Discrimination between $\Lambda \mathrm{CDM}$, quintessence, and modified gravity models using wide area surveys

In the past decade or so observations of supernovae, large scale structures, and the cosmic microwave background have confirmed the presence of what is called dark energy—the cause of the accelerating expansion of the Universe. Its density has also been measured as well as the value of other cosmological parameters according to the concordance $\Lambda \mathrm{CDM}$ model with few percent uncertainties. The next generation of surveys should allow to constrain this model with better precision or to distinguish between a $\Lambda \mathrm{CDM}$ and alternative models such as modified gravity and (interacting)-quintessence models. In this work we parametrize both the homogeneous and anisotropic components of matter density in the context of interacting dark energy models with the goal of discriminating between $f(R)$ modified gravity and its generalizations, and interacting dark energy models, for which we also propose a phenomenological description of energy-momentum conservation equations inspired by particle physics. It is based on the fact that the simplest interactions between particles/fields are elastic scattering and decay. The parametrization of the growth rate proposed here is nonetheless general and can be used to constrain other interactions. As an example of applications, we present an order of magnitude estimation of the accuracy of the measurement of these parameters using Euclid and Planck surveys data, and leave a better estimation to a dedicated work.

Figure 1

$A(z)$ as a function of the redshift. To see how well $A(z)$ can distinguish between various models and how systematic and statistical errors as well as the parametrization affect the reconstructed model, we consider 3 parametrizations as written on the plot above. Note that parametrizations for the plot in the center and on the right are equivalent up to a redefinition of coefficients $w_0$ and $w_1$. We first consider a given value for $w(z)$ at $z=0$ and $z=3$, determine corresponding coefficients $w_{0i}$ and $w_{1i}$ where index $i$ is for initial. Then to simulate systematic errors we plot the following models: $w_0=-1+|w_{0i}+1|$, $w_1=w_{1i}$ (dotted line), $w_0=-1-|w_{0i}+1|$, $w_1=-w_{1i}$ (dot-dash), $w_0=-1+|w_{0i}+1|$, $w_1=-w_{1i}$ (dashed) and $w_0=w_{0i}$ and $w_1=w_{1i}$ (full line). The shaded regions present statistical errors. The uncertainty of $az$ is $1{\sigma }_{A(z=0)}=0.01$ (top row) and $1{\sigma }_{A(z=0)}=0.05$ (bottom row) at $z=0$ and evolves with the redshift as ${\sigma }_A(z)={\sigma }_A(z=0)(1+z{)}^2$. It seems to be possible to distinguish between normal and phantom dark energy models easily, if uncertainties are limited to few percents. Evidently, achieving such a precision is challenging even for space missions such as Euclid.