Separating Dark Physics from Physical Darkness: Minimalist Modified Gravity vs. Dark Energy

0.27 MB·English

by  Dragan Huterer

Separating Dark Physics from Physical Darkness:
Minimalist Modified Gravity vs. Dark Energy
Dragan Huterer1 and Eric V. Linder2
1Kavli Institute for Cosmological Physics and Astronomy and
Astrophysics Department, University of Chicago, Chicago, IL 60637
2Berkeley Lab, University of California, Berkeley, CA 94720
(Dated: February 5, 2008)
The acceleration of the cosmic expansion may be due to a new component of physical energy
density or a modification of physics itself. Mapping the expansion of cosmic scales and the growth
oflargescalestructureintandemcanprovideinsightstodistinguishbetweenthetwoorigins. Using
Minimal Modified Gravity (MMG) – a single parameter gravitational growth index formalism to
parameterize modified gravity theories – we examine the constraints that cosmological data can
placeonthenatureofthenewphysics. Fornextgenerationmeasurementscombiningweaklensing,
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supernovaedistances, and thecosmic microwave background we can extend thereach of physicsto
0
0 allow for fitting gravity simultaneously with the expansion equation of state, diluting the equation
2 ofstateestimationbylessthan25%relativetowhengeneralrelativityisassumed,anddetermining
the growth index to 8%. For weak lensing we examine the level of understanding needed of quasi-
n
and nonlinear structure formation in modified gravity theories, and the trade off between stronger
a
precision but greater susceptibility tobias as progressively more nonlinear information is used.
J

1
3 I. INTRODUCTION approach and its range of validity. §III lays out the cos-

mologicalprobes, fiducial models, and survey data char-
2
acteristics used in our analysis. We examine in §IV the
v Whether the acceleration of the cosmic expansion is
ability of next generation cosmological probes to reveal
1 due to a new physical component or a modification of
8 gravitation, the answer will involve groundbreaking new new gravity simultaneous with fitting cosmological ex-
6 pansion,andthe cosmologicalbiasincurredifweneglect
physics beyond the currentstandardmodels for high en-
8 toallowthepossibilityofbeyondEinsteingravity. Lever-
ergyphysicsandcosmology. Toobtaintheclearest,unbi-
0 age and systematics from the nonlinear regime are dis-
asedpictureofthe fundamentalphysicsweneedtoallow
6
cussed in §V.
0 forthe possibility ofgravitationbeyondEinsteinandsee
/ where the data lead. This article presents simultaneous
h
fitting ofthe cosmicexpansionandthe theoryofgravity.
p
- Direct measurements of the expansion history can be II. GRAVITATION AND GROWTH
o interpreted generally as the equation of state of the uni-
r
t verse;thismayormaynotcorrespondtoaphysicalcom- Attempting to invent a generalprescription for taking
s ponent. Measurementsofthegrowthhistoryofmassfluc- into account the effects of modified gravity is like seek-
a
: tuations combine information on the expansion and the ing a general treatment of non-Gaussianity: there are so
v
theoryofgravity. Thetwoincomplementaritythusallow many ways in which a theory can be “not”, in which a
i
X separation of the possible origins for cosmic acceleration symmetry can be broken, that it seems a hopeless task.
– a physicaldark component, e.g. a new field in high en- However,wedonotseekanall-encompassingdescription
r
a ergy physics, or dark (new) physics, e.g. a modification of modified gravityin all its aspects, but rather its gross
of Einstein gravity. effectsoncosmologicalobservablesbeyondtheexpansion
Within a specific theory of modified gravity one can rate.
attempt to calculate the cosmological observables and As a beginning step, we take a fairly conservative ap-
determine the goodness of fit with data. This model de- proach we call Minimal Modified Gravity (MMG). The
pendent approach must proceed on a case by case basis modifications we consider are small, since general rela-
and moreover suffers from difficulties in computation of tivity givesa predominantlysuccessfuldescriptionofthe
many quantities due to the complexity of the theories. universe and its large scale structure, and homogeneous,
Nor are the modifications necessarily well motivated or i.e.notdependentonenvironment`alachameleonscenar-
completelyfreefrompathologies. Thealternateapproach ios [1]. We assume that structure formationcontinuesto
taken here is phenomenological, using a model indepen- be described adequately by growth from Gaussian den-
dent yet physically reasonable and broad parameteriza- sity perturbations. For weak gravitational lensing, we
tionofthegravitymodificationtogaininsightintotheef- do not have to get too deeply involved in the nonlinear
fect ofgeneralizingEinsteingravity. This is closelyanal- density regime and can concentrate on the growth law;
ogoustothewidelysuccessfulequationofstateapproach weassumechangestothegravitationaldeflectionlaware
to modifications of the expansion history, giving model negligible.
independent constraints and understanding. This approach seeks to build understanding of modi-
In §II we discuss the development of the parametrized fied gravity by taking a modest step away from general
2
relativity. By examining an alteration in the linear per- A key aspect to note is that much of the growth is
turbationgrowth,preservingthestandardmappingfrom determined by the expansion history, even in modified
linear to nonlinear fluctuations, we obtain a clear pic- gravity,soweshouldnotthrowawaythatknowledge. By
ture of a specific effect of modified gravity (see §V for following the effects, treating the expansion in terms of
relaxation of the assumption of standard mapping and the well-developed equation of state formalism (whether
discussion of scale dependence). This serves as a proxy arising from a physical dark energy or a modified Fried-
for presumably more complicated and model dependent mann equation), i.e.
effects.
The form of the growth equation can be written so as H2(z)/H02 = ΩM(1+z)3+δH2(z) (5)
to directly show the influence of the cosmic expansion 1 dlnδH2
and the additional effect of the gravity theory. For a w(z) = −1+ , (6)
3dln(1+z)
matter density perturbation δ ≡δρ/ρ, the linear growth
factor g = δ/a (scaling out the matter dominated uni- andaddinganewparametertoincorporatetheeffectsof
verse behavior δ ∼a) evolves in general relativity as modified gravity specifically on the growth source term,
we render the physics appropriately. This was the mo-
1dlnH2
g′′ + 5+ a−1g′ tivation behind the gravitational growth index formal-
(cid:20) 2 dlna (cid:21) ism of [2], which we follow, calling the ansatz MMG. An
1dlnH2 3 alternate approach is to define wholly separate growth
+ 3+ − Ω (a) a−2g =0, (1)
(cid:20) 2 dlna 2 M (cid:21) variables (see, e.g., [8, 9]).
The gravitational growth index serves as a proxy for
whereaprimedenotesderivativewithrespecttothescale thefullmodifiedgravitytheory. Thelineargrowthfactor
factor a, H = a˙/a is the Hubble parameter, and ΩM(a) is approximated by
isthedimensionlessmatterdensity. Sincetheglobalcos-
mology parameters H and ΩM(a) are, essentially, the g(a)=e 0adlna[ΩM(a)γ−1], (7)
expansion history, we see that the cosmic expansion de- R
termines the structure growth.
where γ is the growth index. This was shown to be ac-
To make the relation between growth probes and ex-
curate to 0.2% compared to the exact solution within
pansion probes such as the luminosity distance-redshift
general relativity for a wide variety of physical dark en-
relationship r (z) even more explicit, we can start from
l ergy equation of state ratios. We verify explicitly that
the growth equation as
for dynamical dark energy where the equation of state
δ¨+2Hδ˙−4πρδ =0, (2) ratio is parametrized as w(a)=w0+wa(1−a), the for-
mulas for γ(w) given in [2] recover g(a) to better than
where a dot denotes time derivative, and use 0.3% when w0+wa < −0.1. That is, the growth index
parametrization of linear growth is extremely robust as
r =a−1 dt/a=a−1 da/(a2H) (3) long as the early matter dominated epoch is not upset.
l
Z Z The gravitational growth index formalism has also
beentestedandfoundaccurateto0.2%forasinglemod-
(for a flat universe to keep the notation simple) to write
ified gravity scenario [2], DGP [10, 11] braneworld grav-
d2δ dδ ity, giving γ =0.68. We conjecture that it may work for
dr2(a−2−Hrl)2− dr (Ha−2+a¨/a)−4πρδ =0. (4) modifiedgravitytheorieswith monotonic,smooth(Hub-
l l ble timescale) evolution in the source term, so long as
Thusingeneralrelativitythegrowthclearlycontainsthe the matterdominatedepochis notdisrupted. Forexam-
same cosmologicalinformation as the distance relation. ple,inDGPthesourcetermreceivesasmoothcorrection
As discussed in [2], we can alter the growth equation 1−(1/3)(1−Ω2 (a))/(1+Ω2 (a)) [11–14]. Preliminary
M M
in modified gravity by changing the matter source term resultsinscalar-tensortheoryalsoindicatesuccessfulap-
(proportional to Ω (a) in Eq. (1) or ρδ in Eq. (2)) or proximation [15]. Future work includes testing this for
M
adding a new source through a nonzero right hand side. other specific models; here we use the growth index as
Greenfunctionsolutionsforsuchmodificationsaregiven an indicator of possible effects of modified gravity, with
in [2]. The matter source term can be written as Qδ, the advantage of knowing at least it is robust for many
and Q = ∇2Φ/δ arises from the equivalent of the Pois- cases beyond ΛCDM.
son equation relating the metric potential Φ to the mat-
ter perturbation δ. One possibility is to make a phe-
nomenological modification to the Poisson equation and III. FIDUCIAL MODEL AND COSMOLOGICAL
investigate its effects on structure growth; this has been DATA
investigatedby [3–6] and we revisitit in §V. An intrigu-
ingapproachofgeneralquadraturerelationsbetweenthe To assess the leverage of cosmological observations
matter perturbations and metric potentials is discussed to reveal dark physics vs. physical darkness, we si-
by [7]. multaneously fit nine cosmological parameters: A, the
3
normalization of the primordial power spectrum at is [20]
k = 0.05hMpc−1; physical matter and baryon den-
fid hγ2 i
nsietuietsrinΩoMmh2assaensdmΩνB,hm2a,ttseprecetnraerlgiynddeexnsnit,ystuomdayofretlhae- Ciκj(ℓ)=Piκj(ℓ)+δij n¯init , (10)
tive to critical ΩM, and parameters describing the ef- where hγ2 i1/2 is the rms intrinsic shear in each compo-
fective dark energy equation of state w and w , where int
0 a nent, taken to be 0.22, and n¯ is the average number of
i
w(a) = w + (1 − a)w . The mass power spectrum
0 a galaxiesintheithredshiftbinpersteradian. Thecosmo-
∆2(k,a)≡k3P(k,a)/(2π2) is written as
logicalconstraintscanthenbecomputedfromtheFisher
matrix
∆2(k,a)= 254ΩA2M (cid:18)kkfid(cid:19)n−1(cid:18)Hk0(cid:19)4g2(a)T2(k)Tnl(k,a) FiWj L =Xℓ ∂∂Cpi Cov−1 ∂∂pCj, (11)
(8) where p are the cosmological parameters and Cov−1 is
where T(k) is the transfer function, and T (k,a) is the i
nl theinverseofthecovariancematrixbetweentheobserved
prescription for the nonlinear power. Modified gravity
power spectra whose elements are given by
enters through the ninth parameter, the gravitational
growth index γ in the linear growth function g(a) from Cov Cκ(ℓ),Cκ(ℓ′) = δℓℓ′ × (12)
Eq. (7), with the growth index γ = 0.55 for the fiducial ij kl (2ℓ+1)f ∆ℓ
(cid:2) (cid:3) sky
cosmologycorrespondingto aflatuniversewith Einstein
Cκ(ℓ)Cκ(ℓ)+Cκ(ℓ)Cκ (ℓ) .
gravity and a cosmological constant. The other fiducial ik jl il jk
(cid:2) (cid:3)
values adopted correspond roughly to the current con- ThefiducialWLsurveyassumes1000squaredegreeswith
cordance cosmology, with Ω = 0.3, w = −1, w = 0, tomographicmeasurementsin10uniformlywideredshift
M 0 a
Ω h2 = 0.023, Ω h2 = 0.14, n = 0.97, m = 0.2 eV bins extending out to z =3. The effective source galaxy
B M ν
(one massive species), and A=2×10−9 (corresponding density is 100 per square arcminute.
to σ ≃ 0.9). While the exact values of some of these WewillalsosometimesconsideraSouthPoleTelescope
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parameters,especially σ , are still a subject of much de- (SPT [21]) type cluster survey with sky coverageof5000
8
bate, we do not expect that different values allowed by deg2 andatotalofabout25000clusters(fortheassumed
current data will change any of our conclusions on the cosmology with σ = 0.9), giving a number-redshift test
8
detectability of non-standard growth. involving the geometric volume and the number density
The linear power spectrum uses the fitting formulae fromgrowthofstructure. For simplicity, we neither con-
of [16]. We always use the linear growth function from sider the additional information provided by masses of
Eq. (7) and account for its dependence on cosmological the clusters, nor degradation of constraints due to the
parameters Ω , w , w and γ when taking the deriva- imperfectlycalibratedmass-observablerelation(herethe
M 0 a
tives for the Fisher matrix. To complete the calcula- observableis the Sunyaev-Zel’dovichflux). Ourprevious
tionofthefullnonlinearpowerspectrumweusethehalo tests have shown that these two effects roughly cancel
model fitting formulae of [17]. out in the final cosmological constraints [22]. Then the
For the cosmological probes, we assume future weak cluster Fisher matrix is
lensing (WL) and Type Ia supernova (SN) data as pro- 1 ∂N(z )∂N(z )
vided by the SNAP experiment [18] as well as cosmic Ficjlus = N(z ) ∂p k ∂p k (13)
microwavebackgroundanisotropy (CMB) data provided Xk k i j
by the upcoming Planck satellite [19]. where N(z ) is number of clusters in kth redshift bin.
k
In this work, for weak lensing we only consider the The SN surveyprovides a luminosity distance-redshift
two point correlation function. The weak lensing shear test,with2800SNedistributedinredshiftouttoz =1.7
power spectrum measures cosmology through both the asgivenby[18],andcombinedwith300localsupernovae
mass power spectrum and distance factors, uniformly distributed in the z = 0.03−0.08 range. We
add systematic errors in quadrature with intrinsic ran-
∞
dz ℓ
Pκ(ℓ)= W (z)W (z)∆2 ,z , dom Gaussian errors of 0.15 mag per SNe. The system-
ij Z r(z)2H(z) i j (cid:18)r(z) (cid:19) aticerrorscreateaneffectiveerrorfloorof0.02(1+z )/2.7
0 i
(9) mag per bin of ∆z =0.1 centered at redshift z .
i
where r(z) is the comoving distance and H(z) is the For the CMB we use the full Fisher matrix predicted
Hubble parameter. The weights Wi are given by for the Planck experiment with polarization informa-
Wi(χ)∞= (3/2)ΩMH02fi(χ)(1 + z) where fi(χ) = tion (W. Hu, private communication). Note that most,
r(χ) χ dχsni(χs)r(χs − χ)/r(χs), χ is the radial co- though not all, information about dark energy is cap-
ordinRate and n is the comoving density of galaxies if χ tured in the distance to the last scattering surface from
i s
fallsinthedistancerangeboundedbytheithredshiftbin theacousticpeaksofthepowerspectrum(e.g.[23]). The
and zero otherwise. We employ the redshift distribution effective precision of the angular diameter distance to
of galaxies of the form n(z)∝ z2exp(−z/z ) that peaks z =1089 from Planck is 0.4% with temperature and po-
0
at 2z =1.0. The observed convergencepower spectrum larization information [24].
0
4
IV. FITTING GRAVITY
The combinationof probes of the expansionhistoryof 0.8
the universe and the growth history of large scale struc- 0.6 allows for
ture tests the nature of the acceleration physics. The modified gravity
expansion history is described by the effective equation 0.4
of state parameters w and w , and the deviation of the
0 a
0.2
growth history from that given by Einstein gravity un-
der that expansion is measured by the growth index γ. wa 0
Almost all cosmological analyses to date, however, have
assumed Einstein gravity or worked within a specific al- -0.2
ternate theory of gravity,rather than fitting for gravity.
-0.4
Ignoringthe possibility ofmodified gravitycreatesthe
neglects modified
risk of the biasing our cosmological conclusions. This -0.6
gravity having ∆γ=0.1
holds not only for “gravitational”parameters but all in-
-0.8
formation. Neglectingpossiblemodificationisequivalent -1.2 -1.15 -1.1 -1.05 -1 -0.95 -0.9 -0.85 -0.8
w
to fixing the gravitationalgrowthindex γ to its Einstein
0
value(e.g.γ =0.55forgeneralrelativityandacosmolog-
icalconstantmodel); howeverthiswillbiasthe otherpa-
FIG. 1: 68% and 95% CL constraints on the expansion his-
rameters due to their covariances with γ (this “gravity’s tory equation of state parameters w0 and wa, marginalized
bias” was illustrated for the linear growth factor alone, overallotherparameters. Thetwoblue(filled)contoursgive
rather than the weak lensing shear power spectrum, in constraintsfromcombiningweaklensing,supernovae,cluster,
Fig. 5 of [25]). and CMB data, while simultaneously fitting for beyond Ein-
Suppose the true value of the growth index differs by stein gravity. The black dot shows the fiducial model. The
∆γ fromits assumedgeneralrelativityvalue. Thisprop- two red (empty) contours show the biased constraints if the
agates through into the weak lensing shear cross power modified gravity growth rate, here with ∆γ =0.1, is ignored
spectrumCκ(ℓ)atmultipoleℓforthepairofredshiftbins (i.e. fixing γ to thegeneral relativity value).
α
α≡{i,j},changing it from the assumed generalrelativ-
ity value of C¯κ(ℓ). Using the Fisher matrix formalism,
α
the bias on any of the P cosmologicalparameters is
must attempt to fit for beyond Einstein deviations.
Including the gravitational growth index as an addi-
δp = F˜−1 Cκ(ℓ)−C¯κ(ℓ) tional parameter, and marginalizing over it to estimate
i ij α α
Xℓ (cid:2) (cid:3) the effective equation of state parameters describing the
∂C¯κ(ℓ) expansion,removesthe biasbutnecessarilydegradespa-
×Cov−1 C¯κ(ℓ),C¯κ(ℓ) β (14) rameter determination. (This would of course become
α β ∂p
(cid:2) (cid:3) j more severe if we required more growth variables than
≈ (∆γ)F˜−1F (15) just γ.)
ij jg
Thedegradationonthe weaklensingshearconstraints
where the last line follows if the finite difference is re- from marginalizing over γ causes a factor ∼ 2 increase
placed by a derivative. Here F˜ is the (P −1)×(P −1) in the contour area. While adding CMB or supernovae
Fisher matrix that specifically does not include the data does not directly constrain the growth, they prove
growthindexγ,F isthefullP×P Fishermatrix,summa- valuable in breaking degeneracies between parameters.
tionsoverj andtheredshiftbinindicesα,β areimplied, AddingCMBtoWLimprovesconstraintsby30-35%,but
and g is the index corresponding to the γ parameter in still suffers the factor of 2 weakeningrelative to fixing γ.
the matrix F. The inclusion of SN is potent in reducing uncertainties.
Biasinthecosmologyfromneglectingthepossibilityof WL+SN+CMB allows for fitting modified gravity, im-
modified gravity can be significant. The red (open) con- proving parameter estimation by 5-7 times and the area
toursinFig.1demonstratethatassumingEinsteingrav- constraint by 40 times over WL alone. Conversely, it
ity in a universe with γ actually higher by 0.1 can shift only dilutes estimation of w0 by 23%, wa by 14%, and
the expansion characteristics (effective equation of state the contour area by 35% relative to the “fixed to Ein-
parameters) by ∼ 2σ. Recall that the DGP braneworld stein” case. This seems a fairly modest price to pay for
model has ∆γ = 0.13 with respect to the standard cos- extending the physics reach to beyond Einstein gravity.
mologicalconstantcase. Notetoothatwithinthissimple The marginalized uncertainties on the key parameters
treatment of modified gravity a shift in γ moves the w - describing the nature of the effective dark physics are
0
w contour along the degeneracy direction, else the bias shown in Table I. Since SN and CMB do not probe
a
wouldbe evenlarger. Given the price ofclosing oureyes growth, we start with WL measurements and add other
to the issue of possible gravitational modifications, we probes in sequence.
5
TABLE I: Fiducial constraints on the gravitational growth
index γ (last column), as well as the two expansion history
parametersw0andwa. Startingwiththeweaklensingsurvey,
we then add the supernova, CMB, and cluster information
consecutively. Note that both WL and cluster surveys are
partially sensitive to nonlinear physics; constraints that only
rely on the linear regime are discussed in §V.
Probe σ(w0) σ(wa) σ(γ)
Weak lensing 0.33 1.16 0.23
+ SNeIa 0.06 0.28 0.10
+ Planck 0.06 0.21 0.044
+ Clusters 0.05 0.16 0.037
Whilemarginalizingoverthegrowthindexallowsusto
account for modified gravity rather than ignoring it, we
also wantto measurethe modificationitself. That is, we
want to extract information quantifying the deviation,
to guide us towardany “darkphysics”, not just say that
there is some inconsistency with general relativity. Fig-
ure 2 show the constraints on the growthindex from the
cosmological probes, in the γ-Ω plane (marginalizing
M FIG.2: 68%CLconstraintsonthegravitationalgrowthindex
over the equation of state and other parameters). γ and the matter density ΩM, marginalizing over the effec-
Weseeanenormousdifferenceintheleverageonmod- tive equation of state (and all other parameters). To fit for
ified gravity as we combine probes, again due to the beyond Einstein gravity as well as the expansion history re-
breakingofdegeneracies. Thedeterminationofγ reaches quiresacombinationofprobes. Othereffectsongrowth,such
0.044 for the combination WL+SN+CMB, representing asneutrinomasses, should betakenintoaccount aswell; the
inner contour of each pair shows the effect of holding this
8% precision with respect to the Einstein value. If we
fixed,most severe for weak lensing in isolation.
do not include supernova data, then the uncertainty in-
creases by a factor 2 (5 with only WL), and furthermore
the overall contour area in the γ-Ω plane increases by
M
11 times (44 for WL only). So complementaritybetween
per ∆z = 0.1 as a function of redshift for our fiducial
WL and SN is quite important for answering the ques-
survey (motivated by the South Pole Telescope), for two
tion of whether we are facing dark physics or physical
values of the growth index (γ = 0.55 and 0.65). Since
darkness.
we normalized the power spectrum at high redshift, the
Complementarityofprobesisimportantforseparation
number density of clusters is independent of the growth
of different physical effects on growth. While inclusion
index at high redshift. As z → 0, the volume element,
of neutrino mass, which can suppress growth (as does
which is independent of the growth index by definition,
increasing γ), broadens the uncertainty area of WL by
dominatesoverthenumberdensityandmakesthecounts
74%, adding other methods immunizes against such a
go to zero for either model. Therefore the biggest differ-
“theory systematic”, with the effect on WL+SN+CMB
ence between the two models is around the peak of the
limited to 10%.
redshift distribution at z ∼ 0.6, as illustrated in the left
panelofFig.3. Alsoshownarenumbercountpredictions
when the equation of state and neutrino mass have been
V. STRUCTURE IN THE NONLINEAR
perturbed from their fiducial values by 0.05 and 0.3 eV
REGIME
respectively. While the strong degeneracy of the growth
index with other cosmological parameters is apparent,
A. Beyond the linear regime it is worth noting that clusters can in principle provide
several other observables to break this degeneracy, chief
The gravitational growth index modifies the linear among them being the mass information. However, in
growth factor, which then propagates into the full non- the nonlinear regime we are at increased risk of the sim-
linear growthof structure. Here we examine some issues pleMMGmodelbreakingdown,possiblyrequiringmodel
relatedtothe nonlinearregime. Inparticular,clustersof dependent simulations for a specific theory of gravity.
galaxies involve nonlinear growth and we might wonder Weak lensing also involves scales in the quasi-linear
whether special sensitivity to modified gravity arises in and nonlinear regimes. The sensitivity of a weak lens-
this regime. TheleftpanelofFig.3showsclustercounts ing survey is shown in the right panel of Fig. 3. For
6
4000
44
γ = 0.55
γ=0.55 -4 γ = 0.65
10
3000 ∆γ=0.1 ) 33
∆dz) z2000 ∆∆wm=ν=00.0.35 eV κπ / (2Pii 22
N/ 1) 10-5
d +
( (l
l
1000
{i,i}=11
0 10-6
0 0.5 1 1.5 2 100 1000 10000
z
l
FIG. 3: Left panel: Cluster counts per ∆z =0.1 as a function of redshift for our fiducial survey, for two values of the growth
index(γ =0.55 and0.65). Alsoshownarecaseswhentheequationofstateandneutrinomasshavebeenperturbedfrom their
fiducialvaluesby0.05 and 0.3 eVrespectively. Rightpanel: Auto-correlation powerspectra of4-binweak lensing tomography
for the two values of thegrowth index,with statistical errors shown for the γ =0.55 model.
simplicity,weconsiderthe samefiducialsurveyasbefore however, very directly depends on the metric potentials
but with 4-bin (instead of 10-bin) tomography, with di- which themselves are likely to be altered in the modi-
visions z = [0,0.5],[0.5,1],[1,1.5],[1.5,3]. We show the fied gravity theory). By including nonlinear scales we
four auto-correlation power spectra with corresponding increasethestatisticaldiscriminationpowerwithrespect
statistical errors for γ =0.55, and the same but without togrowthbutmaybiasthe results asaresultofemploy-
errors for γ = 0.65. The raw signal-to-noise for distin- ing an improper nonlinear prescription for the modified
guishingthetwovaluesofthegrowthindexincreaseswith gravity growth. We now consider the trade off between
redshift,givinganadvantagetoadeepersurvey. Thean- these two trends.
gularscaleatwhichthetwovaluesarebestdistinguished Since weak lensing probes a range of scales, we can
decreaseswith redshift, so the multipole increases,going consider limiting the weak lensing information to scales
fromℓ∼500forthe firstbintoℓ∼5000forthe lastbin, k ≤ k and investigate how the information on the
cut
making advantageous a higher resolution survey. growth index is degraded. This “k-cutting” can remove
Wehavealsocheckedthatthesensitivitytothegrowth thosephysicalscaleswherewelackdependableestimates
index increases as the density of resolvedsource galaxies ofmodifiedgravityeffects. Themoststraightforwardway
ng and their mean distance, parametrized by the mean to implement the cutting is to use weak lensing power
of the redshift distribution, are increased. All these fac- spectrum nulling tomography (see §5 of [26]) where, for
tors indicate that a space-basedweak lensing survey has a given multipole ℓ, only the lens planes with distances
certain definite advantages for testing beyond Einstein r(z) > ℓ/k are allowed to contribute information.
cut
gravity. At this time, however, we have not further pur- Higher ℓ then contribute increasingly less information,
suedthesesensitivitytestsnortriedtodeviseanoptimal and exhaust all information below k . For a fixed k ,
cut cut
strategy to determine the growth index. The reason is we compute the ratio of the error in the growth index
that by far the dominant uncertainty is a theory uncer- relative to the error with k = ∞. The results are
cut
tainty — our ability to predict the nonlinear clustering shownwiththe solidline inFig.4. Restricting the infor-
statistics and associated observables — in a given modi- mation to purely linear scales (k <0.2hMpc−1) leads
cut
fied gravity theory. to degradations in the marginalized error in γ of more
than a factor of 10 relative to the full nonlinear fidu-
cialcase. However,when the quasi-linearscalesare used
B. Uncertainty and bias (k < 1hMpc−1), the degradation is kept to a factor
cut
of a few, and the resulting constraints on the growth in-
So far we have included nonlinear scales when obtain- dex are still interesting. Therefore, even if the nonlinear
ing constraints from measurements involving the growth prediction out to k ≈ 10hMpc−1 (ℓ ≈ 10000) in modi-
of structure. Indeed, none of the cosmological growth fied gravity theories is unfeasible, efforts to understand
probes is solely a linear theory probe, with the possi- predictions on quasi-linear scales are well worthwhile.
bleexceptionoftheIntegratedSachs-Wolfeeffect(which, Since we desire to retain values of k beyond the lin-
7
30 k 3
error bias δ(∆2(k,z))=c(cid:18)k∗(z)(cid:19) (17)
d
σγ, fi 10 c=0.05 where c is a dimensionless constant which represents the
) / fractional error in the 1-halo term. We then use the
γn Fisher formalism to propagate this offset into biases on
s i 3 cosmologicalparameters. (The expansionparametersw0
bia c=0.01 and wa are not appreciably affected by nulling, so we
r focus on γ.)
o
or 1 The less of the nonlinear regime we use, the less ef-
r fect the misestimated 1-halo,or c, term has. The results
r
e
( are shown in Figure 4 where the dashed curves give the
biasinthegrowthindexdividedbythefiducialstatistical
0.3 error (i.e. the statistical error for k = ∞). It is clear
0.1 1 10 100 cut
k (h / Mpc) that,asweperformincreasinglymoredrasticnulling,de-
cut creasing the value of k , the bias/error ratio decreases
cut
significantly. For c = 0.05 (0.01), cutting information
FIG. 4: The solid (black) line shows the degradation in con- beyond k = 1 (6) hMpc−1 gives bias in γ that is be-
straintson thegrowth index γ from aSNAP-typeweak lens- cut
low the statistical error, while increasing the error by a
ing survey (plus SN+CMB) as small-scale weak lensing in-
factor of three (25%). One might expect the bias model
formation is increasingly dropped. For each valueof kcut, we
drop all information from k > kcut following [26] and com- of Eq. (17) to be cut off in the stable clustering regime,
putetheratiooftheconstraintonγ relativetothecasewhen causing bias in γ to level off or decline at k greater than
all information is used, that is, when kcut =∞. The dashed afewtimesknl,e.g.k >1hMpc−1. Thiswouldallowuse
curves (dark (red) and light (orange)) lines show the bias in of more of the nonlinear regime.
thegrowthindexdividedbythekcut =∞statisticalerrorfor
the bias model given in Eq. (17) with c= 0.05 or 0.01. The
optimum choice of kcut, with the least risk, occurs near the C. Scale dependent growth
intersection of theerror and bias curves.
MMG, withthe gravitationalgrowthindex formalism,
has been adopted as the simplest reasonable method of
earregime,fortheir leverage,it is instructiveto make at accounting for the effects of beyond Einstein gravity on
least an approximate estimate of the bias that might be cosmological probes involving the growth of structure.
induced by keeping such data. Within the halo model, One part of its simplicity is that γ acts in a scale inde-
the linear regime corresponds to the 2-halo term of clus- pendentfashion;thisshouldreproduceglobaleffectssuch
tering between dark matter halos, while the nonlinear as time varying gravitational coupling. However many
regimeinvolvesthe1-halotermoftheprofileandconcen- modifications to gravity will introduce scale dependence
trationwithinahalo. Fork≈0.2−1hMpc−1,the1-halo in the growth. For example, in the DGP braneworld we
contributionmaybeapproximatedbyawhitenoiseterm mightexpectchangestothegrowthequationbeyondthe
in the power spectrum [27], due to Poisson fluctuations varying coupling on both small and large scales, due to
in the number of halos. In other words, the Vainshtein radius (where the scalar degrees of free-
dombecomerelevant)andrelativisticeffectsrespectively
(R. Scoccimarro, private communication).
3
k Scale dependence on small scales affects the nonlin-
∆2(k,z)≃∆2 (k,z)+ . (16)
lin (cid:18)k∗(z)(cid:19) ear regime, and this is just what we looked at with the
bias calculations above. On large scales, we may treat
We find a good fit to the full power spectrum in the the modification of the source term mode by mode in
quasi-linear regime for k∗(z) = (5/3)knl(z), where the the linear growth equation. From a harmonic analysis
nonlinear scale k (z) is defined via ∆2(k (z),z)=1. of metric perturbations, [28, 29] found long wavelength
nl nl
Since neither the presence of a 1-halo/2-halo split nor corrections to the Poisson equation; recall from §II this
the Poisson fluctuations in number should rely on the altersthefactorQinthesourcetermQδ. SuchJLWcor-
specific gravity theory, we adopt this approach and con- rectionswouldmultiply Ω (a)by Q∼1+αe−(k/kJLW)2.
M
sider what happens if we improperly estimate the 1-halo We can then either solve the modified differential equa-
effects. This wouldshift the white noiseterm to a differ- tion for the linear growth, or we can retain the growth
ent value (equivalent to changing the scale at which the index approach,but similarly multiply Ω (a) in Eq. (7)
M
2-haloand1-haloterms are comparable),i.e.biasing the by that same factor (this can also be viewed as making
power spectrum by the growth index γ a function of k). The characteristic
8
scale is the horizon scale, k ∼H, and the amplitude morestatisticalpower,beingmoresuitabletonear-future
JLW
α ∼ O(1). We find that effects on the growth are then data, while we think the loss in generality is minimal.
negligible for k >10−3hMpc−1. We also emphasize the advantage in retaining the maxi-
Thus horizonsizescale dependence, asshouldholdfor malphysicsinformationbytreatingtheexpansioneffects
theDGPcaseandscalar-tensortheoriesaswell,willhave on growththrough the effective equation of state, giving
essentially no effect on the weak lensing probe, as the clear separation from deviations in the gravity theory.
WL power spectrum errors on the near-horizon scales HowaccuratelysuchaprogramcanrevealbeyondEin-
are very large due to sample variance; see e.g. Fig. (10) steingravity–darkphysicsvs.physicaldarkness–isthe
in [30]. Weak lensing can be reasonably treated by the main topic of this paper. We have shown that measure-
scaleindependentγ factorover10−3 <∼k/hMpc−1 <∼10. ments from weak gravitational lensing, Type Ia super-
Such scale dependence may however need to be taken novae, and the CMB combined can measure the growth
into accountfor attempts to use the ISW effect to probe index to about 8%, or to ±0.04 around its ΛCDM value
cosmology, as mentioned previously. Only if the gravi- (and galaxy cluster data could potentially reduce this
tation theory possesses a substantially smaller scale, ap- evenfurther). Atthesametime,theconstraintsonother
proachingthenonlinearscale,asinthephenomenological cosmological parameters are not appreciably degraded,
alterationsofthePoissonequationin[3–6],areweforced essentially because the surveys probe a range of scales
to more elaborate parameterizations than γ. and thus their complementarity breaks the degeneracies
Thespecificoptimumofthetradeoffbetweenleverage between parameters.
on cosmological parameter constraints from the added One particular concern in the program of distinguish-
information of smaller scales and increasing risk of bias inggeneralrelativityfrommodifiedgravityisthenonlin-
from gaps in our understanding will depend on the spe- ear density regime of structure formation. Even for the
cific gravity theory. Since this is what we are trying to limitednumber ofwell-definedmodifiedgravitytheories,
obtain insight into, a rational, model independent ap- details of nonlinear clustering are currently unknown.
proach might be to carry out both a wide area survey While in principle the nonlinear structure formation is
to squeeze the most statistical power out of the rela- calculable from N-body simulations of modified gravity,
tively weakly discriminating low ℓ (linear) regime and creating these simulations in practice is extremely diffi-
a deep, high resolution and high number density survey cultexceptforsomeverysimplecases. Thestructureand
to probe the richer high ℓ (quasi- and nonlinear)regime. evolutionofgalaxyclusters,whicharenonlinearobjects,
This indicates possible strong complementarity between isfairlystronglydependentonthenonlinearphysics,and
a ground-based weak lensing survey such as LSST [31] is consequently problematic.
and a space-based survey like SNAP. Weak lensing, on the other hand, probes a range of
scales,and we studied how our results behave if we drop
small-scale (that is, nonlinear) information. Using the
VI. DISCUSSION nulling tomography approachand a reasonably well mo-
tivatedtoymodelforbiasduetouncertaintyinnonlinear
This article presents an approachfor simultaneous fit- structure,wefoundthatcuttingoutthesmallscaleinfor-
ting of the cosmic expansion and the theory of gravity. mation (k >∼ 1hMpc−1) can lead to significant decrease
We have advocated a “minimalist” strategy of distin- in the resulting bias in the growth index, at the expense
guishing modified gravity from dark energy, which con- of increasing the statistical error in it by a factor of a
sists in measuring a single parameter, the growth index few.
γ. In addition to reproducing the linear growthfunction Other cosmological observables exist with sensitivity
for essentially all standard gravity, dark energy models to the growth of fluctuations, and hence can be used to
(parametrized with w and w ), the growth index also constrain MMG, but we have not discussed them in any
0 a
fits the linear growth of a single known modified grav- detail. Forexample,thebispectrumofweakgravitational
ity theory, the DGP braneworld scenario. Therefore, it lensing is a potentially powerful probe [32]; however, it
is reasonable to expect that the growth index can be hasprovedtobeatoughtasktocalibratethebispectrum
used to measure deviationsfrom standardgravity: given even in standard general relativity. The same holds for
themeasurementsofthebackgroundexpansionrate(pa- Lyman-alpha forest observations. The Integrated Sachs-
rameterized, say, by Ω , w and w ), standard gravity Wolfe effect is a potentially strong discriminant of mod-
M 0 a
predicts the value of γ, and a statistically significant de- ified gravity models (see in particular recent predictions
viationfrom this value can in principle be interpretedas oftheISWinDGPmodels[14]);however,themetricpo-
evidence for – and characterizationof – beyond Einstein tentials are particularly sensitive to the structure of the
gravity. modified gravity theory. Neither we nor anyone else has
This is a first step, hence our emphasis in calling it yetsucceededinfinding a genericparametrizationofthe
Minimal Modified Gravity. One could examine more deviations from general relativity for the ISW effect.
complex schemes but these may be more model depen- Ourworkoutlinesafirststepintreatingmodifiedgrav-
dent or employ more parameters and are therefore likely ity models. At the time of this writing, there is hardly
togiveweakerresults. TheMMGapproachthereforehas a single well-defined modified gravity theory that does
9
not look like standard general relativity in terms of ob- By continuing forward with advances in measurements,
servables, but is not already ruled out or disfavored by theory, and computation we can lift the darkness on the
data. Considerable effort is underway to construct such new physics.
theories (e.g. [33–47]; for a review see [48–50]) and test
them experimentally.
Itishearteningthatthestrongcomplementarityincos-
Acknowledgments
mologicalprobessuchasthecombinationofweaklensing,
supernovae, and the CMB does provide important infor-
mationonthequestionofdarkphysicsvs.physicaldark- This work has been supported in part by the Di-
ness. Such data from next generation experiments, here rector, Office of Science, Department of Energy under
calculatedspecificallyforSNAPandPlanck,doesfurnish grantDE-AC02-05CH11231andbyNSFAstronomyand
arealtestnotjustofindividualmodelsbutofthephysics Astrophysics Postdoctoral Fellowship under Grant No.
framework beyond Einstein. We will be able to measure 0401066. We gratefully acknowledge discussions with
the effective equation of state describing the cosmic ex- Wayne Hu (who supplied the full CMB Fisher matrix)
pansion and simultaneously reveal the theory of gravity. and Martin White.
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