The Large Scale Structure of f(R) Gravity

0.25 MB·English

by  Yong-Seon Song

The Large Scale Structure of f(R) Gravity
Yong-Seon Song,1 Wayne Hu,1,2 and Ignacy Sawicki1,3∗
1 Kavli Institute for Cosmological Physics, Enrico Fermi Institute, University of Chicago, Chicago IL 60637
2 Department of Astronomy & Astrophysics, University of Chicago, Chicago IL 60637
3 Department of Physics, University of Chicago, Chicago IL 60637
(Dated: February 3, 2008)
We study the evolution of linear cosmological perturbations in f(R) models of accelerated ex-
pansion in the physical frame where the gravitational dynamics are fourth order and the matter is
minimally coupled. These models predict a rich and testable set of linear phenomena. For each
expansion history, fixed empirically by cosmological distance measures, there exists two branches
of f(R) solutions that are parameterized by B ∝ d2f/dR2. For B < 0, which include most of the
models previously considered, there is a short-timescale instability at high curvature that spoils
agreement with high redshift cosmological observables. For the stable B >0 branch, f(R) models
7
can reduce the large-angle CMB anisotropy, alter the shape of the linear matter power spectrum,
0
and qualitatively change the correlations between the CMB and galaxy surveys. All of these phe-
0
nomenaareaccessible withcurrentandfuturedataandprovidestringent testsof generalrelativity
2
on cosmological scales.

n
a PACSnumbers:
J

5
I. INTRODUCTION Regardless of the outcome of small-scale tests of grav-
1
ity in f(R) models, it is worthwhile to examine the cos-

2 mological consequences of treating f(R) as an effective
v Cosmic acceleration can be explained either by miss- theory valid for a cosmologically appropriate range of
2 ingenergywithanexoticequationofstate,dubbeddark curvatures. At the very least by making concrete pre-
3 energy, or by a modification of gravity on large scales.
dictions ofcosmologicalphenomenain these models, one
5 Indeed the cosmological constant can be considered ei-
gains insight on how cosmology can test gravity at the
0
ther as a constant added to the Einstein-Hilbert action
largest scales.
1
orasvacuumenergy. Non-trivialmodificationswherethe
6 In this Paper, we develop linear perturbation theory
addition is a non-linear function f(R) of the Ricci scalar
0 for predicting cosmological observables such as the Cos-
/ that becomes important only at the cosmologically low mic Microwave Background (CMB) and the large-scale
h
values of R have also been shown to cause acceleration
structureoftheuniverseexhibitedingalaxysurveys. We
p
[1, 2, 3]. They are furthermore free of ghosts and other
- workinthephysicalframewherethematterisminimally
o types of instabilities for a wide range ofinteresting cases
coupled and obeys simple conservation laws.
r [3, 4, 5].
t We begin in II by reviewing the properties of f(R)
s §
a Solar-system tests of gravity provide what is perhaps modelsandtheirrelationshiptothe expansionhistoryof
: the leading challenge to f(R) models as a complete the- the universe. In III, we derive the fourth order pertur-
v §
ory of gravity [6]. The equivalence of f(R) models to bationequationsand,usinggeneralpropertiesdemanded
i
X scalar-tensor theories lead to conflicts with parameter- by energy-momentum conservation [14, 15, 16], recast
r izedpost-Newtonianconstraintsatabackgroundcosmo- themintoatractablesecondorderform. In IVweiden-
a logicaldensity of matter. It is howeverstill controversial tifyashorttimescaleinstabilitythatrenders§awideclass
whether the whole class of f(R) modifications can be off(R)modelsnotviablecosmologically. Wepresentso-
ruled out by this equivalence. Matter in the solar sys- lutions on the stable branch in V and explore their im-
§
tem becomes non-minimally coupled in the transformed pact on cosmological power spectra in VI. We discuss
§
frame leading to non-trivial modifications of the scalar these results in VII.
§
field potential. In the original Jordan—or physical—
frame, it has been shown that the Schwarzschild metric
solves the modified Einstein equations of a wide range II. EXPANSION HISTORY
of f(R) models [7, 8] but this solution is not necessarily
relevant for the solar system [9]. Recent work has also We considera modificationto the Einstein-Hilbertac-
raised the question as to whether solar-system gravity tion of the form [17]
problemsmaybecometractableiff(R)isviewedassim-
ply a first-order correction term to the high R limit of R+f(R)
S = d4x√ g + , (1)
general relativity [10, 11, 12, 13]. − 2µ2 Lm
Z (cid:20) (cid:21)
whereRistheRicciscalar,whichwewillsometimesrefer
to as the curvature, µ2 8πG, and is the matter
m
≡ L
∗Electronicaddress: [email protected] Lagrangian. Variation with respect to the metric yields
2
the modified Einstein equations For convenience, let us define the dimensionless quan-
tities
f
Gαβ +fRRαβ −(2 −(cid:3)fR)gαβ −∇α∇βfR =µ2Tαβ,(2) E = H2 , R =3(4E+E′), y = f . (9)
H2 H2 H2
0 0 0
where f df/dR and likewise f d2f/dR2 below.
R RR
We define≡the metric to include scala≡r linear perturba- Here H H(lna = 0) = h/2997.9 Mpc−1 is the Hub-
0
≡
tions around a flat FRW background in the Newtonian ble constant. The modified Friedmann equation can be
or longitudinal gauge recast into an inhomogeneous differential equation for
y(lna)
ds2 = (1+2Ψ)dt2+a2(1+2Φ)dx2. (3)
− ρ 4E′+E′′
L[y] = µ2 DE , (10)
Given that the expansionhistory and dynamics of linear − H2 E
0 (cid:18) (cid:19)
perturbations are well-tested in the high curvature, high
redshiftlimitbytheCMB,werestrictourconsiderations where the differential operator on the lhs is given by
to models that satisfy [1, 2, 3] 1E′ 4E′′+E′′′
′′ ′
L[y] y 1+ + y
lim f(R)/R 0. (4) ≡ − 2 E 4E′+E′′
R→∞ → 1 (cid:18)4E′+E′′ (cid:19)
+ y. (11)
With this restriction, the main modifications for viable 2 E
(cid:18) (cid:19)
models with stable high curvature limits arise well af-
For illustrative purposes, we take an expansion history
ter the radiationbecomes a negligible contributor to the
thatmatches a darkenergymodel with a constantequa-
stress energy tensor. We can then take it to have the
tion of state w,
matter-dominated form
T00 = −ρ(1+δ), E =(1−ΩDE)a−3+ΩDEa−3(1+w), (12)
T0i = ρ∂iq, and
Ti = 0. (5)
j µ2ρ
DE =Ω a−3(1+w). (13)
The modified Einstein equations with the FRW back- 3 H02 DE
ground metric and δ = q = 0 yields the modified Fried-
SinceEq.(11)isasecond-orderdifferentialequation,the
mann equation
expansionhistorydoesnotuniquely specifyf(R)butin-
1 µ2ρ stead allows a family of solutions that are distinguished
H2 f (HH′+H2)+ f +H2f R′ = . (6) byinitialconditions. Thisadditionalfreedomreflectsthe
R RR
− 6 3
fourth-order nature of f(R) gravity.
Here and throughout, primes denote derivatives with re- To set the initial conditions, take y± to be the two
spect to lna. solutionsofthehomogeneousequationL[y]=0. Athigh
There is sufficient freedom in the function f(R) to re- curvature, these solutions are power laws y± ap± with
∝
produce any desiredexpansionhistoryH. Hence the ex-
pansionhistory alonecannot be usedas a test of general 7 √73
p± = − ± . (14)
relativity though it can rule out specific forms of f(R). 4
The dynamics of linear perturbations on the other hand
Since p− 3.9, stimulation of this decaying mode vi-
do test general relativity as we shall see. ≈ −
olates the condition that f /R 0 at high R. We
R
We therefore seek to determine a family of f(R) func- →
therefore set its amplitude to zero in our solutions (c.f.
tions that is consistent with a given expansion history
[19, 23, 24]). The particular solution in the high curva-
[18, 19, 20, 21, 22]. Without loss of generality, we can
ture limit becomes
parameterizethe expansionhistory interms ofanequiv-
alent dark energy model y = 6ΩDE a−3(1+w). (15)
part 6w2+5w 2
µ2 −
H2 = (ρ+ρDE). (7) ThereforewhennumericallyintegratingEq.(11)wetake
3
y(lna ) = Ay (lna )+y (lna ), (16)
This yields a second order differential equation for f(R) i + i part i
′
y (lna ) = p Ay (lna ) 3(1+w)y (lna ),
i + + i part i
1 µ2ρ −
−fR(HH′+H2)+ 6f +H2fRRR′ =− 3DE , (8) at some initial epoch ai ∼10−2.
Since the modifications to gravity appear at low red-
where H2 and R are fixed functions of lna given the shifts, it is more convenient to parameterize the individ-
matching to the dark energy model. ual solutions in the family by the final conditions rather
3
than the growing mode amplitude of the initial condi- 0
tions. Given that a constant f(R) is simply a cosmo- B =
0
logical constant and a linear one represents a rescaling 3
-0.2
opfheGnoomreµn,aitthisatfRaRre, tuhneiqsueecotnodthdeermivoadtiivfiec,atthioant.cIonntpraorls- R)/R 1
ticular,weshallseethataspecificdimensionlessquantity f( -0.4 0.3
0
f H -0.3
B = 1+RRf R′H′ (17) -0.6 -1 -3 (a) w=-1, W DE=0.76
R
2 1 E ′′ ′4E′′+E′′′ 0 3
= y y ,
3(1+f )4E′+E′′E′ − 4E′+E′′
R (cid:18) (cid:19) -0.2
1
is most closely linked with the phenomenology. B is a R
stronglygrowingfunctioninoursolutionsandinthehigh R)/-0.4 0.3
f( 0
curvature limit has a growth rate -0.6
-0.3
B′ -0.8 -1
p (18) (b) w=-0.9, W =0.73
B ≡ B -3 DE
10 100 1000
given by pB = 3+p+, if the growing mode dominates, R/H02
and p = 3w, if the particular mode dominates.
B
−
We will therefore characterize solutions with a given FIG. 1: Every expansion history that can be parameterized by
expansion history family by B0 B(lna = 0). If B0 = a dark energy model with ρDE(lna) can be reproduced by a one
0 and the background expansio≡n is given by w = 1 parameter familyoff(R) models, indexedbyB0 ∝fRR/(1+fR)
then f(R) = const., B(lna) = 0 and the model has−a at the present epoch (left end point of curves), that approaches
theEinstein-Hilbertactioninthehighcurvaturelimit. (a)ΛCDM
true cosmological constant. More generally B = 0 will expansionhistory(w=−1,ΩDE=0.76,h=0.73). (b)Dynamical
correspond in linear theory to the dynamics of a dark darkenergyexpansionhistory(w=−0.9,ΩDE=0.73,h=0.69).
energycomponentforscalesabovethedarkenergysound
horizon.
InFig.1,weplotthefamilyoff(R)modelsthatmatch
these auxiliary parameters so that their effect vanishes
two representative expansion histories parameterized by
at large scales and early times.
(w,Ω ,h). These models are chosen to be consistent
DE
On superhorizon scales (k/aH 1), the evolution of
with current WMAP CMB data [25] and span a range ≪
metric perturbations must be consistent with the back-
thatisconsistentwithsupernovaeaccelerationmeasures.
ground evolution provided that the background solu-
Given the similarity between these models, we will take
tion is valid, i.e. that fluctuations about it are stable.
the w = 1 ΛCDM expansion history for illustrative
− Bertschinger [16] showed that the familiar conservation
purposes below.
of the curvature fluctuation on comoving hypersurfaces
The linear perturbation analysis that follows does not
(ζ′ = 0) for adiabatic fluctuations in a flat universe ap-
require the matching to the specific expansion histories
plies to any metric based modified gravity model that
parameterized by (w,Ω ,h) here. This reverse engi-
DE obeys energy momentum conservation µT = 0. The
neering is only a device to find observationally accept- ∇ µν
gauge transformation into the Newtonian gauge
able f(R) models. What is required is that the back-
ground solutions provide H(lna) and B(lna). On the ζ =Φ+Hq (19)
other hand, linear perturbation theory does inform the
implies
choice of a background solution. We shall find in IV
that the B <0 branchof the family is unstable to lin§ear ζ′ =Φ′+H′q+Hq′ =0, (k =0), (20)
perturbations in the high curvature regime.
and momentum conservation requires
′
Hq = Ψ (21)
−
III. LINEAR PERTURBATION EQUATIONS
so that
′ ′
ThemodifiedEinsteinequations(2)representafourth- Φ Ψ+H q =0, (k =0). (22)
−
orderset of differentialequations for the two metric per-
CombiningEq.(21)and(22)yieldsasecondorderdiffer-
turbationsΨandΦinthepresenceofmatterdensityand
ential equation for the Newtonian metric perturbations
momentum fluctuations δ and q. To solve this system of
[15]
equations,weintroduce auxiliaryparametersto recastit
as a larger set of second-order differential equations. It ′′ ′ H′′ ′ H′ H′′
Φ Ψ Φ Ψ=0, (k =0).(23)
is numericallyandpedagogicallyadvantageousto choose − − H′ − H − H′
(cid:18) (cid:19)
4
Theevolutionofthemetricfluctuationsmustinthisway The metric fluctuations S and Φ− act as sources to ǫ
be consistent with the expansion history defined by H. which then feed back into their evolution weighted by
Note that this equation applies to any modified gravi- (Bk/aH)2. To complete this system, the dynamics of θ
tational scenario that satisfies the required conditions. are supplied by the derivative of its definition Eq. (24)
The DGP braneworld acceleration model [26] represents combined with Eq. (25)
anothervalidapplication[27]. Whatdoesrequireaspec-
′ ′
3E B E
ificationofatheoryistherelationbetweenΦandΨ. Un- θ′+ 2 + θ = 2 ǫ. (30)
dergeneralrelativityandassumingT takesthematter- − − 2 E B E′
µν (cid:18) (cid:19)
only form of Eq. (5), the closure relationis Φ= Ψ and
− Eqs. (28) and (30) can also be combined to eliminate θ
Eq. (23) in fact applies on all scales. With a dynamical
leaving a second-order differential equation for ǫ. This
darkenergy componentin T , it applies abovethe dark
µν combined relation may alternately be derived directly
energy sound horizon [28].
fromthe trace ofthe ij componentof the Einsteinequa-
To capture the metric evolution of f(R) models for
tions.
k = 0, let us introduce two parameters: θ the deviation
6 As in general relativity, the remaining Einstein equa-
from ζ conservation, Eq. (22)
tions become constraint equations given the dynamical
H′ k 2 variables ∆,Hq,ǫ,θ. The 00 equation may be expressed
′ ′ ′
ζ =Φ Ψ+H q = Bθ, (24) as the modified Poissonequation,
− H aH
(cid:18) (cid:19)
and ǫ the deviation from the superhorizon metric evolu- BE′ E′ µ˜2a2ρ
tion Eq. (23) 2Φ−− 2 E 4E′+E′′(S+3Bǫ)= k2 ∆, (31)
H′′ H′ H′′ where ∆ is the density perturbation in the comoving
′′ ′ ′
Φ Ψ Φ Ψ
− − H′ − H − H′ gauge
(cid:18) (cid:19)
k 2 ∆=δ 3Hq, (32)
= Bǫ. (25) −
aH
(cid:18) (cid:19) and the trace-free ij component becomes
The coefficients in front of the deviation parameters are
2
chosen to bring out the fact that their effect vanishes as Φ+Ψ = (S Φ−)
3 −
k/aH 0andB 0solongasthedynamicsguarantees
stable→behavior of→the parameters themselves (see IV). BE′ k 2
§ = (Hq) B (33)
The Einstein equations can then be recastas a second 2 E − aH
(cid:18) (cid:19)
orderdifferential equationfor ǫ and constraintequations 1 E′ 1E′
for the othermetric variables. Since ǫ itself containssec- × 34E′+E′′(S+3Bǫ)+ 2 E Bθ .
ond derivatives of the fundamental metric perturbations h i
ΦandΨ,theequationsareimplicitlyfourthorder. Itwill NotethatasB 0andµ˜ µtheseconstraintequations
→ →
be convenientto separateouttwo linearcombinationsof become the usual Poisson and anisotropy equations. In
the underlying metric fluctuation particular for B = 0, the dark energy-closure relation
Φ= Ψ is recovered.
1 −
Φ− = (Φ Ψ), S =2Φ+Ψ, (26) Finally,theconservationlawsprovidethedynamicsfor
2 −
the matter fluctuations and are unmodified by f(R)
and a reduced mass scale or rescaling of G
2
k
µ˜2(lna)= µ2 . (27) ∆′ = aH Hq−3ζ′, (34)
1+fR(lna) (cid:18) (cid:19)
Interms of these variables,the 0icomponentofthe Ein-
1
′
stein equations becomes a dynamical equation for ǫ Hq = Ψ= (S 4Φ−). (35)
− −3 −
1 1 BE′
B(ǫ′+G1ǫ) = G2S (S 2Φ−)+ (S 2Φ−) Theimpactofthemodificationtogravitycomesfromthe
3 − 3 − 6 E −
metric evolution. Eq. (24) implies
1E′ 1µ˜2ρ
+ 2 E + 3 H2 Hq (28) ′ H′ k 2
(cid:18) (cid:19) ζ = Bθ. (36)
µ˜2ρ E′′ 1E′ k 2 H (cid:18)aH(cid:19)
+ 4+ θ,
"H2 (cid:18) E′(cid:19)− 2 E (cid:18)aH(cid:19) # In fact directly integrating Eq. (36) and checking for
consistency between Hq defined through Eq. (19) and
where
Eq. (35) tests the numerical accuracy of solutions.
E′ E′′ 4E′′+E′′′ 1E′ B′ Along with initial conditions for each of the fluctu-
G = 1 +2 + B+2 ,
1 − E E′ − 4E′+E′′ 2 E B ations, these equations provide a complete and exact
E′′ B′ description of scalar linear perturbation theory in f(R)
G2 = 4+ E′ + B −G1. (29) gravity for a matter-only universe.
5
IV. STABILITY AT HIGH CURVATURE on the other hand are stable and in fact correspond to
the background expansion histories studied by [23, 33].
The fourth-order nature of the linear perturbation However these expansion histories have gravity modified
equationsderivedinthe previoussectionraisesthe ques- throughout the matter dominated epoch and in partic-
tion of stability in the high-curvature limit to general ular a t1/2. They produce phenomenology at high
∝
relativity [29, 30, 31, 32]. Strongly unstable metric fluc- redshift that would violate constraints from the CMB.
tuationscancreateorderunityeffectsthatinvalidatethe We will omit them from further consideration below.
background expansion history.
Thekeyequationsforstabilityare(28)and(30)which
V. METRIC EVOLUTION SOLUTIONS
describethe evolutionofthe deviationparameters. Con-
sider the high redshift limit of high curvature where
B 0 and wavelengths of interest are well outside the Inthissectionwediscussthenumericalsolutionsofthe
|hor|i→zon k/aH 1. In this limit the evolution equations linearperturbationequationsonthestableB >0branch.
simplify to ≪ Toexposetheunderlyingfeaturesofthesolutions,weex-
amine the relevant limiting cases below. We begin with
7 2 1 the initial epoch where B 1 and the fluctuations are
′′ ′
ǫ + 2 +4pB ǫ + Bǫ= BF(Φ−,S,Hq), (37) superhorizon sized k/aH| |≪1. We then examine large
(cid:18) (cid:19) scale or “superhorizon” m≪odes where B1/2k/aH 1
≪
where F(Φ−,S,Hq) is the source function for the de- whose evolution is completely determined by the back-
viation ǫ and recall p is the growth index of B from ground expansion history and the form of f(R). Finally
B
Eq.(18). The details of F are not importantfor the sta- we track the evolution of small scale or “subhorizon”
bility analysis other than that it provides a source that modesuntilB1/2k/aH 1wheretheirevolutionreaches
≫
isofordertheperturbationparametersthatareitsargu- the simple form implied by quasistatic equilibrium.
ments. Under the assumption that the general relativis-
tic solution is stably recoveredin this limit, it acts as an
external source to the deviations. The stability question A. Initial Conditions
can be phrased as whether ǫ remains self-consistently of
orderthesesourcesorgrowsandpreventstherecoveryof On the stable B > 0 branch, we can set the initial
the solutions. conditions when B 1 and the mode is superhorizon
≪
The stability equation (37) has the peculiar feature sized, k/aH 1. In this case, the initial conditions
≪
that the frequency squared 2/B diverges as B 0 in- forthenormalfluctuationparametersfollowthegeneral-
| | →
dependently of k, resembling a divergent real or imagi- relativistic expectation
narymassterm. Evolutionofǫcanoccuronatimescale
3
much shorter than the expansion time. If B < 0, ǫ is Φi = ζi,
5
highly unstable and deviations will grow exponentially.
Ψ = Φ ,
i i
If B > 0, ǫ is highly stable and is driven to the value −
2
requiredbythesourcefunctionǫ=F/2. Thisshorttime 2 k
∆ = Φ ,
i i
scale behavior can also be seen directly in the 4th order 3 aH
(cid:18) (cid:19)
form of the Einstein equation. The trace of the ij equa- 2
Hq = Φ , (40)
tion or the derivative of the 0i equation have their 4th i i
3
order terms multiplied by the small parameter B.
whereζ =const. istheinitialcomovingcurvature. These
i
Thusdespitetheapparentrecoveryofgeneralrelativity
relations also imply Φ− = Φ and S = Φ with vanishing
intheactionathighcurvatureR,thegeneral-relativistic
first derivatives initially.
solutions to linear perturbationtheory are not recovered
Detailed balance gives the deviation parameters as
for B < 0. In terms of f(R), B f in this limit and
RR
∝ 1
hence models like [1] θ = p Φ ,
i B i
9
M2+2n 3 5
f(R)= , (n>0), (38) ǫ = +p θ , (41)
− Rn i −2 2 B i
(cid:18) (cid:19)
andthe highfrequency termintheir evolutionequations
and [31]
ensures that they stay locked to these relations until B
f(R)= M2exp( R/λM2), (39) becomes non-negligible.
− −
are included in this class of unstable models.
The instability causes any finite patch of a universe B. Superhorizon Evolution
that starts at high curvature to break away from the
background solution into either a low curvature solution Given that θ and ǫ are locked to the initial values of
or a singularity. The low curvature R Gρ solutions Eq. (41) when B 1, their definitions in Eq. (24), (25)
≪ ≪
6
imply thatthey havenegligible effect onthe evolutionof 1
the metric fluctuations Φ, Ψ. This remains true even as
L CDM
the mode evolvesintothe B 1 regimeif k/aH 1. In k/aH =100
0
∼ ≪ 0.9
particular, the anisotropy relation of Eq. (34) becomes
i
/F 10
′
Φ+Ψ=BH q (42) F0.8
1 0
and closes the general relation for superhorizon metric
0.7
fluctuations Eq. (23):
H′′ B′ H′ 100
′′ ′
Φ + 1 + +B Φ (43)
− H′ 1 B H 1.2
(cid:18) − (cid:19)
+ H′ H′′ + B′ Φ=0, (k =0). i 1 10
H − H′ 1 B /F-
(cid:18) − (cid:19)
F0.9 1
TheevolutionofΦiscompletelydeterminedbytheback-
0
groundevolutionandthespecificationoff(R). Formally,
0.8 w=-1, W =0.76, B =1 L CDM
this solution also applies to the unstable branch B < 0 DE 0
at k = 0 but is only valid at finite k for large B . The 0.01 0.1 1
| | a
pointatwhich B =1 is a regularsingularpointfor typi-
cal B(lna) and so Φ evolvessmoothly through it. Φ will
grow on the expansion time scale if
FIG.2: Evolutionofmetricfluctuations Φ(upperpanel)andΦ−
H′ H′′ B′ (dlioffweerernptacnleols)urfeorreBla0ti=ons1oanndsuapeΛrCaDndMsuebxp-haonrsiizoonnhsicsatloersy.forTΨhe,
+ <0. (44)
H − H′ 1 B Eqs.(42)and(46),leadtoqualitativelydifferentevolutionforthe
− twolimitswithatransitionregioninbetween. Φ−,whichcontrols
Growthis typicallya transientphenomenonatthe onset effects in the CMB and enters directly into the Poisson equation,
hasascale-dependentgrowththatmakesitincreasinglylargerthan
of acceleration given that the presence of matter makes
H′/H H′′/H′ positive. For example, if the expansion theΛCDMpredictionathighk. ResultsforothervaluesofB0can
bescaledfromthisfigurebynotingthatthetransitionoccurswhen
−
rate approaches the de Sitter case of a constant in the k/aH ≈B−1/2.
future H′/H 0 and the solution to Eq. (44) becomes
→
C C
1 2 ′ ′
Φ= + da(1 B)H , (H /H 0), (45)
a a − → The Poisson equation then takes the simple form
Z
where C and C are constants. This implies decaying
1 2 1
solutions unless BH′ grows. H′ decays and, since in de k2Φ− = µ˜2a2ρ∆ (47)
2
Sitter R const, B should asymptote to a constant.
→
Eq. (45) then implies Φ 1/a. Note that this stability
∝ and the conservation laws become
analysisdiffersfromtreatmentsthattakeapuredeSitter
expansionwith no matter since that assumptionforces a
k 2
closure relation of Φ=Ψ (c.f. [34, 35]). ∆′ = Hq,
Anexampleofthesuperhorizonevolutionofthemetric aH
(cid:18) (cid:19)
isshowninFig.2(k =0curves)foramodelwithB0 =1. Hq′ = 4Φ−, (48)
WhileΦismonotonicallysmallerthantheB =0ΛCDM 3
prediction,Φ− is monotonicallylargerdueto the closure
relation between Φ and Ψ of Eq. (42). where we have dropped temporal derivatives when com-
paredwithspatialgradientswhereappropriate. Thissys-
tem describes a scale free evolution for Φ− or ∆. The
C. Subhorizon Evolution transition between these two scale-free regimes occurs
when (k/aH) B−1/2. This scale- and time-dependent
∼
transition leads to a scale-dependent growth rate. Un-
For subhorizon scales where k/aH 1, Eqs. (28) and
(30) form an oscillator equation who≫se frequency scales like for Φ, Φ− has monotonically enhanced power as k
increases on the B > 0 branch. Because of the time de-
as k/aH. Therefore the amplitude of ǫ is driven to zero
pendence of the transition, the total growth to z = 0
whencomparedwithΦ−. WhencombinedwiththePois- continues to increase with k even for k/aH B−1/2.
son and anisotropy equation, this requires [31]
≫
InFig.2,weshowthe full numericalsolutionfromthe
3 initial conditions through the super- to the sub-horizon
lim S 0, Ψ 2Φ, Φ− Φ (46)
Bk/aH→∞ → →− → 2 evolution for a few representative modes.
7
VI. POWER-SPECTRA OBSERVABLES that the changes to the power spectrum occur mainly at
the lowest multipoles, WMAP constraints on the ampli-
Thescaledependencesofthelineargrowthrateofmet- tude of the peaks can be directly translated into a nor-
ric anddensity perturbations change predictions for cos- malizationofthepowerspectrumonscalescorresponding
mologicalpowerspectrainthelinearregime. Letusmake to the acoustic peaks. For the ΛCDM expansion history
the usual assumption that the initial spectrum of fluctu- of w = 1, ΩDE =1 Ωm =0.76, h=0.73 the normal-
ations in the comoving curvature is given by a power ization−from WMAP−is δζ = 4.52 10−5 for an optical
×
law. For a starting epoch during matter domination, depthtoreionizationofτ =0.092. Wefurthertakeatilt
this power law is modified by the usual matter-radiation of n=0.958, Ωbh2 =0.0223.
transfer function T(k)
k3P k n−1 total quadrupole
ζi =δ2 T2(k), (49)
2π2 ζ k L CDM
(cid:18) n(cid:19)
103
where δζ is the rms amplitude atthe normalizationscale )2
k .
n
The modifications to the CMB depend on the scale- p (mK2
dependent change in the potential growth rate C/2
6
Φ−(a,k) 102 ISW quadrupole
G(a,k)= , (50)
Φ−(ai,k)
through the Integrated Sachs-Wolfe (ISW) effect. This
effectcomesfromthedifferentialredshiftthatCMBpho-
1 2 3 4
tons suffer as they transit the evolvingpotential. It con- B
0
tributes to the angular power spectrum of temperature
anisotropies as
FIG.3: CMBquadrupolepower6C2/2πcontributedbythemodi-
fiedISWeffect(dashedcurve)andtotal(solidcurve)asafunction
dk 9 k3P
CII =4π [II(k)]2 ζi , (51) of B0 in the ΛCDM expansion history. For reference, the ΛCDM
l k l 25 2π2 total quadrupole is also shown (horizontal line). The change in
Z the growth of the potential causes a near nulling of the ISW ef-
where fect at B0 ≈ 3/2 and a substantial reduction of power between
0.2.B0.2.5.
dG
II(k)=2 dz j(kD). (52)
l dz l
Z
Here D = dz/H is the comoving distance out to red-
shift z.
R
In Fig. 3, we show the quadrupole power as a func-
tion of B0 contributed by the ISW effect as well as the 2K)3000
total quadrupole. Power in the quadrupole arises near
sisschaoalwemsnionifnimkF/uiHmg.02a∼rroeu1dn0udcaeBnsd0it≈sone3ta/hr2e,Bww0eha∼ekre1e.vtohlIeuntIiSfoaWnctoetfffhΦeecr−et pmTT+1)C/2 (l 0 (L BCD0M)
is a negligible contributor to the power, and a substan- l(l 1/2
tial reduction for 0.2 . B . 2.5. Further reduction of 3/2
0
large-scale power can be achieved by changing the ini- 1000
tial power spectrum to simultaneously suppress horizon
scale power in the Sachs-Wolfe effect from recombina-
tion. Hence these models provide the opportunity to 10 100 1000
bringthepredictedensemble-averagedquadrupolecloser l
tothemeasurementsonoursky[25]. ModelswithB &3
0
produce an excess of large-angle anisotropy and exacer- FIG. 4: CMB angular power spectra for the ΛCDM expansion
batethetensionwiththedata. Notehoweverthatdueto historyforB0=0(ΛCDM),1/2,3/2. Powerinthelowmultipoles
is lowered by the reduction in the ISW effect. Power at the high
sample variance, changes in the likelihood will be small.
multipolesoftheacousticpeaksisleftunchanged.
We will address constraints on the models in a separate
work.
We show the full spectrum of temperature anisotropy The WMAP normalization then allows us to predict
CTT inFig.4forafewrepresentativevaluesofB . Given thematterpowerspectrumtoday. Letusdefinetheden-
l 0
8
sity growth rate where
DG(a,k)= ∆(a,k)ai (53) Ilgj(k)= dzDG(a,k)nj(z)bj(z)jl(kD), (56)
∆(a ,k)
i Z
n (z) is the galaxy redshift distribution normalized to
suchthatD =abeforef(R)effects becomeimportant. j
G
dzn =1, and b (z) is the galaxy bias.
In f(R) models the potential and density growth rates j j
ThecrosscorrelationbetweentheCMBISWeffectand
Eq.(50)and(53) candiffer non-triviallydue tothe time
R
galaxies becomes
dependent(1+f )rescalingofGinthePoissonequation.
R
The linear power spectrum then becomes dk 6 k2 k3P
CgjI =4π Igj(k)II(k) ζi . (57)
k3 P (k,a) = 4 D2(a,k) k4 k3Pζi . (54) l Z k l l 25ΩmH02 2π2
2π2 L 25 G Ω2 H4 2π2
m 0 Thecorrelationcoefficientbetweenthetotaltemperature
anisotropy and the galaxies is given by
CgjI
R l (58)
l
≡ CTTCgjgj
l l
q
3h) andis independent of the galaxybias if it is slowlyvary-
c/ ing with redshift. We have neglected galaxy magnifica-
p
M 104 tion bias which leads to an additional source of correla-
k) ( tion.
(L B
P 0 For definiteness, we assume that the galaxy sets come
1
from a net galaxy distribution of
0.1
0.01 n (z) z2e−(z/1.5)2, (59)
0.001 g
∝
103 0 (L CDM)
which is further partitionedby photometric redshift into
0.001 0.01 0.1 several galaxy samples,
k (h/Mpc)
zj−1 z zj z
n (z) n (z) erfc − erfc − (,60)
j g
FIG.5: LinearmatterpowerspectrumforseveralvaluesofB0 ∝ (cid:20) (cid:18) √2σ(z) (cid:19)− (cid:18)√2σ(z)(cid:19)(cid:21)
intheΛCDMexpansionhistory. Thechangeintheamplitude
where erfc is the complementary error function and
ofthepowerathighB0 &0.1isnearlydegeneratewithgalaxy
bias. Smaller values of 0.001.B0 .0.1 change the shapeof σ(z)=0.03(1+z) reflects the effect of photometric red-
the linear power spectrum at a potentially observable level. shift scatter.
All spectra are normalized to the WMAP anisotropy from Fig. 6 shows the correlation coefficient for several val-
recombination. ues of B . We take two redshift bins from Eq. (60) par-
0
titioned by z = 0,0.4,0.8 to achieve effective redshifts
j
of z¯ = 0.2 and 0.6. Current observations constrain the
We showP (k) forseveralchoicesofB inFig.5. De-
L 0 correlation near l 20 corresponding to scales which
spite the large change in amplitude at high k, the high ∼
are an order of magnitude smaller than those contribut-
B models cannot be automatically ruled out by galaxy
0 ing to the ISW quadrupole. Between 1/2 . B . 1
clusteringdatasincethenearlymultiplicativeshiftcanbe 0
the galaxy-ISWcorrelationis substantially reduced. For
mimicked by galaxy bias. Likewise, non-linear measure-
B & 3/2, galaxies are in fact anti-correlated with the
ments of the mass power spectrum through the cluster 0
abundance, Lyman-α forest, and cosmic shear also can- CMB since Φ− grows on the relevant scales. A loose
bound from the observed correlation would therefore be
not be straightforwardly applied. As the local curvature
B .3/2at the significance levels ofthe reporteddetec-
exceedsthebackgroundcurvatureincollapseddarkmat- 0
tions (e.g. [36, 37, 38, 39, 40, 41]) but we expect more
terhalosonewouldexpectthegravitationaldynamicsto
detailed modeling to yield better constraints in the fu-
returntoNewtonian. Forthisreason,ourpredictionsare
restricted to the linear regime at k . 0.1h Mpc−1. We ture. It is likely that a significant reduction of the large
angleanisotropyfromthis mechanismcouldbe excluded
intend to explore these issues further in a future work.
unlessothersources,suchasmagnificationbias,cangen-
Finally the cross correlation between the ISW effect
erate the observed positive correlation.
and the angular power spectra of galaxies is markedly
different in these models and potentially excludes large
B solutions. The angular power spectra of galaxies is
0 VII. DISCUSSION
given in the linear regime by
Cgjgj =4π dk[Igj(k)]2 4 k4 k3Pζi, (55) We have studied the evolution of linear cosmological
l k l 25Ω2 H4 2π2 perturbationsin f(R)models for modifiedgravityin the
Z m 0
9
0.4 with CMB measurements.
(a) z=0.2
For the stable B > 0 branch, f(R) models predict a
B =
0.2 00 (L CDM) rich set of linear phenomena that can be used to test
1/2 such deviations from general relativity. For example,
Rl large B 1 models lower the large-angle anisotropy
1
∼
of the CMB and may be useful for explaining the low
0
quadrupole observedonour sky. They alsopredict qual-
3/2
itatively different correlations between the CMB and
galaxy surveys which may provide the best upper limit
-0.2
on the deviations currently available. Smaller deviations
(b) z=0.6
B = in B are observable at smaller scales through changes to
0
0 (L CDM) theshapeofthelinearpowerspectrum. Inthelimitthat
0.2
1/2 B 0andtheexpansionhistoryisgivenbyΛCDM,lin-
→
Rl 1 ear perturbations in f(R) models approach the general
relativistic predictions exactly. We intend to examine
0 constraints on f(R) models in a future work.
3/2
More generally, this class of phenomenological f(R)
-0.2 models provides insight on the types of deviations that
10 20 30 40 mightbeexpectedfromalternatemetrictheoriesofgrav-
l
ity in the linear regime. Conservation of the matter
stress-energytensorseverelyrestrictstheformofallowed
FIG. 6: Cross correlation coefficient between the CMB and
deviationsonbothsuper-andsub-horizonscales[16,27].
galaxiesintheΛCDMexpansionhistory. Shownaretworep-
Even if these f(R) models prove not to be viable as a
resentativeredshiftbinscenteredaroundz¯=0.2and0.6with
complete alternate theory of gravity that includes solar-
B0 = 0 (ΛCDM), 1/2, 1, 3/2. The cross correlation is sub-
system tests, they may serve as the basis for a “parame-
stantiallyreducedfor1/2.B0 .1andbecomesnegativefor
B0 &3/2. terizedpost-Friedmann”descriptionoflinearphenomena
thatparallelstheparameterizedpost-Newtoniandescrip-
tion of small-scale tests.
physical—or Jordan—frame. Here the gravitational dy-
namicsarefourthorderandthematterisminimallycou-
pled and separately conserved. For models that recover Acknowledgments: We thank Sean Carroll, Mark Hind-
the Einstein-Hilbert action at high curvature R, we find marsh, Michael Seifert, Kendrick Smith, Bob Wald, and
thatforeachexpansionhistoryspecifiedbyH(lna)there the participants of the Benasque Cosmology Workshop
exists two branches of f(R) solutions that are parame- and the Les Houches Summer School for useful conver-
terized by B f , the second derivative of f(R). For sations. This work was supported by the U.S. Dept. of
RR
∝
B <0,whichincludesmostmodelspreviouslyconsidered Energy contract DE-FG02-90ER-40560. IS and WH are
[1,31],thereisashort-timescaleinstabilitythatprevents additionally supported by the David and Lucile Packard
recovery of the general-relativistic expectations at high Foundation. This work was carriedout at the KICP un-
curvature that is important for maintaining agreement der NSF PHY-0114422.
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