Effective dark energy equation of state in interacting dark energy models

TL;DR

The work studies how a non-minimal interaction between dark matter and dark energy, modeled by $Q(z)=\alpha H (1+z)^{-\beta} \rho_w$, biases the reconstruction of the dark energy EoS from CMB data. It derives exact analytic solutions for the IDE background via density-modulation functions $f(z)$ and $g(z)$, with $f(z)=\exp\left(-\frac{\alpha}{\beta}\left[(1+z)^{-\beta}-1\right]\right)$ for $\beta\neq 0$ (and $f(z)=(1+z)^{\alpha}$ if $\beta=0$) and a $g(z)$ governed by $\frac{dg}{dz}=-\alpha \frac{\Omega_{w0}}{\Omega_{m0}} f(z) (1+z)^{\gamma}$, where $\gamma=3w-\beta-1$, expressible via the incomplete Gamma function; densities are $\rho_m=\Omega_{m0} g(z) (1+z)^3$ and $\rho_w=\Omega_{w0} f(z) (1+z)^{3(w+1)}$. A closed-form for the effective DE EoS is $w_{eff}(z)=\dfrac{p_w}{\rho_{weff}}=\dfrac{w}{1+\dfrac{\Omega_{m0}}{1-\Omega_{m0}} \dfrac{\Delta g(z)}{f(z)} (1+z)^{-3w}}$, along with a first-order approximation $w_{eff}(z) \approx w\left(1 - \dfrac{\alpha}{\beta-3w} (1+z)^{-\beta}\right)$ and $w_{eff0}=w\left(1 - \dfrac{\alpha}{\beta-3w}\right)$. The results show that IDE can mimic phantom behavior even when $w>-1$ and highlight that background energy transfer alone does not uniquely determine perturbation-level transfer, underscoring the need for additional assumptions to constrain IDE models.

Abstract

In models where dark matter and dark energy interact non-minimally, the total amount of matter in a fixed comoving volume may vary from the time of recombination to the present time due to energy transfer between the two components. This implies that, in interacting dark energy models, the fractional matter density estimated using the cosmic microwave background assuming no interaction between dark matter and dark energy will in general be shifted with respect to its true value. This may result in an incorrect determination of the equation of state of dark energy if the interaction between dark matter and dark energy is not properly accounted for, even if the evolution of the Hubble parameter as a function of redshift is known with arbitrary precision. In this paper we find an exact expression, as well as a simple analytical approximation, for the evolution of the effective equation of state of dark energy, assuming that the energy transfer rate between dark matter and dark energy is described by a simple two-parameter model. We also provide analytical examples where non-phantom interacting dark energy models mimic the background evolution and primary cosmic microwave background anisotropies of phantom dark energy models.

Figures (4)

• Figure 1: Evolution of $g$ (solid line) and $f$ (dashed line) with redshift $z$ for $\beta=0.5,1,2,4$, $w=-0.99$ and $\Omega_{m0} g_\infty = 0.27$. The functions $f(z)$ and $g(z)$ were computed using Eqs. (\ref{['fexact']}) and (\ref{['gexact']}), respectively, and the values of $\alpha$ were chosen such that $g_\infty=0.8$ for all models.
• Figure 2: Evolution of the effective DE EoS parameter $w_{eff}$ with redshift $z$ computed using the exact solution obtained from Eqs. (\ref{['fexact']}), (\ref{['gexact']}) and (\ref{['waexact']}) for $\beta=0.5,1,2,4$, $w=-0.99$ and $\Omega_{m0} g_\infty = 0.27$ (solid lines) as well as the corresponding evolution obtained using the approximation given in Eq. (\ref{['waapprox']}) for the same values of $\beta$, $w$ and $\Omega_{m0}$ (dashed lines). The values of $\alpha$ were chosen such that $g_\infty=0.8$ for all models.
• Figure 3: Same as Fig. 1 but with $g_{\infty}=1.2$ and $w=-0.95$.
• Figure 4: Same as Fig. 2 but with $g_{\infty}=1.2$ and $w=-0.95$.