The Equation of State Approach to Cosmological Perturbations in f(R) Gravity

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by  Boris Bolliet

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The Equation of State Approach to Cosmological Perturbations in
f(R) Gravity
BorisBolliet
Laboratoire de Physique Subatomique et de Cosmologie, Universit´e Grenoble-Alpes,
CNRS/IN2P3
53, avenue des Martyrs, 38026 Grenoble cedex, France
6 RichardA.Battye
1
Jodrell Bank Centre for Astrophysics, School of Physics and Astronomy, The Universityof
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Manchester, Manchester M13 9PL, U.K.
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b
JonathanA.Pearson
e
F School of Physics and Astronomy, Universityof Nottingham, Nottingham NG7 2RD, U.K.

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Thediscoveryofapparentcosmologicalaccelerationhasspawnedahugenumberofdark

] energyandmodifiedgravitytheories. Thef( )modelsofgravityareobtainedwhenone
c replacestheRicciscalarintheEinstein-HilbRertactionbyanarbitraryfunctionf( ),
q R
gr- S= 21κZ d4x√−g {R+f(R)}+Sm, (1)
[
2 wachteiorne κde≡scr8ibπiGngisthtehestraensdcaalreddmNaetwtetronfi’esldcso.nIsntatnhti,sRwoirsktwheepRriocvciidsecaelxaprreasnsdionSsmfoisrtthhee
v equations of state (EoS) for perturbations which completely characterize the linearized
5 perturbationsinf( )gravity,includingthescalar,vector,andtensormodes. TheEoS
8 formalism1isapowRerfulandelegantparametrizationwherethemodificationtoGeneral
5 Relativityaretreatedasadark-energy-fluid. Theperturbeddark-energy-fluidvariables
7 such as the anisotropic stress or the entropy perturbation are explicitly given in terms
0 ofparametersofthemodelofinterest.
.
1 Keywords:Modifiedgravity;DarkEnergy;Cosmologicalperturbations.
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1 1. Field Equations
:
v
Varying the action (1) with respect to the metric yields the field equations,
i
X
r Gµν =κ(Tµν +Dµν), (2)
a
whereG istheEinsteintensorandT isthestress-energytensorofthestandard
µν µν
matter fields. Allcontributionsdue tof(R)arepackagedintothe extra-termD ,
µν
which we call the stress-energy tensor of the dark sector, explicitly formulated as
κD ≡ 1fg −(R +g (cid:3)−∇ ∇ )f , (3)
µν 2 µν µν µν µ ν R
where R is the Ricci tensor, and f ≡ df . A direct calculation shows that
µν R dR
D is covariantly conserved, that is, ∇µD = 0. The background geometry is
µν µν
assumedto be ds2 =−dt2+a(t)2δ dxidxj, where a(t) is the scalefactor. Instead
ij
of the first and second order time derivative of the Hubble parameter, H, we use
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the dimensionless parametersa
H′ R′
ǫ ≡− , ǫ¯ ≡− , (4)
H H H 6H2
wheretheprimedenotesderivativewithrespecttod/dlna. Thedarksectorcanbe
viewed as a fluid, with energy density ρ ≡ 1 D and pressure P ≡ 1 δijD .
de a2 00 de 3a2 ij
The field equations (2) read
Ω +Ω =1, w Ω +w Ω = 2ǫ −1, (5)
m de m m de de 3 H
where Ω = κρi is the density parameter and w ≡ P/ρ. From (3). For the f(R)
i 3H2 i i i
fluid, they are explicitly given by
f
Ω = − +(1−ǫ )f −f′ , (6a)
de 6H2 H R R
1
w +1= − (2ǫ f +(1+ǫ )f′ −f′′). (6b)
de 3Ω H R H R R
de
2. Dynamics of Linear Perturbations
The dynamics of linear perturbations is presented in Fourier space. Instead of the
coordinate wavenumber, k, a reduced dimensionless wavenumber is introduced,
k
K≡ , (7)
aH
sothatK caneasilyrecognizethe sub-(super)-horizonregimes. Inthe synchronous
gauge,the non-zerometric perturbationsareδg =a2h . Inanorthonormalbasis
ij ij
{kˆ,ˆl,mˆ} in k-space, the spatial matrix h is further decomposed as h = 1hδ +
ij ij 3 ij
h σ +hV·v +hT·e ,wherethenotationshVandhTcontainthetwovectorandthe
k ij ij ij
twotensorpolarizationstatesrespectively,andthedotproducthastobeunderstood
as a sum over the polarization states. Instead of h, we use the combination 6η ≡
h − h. The basis matrices are σ = kˆ kˆ − 1δ for the longitudinal traceless
k ij i j 3 ij
mode, v(1) = 2kˆ ˆl and v(2) = 2kˆ mˆ for the vector modes and e× = 2ˆl mˆ ,
ij (i j) ij (i j) ij [i j]
e+ =ˆl ˆl −mˆ mˆ for the tensor modes. In the conformal Newtonian gauge, δg =
ij i j i j 00
−2a2ψ and t δg = −2a2φδ (the tensor and vector modes remain the same in
ij ij
both gauges). An additional scalar degree of freedom arises at the perturbative
level from the non-vanishing f′ , given byb
R
f′ δR
χ≡− R . (8)
ǫ¯ 6H2
H
This feature is a manifestation of the well-known connection between f(R) theo-
ries and non-minimally coupled scalar-tensor theories. The expression of a generic
aWiththesenotationtheRicciscalarreads =12H2(1 1ǫ ).
bWith the gauge invariant notation the peRrturbed Ricc−i sc2alHar reads δ = 6H2(W +4X
1K2(Y 2Z) ǫ¯ T). The fact that T appears explicitly inthe expressRion of−δ indicates tha−t
3 − − H R
thisisnotagaugeinvariantquantity.
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perturbed stress-energy tensor, Uµ , is
ν
δUµ =(ρδ+δP)uµu +(ρ+P)(u δuµ+uµδu )+δPδµ +PΠµ , (9)
ν ν ν ν ν ν
where the density contrast is δ ≡ δρ/ρ, the Hubble flow is parametrized by u =
ν
(−1,~0) in coordinate time, and δu =(0,δu ) is the perturbed velocity field whose
ν i
scalar mode is θ ≡ ikjδuj.
k2
Our results areare obtainedin both the synchronousandconformalNewtonian
gauges thanks to a new set of variables presented bellow (quantities denoted with
the subscript ‘S’ (‘C’) are evaluated in the synchronous (conformal Newtonian)
gauges respectively).
Symbol Synchronous gauge Conformal Newtonian gauge
h′
T k 0
2K2
Y T′+ǫ T ψ
H
Z η−T φ
X Z′+Y Z′+Y (10)
W X′−ǫ (X +Y) X′−ǫ (X +Y)
H H
Θˆ Θ +3(1+w)T Θ
s c
δˆP δP +P′T δP
s s c
χˆ χ +f′ T χ
s R c
Instead of δ and θ, we make an extensive use of the dimensionless variables
∆≡δ+3(1+w)Hθ, Θ≡3(1+w)Hθ. (11)
The perturbed pressure is packaged into the gauge invariant entropy perturbation,
δˆP dP
wΓ= − ∆−Θˆ . (12)
ρ dρ
(cid:16) (cid:17)
Theanisotropicstressisthespatialtracelesspartofthestress-energytensor. Inthe
samewayasthemetricperturbation,itdecomposesintoonescalar,ΠS,twovector,
ΠV, and two tensor modes, ΠT. The generic perturbed fluid equation, δ(∇µU )=
µν
0, are
∆′−3w∆−2wΠS+g ǫ Θˆ =3(1+w)X,(13a)
K H
w′ w′
Θˆ′+ ǫ − Θˆ − 3w− ∆−2wΠS−3wΓ =3(1+w)Y.(13b)
(cid:18) H 1+w(cid:19) (cid:18) 1+w(cid:19)
The prime denote derivative with respect to d/dlna and g ≡ 1+ K2 . The field
K 3ǫH
equations (2) expanded to linear order in perturbations are
− 2K2Z =Ω ∆ +Ω ∆ , (14a)
3 m m de de
2X =Ω Θˆ +Ω Θˆ , (14b)
m m de de
2W +2X− 2K2(Y −Z)=Ω (δˆP /ρ )+Ω (δˆP /ρ ), (14c)
3 9 m m m de de de
1K2(Y −Z)=Ω w ΠS +Ω w ΠS, (14d)
3 m m m de de de
1hV′′+(1 − 1ǫ )hV′ =Ω w ΠV+Ω w ΠV, (14e)
6 2 6 H m m m de de de
1hT′′+(1 − 1ǫ )hT′+ 1K2hT =Ω w ΠT+Ω w ΠT. (14f)
6 2 6 H 3 m m m de de de
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3. Equation of State for Perturbations
Inthef(R)scenario,theexpansiontofirstorderinperturbationsofthedarksector
stress-energy tensor is
κδD = −f δR + 1fδg + 1g f δR−f R δR
µν R µν 2 µν 2 µν R RR µν
+δ(∇ ∇ f )−((cid:3)f )δg −g δ((cid:3)f ). (15)
µ ν R R µν µν R
This allows us to isolate the perturbed fluid variables for the f(R) dark sector
theory. The tensorandvectorprojectionsof(15)readilyconstitute theEoSforΠV
de
and ΠT,
de
Ω w ΠV =−1f hV′′− 1{(3−ǫ )f +f′ }hV′, (16a)
de de de 6 R 6 H R R
Ω w ΠT =−1f hT′′− 1{(3−ǫ )f +f′ }hT′− 1f K2hT. (16b)
de de de 6 R 6 H R R 6 R
Thetime-timeprojectionprovidesausefulformulaethatenablestowriteχˆinterms
of ∆ ,X and Z,
de
Ω ∆ =−g ǫ χˆ+f′ (X + 1K2Z). (17)
de de K H R 3
The longitudinal spatial traceless projection is
Ω w ΠS =−1K2χˆ− 1f K2(Y −Z). (18)
de de de 3 3 R
Using (18,17)andthe perturbed field equations (14), it is possible to write Γ and
de
ΠS in terms of the other perturbed fluid variables:
de
w Γ =(ζ − 2 ǫ¯H 1 )∆ −ζ Θˆ + Ωmζ (∆ −Θˆ ) (19a)
de de de 3gKǫHfR′ de de de Ωde m m m
w ΠS = gK−1 1 (1+ 1f′ )∆ − 1f′ Θˆ + 1 Ωmf′ (∆ −Θˆ ) (19b)
de de gK 1+fR n 2 R de 2 R de 2Ωde R m m o
where ζ ≡ gKǫH−ǫ¯H − dPi. These expressions constitute the EoS for perturbations
i 3gKǫH dρi
in f(R) gravity. From here, the whole dynamics of the (scalar) perturbations is
providedbythefourperturbedfluidequations(13),plusoneevolutionequationfor
the metric perturbationscd. This approachis as powerfulas elegant: it provides an
efficientwaytosolvethelinearperturbationinf(R)gravity,andthephenomenology
becomestransparentthroughtheinterpretationofthefluidvariables. Herewehave
illustratedthe EoSapproachwith f(R) gravity,butit shouldapply to anymodified
gravitytheory. The EoSapproachcanbe analternative(or complementary)to the
“parametrized post Friedmann framework” developped by Ferreira-Baker-Skordis.
It enables to simply characterize intricate modified gravity theories through their
equations of state for perturbations. In the quest for understanding the nature of
dark energy, this approach seems to us particularly attractive as it links formal
modification to General Relativity to phenomenology in a straightforwardway2.
cZ′=X Y,whereX andY arewrittenintermsoftheperturbedfluidvariables.
dInfact,t−hesystemoffivedifferentialequationsisoverdetermined: whenK=0,thefieldequation
(14a)canbeusedtoexpress∆ intermsof∆ ;andwhenK=0,thesameequationgivesZ in
de m 6
termsofthe∆’s
i
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References
1. R. A. Battye and J. A. Pearson, Parametrizing dark sector perturbations via
equations of state, Phys.Rev. D88 (2013), no. 6 061301, [arXiv:1306.1175](and
references therein).
2. R. A.Battye,B.BollietandJ.A.Pearson,f(R) asadarkenergyfluid, accepted
for publication in Phys.Rev. D, [arXiv:1508.04569](and references therein).