Generalized Swiss-cheese cosmologies: Mass scales

We generalize the Swiss-cheese cosmologies so as to include nonzero linear momenta of the associated boundary surfaces. The evolution of mass scales in these generalized cosmologies is studied for a variety of models for the background without having to specify any details within the local inhomogeneities. We find that the final effective gravitational mass and size of the evolving inhomogeneities depends on their linear momenta but these properties are essentially unaffected by the details of the background model.

Figure 1

The abscissa is the dimensionless fraction $\mathcal{D}\times {10}^{-5}$ and the ordinate gives $r_{\Sigma }(0)$ in units of Mpc. The top curve is the $\Lambda \mathrm{CDM}$ model. The middle curve is $\Lambda \mathrm{CDM}+\mathrm{\text{non interacting}}$ radiation (with the curve just below giving $r_{\Sigma }(100)$ for comparison). The second to bottom curve is the noninteracting matter and radiation without $\Lambda$, and the bottom curve is the $\Lambda \mathrm{CDM}$ with integration upper limit going to $z=1500$. We have adopted the values ${\Omega }_{m_0}=0.27$, $h\equiv H_0/100=0.72$ and $T_0=2.73$.

Figure 2

Variation in the evolution of ${\overline{r}}_{\Sigma }$ with redshift from variations in $\mathcal{D}$ for a $\Lambda \mathrm{CDM}$ background. We have used $h=0.72$ and $T_0=2.73$, and like all “$\overline{}$” quantities we have used ${\Omega }_{m_0}=0.27$ and ${\Omega }_{R_0}=8\times {10}^{-5}$. These surfaces start with ${\overline{r}}_{\Sigma }(z=\infty )=0$. The value of the linear momentum $\mathcal{D}$ determines the final size of these inhomogeneities.

Figure 3

Evolution of $r_{\Sigma }$ with redshift for some values of the matter density ${\Omega }_m$. We have used a $\Lambda \mathrm{CDM}$ background with a total linear momentum of $\mathcal{D}=5\times {10}^{-5}$ and we have extrapolated the sizes of these inhomogeneities to the infinite future ($z=-1$). The dotted line shows the size at the present redshift ($z=0$). We have used values of ${\Omega }_m$ consistent with the 99.9% confidence interval of 20 to show the effect of ${\Omega }_{m_0}$. For comparison, the latest WMAP5 [19] results give ${\Omega }_{m_0}=0.279\pm 0.015$.

Figure 4

Evolution of ratio ${\mathcal{M}}_{\Sigma }/{\overline{\mathcal{M}}}_{\Sigma }$ with redshift for some values of $\mathcal{D}=\overline{\mathcal{D}}$, where $\mathcal{M}$ is given by the $\Lambda \mathrm{CDM}$ with ${\Omega }_m=0.33$ and $\overline{\mathcal{M}}$ is our standard $\Lambda \mathrm{CDM}$ model.

Figure 5

Evolution of ratio ${\mathcal{M}}_{\Sigma }/{\overline{\mathcal{M}}}_{\Sigma }$ for $\Lambda \mathrm{CDM}$ with $\mathcal{D}=5\times {10}^{-5}$ for some values of ${\Omega }_m$.

Figure 6

The same as Fig. 4 where $\mathcal{M}$ now has the constant equation of state (CES) model for the background.

Figure 7

Ratio ${\mathcal{M}}_{\Sigma }/{\overline{\mathcal{M}}}_{\Sigma }$ for $\mathcal{D}=5\times {10}^{-5}$ for different values of $w$ with ${\Omega }_m=0.27$ for the constant equation of state model.

Figure 8

The same as Fig. 7 with the variable equation of state (VES) model as background for $\mathcal{M}$. Only the region $w_a<1$ is of physical interest.

Figure 9

Same as Fig. 7; $\mathcal{M}$ is given for the Dvali-Gabadadze-Porrati (DGP) model.

Figure 10

Ratio $M_{\Sigma }/{\overline{M}}_{\Sigma }$ evaluated at $z=-1$ as a function of $\xi$ for $w=-1$, $\mathcal{D}=5\times {10}^{-5}$ and ${\Omega }_{de}=0.73$ for the interacting dark energy (IDE) model.

Figure 11

Same as Fig. 7, but now $\mathcal{M}$ is given for the cardassian (CAR) model.

Figure 12

Same as Fig. 10 but with $A=0.67$ and 0.78 as a function of the parameter $\gamma$ for the Chaplygin (CHA) gas model.

Figure 13

Same as Fig. 10 but for the affine (AFF) parameter model as a function of $\alpha$.

Figure 14

Same as Fig. 7; $\mathcal{M}$ is given for the new agegraphic dark energy (NADE) model [25].