Dynamics of a spherical object of uniform density in an expanding universe

We present Newtonian and fully general-relativistic solutions for the evolution of a spherical region of uniform interior density ${\rho }_{\mathrm{i}}(t)$, embedded in a background of uniform exterior density ${\rho }_{\mathrm{e}}(t)$. In both regions, the fluid is assumed to support pressure. In general, the expansion rates of the two regions, expressed in terms of interior and exterior Hubble parameters $H_{\mathrm{i}}(t)$ and $H_{\mathrm{e}}(t)$, respectively, are independent. We consider in detail two special cases: an object with a static boundary, $H_{\mathrm{i}}(t)=0$; and an object whose internal Hubble parameter matches that of the background, $H_{\mathrm{i}}(t)=H_{\mathrm{e}}(t)$. In the latter case, we also obtain fully general-relativistic expressions for the force required to keep a test particle at rest inside the object, and that required to keep a test particle on the moving boundary. We also derive a generalized form of the Oppenheimer-Volkov equation, valid for general time-dependent spherically symmetric systems, which may be of interest in its own right.

Figure 1

Model of a spherical object of size $a(t)$ and uniform interior density ${\rho }_{\mathrm{i}}(t)$ residing in an expanding universe with uniform exterior density ${\rho }_{\mathrm{e}}(t)$. The corresponding Hubble parameters of the interior and exterior regions are denoted $H_{\mathrm{i}}(t)$ and $H_{\mathrm{e}}(t)$, respectively.

Figure 2

The pressure profile interior (left panel) and exterior (right panel) to the Sun, modeled as a uniform-density object residing in a concordance $\Lambda \mathrm{CDM}$ universe, at the present cosmological epoch $t=t_0$, and the future times $t=1.2t_0$ and $t=5t_0$. For the interior region, the three profiles are indistinguishable. In both regions, the Newtonian and general-relativistic results are indistinguishable.

Figure 3

The radial pressure profile for an approximate model of cluster of galaxies, taken to be a uniform-density object with excess mass $m={10}^{15}{M}_{\odot }$ and current radius $a(t_0)=5\mathrm{Mpc}$, for (a) $r<5\mathrm{Mpc}$; (b) $3<r<10\mathrm{Mpc}$; and (c) $5<r<1000\mathrm{Mpc}$. The exterior cosmology is assumed to be concordance $\Lambda \mathrm{CDM}$. (a) $p_i$, (b) $p_i$ and $p_e$, (c) $p_e$.

Figure 4

The logarithm of the fractional error in the approximate series solution for the interior quantity $f_{1,\mathrm{i}}$ (Table ), relative to the exact numerical result, for fixed arbitrary values $m=1$, $a(t)=1000$ and $R(t)=10000$. (a) $k=-1$, (b) $k=1$.