Time dependent couplings in the dark sector: from background evolution to nonlinear structure formation

TL;DR

This study extends coupled dark energy models by introducing time-dependent DE–CDM couplings and assessing their impact from background evolution to nonlinear structure formation. By solving the background and linear perturbation equations and performing high-resolution N-body simulations with a modified GADGET-2 code, the authors demonstrate that the coupling form critically controls the fate of cosmic structures: scale-factor–driven couplings mimic $\Lambda$CDM for much of the history but amplify small-scale power at late times, while exponential couplings produce a Growing $\phi$MDE that alters growth and halo properties in distinctive ways. The results show that variable couplings can lower inner halo densities and halo baryon fractions (notably for the EXP010e2 model) and modify halo concentrations, offering a potential route to alleviate small-scale tensions without conflicting with background evolution constraints. Overall, time-dependent couplings emerge as a viable and testable extension of the standard cosmological model with rich phenomenology in both linear and nonlinear regimes.

Abstract

We present a complete numerical study of cosmological models with a time dependent coupling between the dark energy component driving the present accelerated expansion of the Universe and the Cold Dark Matter (CDM) fluid. Depending on the functional form of the coupling strength, these models show a range of possible intermediate behaviors between the standard LCDM background evolution and the widely studied case of interacting dark energy models with a constant coupling. These different background evolutions play a crucial role in the growth of cosmic structures, and determine strikingly different effects of the coupling on the internal dynamics of nonlinear objects. By means of a suitable modification of the cosmological N-body code GADGET-2 we have performed a series of high-resolution N-body simulations of structure formation in the context of interacting dark energy models with variable couplings. Depending on the type of background evolution, the halo density profiles are found to be either less or more concentrated with respect to LCDM, contrarily to what happens for constant coupling models where concentrations can only decrease. However, for some specific choice of the interaction function the reduction of halo concentrations can be larger than in constant coupling scenarios. In general, we find that time dependent interactions between dark energy and CDM can in some cases determine stronger effects on structure formation as compared to the constant coupling case, with a significantly weaker impact on the background evolution of the Universe, and might therefore provide a more viable possibility to alleviate the tensions between observations and the LCDM model on small scales than the constant coupling scenario. [Abridged]

Figures (17)

• Figure 1: Mass correction of CDM particles as a function of redshift for the different interacting DE models under investigation. The three panels refer to models with three different values of the coupling at $z=0$, respectively $\beta _{c}(\phi _{0}) = 0.5$ ( upper panel), $\beta _{c}(\phi _{0}) = 0.75$ ( middle panel), and $\beta _{c}(\phi _{0}) = 1.0$ ( lower panel). The color legend for all the models is given in the upper panel.
• Figure 2: Left Panel: The evolution of the DE fractional energy density $\Omega _{\phi }$ (solid lines) for a few models of variable coupling, and in addition for the constant coupling models RP5 studied in BA10. The "$\phi$MDE" phase is clearly visible for the RP5 model, and is completely absent for the phenomenological coupling models EXP010a2 and EXP015a3, that show no significant difference with respect to a $\Lambda$CDM evolution. The the exponential coupling models have the intermediate behavior of a "Growing $\phi$MDE". Dotted lines represent the value of the analytic "$\phi$MDE" solution Amendola_2000 extended to the case of growing couplings. Right Panel: Equation of state parameter $w_{\phi }$ for the same models as in the left panel. The phenomenological coupling models have always an equation of state parameter very close to the cosmological constant case, while the exponential coupling models follow a similar behavior to the constant coupling model RP5.
• Figure 3: The ratio of the luminosity distance between $z=0$ and $z=3$ ( left panel) and of the angular-diameter distance between $z=10$ and $z=1000$ ( right panel) to the respective $\Lambda$CDM evolution as a function of redshift. The two thick solid black lines in the two plots represent the two limiting cases considered in this work, i.e.$\,$ the constraints given by Bean_etal_2008 and LaVacca_etal_2009. The dark-grey and light-grey shaded regions that lie, respectively, between the $\Lambda$CDM value of $1$ and each of the two limiting scenarios represent the allowed regions according to the selection criterion described in the text. As it can be noticed, only a few models are found to be compatible with the constraints of Bean_etal_2008, while a large fraction of the models considered in the present work are compatible with the bounds of LaVacca_etal_2009.
• Figure 4: This figure shows the evolution as a function of redshift of the coupling strength $\beta _{c}(\phi )$ ( panel a), of the scalar field velocity $x$ as defined in Eqn. \ref{['dimensionless_variables']} ( panel b), and of the friction term coefficient given by $2 x \beta _{c}(\phi )$ ( panel c), for the five coupled DE models studied with our high-resolution N-body simulations and in addition for the constant coupling model RP5 studied in BA10 which is overplotted for comparison. It is worth noticing how the dynamic evolution of the scalar field modulates the effect of the coupling in the friction term: despite a large value of the coupling $\beta _{c}(\phi )$ at low redshifts the friction term is strongly suppressed in the models which feature a too fast decrease of the coupling with redshift due to the absence of a proper "$\phi$MDE" or of a "Growing $\phi$MDE" phase.
• Figure 5: The evolution of the coupling derivative $\beta '_{c}(\phi )$ as a function of the e-folding time $N \equiv \ln a$. The different colors represent the different types of time evolution of the coupling function $\beta _{c}(\phi )$ while the different linestyles correspond to different values of the coupling at the present time. It is important to notice that none of the models presents a value of the coupling derivative larger than ${\cal O}(1)$ at any stage of cosmic history.