Generalized Lemaître-Tolman-Bondi model with inhomogeneous isotropic dark energy: Observational constraints

We consider on-center and off-center observers in an inhomogeneous, spherically symmetric, isocurvature (flat) concentration of dark energy with a typical size of a few Gpc. Such a concentration could be produced e.g. by a recently formed global monopole with core size that approaches the Hubble scale. In this case we would have what may be called “topological quintessence” in analogy with the well-known topological inflation. We show that the minimum comoving radius $r_{0\min }$ of such a dark energy inhomogeneity that is consistent with the Union2 type Ia supernovae data at the $3\sigma$ level is $r_{0\min }\simeq 1.8\mathrm{Gpc}$. As expected, the best-fit fractional dark energy density at the center, ${\Omega }_{X,\mathrm{in}}$, approaches the corresponding $\Lambda \mathrm{CDM}$ value ${\Omega }_{X,\mathrm{in}}=0.73$ for large enough values of the inhomogeneity radius $r_0$ ($r_0>4\mathrm{Gpc}$). Using the Union2 data, we show that the maximum allowed shift $r_{\mathrm{obs}\mathrm{\text{−}}\max }$ of the observer from the center of the inhomogeneity is about $0.7r_0$, which respects the Copernican principle. The model naturally predicts the existence of a preferred axis and alignment of the low CMB multipoles. However, the constraints on $r_{\mathrm{obs}\mathrm{\text{−}}\max }$ coming from the magnitude of the CMB dipole remain a severe challenge to the Copernican principle and lead to $r_{\mathrm{obs}\mathrm{\text{−}}\max }<110\mathrm{Mpc}$, even for an inhomogeneity radius as large as $r_0=7\mathrm{Gpc}$.

Figure 1

The angle $\xi$ between the tangent vector to the geodesic and the unit vector along the axis connecting the center with the observer.

Figure 2

The $1\sigma$, $2\sigma$, and $3\sigma$ contours in the parameter space $r_0-{\Omega }_{M,\mathrm{in}}$ for an on-center observer, obtained from Eq. (3.2). The parameter values $(r_0,{\Omega }_{X,\mathrm{in}})=(3.37\mathrm{Gpc},0.69)$ provide a good fit to the data and will often be assumed when dealing with the off-center case. The corresponding ${\chi }^2$ contours for the usual LTB model (only $\mathrm{\text{matter}}+\mathrm{\text{curvature}}$, ${\Omega }_{M,\mathrm{out}}=1$) are also shown. Note that the relation ${\Omega }_{M,\mathrm{in}}=1-{\Omega }_{X,\mathrm{in}}$ is obviously only valid for the IDE model.

Figure 3

The distance modulus as obtained from Eqs. (2.34, 2.43) in the direction of the center ($\xi =\pi$) and in the opposite direction ($\xi =0$), for various values of the observer shift $r_{\mathrm{obs}}$ and fixed $(r_0,{\Omega }_{X,\mathrm{in}})=(3.37\mathrm{Gpc},0.69)$. The offset ${\mu }_0$ has been set to the best-fit value corresponding to $\xi =\pi$.

Figure 4

Contour plot of ${\chi }_{2\min }^2(l_c,b_c)$ in galactic coordinates obtained after minimization with respect to $r_{\mathrm{obs}}$ and ${\mu }_0$. The ${\chi }^2$ contours shown correspond to $r_0=3.37$ and ${\Omega }_{X,\mathrm{in}}=0.69$, and $\Delta {\chi }^2$ represents the improvement in the fit with respect to the on-center case. The best-fit directions towards the inhomogeneity center for all the $(r_0,{\Omega }_{X,\mathrm{in}})$ values in Fig. 2 are also shown. Finally, the red point indicates the largest best-fit value for the shift of the observer (see the text and Fig. 5).

Figure 5

Contour plot in galactic coordinates of the best-fit value of $r_{\mathrm{obs}}$ as a function of the location of the inhomogeneity center, given by the coordinates $(l_c,b_c)$). The largest best-fit ($r_{\mathrm{obs}}=654\mathrm{Mpc}$) and absolute best-fit [$r_{\mathrm{obs}}=573\mathrm{Mpc}$, corresponding to the minimum value of ${\chi }_2^2$ in Eq. (3.3)] values for the observer’s shift are also shown.

Figure 6

The form ${\chi }^2(r_{\mathrm{obs}})$ after minimization with respect to ${\alpha }_c$, ${\delta }_c$, ${\mu }_0$ for various values of $r_0$, ${\Omega }_{X,\mathrm{in}}$. The fit is statistically acceptable out to values of $r_{\mathrm{obs}}\simeq 0.7r_0$. Clearly, in this case the Copernican principle is respected.

Figure 7

The predicted CMB dipole $a_{10}(r_{\mathrm{obs}})$ for ${\Omega }_{X,\mathrm{in}}=0.69$ and various values of the inhomogeneity scale $r_0$. The dashed line corresponds to the observed value [whose uncertainty is too small to be shown, cf. Eq. (3.4)].

Figure 8

The spatial fraction $f(r_0)\equiv (\frac{r_{\mathrm{obs}\mathrm{\text{−}}\max }}{r_0}{)}^3$, where the observer needs to be confined in order to be consistent with the value of the observed CMB dipole, for $(r_0,{\Omega }_{X,\mathrm{in}})=(3.37\mathrm{Gpc},0.69)$. Notice the rapid increase of the maximum allowed $r_{\mathrm{obs}\mathrm{\text{−}}\max }$ as we increase the inhomogeneity size $r_0$. It is anticipated that as we reach the value $r_0\simeq 13.7\mathrm{Gpc}$ (the distance to the last-scattering surface) the model becomes indistinguishable from the $\Lambda \mathrm{CDM}$ and therefore no constraints are imposed on the value of $r_{\mathrm{obs}}$ using the CMB dipole.

Figure 9

The predicted CMB maps of additional temperature fluctuations in galactic coordinates corresponding to the full map (a), dipole (b), quadrupole (c), and octopole (d). We have selected the direction of the preferred axis to coincide with the direction of the observed CMB dipole and used $(r_0,{\Omega }_{X,\mathrm{in}})=(3.37\mathrm{Gpc},0.69)$, $r_{\mathrm{obs}}=30\mathrm{Mpc}$.

Figure 10

The quadrupole moment $a_{20}$ as a function of the observer shift $r_{\mathrm{obs}}$, for different inhomogeneity sizes $r_0$ and ${\Omega }_{X,\mathrm{in}}=0.69$. The yellow band represents the value measured by WMAP [1], including just the observational uncertainty [cf. Eq. (3.7)]. The 68% and 95% confidence level regions for $a_{20}$ according to the more complete statistical analysis carried out in 6 are also shown (gray and blue shaded regions, respectively). Finally, the vertical dashed lines indicate the maximum value of $r_{\mathrm{obs}}$ (for each $r_0$) compatible with the measured dipole, Eq. (3.4).