1

f(R) THEORIES OF GRAVITY

An informal discussion

Nathalie Deruelle

APC-CNRS, Paris

Paris, October 22nd 2010

VIA lecture

2

Outline of the lecture

1. f(R) gravity and dark energyor : a one-eyed (re-)view of an interesting failure

2. f(R) gravity as scalar-tensor theoriesor : the Jordan vs Einstein frame representations

3

f(R) GRAVITY AND DARK ENERGY: CONTEXT

Space, time and gravity at large are well represented by a Friedmann-Lemaıtre geometrythe scale factor of which started to accelerate recently

No Big Bang

1 2 0 1 2 3

expands forever

-1

0

1

2

3

2

3

closed

recollapses eventually

Supernovae

CMB

Clusters

open

flat

Knop et al. (2003)Spergel et al. (2003)Allen et al. (2002)

Supernova Cosmology Project

Ω

ΩΛ

M

ds2 = gijdxidxj = −dt2 + a2(t)dσ2k

Gij = κ T totalij ; DjT

ij = 0

H2 + ka2 = κ

3ρtotal ; H = 1a

dadt(

HH0

)2

= Ω0rad

(a0a

)4 + Ω0mat

(a0a

)3 + Ω0k

(a0a

)2 + Ω0DE

H0 ≈ 70 ; Ω0rad ≈ 10−4 ; Ω0

mat ≈ 0.3 ; Ω0k ≈ 0

Ω0DE ≈ 0.7 : “Dark Energy” (Turner, 1999)

=⇒ a(t)→ e√

Λ/3 t , Λ ≡ 3Ω0ΛH2

0 ; κTDEij = −Λgij

4

Dark energy as Λ : the pros

• Λ was present in Einstein’s theory right from the beginning (1916)(and in Newton’s... see Calder and Lahav, 2008)

... and its ‘misuses’ never meant that it was zero.

– Einstein’s 1917 “blunder” (Gamow) is not to have used Λbut to have failed to predict the expansion of the universe

– The fall of the steady state theory : not because Λ = 0but because it failed to accomodate the CMB

• Λ-CDM model of structure formation,

from CMB-anisotropies to clustering of galaxies,works remarkably well.

The cosmological constant, set to zero for lack of observations,has, at long last, been measured.

5

Dark energy as Λ : the cons

• Usual objections to identify DE and Λ : why now, and why so small ?– The fact that Λ starts to dominate now is “anti-copernican”– The “weight” of (Planck size) vacuum fluctuations is 10120ρΛ :

there must be a mechanism which renormalizes Λ to zero.• Are these arguments convincing ? Bianchi-Rovelli, 2010 :

– Copernican principle must notbe pushed too far...– No strong probability argumentagainst “now”

– ρvacuum = 10120ρΛ :does not mean that ρΛ = ρvac

does not mean that ρvac|ren = 0

6

If Dark energy is not a cosmological constant, what can it be ?

• An artefact of the averaging process ?Gij(<gkl>) = κ <Tij > instead of <Gij(gkl)>= κ <Tij >Ellis, 1971,..., Buchert, 1989..., Rasanen, 2003...

• A misinterpretation of the fact that we are in a big void ?CG 2010 seminars : Clarkson ; Moffat ; Chul-Moon Yoo ; Romano

• Exotic matter ? (ρΛ + pΛ ≈ 0) e.g. :Quintessence : R.G. plus ϕ with V ∝ 1/ϕn (Steinhardt et al., 1997)

• “Modified” gravity ? e.g. :– MOND, see, e.g., Navarro and Acoleyen, 2005– “Branes” : DGP (2000), Deffayet (2001)– “Horava Gravity” (2008)– f(R) lagrangian, instead of Hilbert’s R (or f(Riem, DRiem))

C.D.T.T. (2003), Capozziello et al. (2003)

7

f(R) GRAVITY: ESSENTIALS

The action is : S[gij] = 12κ

∫d4x√−g f(R) + Sm(Ψ ; gij)

(Weyl 1918, Pauli 1919, Eddington, 1924)

Metric variation yields the following field equations:f ′(R) Gij + 1

2(Rf ′ − f)gij + gijD2f ′ −Dijf

′ = κ Tij

• A “varying gravitational constant” : f ′(R) Gij

• A “potential term” : 12(Rf ′ − f)gij

• Diffeo invariance + minimal coupling to matter: ⇒ DjTij = 0

• a 4th order diff eqn for gij : gijD2f ′ −Dijf

′ =⇒ extra gravity dof

• The trace : 3D2f ′ + (Rf ′− 2f) = κ T =⇒ R no longer “traces” T .

8

f(R) cosmological models of dark energy: the idea

• The Carroll-Duvvuri-Trodden-Turnerand Capozziello-Carloni-Troisi proposal (2003)

f(R) = R− µ2(1+n)

Rn (n > 0) ; µ2 ∼ 10−33eV or µ2 = 1L2

Hubble

Late time Friedmann equation when matter is negligible (R µ2) yields

a(t) ∝ t2

3(1+wDE) with wDE → −1 for large n (2,3,4 is enough)

• f(R) Friedmann equations as a dynamical system =⇒ caveat !Amendola, Polarski, Tsujikawa et al., 2003 onwards

When (dust) matter is non negligiblef(R) = R + βR−n models, n > 1 behave as : a ∝ t1/2, not a ∝ t2/3

f(R) = R + βR+n models with n < 1 do have a proper matter era.

9

f(R) gravity and local tests

• A ninety year old mistake

CDTT, 2003 : “Many solar system tests of gravity theory depend on theSchwarzschild (de Sitter) solution, which Birkhoff’s theorem ensures is theunique, static, spherically symmetric solution (...). Astrophysical tests ofgravity will be unaffected by the modification we have made.”

Weyl (1918), Pauli (1919) and Eddington (1924) had said the same...

Pechlaner-Sexl (1966), Havas (1977) : f(R) field equations are fourthorder differential equations ; they possess extra-runaway solutions : there isno Birkhoff theorem ; the solution outside an extended source is not theSchwarzschild metric and depends on its equation of state.

Rediscovered by Chiba (2003), Olmo (2005), Erickcek et al (2006),Navarro-Acoleyen (2006),...

10

• A scalar curvature “locked” at its cosmological, small, value...

– Equation of motion for the scalar curvature :

3D2f ′ + (Rf ′ − 2f) = κT (not R = −κT as in GR !)

– Linearize around the “cosmological platform” : R = Rc + R1 so that

4R1 −m2cR1 = κT

3f ′′cwith m2

c = (f ′−Rf ′′)|c3f ′′c

For all one-scale models, say, f(R) = R−R2c/R

m2c = O(Rc) = O(1/L2

Hubble) whereas 4 = O(1/L2SS)

therefore : m2c ≈ 0 and R ≈ Rc

(1− GM

r

) κρ

(confirmed numerically, Multamaki-Vilja, Kanulainen et al. (2007)...)

– Conclusion : if mcLSS 1, then R ≈ Rc outside and inside the star.

11

• ... which yields a catastrophic value for the PPN parameter γ

One-scale f(R) models predict a small R out and in the sun.

What about the metric ?

– Metric ansatz : ds2 = −(1 + a(r))dt2 + (1 + b(r))dr2 + r2dΩ2

– Field equations : f ′(R) Gij + 12(Rf ′− f)gij + gijD

2f ′−Dijf′ = κ Tij

Linearize around the “cosmological platform” and find :

ds2 ≈ −(1− 2GM

r

)dt2 +

(1 + 2γGM

r

)d~x2 with γ = 1

2

Erickcek et al (2006)Cassini : γ = 1 to 10−5

Conclusion : all one-scale f(R) models of dark energy are ruled out.

12

• “Evading” Solar System “Constraints”

(a) Heuristic arguments

The idea : give a heavy mass to the scalar curvature

R = Rc + R1 4R1 −m2R1 = κT3f ′′ , m2 = (f ′−Rf ′′)

3f ′′

If mLSS 1 then R1 ≈ −κT and find ≈ Schwarzschild outside the sun.

*

– mcLSS 1 through fine-tuning of f(R), see eg Nojiri-Odintsov, 2003

but... R1 Rc and the linear approximation is no longer valid.– mlocLloc 1 through the introduction of a “local platform” :

Linearize around R ∼ κρloc Rc and NOT around Rc with mLloc 1Starobinski (2007)

13

(b) Galactic platform and the “chameleon mechanism”

Hu and Sawicki, 2007Recall the eom for the scalar curvature :

4f ′ = dVdf ′

The “effective potential” V (R), such thatdVdf ′ = −κ

3ρ− Rf ′−2f3

may have 2 zero at f ′out, ; if, moreover :

f ′out, = f ′(κρout, κρ) andf ′out−f ′GM/r

1

then R is forced to track matterKhoury-Weltman (2003)

Faulkner et al. (2006), Navarro-Acoleyen (2006)Capozziello-Tsujikawa (2007), Tamaki-Tsujikawa (2008), Brax et al. (2008)

The Chameleon effect relies on the existence of a local environment.

14

f(R) cosmological models of dark energy today

f(R) models of dark energy such as :

f(R) = R + λRc

[1

(1+R2/R2c)

n − 1]

; f = R−m2 c1(R/m2)n

c2(R/m2)n+1

Starobinski (2007), Hu-Sawicki (2007), ...evade the Solar System constraints thanks to the chameleon mechanism

*Such models are undistinguishable from Λ-CDM from the matter era on.... However, because the “mass” of the scalar curvature (m2 ≈ 1/

√3f ′′) is

chosen to be high in high matter density environment: R →∞ at finite zStarobinski (2007), Tsujikawa (2007)

... but this problem can be cured too :

f(R) = R− µRc(R/Rc)

2n

(R/Rc)2n+1+ R2

6M2

Appleby, Battye and Starobinsky (2009)

Such “cured viable” f(R) gravity theories are the basis of present sudies ofcosmological perturbations and CMB predictions.

(see e.g. De Felice-Tsujikawa, Liv. Rev., 2010)

15

f(R) models of dark energy, summary

f(R) = R− µRc(R/Rc)

2n

(R/Rc)2n+1+ R2

6M2

Failed attempt? Technically no, observationaly no, ...aesthetically?

... It is clear that f(R) gravity

does not match Λ-CDM model of structure formation

in conceptual simplicity and in explaining observations !

16

Reminder : outline of the lecture

1. f(R) gravity and dark energyor : a one-eyed (re-)view of an interesting failure

2. f(R) gravity as scalar-tensor theoriesor : the Jordan vs Einstein frame representations

17

Isolating the “scalaron” and introducing the Einstein frame

• The scalaron

The action is : S[gij] = 12κ

∫d4x√−g f(R) + Sm(Ψ ; gij)

– introduce R ≡ s as an independent field– the action becomes, φ being a Lagrange multiplier:

S[gij, s, φ] = 12κ

∫d4x√−g [f(s) + φ(R− s)] + Sm(Ψ ; gij)

– Extremization wrt s gives: φ = f ′(s)– Insert this constraint and obtain the “Helmholtz” action

S[gij, s] = 12κ

∫d4x√−g [f ′(s)R− 2U(s)] + Sm(Ψ ; gij)

– the eom become (with U ≡ 12(Rf ′ − f)) :

f ′(s)Gij + gijU(s) + gijD2f ′(s)−Dijf

′(s) = κTij

f ′′(s)(s−R) = 0

18

• The Einstein frame

– To decouple the second derivatives in the EOM, introduce (Higgs 1959) :

g∗ij = f ′(s)gij

g∗ is the “Einstein frame” metric; g ≡ g is the “Jordan frame” metric

The action then becomes:

S[g∗ij, ϕ] = 2κ

∫d4x√−g∗

[R∗

4 −12(∂

∗ϕ)2 − V (ϕ)]

+ Sm[Ψ; gij = e−√

23ϕg∗ij]

where ϕ(s) =√

3 ln√

f ′(s) , V (s) = sf ′(s)−f(s)4f ′2(s)

– In this “Einstein frame” the action for f(R) gravity is that of Einstein’sgravity plus a scalaron, with matter non-minimally coupled to the metric.

hence : f(R) gravity is coupled quintessenceEllis et al. (1989), Damour-Nordvedt-Polyakov (1993), Wetterich (1995),Amendola (1999), Copeland et al. (2006),...

with a potential

19

On the mathematical equivalence of the Jordan and Einstein frames:the various hamiltonian formulations of f(R) theories

In a nutshell : “Jordan frame” action : S[gij] = 12κ

∫d4x√−g f(R).

(a) There are two routes to the ADM formulation

The “Odstrogradsky” route where the extra dof is K (gµν),(Buchbinder-Lyahovich 87, Querella 99, Ezawa et al 99-09)

The “Boulware” route where the extra dof is R (gµν),(Boulware 84, Hawking-Luttrell 84, H.J. Schmidt 94; Demaret-Querella 95)

Both hamiltonians are related by (non-linear) canonical transformations.

(b) Einstein frame formulation

(where S[g∗ij, ϕ] = 2κ

∫d4x√−g∗

[R∗

4 −12(∂

∗ϕ)2 − V (ϕ)])

Both hamiltonians can be canonically transformedto the ADM hamiltonian of GR coupled to a scalar field.

(N.D., Sendouda,Youssef, 09)(ND, Sasaki, Sendouda and Yamauchi, 09)

20

Coupling the “scalaron” to matter: “detuned” f(R) gravity(ND, Sasaki, Sendouda, 2007)

• The idea:

– Introduce the modified “Helmholtz” lagrangian :

S[gij, s] = 12κ

∫d4x√−g [f ′(s)R− (sf ′(s)− f(s)] + Sm(Ψ ; gij = e2C(s)gij)

– Equations of motion :

f ′(s)Gij + 12gij(sf ′(s)− f(s)) + gijD

2f ′(s)−Dijf′(s) = κTij

s = R− 2κC ′(s)T/f ′′(s) (⇒ DjTji = TC ′(s)∂is)

C(s) = 0, g = g, s = R : standard f(R) gravity.

C(s) 6= 0 : “detuned” f(R) gravity

C(s) = ln√

f ′(s), g = g∗ : minimal coupling in Einstein’s frame.

21

• Jordan frame representation of f(R) detuned gravity

The “Jordan frame” is the spacetime, M, with metric gij = e2Cgij towhich matter is minimally coupled (that is : DjT

ij = 0, e.g. ρ ∝ 1/a3).

In this frame the action is a Brans-Dicke type action:

S[gij,Φ] = 12κ

∫d4x√−g

[ΦR− ω(Φ)

Φ (∂Φ)2 − 2U(Φ)]

+ Sm[Ψ; gij]

where Φ(s) = f ′(s)e−2C(s) , U(s) = 12(sf

′(s)− f(s))e−2C(s)

and ω(s) = −3K(s)(K(s)−2)2(K(s)−1)2

with K(s) = d C

d ln√

f ′.

hence : detuned f(R) models are scalar-tensor theories of gravityFor exhaustive presentations : Y. Fuji and K. Maeda, CUP (2007) ;T. Damour and G. Esposito-Farese 1992.

If C = 0, then standard f(R) gravity is Brans-Dicke theory with ω = 0with a potential

22

Recapitulation:

• In the “Jordan frame”, matter is minimally coupled to the metric.

The action is: S[gij] = 12κ

∫d4x√−g f(R) + Sm(Ψ ; gij)

and the associated EOM is fourth order.

• The EOM become second order by promoting s = R or ϕ =√

3 ln√

f ′(R)to the satus of an independent field, the scalaron.

• It is then natural to extend the theories by coupling it to matter (“detuned”f(R) gravity, which is equivalent to scalar-tensor theories with a potential).

• In the “Einstein frame” the action for gravity is Hilbert’s

S[g∗ij, ϕ] = 2κ

∫d4x√−g∗

[R∗

4 −12(∂

∗ϕ)2 − V (ϕ)]

+ Sm[Ψ; gij = e2C−√

23ϕg∗ij]

The price to pay being that matter is non-minimally coupled to gravity.

• Cosmological models of dark energy, solar tests etc can be, and are,studied using one or the other frame.

23

On the “physical” equivalence of the Jordan and Einstein frame :

• an endless debate

– Einstein frame is the “physical” frame :Magnano-Sokolewski (93, 07), Gunzig-Faraoni (98) (but see Faraoni (06))(“DEC does not hold in JF hence no positive energy theorem”)

– Jordan frame is the “physical” frame : Damour Esposito-Farese (92) · · ·“Jordan metric defines lengths and times actually measured by lab rods andclocks (which are made of matter)” (Esposito-Farese Polarski, 2000)

– Jordan and Einstein frames are equivalent (classically) : Flanagan (04);Makino-Sasaki (91), Kaiser (95) (CMB anisotropies) ; Catena et al (06)(cosmo)

– Why this debate ? In all frames test particles fall the same way (WEP),but only in the Jordan frame do they follow geodesicsOnly in the Jordan frame can the EEP be enforced : DjT

ji = 0 ⇒ ∂jT

ji ≈ 0

in local inertial frame Dicke 1961

24

• Jordan vs Einstein frame : the example of Nordstrom’s theory

– “Einstein frame formulation” (Nordstrom, 1912)

dτ2 = −ηijdXidXj , S = −1κ

∫d4X ∂iΦ∂iΦ−

∑m

∫(1 + Φ)dτ

EOM: Dui

dτ = − 11+Φ

(∂iΦ + uiuj

c2 ∂jΦ)

WEP but no geodesic motion !

Hydrogen atom : S = −∑∫

m dτ + e∫

AidXi with m = (1 + Φ)m

The frequency of an atomic transition depends on where it is measured.

ν(P ) =(

1n′2 −

1n2

) m(P )q4

2 hence ν =√

Φ(P )Φ(P0)

ν0 .

ν is the frequency of transition measured “there and then”;ν0 is the frequency of the same transition measured “here and now”

–“Jordan frame” formulation (Einstein-Fokker, 1914) :

EEP : gij = (1 + Φ)2ηij and EOM become : R = 3κTm , DjTijm = 0

m = const. is the mass of the electron in all freely falling frames : ν = ν0

25

• Jordan vs Einstein frames : an f(R) example

Consider FRW metric in JF : ds2 = −dt2 + a2(t)dx2

Define H(t) ≡ aa , z(t) ≡ a0

a − 1

In the EF : ds∗2 = f ′(a)f ′0

ds2 ≡ −dt2∗ + a2∗dx2

Define H∗(t) ≡ 1a∗

da∗dt∗

, z∗(t) ≡ a0a∗(t)

− 1

– The H(z) and H∗(z∗) functions are different. (Correct.)

– H(z) and H∗(z∗) are the “Hubble law” in the J. and E. frames, hence :

“the Jordan and Einstein frames are physically inequivalent”.Capozziello et al., 2010

Wrong.

26

First, relate observable variables :

redshift (z = ν0ν0− 1) vs luminosity (D =

√L

4πl with L = Nhν20)

– ν0 is the frequency of some atomic transition “here and now” ;– ν0 and l are the observed frequency and apparent luminosity.

In the JF :The frequency ν of the atomic transition “there and then” is the sameas in the lab now : ν = ν0. The difference between ν0 and ν0 is due tothe expansion a(t). Hence, as in GR :

z = a0a − 1, D = (1 + z)

∫ z

0dzH

In the EFthe frequency ν “there and then” is NOT the frequency measured inthe lab now :

√f ′ν0 =

√f ′0 ν. But the expansion rate is not the same

either : a =√

f ′ a

The two compensate and the relationship between z and D is unchanged.(Catena et al., 2006)

Relationships between observables do not depend on the frame.(for more on the subject, see ND Sasaki, 2010)

27

Conclusion

1. f(R) gravity and dark energyAn interesting failure.Aspects of chameleon mechanism still to be explored

2. f(R) gravity as scalar-tensor theories“Detuned” f(R) gravity is a scalar-tensor theory WITH a potentialThe Jordan and Einstein frame equivalence to understand the EEP

3. A remark on Black Holes in f(R) theories:Their “inertial” and gravitational masses are differentThey violate the SEP (they do not fall like test particles)Consequences ?