# Theoretical Priors On Modified Growth Parametrisations 

$^{1}$Yong-Seon Song, 1 , $^{2}$Lukas Hollenstein, $^{1}$Gabriela Caldera-Cabral and $^{1}$Kazuya Koyama 

$^{1}$Institute of Cosmology & Gravitation, Dennis Sciama Building, University of Portsmouth, Portsmouth, PO1 3FX, United Kingdom 

$^{2}$D´ epartement de Physique Th´ eorique, Universit´ e de Gen` eve, 24 Quai Ernest Ansermet, 1211 Gen` eve 4, Switzerland 

(Dated: October 30, 2018) 

Next generation surveys will observe the large-scale structure of the Universe with unprecedented accuracy. This will enable us to test the relationships between matter over-densities, the curvature perturbation and the Newtonian potential. Any large-distance modification of gravity or exotic nature of dark energy modifies these relationships as compared to those predicted in the standard smooth dark energy model based on General Relativity. In linear theory of structure growth such modifications are often parameterised by virtue of two functions of space and time that enter the relation of the curvature perturbation to, first, the matter over- density, and second, the Newtonian potential. We investigate the predictions for these functions in Brans-Dicke theory, clustering dark energy models and interacting dark energy models. We find that each theory has a distinct path in the parameter space of modified growth. Understanding these theoretical priors on the parameter- isations of modified growth is essential to reveal the nature of cosmic acceleration with the help of upcoming observations of structure formation. 

Our understanding of the laws of physics we gained on Earth has already been challenged by cosmological obser- vations of the formation of galaxies and the accelerated expansion of the Universe [1, 2]. In order to establish a cosmological standard model without inconsistencies be- tween observations and known physics, it is unavoidable either to assume the presence of as yet undetected dark materials filling 95% of the energy of the Universe, or to modify some of the fundamental laws of physics such as Einstein’s General Relativity. The information on the underlying physics of the Universe is provided by look- ing at the formation of the large scale structure (LSS) of matter in the Universe. While current observations have revealed the breakdown of our knowledge of physics on cosmological scales, the future observation of the LSS will provide a clue to reveal the nature of the late-time cosmic acceleration; which part of our physical picture should be modified - matter or gravity, and how it is modified [3–32]. 

The current standard cosmological model was estab- lished based upon unknown dark materials and the va- lidity of General Relativity (GR) on cosmological scales. We call this standard model smooth dark energy (sDE). It consists of the following energy components: 5% baryons as described by the standard model of particle physics and 95% of unobserved dark particles – 20% of collision- less dark matter, fitting galaxy formation and the peaks in the spectrum of cosmic microwave background (CMB) anisotropies, and 75% dark energy explaining the cosmic acceleration, described by a perfect fluid that is homo- geneous (smooth) on sub-horizon scales. It is difficult to accept dark energy as a standard building block of energy components of the Universe as it is not yet supported 

PACS numbers: draft 

## I. INTRODUCTION 

neither phenomenologically nor theoretically. Gravity on cosmological scales can be modified to accelerate the cosmic expansion without dark energy. The degeneracy between GR and modified gravity (MG) models in the background expansion history is broken by the growth history of structure formation. Any departure from the growth history in sDE is probed by two functions ex- pressing new degrees of freedom: the mass screening, Q , changes the relationship between the curvature pertur- bation and the matter over-densities, while the effective anisotropic stress, η , alters the relationship between the curvature perturbation and the Newtonian potential. 

Here we study whether the possible detection of non- trivial Q and η can be considered as a probe of the break- down of the sDE model at cosmological scales. We show that non-trivial Q and η functions are also induced in other GR models via clustering of dark energy or inter- actions between dark energy and dark matter. GR will be confirmed (or excluded) by looking at the relation be- tween Q and η , not merely by a departure of Q and/or η from unity. We find that both MG models [33, 34] and exotic GR models [35] provide unique distinguish- able trajectories that are well presented in the plane of two transformed parameters: Σ ≡ Q (1 + 1 /η ) / 2 param- eterises the relation between the lensing potential and matter over-densities and µ ≡ Q/η represents the re- lation between the Newtonian potential and the matter over-densities. We examine these parameters in various models and show that each theory has a distinct path on the plane of those parameters. The understanding of these theoretical priors is essential to reveal the nature of the cosmic acceleration from future structure formation observations. 

This paper is organised as follows. In section II we introduce the smooth dark energy (sDE) model and the modified growth parameters Q and η . Then we investi- 

