XX 2018

# Cosmological Tests of Gravity

Pedro G. Ferreira1 1Astrophysics, University of Oxford, Oxford OX1 3RH, UK;
email<EMAIL_ADDRESS>
###### Abstract

Cosmological observations are beginning to reach a level of precision that
allows us to test some of the most fundamental assumptions in our working
model of the Universe. One such as assumption is that gravity is governed by
the General Theory of Relativity. In this review we discuss how one might go
about extending General Relativity and how such extensions can be described in
a unified way on large scales. This allows us to describe the phenomenology of
modified gravity in the growth and morphology of the large scale structure of
the Universe. On smaller scales we explore the physics of gravitational
screening and how it might manifest itself in galaxies, clusters and, more
generally, in the cosmic web. We then analyse the current constraints from
large scale structure and conclude by discussing the future prospects of the
field in light of the plethora of surveys currently being planned.

The key results are:

  * •

there are a plethora of alternative theories of gravity which are
restricted by fundamental physics considerations;

  * •

there is now a well established formalism for describing
cosmological perturbations in the linear regime for general
theories of gravity;

  * •

gravitational screening can mask modifications to general relativity
on small scales but may, itself, lead to distinctive signatures in the
large scale structure of the Universe;

  * •

current constraints on both linear and non-linear scales may be
affected by systematic uncertainties which limit our ability to rule
out alternatives to General Relativity;

  * •

the next generation of cosmological surveys will dramatically improve
constraints on General Relativity, by up to two orders of magnitude.

###### keywords:

General Relativity, Cosmology, Large Scale Structure, Cosmic Microwave
Background, Early Universe

††journal: Annu. Rev. Astron. Astrophys.

###### Contents

  1. 1 Why test gravity?
    1. 1.1 The success and the uniqueness of General Relativity.
    2. 1.2 Hints from the late and early Universe.
    3. 1.3 The (almost) strong field regime.
    4. 1.4 Modified gravity in cosmology: a roadmap.
  2. 2 Modern Cosmology.
    1. 2.1 The expanding Universe.
    2. 2.2 Linear cosmological perturbation theory.
    3. 2.3 The observables: power spectra, growth rate and cosmic shear.
    4. 2.4 Small scales and non-gravitational physics.
    5. 2.5 $\Lambda$CDM
  3. 3 Modified gravity and cosmology.
    1. 3.1 What is General Relativity?
    2. 3.2 Modified gravity as extra degrees of freedom.
    3. 3.3 Alternatives: higher derivatives, higher dimensions and non-locality.
    4. 3.4 The speed of gravitational waves and GW170817
    5. 3.5 General theories of gravity on cosmological scales.
    6. 3.6 The quasi-static regime.
    7. 3.7 The phenomenology of modified gravity.
  4. 4 Fifth forces and gravitational screening.
    1. 4.1 Gravitational regimes and the observational desert.
    2. 4.2 Gravitational screening.
    3. 4.3 The phenomenology of screening.
    4. 4.4 Observational consequences.
  5. 5 Constraints from Large Scale Structure.
    1. 5.1 The evidence for $\Lambda$CDM
    2. 5.2 Minor anomalies.
    3. 5.3 Parameterizing gravity.
    4. 5.4 Current constraints: the CMB
    5. 5.5 Current constraints: galaxy weak lensing and growth rate.
    6. 5.6 Model specific constraints: scalar-tensor theories.
  6. 6 Alternative constraints - clusters, galaxies and voids.
    1. 6.1 Cluster masses.
    2. 6.2 Screened fifth forces in galaxies.
    3. 6.3 Cosmic voids.
  7. 7 Future prospects.
    1. 7.1 Theoretical status.
    2. 7.2 Observational status.
    3. 7.3 Planned surveys and prospects.
    4. 7.4 Synergies with other gravitational probes.

##  1 Why test gravity?

###  1.1 The success and the uniqueness of General Relativity.

The success of the standard model of fundamental physics is unquestionable and
unassailable. There is a complete model for the strong and electro-weak forces
(dubbed the standard model – SM) which has passed every single test to which
it has been subjected. While some minor anomalies have cropped up, they are
not sufficiently significant to overhaul the main picture: a gauge theory of
interactions between fermions where electro-weak symmetry is spontaneously
broken by a scalar field. The remaining force, gravity, is perfectly described
by the general theory of relativity (GR), a theory of a dynamical space time
where general covariance (also called general coordinate invariance or
diffeomorphism invariance) plays a fundamental role. Again, observations
repeatedly confirm that GR works in a number of regimes. Observations and
experiments on the scale of the Solar System using lunar laser ranging,
satellite missions such as Cassini or exquisite measurements of planetary
orbits compete with measurements of individual and binary millisecond pulsars
as well as triple systems to obtain exquisite constraints of gravity (Will,
2014). With the advent of gravitational wave astronomy and black hole imaging,
it will be possible to, for the first time, find constraints in the strong
gravity regime.

When extrapolated to the largest, cosmological, scales, the SM and GR seem to
accurately describe the history and evolution of the universe, as well as the
large scale structure of space-time and matter. In that case, the SM and GR
have been extrapolated to length scales and energy scales which are well
beyond any current experimentally (or astrophysically) acessible regime. In
other words, two theories which pass tests in the laboratory and in the Solar
System with flying colours are used to predict what is observed on length
scales which are up to fifteen orders of magnitude greater. And the
predictions match observations with remarkable consistency: the expansion rate
of the universe, the morphology of large scale structure, the abundance of
different types of particles, can all be consistently modelled with the
current cosmological model. Furthermore, it allows us to predict that there is
a dark sector to the Universe, i.e. that in the context of SM and GR, we
should expect $95\%$ of the energy density of the Universe to consist of some
exotic form of material – dark matter and dark energy. The simplest
formulation of this cosmological model has been called $\Lambda$\- Cold Dark
Matter ($\Lambda$CDM) where $\Lambda$ represents the cosmological constant. It
is now customary to express cosmological results in terms of constraints on
$\Lambda$CDM, some of which can reach sub-percent accuracy.

If one delves into the structure of gravity, there are compelling reasons to
believe that GR is the unique theory that can explain it. Assuming that
gravity is mediated by a spin-2 carrier – a transverse, traceless field –
there are a number of arguments that show that GR is the non-linear theory
that can correctly explain its dynamics. In a series of lectures and papers
(Feynman et al., 1995; Weinberg, 1964; Deser, 1970) it was shown that,
beginning with the massless, Fierz-Pauli action for a spin-2 field,
$h_{\alpha\beta}$ (Fierz & Pauli, 1939), the requirement of general covariance
implies that one needs to consider non-linear corrections and that, when all
of them are correctly taken into account, means that the fundamental action is
unique and is that of GR. An alternative, and more classical, viewpoint
(Lovelock, 1972) shows that the only second-order, local gravitational field
equations derivable from an action containing solely the 4D metric tensor
(plus related tensors) are the Einstein field equations with a cosmological
constant. This is backed up by a similar conclusion from the geometrodynamic
point of view (Hojman et al., 1976).

###  1.2 Hints from the late and early Universe.

While the evidence for the SM and GR as an integral component of the current
working model of the Universe is extremely strong, it is important to step
back and ponder the way forward. Indeed, with these core theories we can make
remarkable predictions on cosmological scales and, in particular, it leads to
extremely strong evidence for a dark sector. The evidence is so compelling
that the quest to understand this dark sector is the scientific driver of a
number of large scale observational programmes. Nevertheless, given that there
is no other (non-gravitational) evidence for the dark sector, it is a matter
of common sense to question some of the fundamental assumptions that go into
the evidence. And the main assumption is that GR is the underlying theory of
gravity. An alternative point of view then, is that evidence for the dark
sector may signal a break down of GR on cosmological scales, at late time.

There is tantalizing evidence for deviation from general relativity if one
attempts to place constraints on the physics of the very early universe. From
observations of the anisotropy of the CMB, it is now possible to place
extremely stringent constraints on the statistics of primordial fluctuations.
The simplest assumption is that these fluctuations can be characterized in
terms of an overall amplitude for both scalar and tensor components along with
spectral indices for their spatial morphology. Current constraints on the
spectral index place it close to, but not exactly at, scale invariance; they
also place an upper-bound on the amplitude of tensor fluctuations (in the form
of primordial gravitational waves) relative to scalar fluctuations (in the
form of density perturbations) (Planck, 2018b). The most favoured theory that
can explain these observations is inflation: a period of accelerated expansion
in the early universe, amplifying microscopic quantum fluctuations into
macroscopic cosmological perturbations. Constraints from the CMB greatly
restrict the range of models which are observationally viable. Remarkably,
those that are favoured involve some modification to GR at very early times,
either through a non-minimal coupling between the inflaton scalar field and
the Ricci tensor or through the inclusion of higher order powers of the Ricci
tensor in the fundamental action (Kaiser & Sfakianakis, 2014).

###  1.3 The (almost) strong field regime.

To some extent it is unsurprising that modifications should arise in a regime
of high curvature. Purely from an effective field theory point of view
(Donoghue, 1994; Burgess, 2003), one should expect corrections to GR to emerge
in this regime – the Einstein-Hilbert action should have corrections
proportional to $R^{2}$, $R^{\mu\nu}R_{\mu\nu}$ and so one. Furthermore, in
such high curvature regimes, any extra fields (or degrees of freedom) which
are frozen at low energies will, most likely, be dynamical and contribute to
any non-standard gravitational dynamics. It is then natural to look for
modifications away from the weak field regime: while it is practically
impossible to access high curvature regimes (apart from the early Universe or
in the cores of black holes) one can be slightly less ambitious and look at
the regime where the gravitational potential, $\Phi$ is of order unity. And,
as luck would have it, two brand new windows on gravitational physics have
opened up in this regime: gravitational wave detection and black hole imaging.

The advanced Laser Interferometric Gravitational Observatory (aLIGO) has
observed a number of coalescing compact objects through their gravitational
wave emission (LIGO, 2016a). A number of merging black hole and neutron star
binaries have been used to place stringent constraints on gravitational
physics – thus far these constraints have focused on internal consistency
tests in the waveform models where the signal is dominated by the weak field
regime during the inspiral and after the merger (LIGO, 2016b). But as the
number of events build up, and analytic understanding of the merger improves,
it should be possible to dig into the merger where the gravitational potential
becomes appreciable. It should also be possible to extract information from
the ring-down of the merger where signatures of deviation from GR may become
evident in the quasi-normal mode spectrum. From this spectrum it will be
possible, for example, to test the no-hair theorem or probe for the presence
of extra degrees of freedom (Berti et al., 2015). Alternatively, by studying
mergers which are associated with electromagnetic counterparts, it is possible
to place constraints on the propagation of gravitational waves and place
constraints on the gravitational sector. Indeed, that has already been done to
great effect (Baker et al., 2017; Creminelli & Vernizzi, 2017; Ezquiaga &
Zumalacarregui, 2017) with the recently detected binary neutron star merger
GW170817 (LIGO, 2017a).

A coordinated effort by the consortia of telescopes that make the Event
Horizon Telescope to observe Sgr A∗ at the center of the Galaxy, will lead to
the first image of a black hole event horizon
(https://eventhorizontelescope.org) . By analyzing the structure of the
accretion flow and how it wraps around the dark inner shadow, it will be
possible to probe the structure of space-time close to the Schwarzschild
radius and test whether it is consistent with GR, i.e. whether it is has a
Kerr geometry. Preliminary measurements have already picked up structure on
the scale of the event horizon (Doeleman et al., 2008) and, while one should
expect the intricacies of the physics of accretion flows to complicate the
analysis, it will be possible, for the first time, to look for evidence for
deviations to GR close to (but not in) the strong gravity regime (Johannsen &
Psaltis, 2010; Broderick et al., 2013).

###  1.4 Modified gravity in cosmology: a roadmap.

Let us now turn our sights onto cosmological scales at late times. A first
guess would be that deviations from GR on these scales would be unexpected.
While the expanding Universe is, in some sense, a core result of GR or any
relativistic form of gravity, and can be thought of as in the strong field
regime, any fluctuations away from the homogeneous and isotropic space-time
are extremely weak. Naively one would expect that it would be very difficult
to construct modifications to GR in the regime where the curvature is small –
in fact one might even think it problematic given that Minkowski space is the
limit of zero curvature and we believe it to be an excellent approximation in
a number of classical and quantum scenarios.

The past few decades have shown that naive expectations about the cosmological
regime don’t necessarily bear out. For a start, there has been an explosion of
theoretical ideas leading to models that modify gravity in the infra-red while
leaving it consistent with GR on astrophysical scales (Clifton et al., 2012).
These have involved exploring the various loop-holes to Lovelock’s theorem –
higher dimensions and derivatives, extra fields and non-locality. The various
proposals can all be rephrased in terms of extra fields which couple non-
minimally to gravity and lead to new fifth forces. Non-linear, classical
effects can lead to screening of gravitational fifth forces in regimes of high
gravitational potential and curvature while leaving them unscreened on large
scales and in regimes of low curvature (Vainshtein, 1972; Khoury & Weltman,
2004; Hinterbichler & Khoury, 2010). Constraints on fifth forces are
remarkable (Adelberger, Heckel & Nelson, 2003) but are restricted to small
scales (compared to cosmology). Hence, looking for modifications to general
relativity on large scales is tantamount to looking for the effects of fifth
forces in an unscreened regime. This is uncharted territory and one of the few
pristine arenas where we might find evidence for new physics.

The search for dark energy has also become a search for modifications to
general relativity on large scales. All the main observational programmes
currently being planned have, as one part of their science cases, the goal of
testing gravity using a range of different observables with a variety of
tracers of large scale structure. The purpose of this review is to establish
the core ideas underpinning such a programme. I will describe the basic ideas
and tools that go into extracting meaningful constraints from cosmological
data.

The structure of this review is as follows. In Section 2 I briefly review
current cosmology, describing the key aspects of gravitational physics that
play a role in the evolution of the Universe. In Section 3 I discuss how
gravity can be modified on cosmological scales, with a particular emphasis on
how this involves extra degrees of freedom and fields. In Section 4 I work
through the basics of gravitational screening and how it might mask
modifications to gravity in certain regimes. In Section 5 I talk through the
various, existing constraints on gravity with large scale observables using
existing surveys while in Section 6 I focus on constraints involving the non-
linear regime and environmental effects. Finally, in Section 7, I map out the
future of the field.

##  2 Modern Cosmology.

###  2.1 The expanding Universe.

The evolution of the Universe can be predicted from the field equations of GR,
under the assumption that the metric of space-time, $g_{\alpha\beta}$ is
homogeneous and isotropic, $g_{\alpha\beta}={\rm diag}(-1,a^{2}\delta_{ij})$
where $a$ is the scale factor and only depends on time (for simplicity we will
assume spatial flatness throughout). The dynamics of $a$ is given by

$\displaystyle H^{2}\equiv\left(\frac{\dot{a}}{a}\right)^{2}=\frac{8\pi
G}{3}{\bar{\rho}}$ (1)

where overdot denotes a derivative with regards to time and ${\bar{\rho}}$ is
the mean density of the Universe. The acceleration of the scale factor is
given by

$\displaystyle\frac{\ddot{a}}{a}=-\frac{4\pi G}{3}({\bar{\rho}}+3{\bar{P}})$
(2)

where ${\bar{P}}$ is the total mean pressure of the Universe. Equation 1 can
be rewritten in terms of the fractional energy densities of the various
components, $\Omega_{X}={8\pi G\bar{\rho}}_{X}/(3H_{0}^{2})$ (where $X$ can
be: radiation, R, baryons, B, cold dark matter, C, dark energy, DE) such that

$\displaystyle H^{2}(a)=H^{2}_{0}\left[\frac{\Omega_{\rm
R}}{a^{4}}+\frac{\Omega_{\rm B}+\Omega_{\rm C}}{a^{3}}+\frac{\Omega_{\rm
DE}}{a^{3(w+1)}}\right]$ (3)

(where for simplicity we have assumed here a constant $w$). Clearly, a
measurement of $H$ as a function of time or scale factor, $a$, can be used to
place constraints on the $\Omega_{X}$ and on the equation of state of dark
energy, $w={\bar{P}}_{\rm DE}/{\bar{\rho}}_{\rm DE}$.

Distant objects will have their light redshifted: $\nu_{\rm emitted}/\nu_{\rm
observed}=a_{0}/a\equiv 1+z$ where the ratio of the emitted and observed
frequencies depend on the scale factor at emission (a) and at the observer
$a_{0}$, and where we have defined the redshift $z$. The relation between
distance and redshift can be reconstructed through Equation 1. For example,
for a Euclidean universe, the angular-diameter distance, $D_{A}$, (i.e. the
distance reconstructed from knowing the physical size and measuring the angle
it subtends in the sky) is given by

$\displaystyle D_{A}(z)=\frac{1}{1+z}\int_{0}^{z}\frac{dz}{H(z)}$ (4)

and the corresponding luminosity distance is given by $D_{L}=(1+z)^{2}D_{A}$.
By measuring redshifts and distances of stars, galaxies, supernovae and
matching them to $D_{A}(z)$ or $D_{L}(z)$ at $z$ close to $0$ we can estimate
$H(z=0)\equiv H_{0}$; by measuring more distant objects, we can infer other
cosmological parameters such as the $\Omega_{X}$’s and $w$.

The expansion of the Universe has an impact on matter and radiation.
Throughout its history, the Universe has undergone a series of transitions as
the overall temperature decreases. One of the most striking transitions occurs
when the temperature is of a few thousand degrees Kelvin, corresponding to a
few tenths of an electron-Volt. At that time, when the Universe was about
380,000 years old, at a redshift, $z\simeq 1100$, free electrons and protons
combined to form neutral Hydrogen. As a result, the ambient medium which was,
until then, optically thick, became transparent; photons which last scattered
with residual electrons and protons, remained free to propagate along null
geodesics. As a result, we are constantly bombarded by a relic bath of
radiation with black-body distribution with temperature, ${\bar{T}}=2.735^{o}$
Kelvin and an almost perfectly isotropic distribution. Indeed, it is useful to
think of these relic photons as having been emitted at time $t_{*}$ from the
surface of sphere – the surface of last scatter – centred on us.

###  2.2 Linear cosmological perturbation theory.

It is, of course, with the study of large scale structure that cosmology has
really taken off. Going beyond a homogeneous and isotropic universe, one needs
to include inhomogeneities. This can be done by perturbing the metric,
$g_{\alpha\beta}={\rm diag}[-(1+2\Phi),a^{2}(1-2\Psi)\delta_{ij})]$ (in the
Newtonian gauge), the various densities,
$\rho_{X}={\bar{\rho}}_{X}(1+\delta_{X})$ and pressures,
$P_{X}={\bar{P}}_{X}(1+\delta_{PX})$ and the temperature of the CMB,
$T={\bar{T}}(1+\frac{\delta T}{\bar{T}})$; note that $\Phi$, $\Psi$,
$\delta_{X}$ and $\delta_{PX}$ are functions of time, $t$ and position,
${\vec{x}}$ while $\frac{\delta T}{\bar{T}}$ is function of $t$, ${\vec{x}}$
and direction ${\hat{n}}$.

The evolution of the perturbations can be found through cosmological
perturbation theory: taking the full equations for gravity (the Einstein field
equations) along with the equations for conservation of mass and momentum (the
relativistic Euler equation) and perturbing to linear order. So, for example,
in Fourier space, a combination of the $00$ and $0i$ component of the Einstein
field equations yields something akin to the relativistic Newton-Poisson
equation,

$\displaystyle-k^{2}\Phi=4\pi G\Delta\rho$ (5)

with
$\Delta\rho=\sum_{X}{\bar{\rho}}_{X}[\delta_{X}+3(1+w_{X})\frac{H}{a}\frac{\theta_{X}}{k^{2}}]$
and $\theta_{X}={\vec{\nabla}}\cdot{\vec{v}}_{X}$ where ${\vec{v}}_{X}$ is the
peculiar velocity of component $X$ and $k$ labels the Fourier mode. And the
traceless, transverse component (in the absence of anisotropic stress) of the
Einstein field equation give

$\displaystyle k^{2}(\Phi-\Psi)=0$ (6)

The perturbed conservation equations (i.e. the linearized
$\nabla_{\mu}T^{\mu\nu}=0$ equation, where $\nabla$ is the covariant
derivative and $T^{\mu\nu}$ is the energy momentum tensor) can be used find
additional evolution equations for $\delta_{X}$ and $\theta_{X}$. With a
suitable definition of the sound speed, $c^{2}_{X}$ one can replace
$\delta_{PX}$.

###  2.3 The observables: power spectra, growth rate and cosmic shear.

When studying the perturbations around a homogeneous and isotropic background,
statistics is paramount. First of all, and given that the (ensemble) average
(or mean) of all linear perturbations is zero, variance plays a key role. Thus
the relevant quantities that need to be considered are the power spectrum of
density fluctuations, $P(k)$, and the angular power spectrum of CMB
fluctuations, $C_{\ell}$, given by

$\displaystyle\langle\delta^{*}_{M}({\bf k}^{\prime})\delta_{M}({\bf
k})\rangle$ $\displaystyle=$ $\displaystyle(2\pi)^{3}P(k)\delta^{(3)}({\bf
k}-{\bf k}^{\prime})$ $\displaystyle\langle
a^{*}_{\ell^{\prime}m^{\prime}}a_{\ell m}\rangle$ $\displaystyle=$
$\displaystyle C_{\ell}\delta_{\ell\ell^{\prime}}\delta_{mm^{\prime}}$ (7)

where $\delta_{M}({\bf k})$ is the Fourier transform of
$\delta_{B}+\delta_{C}$, and $a_{\ell m}$ is the spherical harmonic transform
of $T({\bf n})$ (or of any two-dimensional observable on a sphere). The
various power spectra (and cross-spectra, i.e. the cross correlation between
different fields) are at the heart of the bulk of cosmological analysis.

Of particular importance is the rate of growth of cosmic structure. The linear
density contrast can be factorized, $\delta_{M}({\bf k},t)=D(t)\delta({\bf
k},t_{0})$ where $t_{0}$ is the physical time today. In the case of a matter
dominated Universe (or a Universe with a cosmological constant), the fact that
the time dependence of the evolution is scale-independent is strictly
satisfied. One can then define two useful quantities, the growth rate,

$\displaystyle f\equiv\frac{d\ln D}{d\ln a}$ (8)

and the ”density weighted growth rate”, $f\sigma_{8}$, where
$\sigma^{2}_{8}=\langle(\delta M)^{2}/{\bar{M}}^{2}\rangle|_{8h^{-1}{\rm
Mpc}}$, i.e. the mass-variance on $8h^{-1}{\rm Mpc}$ scales. These quantities
map out the time dependence of gravitational collapse and will play a key role
in cosmological tests of gravity. A fair representation of the current
observational status of the growth can be found in Figure 1: the shaded line
shows the expected growth rate from CMB constraints (dominated by density and
potential fluctuations at $z\simeq 1100$) and projected forward assuming
$\Lambda$CDM, which should be compared with individual and independent
constraints (i.e. the various dots with errors bars) from galaxy surveys which
are (necessarily) at late redshift.

Figure 1: The density weighted growth weight as a function of redshift
(Planck, 2018a). The shaded line is the extrapolated value for $\Lambda$CDM,
marginalized over the parameters subjected to the Planck 2018 data. The shapes
with error bars are a selection of independent measurements from large scale
structure surveys.

The anisotropies in the CMB contain information about the radiation density
contrast, $\delta_{\rm\gamma}$, the baryon velocity, ${\vec{v}}_{\rm B}$ and
potential fluctuations, $\Phi$ at last scattering (known as the Sachs-Wolfe
effect) as well as potential fluctuations from last scattering until now
(known as the Integrated Sachs-Wolfe effect) through:

$\displaystyle\frac{\delta
T}{T}({\hat{n}})=\frac{1}{3}\delta_{\gamma}|_{*}-{\vec{v}}_{\rm
B}\cdot{\hat{n}}|_{*}+\Phi|_{*}-\int_{t_{*}}^{t_{0}}({\dot{\Phi}}+{\dot{\Psi}})\frac{dt}{a}$
(9)

where subscript $*$ means evaluated at the time of last scattering, $t_{*}$.
Furthermore, the CMB will be slightly polarized due to the finite thickness of
the surface of last scattering and by the reionized electrons at later times.
In the left hand panel of Figure 2 we show the angular power spectra of the
anisotropy in CMB intensity and polarization; one can see distinctive features
such as a series of oscillations due to the sound waves in the baryon-photon
plasma before recombination as well as the damping on small scales (i.e. large
$\ell$) that arises from the thickness of the surface of last scattering.

The CMB will also be affected by the intervening structure, from the surface
of last scatter until now. As the photons propagate through the perturbed
space-time, they will be deflected and the structure of hot and cold spots
will be distorted and deformed – the CMB will be lensed. One way the effect
will manifest itself is be distorting the angular power spectrum of the CMB;
specifically it will very slightly smooth out the peaks and troughs arising
from the baryon-photon oscillations generated before recombination. Another
way lensing affects the CMB is by modifying the statistics of $\frac{\delta
T}{T}$; what was originally a multivariate Gaussian will become non-Gaussian
due to lensing and by measuring the 4-point function (or equivalently the
trispectrum) and seeing how it deviates from the Gaussian value, it is
possible to reconstruct the angular power spectrum of the lensing potential,
$C^{\phi\phi}_{\ell}$. In the right hand panel of Figure 2 we show the angular
power spectrum of the reconstructed lensing potential, i.e. of
$C^{\phi\phi}_{\ell}$.

Figure 2: Left hand panel: (top) the angular power spectrum of anisotropies of
the CMB from the Planck 2018 data release (Planck, 2018a) and (bottom) the
difference between the data and the best fit theoretical model ($\Lambda$CDM).
Right hand panel: the angular power spectrum of the CMB lensing potential
(orange) from Planck and a selection of best fit models to different
combinations of additional data sets.

Distant galaxies will also be lensed by intervening structure. Measurements of
distortions of the shapes of galaxies, i.e. changes in the projected
ellipticities, and fluctuations in magnitudes, can be directly related to the
distortion tensor

$\displaystyle\left(\begin{array}[]{cc}1-\kappa-\gamma_{1}&-\gamma_{2}\\\
-\gamma_{2}&1-\kappa+\gamma_{1}\end{array}\right)=\delta_{ij}+\int_{0}^{\chi}d\chi^{\prime}(\Phi+\Psi)_{ij}[{\vec{x}}(\chi^{\prime})]\chi^{\prime}\left(1-\frac{\chi^{\prime}}{\chi}\right)$
(12)

where the $\gamma_{i}$ are the shears and $\kappa$ is the convergence,
$\chi=\int_{t_{0}}^{t}\frac{dt^{\prime}}{a(t^{\prime})}$ is the conformal
distance and ($i$,$j$) are directions perpendicular to the direction of the
light ray. One can attempt to reconstruct the gravitational potential probed
by the incoming light rays by inverting equation 12 and, for example, relating
it to underlying mass distribution through the Poisson equation. This method
has been successfully used for reconstructing the mass distribution of
clusters of galaxies as well as for mapping out the large scale distribution
of dark matter. Alternatively, one can study the auto and cross correlations
of $\kappa$ and $\gamma_{i}$ as well as cross correlations with other measures
of the galaxy density fluctuations.

###  2.4 Small scales and non-gravitational physics.

While much of what has been discussed is primarily applicable to the largest
scales, where linear perturbation theory is applicable, the observables can
also be used on smaller scales. Furthermore, we expect the smallest scales to
have a larger statistical weight – there are more modes contributing to any
particular quantity. But while the statistical weight will improve, the
underlying physics becomes more complex as non-linear gravitational collapse
plays a role, allied with non gravitational physics. Baryons, through the role
of gas and stars or highly energetic processes like supernovae and active
galactic nuclei can play a significant role in moulding structure on scales of
Megaparsecs and below (Chisari et al., 2018).

There has been considerable progress in developing both analytic techniques
(through different versions of perturbation theory (Bernardeau et al., 2002)
and effective field theory (Carrasco et al., 2012)) and numerical algorithms
(with N-body simulations, combined N-body and hydrodynamic simulations and,
more recently, machine learning algorithms). In this, more complex regime, one
can look at a more varied set of statistics, above and beyond the variances
described above. For example, three point and four point statistics (i.e.
which are cubic or quartic in the perturbed fields) will have additional
information about the physical processes at play.

A different point of view is to look at what seem to be the building blocks of
large scale structures, i.e. galaxies and clusters of galaxies or, from the
CDM paradigm, halos of dark matter. This more ”granular” view of the
distribution of matter can be used to constrain a self consistent model – the
halo model – which quantifies the structure of the universe in terms of the
density profile of individual halos, $\rho(r)$, the number density of halos of
a give mass, $n(M)$ and the linear powerspectrum of fluctuations on the
largest scales (Cooray & Sheth, 2002).

While there has been tremendous progress in developing efficient quantitative
methods for the non-linear regime, and they, arguably, may be of sufficient
accuracy for the analysis of current data sets, much work still needs to be
done to develop a fully accurate and general approach to this regime.

###  2.5 $\Lambda$CDM

Putting all the pieces together, it is possible to come up with a consistent
picture of what the Universe looks like on large scales. It is accurately
described by GR with a cosmological constant ($\Lambda$) making up roughly
$70\%$ of the energy budget, pressureless cold dark matter (CDM) making up
roughly $25\%$ of the energy budget and baryons making up $5\%$. The evidence
for this model has accumulated over two decades. The first compelling evidence
came from measurements of the luminosity distance through observations of
distant supernovae IA (SCP, 1998; Riess et al., 1998). Crucially, a few years
later, measurements of the angular power-spectrum of the CMB from ground-based
and balloon borne experiments found the first constraints on the geometry of
the Universe (BOOMERanG, 2000; MAXIMA, 2000; Miller et al., 2002), ruling out
curved models and singling out a flat Universe with a cosmological constant.
Constraints from the abundance of clusters (SDSS, 2002) placed constraints on
the fractional energy density in dark matter, $\Omega_{M}$, further cementing
$\Lambda$CDM as the preferred model for describing the Universe. Subsequent
cosmological observations, which we will allude to in Section 5 have led to
precision constraints (in some case at the sub-percent level) on the
parameters of $\Lambda$CDM.

##  3 Modified gravity and cosmology.

###  3.1 What is General Relativity?

Before we delve into what it means to modify gravity, let us establish what we
mean by standard gravity. We take it to be Einstein’s General Theory of
Relativity, where the gravitational field is a metric, $g_{\alpha\beta}$ whose
dynamics is described by the Einstein-Hilbert action, and which is minimally
coupled to matter. In other words, gravity is encapsulated in the action

$\displaystyle\frac{1}{16\pi G}\int d^{4}x\sqrt{-g}(R-2\Lambda)+\int
d^{4}x\sqrt{-g}{\cal L}_{M}(g_{\alpha\beta},\cdots)$ (13)

where $G$ is Newton’s constant, $g$ is the determinant of $g_{\alpha\beta}$,
$R$ is the Ricci scalar, $\Lambda$ is the cosmological constant and ${\cal
L}_{M}$ is the Lagrangian density for the matter and radiation fields which
are minimally coupled to the metric. It is often convenient to work with the
reduced Planck mass, $M^{2}_{\rm Pl}=1/8\pi G$. The equations of motions are
then given by

$\displaystyle G_{\mu\nu}\equiv R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=-\Lambda
g_{\mu\nu}+8\pi GT_{\mu\nu}.$ (14)

where we have defined the Einstein tensor, $G_{\mu\nu}$.

GR is an incredibly special theory, as we hope we made clear in Section 1.
Before we move venture into the unknown, it is useful to signal potential
pitfalls and problems that arise when one tries to consider extensions. To
begin with, GR is endowed with properties that make it a predictive theory.
The equations are highly non-linear and, a-priori, there is no reason to
assume that it allows a well-posed initial value problem. In other words,
given a set of initial conditions, is there a unique and well-defined
evolution for the metric, given by equations 14? It turns out that there is
(Choquet-Bruhat & Geroch, 1969) and this means that it is possible to trust
numerical solutions to the evolution equations, even in the mildly strong
field regime such as, for example, in binary black hole mergers (Lehner &
Pretorius, 2014). The same cannot be said for the types of extensions we will
see below (Allwright & Lehner, 2018).

Furthermore, as we shall see, extensions to GR almost universally involve
considering new degrees of freedom which, quite easily, may be unstable in one
form or another. If one considers equations of motions that have higher
derivatives, these may suffer from the Ostragradski instability (Woodard,
2007); they may also have ghosts or tachyonic instabilities that may render
these theories phenomenologically unviable. Any attempt at venturing beyond GR
must bear these serious problems in mind.

###  3.2 Modified gravity as extra degrees of freedom.

In the introduction, we stated Lovelock’s theorem and we mentioned that it was
a useful guide to how we might deform gravity away from Equation 13. Let us
now use it to explore a (by no means exhaustive) set of alternatives. For a
start, let us consider adding an extra, scalar degree of freedom. One of the
most iconic alternatives to GR is Jordan-Brans-Dicke (JBD) gravity (Brans &
Dicke, 1961) in which the Einstein-Hilbert term is transformed as
$\frac{M^{2}_{\rm Pl}}{2}R\rightarrow\varphi R$ and the scalar field is
endowed with dynamics with a kinetic energy of the form
$\omega(\partial_{\mu}\varphi\partial^{\mu}\varphi)/\varphi$. In practice this
involves making the Planck mass (or Newton’s constant) dynamical, with one
free parameter, $\omega$. JBD gravity is a particular example of a scalar-
tensor theory for which there is an (almost) complete description if one
requires (as in Lovelock’s theorem) that the equations of motion are $2^{\rm
nd}$ order. Known as the Horndeski (or Covariant Galileon) action (Horndeski,
1974; Deffayet et al., 2009; Koboyashi et al., 2011), it is given by:

$\displaystyle S=\int d^{4}x\sqrt{-g}\left\\{\sum_{i=2}^{5}{\cal
L}_{i}[\varphi,g_{\mu\nu}]+{\cal L}_{M}[g_{\mu\nu},\cdots]\right\\},$ (15)

and the building blocks of the scalar field lagrangian are

$\displaystyle{\cal L}_{2}$ $\displaystyle=$ $\displaystyle K,$
$\displaystyle{\cal L}_{3}$ $\displaystyle=$ $\displaystyle-G_{3}\Box\varphi,$
$\displaystyle{\cal L}_{4}$ $\displaystyle=$ $\displaystyle
G_{4}R+G_{4X}\left\\{(\Box\varphi)^{2}-\nabla_{\mu}\nabla_{\nu}\varphi\nabla^{\mu}\nabla^{\nu}\varphi\right\\},$
$\displaystyle{\cal L}_{5}$ $\displaystyle=$ $\displaystyle
G_{5}G_{\mu\nu}\nabla^{\mu}\nabla^{\nu}\varphi-\frac{1}{6}G_{5X}\big{\\{}(\nabla\varphi)^{3}-3\nabla^{\mu}\nabla^{\nu}\varphi\nabla_{\mu}\nabla_{\nu}\varphi\Box\varphi$
(16)
$\displaystyle+2\nabla^{\nu}\nabla_{\mu}\varphi\nabla^{\alpha}\nabla_{\nu}\varphi\nabla^{\mu}\nabla_{\alpha}\varphi\big{\\}}\,.$

We have that $K$ and $G_{A}$ are functions of $\varphi$ and
$X\equiv-\nabla^{\nu}\varphi\nabla_{\nu}\varphi/2$, and the subscripts, $X$
and $\varphi$, denote derivatives. The four functions, $K$ and $G_{A}$
completely characterize this class of theories. Further extensions have been
found that preserve the $2^{\rm nd}$ order nature of the equations of motion
(Zumalacarregui & Garcia-Bellido, 2014; Gleyzes et al., 2015a).

Of particular interest are theories in which additional symmetries are imposed
and which will, through naturalness arguments, be protected at the quantum
level. Of particular note are Galileon theories (Nicolis et al., 2009) where
the action is invariant under field redefinitions of the form
$\varphi\rightarrow\varphi+c_{\mu}x^{\mu}+d$. Other possibilities are scale
invariant theories (Blas et al., 2011; Garcia-Bellido et al., 2011; Ferreira
et al., 2016), invariant under joint field/metric transformations of the form
$\varphi\rightarrow\lambda^{-1}\varphi$ and
$g_{\alpha\beta}\rightarrow\lambda^{2}g_{\alpha\beta}$.

There are also other possible choices for extra fields. So, for example, one
can opt for a four-vector, $A_{\mu}$, leading to a different class of
theories. Of particular note are Einstein-Aether theories (Jacobson &
Mattingley, 2000) (and their generalizations –Zlosnik et al. (2007)) in which
one imposes a time-like constraint, $A_{\mu}A^{\mu}+1=0$; such theories are
used as models for spontaneously broken Lorentz symmetry. Other possibilities
are extensions of the Maxwell-Proca action, in which the vector field is given
an explicit mass (Heisenberg, 2014). It is possible to go yet further and
include another tensor field, i.e. another metric $f_{\alpha\beta}$, as the
extra degree of freedom. Such theories have garnered a fair amount of
attention recently: by constructing a potential, $V(g^{-1}f)$ one can obtain a
theory of a massive graviton which isn’t plagued by instabilities, most
notably ghosts (Hinterbichler, 2012; De Rahm et al., 2011; Hassan & Rosen,
2012). There are, of course, a myriad of possibilities, adding multiplets of
fields or a combination of fields with different spins (Clifton et al., 2012).

###  3.3 Alternatives: higher derivatives, higher dimensions and non-locality.

Instead of explicitly adding extra fields, it is possible to do so implicitly
in a variety of different ways. One of the simplest ways is to replace $R$ in
the Einstein-Hilbert action by a function of other curvature invariants,
$R\rightarrow
f(R,R^{2},R_{\alpha\beta}R^{\alpha\beta},R_{\alpha\beta\mu\nu}R^{\alpha\beta\mu\nu},\cdots)$
where $R_{\alpha\beta}$ and $R_{\alpha\beta\mu\nu}$ are the Ricci and Riemann
tensors. In doing so, one introduces higher derivative terms in the action
which is tantamount to adding new solutions to the equation of motion or,
equivalently, adding new degrees of freedom. This can be made clear in the
particular case where the $f$ depends solely on the Ricci scalar (Sotiriou &
Faraoni, 2010); by defining a new scalar field, $\varphi\sim\ln df/dR$, the
theory can be re-expressed as scalar-tensor theory, a subset of the theories
discussed above.

One approach which emerges from attempts at unification, such as string
theory, is to enlarge the number of dimensions. A notable example is the
Kaluza-Klein model (Clifton et al., 2012) in which there is one extra
dimension and the dynamics is given by the 5-dimensional Einstein-Hilbert
action for the metric $g_{AB}$ where $A$, $B$ take values from $0$ to $4$. By
redefining the metric as
$g_{AB}=e^{\varphi/\sqrt{3}}g_{\mu\nu}+e^{-\varphi/\sqrt{3}}A_{\mu}A_{\nu}$
when $A$, $B$ take values from $0$ to $3$, $g_{\mu
4}=e^{-\varphi/\sqrt{3}}A_{\mu}$ and $g_{44}=e^{-\varphi/\sqrt{3}}$, it is
possible to trivially equate a higher dimensional theory with a lower
dimensional theory with extra fields (albeit in which the fields have an extra
coordinate dependence on $x^{4}$). By compactifying the extra dimension or by
confining the dynamics to a 3-space (Arkani-Hamed et al., 1998; Randall &
Sundrum, 1999), it is possible to obtain an approximate, but not exactly,
$3+1$ universe with modified gravitational dynamics (given by extra fields). A
particularly popular model – the DGP model – was proposed in (Dvali et al.,
2000) of a three dimensional ”brane” living in a five dimensional space.
Through a judicious choice of the brane geometry and tension, vis-a-vis the
background space-time it is possible to obtain a ”self”-accelerating Universe
which mimics one with a cosmological constant (or with dark energy).
Furthermore, deviations from GR emerge on large scales (and not on small
scales as one expects in ”conventional” higher dimensional universes).

The final, more exotic, direction in which we can take this is by considering
non-locality (Deser & Woodard, 2007; Maggiore & Mancarella, 2014). It is
possible to, for example, include terms in the action of the form $R\times
R/\Box$ where the inverse D’Alembertian can now be interpreted as a Greens
function (which is by definition non-local). Interesting effects arise in a
cosmological context due to the non-locality in time and it can be shown that
such theories are equivalent to scalar tensor theories by replacing
$\Box\varphi\equiv R$ and imposing strict constraints on the initial
conditions in such a way as to kill all spurious modes arising from the
homogenous solutions to this equation.

###  3.4 The speed of gravitational waves and GW170817

Before I delve into the cosmological regime of gravitational theories, it is
important to briefly highlight a recent event. The discovery of a binary
neutron star (BNS) merger with both gravitational waves and a multitude of
electromagnetic signals has heralded the advent of multi-messenger astronomy
(LIGO, 2017a). One of the remarkable bi-products has been the ability to place
stringent constraints on the speed of gravitational waves: it is within
$10^{-15}$ of the speed of electromagnetic waves. This fact allows us to
severely restrict the range of allowed modified gravity theories that have an
impact on cosmological scales (Baker et al., 2017; Creminelli & Vernizzi,
2017; Ezquiaga & Zumalacarregui, 2017) – the extra degrees of freedom will
serve as a medium through which the gravitational waves propagate and,
depending on their coupling to gravity, may affect their speed. It has been
shown that, for example, scalar-tensor theories are strongly constrained by
the multi-messenger detection of the BNS, as is a broad swathe of vector-
tensor theories.

###  3.5 General theories of gravity on cosmological scales.

I briefly covered a variety of ways in which one can extend GR and we have
seen that, in general, it can be seen as adding extra degrees of freedom. But
we have also seen that the landscape of possible extensions is vast and
resides in large functional space of possibilities. This will give pause for
thought: given that there is a restricted amount of observational data, the
task of comprehensively constraining GR seems difficult at best. The key to
progress is to realize that, in this endeavour, one should focus on specific
regimes in which certain characteristics of the theory may simplify. Indeed,
this is standard practice in high-energy physics, when one focuses on the
relevant aspects of a theory at a given energy scale (for example, at the beam
luminosity of a particular accelerator). Furthermore, this approach is already
standard practice in, for example, the analysis of gravitational data on the
scale of the Solar System. There, an efficient formalism for tackling weak
field, non-relativistic systems – the Parametrized Post Newtonian (PPN)
formalism – has been used to characterize (non)-deviations from GR of various
forms (Thorne & Will, 1971). It would make sense, then, to apply the same
rationale to cosmology.

At its most basic level, cosmology is about the expansion of the Universe,
i.e. a detailed description of the time evolution of the scale factor, $a(t)$.
As we have seen in GR, this allows us to disentangle the different
constituents of the Universe, by constraining the $\Omega_{X}$ in the Friedman
equation. Of particular interest is the fact that we may be able to constrain
the equation of state of the dark energy component, $w_{\rm DE}$. In other
words, observables such as $H(z)$ or $D_{A}(z)$ can be converted into
constraints on $w_{\rm DE}(z)$ which in turn can be used to extract the
fundamental nature of the dark energy (i.e. if it is a cosmological constant,
if it is a scalar field with a particular form of the potential, $V(\varphi)$,
etc).

How does the situation change if one now wishes to constrain gravity itself?
Unfortunately, not very much. To see why this is so, we need to realize that
any modification to GR can be written in the form

$\displaystyle G_{\mu\nu}=-\Lambda g_{\mu\nu}+8\pi
GT_{\mu\nu}+U_{\alpha\beta}$ (17)

where $U_{\alpha\beta}$ can be a functional of $g_{\alpha\beta}$, the matter
fields and any extra degree of freedom (such as scalar, vector and tensor
fields). If we now impose homogeneity and isotropy, we have that
$U_{\alpha\beta}={\rm diag}(8\pi G\rho_{U},8\pi GP_{U}\delta_{ij})$ where
$(\rho_{U},P_{U})$ are two free functions of time; note that we have chosen a
notation that shows that, at the background level, modifying GR is essentially
equivalent to adding a new energy-momentum tensor, which can also be
characterized by an equation of state, $w_{\rm U}$. This means that, at the
background level, modifying GR is indistinguishable from adding in a dark
energy fluid (Kunz, 2012) and it is impossible to distinguish between the two
with observables such as $H(z)$ or $D_{A}(z)$.

Given the paucity of information in the background, we need to delve a level
deeper in complexity and look at linear perturbations of homogeneous and
isotropic background space-time. As we saw, in GR, this leads to a series of
$2^{\rm nd}$ order coupled differential equation (as well as constraints) for
the perturbed variables $\delta_{X}$, $\theta_{X}$, $\Phi$ and $\Psi$. If we
now go beyond GR, but remain in the arena of linear perturbations, we can
easily guess what the structure of these ”extended” equations might be (Baker
et al., 2012). They will be linear and (for now) $2^{\rm nd}$ order but they
will include new, extra degrees of freedom. So, for example, if the extra
degree of freedom arises from a scalar field, $\varphi$, we will now need to
include its perturbation, $\delta\varphi$.

Let us now take two of the perturbed Einstein field equations and generalize
them. We can modify Equation 5 to

$\displaystyle-k^{2}\Phi=4\pi
G\Delta\rho+A_{0}k^{2}\Phi+F_{0}k^{2}+\alpha_{0}k^{2}\delta\varphi+\alpha_{1}k\delta{\dot{\varphi}}+\cdots\Gamma$
(18)

and Equation 6 to

$\displaystyle
k^{2}(\Phi-\Psi)=D_{0}k^{2}\Phi+D_{1}k{\dot{\Phi}}+K_{0}k^{2}\Gamma+K_{1}k{\dot{\Gamma}}+\epsilon_{0}k^{2}\delta\varphi+\epsilon_{1}k\delta{\dot{\varphi}}+\epsilon_{2}\delta{\ddot{\varphi}}$
(19)

where $k\Gamma=(\dot{\Phi}+{\cal H}\Psi)$ and $A_{0}$, $D_{0}$, $D_{1}$,
$F_{0}$, $K_{0}$, $K_{1}$, $\alpha_{0}$, $\alpha_{1}$, $\epsilon_{0}$,
$\epsilon_{1}$ and $\epsilon_{2}$ are free functions of time and $k$; setting
these free functions to zero recovers GR but, in principle, one can span the
most general class of gravitational theories which involve an extra scalar
field. The generality of this approach is appealing yet one is clearly faced
with a challenge: is it feasible to, with a finite amount of cosmological
data, constrain these multiple functions of time and space?

We have been working at the level of the equations of motion but, in almost
full generality, alternative theories to GR will obey an action principle.
This means that it would make more sense to extend the action itself. We have
already seen that at the full, non-linear level, with the Horndeski action
presented in Equation 15. But if one limits oneself to the most general theory
at the linear level, the exercise simplifies greatly: one simply has to
construct the most general action, quadratic in the perturbation variables and
which satisfies any desired symmetry principle, the most important of which is
linearized diffeomorphism invariance, i.e. invariance under transformations of
the type $x^{\alpha}\rightarrow x^{\alpha}+\xi^{\alpha}$.

There are a few ways in which one can go about constructing the extend linear
action. A systematic way is to write the most general quadratic action and
then impose symmetries to reduce the number of terms (Battye & Pearson, 2011;
Lagos at al, 2016; Lagos & Ferreira, 2016; Lagos at al, 2016). This has been
shown to be general and applicable to any type of extra degree of freedom:
scalars, vectors and tensors. If one restricts oneself to the case where the
extra degree of freedom is a scalar field, a simpler, more efficient approach
can be taken with what has been dubbed the Effective Field Theory (EFT)
approach (Creminelli et al., 2009; Gubitosi et al., 2012; Bloomfield et al.,
2012; Gleyzes et al., 2013, 2015b): starting with a general action which only
depends on metric perturbations, one then performs a linearized diffeomorphism
transformation, in which $\xi^{0}=\delta\varphi$ and, in such a way one
generates the most general linearized scalar-tensor action.

It turns out that, in the case of scalar-tensor theories, there is a
straightforward approach of obtaining the most general action for linear
perturbations: one takes the Horndeski action of Equation 15 and Taylor
expands to $2^{\rm nd}$ order in the perturbations. If one does that, one
finds that, instead of depending on a multitude of free functions of time and
scale, as we saw above, the resulting action merely depends on four functions
of time, $\alpha_{A}(t)$ with $A=K,B,M,T$ which can be directly related to the
original action (Bellini & Sawicki, 2014). These functions are

$\displaystyle M^{2}_{*}$ $\displaystyle\equiv$ $\displaystyle
2\left(G_{4}-2XG_{4X}+XG_{5\varphi}-{\dot{\varphi}}HXG_{5X}\right),$
$\displaystyle HM^{2}_{*}\alpha_{M}$ $\displaystyle\equiv$
$\displaystyle\frac{d}{dt}M^{2}_{*},$ $\displaystyle H^{2}M^{2}_{*}\alpha_{K}$
$\displaystyle\equiv$ $\displaystyle
2X\left(K_{X}+2XK_{XX}-2G_{3\varphi}-2XG_{3\varphi X}\right)$
$\displaystyle+12\dot{\varphi}XH\left(G_{3X}+XG_{3XX}-3G_{4\varphi
X}-2XG_{4\varphi XX}\right)$
$\displaystyle+12XH^{2}\left(G_{4X}+8XG_{4XX}+4X^{2}G_{4XXX}\right)$
$\displaystyle-12XH^{2}\left(G_{5X}+5XG_{5\varphi X}+2X^{2}G_{5\varphi
XX}\right)$
$\displaystyle+14\dot{\varphi}H^{3}\left(3G_{5X}+7XG_{5XX}+2X^{2}G_{5XXX}\right),$
$\displaystyle HM^{2}_{*}\alpha_{B}$ $\displaystyle\equiv$ $\displaystyle
2\dot{\varphi}\left(XG_{3X}-G_{4\varphi}-2XG_{4\varphi X}\right)$
$\displaystyle+8XH\left(G_{4X}+2XG_{4XX}-G_{5\varphi}-XG_{5\varphi X}\right)$
$\displaystyle+2\dot{\varphi}XH^{2}\left(3G_{5X}+2XG_{5XX}\right),$
$\displaystyle M^{2}_{*}\alpha_{T}$ $\displaystyle\equiv$ $\displaystyle
2X\left[2G_{4X}-2G_{5\varphi}-\left(\ddot{\varphi}-\dot{\varphi}H\right)G_{5X}\right]\,.$
(20)

Each of these functions are linked to specific physical properties of the
theory: $M^{2}_{*}$ and $\alpha_{M}$ are related to time variations in the
background Newton’s constant, $\alpha_{K}$ to the generalized canonical
kinetic term of simple DE models, $\alpha_{B}$ quantifies kinetic mixing
between $\varphi$ and the scalar perturbations of the metric and $\alpha_{T}$
is associated to modifications to the speed of propagation of tensor modes.

The fact that general theories of gravity can, at the level of linearized
cosmological perturbations, be completely characterized by a handful of free
functions is true in general, including, for example, the case where the extra
degrees of freedom are 4-vectors or tensors. And it gives hope to the idea
that it might be feasible to constrain gravity on large scales.

###  3.6 The quasi-static regime.

It turns out that there is a further simplification if one restricts oneself
to sufficiently sub-horizon scales, i.e. scales in which $k/(aH)\sim k\eta\gg
1$ where $ad\eta=dt$ allows us to define the conformal time $\eta$. Known as
the ”quasi-static” regime (the cosmological equivalent of the Newtonian
limit), this covers scales up to hundreds of Megaparsecs and thus encompasses
most of the current and future large scale structure surveys under
consideration. In that regime, it is an excellent approximation to effectively
freeze out the dynamics of the extra degree of freedom; for example, in the
case of a scalar field, this means approximating
$\Box\varphi\propto{\rm"source"}$ by $-\nabla^{2}\varphi\propto{\rm"source"}$.
Replacing the extra degree of freedom in the generalized Einstein field
equations leads to modified equations 5 and 6:

$\displaystyle-k^{2}\Phi$ $\displaystyle=$ $\displaystyle 4\pi G\mu\Delta\rho$
(21) $\displaystyle\Phi$ $\displaystyle=$ $\displaystyle\gamma\Psi$ (22)

where $\mu$ is the modified, effective Newton’s constant (and can be rewritten
as $\mu=G_{\rm eff}/G_{0}$) and $\gamma$ is often called the gravitational
slip (Hu & Sawicki, 2007; Bertschinger & Zukin, 2008; Amin et al., 2009). The
quasi-static parameters $\mu$ and $\gamma$ are generally functions of time and
scale, $k$; typically the scale dependence either kicks in on very small
scales (i.e. deeply in the non-linear regime) or naturally, close to the
cosmological horizon when the quasi-static approximation breaks down. This
means, that, effectively, when constraining theories of gravity on
cosmological scales, one is constraining two free functions of time. An
approximate approach is to extend this parametrization to all scales,
including super-horizon scales, greatly simplifying any attempts at
cosmological parameter estimation.

Note that, it is customary to assume that the quasi-static regime is
synonymous with the use of $\mu$ and $\gamma$ on subhorizon scales – we will
do so in this review. But we need to point out that this is not entirely
general and that in Beyond Horndeski theories (Mancarella, 2017), there will
be additional terms in equations 21 and 22 proportional to the matter velocity
which also have to be taken into account; this means one would need to extend
the number of functions in the quasi-static regime from two to four.

Over the past few years, quite a few variants of $\mu$ and $\gamma$ have been
proposed, it is worth our while to briefly cover them. Some authors choose the
following notation

$\displaystyle-k^{2}\Psi$ $\displaystyle=$ $\displaystyle 4\pi G\mu\Delta\rho$
(23) $\displaystyle\Phi$ $\displaystyle=$ $\displaystyle\eta\Psi$ (24)

If we redefine the $\mu\rightarrow{\tilde{\mu}}$ we have
${\tilde{\mu}}=\mu/\gamma$ (where $\mu$ is defined in Equation 21) and
$\eta=\gamma$. ${\tilde{\mu}}$ is of particular use as (we shall see) it
sources the growth rate of structure. In the same way, we can define a
parameter that sources weak lensing

$\displaystyle\Sigma=\mu\left(1+\frac{1}{\gamma}\right)={\tilde{\mu}}(1+\eta)$
(25)

Some authors often define $G_{M}$ and $G_{L}$ (where $M$ stands for ”Matter”
and $L$ stands for ”Light”) such that

$\displaystyle G_{M}={\tilde{\mu}}\ \ \ G_{L}=\Sigma.$ (26)

The quasi-static parameters can be directly connected to more fundamental
parameters in underlying theories. For example, in the case of scalar-tensor
theories arising from the Horndeski action, these parameters have a specific
structure (De Felice et al., 2011)

$\displaystyle\mu$ $\displaystyle=$
$\displaystyle\frac{1+h_{1}k^{2}}{h_{2}+h_{3}k^{2}}$ $\displaystyle\gamma$
$\displaystyle=$ $\displaystyle\frac{h_{2}+h_{3}k^{2}}{h_{4}+h_{5}k^{2}}$

where $h_{i}$, $i=1,\cdots 5$ are functions of time and assembled from the 4
$\alpha_{X}$ parameters from equations 20; thus constraints on the quasi-
static parameters lead to constraints on slices of the functional space of
free parameters, $\alpha_{X}$. And vice versa, by restricting one self to
scalar tensor theories, one can establish a restricted set of priors on $\mu$,
$\gamma$ or $\Sigma$ (Pogosian & Silvestri, 2016; Peirone et al., 2018)

###  3.7 The phenomenology of modified gravity.

The dynamics of linear perturbations in modified versions of GR is
sufficiently well understood that there are now codes that accurately model
the evolution of perturbations in such theories, calculating cosmological
observables. The basic framework is a code that integrates the perturbed
Einstein field equations, the coupled set of matter (and other fluid like)
conservation equations and the Boltzman equations for radiation and neutrinos.
The main, general purpose, Einstein-Boltzman solvers are EFTCAMB (Hu et al.,
2013) and MGCAMB (Hojiati et al., 2011) and ISITGR (Dossett et al., 2011)
(derived from CAMB (Lewis et al., 1999)) and HiCLASS (Zumalacarregui et al.,
2017) derived from CLASS (Blas et al., 2011) although over a dozen more
specific Einstein-Boltzman solvers have been written. Most of these codes have
been cross calibrated and agree at the sub-percent level (Bellini et al.,
2017).

While we have focused on how we can accurately model general deviations from
GR, we now discuss how to constrain these deviations. The most fundamental
constraint will come from measurements of the growth rate: from redshift space
distortion (RSD), peculiar velocities, the integrated Sachs-Wolfe effect and
from tomographic measurements of weak lensing. All of these will constrain the
time evolution of $\delta_{\rm M}$ through $f$ or $f\sigma_{8}$. To see how
the growth rate is affected by deviations from GR, it is instructive to focus
again on the quasi-static regime (Baker et al., 2014). There, the evolution
equation is now given by

$\displaystyle\frac{df}{d\ln a}+q(a)f+f^{2}=\frac{3}{2}\Omega_{\rm
M}(a)\frac{\mu}{\gamma}$ (27)

where $q(a)=\frac{1}{2}[1-3w_{\rm U}(a)(1-\Omega_{\rm M}(a))]$. We can clearly
see that the growth rate will be affected by both modifications to the
background (which is encapsulated here in the equation of state of the
effective fluid driving the acceleration, $w_{\rm U}$) and the modified
perturbed equations through the quasi-static parameters. As a result there
will be a degeneracy between $w_{\rm U}$ and $\mu/\gamma$ (Simpson & Peacock,
2010) which can only be broken with measurements of cosmological distances. A
particular parametrization of the growth is often used, of the form

$\displaystyle f=\Omega_{M}^{\gamma_{G}}$ (28)

where $\gamma_{G}$ is approximately a constant for any specific theory over a
narrow range of redshifts (Linder, 2005). For $\Lambda$CDM, it can be shown
that $\gamma_{G}=6/11$.

Weak lensing will also probe the growth rate – via the modified Newton-Poisson
equation it measures the change in the density contrast (Amendola et al.,
2008). But it goes further; if we look at equation 12 we can see that it
depends on both $\Phi$ and $\Psi$. If we now focus on convergence we have that
$\kappa\propto\int d\chi\nabla^{2}(\Phi+\Psi)\sim\int
d\chi\nabla^{2}(1+\frac{1}{\gamma})\Phi\sim\int
d\chi\frac{3}{2}(aH)^{2}\Omega_{\rm M}\mu(1+\frac{1}{\gamma})\delta_{\rm M}$
where we used the Friedman equation to simplify the result. In other words, as
well as depending on the growth rate (via $\delta_{M}$) it probes one of the
derived quasi-static parameters described above: $\Sigma$.

Extending the set of cosmological parameters to include extra parameters (such
as $\mu$ and $\gamma$) may lead to degeneracies – we have already seen such a
case between $w_{\rm U}$ and $\mu/\gamma$ in the case of the growth rate. A
particularly problematic parameter is galaxy bias, i.e. the mapping between
the galaxy density contrast, $\delta_{\rm G}$ and the underlying matter
density contrast, $\delta_{\rm M}$. A simple approximation is that
$\delta_{\rm G}\simeq b_{\rm G}\delta_{\rm M}$ where $b_{\rm G}$ is scale and
time dependent (although there are now, far more sophisticated models of bias
(Desjacques et al., 2016)). Given that many direct probes of the growth rate
are in terms of the galaxy distribution, one often finds that resulting
constraints are on $f/b_{\rm G}$.

A proposal to circumvent the dependence on bias is to consider combinations of
statistics that, to some extent, ”divide out” the dependence on bias (Zhang et
al., 2007). A notable example is

$\displaystyle E_{G}(R)\propto\frac{\xi_{\rm GL}(R)b_{\rm G}}{\xi_{GG}(R)}$
(29)

where $\xi_{\rm GG}(R)$ and $\xi_{GL}(R)$ are the 2-D galaxy-galaxy and
galaxy-lensing correlation functions and $b_{G}$ is an independent measurement
of the galaxy bias of the same sample. This statistic was originally
formulated in Fourier space but is implemented in real space where a number of
practical issues need to be considered (Leonard et al., 2015). Alternatively
one can consider the full suite of correlation functions (including the
lensing-lensing correlation function) which, combined, can break the
degeneracy between $b_{\rm G}$ and other parameters.

##  4 Fifth forces and gravitational screening.

###  4.1 Gravitational regimes and the observational desert.

In the previous section we focused on what one might consider to be a ”clean”
regime, where dynamics is well described by linear cosmological perturbation
theory, and showed how it was relatively straightforward to parameterise
general theories of gravity. This regime benefits from the fact that the
dynamics is perfectly understood and it is possible to solve the evolution
equations with arbitrarily high accuracy. As we saw in Section 2, it is where
modern cosmology has had a resounding success. But it is also a regime with
severe limitations: the larger the scales we have, the less statistical power
to constrain parameters. We observe one universe and hence there is a finite
amount of information to be extracted from its observation; this cosmic
variance is largest for the largest scale modes. It would be desirable to look
at smaller scale modes but, as we have seen, non-linear physics (both
gravitational and baryonic) play an unavoidable role there.

Figure 3: Gravitational systems as a function of the typical values of their
gravitational potential and curvature (Baker & Bull, 2015). The Solar System
is on the middle left, while black holes are on the top right. Cosmological
structures are at the bottom.

There is another way of looking at the large scale, cosmological regime in
comparison to other regimes (Baker et al., 2015) and this is best illustrated
in Figure 3. On the horizontal axis one has the typical gravitational
potential, $\Phi$ (in this plot labelled by $\epsilon$) of a system while on
the vertical axis one has the typical curvature (labelled by $\xi$), here
expressed in terms of the Kretschmann scalar (i.e. a quadratic invariant
constructed from the Riemann tensor). Scattered on the plot are a number of
different systems (or regimes in which gravity can be constrained), from the
Solar System and pulsars to black hole mergers and imaging. Cosmology stands
apart in a regime of low curvature.

A striking feature of Figure 3 is that there seems to be a ”desert”, a dearth
of observation between cosmology and other probes, for values of $\xi$ between
$10^{-50}$ and $10^{-42}$. The typical systems found in this desert are
galaxies and clusters where the interplay of non-linear gravitational and
baryonic physics as well as highly non-linear feedback mechanisms (such as
supernovae and Active Galactic Nuclei - AGNs) make them particularly hard to
model. Indeed we can see this desert as the transition between two easy
regimes: the high curvature part where one is studying isolated systems and
the low curvature part where linear theory on a homogeneous background
applies.

There is another interesting feature that can be observed. In the high
curvature regime, there are strong constraints on gravity - in the case of the
Solar System and millisecond pulsars, exceptionally so. One is guaranteed to
have modifications to GR in the top right-hand side as one delves into the
regime where gravity becomes much stronger. But the current quest for
modifications are in the bottom half of the regime where, we argued in the
introduction, it is somewhat unnatural for deviations to GR to arise. If it
turns out that GR is strictly true in the top half of the plane and violated
in the bottom half of the plane, it would seem that some mechanism is
shielding regions of high potential from deviations, i.e. from extra fifth
gravitational forces and that the transition occurs somewhere in the desert.

###  4.2 Gravitational screening.

While the description above is somewhat qualitative, it turns out that there
are mechanisms of what is known as gravitational screening which can restore
GR in regions of high density, potential or curvature but which allow for the
emergence of fifth forces elsewhere (Joyce et al., 2014). A useful,
alternative, way of thinking about screening is to realize that a fifth force
is always associated to a ”charge” and that the strength of the fifth force is
proportional to the charge. One can then envisage a mechanism such that in the
screened regime the charge is surpressed and the fifth force is absent while
in the unscreened regime, the force has full effect.

To understand how screening can work, let us revisit one of the simplest
scenario, a non-minimally coupled scalar field with a potential (Khoury &
Weltman, 2004; Burrage & Sakstein, 2017). It is always possible to transform
such a system into a frame in which the scalar field becomes minimally coupled
to gravity but non-minimally coupled to the matter sector. In the simplest
case, the background scalar field obeys an equation of the form

$\displaystyle\Box\varphi+\frac{dV_{\rm eff}}{d\varphi}=0$ (30)

where the effective potential, $V_{\rm eff}=V-{\bar{\rho}}\varphi$, i.e. the
minimum of the potential will depend on the local matter density. If we solve
for $dV_{\rm eff}/{d\varphi}(\varphi_{0})=0$, we can then expand
$\varphi=\varphi_{0}+\delta\varphi$ and we find that, in the quasi-static
regime, the perturbations in the scalar field obey an equation of the form

$\displaystyle\nabla^{2}\delta\varphi+m^{2}(\varphi_{0})\delta\varphi\propto"{\rm
source}"$

where $m^{2}(\varphi_{0})=d^{2}V_{\rm eff}/{d\varphi}^{2}$. If we assume a
localized mass, $M$ this means that

$\displaystyle\delta\varphi=\beta\frac{Me^{-m(\varphi_{0})r}}{r}$ (31)

where $\beta$ is a constant of proportionality set by the coupling between the
scalar field to matter and we see that $m(\varphi)$ functions as a cutoff,
setting the range of the fifth force. Furthermore, the cut-off depends on the
environment so that, if we choose a $V\propto\varphi^{-n}$, we have that for
large $\rho$, $m(\varphi_{0})$ is large and the force is short range but if
$\rho$ is small, $m(\varphi_{0})$ will be small and the force can be long
range. Thus we have that the fifth force can strongly satisfy local
constraints yet have an impact on cosmological scales.

We have discussed one particular type of screening, known as Chameleon
screening. Interestingly, it has been shown that, if Chameleon screening is to
arise in a cosmological context, it has to be on scales less than a
Megaparsec, i.e. deep in the non-linear regime; furthermore, the screening
scalar field cannot be responsible for accelerated expansion directly through
gravitational effects and only as a conventional source of dark energy (Wang
et al., 2012). There are, however, other forms of screening that we should
mention. The first one arises in theories in which the scalar mediator of the
fifth force has a non-trivial kinetic term, as in, for example bimetric
theories of massive gravity or scalar-tensor theories constructed from shift-
symmetric term – Galilleons (Nicolis et al., 2009) – or bimetric theories such
as massive gravity. Known as the Vainshtein mechanism (Vainshtein, 1972), the
modified kinetic term now transforms the overall strength of the fifth force,
the $\beta$ in equation 31, into a function of $\varphi_{0}$ which in turn can
depend on the source and environment. Typically one find that near very
massive objects, $\beta\rightarrow 0$ while around lighter or more diffuse
objects, $\beta$ can be non-negligible. Another variant is Symmetron
(Hinterbichler & Khoury, 2010) where an environmental dependence of the
coupling between the scalar field and matter leads to a similar effect on
$\beta$.

###  4.3 The phenomenology of screening.

Screening adds more complexity on to what is already a particularly difficult
regime to model. While the payoff for looking at smaller scales is greatly
enhanced statistical power (there are more independent samples on smaller
scales), one is faced with the non-linear aspects of gravity which are far
less well understood than the clean linear regime discussed in the previous
section. Furthermore, baryonic physics can now have a significant impact on
attempts at making predictions.

There have been some attempts at developing analytic techniques for studying
the impact of screening on the non-linear regime. Working in the quasi-static
regime, it is possible to extend or modify higher order perturbation theory
methods to included the effect of fifth forces (Koyama et al., 2009; Cusin et
al., 2018) although the effectiveness of such methods has been somewhat
limited. More progress has been made on developing enhanced N-body codes which
incorporate the effect of fifth forces (Winther et al., 2015). There, again
using the quasi-static approximation, it is possible to add corrections to the
gravitational forces which captures the effect of environment dependent fifth
forces. Furthermore, it is possible to couple the modified gravitational
evolution with hydrodynamic equations to allow a first foray into the impact
of baryonic physics and how it feeds back into gravitational collapse.

Currently, the main codes used for studying the non-linear regime and its
interplay with baryons are ECOSMOG (Li et al., 2012) and ISIS (Llinares et
al., 2014) which are modified versions of RAMSES (Teyssier, 2002), an
adaptive-mesh-refinement code, and MG-GADGET (Puchwein et al., 2013) which is
a modified version of GADGET (Springel, 2005), an SPH code. There have also
been attempts at writing faster N-body solvers with fifth forces, based on
particle-mesh (PM) solvers, most notably modified versions of COLA
(Valogiannis & Bean, 2017; Winther et al., 2017). As with the Einstein-
Boltzman solvers, there has been a concerted effort to cross-calibrate various
codes so that they agree for a range of models on a range of scales (Winther
et al., 2015).

###  4.4 Observational consequences.

If gravitational screening is responsible for the qualitative difference
between different regimes, then one needs to take a more nuanced view when
undertaking a statistical analysis of large scale structure. In particular,
one needs to look at the bigger picture, not only focusing on strictly local
values of correlation functions or number counts, but including information
about the environment in which these quantities are being estimated. One
approach is almost geographic in nature: to construct a gravitational map of
the Universe which allows us to identify regimes in which systems may be
unscreened.

There have been attempts at constructing a gravitational map of the local
Universe (i.e. which extends out to approximately $200h^{-1}$ Mpc) (Cabre et
al., 2012; Desmond et al., 2018b). The basic idea in Desmond et al. (2018b) is
to use a catalogue of galaxies which is as complete as possible in that region
– the 2M++ catalogue. One then uses abundance matching to identify the
possible dark matter halos associated with those galaxies and to fill in for
any smaller halos which will not have been picked up by the survey. Finally,
one uses a constrained realization of the long wavelength modes to fill in the
rest of the dark matter on large scales. From the combination of all these
terms it is possible to construct a map of the gravitational potential, tidal
forces and local curvature. It is also possible to model the uncertainty in
all the steps leading to an overall local error budget on the map. This map
can then be used to distinguish between the regions that are unscreened and
screened.

Once one can identify qualitatively different regions, it is then possible to
look for signatures of the unscreened phase, i.e. for evidence of the
unsupressed fifth force associated with deviations from GR. The way to go
about doing this is to try and identify objects which might behave differently
in unscreened environments: some object will either be sufficiently dense,
compact or large that they will self screen themselves (i.e. their fifth force
charge will be effectively zero) while others will be too diffuse or light to
be self screened. The hall mark of the fifth force in unscreened regions will
be what seem like effective violations of the equivalence principle.

In this context, there are a number of interesting effects that one might look
for (Jain & Vanderplas, 2011). For a start, the different components of a
galaxy will behave in different ways. For example, in a theory with chameleon
(or symmetron screening) the stars in a galaxy will self-screen while the gas
will be too diffuse and will not be screened. The net result is that there
should be an offset between the optical and radio components of a galaxy if
there is screening. By looking for this effect it should be possible to place
constraints on the amplitude and range of the a screened fifth force.

Another possibility is by looking at how a galaxy is deformed as it sits in a
dark matter halo. The stellar disk might lag behind the halo centre inducing a
gradient in the gravitational potential. As a result the galaxy will be warped
into a cup shape where the amount of deformation will depend on the strength
of the fifth force. The effect will be strongest if the disk is completely
perpendicular to the external unscreened force field.

A more straightforward effect will be that the unscreened component of the
galaxy – the gas – will have a faster rotation curve relative to the stellar
disk as a result of the additional fifth force having the same direction as
the normal gravitational force. Furthermore, in the case of edge-on infall,
the stellar and gas disks (and rotation curves) will have greater differences
or assymetries with the edges which are closest to the centre of the halo
being more compressed than the far side.

Finally, a similar and striking signature may be the relative offset of the
galaxy centroid and the super massive black hole residing at its centre (Hui &
Nicolis, 2012). If a galaxy falls in an unscreened potential, much of the
galaxy will feel a fifth force while the black hole, as a result of the no-
hair theorem, will be shielded and won’t be subjected to it.

Another line of attack is to roll out suitably tailored versions of the usual
n-point statistics which are used in standard cosmological analysis. In
particular, it makes sense to correct estimators so that they downweight
screened regions and enhance unscreened regions (White, 2016; Valogiannis &
Bean, 2018). In the case of chameleon screening, this can easily be achieved
by downweighting regions of high density, by ”clipping” high density peaks or
by introducing some form of nonlinear function of the density field
${\tilde{\delta}}[\delta_{M}]$ such that high density values are damped.
Examples of such transformations are ${\tilde{\delta}}=\ln(\delta+1)$ and
${\tilde{\delta}}=[(\rho_{*}+1)/(\rho_{*}+\rho_{m})]^{p}$ where $\rho_{*}$ and
$p$ are free parameters.

An approach that has been gaining traction is to focus on what are, for sure,
the most unscreened parts of the Universe: voids. Voids have a scale of about
$20-100h^{-1}$ Mpc and are typically surrounded by outflows; they are seen as
a new and interesting arena in which to do cosmology, parts of the density
field where many of the linear and very mildly linear results still hold.
Furthermore, with the vast increase in the size and depth of cosmological
surveys, it is now possible to construct meaningful statistics with void
catalogues in the same way as one does with galaxy clusters. And given that
voids are particularly prone to be affected by unscreened fifth forces, it
makes sense to focus on them if one wants to test gravity.

As yet, the exact definition of a void has not yet been completely established
and agreed on (Sutter et al., 2012; Nadathur et al., 2015). A number of
different factors come into play, from dimensionality to the type and number
density of the tracers used, as well as, operationally, the type of void
finding algorithm that is used. As a result, void catalogues can be quite
different from each other and the resulting statistics can disagree.

Nevertheless, if one settles on one particular catalogue (or better, considers
a range of catalogues) one can proceed to assess a number of statistics. In
particular, one looks at the usual statistics applied in cosmological analysis
except now in this very particular environment. So, for example, one can
assess lensing of background galaxies by the void (Barreira et al., 2015;
Melchior et al., 2014; Baker et al., 2018). One can also measure redshift
space distortions in the void and compare them to measurements outside the
void (Hamaus et al., 2015). Another approach is to cross correlate the void
profile with the CMB (Cai et al., 2014) in the same direction and in doing so,
pull out the integrated Sachs-Wolfe term (the last part of equation 9).
Finally, one can approach voids in the same way one approaches clusters and
quantify their abundance as a function of scale and redshift (Clampitt & Cai,
2012).

##  5 Constraints from Large Scale Structure.

###  5.1 The evidence for $\Lambda$CDM

We seem to be living in charmed times, at least from the point of view of
cosmological observations. With observations of the CMB – with the COBE, WMAP
and PLANCK satellite missions and a host of ground-based experiments –
complemented by the burgeoning precision probes of LSS, we now have an
accurate mathematical model of the origin and evolution of the Universe in
which the six parameters that govern it have been measured to exquisite
precision (Planck, 2018a). Based on GR with a cosmological constant (and
described in Section 2), $\Lambda$CDM fits almost all the data with remarkable
consistency.

The observational apparatus that we have to support $\Lambda$CDM is
formidable. As well a the CMB experiments, the main observations to contribute
to cosmological constraints are: spectroscopic surveys – BOSS (Alam et al.,
2016) and WiggleZ (WiggleZ, 2011) – measuring galaxy spectra (i.e. redshifts)
and angular positions; photometric and weak lensing surveys – KIDS (KIDS,
2016), HSC (HSC, 2018) and DES (DES, 2018a) – measuring galaxy images and
shapes, angular positions and a crude measurement of redshifts. There are
preliminary attempts at line-intensity mapping (for example measuring the 21
cm spin flip transition of neutral hydrogen) but nowhere near at the level
required to do cosmology (Switzer et al., 2013).

Just to belabour the point, it is useful to see how well some of the
fundamental parameters of $\Lambda$CDM are currently constrained (Planck,
2018a). The initial conditions of fluctuations are characterized by a spectal
scalar index, found to be $n_{S}=0.965\pm 0.004$, i.e. almost but not exactly
scale invariant; the difference from scale invariance is $8-\sigma$,
consistent with (and, indeed, some might say strongly indicative of) a period
of primordial inflation. This constraint is supplemented with an upper bound
on the tensor to scalar ratio (i.e a measure of the amount of gravitational
waves left over from inflation) of $r<0.07$. Another notable constrain is on
the geometry of the universe: the current, tightest constraint is
$\Omega_{K}=-0.001\pm 0.002$. The Hubble parameter is constrained, from
cosmological data, to be $H_{0}=67.4\pm 0.5$ km s-1 Mpc.

A key incentive to consider modifications of gravity is the evidence for
accelerated expansion. While $\Lambda$CDM is a remarkable fit to the data, it
is instructive to consider constraints on the equations of state of the form
$w_{\rm DE}(a)=w_{0}+w_{a}(1-a)$. The most complete analysis using a
comprehensive array of data sets (Planck, 2018a) find that for a constant
equation of state (i.e. $w_{a}=0$), $w_{0}=-1.028\pm 0.032$ if one uses the
Planck CMB data combined with distance measures from supernovae and the BAO in
the matter power spectrum. If one allows for $w_{a}=0$, the same collection of
data sets lead to $w_{0}=-0.961\pm 0.077$ and $w_{a}=-0.28^{+0.31}_{-0.27}$
while using the Planck data, the BAO and redshift space distortion
measurements of growth from galaxy surveys and galaxy weak lensing data lead
to $w_{0}=-0.76\pm 0.2$ and $w_{a}=-0.72^{+0.62}_{-0.54}$. There is clearly
some lee-way in terms of an evolving equation of state but, as discussed
above, this can’t necessarily be used to find any evidence for modifications
to General Relativity.

###  5.2 Minor anomalies.

Given these remarkable constraints and the success of $\Lambda$CDM, it is
useful to look at the (minor) inconsistencies and anomalies. While I do not
yet believe they are at the level that call into question the success of the
standard model, they will be useful in interpreting constraints on gravity
later on in this section. There have been a number of small internal anomalies
flagged in the current CMB data: large scale surpression of power and evidence
for mild anisotropy in the low quadropoles (Schwarz et al., 2015), mild
inconsistency between large angle and small angle angular power spectrum
constraints of cosmological parameters (Spergel et al., 2015), a discrepancy
in the constraint on $\sigma_{8}$ from primordial and cluster abundances from
Sunyaev-Zeldovich measurements (Planck, 2014), etc.

A persistent anomaly is the discrepancy in the constraint on $H_{0}$ from
local measurements, such as those using Cepheids in (Riess et al., 2016) as
compared to cosmological measurements from, for example, the CMB (Planck,
2018a). While both groups claim percent level constraints on the expansion
rate, the discrepancy between the two results is much greater, at a level of
reasonable significance. Most attempts at coming up with an explanation for
this discrepancy from exotic physics do not explore deviations from GR and I
will therefore not pursue this anomaly here.

The only anomaly I would highlight here is the inconsistency between the
amplitude of lensed CMB fluctuations as inferred directly from the angular
power spectrum of temperature fluctuations and the power spectrum of the
lensing potential reconstructed from the maps (Planck, 2018a) . In particular,
if ones free up the amplitude, $A_{L}$ (such that $A_{L}=1$ is the
$\Lambda$CDM value) of the lensing contribution that smooths the the angular
power spectrum of temperature fluctuations (as described in Section 2),
current constraints from Planck give $A_{L}=1.243\pm 0.095$ at the $68\%$
confidence level. Paradoxically, if one attempts to reconstruct the power
spectrum of the lensing power spectrum, $C^{\phi\phi}_{\ell}$, directly from
the maps one finds that it is consistent with what one would expect from
$\Lambda$CDM; i.e. two different methods for estimating the lensed CMB from
the same data set give different results.

As mentioned in Section 3, a key observable to test gravity is the growth rate
of structure, i.e. the speed with which gravitational collapse ensues. The
main method for measuring the growth rate is with Redshift Space Distortions
(RSD). RSDs can be seen in the redshift space correlation function (or power
spectrum): along the line of sight, the proxy for distance – the redshift –
will have a correction from peculiar velocities due to the local gravitational
potentials (Kaiser, 1987). This means that correlations along the line of
sight will be deformed relative to correlations tangential to the line of
sight, i.e. the correlation function (or power spectrum) will be anisotropic.
On linear scales – scale of tens of Megaparsecs – the correlation function, or
power spectrum is squashed while on smaller, non-linear scales, they are
stretched.

A mild (and arguable) anomaly is the evidence for low growth rate coming from
RSDs. First highlighted in (Macaulay et al., 2013) but also thoroughly refuted
in (Samushia et al., 2014), the discrepancy can best be assessed in Figure 1
where we can see the predicted, density weighted growth rate for $\Lambda$CDM,
marginalized over a host of cosmological parameters and constrained by the
Planck 2018 data (Planck, 2018a). Two things are of note: first, that current
low redshift constraints (from RSDs) are weak compared to predicted CMB
constraints and second, that many data points seem to be below the predicted
line. I emphasize that the evidence for the low growth rate is not yet
significant and, for some, not even indicative.

Figure 4: Constraint on $S_{8}$ from the Planck 2015 data (red contours) and
the KIDS-450 data (blue-contours) from KIDS (2018).

A much more significant inconsistency arises in the comparison between galaxy
weak lensing constraints and the CMB. If we focus on
$S_{8}=\sigma_{8}(\Omega/0.3)^{1/2}$, we can see in Figure 4, that the galaxy
weak lensing measurements from the KIDS survey (KIDS, 2018) leads to a
somewhat lower value of $S_{8}$ than inferred from the Planck CMB
measurements. This discrepancy is present in the DES data (DES, 2018a)
although at a milder level; it is possible to further soften the difference
with a more elaborate likelihood which, one hopes, can capture the statistical
and systematic uncertainties more accurately. But, crucially, these two,
completely independent, imaging surveys both find this inconsistency and
combining them, or including more data may only reinforce this discrepancy.

So, in summary, current data consistently points to $\Lambda$CDM, which is
firmly rooted in GR. There are some mild inconsistencies which are not
statistically very significant but might lead one to explore deviations from
the standard model. We can now look at what current data has to say about
extension of GR.

###  5.3 Parameterizing gravity.

In Section 3 we discussed how to parametrize deviations from GR, in the linear
regime, on large scales. In the case of scalar-tensor theories described by
the Horndeski action, this is straightforward to do in terms of the five
$\alpha_{X}$ parameters on both super-horizon and sub-horizon scales. It is
also true that most of the statistical power will come from smaller scales
where, generally, the quasi-static approximation can be used. This greatly
simplifies the analysis as one is restricted to two free function, $\mu$ and
$\gamma$. In most of what follows, we will present constraints in terms of
these parameters or combinations of them. But, before we do, a few comments
are in order.

Most parameter constraints using $\mu$ and $\gamma$ assume time dependent, but
scale free, functions which are valid from sub-horizon to super-horizon
scales. This is clearly not correct, in the sense that it does not naturally
emerge from the approach described in Section 3: one should, at least, expect
a scale dependence at or greater than the horizon scale but more generally,
the quasi-static approximation breaks down and one needs to, correctly,
include the dynamics of the extra degree of freedom (which will, for example,
depend on its initial conditions). Having said that, most constraints from LSS
are, effectively, on sub-horizon scales (where there is more statistical
power) and so one hopes that any inconsistencies in modelling very large scale
deviations will be vastly supressed by the data on smaller scales. Effects on
the scale of the horizon may be important for large enough surveys (although
see Baker & Bull (2015) for how insignificant the constraining power of
horizon scale modes is) or in the case of the primordial CMB (although these
occur at $z\simeq 1000$ and thus are unaffected by late time deviations from
GR). The advantages of focussing on $\mu$, $\gamma$ outweigh the problems just
described. Nevertheless, later on we will look at constraints on scalar-tensor
theories where the large scale behaviour is correctly taken into account.

Another, important, point needs to be highlighted. If we restrict ourselves to
scale free $\mu$ and $\gamma$, one still needs to choose an appropriate time
dependence. Strictly speaking, these function will depend on $a$, ${\dot{a}}$
and, in the case of scalar-tensor theories, the background scalar field value,
$\varphi_{0}$, all of which depend on $t$. A common approach has been to
assume that deviations from GR are tied to the onset of accelerated expansion
which can be encapsulated in the effective value of the background energy
density, $\Omega_{\rm DE}$ or $\Omega_{U}$ (Ferreira & Skordis, 2010) in the
notation introduced in Section 3. So, for example, one can take
$\mu=1+\alpha\times(\Omega_{\rm DE}/\Omega_{\rm DE0})$ where $\Omega_{\rm
DE0}$ is the value of $\Omega_{\rm DE}$ today, $\alpha$ is a constant and one
often refers to $\mu_{0}\equiv\mu(z=0)=1+\alpha$ (and likewise for $\gamma$).
A more general approach is to divide the temporal dependence of $\mu$ and
$\gamma$ up into uncorrelated redshift bins – for example, there will be a low
redshift bin starting off at $z=0$ followed by a succession of bins. In fact,
this approach is often generalized to include scale dependence, i.e. bins in
the fourier mode, $k$, as well as $z$.

There are a few drawbacks to any choice of time dependence. First of all, it
has been argued that the form $\mu=1+\alpha(\Omega_{\rm DE}/\Omega_{\rm DE0})$
(or of any parameters dependent on $\Omega_{\rm DE}$) does not necessarily
reflect the evolution of $\mu$ and $\gamma$ one finds in specific models
(Linder et al., 2016). While this may be true in some specific cases, this
time dependence is, generally, a rough approximation of what one might find in
theories in which the accelerated expansion is tied to deviations from GR. On
the other hand, dividing up the temporal in redshift bins often leads an ill-
posed problem (i.e. to many parameters to fit, given the data) and for which
results are difficult to determine. Both of theses approximations can be
sharpened if, for example, one considers a more systematic expansion of $\mu$
and $\gamma$ in terms of some basis functions, $f_{n}(\Omega_{\rm DE})$ for
example, that can capture more elaborate time dependence (for example choosing
$f_{n}(x)=x^{n}$ or $f_{n}(x)=j_{n}(x)$ where the $j_{n}$ are spherical
Bessell functions) or by considering some form of Principal Component Analysis
of the time dependence of these functions. One then has to be careful in
choosing priors that adequately reflect the physical behaviour of this
functions (Crittenden et al., 2012; Espejo et al., 2018)

A second drawback then emerges in that the results one finds are, for current
data, strongly dependent on the parametrization one considers (or for example
the number of free functions one incorporates into the parametrization). For
example, with current data it is possible to constrain a constant $\mu$ at the
percent level while $\mu=1+\alpha(\Omega_{\rm DE}/\Omega_{\rm DE0})$ is
constrained, at best, at ten to twenty percent level (an example of this will
be seen later on, when we focus on the BOSS data). Having declared the
problems, we will stick with the conventional parametrization in the
discussion that follows.

Note that most current analysis combine multiple data sets – e.g. CMB with
galaxy weak lensing – so the division between the different results that
follow us somewhat artificial. Nevertheless I choose to do so to highlight the
impact of the different observables, and their specific effects, on
constraints on gravity.

Figure 5: Plot from (Planck, 2018a). Constraints on $\mu_{0}$ and $\eta_{0}$
(${\tilde{\mu}}$ and $\gamma$ in this review). On the left hand plot we can
see the effect of the $A_{L}$ anomaly leading to a discrepancy with GR. On the
right hand plot, with the inclusion of lensing estimated from the maps, and
galaxy weak lensing from DES, consistency with GR is restored.

###  5.4 Current constraints: the CMB

If I take a selection of constraints on deviations from GR, I naturally begin
with the analysis of the Planck data (combined with other data sets). In
(Planck, 2016) and in (Planck, 2018a), a comprehensive analysis of different
parametrizations was undertaken with the 2015 and 2018 Planck data releases;
the overall take home message was that current cosmological data is broadly
consistent with GR. But of particular interest (and representative of the
analysis as a whole) are the features seen in Figure 5 from Planck (2018a),
where constraints on ${\tilde{\mu}}$ (here labelled $\mu_{0}$) and $\eta$ are
presented. To begin with we can see that there is a degenerate direction along
which constraints are much broader - this direction is set by constraints on
the $\Phi+\Psi$ from CMB lensing and thus on $\Sigma$ (if one transforms to
the $\mu$, $\Sigma$ parametrization, one can, to some extent decorrelate the
two parameters being constrained).

Second, we note that the error bars on these parameters are of order $1$.
While these may seem very weak compared to the precision one obtains on
astrophysical scale (for example, the constraints on the analogous parameter
to $\gamma$ from the Solar System is of order $10^{-5}$) they are obtained in
a completely different regime, on scales which are fifteen orders of magnitude
greater and where the curvature scale is twenty orders of magnitude smaller
(see Figure 3).

Finally, and most striking, is that fact that there seems to be a detection of
deviations from GR, away from $\mu_{0}=\gamma_{0}=1$. This push away from GR
from the Planck data alone is due to the enhanced amplitude of CMB lensing
anomaly (the ”$A_{L}$ anomaly”) previously discussed in this section, as can
be seen in the left hand panel of Figure 5. In the right hand panel we see
that, if one then includes the measurement of the lensing potentials from the
maps – $C^{\phi\phi}_{\ell}$ – (which, recall from the discussion above, is an
independent measurement to the one used to determine $A_{L}$), the constraints
are pushed towards GR. Including galaxy weak lensing (in this case from DES)
which, as we saw, favours a lower amplitude in the lensing potentials, further
pulls the constraints towards GR. Adding the RSD constraints has a small
effect in the orthogonal direction.

Is this a significant detection of deviations from GR? Not if one properly
accounts for the increase in complexity of the model, even though there is an
improvement in the goodness of fit. GR would have to be excluded at a much
higher significance for the detection to be noteworthy.

Figure 6:  Constraints from the KIDS-450 data combined with RSD measurements
from 2dFLenS and BOSS (in the overlap region with KIDS-450) and other data
sets. The right hand plot shows constraints on $Q_{2}$ (defined as
${\tilde{\mu}}$ this review) and $\Sigma$. The left hand plot shows
constraints on $\Omega_{\rm m}$ and $\sigma_{8}$; adding ${\tilde{\mu}}$ and
${\Sigma}$ alleviates the discrepancy in $S_{8}$.

###  5.5 Current constraints: galaxy weak lensing and growth rate.

We now turn to a different set of constraints, obtained from a combination of
galaxy weak lensing from the KIDS-450 with the CMB and RSDs (KIDS, 2018) . In
this case, the parameters being constrained are ${\tilde{\mu}}$ (here dubbed
$Q$) and $\Sigma$, binned in both redshift and scale. If we look at the
constraints for the lowest redshift bin (labelled by a ”2”), on largest scales
(assume that we are marginalizing over the other bins) and focus on the case
of CMB alone (the red dashed lines in the right hand panel) we see, again, a
mild inconsistency with GR; in this case it is milder than in the Planck
analysis and comes tied to vastly inflated uncertainties (as is expected from
our discussion above, given the number of free parameters – or bins – now
included). Weak lensing and RSD data (without CMB) leads to tighter
constraints, centered on the GR value. To combine them with CMB data, the
authors have introduce a large scale cut to mitigate the effect of non-linear
scales. In that case, the constraints including the CMB are completely
consistent with GR; again the low lensing potentials from galaxy weak lensing
compensate the high $A_{L}$ constraint from the CMB. An interesting effect of
enlarging the parameter constraints to include deviations from GR is that the
”$S8$” discrepancy can be substantially mitigated. In the left hand panel of
Figure 6 one can see that what were orginally disjunct contours (the dashed
lines) can be rendered overlapping (the teal and red shaded contours) by
adding in $\mu$ and $\gamma$. We note that the Dark Energy Survey (DES) has
undertaken an analysis of their first year of data DES (2018b) to come up with
constraints on $\mu$ and $\Sigma$. Their constraints are of the same order of
magnitude as the KIDS-450 constraints although marginally more consistent with
the GR values.

Figure 7: Constraints of $G_{M}$ and $G_{L}$ using BOSS measurements of the
growth rate combined with other data sets (figure kindly produced by E. M.
Mueller). A parameterization of the form $G_{X}=1+(G_{X}^{(s)}-1)a^{s}$ is
assumed with $s=0,1,3$.

We now briefly look at the SDSS-III BOSS constraints in (Mueller et al., 2018)
where the authors concentrate on $G_{M}$ and $G_{L}$ which, we know from
Section 3, can be directly related to $\mu$ and $\gamma$. Assuming a time
dependence for these quantities of the form $G_{X}=1+(G_{X}^{(s)}-1)a^{s}$
where $X=M,L$ and, in their analysis, $s\in\\{0,1,3\\}$. In Figure 7 we see
how a time independent choice ($s=0$) leads to very strong constraints while
$s=1$ (and even more so with $s=3$) leads to weaker constraints as deviations
from GR are more suppressed at high redshift. Constraints on the growth index
are found to be $\gamma=0.566\pm 0.058$ which, again, show strong consistency
with GR.

It is important to note that the analysis of data to constrain $f\sigma_{8}$
involves a number of steps in which a particular cosmology and theory of
gravity is assumed. A concern, then, is that estimates of $f\sigma_{8}$ may be
biased by these assumptions. Attempts at estimating the size of this bias
(Barreira et al., 2016; Bose et al., 2017) show that it is insignificant for
current surveys but may be an issue when their statistical power greatly
improves.

Figure 8: State of the art constraints on $E_{G}$ using both CMB and galaxy
weak lensing (figure kindly produce by S. Singh).

A parallel line of attack has been to determine constraints on GR from $E_{\rm
G}$. There have been a number of attempts at measuring $E_{\rm G}$ using
galaxy-galaxy lensing (Reyes, R. et al., 2010; Amon et al., 2018) and CMB
lensing combined with Galaxy velocities (Pullen et al., 2016) finding this
statistic to be generally consistent with GR, but with a few minor
complications (for a thorough analysis (Singh et al., 2018)). First of all, in
the case of Pullen et al. (2016), the large angle contribution of CMB lensing
leads to a 2.6$\sigma$ inconsistency with GR. Second, the choice of background
cosmology has a strong impact on the estimate of $E_{G}$; for a comfortable
consistency with $\Lambda$CDM, one needs a slightly lower $\Omega_{M}$ than
normally assumed. The current, state of the art, constraints on $E_{G}$ are
given in Figure 8.

It is not clear whether $E_{G}$ is a particularly powerful statistic for
testing GR and whether it adds anything more than the other, more standard
likelihood analysis currently being undertaken. While, in principle, it should
mitigate the contaminating effect of galaxy bias, in practice and due to the
way it is implemented, one is still at the mercy of one’s best bias model.
Furthermore, a proper, full likelihood analysis of all cosmological data will
necessarily include cross-correlations between lensing and density
measurements and will inevitably work towards breaking any degeneracy between
nuisance parameters such as bias and the GR parameters one is attempting to
constrain.

###  5.6 Model specific constraints: scalar-tensor theories.

Instead of using a general, parametrized approach to constraining GR, it is
sometimes useful to focus on specific models. This makes it possible to, to
some extent, span different regimes and compare, for example, the constraining
power of Solar System probes versus cosmological probes. The workhorse for
such constraints is Jordan-Brans-Dicke gravity, the simplest scalar-tensor
theory described in Section 3 with its one free, dimensionless parameter,
$\omega_{\rm BD}$. Current astrophysical constraints of $\omega_{\rm BD}$ are
remarkable. From the Cassini time delay it is possible to constrain
$\omega_{\rm BD}>40,000$ (Bertotti et al., 2003) while from milli-second
pulsars the constraints are around $\omega_{\rm BD}>20,000$ (Freire et al.,
2012). Current cosmological constraints, using the Planck data are much
weaker: $\omega_{\rm BD}>900$ and come primarily from the effect of the extra
scalar field on the background expansion and thus on the angular diameter
distance (Avilez & Skordis, 2014). Measurements of the growth rate are not yet
up to the level of precision required to lead to competitive constraints on
$\omega_{\rm BD}$.

Jordan-Brans-Dicke gravity is not screened on astrophysical scales but more
general scalar-tensor theories will be. For example Galileon theories may have
Vainshtain screening while conformally coupled theories with potentials may
have Chameleon or symmetron screening. If that is the case, then the only way
of obtaining tight constraints may be in the cosmological regime (although see
Sections 4 for other approaches). It would make sense then to look at
constraints on a general scalar tensor theory such as the one described by the
Horndeski action. Furthermore, as we saw above, the $\mu$, $\gamma$
parametrization shouldn’t be extended to horizon and superhorizon scales
(although these scales won’t have a strong statistical weight and will
contribute little to the final results). The Horndeski action allows a
completely self consistent parametrization on all scales in terms of the
$\alpha$ parameters.

The most comprehensive analysis of the Horndeski action was undertaken in
(Bellini et al., 2016), where the authors used CMB data (a combination of the
2013 Planck power spectra and WMAP polarization), galaxy redshift survey data
in the form of the power spectrum (from WiggleZ), the BAO (from BOSS, WiggleZ
and 6DFGS) and RSDs (from SDSS, BOSS, WiggleZ, 6DFGS and VIPERS) to place
constraints on the $\alpha$s. In their analysis, the authors used a
parametrization of the form $\alpha_{X}=c_{X}\Omega_{\rm DE}$ (where $c_{X}$
is a constant), along the lines of the one use for $\mu$, $\gamma$. On the
scales being constrained, $\alpha_{K}$ was found to be irrelevant and could be
fixed to fiducial values without affecting the final results. The typical
variance on $c_{X}$ was found to be ${\cal O}(1)$ and the parameters were
found to be mostly uncorrelated.

Interestingly, the analysis of (Bellini et al., 2016) found that the best fit
values of $\alpha_{B}$ and $\alpha_{T}$ were away from GR and, for some
combination of data sets, at the $95\%$ confidence level. While for CMB alone,
the results were perfectly consistent with GR, the inclusion of RSDs pulled
the preferred values in such a way that $c_{T}$ was preferentially less than
zero and $c_{B}$ was preferentially greater than zero. This is somewhat at
odds with the analysis of $\mu$, $\gamma$ in Planck (2018a) where it is the
$A_{L}$ anomaly that is pushing the constraints away from GR and where the
RSDs do not play a significant role. The authors do not, however, claim a
significant detection of deviations from GR on the grounds that, having
introduced more parameters, the evidence (which they compare to GR using the
Bayes factor) is not sufficiently significant. In other words, the increase in
complexity in choosing Horndeski is not compensated by the improvement in the
fit to the data.

A restricted analysis of Horndeski gravity taking into account the constraints
from GW170817 on the speed of gravitational waves has been undertaken in
Kreisch & Komatsu (2017) . The authors found that, varying $\alpha_{B}$, the
values of $\alpha_{M}$ were discrepant with GR at the $95\%$ confidence level
(but fully consistent if $\alpha_{B}=0$) and they argue that this is due to
the fact that, with the specific choice of kineticity $\alpha_{k}=0.1$, an
artificial prior is induced which biases the results. More recently. the
analysis of the Planck 2018 data (Planck, 2018a), for the parameter subspace
where $\alpha_{B}=-\alpha_{M}$ found slight preference for $\alpha_{M}<0$
which disappeared once RSD and WL data was included, consistent with their
analysis in term of $\mu$, $\gamma$.

Of particular interest are the current constraints on Galileon theories. The
constraints on the speed of gravitational waves from GW170817 rules out
theories with quartic or quintic terms in the action of equation 15.
Constraints from large scale structure and, in particular, the Integrated
Sachs Wolfe effect in equation 9 can be used to show that the cubic terms lead
to a grossly discrepant effect (Renk et al., 2017). This means that
cosmological Galileon gravity is effectively ruled out.

As we have seen, they current situation is that constraints on different
parameterizations of deviations from GR on cosmological scales are still weak.
Yet, in particular situations there is mild evidence for non GR behaviour. In
one case, this evidence is primarily driven by the high $A_{L}$ anomaly in the
angular power spectrum of the CMB and can be compensated by the low amplitude
of galaxy weak lensing or RSDs. In another analysis, in this case of Horndeski
theories, it is the RSDs that are pulling the constrained values of the
$\alpha$s away from GR. Furthermore, the results depend quite significantly on
the time dependence assumed or the cuts to which the data sets have been
subjected. Given that the attempts at constraining deviations from GR are so
disparate in methodology, as well as few and far between, there is a serious
concern that a complete and final set of reliable, calibrated results is still
lacking and that can be taken as a basis for more definitive statement about
the status of GR on cosmological scales. Furthermore, the overriding consensus
is that there is no evidence for deviations from GR from a Bayesian evidence
point of view: the complexity of the extensions outweighs the improvement in
the fits.

Nevertheless, these various analysis’ have shown the way forward in how to
constrain gravity on cosmological scales. A clear approach to the analysis is
emerging and, more importantly, a sense of the systematic problems and
pitfalls that must be dealt with if one is to obtain significant constraints
of GR with current and future data.

##  6 Alternative constraints - clusters, galaxies and voids.

As we have seen, there has been some progress in applying the techniques for
constraining GR on large scales to existing data, with mixed results. The
situation is less developed on smaller scales where, on the one hand, non-
linear gravitational collapse and baryonic physics can play a significant role
and, on the other hand, the distinctive signatures of screening can emerge.
Existing constraints are fewer, more scattered and, unlike on large scales,
very model specific. In this section we will undertake a brief survey of the
more durable results.

###  6.1 Cluster masses.

To begin with, and as we saw discussed in Sections 3 and 4, it makes sense to
look for evidence for gravitational slip by comparing the way relativistic
particles (i.e. photons) and non-relativistic particles (e.g. baryons or dark
matter) respond to a gravitational potential and mass. One such approach is to
compare different ways of measuring cluster masses: X-ray, Sunyaev-Zeldovich
and weak lensing observations. Each one of these observations measures
different things, from temperature and pressure profiles to gravitational
potentials, and it is necessary to construct a model (using, for example the
fluid equations and assuming hydrostatic equilibrium) to bring them all
together. Such an approach has been undertaken in Terukina et al. (2014) and
used to place constraints on the different free parameters in an individual
cluster, including one that controls deviations from gravity. The calculation
is done in the non-linear regime and has been done for one particular model,
$f(R)$; the authors found $df/dR\lesssim 5\times 10^{-5}$. This method has
also been applied to constraints on the Vainshtein mechanism where no evidence
for screening was found (Sakstein et al., 2016).

Instead of looking at individual clusters, one can look at the statistics of
ensembles of clusters and, for example, the cluster mass function, $n(M)$.
Arguably, this statistic will not be overwhelmingly contaminated by baryonic
physics given the size of the objects and the scales probed. Attempts at
constraining this statistic necessarily involve N-body simulations (although
progress has been made developing semi-analytic methods) and are, again, model
specific. Again, most effort has gone into testing $f(R)$ (and some other
alternatives) using estimates of cluster abundances from a range of cluster
catalogues (Schmidt et al., 2011; Lombriser et al., 2012; Cataneo et al.,
2015). The constraints are at the level of $df/dR\lesssim 1-10\times 10^{-5}$.

###  6.2 Screened fifth forces in galaxies.

Some progress has been made in looking for screening within galaxies. The
latest, most detailed search for chameleon screening has, arguably, lead to
evidence of a screened fifth force of the form $F=-\Delta Ge^{-mr}/r^{2}$ with
a strength of $\Delta G/G\simeq 0.01$ on scales of $1/m\sim 1$ Mpc. This
evidence comes from two very different types of analysis. In the first case, a
systematic offset between the screened stellar component and the unscreened HI
component of a selection of galaxies is consistent with the presence of a weak
fifth force (Desmond et al., 2018b, c). A number of systematic effects have
been included and are taken into account in the error budget although two
caveats most be considered. If one considers a (overly) conservative error on
the centre of the HI gas (effectively doubling the uncertainty that is
presented in the data set), the detection goes away and one obtains a strong
upper bound, $\Delta G/G<0.01-0.1$ depending on the screening scale. But, and
more importantly, little is know about how baryonic effects may lead to a
signal that mimics the fifth force. While the signature that is constrained is
very specific, a detailed analysis of ultra-high resolution simulations of
galaxies in a large enough volume is required to completely confirm or rule
out the detection.

The second type of the analysis looks at another signature described in
Section 4: warped stellar disks as the galaxies fall into their (displaced)
dark matter halos (Desmond et al., 2018). There the authors used r-band images
of mostly late time galaxies in the NASA Sloan Atlas to compile a catalogue of
approximately $10^{3}$ warps with a specific focus on the ”cup” shaped
signature of a screened fifth force. Again, they found a strong indication of
a fifth force with $\Delta G/G\simeq 0.01$ on scales of around $2$ Mpc,
surprisingly consistent with the measurement of star-gas offsets. Remarkably,
this new analysis is with an altogether different data set and involving a
very different signature. Nevertheless it is potentially plagued by its own,
as yet, unquantified systematics: the effects of baryonic physics is poorly
understood but more importantly, there is no clear idea yet of how warps arise
in galaxy. While there is a consensus that warps are induced by environmental
effects, the fact that warps have been found in isolated galaxies calls this
environmental origin into question. A better understanding of how warps form
would greatly help in firming up this detection of a screened fifth force.

A different, but related, constraint on screened forces has been found by
attempting an Eotvos like test on galactic/cosmological scales. In essence,
the idea is to compare the motion of a ”small” stellar system and a
supermassive black hole in the gravitational potential of another, larger
system. A black hole is chosen as one of the test bodies because in some
theories (Hui & Nicolis, 2012), it will satisfy a no-hair theorem and
therefore will be unscreened (the authors are inspired by, and very
specifically address, the case of Galileon theories which are endowed with
Vainshtein screening). As a result, and for a reasonable choice of the
screening parameters, one should expect an offset of ${\cal O}({\rm kpc})$
between a black hole and its host galaxy.

In (Asvathaman et al., 2017) and (Sakstein et al., 2017), the authors looked
at the black hole centred on M87 as it falls under the influence of a sub-
clump of the Virgo cluster around M84 and M86. They found that the fifth force
field strength of the Galileon must be less than $10^{3}(km/s)^{2}/kpc$. The
authors had to make a number of assumptions about the mass to light ratio,
concentration, halo profile, etc, but ultimately found that their constraint
is stronger than current constraints on Galileons from the Solar System.
Again, much is to be done to better understand the systematics and it is clear
that more robust constraints will be obtained with more objects but the route
is promising for constraining (or detecting) Vainshtein screening.

###  6.3 Cosmic voids.

The arena, par excellence, where constraints on the screening mechanism should
be most effective, is in cosmic voids. As explained in section 4, one should
expect forces to be unscreened as they are regions of low density, curvature
or potential. There has been a reasonable amount of effort in thinking about
what signatures to look for, primarily based on analytic calculations and
numerical simulations. The use of voids as a tool of more standard
cosmological analysis is also growing with first attempts at measuring the
void-correlation function already in hand.

Surprisingly, there are no de facto constraints on deviations from GR with
current data. The only attempt, thus far, is embedded in efforts to constrain
cosmological parameters from the void correlations function. The authors of
(Hamaus et al., 2016) have found a constraint on the linear $f/b=0.417\pm
0.089$ at median redshift, ${\bar{z}}=0.57$, which is entirely consistent with
$\Lambda CDM$. Thus, they claim, there is no evidence for deviations from GR
within voids.

##  7 Future prospects.

###  7.1 Theoretical status.

Testing gravity with cosmology has come a long way over the last decade. And,
in some aspects, one might consider it a mature field. For a start, we have a
much clearer understanding of what it means to construct alternatives to
General Relativity. Primarily it involved adding extra degrees of freedom but
do so one has to be careful that instabilities, ghosts or other pathologies
aren’t unwittingly introduced. And even if they are, they may turn out to be
benign. So, while in the past, there were a set prejudices and a disparate set
of rules of what one could or could not do when tampering with GR, there is
now a much more coherent and sensible approach. One of the by products is that
we have a clearer understanding of the landscape of gravitational theories and
how they link together. For example, scalar-tensor theories take pride of
place and are consumately well understood even though that understanding is
not necessarily complete. And their structure can be mimicked in vector-tensor
theories leading to a burgeoning exploration of this different part of the
landscape. Tensor-tensor theories – massive gravity or bigravity theories –
will reduce, in a particular sector to a type of scalar tensor theories but
also have properties that are uniquely their own. And so on as one moves into
higher dimensions and non-locality.

From the point of view of cosmology, and in particular on the largest scales,
there is now a completely well understood theory of a linearly perturbed
expanding universe. So much so that it can be (and has been) implemented in
precise Einstein-Boltzman solvers that allow us to predict (linear)
cosmological observables with exquisite precision. To a lesser extent, we also
have a good idea of how smaller scales may be subjected to gravitational
screening mechanisms that can mask fifth forces; we have a reasonable complete
view of the different possible mechanisms and have numerical implementation of
some of them.

###  7.2 Observational status.

When it comes to obtaining constraints from current data, the situation is
more uncertain. The first steps have been taken and there are now a number of
pipelines that can take current observations of the large scale structure of
the universe – the CMB, galaxy and weak lensing surveys – and find a set of
constraints on gravitational parameters, i.e. parameters that encapsulate
deviations from GR on linear scales. The results are intriguing in that
anomalies in the CMB power spectrum due to what seems like excess
gravitational lensing push the constraints away from GR. Paradoxically, the
low amplitude of lensing in galaxy gravitational lensing surveys can
compensate when combined with the CMB data. Measurements of growth seem
marginally lower than what is expected from GR yet do not yet have enough
statistical power to play significant role.

An overall concern is the disparate set of methods and parametrizations being
used by the (few) different groups which makes it difficult to compare and,
more importantly, cross calibrate the different results. This differs from
cosmological constraints of parameters within $\Lambda$CDM; there, a large
number of groups have found constraints on, for example, the fractional
densities of the different energy densities, the expansion rate, the spectral
index of fluctuations, etc and there is a very clear and accurate consensus of
what they are. Until this level of rigor is applied to the gravitational
parameters, it is difficult to interpret what the current constraints actually
mean or whether they are robust.

Nevertheless, where there are constraints, they are still weak, roughly of
${\cal O}(1)$. This means that, even in the case where there is a 3$\sigma$
discrepancy with GR, the increase in complexity by including extra
gravitational parameters still outweighs the improvement in the likelihood.
This is typically characterized in terms of Bayes factors, comparing the
Bayesian evidence between the two different scenarios. The fact that, from
this point of view, deviations from GR are disfavoured relative to GR
parallels other attempts at extending $\Lambda$CDM and merely confirms that
the base model is still the best candidate for explaining the Universe.

On smaller scales, deeply into the non-linear regime where more complex,
baryonic physics can play a role, tests of gravity are still in their infancy
(Cataneo et al, 2018). While it is an extraordinarily difficult regime to work
in and prone to countless systematics, it is also, potentially immensely
rewarding. Data on these scales is accruing at a phenomenal rate and, if
systematics are properly modelled and under control, they should allow for
constraints (or detections) with a high statistical power. The first steps
have been taken with a few detections and constraints of an altogether new
type of effect – gravitational screening – which have emerged. There is
clearly a vast and uncharted territory to be explored.

###  7.3 Planned surveys and prospects.

Although current constraints are weak and inconsistent, the hope is that with
the new generation of surveys, the situation will change dramatically. Quite
possibly, over the next decade, we will transition from ${\cal O}(1)$
constraints on deviations from GR to ${\cal O}(10^{-1})$ or even ${\cal
O}(10^{-2})$. A number of planned experiments dominate the landscape. The
Euclid satellite (http://sci.esa.int/euclid/), primarily funded by the
European Space Agency, will survey 15,000 square degrees of the sky, building
up a catalogue with spectra of upto $10^{7}$ galaxies. In parallel, a ground-
based spectroscopic survey, DESI (http://desi.lbl.gov), based in Arizona will
collect spectra of galaxies of a similar number and area. The Large Synoptic
Survey Telescope, LSST, (http://www.lsst.org) based in the Atacama Desert will
undertake a deep, imaging, survey ($r>27$) with an area of up to 20,000 square
degrees. The Square Kilometre Array, SKA, (https://www.skatelescope.org) will
produce a radio survey of HI emission and galaxies in the frequency range
$50-1760$ MHz, mapping structure out to $z\simeq 3-4$. The Simons observatory
(https://simonsobservatory.org/index.php), also in the Atacama Desert will map
the CMB to exceptionally high resolution and sensitivity, while the Stage IV
CMB survey (https://cmb-s4.org) will, hopefully, be a coordinated set of
multiple ground based observatories which will cover upto $40\%$ of the sky.

One can forecast the precision with which one will be able to constrain the
gravitational parameters. For example, with Euclid or LSST, we expect to
reduce the uncertainty in $\mu$ and $\gamma$ from ${\cal O}(1)$ today down to
${\cal O}(10^{-2})$ . If we restrict ourselves to scalar-tensor theories
described by the Horndeski action, again, and consider combinations of LSST,
SKA and Stage IV experiments, we find the uncertainty in the $\alpha_{X}$ is
reduced to ${\cal O}(10^{-1})$ to ${\cal O}(10^{-2})$ (Alonso et al., 2017).
An interesting case is that of JBD gravity where we found that current
constraints place $\omega_{\rm BD}>10^{3}$; future constraints, using
combinations of LSST, SKA and Stage IV will push $\omega_{\rm BD}>{\rm few}\
\times\ 10^{4}$, comparable with constraints on the Solar System and
astrophysical scales. Once we reach this level, we will truly have entered a
new age in gravitational physics, with constraints on cosmological scales
playing a significant role in understanding gravity beyond GR. In fact, if the
values of $\mu$ and $\gamma$ (or equivalently the current values of the
$\alpha_{X}$) which current data seem to find discrepant with GR hold up, then
there will be uncontrovertible evidence of new physics.

Future surveys will also allow us to greatly improve searches for screening,
or fifth forces more generally, in galaxies and other astrophysical
structures. For example, with the SKA it will be possible to obtain constrains
on $\Delta G/G\simeq 10^{-9}$ using HI-optical offsets of galaxies centres
(Desmond et al., 2018b, c).

###  7.4 Synergies with other gravitational probes.

With the parallel development of observational cosmology, gravitational wave
physics and black hole imaging, interesting (and inevitable) synergies have
developed. For a start, and as more and better detections of compact object
mergers are recorded, it will become possible to undertake cosmological
constraints with such events. A first attempt with GW170817 at constraining
the Hubble constant has shown the tremendous effectiveness of this new window
on the expansion rate of the Universe (LIGO, 2017b). The constraint on the
speed of gravitational waves has already had a devastating impact on the fauna
of modified gravity theories, leading to a severe cull of acceptable models
(Baker et al., 2017; Creminelli & Vernizzi, 2017; Ezquiaga & Zumalacarregui,
2017). Furthermore, the absence (or presence) of friction in the propagation
of gravitational waves can have an impact on what deviations from GR are
allowed (Belgacem et al., 2018; Amendola et al., 2018). And the promise of
detecting a stochastic background of gravitational waves is leading to renewed
scrutiny of novel effects that might arise in deviations from GR.

A particular example of synergy between cosmological and gravitational wave
physics has recently been invoked in Tattersall et al. (2018) where it was
shown that many (primarily scalar-tensor) theories in which the extra fields
had a cosmological impact, yet in which the speed of gravitational waves was
equivalent to that of light, would obey no-hair theorems to some degree. This
means that, for example, any smoking gun for deviations of gravity would have
to be on large, cosmological, scales or in the, still dynamical, regime post-
merger of binary collision events. I.e. these theories would not manifest
themselves during the inspiral or in black hole imaging.

This emerging and exciting synergy is the hallmark of what I believe is a new
and exciting phase in gravitational physics, a new ”golden age” for general
relativity. Until now the focus, and quest, for new physics has been in the
high energy regime, of particle accelerators and direct (and indirect) dark
matter searches. With the multiple new windows on the gravitational universe,
in which cosmology takes pride of place, one would hope that new forces and
phenomena are on the verge of discovery. In the very least, we will end up
with a cast iron theory for gravity, tested over an enviable range of scales
and regimes.

## DISCLOSURE STATEMENT

The author is not aware of any affiliations, memberships, funding, or
financial holdings that might be perceived as affecting the objectivity of
this review.

## ACKNOWLEDGMENTS

Much of my understanding of this field was developed with a wide number of
collaborators, to all of whom I am grateful. I would like to thank specific
input from Alex Barreira, Harry Desmond, Shahab Joudaki, Kazuya Koyama, Martin
Kunz, Macarena Lagos, Danielle Leonard, Antony Lewis and Alessandra Silvestri.
I am also extremely grateful to Eva-Maria Mueller, Shahab Joudaki and Sukhdeep
Singh for producing ”bespoke” images for this review. This work was supported