Indications of a Late-Time Interaction in the Dark Sector

We show that a general late-time interaction between cold dark matter and vacuum energy is favored by current cosmological data sets. We characterize the strength of the coupling by a dimensionless parameter $q_V$ that is free to take different values in four redshift bins from the primordial epoch up to today. This interacting scenario is in agreement with measurements of cosmic microwave background temperature anisotropies from the Planck satellite, supernovae Ia from Union 2.1 and redshift space distortions from a number of surveys, as well as with combinations of these different data sets. Our analysis of the 4-bin interaction shows that a nonzero interaction is likely at late times. We then focus on the case $q_V\ne 0$ in a single low-redshift bin, obtaining a nested one parameter extension of the standard $\mathrm{\Lambda }\mathrm{CDM}$ model. We study the Bayesian evidence, with respect to $\mathrm{\Lambda }\mathrm{CDM}$, of this late-time interaction model, finding moderate evidence for an interaction starting at $z=0.9$, dependent upon the prior range chosen for the interaction strength parameter $q_V$. For this case the null interaction ($q_V=0$, i.e., $\mathrm{\Lambda }\mathrm{CDM}$) is excluded at 99% C.L.

Figure 1

Two-dimensional ${\mathrm{\Omega }}_c{h}^2$-$q_i$ contours at 68% and 95% C.L. (left) and one-dimensional $q_i$ probability distributions (right) from Planck (black), $\text{Planck}+\mathrm{SN}$ (pink), and $\text{Planck}+\mathrm{RSD}$ (purple). The addition of the RSD data sets breaks the degeneracy between the two parameters, ${\mathrm{\Omega }}_c{h}^2$ and $q_i$, and narrows the probability distributions of $q_3$ and $q_4$ in particular. A null interaction at low redshift is excluded with high significance.

Figure 2

RSD measurements [28, 29, 30, 31, 32, 33] plotted against the theoretical predictions from the best-fit iVCDM model with $q_{34}=-0.128$ (blue) and a $\mathrm{\Lambda }\mathrm{CDM}$ model ($q_{34}=0$) with the same values of cosmological parameters (black).

Figure 3

Bayesian evidence as a function of the prior width, expressed in terms of standard deviations from the mean value of the nested parameter. In purple (solid line) the $q_{34}$ model ($z_{\mathrm{in}}$=0.9) and same model with different choices of $z_{\mathrm{in}}$. In grey (dashed line) the $m_{\nu }$-$\mathrm{\Lambda }\mathrm{CDM}$ model. On the right we report the empirical Jeffreys’ scale defined in Ref. [39].

Figure 4

${\mathrm{\Omega }}_m$-${\sigma }_8$ contours at 68% and 95% C.L. from Planck experiment (black) and $\text{Planck}+\mathrm{RSD}$ (purple) for three theoretical models. The tension between the Planck and RSD data sets that arises in the $\mathrm{\Lambda }\mathrm{CDM}$ model (left) is resolved in the $q_{34}$ interacting vacuum model (middle). Also, in the $\mathrm{\Lambda }\mathrm{CDM}$ model with massive neutrinos (right), this tension with RSD is alleviated (in contrast to the tension that arises when considering nonlinear probes of LSS growth [40]).

Figure 5

Effects of varying $z_{\mathrm{in}}$ in the single bin iVCDM model. Left: $q_V$-$z_{\mathrm{in}}$ contours at 68% and 95% C.L.; $z_{\mathrm{in}}$ is poorly constrained but the degeneracy with $q_V$ is weak. Right: $q_V$ posterior distribution when $z_{\mathrm{in}}$ is either fixed at $z_{\mathrm{in}}=0.9$ ($q_{34}$ model) or marginalized. The posterior when considering a log prior, ${\log }_{10}|q_V|\in [-2,2]$ with fixed $z_{\mathrm{in}}$ is shown by a dashed line.