## Testing gravity in the local universe

##### Abstract

General relativity (GR) has stood as the most accurate description of gravity for
the last 100 years, weathering a barrage of rigorous tests. However, attempts
to derive GR from a more fundamental theory or to capture further physical
principles at high energies has led to a vast number of alternative gravity theories.
The individual examination of each gravity theory is infeasible and as such a
systematic method of examining modified gravity theories is a necessity.
Studying generic classes of gravity theories allows for general statements about
observables to be made independent of explicit models. Take, for example,
those models described by the Horndeski action, the most general class of scalar-tensor
theory with at most second-order derivatives in the equations of motion,
satisfying theoretical constraints. But these constraints alone are not enough for
a given modified gravity model to be physically viable and hence worth studying.
In particular, observations place incredibly tight constraints on the size of any
deviation in the solar system. Hence, any modified gravity would have to mimic
GR in such a situation. To accommodate this requirement, many models invoke
screening mechanisms which suppress deviations from GR in regions of high
density. But these mechanisms really upon non-linear effects and so studying
them in complex models is mathematically complex.
To constrain the space of actions of Horndeski type to those which pass solar-system
tests, a set of conditions on the four free functions of the Horndeski
action are derived which indicate whether a specific model embedded in the action
possesses a GR limit. For this purpose, a new and surprisingly simple scaling
method is developed, identifying dominant terms in the equations of motion by
considering formal limits of the couplings that enter through the new terms in the
modified gravity action. Solutions to the dominant terms identify regimes where
nonlinear terms dominate and Einstein’s field equations are recovered to leading
order. Together with an efficient approximation of the scalar field profile, one can
determine whether the recovery of Einstein’s field equations can be attributed to
a genuine screening effect.
The parameterised post-Newtonian (PPN) formalism has enabled stringent tests
of static weak-field gravity in a theory-independent manner. This is through
parameterising common perturbations of the metric found when performing a
post-Newtonian expansion. The framework is adapted by introducing an effective
gravitational coupling and defining the PPN parameters as functions of position.
Screening mechanisms of modified gravity theories can then be incorporated
into the PPN framework through further developing the scaling method into
a perturbative series. The PPN functions are found through a combination of
the scaling method with a post-Newtonian expansion within a screened region.
For illustration, we show that both a chameleon and cubic galileon model have
a limit where they recover GR. Moreover, we find the effective gravitational
constant and all PPN functions for these two theories in the screened limit.
To examine how the adapted formalism compares to solar-system tests, we also
analyse the Shapiro time delay effect for these two models and find no deviations
from GR insofar as the signal path and the perturbing mass reside in a screened
region of space. As such, tests based upon the path light rays such as those done
by the Cassini mission do not constrain these theories.
Finally, gravitational waves have opened up a new regime where gravity can be
tested. To this end, we examine how the generation of gravitational waves are
affected by theories of gravity with screening to second post-Newtonian (PN)
order beyond the quadrupole. This is done for a model of gravity where the
black hole binary lies in a screened region, while the space between the binary’s
neighbourhood and the detector is described by Brans-Dicke theory. We find
deviations at both 1.5 and 2 PN order. Deviations of this size can be measured by
the Advanced LIGO gravitational wave detector highlighting that our calculation
may allow for constraints to be placed on these theories. We model idealised data
from the black hole merger signal GW150914 and perform a best fit analysis.
The most likely value for the un-screened Brans-Dicke parameter is found to be
ω = -1:42, implying on large scales gravity is very modified, incompatible with
cosmological results.