflated, what is on the surface can not go out from it, but is forced to follow thecontraction, even to a point.

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CHAPTER 2Standard Comsmology

2.1 Friedmann-Lemaitre-Robertson-Walker metricAfter the birth of General Relativity and the development of differential geometry,it was possible to begin to study the properties of the universe through a formaltheory that would lead to theoretical results in agreement with experimental data.

Possible geometries of an homogeneous and isotropic universeHomogeneous and isotropic spaces have the largest possible symmetry group; inthree dimensions there are three independent translations and three rotations.These symmetries strongly restrict the admissible geometry for such spaces. Thereexist only three types of homogeneous and isotropic spaces with simple topology:

• flat space

• a three-dimensional sphere of constant positive curvature

• a three-dimensional hyperbolic space of constant negative curvature.

It is possible to work with such spaces by embedding them in a higher dimensionalspace; this means describing their geometry with some appropriate coordinatesin the new space. For instance, the first two spaces can be embedded in a four-dimensional Euclidean space with Cartesian coordinates; in other words it ispossible to "visualize" them (e.g., a 2-sphere is easy to imagine and in fact it canbe embedded in a three-dimensional Euclidean space). The latter, instead, becauseof a negative curvature, cannot be embedded, i.e. visualized, in a four-dimensionalEuclidena space. (e.g. a 2-sphere with imaginary radius is hard to imagine, though

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such space, known as Lobachevski space, can be visualized as a hyperboloid inLorentzian three-dimensional space). The geometry of these spaces is described byinfinitesimal interval via the metric in the following way:

dl2 = gijdxidxj (2.1)

If one performs a an infinitesimal coordinate transformation xk → xk + ξk, themetric changes according to

δgij = −(∇iξj +∇jξi) (2.2)

If δgij = 0, then the metric is called form-invariant under the coordinate trans-formation, which takes the name isometry, and ξ is called Killing vector, whichmeans that satisfies the Killing equation

∇iξj +∇jξi = 0 (2.3)

This invariance is strongly related to a certain symmetry of the metric and, thus,of the space. Any given space of dimension n cannot have more than n(n+ 1)

2Killing vectors. The more the Killing vectors, the more the metric is invariantunder independent coordinate transformations. When the number of Killing vectorsis highest, then the space is said to be maximally symmetric and it has the mostpossible symmetry properties for the given dimension. It can be demonstratedthat if a space has the maximum number of Killing vectors then it is homogeneousand isotropic and viceversa. Moreover, maximally symmetric spaces are uniquelyspecified by a "constant curvature" K, which can be positive, negative or null. Tothese three cases the associated metric describes respectively an hypersphere Sn,an hyperbolic space Hn and an Euclidean space Rn.

The metric of a three-dimensional homogeneous and isotropic universeNow consider a space with dimension n = 3 with 6 Killing vectors, i.e. anhomogeneous and isotropic spatial universe. One can derive the metric of such aspace.

Consider the analogous two-dimensional homogeneous, isotropic surface. Thegeneralization to three dimensions is straightforward. Two well known cases ofhomogeneous, isotropic surfaces are the plane and the 2-sphere. They both can beembedded in three-dimensional Euclidean space with Cartesian coordinates x, y, z.The equation describing the embedding of a two-dimensional sphere is

x2 + y2 + z2 = a2 (2.4)

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where a is the radius of the sphere. Differentiating this equation, it follows that fortwo infinitesimally close points on the sphere

dz = −xdx+ ydy

z= ± xdx+ ydy√

a2 − x2 − y2 (2.5)

Substituting this expression into the three-dimensional Euclidean metric

dl2 = dx2 + dy2 + dz2 (2.6)

gives

dl2 = dx2 + dy2 + (xdx+ ydy)2

a2 − x2 − y2 (2.7)

In this way, the distance between a pair of points located on the 2-sphere isexpressed entirely in terms of two independent coordinates x and y. It is convenientto introduce instead of x and y the angular coordinates r′,φ defined in the standardway:

x = r′ cosφ, y = r′ sinφ. (2.8)

Differentiating the relation x2 + y2 = r′2 gives

xdx+ ydy = r′dr′ (2.9)

Combinig this withdx2 + dy2 = dr2 + r′2dφ2 (2.10)

the metric in (2.7) becomes

dl2 = dr′2

1− (r′2/a2) + r′2dφ2 (2.11)

The limit a2 → ∞ corresponds to a (flat) plane. a2 can be formally taken to benegative and then metric (2.11) describes a homogeneous, isotropic two-dimensionalspace with constant negative curvature, known as Lobachevski space. Unlike theflat plane or the two-dimensional sphere, Lobachevski space cannot be embeddedin Euclidean three-dimensional space because the radius of the “sphere” a isimaginary (this is why this space is called a pseudo-sphere or hyperbolic space).Of course, this does not mean that this space cannot exist. Any curved spacecan be described entirely in terms of its internal geometry without referring to itsembedding. Introducing the rescaled coordinate r = r′/

√|a2|, themetric (2.11) can

be recast asdl2 = |a2|

(dr′2

1− kr′ + r′2dφ2)

(2.12)

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where k = +1 for the sphere (a2 > 0), k = −1 for the pseudo-sphere (a2 < 0) andk = 0 for the plane (two-dimensional flat space). The generalization of the abovemetric to four dimensions is straightforward: add the third spatial dimension sothat

dφ2 −→ dθ2 + sin θ2dφ2 (2.13)

and the temporal dimension −dt2. What follows is themaximally spatially symmetric Friedmann-Lemaitre-Robertson-Walker line element1

ds2 = −dt2 + a2(t)[

dr2

1− kr2 + r2(dθ2 + sin2 θdφ2)]

(2.14)

This metric corresponds to a particular gauge, i.e. a choice of coordinates, inwhich the universe appears as spatially homogeneous and spatially isotropic. Thecoordinates xµ = (t, r, θ, φ) are called co-moving coordinates2. An object at restwith respect to these coordinates will remain at rest with time: it moves along withthe expansion of the space. If an object has velocity with respect to co-movingcoordinates (caused by gravitational intercation for instance), then it has a peculiarvelocity, which is different from the co-moving velocity, that is the one governed bythe Hubble law (the sum of the two velocities is called recessional velocity and isthe one measured by observations with Doppler shift in spectral lines).

It is worth to introduce new variables to explicit the FLRW metric in such away that its structure is easier to manipulate mathematically. These variables arecalled conformal3 and read

dχ2 = dr2

1− kr2 and dη = dt

a(t) (2.15)

In this way (2.14) becomes

ds2 = a2(t)[−dη2 + dχ2 + φ2k(χ)(dθ2 + sin2θdφ2)] (2.16)

where

φk(χ) =

sinh2 χ, k = +1χ2, k = 0sin2 χ k = −1

(2.17)

1adding the term spatially is not for fun; in fact it is worth to emphasise that homogeneityand isotropy hold for space and not for spacetime, that is only for the three spatial dimensions.

2Only a co-moving observer will think that the universe looks isotropic; in fact on Earth weare not quite co-moving, and as a result we see a dipole in the cosmic microwave background(CMB) as a result of the conventional Doppler effect.

3The name embodies the proportionality to the Minkowski metric with t→ η and r → η

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One example of mathematical simplicity coming from these coordinates is theequation for light geodesics. The condition for a geodesics is

ds2 = 0

and by symmetry the radial trajectory θ, φ = const corresponds to a geodesics.Thus it follows that

dη2 − dχ2 = 0 (2.18)so, radial geodesics are described by

χ(η) = ±η + const (2.19)

and correspond to straight lines at 45 degrees in the η − χ plane.

Hubble LawThe Hubble law is the unique expansion law compatible with homogeneity andisotropy. Given two co-moving points A and B in a two-dimensional sphere, thedistance on the surface of the sphere is given by

rAB = a(t)θAB (2.20)

where a(t) is the radius of the two-sphere and θAB is the angle subtended by theradii of the points A and B. Differentiating (2.20) with respect to time we obtain

rAB = vAB = a(t)θAB = a(t)a(t)rAB (2.21)

Defining now the Hubble parameter

H = a

a(2.22)

and passing to three-dimensional vectors Hubble law takes the form

vAB = HrAB (2.23)

This is the most used version of Hubble Law, which relates the "distance" of anobject to its "velocity".

However there are some considerations that have to be done, for this formulahas some limitations. At large scales General Relativity is well experimentallytested, thus it can be used to improve the notion of distance. Distance is definedas follows:

dl2 = γµνdxµdxν (2.24)

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whereγµν = −gµν + g0µg0ν

g00(2.25)

is the three-dimensional metric tensor which defines the metric, i.e. the geometricalproperties of the space. It is important to point out that components gµν depend,generally, on x0, so that spatial metric (2.24) varies with time. For this reason,it is meaningless to integrate dl: such an integral would depend on the worldline between the two given spatial points. Consequently, in General Relaticity,the notion of distance determined by two bodies has not, in general, meaning: itjust has a local meaning. The only case in which distance can be defined evenin finite regions of space is that of reference systems where gµν do not depend ontime, and, consequently, the integral

∫dl gains an actual meaning. This is not the

case for FLRW metric, where the scale factor a = a(t) depends on time, thereforeit is possible to talk about distances only on a local scale, which is around 109

lightyears.4 Nevertheless, it is possible to define different kinds of "distances" whichcan be used depending on the physical conditions of the system under investigation.Such distances are: luminosity distance dL, proper motion distance dM , angulardiameter distance dA, related to each other by a rather simple relation (if FLRWmetric holds for the universe)

dL = (1 + z)dM = (z)2dA (2.26)

where z is the cosmological redshift, which will be defined later.To summarize, when regular Hubble law is used, it is imperative to remember thatit is valid in an approximation.

From Hubble law it is possible to evaluate an important quantity: the Hubbleradius. The latter is the radius at which an object reaches the recessional velocityequal to that of light

RH = c

H(2.27)

Physical objects farther than this radius have velocity v > c. This is not in contrad-diction with Special Relaticity, because the quantities used to define recessionalvelocity, distance and time, do not belong to the same inertial frame.

4Recall that, generally, scales which are interesting to study are between 100-3000 Mpc, wherethe Cosmological Principle is valid: some of this interval is included in 109 lightyears. It isimportant to note that if the scale factor a(t) were a constant, say a0, then it would be possibleto define distances for any scale: this is possible only in a static universe, which is not the case forthe universe we live in. Or otherwise, in expanding or contracting universes, by taking fixed timespatial hyperfurfaces: for the FLRW metric

∫ √ds2t=constant ∝ a(t), although then distance is no

more physical. Indeed, try to use a ruler to istanteneously measure a distance of some light years.

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Cosmological RedshiftThe expansion of the universe leads to a redshift of the photon wavelength. Toanalyse this effect, let us consider a source of radiation with co-moving coordinateχem, which at time ηem emits a signal of short conformal duration ∆η. Accordingto equation of light geodesics, the trajectory of the signal is

χ(η) = χem − (η − ηem) (2.28)

and it reaches a detector located at χobs = 0 at time ηobs = ηem + χem. Theconformal duration of the signal measured by the detector is the same as at thesource, but the physical time intervals are different at the points of emission anddetection. They are equal to

∆tem = a(ηem)∆η and ∆tobs = a(ηobs)∆η (2.29)

respectively. If ∆t is the period of the light wave, the light is emitted withwavelength λem = ∆tem but is observed with wavelength λobs = ∆tobs, so that

λobs

λem= a(ηobs)a(ηem) (2.30)

Thus, the wavelength of the photon changes in proportion to the scale factor,λ(t) ∝ a(t), and its frequency, ω ∝ 1/λ, decreases as 1/a.The redshift parameter is defined as the fractional shift in wavelength of a photonemitted by a distant galaxy at time tem and observed on Earth today:

z = λobs − λem

λem(2.31)

According to (2.30), the ratio λobs/λem is equal to the ratio of the scale factors atthe corresponding moments of time, and hence

1 + z = a0

aem(2.32)

where a0 is the present value of the scale factor. The light detected today wasemitted at some earlier time tem and, according to (2.32), there is a one-to-onecorrespondence between z and tem. Therefore, the redshift z can be used insteadof time t to parameterize the history of the universe. A given z corresponds toa time when our universe was 1 + z times smaller than now. It is important toremark that this redshift is not the Doppler effect due to relative velocities, but itis due to the space expansion. Within the approximation of short distances (smallcompared to the Hubble radius) they look the same, thus it is possible to relatevelocity and redshift parameter by v = cz, where c is the speed of light.

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HorizonsAn important physical notion in cosmology is that of horizon, which is very usefulin understanding what is the amount of information available to experimental ob-servation from any physical source. In fact, an horizon delimits a region, accordingto some rule, and usually what is inside this region is of direct interest.

Particle Horizon:This horizon corresponds to a boundary of a region of space which encloses allphysical phenomena that can have causal contact with the observer at a certain timeand it is finite if the age of the universe is finite.5 More specifically, it is a sphere ofradius R = cttravelled and this radius grows at the speed of light. An observer cannotreceive any information at a certain conformal time from phenomena that haveoccured outside the odiern particle horizon at the beginning of the universe. Thiscan be seen finding of the maximum co-moving distance that light can propagate

χp(η) = η − ηi =∫ t

ti

dt

a(2.33)

At time η information from events at χ > χp is inaccessible to an observer locatedat χ = 0.

Optical Horizon:The initial time from which light has travelled is trecombination, which is the momentwhen the universe became opaque to light, that is when light radiation decoupledfrom the rest. This time is roughly trecombination = 300000yrs after the Big Bangand the light reaching us from this moment is the CMB6. The physical distance ofoptical horizon is obtained multiplying χopt by the scale factor:

dopt = a(η)(η − ηrecombination (2.34)

Hubble Horizon:This horizon is really the Hubble scale, 1/H. In some circumstances it is of theorder of the particle horizon, that is why it shares the term horizon. Althoughthis is not a true horizon, it is rather important when talking about cosmologicalperturbations. It ishould be called curvature scale, because it is a dynamical scale

5Information cannot exceed the speed of propagation of light; since the latter is discrete, c,information travels a finite distance in a finite time interval.

6Information before this time is unavailable through light radiation. Thus, decoupling is thekey for understanding the evolution of the first stages of the universe: detecting some phenomenathat decoupled before light means having access to information about an earlier moment in timeof the universe. Candidates to this role are: gravitational waves and primordial neutrinos.

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that describes the rate of expansion of the universe and states the size of the localinertial frame.

Event Horizon:This horizon is the complement of the particle horizon. It encloses the region ofspace from which signals sent at a given moment in time η will never reach anobserver in the future. In co-moving coordinates, the distance from the observerthat satisfies the latter condition is

χ > χe(η) =∫ ηmax

ηdη = ηmax − η (2.35)

The physical size is then given by

de(t) = a(t)∫ tmax

t

dt

a(2.36)

and its convergence depends on the shape of the scale factor7.

Friedmann equationsEnergy-momentum Tensor An important assumption when studying the evolutionof the scale factor a(t) is to choose matter and energy to be a perfect fluid, that isat rest in co-moving coordinates. Then, the 4-velocity reads

uµ = (1, 0, 0, 0) (2.38)

From the property of homogeneity and isotropy of a perfect fluid, the energy-momentum tensor

Tµν = (ρ+ p)uµuν + pgµν (2.39)

becomes diagonal,T µν = (−ρ, p, p, p) (2.40)

Conservation of Energy Moreover, it is useful to introduce the conservation ofenergy. From the first principle of thermodynamics follows

dE = −pdV (2.41)7In a De Sitter universe, for example, the event horizon is constant, even if tmax =∞, because

the universe undergoes an exponential expansion

a(t) = e−Ht =⇒ de(t) = eHt∫ ∞t

e−Htdt = H−1 (2.37)

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Since V ∝ a3, we can rewrite this conservation law as

d(ρV ) = −pdV (2.42)d(ρa3) = −pd(a3) (2.43)

dρa3 + 3ρa2da = −3pa2da (2.44)

dρ = −3(ρ+ p)daa

(2.45)

from which, deriving with respect to time,

ρ = −3H(ρ+ p) (2.46)

Equation fo state Also, it is useful to proceed with a definition of an equation ofstate, which relates energy density ρ to pressure p. From a perfect fluid follows

p = ωρ (2.47)

If ω is not a constant it would not be really legitimate to call this an "equation fostate", neverthelss it is used very powerfully because of its simplicity. Using theequation of state and considering ω as a constant, the energy equation becomes

ρ

ρ= −3H(1 + ω) (2.48)

It follows thatρ ∝ a−3(1+ω) (2.49)

This equation describes the evolution of an energy density with respect to the scalefactor.

Friedmann equations Friedmann equations can be derived fron Einstein Equations(EE)

Rµν = 8πG(Tµν −12gµνT ) (2.50)

using the FLRW metric (2.14).From the component µν = ij one gets

H2 + k

a2 = 8πG3 ρ (2.51)

From the component µν = 00 gives

a

a= −4πG

3 (ρ+ 3p) (2.52)

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These two equations (2.51) and (2.52) are called Friedmann equations and areevolution laws for the scale factor a(t). The combination of (2.51) and either theconservation law (2.46) or the acceleration equation (2.52), supplemented by theequation of state p = p(ρ), forms a complete system of equations that determinesthe two unknown functions a(t) and ρ(t). The solutions, and hence the future ofthe universe, depend not only on the geometry but also on the equation of state.

There is a version of Friedmann equations that includes a cosmological constant,which in literature is represented by the letter Λ. It is sufficient to proceed asbefore from Einstein Equations given as follows:

Rµν = 8πG(Tµν −

12gµνT

)+ Λgµν (2.53)

Then, Friedmann equations in a more complete form are

H2 + k

a2 = 8πG3 ρ+ Λ

3 (2.54)

a

a= −4πG

3 (ρ+ 3p) + Λ3 (2.55)

This form of Friedmann equations are not always used and in these particular casesone prefers to set Λ = 0, having then (2.51) and 1eqrefeq:friedmann2.

Evolution of scale factorConsidering various ωi corresponding to different forms of energy, it is possible toobtain a scheme for the relation between density and scale factor and thus derivewhat kind of energy density is dominant in an interval of time, called era. Recasting(2.49) as

ρ ∝ a−ni (2.56)

where ni = 3(1 + ωi), then it follows table 2.1 for the various energy densities ρi.From table 2.1 it is clear that when the scale factor approaches zero, then

ωi nimatter 0 3radiation 1

3 4curvature -1

3 2vacuum -1 0

Table 2.1: Values of ωi and evolution power of energy densities.

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the radiation density is dominat over the othe densities and there is a radiation-dominated era (RD), whereas letting the scale factor increase there is a matter-dominated era (MD). It is very important to notice that if ρΛ, that is a vacuumenergy density, is not null, then it will dominate on all other forms of energy astime goes on.8 Also, the energy density related to the curvature, ρc, is fictitious inthe sense that it is non physical; it contributes, but it is not dominant in any ageof the universe.

An important quantity to define is the parameter Ω defined as follows

Ω = ρ

ρc, ρc = 3H2

8πG (2.57)

where ρc is the critical density. This parameter states the fate of the universe. Infact, according to its value, consequently the sign of k is determined and thus thegeometry of the universe. There is a very simple rule related to this parameter.Equation (2.54), dividing by H2 and rearring terms, can be recast as

Ω− 1 = k

a2H2 + Λ3H2 (2.58)

Considering Ω as referring to any kind of energy density, i.e. Ωi = ρi/ρ, and takinginto account that the two quantities on the right hand side of (2.58) are Ωk andΩΛ, then follows the golden rule of cosmology

∑Ωi = 1 (2.59)

Finally, it is possible to consider how the scale factor a(t) evolves in some era. ForΛ = 0 and any value of ωi such that ωi 6= −1, there is a simple realtion which isderived by integrating the first Friedmann equation (2.51),

a(t) ∝ t2

3(1+ωi) (2.60)

For Λ 6= 0 and ω = −1, solution for the scale factor is quite different. In thiscase the energy density is constant and so is the Hubble parameter H, from (2.54).Then

H(t) = a

a= const = H0 =⇒ a(t) = eH0t (2.61)

8Experimental observations suggest that nowadays we are experiencing a vacuum-dominatedera.

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2.2 Shortcomings of standard Big-Bang TheoryStandard Cosmology predicts almost all the backward history of the universe. Still,there are unsolvable puzzles. It is when Genral Relativity breaks down that thereexist a time limit in the past where predictions are no more solid and new puzzlesarise. This is the time of the domain of Quantum Gravity and is of the order ofthe Planck time,

tPl = 10−43s. (2.62)The misteries about the birth of our universe are within a time interval of suchunbelievable shortness. In other words, the whole problem of Standard Cosmologyis represented by the initial conditions of the universe. There are two independentsets of initial conditions characterizing matter:

1. spatial distribution9

2. initial field of velocities

Let us see what the puzzles arise from these initial conditions.

Horizon ProblemThe present homogeneous domain is at least as large as the present horizon, whichis roughly ct0 = 1026m, since the universe is roughly 15 billion years old. At thePlanck time, tPl, this domain was

lPl ∼ ct0aPla0

(2.63)

It is possible to compare this value with that of the causal domain lc = ctPl:

lPllc∼ t0tP

aPla0

(2.64)

Since the ratio of scale factors can be estimated from the ratio of temperatures by

aPla0∼ T0

TPl∼ O(1)

1032 ∼ 10−32 (2.65)

it follows thatlPllc∼ 1028 (2.66)

9Spatial distribution is related to the energy density ρ(x). Note that here there is a dependanceon points in three-dimensional space, whereas when studying large scales this dependace is droppedassuming the Cosmological Principle.

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This means that the scale of the universe at the Planck time exceeded the scaleof causal domain by 28 orders of magnitude, which correspond to 1084 causallydisconnected regions. Since none physical information could have been transmittedbetween these regions, it is unclear why they had such a fine-tuned matter distribu-tion10. Homogeneity scale must have been always larger than the scale of causality.This is why this problem is called horizon problem.

Flatness ProblemThere are two approches to this topic. The first starts directly from the initialfield of velocities and considers the fine-tuning of energies at the Planck time. Thesecond one concerns the cosmological parameter Ω.

Initial field of velocities:Velocities must all be spcified along with matter distribution to have the Cauchyproblem completely posed and thus obtaining evolution of the universe unam-bigously. These velocities must obey Hubble law, otherwise homogeneity is notpreserved11. This must occur in any region, even if causally disconnected. It ispossible to see how accurate velocities have to be in order to satisfy observableconditions today. Energy is conserved, thus the sum of the kinetic energy andgravitional self-interacting potential energy remains constant in time:

Etot = EkP l + Ep

P l = Ek0 + Ep

0 (2.67)

Considering the ratio of total energy and kinetic energy for a distribution of mattergives

EtotP l

EPl= Ek

P l + EpP l

EkP l

= Ek0 + Ep

0Ek

0

(a0

aPl

)2(2.68)

Since Ek0 ∼ |E

p0 | and a0/aPl ≤ 10−28 we find

EtotP l

EkP l

≤ 10−56 (2.69)

This means that the velocities at the Planck time must be fine-tuned so that thetwo energies compensate so precisely that the ratio does not exceed an error of10−54%. If it exceeds it, then the universe either recollapses or becomes “empty”too early.

10the question is: why should these regions have almost the exact same matter distribution ifthey could not communicate to go to thermal equilibrium?

11Recall that Hubble law is the only expansion law compatible with homogeneity.

20

Cosmological Parameter:Setting Λ = 0, (2.58) reads

Ω− 1 = k

a2H2 (2.70)

If k = 0, then Ω = 1 identically at any time. More generally k is a constant, hence

(ΩPl − 1)(Ha)2Pl = (Ω0 − 1)(Ha)2

0 (2.71)

from which it follows that

(ΩPl − 1) = (Ω0 − 1) (Ha)20

(Ha)2Pl

= (Ω0 − 1)(a0

aPl

)2≤ 10−56 (2.72)

It is clear that ΩPl must be extremely close to unity

ΩPl ∼ 1 (2.73)

This a fine-tune problem; indeed if k 6= 0

Ω− 1 ∝ a2 RD era (2.74)Ω− 1 ∝ a MD era (2.75)

and thus Ω would depart from unity if not extremely close to it.

Initial Perturbation ProblemOne further problem is the origin of primordial inhomogeneities, needed to explainthe large-scale structure of the universe. They must be initially of the order ofδρ/ρ ∼ 10−5 on galactic scales. This is somehow incompatible with homogeneity andisotropy. A successful theory must be able to take in account both inhomogeneitiesand the Cosmological Principle.

Since the discovery of these problems, Cosmology has produced a great numberof theories in order to explain initial condition in a natural way. Inflation is oneof them and it actually solves the puzzles of Standard Cosmology is an easy andelegant way.

21

CHAPTER 3Inflationary Cosmology

3.1 InflationMany attempts have been made to successfully overcome the shortcomings of theStandard Cosmology and many theories have been proposed. However, not alltheories have the same value and it seems reasonable that those which involveprinciples that are very general and can be tested by experiments and observationsare more scientifically suitable. This is exactly the reason why Standard Cosmologyfails when trying to describe initial conditions: such conditions are too fine-tuned.

Why InflationNowadays there is a theory among all others that is a candidate for describinginitial conditions, without too many restrictions. This theory is called Inflation.The etimology of this name resembles the term blow-up, which is very characteristicof this theory. in fact, inflation predicts a stage of very fast expansion of spaceitself. To understand better this point, let us consider the ratio

aia0

(3.1)

This ratio enters in both the flatness and horizon problems. Since it is known thatgravity is an attractive force and therefore it must decelrate the expansion of theuniverse, there is no way to have this ratio smaller than unity. But it turns outthat to explain why regions of our universe have the same homogeneities whilenot having been in causal contact in the past is easily done by assuming that theywere in contact in a primordial epoch before an expasion which stretched comoving

23

lenght scales faster than the growing of the Hubble radius so that particles wouldreach causally disconnected regions; in other words, in a certain period of time theratio (3.1) must have been smaller than unity.

This can be done only assuming that during some period of expansion gravityacted as a “repulsive” force. Let us then give the general definition of Inflation:

Inflation is a stage of accelerated expansion of the universe when gravity actsas a repulsive force.

This can be formally stated in the followng way:

Inflation ⇐⇒ a > 0 (3.2)

The first model of Inflation, known as old inflation, was built by Alan Harvey Guthin 1981. Guth realized that such a stage of rapid expansion could easily solve theproblems coming from Standard Cosmology. But let us come back to the definitionof inflation and let us see what is the consequent condition on energy density andpressure of a fluid that can be responsible for such as stage. Consider the secondFriedmann equation

a

a= −4πG

3 (ρ+ 3p) (3.3)

If a < 0 thenρ+ 3p > 0 (3.4)

This is known as strong energy dominance condition, which is an intuitive condition,because both pressure and energy density are positive. Instead, to fulfill theinflation condition a > 0, the strong energy domincance condition must be violated,that is

ρ+ 3p < 0 (3.5)which implies that if energy density is positive, then pressure is negative. Thisresult raises a question: what is a good candidate to this condition? There canbe many, in the sense that theoretically it is possible to create such candidatesfrom various fields of physics. For instance, a positive cosmological constant Λ,for which the equation of state is pΛ = −ρV fulfills the required condition. Thisexample is relevant; in fact, it can solve the shortcomings of Standard Cosmologyfairly easily. But first, let us see what properties distinguish a stage governed by apositive cosmological constant.

De Sitter stageThis stage is solution of Einstein Equations when the perfect fluid describing matterin the universe is largely represented by a cosmological constant. The conditions

24

on Friedmann equations are:

ρ = constant (3.6)H = constant (3.7)

The evolution of the scale factor a(t) is given by

a(t) = atieH(t−ti) (3.8)

that is, it grows exponentially with time. This is a very useful relation; with it, itis possible to solve the problems of intial conditions. However, the exact de Sittersolution fails to satisfy all necessary conditions for successful inflation: namely,it does not possess a smooth graceful exit into the Friedmann stage1. Still, itrepresents a very good approximation of realistic inflationary models. Let us seenow how this stage of expansion can solve the shortcomings of standard Big-bangTheory.

Horizon Problem in Inflationary CosmologyIn order to solve this problem let us ask that the largest scale that we observe today,the present horizon H−1

0 , was reduced to a lenght scale λH0(tI), smaller than theHubble radius H−1

I , where subscript I refers to inflation. This is a way of askingthat a causally disconnected region today was a causally connected region beforeinflation. This leads to the time interval needed for inflation to occur in order toexplain the homogeneity of the universe. Defining N as the number of e-foldings2

N = H(tend − tI) (3.9)

where tend is the time at which inflation ends. N is the exponent in the evolutionrule for the scale factor during inflation according to (3.8). The ratio between thescale factor at the beginning of inflation and a t end of inflation is

atIatend

= atIatIe

H(tend−tI) = e−N (3.10)

It follows that λH0(tI) reads

λH0(tI) = H−10

(atIat0

)= H−1

0

(atend

at0

)(atIatend

)= H−1

0

(T0

Tend

)e−N (3.11)

1Daring to present a theory about the birth of the universe that is not consistent with a FLRWuniverse will not bring a Nobel Prize, since Standard Cosmology is well rooted on experimentalobservations and valid, as said, until the Planck Era.

2The number of e-foldings (or e-folds) N is the time interval in which an exponentially growingquantity increases by a factor e. Indeed, dN = Hdt gives dN = dlna, from which a(t) = eHt.When t = N/H, a(t) = eN .

25

Imposing this quantity to be smaller than Hubble radius, i.e.

H−10

(T0

Tend

)e−N ≤ H−1

I (3.12)

gives an equation for the number of e-folsings N , whose result is

N ≥ ln(T0

H0

)+ ln

(HI

Tend

)(3.13)

Neglecting the second logarithm, the result for N is

N ≥ 60 (3.14)

Flatness Problem in Inflationary CosmologyIn Standard Cosmology the fine-tuning problem was given by (2.72), where thevalue of ΩPl needed to be extremely close to unity. To solve this problem it issufficient to request that the cosmological parameter Ω is very close to unity at theend of inflation. Since during inflation the Hubble rate is constant

Ω− 1 = k

a2H2 ∝1a2 (3.15)

Consider then the following ratio

(Ω− 1)t=tend

(Ω− 1)t=tI=(aI

aend

)2= e−2N (3.16)

Setting (Ω − 1)t=tI of order unity without loss of generality, the value N = 60satisfies the request. For any number of e-foldings larger than 60 the cosmologicalparameter is extremely close to unity. Therefore, not only this solves the flatnessproblem, but gives a straightforward prediction of the theory of inflation3 whichreads

Inflation =⇒ Ω0 = 1 (3.17)It is clear how easily inflation solves both the horizon problem and the flatness

problem by means of a sufficient number of e-foldings N . However, a crucialquestion remains: what is driving inflation? There can be many candidates to thisrole and therefore many different theories of inflation, but neverless one of the mostinteresting right now, because of experimental observations done by BICEP2 andPlanck, is surely the inflaton.

3It has to be specified that Ω is close to unity not because of a change in the geometry of theunvierse, say from a sphere or pseudo-sphere to flat plane, but rather because the radius of thespace is stretched so much that locally flatness is experienced.

26

3.2 Inflaton field

Candidate for inflationOne of the best candidates for inflation is a scalar field φ, which is called inflaton.To explain how an expansion stage of the universe is possible, Quantum FieldTheory is required. However, a first attempt to see that a scalar field φ is effectivelya good candidate can be done using classical Field Theory. This is due to the factthat the so called quantum fluctuations of the field are negligebale with respect toits classical evolution of the field, i.e. its mean value. The main condition that hasto be satisfied is (3.5), thus breaking the strong energy dominance condition. Letus define the inflaton field by its action

S =∫d4x√−gL =

∫d4x√−g

[−1

2∂µφ∂µφ− V (φ)

](3.18)

Here √−g = a3 for the FLRW metric. The scalar field obeys Eulero-Lagrangeequations

∂µδ(√−gL)δ∂µφ

− δ(√−gL)δφ

= 0 (3.19)

It follows that

φ+ 3Hφ− ∇2φ

a2 + V ′(φ) = 0 (3.20)

It is important to notice the presence os the friction term 3Hφ: a scalar field rollingdown its potential suffers a friction due to the expansion fo the universe. Moreover,the energy-momentum tensor of the scalar field reads

Tµν = ∂µφ∂νφ+ gµνL (3.21)

The corresponding energy density ρφ and pressure density Pφ are

ρφ = T00 = φ2

2 + V (φ) + (∇φ)2

2a2 (3.22)

Pφ = T ii3 = φ2

2 − V (φ)− (∇φ2)6a2 (3.23)

Notice that, if the gradient were dominant, then Pφ = −ρφ/3, which is not asufficient condition to drive inflation. Let us now split the scalar field into theclassical and quantum components

φ(t) = φ0(t) + δφ(x, t) (3.24)

27

where φ0 is the classical field with infinite wavelenght, that is the expectation valueof the inflaton field on the initial isotropic and homogeneous state, while δφ(x, t)represents the quantum fluctuation around φ0. Considering only the classicalcomponent of the scalar field

ρφ = T00 = φ2

2 + V (φ) (3.25)

Pφ = T ii3 = φ2

2 − V (φ) (3.26)

it follows that ifV (φ0) φ2 (3.27)

then the condition on the equation of state holds

Pφ = −ρφ (3.28)

Therefore, a scalar field whose energy is dominant in the universe and whose poten-tial energy dominates over the kinetic term drives inflation. In this approximation,the energy of the inflaton field corresponds to a vacuum energy, for which theequation of state is P = −ρ. however, because Pφ = −ρφ + φ2, the deviation of theequation of state from that for the vacuum is entirely characterized by the kineticenergy, φ2, which must be much smaller than the potential energy V (φ). As longas this condition is satisfied, inflation is occurring; when the kinetic part has grownbig enough to compensate the potential energy, then inflation has ended. Thisdepends on the shape of V (φ).

Slow-roll approximationAlso known as Slow-roll conditions, this approximation is useful to solve theequation of motion for the scalar field (3.20) for the classical part of the field givena general potential V (φ). The gradient term disappears, since the classical part ofthe field does not depend on space coordinates, because it is not of interest in thestage where the scalar field must drive inflation. Then

φ+ 3Hφ+ V ′(φ) = 0 (3.29)

This equation has the same form as the equation of a body falling down on a hill.That is why it is said that the scalar field rolls down its potential. If the followingconditions hold

|φ|2 V (φ), |φ| 3Hφ ∼ |V ′(φ)| (3.30)

then slow-roll approximation is valid. These conditions lead to some importantresults. From the first condition it follows that V (φ) must be flat, i.e. constant.

28

This must hold for a certain interval of time, otherwise inflation would last forever.Instead, at a certain moment the potential starts becoming steep (φ2 grows up)and then it gets to a minimum, where inflation ends. Friedmann equation reads

H2 = 13

(12 φ

2 + V (φ))

(3.31)

and in slow-roll approximation (first condition) gives

H2 ∼ V (φ) ∼ constant (3.32)

which represents a quasi De Sitter stage, thus an exponential growth of the scalefactor a(t) = eHt. In other words, in slow-roll approximation the necessary conditionof exponetial expansion for inflation is satisfied by the scalar field inflaton. Usingthe second slow-roll condition, equation (3.29) can be recast as

3Hφ0 = −V ′(φ) (3.33)

which gives φ0 as function of V ′(φ). It is useful to define Hubble slow-roll parameters

ε = − H

H2 = 4πG φ2

H2 = 12φ2

H2 (3.34)

δ = − φ

Hφ(3.35)

The parameter ε quantifies how much the Hubble rate H changes with time duringinflation. Since the acceleration equation for the scale factor (2.52) can be writtenas

a

a= H +H2 = (1− ε)H2 (3.36)

it follows that inflation can be attained if ε < 1. This is another important propertyof inflation:

Inflation ⇐⇒ ε < 1 (3.37)The parameter δ quantifies the length of the time interval in which inflation occurs.In order to have a sufficiently long accelerated expansion, it is required that |δ| < 1.These two conditions can be recast in terms of the potential V (φ) itself:(

V ′(φ)V (φ)

)2

1,∣∣∣∣∣V ′′(φ)V (φ)

∣∣∣∣∣ 1 (3.38)

Whit these approximations, it is possible to compute the number of e-foldingsbetween the beginning and the end of inflation and see if it coincides with thatneeded to solve both horizon and flatness problems. For a general potential V (φ)

N = ln(aend

aI

)=∫ tend

tIHdt =

∫ φend

φI

H

φdφ ∼

∫ φI

φend

V

V ′dφ (3.39)

29

The shape of the potential V (φ) determines the properties of the evolution of thescalar field. Different potentials lead to different results. There is a large variety ofinflationary models, each of which is built upon a certain potential V (φ).

Preheating and ReheatingInflation ends when V (φ) approaches φ2. At this point everything has been inflatedaway and the energy of the universe is in the form of oscillations of the inflatonfield around the minimum of the potential V (φ); in other words, it is like a bodythat has fallen from a hill and now oscillates on the plain. The universe is frozen.There must be a tremendous exchange of heat coming from the inflaton field inorder to produce a huge amount of entrorpy. This stage links inflation to theFriedmann universe, namely the radiation dominated era. There are two mainphases: preheating and heating.They differ in how fast and how efficiently energy and entropy are transferredto particles, namely bosons. The main parameter associated to this stage is thetemperature of reheating, TRH, which must be high enough in order to inducethermal baryogenesis.

30

CHAPTER 4Cosmological Perturbations

In order to explain the structure that we observe in our universe today, such asgalaxies, the evolution of primordial inhomogeneities must be explained. In orderto do so, two contributions must be taken into account: Gravity, as a sourceof instability at large scales, and Quantum Mechanics as a source of quantumflucatuations at the Planck scale. Whereas in Standard Cosmology small seedsof inhomogeneities must be put by hand into the theory, Inflation predicts theirpresence at a cosmic slace and also their spectrum. Because these inhomogeneitiesare governed at first by Quantum Mechanics and are then amplified to cosmicscales by inflation, the observation of such inhomogeneities today is the largestmacroscopic evidence of quantum effects that physics has ever encountered.

4.1 Gravitational InstabilityGravitational instability is a natural property of gravity. The complete set ofclassical equations which in pricnciple should determine the unknown density ρ,velocity v, entropy S, gravitational potential φ and pressure p reads

∂ρ

∂t+∇(ρv) = 0 continuity equation (4.1)

∂v

∂t+ (v · ∇)v + ∇p

ρ+∇φ = 0 Euler equations (4.2)

dS(x(t), t)dt

= ∂S

∂t+ (v · ∇)S = 0 conservation of entropy (4.3)

∇2φ = 4πGρ Poisson equation (4.4)p = p(ρ, S) equation of state (4.5)

31

Jeans TheoryFormalism around this subject has been first developed by Jeans, who described theevolution of inhomogeneities using the Newtonian theory of gravity in a stationaryuniverse. This led him to find the possible perturbations in homogeneous, isotropicmedia. Slightly disturbing the matter distribution, the five unknown functions are

ρ(x, t) = ρ0 + δρ(x, t). v(x, t) = v0 + δv(x, t). (4.6)φ(x, t) = φ0 + δφ(x, t). S(x, t) = S0 + δS(x, t). (4.7)p(x, t) = p(ρ0 + δρ, S0 + δS) = p0 + δp(x, t) (4.8)

= p0 + ∂p

∂ρδρ+ ∂p

∂SδS = p0 + c2

sδρ+ σδS. (4.9)

where cs is the speed of sound and σ = (∂p/∂S)ρ. Expanding the complete set ofequations (4.1) - (4.5) and keeping terms at first order gives

∂δρ

∂t+ ρ0∇(δv) = 0. (4.10)

∂δv

∂t+ c2

s

ρ0∇δρ+ σ

ρ0∇δS +∇δφ = 0. (4.11)

∂δS

∂t= 0. (4.12)

∇2δφ = 4πGδρ. (4.13)

From these equations, by Fourier transforming, it follows that there are five possibleindependet modes: two adiabatic modes, two vector modes and one entropy mode.The first two and the latter are found respectively imposing δS = 0 and δS 6= 0.The remaining two, vector perturbations, are found imposing δρ = 0 and δS = 0,but do not disturb the energy density, so they are not of much interest at a firstsight. The most interesting are the two adiabatic modes, one of which describesgravitational instability. Let us then find them. Taking the divergence of (4.11)and imposing δS = 0,

∂2δρ

∂t2− c2

s∇2δρ− 4πGρ0δρ = 0 (4.14)

Taking the Fourier transform of the latter,

δρk + (k2c2s − 4πGρ0)δρk = 0. (4.15)

This equation has two independet solutions

δρk ∝ e±iω(k)t (4.16)

32

whereω(k) =

√k2c2

s − 4πGρ0 (4.17)

Now it is crucial to define the Jeans wavelength. This length fixes the limit wherethe quantity under square root changes sign, thus when the solution can be agrowing exponential. The Jeans wavelength is

λj = 2πkj

= cs

(π

Gρ0

)1/2

(4.18)

For λ < λj solutions are oscillating,

δρ ∝ sin(ωt+ k · x) (4.19)

Instead, for λ > λj, solutions are exponential

δρk ∝ e±|ω|t (4.20)

One of this two solutions describes a growing perturbation. So, at large scalesgravity dominates and initial inhomogeneities grow exponentially.

Instability in an expanding universeThe co-moving energy density is a funtction of time and co-moving velocities obeyHubble law

ρ = ρ0(t). v = v0 = H(t)x (4.21)

Ignoring entropy and expanding the set of equations (4.1), (4.2) and (4.4) withrespect to small perturbations of variables,

∂δρ

∂t+ ρ0∇δv +∇(δρ · v0) = 0. (4.22)

∂δv

∂t+ (v0 · ∇)δv + (δv · ∇)v0 + c2

s

ρ0∇δρ+∇δφ = 0. (4.23)

∇2δφ = 4πGδρ. (4.24)

Let us express these equations with respect to the new variable q, given by x = a(t)qto avoid mathematical problems when passing to the frequency domain and let usconsider the evolution of the amplitude of the density perturbations δ = δρ/ρ0

1.1δ is used to describe variations of a quantity, when it is followed by the quantity itself (e.g.,

δv). Instead, if it is found alone, δ corresponds to the ratio just defined.

33

The latter equations become∂δ

∂t+ 1a∇δv = 0. (4.25)

∂δv

∂t+Hδv + c2

s

a∇δ + 1

a∇δφ = 0. (4.26)

∇2δφ = 4πGa2ρ0δ. (4.27)

Taking the divergence of (4.26) and using continuity and Poisson equations toexpress ∇δv and ∇2δφ in terms of δ,

δ + 2Hδ − c2s

a2∇2δ − 4πGρ0δ = 0. (4.28)

which in the frequency domain corrersponds to

δk + 2Hδk +(c2sk

2

a2 − 4πGρ0

)δk = 0. (4.29)

This the newtonian evolution for the spectrum of matter inhomogeneities. Whereasin a static universe gravitation is very efficient in increasing perturbation amplitudes,here it can be demonstrated that it is not the case. Therefore, to have certaininhomogeneities, initial conditions are required to be fine-tuned. Inflation providesthe necessary growth of initial inhomogeneities without fine-tuning. It is interestingto study vector perturbations in an expanding universe. Setting δ = 0, equations(4.22), (4.23) are

∇δv = 0. ∂δv

∂t+Hδv = 0. (4.30)

From the first it follows the peculiar velocity δv is perpendicular to the wavenumberk. The second can be recast as

δvk + a

aδvk = 0. (4.31)

which has the solutionδvk ∝

1a. (4.32)

Thus, vector perturbations decay as the universe expands. This result will be stillvalid in a relativistic treatment, using General Relativity. These perturbationscan have significant amplitudes at present only if their initial amplitudes were solarge that they completely spoiled the isotropy of the very early universe. In aninflationary universe there is no room for such large primordial vector perturbationsand they do not play any role in the formation of the large-scale structure of theuniverse. Vector perturbations, however, can be generated at late times, afternonlinear structure has been formed, and can explain the rotation of galaxies.

34

Instability in General RelativityNewtonian description fails for perturbations larger than the hubble radius. More-over, it cannot describe a relativistic fluid, which is key in the evolution of theuniverse. General Relativity must be taken into account for both short-wavelengthand long-wavelength perturbations. However, there are complications in the use ofthis theory. The main problem is the freedom in the choice of coordinates, gaugefreedom, used to describe perturbations. This problem is known as issue of gaugeinvariance. There are no obvious preferable coordinates and some of them lead tofictitious perturbations and some others remove real perturbations. Here are twocases:

• Consider a time transformation t→ t′ = t+δt(x, t). In addition, ρ(x, t) = ρ(t)for an homogeneous and isotropic universe. Then

ρ(t) = ρ(t′ − δt(x, t)) ∼ ρ(t′)− ∂ρ

∂tδt = ρ(t′) + δρ(x, t′) (4.33)

The second term describes a linear perturbation. It is a fictitious perturbationgiven by simply perturbing the coordinates, since the universe was assumedto be homogeneous and isotropic.

• Consider a universe with inhomogeneities δρ. Now take the hypersurfaces ofconstant time to be the same as the hypersurfaces of constant energy. Thenδρ = 0 in spite of the presence of real inhomogeneities.

To avoid these problems, it is mandatory to define gauge-invariant variableswith physical meaning which then are used to describe perturbations of both thematter field and the metric2.

Metric perturbations. Let us study metric perturbations. The latters can bedecomposed and classified according to the symmetry property of the homogeneous,isotropic background, which at a given moment of time is invariant with respect tothe group of spatial rotations and translations. Let us assume to be in a FLRWuniverse. Then, the metric with small perturbations can be written as

ds2 =[(0)gµν + δgµν(xρ)

]dxµdxν (4.34)

2General Relativity can be summarised in the statement

matter tells spacetime how to curve, spacetime tells matter how to move

It is impossible to separate the matter field and the metric. Inflation driven by the inflatonpredicts metric perturbations, since the infalton field generates energy density inhomogeneities.

35

where |δgµν | |(0)gµν | and (0)gµν is the background metric. Using conformal time,the background metric becomes

(0)gµνdxµdxν = a2(η)(dη2 − δijdxidkj) (4.35)

The δg00 component behaves as a scalar under these rotations and hence

δg00 = 2a2φ (4.36)where φ is a 3-scalar.

The spacetime components δg0i can be decomposed into the sum of the spatialgradient of some scalar B and a vector Si with zero divergence:

δg0i = a2(B,i + Si) (4.37)

where the comma denotes the derivation with respect to the spatial coordinatecooresponding to the value of the index. The vector Si has the constraint Si,i = 0and therefore has two independent components.

The components δgij , which behave as a tensor under 3-rotations, can be writtenas a sum of irreducible pieces:

δgij = a2(2ψδij + 2E,ij + Fi,j + Fj,i + hij) (4.38)

Here ψ and E are scalar functions, vector Fi has zero divergence (F ii = 0) and the

3-tensor hij satisfies the four constraintshii = 0. hij,i = 0. (4.39)

so it is traceless and transverse. Let us count the number of functions introduced todescribe gµν . There are four functions for the scalar perturbations, four functionsfor the vector perturbations (two 3-vectors with one constraint each), and twofunctions for the tensor perturbations (a symmetric 3-tensor has six independentcomponents and there are four constraints). We have ten functions altogether andthis number coincides with the number of independent components of δgµν .

Scalar perturbations are characterized by the four scalar functions φ, ψ,B,E.They are induced by energy density inhomogeneities. These perturbations aremost important because they exhibit gravitational instability and may lead to theformation of structure in the universe.

Vector perturbations are described by the two vectors Si and Fi and are relatedto the rotational motions of the perfect fluid considered. As in Newtonian theory,they decay very quickly and are not very interesting from the point of view ofcosmology.

Tensor perturbations hij have no analog in Newtonian theory. They describegravitational waves, which are the degrees of freedom of the gravitational field itself.In the linear approximation the gravitational waves do not induce any perturbationsin the perfect fluid.

36

Gauge transformations and gauge-invariant variables. Gauge freedom has one benefit,that is providing the possibility of choosing suitable coordinates systems where tocompute physical quantities in an easier way. Nevertheless the most importantquantities to use are gauge-invariant quantities. it is possible to find them. Giventhe coordinate transfrmation

xµ → xµ = xµ + ξµ (4.40)

and decomposing the spatial components ξµ as

ξi = ξi⊥ + ζ ,i (4.41)

it is possible to find the components of the new matric perturbations δgµν of theFLRW metric. They read3

δg00 = δg00 − 2a(aξ0)′ (4.42)δg0i = δg0i + a2

[ξ′⊥i + (ζ ′ − ξ0),i

](4.43)

δgij = δgij + a2[2a′

aδijξ

0 + 2ζ,ij + (ξ⊥i,j + ξ⊥j,i)]

(4.44)

Given these laws of transformation, let us find in which way functions related toscalar, vector and tensor perturbations change under these coordinates transforma-tion.

Scalar perturbations The metric takes the form

ds2 = a2[(1 + 2φ)dη2 + 2B,idx

idη − ((1− 2ψ)δij − 2E,ij)dxidxj]

(4.45)

The functions φ, ψ,B,E transform as

φ→ φ = φ− 1a

(aξ0)′ B → B = B + η′ − ξ0 (4.46)

ψ → ψ = ψ + a′

aξ0 E → E = E + η (4.47)

It follows that only ξ0 and η contribute to the transformations of scalar perturbationsand by choosing them appropriately we can make any two of the four functionsφ, ψ,B,E vanish. What it is now important is to construct two gauge-nvariantquantities, which span the two-dimensional space of the physical perturbations,

3a′ means derivation with respect to the conformal time η, whereas a means derivation withrespect to the universal time t.

37

using a linear combination of these functions. There is an infinite number of suchquantities, therefore it is smart to choose the simplest ones. Let us then infer

Φ = φ− 1a

[a(B − E ′)]′ Ψ = ψ + a′

a(B − E ′) (4.48)

If Φ and Ψ are both zero, then the metric perturbations, if any, are fictitious andcan be removed by a change of coordinates.

Vector perturbations The metric takes the form

ds2 = a2[dη2 + 2Sidxidη − (δij − Fi,j − Fj,i)dxidxj

](4.49)

The functions Si, Fi transform as

Si → Si = Si + ξ′⊥i Fi → Fi = Fi + ξ⊥i (4.50)

The gauge-invariant quantity that can be obtained is

Vi = Si − F ′i (4.51)

Tensor perturbations The metric takes the form

ds2 = a2[dη2 − (δij − hij)dxidxj

](4.52)

and hij does not change under coordinate transformations. It already describes thegravitational waves in a gauge-invariant manner.

Finding gauge-invariant quantities allows to build the equations for cosmologicalperturbations. For instance, it is possible to write perturbed Einstein Equations interms of Φ and Ψ, terefore in a gauge-invariant way4.

4.2 Quantum fluctuationsAccording to Quantum Field Theory, empty space is not entirely empty. There exista vacuum state, which is defined through the action of creation and destructionoperators a†, a upon it, namely

a|Ω >= 0 (4.53)4Other perturbed quantities that can be computed are: affine connection Γρµν , Ricci tensor

Rµν , Einstein tensor Gµν , energy-momentum tensor Tµν .

38

This state has infinite energy, indeed it can be represented as an infinite number ofharmonic oscillator oscillating in all directions. The associated hamiltonian is

H =∑i

~ωi(a†a+ 1

2

)(4.54)

Since in physics only differences in energy are appreciated, the infinite energy comingfrom the term 1/2 is negligible. These oscillators are called quantum fluctuationsand can be regarded as waves of physical fields with all possible wavelenghts, movingin all possible directions5. The scale at which these phenomena are dominating isthe Planck scale.

During inflation the wavelenghts of all vacuum fluctuations of the inflatonfield φ grow exponentially. When a wavelength becomes greater than the Hubbleradius, the fluctuation stops propagating and its amplitude freezes, since it obeysan eqaution like (4.29). However, its wavelength continues to grow exponentially.This makes the frozen fluctuation equivalent to a classical field that does not vanishwhen averaged over a macroscopic interval of time. Once inflation has ended, allthe fluctuations may reenter the horizon, giving rise to wavelengths accessible tocosmological observations. These spectra provide a distinctive signature of inflation.Let us now study the behaviour of inflaton perturbations in a quasi De Sitter stage.

Quantum fluctuations of the inflaton field in a quasi De Sitter stage

Let us introduce the power spectrum of a function f , a useful quantity whichcharacterizes the properties of the perturbations. Such quantity is defined in thefollowing way

Pf (k) = k3

2π2 |fk|2 (4.55)

This function will be used to define spectral indices, that can then be measuredthrough cosmological observations.

Let us look at the evolution of the inflaton field φ fluctuations during inflation.To do so, let us start from a massless scalar field χ in a De Sitter stage, continuewith a massive scalar field χ in a De Sitter stage and then finish with a massivescalar field χ in a quasi De Sitter stage. The inflaton field φ is a massive scalarfield and the evolution of its fluctuations occur in a quasi De Sitter stage, so theresults from the third case result will apply to it.

5If the values of these fields, averaged over some macroscopically large time, vanish then thespace filled with these fields seems to us empty and can be called the vacuum.

39

Massless scalar field in a De Sitter stage Consider a perturbation of the field χ inthe momentum space, δχk. Consider the new variable δσk defined through

δχk = δσka

(4.56)

Such new perturbation obeys the equation

δσ′′k +(k2 − a′′

a

)δσk = 0. (4.57)

which is similar to a Klein-Gordon equation for a scalar field, but with a negativemass term −a′′/a6. Let us consider the two cases of sub-Hubble scales and super-Hubble scales:

• On sub-Hubble scales the following condition holds

k2 a′′

a(4.58)

Thus, equation (4.57) reduces to

δσ′′k + k2δσk = 0 (4.59)

which has solutionsδσk = e−ikη√

2k(4.60)

It follows that fluctuations with wavelength within the horizon oscillate.

• On super-hubble scales the following condition holds

k2 a′′

a(4.61)

Thus, equation (4.57) reduces to

δσk = B(k)a (4.62)

where B(k) is a constant of integration. Matching the sub-Hubble scalesolution with the latter in k = aH, B(k) can be determined

|B(k)|a = 1√2k⇒ |B(k)| = 1

a√

2k= H√

2k3(4.63)

Then δχk turns out to be constant

|δχk| ∼H√2k3

(4.64)

6Recall that a > 0 during a quasi De Sitter stage, so that also a′′ > 0.

40

Massive scalar field in a De Sitter stage Taking into account the mass term mχ,equation (4.57) has a more complicated form, but it can be solved. It turns outthat on super-Hubble scales the fluctuations δχk are no more constant, but acquirea tiny dependence on time

|δχk| ∼H√2k3

(k

aH

) 32−νχ

(4.65)

whereν2χ =

(94 −

m2χ

H2

)(4.66)

In analogy with slow-roll approximation, defining

ηχ =(m2χ

3H2

) 1 (4.67)

it follows that32 − νχ ∼ ηχ (4.68)

Massive scalar field in a quasi De Sitter stage Let us now consider that H is notconstant, but changes according to

H = −εH2 (4.69)

The scale factor, for small values of ε, becomes

a(η) = − 1Hη1+ε (4.70)

The mass term in (4.57) changes again, but the result for fluctuations is the sameas before, taking into account that

νχ ∼32 + ε− ηχ (4.71)

Let us now consider the inflaton field φ and compute its power spectrum forfluctuations δφk.

Pδφ = k3

2π |δφk|2 =(H

2π

)2 ( k

aH

)3−2νφ(4.72)

Let us now define the spectral index as follows

nδφ − 1 = d lnPδφd ln k = 3− 2νφ = 2ηφ − 2ε (4.73)

41

This index has important properties. If nδφ > 1, then the spectrum is said to beblue and there is more power in the ultraviolet; if nδφ > 1 then the spectrum is redand there is more power in the infrared.

Let us conclude this section by saying that the inflaton energy is dominant inthe unverse when inflation occurs and as a consequence its perturbations inducemetric perturbations. This is another important prediction of inflation:

δφ⇐⇒ δgµν (4.74)

If metric perturbations are detected, then there will be a strong evindence that thetheory of inflation driven by a single scalar field is valid. Here is where gravitationalwaves, which are tensor perturbations of the metric, play a role.

4.3 The consistency relationThere is a prediction of inflation driven by a single scalar field regarding metricperturbations, namely scalar and tensor perturbations, which are dominant overvector perturbations. The first are related to energy and matter density and thesecond are gravitational waves. Let us compute their power spectrum and introducethe consistency relation.

Scalar perturbationsFirst we define the gauge-invariant quantity

R = Ψ +Hδφ

φ(4.75)

This is called co-moving curvature perturbation and depends on perturbations ofthe inflaton field. Passing to the momentum domain,

Rk = Ψk +Hδφk

φ(4.76)

The gauge-invariant variable Ψk satisfies the equation

Ψk +HΨk = εH2 δφ

φ(4.77)

In the slow-roll approximation the first term on the left hand side behaves like

Ψk ∼ (slow-roll parameters) ·Ψk (4.78)

42

and thus it is negligible with respect to Ψk in (3.20). Then

Ψk ∼ εH2 δφ

φ(4.79)

This result can be used in Rk to obtain

Rk = Ψk +Hφk

φ∼ (1 + ε)Hδφk

φ∼ H

φk

φ(4.80)

It is possible now to define the power spectrum of scalar perturbations as the powerspectrum of Rk. Using the relation just found between Rk and δφk, it reads

PR(k) = k3

2π2 |Rk|2 = k3

2π2H2

φ2|δφk|2 (4.81)

From the definition of slow-roll parameter ε in (3.34) and the definition of reducedPlanck mass,

H2

φ2= 4πG

εMPl =

√~c

8πG (4.82)

and taking into account that Pδφ(k) is known from (4.72), it follows that

PR(k) = 12M2

Plε

(H

2π

)2 ( k

aH

)nR−1

(4.83)

wherenR − 1 = 3− 2νφ (4.84)

Let us better rewrite PR(k) in the following way

PR(k) = A2R

(k

aH

)nR−1

(4.85)

This is the power spectrum of the scalar perturbations on super-Hubble scales.

Tensor perturbationsThese perturbations are gravtiational waves hij. The action for the gravitationalwaves can be derived by expanding the Einstein action

S =∫ 1

2R√−gd4x (4.86)

43

up to the second order in transverse, traceless metric perturbations hij.The result is

S = 164

∫a2(hi′j h

j′

i − hij,lhj,li )dηd3x (4.87)

Substituting the expansion

hij(x, η) = 1(2π)3/2

∫hk(η)eij(k)eikxd3k (4.88)

where eij(k) is the polarization tensor, into (4.87),

S = 164

∫a2eije

ji (h′kh′−k − k2hkh−k)dηd3k (4.89)

To compute the power spectrum, let us introduce the gaguge-invariant amplitudedefined as

vk =

√√√√eijeji

32π ahk (4.90)

The action becomes

S = 12

∫ (v′kv′−k −

(k2 − a′′

avkv−k

))dηd3k (4.91)

The resulting equation of motion is

v′′k +(k2 − a′′

a

)vk = 0 (4.92)

The latter equation describes the motion of a massless scalar field in a quasi DeSitter stage. Therefore, previous results on evolution of perturbations of masslessscalar fields can be applied and the tensor modes read√

k3

2π2 |vk| =(H

2π

)(k

aH

) 32−νT

(4.93)

whereνT ∼

32 − ε (4.94)

The power spectrum is correpondigly

PT (k) = 8M2

Pl

(H

2π

)2 ( k

aH

)nT(4.95)

44

Rewriting the latter equation with respect to the tensor amplitude, the final resultreads

PT (k) = A2T

(k

aH

)nT(4.96)

wherenT = 3− 2νT = −2ε (4.97)

This is the power spectrum of the tensor perturbations on super-Hubble scales. Theamplitude of the tensor perturbations depends only on the Hubble parameter duringinflation. This amounts to saying that it depends on the energy scale associatedto the inflaton potential. A detection of gravitational waves from inflation will betherefore a direct measurement of the energy scale associated to inflation.

Consistency RelationThe results obtained on scalar and tensor perturbations allow to predict a consis-tency relation which holds for models of inflation driven by one single scalar field φ.Let us define the tensor-to-scalar ratio

r =1

100A2T

425A

2R

=1

1008(

H2πMPl

)2

425(2ε)−1

(H

2πMPl

)2 = ε (4.98)

Using (4.94), let us introduce the consistency relation for models of inflation drivenby a single scalare field φ

r = −nT2 (4.99)

The ratio between the amplitude of tensor modes and that of the curvatureperturbations depends on the spectral index of tensor perturbations. This is thequantity whcih is measured by experimental observations.

Holy Grail of inflationThe cosnistency relation can be verified experimentally and thus it gives Cosmologythe status of a physical theory.Nowadays, two of the major instruments used to scan the sky and are:

• the satellite Planck

• the telescope BICEP2 (Background Imaging of Cosmic Extragalactic Polar-ization 2)

45

Both of these intruments have looked at the CMB anisotropies and sought forE-modes and B-modes, in order to estimate the ratio r.

E-modes, named in analogy with the curl-free electric field E, are associated toscalar perturbations.

B-modes, named in analogy with the divergence-free magnetic field B, areassociated to tensor perturbations.Finding the presence of gravitational waves in the CMB would have major conse-quences:- It would be the first time that they are detected.- These perturbations should be produced by quantum fluctuations and stretchedduring inflation, so their detection would imply that gravity must be described bya quantum theory.-Gravitational waves would decouple at the end of inflation and not interact somuch with matter during the evolution of the universe, so they would be the firstphenomenom that brings us information before recombination, giving a trace ofphysics in action at times that are close to 10−36, something like 49 orders ofmagnitudes before recombination.

Planck, in 2013, has put an upper limit to r [7],

r < 0.11 (4.100)

On the other hand BICEP2, after three years of data collection, has released inMarch 2014 a value for r [2],

r = 0.2+0.07−0.05 (4.101)

suggesting the existence of gravitational waves. This value is in contraddiction withthat of Planck, which months later has published new measurements on the galacticdust attesting that there may be more dust in the Milky Way than what BICEP2has considered. BICEP2 does not exclude the possibility of data contaminationdue to the cosmic dust [1].

Although it may seem a failure, there is a chance that BICEP2, or a subsequentmeasurement of the polarization of the CMB, opens a new window on the universe,leading us back to the origin of time and to distances and phenomena which couldembarass the impressive forward leap accomplished by physics in XX century.

46

Bibliography

[1] BICEP2 Collaboration, 2014, Detection of B-mode Polarization at DegreeAngular Scales by BICEP2, Phys. Rev. Lett. 112, 241101.

[2] BICEP2 Collaboration, 2014, BICEP2 I: Detection of B-mode Polarization atdegree angular scales, arXiv:1403.3985v1 [astro-ph.CO]

[3] Carroll, S., 2004, Spacetime and Geometry, Addison Wesley.

[4] Cognola, G., 2014, course on General Relativity and Cosmology, GeneralRelativity and Cosmology, Trento.

[5] Landau, L. D., Lifshitz, E. M., 1971, The Classical Theory of Fields, PergamonPress.

[6] Mukhanov, V., 2005, Physical Foundations of Cosmology, Cambridge.

[7] Planck Collaboration, 2014, Planck 2013 results. XVI. Cosmological parameters,arXiv:1303.5076 [astro-ph.CO].

[8] Riotto, A., 2002, Inflation and the Theory of Cosmological Perturbations,lectures given at Summer School on Astroparticle Physics and Cosmology,Trieste.

[9] Riotto, A., 2013, Lecture Notes on Cosmology, Départment de PhgysiqueThéorique, Université de Genève, Switzerland.

[10] Weinberg, S., 1972, Gravitation and Cosmology: Principles and Applicationsof the General Theory of Relativity, John Wiley & Sons, Inc.

47

Acknowledgements

I want to thank Prof. Gianluca Grignani for stimulating the interest in studyingdifferent fields of physics in order to better understand the topics of this thesis.

Vorrei ringraziare il Prof. Gianluca Grignani per aver stimolato l’interesse nellostudio di diversi campi della fisica al fine di comprendere gli argomenti trattati inquesta tesi.Ringrazio la mia famiglia per la fiducia e il sostegno in momenti di panico totale.Ringrazio i cari amici che hanno sostenuto conversazioni utili alla materializzazionee compresione dei concetti esposti in questa tesi. Ringrazio infine la dottrina delBrodawey, che ispira anima e corpo ogni giorno, nonché la grande persona insiemealla quale è stata sviluppata in un’esperienza di vita di ordinaria follia.