## Cosmological tests of gravity

The discovery of the accelerated expansion of the Universe is one of the most astonishing and mysterious findings ever in the history of Physics. This discovery was made after astronomers noted that the light emitted by distant explosions of stars, called supernovae, was fainter than expected -- the extra dimming of the light, it turns out, is due to the fact that the galaxies that host these supernovae are accelerating away from us. This is opposite to the decceleration that one would expect from the gravitational pull of matter. This discovery shook our knowledge of fundamental physics and it made clear that our understanding of the cosmos was in needed of serious and profound revisions. There are currently two main explanations for the phenomenon. The first is a new and exotic form of energy called ''dark energy'' or a cosmological constant that is added to Einstein's field equations of General Relativity (GR), and that is capable to accelerate the expansion of the Universe via a property of it that can be loosely referred to as ''negative gravity''. The second explanation does not involve dark energy, and consists of noting that GR is not very well tested on large, cosmological scales, and so the acceleration could be simply due to a different law of gravity. These two competing explanations gave rise to a very active field of research in cosmology and astrophysics, both at the level of the development of new theoretical models, as well as at the level of the design of novel observational tests.

I am a member of the ''Novel Probes'' project, which is a collaboration that combines review articles on modified gravity and dark matter interactions with discussion forums hosted on Slack. Recently, we have published a review article entitled ''The Novel Probes Project -- Tests of Gravity on Astrophysical Scales'', which presents an overview of the state-of-the-art in testing gravity on cosmological and astrophysical scales [12]. A few selected highlights from my own past work on this area are described next.

### Testing gravity with void lensing

Any theory of gravity beyond Einstein's GR (or ''modified gravity'' theories, as they are popularly called) that hopes to stand a chance at being a viable theory, should be effectively indistinguishable from GR on scales of the size of the Solar System, where GR does an impeccable job at describing the law of gravity. On the other hand, the same theory should exhibit stronger departures from GR on the largest observable scales, if the theory is to be able to explain the accelerated expansion of the Universe. These seemingly mutually exclusive requirements are made compatible by so-called ''screening mechanisms'', which is the ability, enabled by nonlinear equations of motion, for the departures from GR to be large where the density fluctuations are low (like on large scales), but small where the density fluctuations and gravitational potentials become large (like in the Solar System).

The existence of screening mechanisms in modified gravity theories motives shifting the attention to the lowest density regions of the Universe to test gravity. This is what I have done in [14] where I looked at the imprint of modified gravity on the gravitational lensing effect due to voids. Voids are the large, and fairly empty spaces that exist in between all of the structures that cluster together in the Universe. Using N-body simulations of a model called Galileon gravity [13], I have looked at the lensing effects the voids in the simulation would produce, and compared them to the effect one would expect in GR. The main finding was that, indeed, in voids there is no evidence for any action of the screening mechanisms to suppress the modifications to GR, and hence, the Galileon model displayed a very different lensing signal; see Fig.1. This demonstrated the very interesting prospect to use void-related observations, such as their lensing signal, to test the law of gravity on large scales.

### Testing gravity with the growth rate of structure

Modifications to the law of gravity on large scales alter the acceleration of galaxies moving in gravitational potentials, and consequently, their velocities. For example, if gravity gets stronger, then galaxies falling towards the bottom of gravitational potentials will do so at higher speeds. The velocity of galaxies can be probed using an effect known as ''redshift space distortions'' (RSD): the component of the galaxy's velocity that is parallel to the line-of-sight will, via the Doppler effect, alter the inferred redshift of the galaxy, and consequently its inferred radial distance. This results in a distinctive anisotropic pattern in the clustering of galaxies that allows us to infer a parameter known as the ''growth rate'' of structure, which measures how fast galaxies are falling inside the gravitational potentials. Different theories of gravity make different predictions for the growth rate, which can therefore be used to distinguish between them.

Together with collaborators, in [15] I used the final measurements of the growth rate of structure from the BOSS galaxy survey to constrain a popular theory of gravity called the Dvali-Gabadadze-Porrati (DGP) model; see Fig.2. An important and crucial aspect of our analysis relative to others in the literature is that we have first validated the RSD analysis pipeline used in the BOSS survey using galaxy mock catalogues constructed out of $N$-body simulations of the DGP model. This is important because the growth rate is not directly observable, but it is instead a parameter of a RSD modeling framework that is not necessarily valid for all theories of gravity. The default BOSS analyses had validated the RSD model using GR simulations, and in [15] we have crucially repeated the validation for DGP simulations. Only after this crucial validation step is one allowed to self-consistently take the inferred growth rate estimates from the data and contrast them with the DGP model predictions. This is what we have done in this work, which resulted in the tightest constraints ever on the DGP model. This neatly illustrated the strong power of growth rate and RSD analyses to constrain gravity in cosmology.

### The rise and fall of the Galileon model

The ''Galileon model'' is a popular theory of gravity that for some time stood out from the rest of the theory space as a particularly useful and interesting model to study. This model was first discovered by Nicolis, Rattazzi and Trincherini in 2009 by working out the most general Lagrangian involving a single Galileon-invariant scalar field with second-order field equations of motion. This construction is formally only valid in flat Minkowskii space, but ''covariantized'' versions of the model were soon developed to make predictions for cosmology. From a phenomenological and observational perspective, this model was interesting not only because it could explain the acceleration of the expansion of the Universe and satisfy Solar System tests of gravity, but also because it comprised a very rich and general phenomenology that made it a perfect work-horse with which to explore and learn about the different imprints that departures from GR can leave on cosmological observations.

The parameter space of the Galileon model can be divided into three main corners known as (in order of increased complexity) the Cubic, Quartic and Quintic Galileons. In my PhD work, I carried out the first complete and self-consistent constraint analyses of this model using CMB data, both from the WMAP [17] and Planck [18] satellites. The main finding of these works was that the Galileon model could fit the data with the same goodness-of-fit as the standard $\Lambda$CDM+GR model, albeit with very different values of the cosmological parameters. For example, while constraints with GR resulted in neutrino mass values compatible with zero, the Galileon model predicted non-zero neutrino masses with over 6$\sigma$ significance, which opened the door for future Earth-based neutrino mass experiments to play a role in discriminating between these two theories of gravity. The Galileon model also predicted a value of the Hubble expansion rate today, $H_0$, that was compatible with estimates made in the local Universe -- thus, the popular $H_0$ tension was naturally solved in Galileon gravity cosmologies.

The beginning of the end of the Galileon model came in 2017 with the first simultaneous detection of the electromagnetic radiation and gravitational waves emitted by the merging of two neutron stars -- this constrained the difference between the speed of light and speed of gravitational waves to be the same to within 1 part in $10^{16}$. The parts of the parameter space of the Quartic and Quintic models that fit the CMB data predict a different propagation speed, and so in one blow, these two corners of the model were ruled out. The Cubic Galileon remained ''alive'' as it predicts the same propagation speed for light and gravitational waves. This model ended up being ruled out however in a work I have collaborated, led by Janina Renk [16], in which we have found that the Cubic Galileon fits very poorly data from the integrated Sachs-Wolfe (ISW) effect: see Fig.3. This data effectively probes the time-evolution of the gravitational potentials responsible for lensing in the second half of the Universe's lifetime: the observations are compatible with gravitational potentials that decay with time, whereas the Galileon model predicts very rapidly growing potentials. This was the ''last nail in the coffin'' of the Galileon model as a viable theory of gravity.

The story of the Galileon model is a beautiful example of how advanced, rich and multifaceted the field of observational tests of gravity has become. The in-depth analyses of the phenomenology of models like Galileon gravity taught us a great deal about the observational imprints to look out for, and the community is equipped now with a large number of different and complementary tests that can and have been used to slash away various parts of the theory space of modified gravity. Einstein's GR has insofar been able to pass all these tests with flying colors -- the future will tell whether or not this will remain the case, with improved and larger cosmological data sets.

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