Topological signatures in CMB temperature anisotropy maps

We propose an alternative formalism to simulate cosmic microwave background (CMB) temperature maps in $\Lambda \mathrm{CDM}$ universes with nontrivial spatial topologies. This formalism avoids the need to explicitly compute the eigenmodes of the Laplacian operator in the spatial sections. Instead, the covariance matrix of the coefficients of the spherical harmonic decomposition of the temperature anisotropies is expressed in terms of the elements of the covering group of the space. We obtain a decomposition of the correlation matrix that isolates the topological contribution to the CMB temperature anisotropies out of the simply connected contribution. A further decomposition of the topological signature of the correlation matrix for an arbitrary topology allows us to compute it in terms of correlation matrices corresponding to simpler topologies, for which closed quadrature formulas might be derived. We also use this decomposition to show that CMB temperature maps of (not too large) multiply connected universes must show “patterns of alignment,” and propose a method to look for these patterns, thus opening the door to the development of new methods for detecting the topology of our Universe even when the injectivity radius of space is slightly larger than the radius of the last scattering surface. We illustrate all these features with the simplest examples, those of flat homogeneous manifolds, i.e., tori, with special attention given to the cylinder, i.e., $T^1$ topology.

Figure 1

Shapes of the integrands inside integrals of the type in (37) for $\ell ={\ell }^{\prime }=2$, and compactification scales $L=1$ and $L=2$ in units of the radius of the last scattering surface.

Figure 2

Shapes of the integrands inside integrals of the type in (37) for $\ell ={\ell }^{\prime }=5$, and compactification scales $L=1$ and $L=2$ in units of the radius of the last scattering surface.

Figure 3

Topological signature of the power spectrum of (a) a cylinder, and (b) a chimney with square base, for the first four $\ell$ modes, normalized w.r.t. $C_{\ell }^{\mathrm{s}.\mathrm{c}.}$, and as a function of the scale of compactification $L$. Note that, for each multipole, one can have a suppression or an excess of power depending on the value of $L$. For typographical reasons, here we write ${\Gamma }^{*}$ instead of $\widehat{\Gamma }$.

Figure 4

Simulated CMB temperature anisotropy map at low resolution ($2\le \ell \le 10$) for a universe with a $T^1$ topology and size $L=2$. Also shown are the lowest multipoles, all of which present clear alignments near the polar direction.

Figure 5

Plots of the topological signatures of the squares $⟨|a_{\ell m}{|}^2{⟩}^{\widehat{\Gamma }}$, normalized w.r.t. $C_{\ell }^{\Gamma }$, for low multipoles ($2\le \ell \le 5$), in a universe with cylindrical topology aligned with the polar axis, and as a function of the scale of compactification $L$. For typographical reasons, here we write ${\Gamma }^{*}$ instead of $\widehat{\Gamma }$.

Figure 6

Dispersion of the $⟨|a_{\ell m}{|}^2{⟩}^{\Gamma }$, normalized w.r.t. $C_{\ell }^{\Gamma }$, for low multipoles ($2\le \ell \le 5$), in a universe with cylindrical topology aligned with the polar axis, and as a function of the scale of compactification $L$.

Figure 7

Dispersion of the $⟨|a_{\ell m}{|}^2{⟩}^{\Gamma }$, normalized w.r.t. $C_{\ell }^{\Gamma }$, for low multipoles ($2\le \ell \le 5$), in a universe with cylindrical topology, as a function of its orientation relative to the polar axis, and for different values of the scale of compactification $L$.