Analytic study of the effect of dark energy-dark matter interaction on the growth of structures

TL;DR

This work derives an analytic framework for the growth of cosmic structures in a dark-energy–dark-matter interacting model in which the energy-momentum exchange scales with the dark-energy density. By expanding the growth equation around $\Omega_{\text{DE}}=0$ and incorporating a modified continuity equation, the authors obtain explicit expressions for the growth-index coefficients $\gamma_0$ and $\gamma_1$ as functions of the coupling $\xi$ and the dark-energy EOS parameters, together with a backward-propagation function for $\sigma_8(z)$. They validate the analytic growth against CAMB and fit to $f\sigma_8$ data, finding that current observations prefer a weak coupling and that the analytic approach is fastest within a restricted parameter range. The results indicate that while the coupling imprints on $f(z)$, $\sigma_8(z)$, and $\gamma(z)$, tighter constraints will require complementary data from future surveys to robustly detect or rule out dark sector interactions.

Abstract

Large-scale structure has been shown as a promising cosmic probe for distinguishing and constraining dark energy models. Using the growth index parametrization, we obtain an analytic formula for the growth rate of structures in a coupled dark energy model in which the exchange of energy-momentum is proportional to the dark energy density. We find that the evolution of $f σ_8$ can be determined analytically once we know the coupling, the dark energy equation of state, the present value of the dark energy density parameter and the current mean amplitude of dark matter fluctuations. After correcting the growth function for the correspondence with the velocity field through the continuity equation in the interacting model, we use our analytic result to compare the model's predictions with large-scale structure observations.

Figures (8)

• Figure 1: Comparison between analytical and numerical computations of $f(z)$ for the model. In the left panel, the modulus of the relative differences, in logarithmic scale; in the right panel, the dashed lines represent the numerical results for $f(z)$, while the solid lines show our analytical results. Values of $|\xi|$ as big as $0.1$ give large discrepancies and should be avoided. We use thin lines to represent them.
• Figure 2: Histograms for the values of the density parameter and dark matter fluctuation today at the scale of $8\,h^{-1}\,M\parsec$ in the $\Lambda$CDM model. The vertical thin lines mark the best-fit values, and the grey area under the histograms show the $1\sigma$ . In the 2D histogram, the colors map the parameter space points to their unnormalized posterior values, from white (lowest values) to black (highest values), with shades of orange representing intermediate values. The white cross marks the best-fit point.
• Figure 3: Evolution of the growth of structures in the coupled model for varying values of the coupling $\xi$. The negative values (black lines) correspond to the CQDE model and the positive values (green lines) to the CPDE model. In both cases we use $w_0 = -1$ for simplification, since we are interested in seeing the effect of the coupling only. The red line is the $\Lambda$CDM result. The data from table \ref{['tab:data']} are also plotted in (b).
• Figure 4: Histograms for the free parameters of CPDE. The vertical thin lines mark the best-fit values, and the grey area under the histograms show the $1\sigma$ . In the 2D histograms, the colors map the parameter space points to their unnormalized posterior values, from white (lowest values) to black (highest values), with shades of orange representing intermediate values. The white crosses mark the best-fit point. Due to the large uncertainties in the measurements, the data could not constrain the interaction and the parameter.
• Figure 5: Marginalized posterior distributions for (a) CQDE and (b) $w$CQDE models. The vertical thin lines mark the best-fit values, while the grey areas under the histograms in the diagonal show the $1\sigma$ . In the 2D histograms, the colors map the parameter space points to their unnormalized posterior values, from white (lowest values) to black (highest values), with shades of orange representing intermediate values. The white crosses mark the best-fit point. As we can see from the results of $w$CQDE, fixing the parameter is not sufficient to constrain the interaction coupling in the already too tight prior.