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# New measurements of $E_{G}$:
Testing General Relativity with the Weyl potential and galaxy velocities

Nastassia Grimm<EMAIL_ADDRESS>Département de Physique Théorique
and Center for Astroparticle Physics, Université de Genève, Quai E. Ansermet
24, CH-1211 Geneve 4, Switzerland  Camille Bonvin<EMAIL_ADDRESS>Département de Physique Théorique and Center for Astroparticle Physics,
Université de Genève, Quai E. Ansermet 24, CH-1211 Geneve 4, Switzerland
Isaac Tutusaus<EMAIL_ADDRESS>Institut de Recherche en
Astrophysique et Planétologie (IRAP), Université de Toulouse, CNRS, UPS, CNES,
14 Av. Edouard Belin, 31400 Toulouse, France

(August 28, 2024)

###### Abstract

We combine measurements of galaxy velocities from galaxy surveys with
measurements of the Weyl potential from the Dark Energy Survey to test the
consistency of General Relativity at cosmological scales. Taking the ratio of
two model-independent observables – the growth rate of structure and the Weyl
potential – we obtain new measurements of the $E_{G}$ statistic with precision
of $5.8-10.7\%$ at four different redshifts. These measurements provide a
considerable improvement to past measurements of $E_{G}$. They confirm the
validity of General Relativity at three redshifts, while displaying a tension
of $2.5\sigma$ at $z=0.47$ as a consequence of the tension found in the
measurements of the Weyl potential. Contrary to conventional methods that rely
on a common galaxy sample with spectroscopic resolution to measure two types
of correlations, we directly combine two observables that are independent of
the galaxy bias. This provides a novel approach to testing the relation
between the geometry of our Universe and the motion of galaxies with improved
precision.

##  I Introduction

The observation of the accelerated expansion of our Universe in 1998 [1, 2]
has called our understanding of the cosmos into question. This unexpected
behaviour can either be explained by the presence of a new form of energy with
negative pressure, called dark energy; or by changing the laws of gravity at
cosmological scales. To determine which of these paradigms is correct, it is
necessary to test the theory of gravity. Gravity determines, on one hand, how
space and time are bent by light and matter; and, on the other hand, it
governs how light and matter move in this distorted geometry. In General
Relativity (GR) these two concepts are directly linked by Einstein’s equations
and conservation equations. If gravity is modified, however, the relation
between the content of the Universe and its geometry typically changes, see
e.g., Ref. [3] for a review.

One powerful way of testing deviations from GR consists therefore in comparing
the distortion of space-time with the velocities of astrophysical objects, for
example galaxies. Such a test has been proposed in 2007 by Zhang et al. [4],
who built the so-called $E_{G}$ statistic. The idea of this test is to measure
the ratio of the Weyl potential, i.e., the sum of the two gravitational
potentials $\Psi_{W}=(\Phi+\Psi)/2$, and the galaxy peculiar velocity $V$. In
GR, this ratio takes a specific value, while a modified theory of gravity
generally yields a different prediction. Hence, any deviation from the
expected $E_{G}$ value would indicate a departure from GR. The strength of
this test lies in the fact that it does not require any modelling of theories
beyond GR: one simply compares two observable quantities to test the
consistency of GR.

Observationally, we cannot access the Weyl potential and galaxy velocities
directly. Instead, galaxy surveys measure correlation functions involving
these quantities, averaged over the sky. The current method to constrain
$E_{G}$ therefore consists in measuring correlations of the Weyl potential, as
well as of the galaxy velocities, with a common galaxy sample. Then, by taking
the ratio between these two sets of correlation functions, the contribution
from galaxy clustering cancels out which leads to the desired $E_{G}$
quantity. In practice, the situation is slightly more complex, since direct
measurements of galaxy velocities only reach up to $z\simeq 0.02$, see e.g.,
Refs. [5, 6, 7, 8, 9, 10]. Velocity-density correlations are therefore
reconstructed from the product of galaxy clustering and the quantity
$\beta\equiv f/b$, which can be directly measured from redshift-space
distortions [11, 12] (here $f$ is the growth rate and $b$ the galaxy bias).
More precisely, $E_{G}$ can be obtained via the ratio,

$E_{G}=\Gamma\frac{C_{l}^{\kappa g}}{\beta C_{l}^{gg}}\,,$ (1)

where $C_{l}^{gg}$ is the galaxy clustering angular power spectrum, and
$C_{l}^{\kappa g}$ denotes the cross-correlation of galaxy clustering with
gravitational lensing. $\Gamma$ is a normalisation pre-factor, see e.g., Ref.
[13] for detail.

This technique has been successfully used with various data sets [14, 15, 16,
17, 13, 18], providing measurements of $E_{G}$ over a wide range of redshifts.
While some of the measurements are in perfect agreement with the predictions
from GR [14, 15, 16, 17, 18], others reveal some tensions [15, 16, 13].
Recently, Ref. [19] showed that 21 cm intensity mapping will provide a robust
way to measure $E_{G}$ with the coming generation of surveys.

In this letter, we present a novel method to measure $E_{G}$ combining 22
measurements of the growth rate of structure, $f\sigma_{8}$ ($\sigma_{8}$
denotes the clustering amplitude), from various spectroscopic surveys [5, 6,
7, 8, 9, 20, 21, 22, 23, 24, 25, 26], with novel measurements of the Weyl
potential from the Dark Energy Survey (DES) [27]. Contrary to previous
methods, our measurements of $E_{G}$ do not rely on a common spectroscopic
data set. Instead, we use model-independent measurements of the evolution of
galaxy velocities (encoded in $f\sigma_{8}$) and the evolution of the Weyl
potential (encoded into a function $\hat{J}$, called Weyl evolution), and
combine them to obtain $E_{G}$. This allows us to benefit, at the same time,
from the full precision of spectroscopic samples to measure velocities and
from the full statistical power of photometric samples to measure the Weyl
potential, which significantly improves the precision of the $E_{G}$
measurements. We obtain values of $E_{G}$ at four redshifts, with precision of
$5.8-10.7\%$, providing a considerable improvement to previous measurements
with $13.3-30.2\%$ precision. We find that in three of the four redshift bins,
$E_{G}$ is fully compatible with GR. In the second redshift bin, however,
$E_{G}$ is 2.5$\sigma$ below the GR prediction. This mild tension is explained
by the value of the Weyl potential, which, as shown in Ref. [27], is
3.1$\sigma$ below GR in the respective bin.

##  II Methodology

Galaxy surveys measure the galaxy clustering correlation function in redshift-
space. Since galaxy peculiar velocities break statistical isotropy, the
correlation function can be decomposed into multipoles, namely a monopole,
quadrupole, and hexadecapole. By combining measurements of these different
multipoles, one can measure the quantity $\hat{f}(z)\equiv f(z)\sigma_{8}(z)$
at redshift $z$ in a model-independent way, i.e., without assuming a specific
theory of gravity. Here, $f(z)=\mathrm{d}\ln\delta/\mathrm{d}\ln a$ is the
growth rate of structure, which encodes the evolution of the galaxy peculiar
velocities, and $\sigma_{8}(z)$ is the clustering amplitude in spheres of 8
Mpc/$h$.

Recently it was shown that weak lensing data can be used in a similar spirit,
to obtain model-independent measurements of the evolution of the Weyl
potential [27]. The evolution of the Weyl potential with redshift can be
encoded in a function $\hat{J}$, defined through [28]

$\displaystyle\Psi_{W}(k,z)=\left(\frac{\mathcal{H}(z)}{\mathcal{H}(z_{*})}\right)^{2}\hat{J}(z)\,\sqrt{\frac{B(k,z)}{B(k,z_{*})}}\frac{\Psi_{W}(k,z_{*})}{\sigma_{8}(z_{*})}\,.$
(2)

Here, $\mathcal{H}$ is the Hubble parameter in conformal time, and $z_{*}$ is
a reference redshift chosen to be well in the matter-dominated era, before
acceleration started. Moreover, the boost factor $B(k,z)$ encodes the non-
linear evolution of matter density perturbations at small scales. As shown in
Ref. [27], $\hat{J}$ can be directly measured by combining galaxy-galaxy
lensing correlations with galaxy clustering correlations. These measurements
are model-independent, i.e., they do not assume any theory of gravity.
Similarly to the measurements of $\hat{f}$ they rely however on one
assumption: that at $z_{*}$, before the phase of accelerated expansion, GR is
recovered. This is a reasonable assumption, given that the main motivation to
search for theories beyond GR is the late-time accelerated expansion of the
Universe, while early-time cosmological observations are compatible with GR.
As constraints from the Cosmic Microwave Background [29] and from
nucleosynthesis [30] largely restrict the scope of allowed theories, it is a
common approach to assume that deviations from GR only occur at late time
[31], driving the accelerated expansion of the Universe.

Given measurements of $\hat{f}$ and $\hat{J}$, $E_{G}$ can then be obtained by
taking the ratio,

$\displaystyle
E_{G}(z)\equiv\left(\frac{\mathcal{H}(z)}{\mathcal{H}_{0}}\right)^{2}\frac{1}{1+z}\,\frac{\hat{J}(z)}{\hat{f}(z)}\,.$
(3)

Note that here we have normalised the ratio by the Hubble parameter, to
recover the standard expression for $E_{G}$ defined in Ref. [4], such that we
can compare with the values previously measured in the literature. In GR,
using that $\Phi=\Psi$ and that $\Phi$ is related to the density via the
Poisson equation, $\hat{J}$ becomes $\hat{J}(z)=\Omega_{\rm
m}(z)\sigma_{8}(z)$, where $\Omega_{\rm m}(z)$ is the matter density contrast,
and $E_{G}$ reduces to $E_{G}=\Omega_{\rm m,0}/f(z)$. In modified theories of
gravity, however, $\Phi$ usually differs from $\Psi$ and the Poisson equation
is typically modified, leading to different values of $E_{G}$. Note that
without the normalisation in front of the ratio $\hat{J}/\hat{f}$ in Eq. (3),
we would instead obtain $E_{G}=\Omega_{\rm m}(z)/f(z)$ in GR.

Table 1: List of $\hat{f}(z)$ measurements used in this work. Dataset | $z$ | $\hat{f}(z)$ | Ref.
---|---|---|---
2MTF | $0.001$ | $0.505\pm 0.085$ | [5]
6dFGS+SNIa | $0.02$ | $0.4280\pm 0.0465$ | [6]
IRAS+SNIa | $0.02$ | $0.398\pm 0.065$ | [7, 8]
2MASS | $0.02$ | $0.314\pm 0.048$ | [7, 9]
2dFGRS | $0.17$ | $0.510\pm 0.060$ | [20]
GAMA | $0.18$ | $0.360\pm 0.090$ | [21]
GAMA | $0.38$ | $0.440\pm 0.060$ | [21]
SDSS-IV (MGS) | $0.15$ | $0.53\pm 0.16$ | [22]
SDSS-IV (BOSS Galaxy) | $0.38$ | $0.497\pm 0.045$ | [22]
SDSS-IV (BOSS Galaxy) | $0.51$ | $0.459\pm 0.038$ | [22]
SDSS-IV (eBOSS LRG) | $0.70$ | $0.473\pm 0.041$ | [22]
SDSS-IV (eBOSS ELG) | $0.85$ | $0.315\pm 0.095$ | [22]
WiggleZ | $0.44$ | $0.413\pm 0.080$ | [23]
WiggleZ | $0.60$ | $0.390\pm 0.063$ | [23]
WiggleZ | $0.73$ | $0.437\pm 0.072$ | [23]
Vipers PDR-2 | $0.60$ | $0.550\pm 0.120$ | [24]
Vipers PDR-2 | $0.86$ | $0.400\pm 0.110$ | [24]
FastSound | $1.40$ | $0.482\pm 0.116$ | [25]
SDSS-IV (eBOSS Quasar) | $0.978$ | $0.379\pm 0.176$ | [26]
SDSS-IV (eBOSS Quasar) | $1.230$ | $0.385\pm 0.099$ | [26]
SDSS-IV (eBOSS Quasar) | $1.526$ | $0.342\pm 0.070$ | [26]
SDSS-IV (eBOSS Quasar) | $1.944$ | $0.364\pm 0.106$ | [26]

We list the full set of $\hat{f}$ measurements used in this work in Table 1.
This data set, containing 22 values between $z=0.001$ and $z=1.944$, is based
on the Gold-2017 compilation of robust and independent measurements [32],
being updated with more recently released data from SDSS-IV [22, 26] as well
as a very low redshift value from the 2MTF survey [5]. From Eq. (3), we see
that to obtain $E_{G}$ at redshift $z$, measurements of $\hat{f}$ and of
$\hat{J}$ at the same redshift are needed. Since $\hat{f}$ is measured from
spectroscopic galaxy surveys, while $\hat{J}$ is obtained from lensing
surveys, the redshifts typically differ. It is therefore necessary to
propagate the measurements of $\hat{f}$ to the desired redshifts, where
measurements of $\hat{J}(z)$ are available. More precisely, we aim to
reconstruct $\hat{f}$ at the four effective redshifts of the DES MagLim lens
sample, where measurements of $\hat{J}$ have been obtained in Ref. [27] as
summarised in the first two columns of Table 2. Note that in principle we
could instead reconstruct the Weyl evolution, $\hat{J}$, at the redshifts
where we have measurements of $\hat{f}$. This would however lead to less
precise measurements of $E_{G}$ since the Weyl evolution is only measured at
four redshifts, while $\hat{f}$ is measured at 22 redshifts, allowing for a
better reconstruction.

A model-independent method to reconstruct $\hat{f}$ is provided by Gaussian
processes. However, as shown in Ref. [33], current data are not precise enough
to allow for a robust reconstruction with this method, as the results are
dominated by the prior range on the kernel hyperparameters. Therefore, we
instead use spline interpolation, treating the values of $\hat{f}$ at the four
DES MagLim effective redshifts as four free parameters. We interpolate between
these free parameters using cubic splines and minimise the difference between
the interpolated curve and the measurements of $\hat{f}$. We have examined
choices for the knots in the spline interpolation other than the MagLim
effective redshifts, such as linearly spaced values along the redshift range
of the measured data, which affects the reconstruction only marginally (at
sub-percent level for the mean values and percent level for the errors at the
MagLim redshifts). Moreover, we have verified that four redshift knots provide
the ideal number of parameters following the Akaike information criterion
[34].

Table 2: We list the first four effective redshifts of the DES MagLim sample along with the respective values of $\hat{J}(z)$ obtained in Ref. [27] (using CMB priors and standard scale cuts), and the values of $\hat{f}(z)$ and $E_{G}(z)$ obtained in this work. $z$ | $\hat{J}(z)$ | $\hat{f}(z)$ | $E_{G}(z)$
---|---|---|---
$0.295$ | $0.326\pm 0.013$ | $0.459\pm 0.019$ | $0.448\pm 0.026$
$0.467$ | $0.331\pm 0.017$ | $0.467\pm 0.020$ | $0.377\pm 0.025$
$0.626$ | $0.387\pm 0.025$ | $0.461\pm 0.021$ | $0.396\pm 0.031$
$0.771$ | $0.354\pm 0.033$ | $0.448\pm 0.024$ | $0.345\pm 0.037$

##  III Results

Figure 1: Literature values of $\hat{f}(z)$ (blue points) along with our
reconstruction at the DES MagLim effective redshifts (red points). In grey, we
show the $1\sigma$ error bands of the spline reconstruction, while the black
line shows the prediction assuming Planck 2018 cosmological parameters [35]
and no deviations from GR.

In Fig. 1, we show in blue the 22 measurements of $\hat{f}$ summarised in
Table 1, and in red we show the values and error bars inferred at the four DES
redshifts, which we list in the third column of Table 2. We note that,
similarly to Refs. [36, 37], we take the covariance matrix of the eBOSS Quasar
and WiggleZ measurements into account. All remaining measurements of $\hat{f}$
have negligible correlations [22, 32]. We see that $\hat{f}$ is well
reconstructed with precision between $4.3\%$ and $5.4\%$. This is not
surprising, as we are using the knowledge of $\hat{f}$ measured at 22
redshifts between 0.001 and 1.944, to obtain the values of $\hat{f}$ at four
new redshifts. As such, each red data point combines information from the
whole set of measurements. We also note that our reconstruction is, within
$1\sigma$, well-compatible with the GR prediction (black line) along the whole
redshift range.

Figure 2: We show our measurements of $E_{G}$ (red) compared to various
literature values.

In Fig. 2, we show in red the values of $E_{G}$ that we obtain with our
method. We note that, to obtain the error bars, we have taken the full
covariance of the $\hat{J}$ measurement from Ref. [27] into account, as well
as the full covariance of the four $\hat{f}$ values obtained with our
interpolation method. For comparison, we also show values previously obtained
in literature (summarised in Table 3), either using cosmic shear [17, 16, 14,
15] or CMB lensing [13, 18]. As we see from Eq. (3), to obtain $E_{G}$ values
that can be compared with the literature, we need to normalise the ratio
$\hat{J}/\hat{f}$ by the Hubble parameter. We assume here that the evolution
of $\mathcal{H}(z)$ obeys $\Lambda$CDM predictions with cosmological
parameters given by Planck [35]. Note that this is not a limitation of our
methodology: we could easily measure directly the ratio $\hat{J}/\hat{f}$ to
test for deviations from GR instead of using Eq. (3). The normalisation is
only needed to compare with previous measurements of $E_{G}$.

Table 3: List of literature values of $E_{G}$. $z$ | $E_{G}$ | Ref.
---|---|---
0.37 | $0.392\pm 0.065$ | [14]
0.32 | $0.48\pm 0.10$ | [15]
0.57 | $0.30\pm 0.07$ | [15]
0.27 | $0.43\pm 0.13$ | [16]
0.31 | $0.27\pm 0.08$ | [16]
0.55 | $0.26\pm 0.07$ | [16]
0.57 | $0.420\pm 0.056$ | [17]
0.57 | $0.24\pm 0.06$ | [13]
0.316 | $0.40^{+0.11}_{-0.09}$ | [18]
0.555 | $0.36^{+0.06}_{-0.05}$ | [18]

We see that the values that we obtain in the first, third and fourth redshift
bin are in good agreement with the GR prediction, shown by the black line in
Fig. 2. In the second bin, we find that our value is $2.5\sigma$ below the GR
prediction. This tension can be traced back to the value of $\hat{J}$ in that
bin, which is $3.1\sigma$ below the GR value [27]. Note that, in the first
redshift bin, $\hat{J}$ is also below the GR value (by 2.3$\sigma$). However,
dividing by $\hat{f}$ with a best-fit value slightly below GR leads to a
measurement of $E_{G}$ in agreement with GR. This shows that, while $E_{G}$
provides an interesting consistency check of GR, it may be insensitive to some
specific types of modifications of gravity. More specifically, models that
lead to a growth rate of structure and a Weyl potential both lower or higher
than in GR may not impact $E_{G}$ significantly.

We summarise our results for $E_{G}$ in the last column of Table 2, and we
find that $E_{G}$ is measured with a precision between $5.8\%$ and $10.7\%$,
compared to $13.3-30.2\%$ precision for the literature values in Table 3. On
one hand, this is due to the fact that our method does not rely on the
correlation of cosmic shear with the same spectroscopic sample that is used to
measure $\hat{f}$. This is contrary to common methods to obtain $E_{G}$, where
the use of the same sample for these two measurements is crucial in order to
cancel the impact of the galaxy bias [4]. With our method, instead, we measure
directly the Weyl evolution $\hat{J}$, which does not depend on the bias.
Naturally, the Weyl evolution is in fact degenerated with the bias in the
galaxy-galaxy lensing signal. However, as shown in Ref. [27], by combining
galaxy-galaxy lensing with galaxy clustering (i.e., the clustering of the
lenses), both $\hat{J}$ and the galaxy bias can be measured. Our method allows
us therefore to use the full statistical power of the photometric DES MagLim
sample, containing 7.6 million galaxies in the first four bins [38], which is
$10-100$ times more than the spectroscopic samples used for previous $E_{G}$
measurements [14, 15, 16, 17, 13, 18].

The second reason for the high level of precision of our results for $E_{G}$
is due to the fact that we combine measurements of $\hat{f}$ over a wide range
of redshift, to infer $\hat{f}$ at the four redshifts where $\hat{J}$
measurements are available. We note that, since previous measurements have
employed information from surveys such as BOSS and GAMA, we expect some mild
correlations between our measurements and previous measurements of $E_{G}$.
However, we employ a much larger wealth of data for $f\sigma_{8}$, leading to
more precise measurements of $E_{G}$.

Finally, we emphasise that our method has the advantage of taking into account
contaminations such as magnification bias [39] and redshift-space distortions
in the galaxy-galaxy lensing and photometric galaxy clustering correlations.
As explained in Ref. [27] and following the DES analysis [38], magnification
bias and redshift-space distortions are consistently included in the
modelling. Since these effects are subdominant, it is enough to model them in
GR to obtain robust measurements of $\hat{J}$ (changing the modelling of these
effects only impacts $\hat{J}$ very mildly). On the contrary, as shown in Ref.
[40], standard methods are affected by lensing magnification, which may spoil
the bias-independence of $E_{G}$.

##  IV Conclusion

In this letter, we have presented new measurements of $E_{G}$ at four
different redshifts. These measurements are probing the ratio between the Weyl
potential and the motion of galaxies. They therefore provide a direct and
robust test of the consistency of GR, which links the motion of astrophysical
objects to the distorted geometry of the Universe. We have found that at three
of the four redshifts, $E_{G}$ is consistent with GR predictions. Solely at
the second redshift bin, we find a tension of $2.5\sigma$, arising from the
$3.1\sigma$ disagreement with GR found in the measurement of $\hat{J}$ in Ref.
[27]. Our measurements are significantly more precise than previous
measurements, providing more stringent constraints on deviations from GR.

Interestingly, previous measurements of $E_{G}$ have been inconclusive
regarding the compatibility with GR, in particular at redshifts between the
second and third bin of our analysis. While Refs. [17, 18] show agreement with
GR, Refs. [16, 13, 15] show mild tensions reaching up to $2.6\sigma$. Our
method will be vital to further investigate the evolution of $E_{G}$ with
redshift, once more measurements of $\hat{J}$, covering especially the range
between the second and third bin, will be available from future surveys.

###### Acknowledgements.

We thank Maria Berti for useful discussions. C.B. and N.G. acknowledge support
from the European Research Council (ERC) under the European Union’s Horizon
2020 research and innovation program (grant agreement No. 863929; project
title “Testing the law of gravity with novel large-scale structure
observables”). C.B. acknowledges support from the Swiss National Science
Foundation.

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