Interaction in the dark sector

It may well happen that the two main components of the dark sector of the Universe, dark matter and dark energy, do not evolve separately but interact nongravitationally with one another. However, given our current lack of knowledge of the microscopic nature of these two components, there is no clear theoretical path to determine their interaction. Yet, over the years, phenomenological interaction terms have been proposed on mathematical simplicity and heuristic arguments. In this paper, based on the likely evolution of the ratio between the energy densities of these dark components, we lay down reasonable criteria to obtain phenomenological, useful, expressions of the said term independent of any gravity theory. We illustrate this with different proposals which seem compatible with the known evolution of the Universe at the background level. Likewise, we show that two possible degeneracies with noninteracting models are only apparent as they can be readily broken at the background level. Further, we analyze some interaction terms that appear in the literature.

Figure 1

The first curve (from left to right) shows the evolution of ratio ${\rho }_m/{\rho }_x$ corresponding to the expression (5) where we have used $\alpha =2.70$ and $\beta =89.6$, consequently, $r_{+}=32.36$. The same can be said for the next curve but with $\alpha =2.76$, $\beta =279$, and $r_{+}=100$. The right-most curve depicts for comparison the evolution of the ratio for the concordance $\mathrm{\Lambda }\mathrm{CDM}$ model, ${\rho }_m/{\rho }_{\mathrm{\Lambda }}=r_0{a}^{-3}$. In all three cases, we have taken $r_0=25/70$. As it is apparent, the interaction $Q>0$ mitigates the coincidence problem.

Figure 2

The curve on the right specializes Eq. (9) for $\alpha =2.5$, $r_{+}=100$, and $\beta =697/3$. The curve on the left results from dividing the previous one by $r$, as given by Eq. (5).

Figure 3

The first two curves from left to right, with $\lambda =8$ and 10, respectively, and $r_0=25/70$, depict the ratio ${\rho }_m/{\rho }_x$ described by Eq. (10). Though not apparent in the figure, these ratios do not vanish asymptotically; they tend to $r_{-}=(25/70)e^{-4}$ and $r_{-}=(25/70)e^{-5}$, respectively. The third curve shows for comparison the evolution of the ratio for the $\mathrm{\Lambda }\mathrm{CDM}$ model, namely, ${\rho }_m/{\rho }_{\mathrm{\Lambda }}=r_0{a}^{-3}$ with an identical value for $r_0$. Again, the interaction is seen to alleviate the coincidence problem.

Figure 4

Left panel: Numerical solution to Eq. (12) in terms of the dimensionless quantity ${\overline{\rho }}_m\equiv \kappa {\rho }_m/(3H_0^2)$ with $\kappa \equiv 8\pi G$, for the choice $\lambda =8$. Also shown is the curve of ${\overline{\rho }}_x={\overline{\rho }}_m/r$. Right panel: The same but for the choice $\lambda =10$.

Figure 5

Plot of the ratio ${\rho }_m/{\rho }_x$ as a function of the scale factor as given by Eq. (13). The two first curves from left to right are for $\lambda =8$, $\lambda =10$, respectively, with $r_0=25/70$. Also shown for comparison is the analogous curve for the $\mathrm{\Lambda }\mathrm{CDM}$ model with the same value for $r_0$.

Figure 6

Numerical solution of Eq. (15) in terms of the dimensionless quantity ${\overline{\rho }}_m\equiv \kappa {\rho }_m/(3H_0^2)$ with $\kappa \equiv 8\pi G$ for the choice $\lambda =9.89$ and $r_{-}=0.031334$. Also shown is the corresponding curve of ${\overline{\rho }}_x$.

Figure 7

The parameter $\beta$ in terms of $\alpha$ as imposed by the constraint $q(a=0.57)=0$ on the first parametrization Eq. (5), with $r_{-}=0$.

Figure 8

Evolution of the deceleration parameter in terms of the scale factor for the first, second, and third parametrizations. In drawing the curves, we used $w=-1$, $r_0=25/70$, and ${\rho }_{b0}/{\rho }_{m0}=0.2$.

Figure 9

Hubble history corresponding to the three parametrizations, Eqs. (5), (10), and (13). For the curve corresponding to the first parametrization Eq. (5), we have assumed $\alpha =2.5$ and $\beta =697/3$, for the second parametrization Eq. (10) $\lambda =8$, and for the third parametrization Eq. (13) $\lambda =9.89$. In drawing the curves, we have taken $w=-1$, $r_0=25/70$, and ${\rho }_{b0}/{\rho }_{m0}=0.2$.

Figure 10

Plot of the ratio $H(z)/(1+z)$ assuming $H_0=68\mathrm{Km}/(\mathrm{s}\mathrm{Mpc})$. Black, blue, and red lines correspond to the first, second, and third parametrizations [Eqs. (5), (10), and (13)], respectively. The data error bars indicate $1\sigma$ confidence level. The $H(z)$ data were borrowed from Refs. [23, 24, 25, 26, 27, 28, 29].

Figure 11

The same as Fig. 10 but using $H_0=73.8\mathrm{Km}/(\mathrm{s}\mathrm{Mpc})$.

Figure 12

Evolution of the ratio (21) with expansion for the first parametrization, Eq. (5). The larger the scale factor, the smaller $\zeta$. In drawing the curve, we have used $r_0=25/70$, $\beta =111.33$, and $\alpha =2.8$.

Figure 13

Evolution of the EOS corresponding to the second parametrization Eq. (10), assuming $Q=0$. In drawing the curve, we have taken $\lambda =11.25$ and $r_{-}=0.0013$ (the best-fit values found above; see text).

Figure 14

Evolution of the Hubble factor normalized to its present value, $H_0$. The solid line corresponds to Eq. (24) with $w(a)$ given by (23). The dashed line depicts the Hubble factor associated to the case of the same $r(a)$ but with $Q$ given by (11) and $w=-1$. In drawing both curves, we assumed ${\mathrm{\Omega }}_{b0}=0.05$, ${\mathrm{\Omega }}_{m0}=0.25$, ${\mathrm{\Omega }}_{x0}=1-{\mathrm{\Omega }}_{b0}-{\mathrm{\Omega }}_{m0}$, $\lambda =11.25$, and $r_{-}=0.0013$.

Figure 15

Evolution of the deceleration parameter for the model with $Q=0$ (solid line) and for the model with $Q$ given by (11) (dashed line). Both models share the same ratio $r$, Eq. (10), but their respective EOS is different; see text. In drawing the curves, we took ${\rho }_{b0}/{\rho }_{m0}=0.2$, $\lambda =11.25$, and $r_{-}=0.0013$.

Figure 16

Evolution of function $\chi$, Eq. (26), in terms of the scale factor for the case of the first parametrization, Eq. (5). As is apparent, $\chi >0$ only for $a>0.4$, thereby no EOS of any noninteracting model can reproduce the Hubble history $H(z)$ of the model corresponding to the first parametrization. In plotting the graph, we have used ${\mathrm{\Omega }}_{b0}=0.05$, ${\mathrm{\Omega }}_{m0}=0.25$, $r_{-}=0$, $r_{+}=40.12$, $\alpha =2.8$, and $\beta =111.33$.

Figure 17

Solid line: Evolution of function $\chi$, Eq. (26), in terms of the scale factor for the case of the second parametrization, Eq. (10). Dashed line: The same but for the second parametrization, Eq. (13). As is seen, $\chi (a\lesssim 0.32)<0$ and $\chi (a\lesssim 0.30)<0$ for the second and third parametrizations, respectively. In drawing the graph, we have used the value of the parameter $\lambda$ that best fits the observational data [$q(a=0.57)=0$], i.e., 11.25 and 9.89 for the second and third parametrizations, respectively. In both cases, we have assumed ${\mathrm{\Omega }}_{b0}=0.05$ and ${\mathrm{\Omega }}_{m0}=0.25$.

Figure 18

Solid line: Evolution of function $\chi$, Eq. (26), in terms of the scale factor for the case of parametrization (30). Dashed line: The same but for parametrization (32). For both parametrizations, $\chi$ is negative in a considerable interval of the scale factor. In drawing the graph, we have considered $r_0=0.25/0.70$, $w=-1$, and $\xi =0.01$. In both cases, we have assumed ${\mathrm{\Omega }}_{b0}=0.05$ and ${\mathrm{\Omega }}_{m0}=0.25$.

Figure 19

Evolution of the ratio $\eta \equiv {\rho }_m(z)/{\rho }_{m0}(1+z{)}^3$, where ${\rho }_m(z)$ is the dark matter density in the case of the second parametrization. The solid line corresponds to $\lambda =10$; the dashed one to $\lambda =8$. In drawing both curves, $r_0=25/70$ was assumed.

Figure 20

Evolution of ${\rho }_m/{\rho }_x$ for $\xi =0.1$ and $\xi =0.01$ according to Eq. (30). In drawing the curves, we have assumed $w=-1$ and $r_0=25/70$. These overlap for $a\ge 0.75$.

Figure 21

The same as Fig. 20 but for Eq. (32).

Figure 22

Two numerical solutions to (33). The lower and upper curves correspond to $\xi =0.1$ and $\xi =0.01$, respectively. In drawing the curves, we have taken $r_0=25/70$ and $w=-1$.

Figure 23

Evolution of the ratio ${\rho }_m/{\rho }_x$ vs the scale factor, according to (34), for $\xi =0.1$ and $\xi =0.01$. In drawing the curves, we have assumed $r_0=25/70$ and $w=-1$.

Figure 24

Ratio ${\rho }_m/{\rho }_x$ vs the scale factor according to Eq. (37). The upper and lower curves correspond to $\xi =0.1$, and $\xi =0.01$. In drawing the curves, we have used $r_0=25/70$ and $w=-1$.