A discriminating probe of gravity at cosmological scales
Pengjie Zhang,1, 2 Michele Liguori,3 Rachel Bean,4 and Scott Dodelson5, 6
1Shanghai Astronomical Observatory, Chinese Academy of Science, 80 Nandan Road, Shanghai, China, 200030
2Joint Institute for Galaxy and Cosmology (JOINGC) of SHAO and USTC
3Department of Applied Mathematics and Theoretical Physics,
Centre for Mathematical Sciences, University of Cambridge,
Wilberforce Road, Cambridge, CB3 0WA, United Kingdom
4Department of Astronomy, Cornell University, Ithaca, NY 14853
5Center for Particle Astrophysics, Fermi National Accelerator Laboratory, Batavia, IL 60510-0500
6Department of Astronomy & Astrophysics, The University of Chicago, Chicago, IL 60637-1433∗
The standard cosmological model is based on general relativity and includes dark matter and
dark energy. An important prediction of this model is a fixed relationship between the gravitational
potentials responsible for gravitational lensing and the matter overdensity. Alternative theories of
gravity often make different predictions for this relationship. We propose a set of measurements
which can test the lensing/matter relationship, thereby distinguishing between dark energy/matter
models and models in which gravity differs from general relativity. Planned optical, infrared and
radio galaxy and lensing surveys will be able to measure EG, an observational quantity whose
expectation value is equal to the ratio of the Laplacian of the Newtonian potentials to the peculiar
velocity divergence, to percent accuracy. We show that this will easily separate alternatives such as
ΛCDM, DGP, TeVeS and f(R) gravity.
PACS numbers:
Introduction.— Predictions based on general relativ-
ity plus the Standard Model of particle physics are at
odds with a variety of independent astronomical obser-
vations on galactic and cosmological scales. This failure
has led to modifications in particle physics. By introduc-
ing dark matter and dark energy, cosmologists have been
able to account for a wide range of observations, from
the overall expansion of the universe to the large scale
structure of the early and late universe [1]. Alternatively,
attempts have been made to modify general relativity at
galactic [2] or cosmological scales [3, 4]. A fundamental
question then arises: Can the two sets of modifications
be distinguished from one another?
The answer is “No” if only the zero order expansion
of the universe is considered. By allowing the dark
energy equation of state wDE to be a free function,
the expansion history H(z) produced by any modified
gravity can be mimicked exactly. Fortunately, struc-
ture formation in modified gravities in general differs
[5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16] from that in
general relativity. The difference we focus on here is
the relationship between gravitational potentials respon-
sible for gravitational lensing and the matter overden-
sity. Lensing is sensitive to ∇2(φ − ψ) along the line
of sight where φ and ψ are the two potentials in the
perturbed Friedman-Robertson-Walker metric: ds2 =
(1 + 2ψ)dt2 − a2(1 + 2φ)dx2 and a is the scale factor.
In standard general relativity (GR), in the absence of
anisotropic stresses, φ = −ψ, so lensing is sensitive to
∗Electronic address: pjzhang@shao.ac.cn
∇2φ. The Poisson equation algebraically relates ∇2φ to
the fractional overdensity δ, so lensing is essentially de-
termined by δ along the line of sight. This is a prediction
of the standard, GR-based theory that is generally not
obeyed by alternate theories of gravity.
Testing this prediction is non-trivial. Astronomers of-
ten use the galaxy overdensity as a probe of the underly-
ing matter overdensity, but the two are not exactly equal.
Here we propose a test of this prediction which is rela-
tively insensitive to the problem of galaxy bias. The basic
idea is simple:
• Extract the matter overdensity at a given redshift
by measuring the velocity field. Matter conser-
vation relates velocities to the overdensities. The
measurement of the velocity field can be accom-
plished by studying the anisotropy of the galaxy
power spectrum in redshift space.
• Extract the lensing signal at this redshift by cross-
correlating these galaxies and lensing maps recon-
structed from background galaxies.
More quantitatively, the galaxy-velocity cross power
spectrum Pgθ ≡ −〈δg(k)θ(−k)〉 can be inferred from
redshift distortions in a galaxy distribution. Here, θ ≡
∇ · v/H(z) and v is the comoving peculiar velocity. In
the linear regime, matter conservation relates θ to δ by
θ = −δ̇/H = −βδ, where β ≡ d lnD/d lna and D is
the linear density growth factor. So, Pgθ = βPgδ , sat-
isfying the first goal above. Cross correlating the same
galaxies with lensing maps constructed from galaxies at
higher redshifts, P∇2(φ−ψ)g can be measured. The ra-
tio of these two cross-spectra therefore is a direct probe
http://arxiv.org/abs/0704.1932v3
mailto:pjzhang@shao.ac.cn
of ∇2(φ − ψ)/(βδ). It does not depend on galaxy bias
or on the initial matter fluctuations, at least in the lin-
ear regime. Modifications in gravity will in general leave
signatures in either β and/or the Poisson equation.
Galaxy-Velocity Cross-correlation.— A galaxy’s
peculiar motion shifts its apparent radial position from
xz to x
z = xz+vz/H(z) in redshift space, where vz is the
comoving radial peculiar velocity. The coherent velocity
component changes the galaxy number overdensity from
δg to δ
g ≃ δg−∇zvz/H(z). Galaxy random motions mix
different scales and damps the power spectrum on small
scales. The redshift space galaxy power spectrum there-
fore has the general form ([17] and references therein)
P sg (k) =
Pg(k) + 2u
2Pgθ(k) + u
4Pθ(k)
k2u2σ2v
H2(z)
where u = k‖/k is the cosine of the angle of the k vector
with respect to radial direction; Pg, Pgθ , Pθ are the real
space galaxy power spectra of galaxies, galaxy-θ and θ,
respectively; σv is the 1D velocity dispersion; and F (x)
is a smoothing function, normalized to unity at x = 0,
determined by the velocity probability distribution. This
simple formula has passed tests in simulations on scales
where δ <∼ 1 [17]. The derivation of Eq. (1) is quite
general, so it should be applicable even when gravity is
modified.
The distinctive dependence of P sg on u allows for si-
multaneous determination of Pg, Pgθ and Pθ [18]. The
parameters we want to determine are the band powers of
Pgθ(k)
1 defined such that P (k) = Pα if kα ≤ k < kα+1,
where k1 < k2 < · · · < kα < · · · . We denote P
as the band power of Pgθ. For a ki in each k bin,
we have a measurement of P sg , which we denote as Pi.
The unbiased minimum variance estimator of P
1 Distance D and H are required to translate the observed galaxy
angular and redshift separation to k. In general, errors in D and
H measurements cause both horizontal and vertical shifts in the
EG plot. Both D and H will be measured by methods like type
Ia supernovae and baryon acoustic oscillations with 1% accuracy,
much smaller than the k bin size adopted, so the horoziontal
shift is negligible. Errors in D show up in both Pgθ and the
Cκg → P∇2(ψ−φ)g inversion through l = kD and thus largely
cancel in evaluating EG. Errors in H(z) only show up in Pgθ
measurement and thus cause a net shift in the value of EG. For
1% error in H, the fractional error in EG is ≤ (neff + 3)1% ≤
3%. Here, neff is the effective power index of the corresponding
power spectra. For the fiducial ΛCDM cosmology, it is negative
in relevant k range. Thus errors induced by uncertainties in
D(z) and H(z) measurement will be sub-dominant, except for
SKA, which requires better control over systematic errors in D
and H measurement. For simplicity, we neglect this potential
error source. Measuring Pgθ also requires to marginalize over
σv. However, in the linear regime k <∼ 0.2h/Mpc, k
2σ2v/H
2 ≪ 1
and F (k2u2σ2v/H
2) ≃ 1, for typical value σv ∼ 300 km/s. Thus
the exact value of σv is not required for our analysis. Without
loss of generality, we adopt σv = 300 km/s.
WiPi, where Wi =
(λ1 + λ2u
i + λ3u
i ). Here,
Fi ≡ F (kuiσv/H), σ
i is the variance of Pi and the three
Lagrange multipliers λα (α = 1, 2, 3) is determined by
λ = (0,
, 0) ·A−1 ; Amn =
2(m+n−2)
. (2)
Galaxy-galaxy lensing.— Weak lensing is sensitive
to the convergence κ, the projected gravitational poten-
tial along the line of sight:
∇2(ψ − φ)W (χ, χs)dχ . (3)
Here, W is the lensing kernel. For a flat universe, χ, χs
are the comoving angular diameter distance to the lens
and source, respectively. Eq. 3 is a pure geometric result
and can be applied to any modified gravity models where
photons follow null geodesics.
A standard method to recover the lens redshift infor-
mation is by the lensing-galaxy cross correlation. For
galaxies in the redshift range [z1, z2], the resulting cross
correlation power spectrum under the Limber’s approxi-
mation is
Cκg(l) =
ng(χ)dχ
W (χ, χs)ng(χ)P∇2(ψ−φ)g(
, z)χ−2dχ
W (χ̄, χs)
∫ l/χ1
P∇2(ψ−φ)g(k, z̄)dk
fα(l)P
Here, χ1,2 are the comoving angular diameter distance to
redshift z1,2 and χ̄ is the mean distance. The band power
α of P∇2(ψ−φ)g is defined at the same k range as P
In practice, we measure the band power Cκg(l,∆l), cen-
tered at l with band width ∆l. The weighting fα(l,∆l)
is defined correspondingly. For each l, only a fraction of
α having fα(l,∆l) 6= 0 contribute.
A discriminating probe of gravity.— With the
above measurements, one can construct an estimator
ÊG =
Cκg(l,∆l)
3H20a
α fα(l,∆l)P
, (5)
whose expectation value is
〈ÊG〉 =
∇2(ψ − φ)
−3H20a
∇2(ψ − φ)
3H20a
≡ EG .
The fractional error on ÊG is
〈∆E2G〉
α〈(δP
α fαP
, (7)
0.4<z<0.6
LAMOST/AS2+LSST
0.85<z<1.15SKA
0.01 0.1
ADEPT+LSST
1.3<z<1.7
0.01 0.1
1.8<z<2.2SKA
FIG. 1: EG as a smoking gun of gravity. Error estimation is
based on ΛCDM and error bars are centered on the ΛCDM
prediction (black solid straight line). We only show those k
modes well in the linear regime. For clarity, we shift the error
bars of LAMOST/AS2+LSST and ADEPT+LSST slightly
rightward. Irregularities in the error-bars are caused by irreg-
ularities in the available discrete k modes of redshift distor-
tion. Dotted lines are the results of a flat DGP model with
Ω0 = 0.2. Dashed lines are for f(R) = −λ1H
0 exp(−R/λ2H
with λ2 = 100. Differences in expansion histories of these
models are of percent level at z < 2 and are not the main
cause of differences in EG. Solid lines with wiggles are for
TeVeS with KB = 0.08, 0.09, 0.1, where the lines with most
significant wiggles have KB = 0.1.
where ∆C2 = [C2κg + (Cκ + C
κ )(Cg + C
g )]/(2l∆lfsky).
Here, Cκ, C
κ , Cg, C
g are the power spectra of weak
lensing convergence, weak lensing shot noise, galaxy and
galaxy shot noise, respectively, and fsky is the fractional
sky coverage. Errors on EG at any two adjacent bins are
correlated, since they always share some same k modes.
However, by requiring lα/χ1 = lα+1/χ2, where l1 < l2 <
· · · < lα < · · · and kα = lα/χ2, EG measurement at each
l bin only involves two k bins and thus only errors in
adjacent bins are correlated.
We choose ongoing/proposed spectroscopic surveys
LAMOST, AS2, ADEPT and SKA as targets of redshift
distortion measurements, and LSST and SKA as targets
of lensing map reconstruction. SKA lensing maps can
be constructed through cosmic magnification utilizing its
unique flux dependence, with S/N comparable to that of
LSST through cosmic shear [19]. Survey specifications
are summarized in TABLE I. The fiducial cosmology
adopted is the ΛCDM cosmology, with the WMAP best
TABLE I: Summary of target surveys.
redshift deg2 Ngal band operation
LAMOSTa z< 0.8 10,000 ∼ 106 optical 2008
AS2b z< 0.8 10,000 ∼ 106 optical ≥ 2009
ADEPTc 1 < z < 2 28,600 ∼ 108 infrared ≥ 2009
SKAd z <∼ 5 22, 000 ∼ 10
9 radio 2020
LSSTe z <∼ 3.5 10,000 ∼ 10
9 optical 2012
ahttp://www.lamost.org/en/
bPrivate communication with Daniel Eisenstein
chttp://www7.nationalacademies.org/ssb/BE Nov 2006 bennett.pdf
dhttp://www.skatelescope.org/
ehttp://www.lsst.org
fit parameters Ω0 = 0.26,ΩΛ = 1 − Ω0, h = 0.72, σ8 =
0.77 and ns = 1. The result is shown in figure 1. In gen-
eral, at k < 0.1h/Mpc, cosmic variance in Cκg and Pgθ
measurements dominates the EG error budget, result-
ing in decreasing error-bars toward larger k. This makes
fsky and the lensing source redshifts the two most rele-
vant survey parameters for EG error estimation. Since
systematic errors in LSST photometric redshifts can be
controlled to better than 1%, errors in EG measurements
of LAMOST/AS2+LSST and ADEPT+LSST caused by
source redshift uncertainties are sub-dominant.
We restrict our discussion to sub-horizon scale pertur-
bations and express equations hereafter in the Fourier
form. Four independent linear equations are required
to solve for four perturbation variables δ, θ, ψ and φ.
The mass-energy conservation provides two: δ̇ +Hθ = 0
and Ḣθ + 2H2θ − k2ψ/a2 = 0. For at least ΛCDM,
quintessence-CDM, DGP and f(R) gravity, the other two
takes the general form
φ = −η(k, a)ψ ,
k2(φ− ψ) = 3H20Ω0a
−1δG̃eff(k, a) . (8)
Here Ω0 is the cosmological matter density in unit of the
critical density ρc ≡ 3H
0/8πG. Refer to [14, 15, 16] for
other ways of parameterizations. MOND has extra scalar
and vector perturbations and does not follow the general
form of Eq. 8 [6, 7].
(1) ΛCDM: η = 1, G̃eff = 1 and EG = Ω0/β. Dynam-
ical dark energy will have large-scale fluctuations [20].
Furthermore, it may also have non-negligible anisotropic
stress and is thus able to mimic modifications in gravity
[21]. But, for models with large sound speed and neg-
ligible anisotropic stress, such as quintessence, these are
negligible at sub-horizon scales and Eq. 8 still holds.
(2) Flat DGP: η = [1 − 1/3βDGP]/[1 + 1/3βDGP],
G̃eff = 1 [9] and EG = Ω0/β, where βDGP = 1−2rcH(1+
Ḣ/3H2) < 0 and rc = H0/(1−Ω0). Ω0 differs from that
of ΛCDM, in order to mimic H(z) of ΛCDM.
(3) f(R) gravity: in the sub-horizon limit, G̃eff =
(1+fR)
−1 [11] and η = 1 [12], with fR ≡ df/dR|B where
B denotes the FRW background. This falls naturally out
of a conformal transformation of the expression for EG
in the Einstein frame into the Jordan frame, noting that
Einstein frame scalar field fluctuations are negligible on
sub-horizon scales [12]. We numerically solve the full per-
turbation equations in the Einstein frame since it is com-
putationally simpler [12] and then conformally transform
to the Jordan frame, which we choose as the physical
frame, evaluating β such that EG = Ω0/(1+fR)β. In the
limit that fR → 0, e.g. for f(R) ∼ λ1H
0 exp(−R/λ2H
[11] with λ1 ≪ λ2, the evolution is observationally equiv-
alent to ΛCDM. For modes that entered the horizon prior
to matter-radiation equality, as we consider here, β, and
therefore EG, is scale invariant for IR modifications to
gravity, with fR > 0.
2 The scale independence of EG
holds in ΛCDM, Quintessence-CDM and DGP. An ob-
served scale-independent deviation in EG from ΛCDM
could signify a special class of modified gravity, as shown
in Fig. 1.
(4) TeVeS/MOND. Besides the gravitational met-
ric, TeVeS [2] contains a scalar and a vector field. These
new fields act as sources for the gravitational potential
φ in the modified Poisson equation and can change the
evolution of cosmological perturbations with respect to
standard gravity [6, 7]. We considered a TeVeS model
with Ωb = 0.05, Ων = 0.17, ΩΛ = 0.78 and we adopted a
choice of the TeVeS parameters that produces a signifi-
cant enhancement of the growth factor. The TeVeS EG
is significantly different from the standard EG (Fig. 1).
It exhibits scale dependence with accompanying baryon
acoustic wiggles. Both features are due to the vector
field fluctuations, which play a significant role in struc-
ture formation [7]. These fluctuations decrease toward
small scales and cause the scale dependency of EG. We
also checked that they affect the final shape of the acous-
tic oscillations of the other components significantly. As
a result, oscillations in φ, ψ and δ do not cancel out per-
fectly in TeVeS when we take the ratio, thus producing
the wiggles in EG.
For the four gravity models investigated, differences
in EG are much larger than observational statistical un-
certainties. Planned surveys are promising to detect per-
cent level deviation from GR and should distinguish these
modified gravity models unambiguously.
2 Scales larger than the horizon at matter-radiation equality are
suppressed [12] and, if measurable, would have a scale dependent
increase in the value of EG in comparison to the small scale value.
3 To simplify the numerical treatment of the TeVeS perturbations
equations while retaining a good qualitative description of all the
significant physical effects at the same time, we introduced sev-
eral approximations. Namely we assumed instantaneous recom-
bination and employed the tight coupling approximation between
baryons and photons at all scales before decoupling; moreover
we evolved perturbations in the massive neutrino component in
a simplified way by switching off neutrinos perturbations when
they were below the free steaming scale and treating them as a
fluid above the free streaming scale.
At large scales, gravity is the only force determining
the acceleration of galaxies and dark matter particles. So
we assumed no galaxy velocity bias. As statistical errors
in EG measurements reach the 1% level (Fig. 1), sev-
eral systematics, besides the one discussed in footnote 1,
may become non-negligible. One is the accuracy of the
redshift distortion formula (Eq. 1), which may be prob-
lematic for those modes with large u, even at very linear
scales [17]. A remedy is to exclude them when extracting
Pgθ, at the expense of statistical accuracy. As discussed
before, accuracy of EG measurement is dominated by
accuracy of P∇2(ψ−φ)g measurements and is thus less af-
fected. A less severe one is the nonlinear evolution, which
becomes non-negligible where the matter power spectrum
variance ∆2m
∼ 0.1. In general relativity, nonlinear cor-
rections to density and velocity differ (Fig. 12, [22]). A
direct consequence is that EG develops a dependence on
the matter power spectrum. Similar effects in modified
gravity models are expected. This can be corrected by
high order perturbation calculations, which should work
well where ∆2m
∼ 0.2.
We thank R. Caldwell, D. Eisenstein, B. Jain, M.
Kunz, J. Ostriker and J.P. Uzan for useful discussions
and the anonymous referees for useful suggestions. PJZ
is supported by the National Science Foundation of China
grant 10533030 and CAS grant KJCX3-SYW-N2. RB’s
work is supported by the National Science Foundation
grants AST-0607018 and PHY-0555216. SD is supported
by the US Department of Energy.
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  The standard cosmological model is based on general relativity and includes
dark matter and dark energy. An important prediction of this model is a fixed
relationship between the gravitational potentials responsible for gravitational
lensing and the matter overdensity. Alternative theories of gravity often make
different predictions for this relationship. We propose a set of measurements
which can test the lensing/matter relationship, thereby distinguishing between
dark energy/matter models and models in which gravity differs from general
relativity. Planned optical, infrared and radio galaxy and lensing surveys will
be able to measure $E_G$, an observational quantity whose expectation value is
equal to the ratio of the Laplacian of the Newtonian potentials to the peculiar
velocity divergence, to percent accuracy. We show that this will easily
separate alternatives such as $\Lambda$CDM, DGP, TeVeS and $f(R)$ gravity.

Introduction.— Predictions based on general relativ-
ity plus the Standard Model of particle physics are at
odds with a variety of independent astronomical obser-
vations on galactic and cosmological scales. This failure
has led to modifications in particle physics. By introduc-
ing dark matter and dark energy, cosmologists have been
able to account for a wide range of observations, from
the overall expansion of the universe to the large scale
structure of the early and late universe [1]. Alternatively,
attempts have been made to modify general relativity at
galactic [2] or cosmological scales [3, 4]. A fundamental
question then arises: Can the two sets of modifications
be distinguished from one another?
The answer is “No” if only the zero order expansion
of the universe is considered. By allowing the dark
energy equation of state wDE to be a free function,
the expansion history H(z) produced by any modified
gravity can be mimicked exactly. Fortunately, struc-
ture formation in modified gravities in general differs
[5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16] from that in
general relativity. The difference we focus on here is
the relationship between gravitational potentials respon-
sible for gravitational lensing and the matter overden-
sity. Lensing is sensitive to ∇2(φ − ψ) along the line
of sight where φ and ψ are the two potentials in the
perturbed Friedman-Robertson-Walker metric: ds2 =
(1 + 2ψ)dt2 − a2(1 + 2φ)dx2 and a is the scale factor.
In standard general relativity (GR), in the absence of
anisotropic stresses, φ = −ψ, so lensing is sensitive to
∗Electronic address: pjzhang@shao.ac.cn
∇2φ. The Poisson equation algebraically relates ∇2φ to
the fractional overdensity δ, so lensing is essentially de-
termined by δ along the line of sight. This is a prediction
of the standard, GR-based theory that is generally not
obeyed by alternate theories of gravity.
Testing this prediction is non-trivial. Astronomers of-
ten use the galaxy overdensity as a probe of the underly-
ing matter overdensity, but the two are not exactly equal.
Here we propose a test of this prediction which is rela-
tively insensitive to the problem of galaxy bias. The basic
idea is simple:
• Extract the matter overdensity at a given redshift
by measuring the velocity field. Matter conser-
vation relates velocities to the overdensities. The
measurement of the velocity field can be accom-
plished by studying the anisotropy of the galaxy
power spectrum in redshift space.
• Extract the lensing signal at this redshift by cross-
correlating these galaxies and lensing maps recon-
structed from background galaxies.
More quantitatively, the galaxy-velocity cross power
spectrum Pgθ ≡ −〈δg(k)θ(−k)〉 can be inferred from
redshift distortions in a galaxy distribution. Here, θ ≡
∇ · v/H(z) and v is the comoving peculiar velocity. In
the linear regime, matter conservation relates θ to δ by
θ = −δ̇/H = −βδ, where β ≡ d lnD/d lna and D is
the linear density growth factor. So, Pgθ = βPgδ , sat-
isfying the first goal above. Cross correlating the same
galaxies with lensing maps constructed from galaxies at
higher redshifts, P∇2(φ−ψ)g can be measured. The ra-
tio of these two cross-spectra therefore is a direct probe
http://arxiv.org/abs/0704.1932v3
mailto:pjzhang@shao.ac.cn
of ∇2(φ − ψ)/(βδ). It does not depend on galaxy bias
or on the initial matter fluctuations, at least in the lin-
ear regime. Modifications in gravity will in general leave
signatures in either β and/or the Poisson equation.
Galaxy-Velocity Cross-correlation.— A galaxy’s
peculiar motion shifts its apparent radial position from
xz to x
z = xz+vz/H(z) in redshift space, where vz is the
comoving radial peculiar velocity. The coherent velocity
component changes the galaxy number overdensity from
δg to δ
g ≃ δg−∇zvz/H(z). Galaxy random motions mix
different scales and damps the power spectrum on small
scales. The redshift space galaxy power spectrum there-
fore has the general form ([17] and references therein)
P sg (k) =
Pg(k) + 2u
2Pgθ(k) + u
4Pθ(k)
k2u2σ2v
H2(z)
where u = k‖/k is the cosine of the angle of the k vector
with respect to radial direction; Pg, Pgθ , Pθ are the real
space galaxy power spectra of galaxies, galaxy-θ and θ,
respectively; σv is the 1D velocity dispersion; and F (x)
is a smoothing function, normalized to unity at x = 0,
determined by the velocity probability distribution. This
simple formula has passed tests in simulations on scales
where δ <∼ 1 [17]. The derivation of Eq. (1) is quite
general, so it should be applicable even when gravity is
modified.
The distinctive dependence of P sg on u allows for si-
multaneous determination of Pg, Pgθ and Pθ [18]. The
parameters we want to determine are the band powers of
Pgθ(k)
1 defined such that P (k) = Pα if kα ≤ k < kα+1,
where k1 < k2 < · · · < kα < · · · . We denote P
as the band power of Pgθ. For a ki in each k bin,
we have a measurement of P sg , which we denote as Pi.
The unbiased minimum variance estimator of P
1 Distance D and H are required to translate the observed galaxy
angular and redshift separation to k. In general, errors in D and
H measurements cause both horizontal and vertical shifts in the
EG plot. Both D and H will be measured by methods like type
Ia supernovae and baryon acoustic oscillations with 1% accuracy,
much smaller than the k bin size adopted, so the horoziontal
shift is negligible. Errors in D show up in both Pgθ and the
Cκg → P∇2(ψ−φ)g inversion through l = kD and thus largely
cancel in evaluating EG. Errors in H(z) only show up in Pgθ
measurement and thus cause a net shift in the value of EG. For
1% error in H, the fractional error in EG is ≤ (neff + 3)1% ≤
3%. Here, neff is the effective power index of the corresponding
power spectra. For the fiducial ΛCDM cosmology, it is negative
in relevant k range. Thus errors induced by uncertainties in
D(z) and H(z) measurement will be sub-dominant, except for
SKA, which requires better control over systematic errors in D
and H measurement. For simplicity, we neglect this potential
error source. Measuring Pgθ also requires to marginalize over
σv. However, in the linear regime k <∼ 0.2h/Mpc, k
2σ2v/H
2 ≪ 1
and F (k2u2σ2v/H
2) ≃ 1, for typical value σv ∼ 300 km/s. Thus
the exact value of σv is not required for our analysis. Without
loss of generality, we adopt σv = 300 km/s.
WiPi, where Wi =
(λ1 + λ2u
i + λ3u
i ). Here,
Fi ≡ F (kuiσv/H), σ
i is the variance of Pi and the three
Lagrange multipliers λα (α = 1, 2, 3) is determined by
λ = (0,
, 0) ·A−1 ; Amn =
2(m+n−2)
. (2)
Galaxy-galaxy lensing.— Weak lensing is sensitive
to the convergence κ, the projected gravitational poten-
tial along the line of sight:
∇2(ψ − φ)W (χ, χs)dχ . (3)
Here, W is the lensing kernel. For a flat universe, χ, χs
are the comoving angular diameter distance to the lens
and source, respectively. Eq. 3 is a pure geometric result
and can be applied to any modified gravity models where
photons follow null geodesics.
A standard method to recover the lens redshift infor-
mation is by the lensing-galaxy cross correlation. For
galaxies in the redshift range [z1, z2], the resulting cross
correlation power spectrum under the Limber’s approxi-
mation is
Cκg(l) =
ng(χ)dχ
W (χ, χs)ng(χ)P∇2(ψ−φ)g(
, z)χ−2dχ
W (χ̄, χs)
∫ l/χ1
P∇2(ψ−φ)g(k, z̄)dk
fα(l)P
Here, χ1,2 are the comoving angular diameter distance to
redshift z1,2 and χ̄ is the mean distance. The band power
α of P∇2(ψ−φ)g is defined at the same k range as P
In practice, we measure the band power Cκg(l,∆l), cen-
tered at l with band width ∆l. The weighting fα(l,∆l)
is defined correspondingly. For each l, only a fraction of
α having fα(l,∆l) 6= 0 contribute.
A discriminating probe of gravity.— With the
above measurements, one can construct an estimator
ÊG =
Cκg(l,∆l)
3H20a
α fα(l,∆l)P
, (5)
whose expectation value is
〈ÊG〉 =
∇2(ψ − φ)
−3H20a
∇2(ψ − φ)
3H20a
≡ EG .
The fractional error on ÊG is
〈∆E2G〉
α〈(δP
α fαP
, (7)
0.4<z<0.6
LAMOST/AS2+LSST
0.85<z<1.15SKA
0.01 0.1
ADEPT+LSST
1.3<z<1.7
0.01 0.1
1.8<z<2.2SKA
FIG. 1: EG as a smoking gun of gravity. Error estimation is
based on ΛCDM and error bars are centered on the ΛCDM
prediction (black solid straight line). We only show those k
modes well in the linear regime. For clarity, we shift the error
bars of LAMOST/AS2+LSST and ADEPT+LSST slightly
rightward. Irregularities in the error-bars are caused by irreg-
ularities in the available discrete k modes of redshift distor-
tion. Dotted lines are the results of a flat DGP model with
Ω0 = 0.2. Dashed lines are for f(R) = −λ1H
0 exp(−R/λ2H
with λ2 = 100. Differences in expansion histories of these
models are of percent level at z < 2 and are not the main
cause of differences in EG. Solid lines with wiggles are for
TeVeS with KB = 0.08, 0.09, 0.1, where the lines with most
significant wiggles have KB = 0.1.
where ∆C2 = [C2κg + (Cκ + C
κ )(Cg + C
g )]/(2l∆lfsky).
Here, Cκ, C
κ , Cg, C
g are the power spectra of weak
lensing convergence, weak lensing shot noise, galaxy and
galaxy shot noise, respectively, and fsky is the fractional
sky coverage. Errors on EG at any two adjacent bins are
correlated, since they always share some same k modes.
However, by requiring lα/χ1 = lα+1/χ2, where l1 < l2 <
· · · < lα < · · · and kα = lα/χ2, EG measurement at each
l bin only involves two k bins and thus only errors in
adjacent bins are correlated.
We choose ongoing/proposed spectroscopic surveys
LAMOST, AS2, ADEPT and SKA as targets of redshift
distortion measurements, and LSST and SKA as targets
of lensing map reconstruction. SKA lensing maps can
be constructed through cosmic magnification utilizing its
unique flux dependence, with S/N comparable to that of
LSST through cosmic shear [19]. Survey specifications
are summarized in TABLE I. The fiducial cosmology
adopted is the ΛCDM cosmology, with the WMAP best
TABLE I: Summary of target surveys.
redshift deg2 Ngal band operation
LAMOSTa z< 0.8 10,000 ∼ 106 optical 2008
AS2b z< 0.8 10,000 ∼ 106 optical ≥ 2009
ADEPTc 1 < z < 2 28,600 ∼ 108 infrared ≥ 2009
SKAd z <∼ 5 22, 000 ∼ 10
9 radio 2020
LSSTe z <∼ 3.5 10,000 ∼ 10
9 optical 2012
ahttp://www.lamost.org/en/
bPrivate communication with Daniel Eisenstein
chttp://www7.nationalacademies.org/ssb/BE Nov 2006 bennett.pdf
dhttp://www.skatelescope.org/
ehttp://www.lsst.org
fit parameters Ω0 = 0.26,ΩΛ = 1 − Ω0, h = 0.72, σ8 =
0.77 and ns = 1. The result is shown in figure 1. In gen-
eral, at k < 0.1h/Mpc, cosmic variance in Cκg and Pgθ
measurements dominates the EG error budget, result-
ing in decreasing error-bars toward larger k. This makes
fsky and the lensing source redshifts the two most rele-
vant survey parameters for EG error estimation. Since
systematic errors in LSST photometric redshifts can be
controlled to better than 1%, errors in EG measurements
of LAMOST/AS2+LSST and ADEPT+LSST caused by
source redshift uncertainties are sub-dominant.
We restrict our discussion to sub-horizon scale pertur-
bations and express equations hereafter in the Fourier
form. Four independent linear equations are required
to solve for four perturbation variables δ, θ, ψ and φ.
The mass-energy conservation provides two: δ̇ +Hθ = 0
and Ḣθ + 2H2θ − k2ψ/a2 = 0. For at least ΛCDM,
quintessence-CDM, DGP and f(R) gravity, the other two
takes the general form
φ = −η(k, a)ψ ,
k2(φ− ψ) = 3H20Ω0a
−1δG̃eff(k, a) . (8)
Here Ω0 is the cosmological matter density in unit of the
critical density ρc ≡ 3H
0/8πG. Refer to [14, 15, 16] for
other ways of parameterizations. MOND has extra scalar
and vector perturbations and does not follow the general
form of Eq. 8 [6, 7].
(1) ΛCDM: η = 1, G̃eff = 1 and EG = Ω0/β. Dynam-
ical dark energy will have large-scale fluctuations [20].
Furthermore, it may also have non-negligible anisotropic
stress and is thus able to mimic modifications in gravity
[21]. But, for models with large sound speed and neg-
ligible anisotropic stress, such as quintessence, these are
negligible at sub-horizon scales and Eq. 8 still holds.
(2) Flat DGP: η = [1 − 1/3βDGP]/[1 + 1/3βDGP],
G̃eff = 1 [9] and EG = Ω0/β, where βDGP = 1−2rcH(1+
Ḣ/3H2) < 0 and rc = H0/(1−Ω0). Ω0 differs from that
of ΛCDM, in order to mimic H(z) of ΛCDM.
(3) f(R) gravity: in the sub-horizon limit, G̃eff =
(1+fR)
−1 [11] and η = 1 [12], with fR ≡ df/dR|B where
B denotes the FRW background. This falls naturally out
of a conformal transformation of the expression for EG
in the Einstein frame into the Jordan frame, noting that
Einstein frame scalar field fluctuations are negligible on
sub-horizon scales [12]. We numerically solve the full per-
turbation equations in the Einstein frame since it is com-
putationally simpler [12] and then conformally transform
to the Jordan frame, which we choose as the physical
frame, evaluating β such that EG = Ω0/(1+fR)β. In the
limit that fR → 0, e.g. for f(R) ∼ λ1H
0 exp(−R/λ2H
[11] with λ1 ≪ λ2, the evolution is observationally equiv-
alent to ΛCDM. For modes that entered the horizon prior
to matter-radiation equality, as we consider here, β, and
therefore EG, is scale invariant for IR modifications to
gravity, with fR > 0.
2 The scale independence of EG
holds in ΛCDM, Quintessence-CDM and DGP. An ob-
served scale-independent deviation in EG from ΛCDM
could signify a special class of modified gravity, as shown
in Fig. 1.
(4) TeVeS/MOND. Besides the gravitational met-
ric, TeVeS [2] contains a scalar and a vector field. These
new fields act as sources for the gravitational potential
φ in the modified Poisson equation and can change the
evolution of cosmological perturbations with respect to
standard gravity [6, 7]. We considered a TeVeS model
with Ωb = 0.05, Ων = 0.17, ΩΛ = 0.78 and we adopted a
choice of the TeVeS parameters that produces a signifi-
cant enhancement of the growth factor. The TeVeS EG
is significantly different from the standard EG (Fig. 1).
It exhibits scale dependence with accompanying baryon
acoustic wiggles. Both features are due to the vector
field fluctuations, which play a significant role in struc-
ture formation [7]. These fluctuations decrease toward
small scales and cause the scale dependency of EG. We
also checked that they affect the final shape of the acous-
tic oscillations of the other components significantly. As
a result, oscillations in φ, ψ and δ do not cancel out per-
fectly in TeVeS when we take the ratio, thus producing
the wiggles in EG.
For the four gravity models investigated, differences
in EG are much larger than observational statistical un-
certainties. Planned surveys are promising to detect per-
cent level deviation from GR and should distinguish these
modified gravity models unambiguously.
2 Scales larger than the horizon at matter-radiation equality are
suppressed [12] and, if measurable, would have a scale dependent
increase in the value of EG in comparison to the small scale value.
3 To simplify the numerical treatment of the TeVeS perturbations
equations while retaining a good qualitative description of all the
significant physical effects at the same time, we introduced sev-
eral approximations. Namely we assumed instantaneous recom-
bination and employed the tight coupling approximation between
baryons and photons at all scales before decoupling; moreover
we evolved perturbations in the massive neutrino component in
a simplified way by switching off neutrinos perturbations when
they were below the free steaming scale and treating them as a
fluid above the free streaming scale.
At large scales, gravity is the only force determining
the acceleration of galaxies and dark matter particles. So
we assumed no galaxy velocity bias. As statistical errors
in EG measurements reach the 1% level (Fig. 1), sev-
eral systematics, besides the one discussed in footnote 1,
may become non-negligible. One is the accuracy of the
redshift distortion formula (Eq. 1), which may be prob-
lematic for those modes with large u, even at very linear
scales [17]. A remedy is to exclude them when extracting
Pgθ, at the expense of statistical accuracy. As discussed
before, accuracy of EG measurement is dominated by
accuracy of P∇2(ψ−φ)g measurements and is thus less af-
fected. A less severe one is the nonlinear evolution, which
becomes non-negligible where the matter power spectrum
variance ∆2m
∼ 0.1. In general relativity, nonlinear cor-
rections to density and velocity differ (Fig. 12, [22]). A
direct consequence is that EG develops a dependence on
the matter power spectrum. Similar effects in modified
gravity models are expected. This can be corrected by
high order perturbation calculations, which should work
well where ∆2m
∼ 0.2.
We thank R. Caldwell, D. Eisenstein, B. Jain, M.
Kunz, J. Ostriker and J.P. Uzan for useful discussions
and the anonymous referees for useful suggestions. PJZ
is supported by the National Science Foundation of China
grant 10533030 and CAS grant KJCX3-SYW-N2. RB’s
work is supported by the National Science Foundation
grants AST-0607018 and PHY-0555216. SD is supported
by the US Department of Energy.
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