Topological quintessence

A global monopole (or other topological defect) formed during a recent phase transition with core size comparable to the present Hubble scale, could induce the observed accelerating expansion of the Universe. In such a model, topological considerations trap the scalar field close to a local maximum of its potential in a cosmologically large region of space. We perform detailed numerical simulations of such an inhomogeneous dark energy system (topological quintessence) minimally coupled to gravity, in a flat background of initially homogeneous matter. We find that when the energy density of the field in the monopole core starts dominating the background density, the spacetime in the core starts to accelerate its expansion in accordance to a $\Lambda \mathrm{CDM}$ model with an effective inhomogeneous spherical dark energy density parameter ${\Omega }_{\Lambda }(r)$. The matter density profile is found to respond to the global monopole profile via an anticorrelation (matter underdensity in the monopole core). Away from the monopole core, the spacetime is effectively Einstein-de Sitter (${\Omega }_{\Lambda }(r_{\mathrm{out}})\to 0$) while at the center ${\Omega }_{\Lambda }(r\simeq 0)$ is maximum. We fit the numerically obtained expansion rate at the monopole core to the Union2 data and show that the quality of fit is almost identical to that of $\Lambda \mathrm{CDM}$. Finally, we discuss potential observational signatures of this class of inhomogeneous dark energy models.

Figure 1

Evolution of the scalar field profile for $\eta =0.1$ (top panel) and $\eta =0.6$ (bottom panel) from the time $t_0$ up to the present time $t_p$. The profiles shown correspond to $t_0$ (dotted line), $t_p/2$ (dot-dashed line), $2t_p/3$ (dashed line), $t_p$ (solid line) and the profile collapses as time increases. Notice that the evolution gets much slower as we increase $\eta$ because the exponent of Eq. (2.30) decreases as ${\eta }^{-2}$ for fixed monopole core density.

Figure 2

Evolution of the scale factors $A(r,t)$ and $B(r,t)$ profile for $\eta =0.1$ (black lines) and $\eta =0.6$ (blue lines) from the time $t_0$ up to the present time $t_p$. A mesh of dots is superimposed to the curve corresponding to $\eta =0.1$, $t=t_p$ to distinguish it from the curve obtained for $\eta =0.6$, $t=t_p$. The profiles shown correspond to $t_0$ (dotted line), $t_p/3$ (dot-dashed line), $2t_p/3$ (dashed line), $t_p$ (solid line). Higher curves correspond to more recent times. Notice that even though the evolution of the scalar field changes significantly as we change $\eta$ from $\eta =0.1$ to $\eta =0.6$ (see Fig. 1) the corresponding evolution of the scale factors is only slightly changed. The (rescaled) region of accelerated expansion is slightly smaller in the case of smaller $\eta$ as expected due to the more rapidly shrinking core seen in Fig. 1.

Figure 3

Ratios $H_A(r,z)/H_A(r,0)$ and $H_B(r,z)/H_B(r,0)$ for $r=0$ (lower solid line), $r=0.5$ and $r=5$ (upper solid lines) along with the corresponding $H_{\Lambda }(r,z)/H_0$ for ${\Omega }_{\Lambda }=0.73$ (thick, dashed line) and for ${\Omega }_{\Lambda }=0$ (dot-dashed line). Also included for comparison $H_{\Lambda }(r,z)/H_0$ for the best fit values of ${\Omega }_{\Lambda }(r)$ shown in Fig. 4: ${\Omega }_{\Lambda }(0.5)\simeq 0.27$ (top panel) and ${\Omega }_{\Lambda }(0.5)\simeq 0.54$ (bottom panel). We have set $\eta =0.1$.

Figure 4

Values of ${\Omega }_{\Lambda }(r)$ that provide the best fit to $H_A(r,z)/H_A(r,0)$ (solid line) and to $H_B(r,z)/H_B(r,0)$ (dashed line).

Figure 5

Evolution of the dark energy (top panel) and properly normalized matter (bottom panel) density profiles. The profiles shown correspond to $t_0$ (dotted line), $t_p/3$ (dot-dashed line), $2t_p/3$ (dashed line), $t_p$ (solid line). Notice the matter underdensity that develops in the monopole core while the monopole density appears to slowly collapse to the center in comoving coordinates.

Figure 6

Top panel: Predicted distance moduli curves (residual with respect to $\Lambda \mathrm{CDM}$ best fit) based on the expansion rate for various values of $r$ are shown in the left panel superposed on the Union2 data. The distance moduli curves correspond to $r=0$ [best fit to Union2; red (upper) line], $r=0.25$, $r=0.5$, $r=5$ (worst fit corresponding to Einstein-de Sitter matter dominated universe away from the monopole core). Bottom panel: The quality of fit of the expansion rate $H_B(r,z)$ to the Union2 data expressed through ${\chi }^2(r)/\mathrm{d}.\mathrm{o}.\mathrm{f}.$. The dashed line corresponds to $\Lambda \mathrm{CDM}$ in both panels.