Entropic cosmology in a dissipative universe

The bulk viscosity of cosmological fluid and the creation of cold dark matter both result in the generation of irreversible entropy (related to dissipative processes) in a homogeneous and isotropic universe. To consider such effects, the general cosmological equations are reformulated, focusing on a spatially flat matter-dominated universe. A phenomenological entropic-force model is examined that includes constant terms as a function of the dissipation rate ranging from $\tilde{\mu }=0$, corresponding to a nondissipative $\mathrm{\Lambda }\mathrm{CDM}$ (lambda cold dark matter) model, to $\tilde{\mu }=1$, corresponding to a fully dissipative CCDM (creation of cold dark matter) model. A time-evolution equation is derived for the matter density contrast in order to characterize density perturbations in the present entropic-force model. It is found that the dissipation rate affects the density perturbations even if the background evolution of the late universe is equivalent to that of a fine-tuned pure $\mathrm{\Lambda }\mathrm{CDM}$ model. With increasing dissipation rate $\tilde{\mu }$, the calculated growth rate for the clustering gradually deviates from observations, especially at low redshifts. However, the growth rate for low $\tilde{\mu }$ (less than 0.1) is found to agree well with measurements. A low-dissipation model predicts a smaller growth rate than does the pure $\mathrm{\Lambda }\mathrm{CDM}$ model (for which $\tilde{\mu }=0$). More detailed data are needed to distinguish the low-dissipation model from the pure $\mathrm{\Lambda }\mathrm{CDM}$ one.

Figure 1

Evolution of the effective equation of state parameter $w_e$ for the indicated dissipation rates $\tilde{\mu }$. Note that $\tilde{\mu }=0$ corresponds to $\mathrm{\Lambda }\mathrm{CDM}$ models, whereas $\tilde{\mu }=1$ corresponds to CCDM models. The background evolution of the universe in each case is equivalent to that in the standard $\mathrm{\Lambda }\mathrm{CDM}$ model because ${\tilde{\mathrm{\Omega }}}_{\mathrm{\Lambda }}={\mathrm{\Omega }}_{\mathrm{\Lambda }}=0.685$.

Figure 2

Evolution of the density perturbation growth factor $\delta$ for various dissipation rates $\tilde{\mu }$. The initial conditions are $\delta ({\tilde{a}}_i)={\tilde{a}}_i$ and ${\delta }^{\prime }({\tilde{a}}_i)={\tilde{a}}_i$ where ${\tilde{a}}_i=a_i/a_0={10}^{-3}$. The background evolution of the universe is the same in every case because ${\tilde{\mathrm{\Omega }}}_{\mathrm{\Lambda }}=0.685$.

Figure 3

Evolution of the growth rate $f_c(z)$ for clustering for various dissipation rates $\tilde{\mu }$. The closed circles with error bars are the observed data points summarized in Ref. [23]. The original data are from Refs. [70, 71, 72, 73, 74, 75, 76]. The background evolution of the universe for each value of $\tilde{\mu }$ is the same because ${\tilde{\mathrm{\Omega }}}_{\mathrm{\Lambda }}=0.685$.

Figure 4

The normalized likelihood $L$ as a function of the dissipation rate $\tilde{\mu }$. A pure $\mathrm{\Lambda }\mathrm{CDM}$ model corresponds to $\tilde{\mu }=0$. The maximum value of $L$ (corresponding to minimum ${\chi }^2$) is obtained for $\tilde{\mu }=0.026$, upon sampling $\tilde{\mu }\in [0.020,0.030]$ in steps of 0.001, with ${\tilde{\mathrm{\Omega }}}_{\mathrm{\Lambda }}=0.685$.

Figure 5

Evolution of the density perturbation growth factor $\delta$ for various values of ${\mathrm{\Omega }}_m/{\mathrm{\Omega }}_{\mathrm{\Lambda }}$ in a pure $\mathrm{\Lambda }\mathrm{CDM}$ model. Bold lines (in color) represent the pure $\mathrm{\Lambda }\mathrm{CDM}$ model (for which $\tilde{\mu }=0$), whereas (black) thin lines represent a low-dissipation model; i.e., $\tilde{\mu }=0.05$ and 0.1 with ${\tilde{\mathrm{\Omega }}}_{\mathrm{\Lambda }}=0.685$. In the pure $\mathrm{\Lambda }\mathrm{CDM}$ model, ${\mathrm{\Omega }}_m/{\mathrm{\Omega }}_{\mathrm{\Lambda }}=0.1$, 0.2, 0.3, and 0.46 are approximately equivalent to ${\mathrm{\Omega }}_{\mathrm{\Lambda }}=0.909$, 0.833, 0.769, and 0.685, respectively. ${\mathrm{\Omega }}_m/{\mathrm{\Omega }}_{\mathrm{\Lambda }}=0.46$ corresponds to a fine-tuned standard $\mathrm{\Lambda }\mathrm{CDM}$ model, which is equivalent to $\tilde{\mu }=0$ shown in Fig. 2. The background evolution of the universe for each value of ${\mathrm{\Omega }}_m/{\mathrm{\Omega }}_{\mathrm{\Lambda }}$ is different because ${\mathrm{\Omega }}_{\mathrm{\Lambda }}$ is different. In contrast, the background evolutions of the universe for ${\mathrm{\Omega }}_m/{\mathrm{\Omega }}_{\mathrm{\Lambda }}=0.46$, $\tilde{\mu }=0.05$, and $\tilde{\mu }=0.1$ are the same because ${\mathrm{\Omega }}_{\mathrm{\Lambda }}={\tilde{\mathrm{\Omega }}}_{\mathrm{\Lambda }}=0.685$.

Figure 6

Evolution of the growth rate $f_c(z)$ for clustering for various values of ${\mathrm{\Omega }}_m/{\mathrm{\Omega }}_{\mathrm{\Lambda }}$ in a pure $\mathrm{\Lambda }\mathrm{CDM}$ model. The closed circles with error bars are the observed data points summarized in Ref. [23]. ${\mathrm{\Omega }}_m/{\mathrm{\Omega }}_{\mathrm{\Lambda }}=0.46$ (i.e., ${\mathrm{\Omega }}_{\mathrm{\Lambda }}=0.685$) corresponds to a fine-tuned standard $\mathrm{\Lambda }\mathrm{CDM}$ model, which is equivalent to $\tilde{\mu }=0$ shown in Fig. 3. The background evolution of the universe for each value of ${\mathrm{\Omega }}_m/{\mathrm{\Omega }}_{\mathrm{\Lambda }}$ is different. See the caption of Fig. 5.

Figure 7

The contours of the normalized likelihood $L$ in the $({\tilde{\mathrm{\Omega }}}_{\mathrm{\Lambda }},\tilde{\mu })$ plane. To calculate a likelihood function in the $({\tilde{\mathrm{\Omega }}}_{\mathrm{\Lambda }},\tilde{\mu })$ plane, Eqs. (67) and (68) are used. For this purpose, ${\chi }^2(\tilde{\mu })$ and $f_c^{\text{cal}}(z_i,\tilde{\mu })$ in Eq. (67) are replaced by ${\chi }^2({\tilde{\mathrm{\Omega }}}_{\mathrm{\Lambda }},\tilde{\mu })$ and $f_c^{\text{cal}}(z_i,{\tilde{\mathrm{\Omega }}}_{\mathrm{\Lambda }},\tilde{\mu })$, respectively. For the likelihood analysis, ${\tilde{\mathrm{\Omega }}}_{\mathrm{\Lambda }}$ and $\tilde{\mu }$ are sampled in the range [0,1] in steps of 0.005. The likelihood function is normalized, using the maximum value, which is obtained for $({\tilde{\mathrm{\Omega }}}_{\mathrm{\Lambda }},\tilde{\mu })=(0.250,0.710)$. The contours of $L=0.9$, 0.8, 0.7, 0.6, and 0.5 are plotted.