More on crinkles in the last scattering surface

TL;DR

The study investigates second-order CMB anisotropies arising from inhomogeneous recombination, where electron-density perturbations $\delta_e$ can be several times larger than baryon-density perturbations. It demonstrates the equivalence of solving the full second-order Boltzmann equation with the approach of solving the combined first- and second-order equations and extracting the second-order part, and it derives approximate solutions in the high-optical-depth regime that reveal cancellations suppressing the second-order monopole, dipole, and quadrupole. Numerically, the dominant bispectrum contribution comes from the KW09 source terms, with the additional second-order terms yielding only a small modification to $f_{NL}$ (approximately $-0.02$ at $\ell_{\max}=2500$), implying a temperature-only signal is marginally detectable ($S/N \sim 1$) and that Planck-like sensitivity is insufficient to discern this effect. The analysis underscores that perturbing $\delta_e$ in first-order tight-coupling solutions misrepresents the true second-order dynamics and emphasizes the potential, albeit modest, role of polarization in enhancing detectability of inhomogeneous recombination signatures.

Abstract

Inhomogeneous recombination can give rise to perturbations in the electron number density which can be a factor of five larger than the perturbations in baryon density. We do a thorough analysis of the second order anisotropies generated in the cosmic microwave background (CMB) due to perturbations in the electron number density. We show that solving the second order Boltzmann equation for photons is equivalent to solving the first + second order Boltzmann equations and then taking the second order part of the solution. We find the approximate solution to the photon Boltzmann hierarchy in l modes and show that the contributions from inhomogeneous recombination to the second order monopole, dipole and quadrupole are numerically small. We also point out that perturbing the electron number density in the first order tight coupling and damping solutions for the monopole, dipole and quadrupole is not equivalent to solving the second order Boltzmann equations for inhomogeneous recombination. Finally we confirm our result in a previous paper that inhomogeneous recombination gives rise to a local type non-Gaussianity parameter f_{NL}~ -1. The signal to noise for the detection of the temperature bispectrum generated by inhomogeneous recombination is ~ 1 for an ideal full sky experiment measuring modes up to l_{max}=2500.

Figures (8)

• Figure 1: $\sin\left[k(r_s(\eta)-r_s(\eta'))\right]$ for $k=0.25$ as a function of $\eta'$ for different values of $\eta$. All curves end at $\eta'=\eta$
• Figure 2: $3\Theta_1^{(1)}-iv^{(1)}$ as a function of conformal time $\eta$ for wavenumber $k=0.2,0.3,0.4\space\rm{Mpc}^{-1}$. Note that it becomes almost monotonically increasing at large $\eta$ when photon free streaming becomes important.
• Figure 3: $\Theta_0^{(1)}$, $|3\Theta_1^{(1)}-iv^{(1)}|$ and $|\Theta_2^{(1)}|$ as a function of $\eta$ for wavenumber $k=0.001\space\rm{Mpc}^{-1}$. Also shown is the visibility function $g(\eta)\equiv -\dot{\tau}e^{-\tau}$.
• Figure 4: $\Theta_0^{(1)}$, $|3\Theta_1^{(1)}-iv^{(1)}|$ and $|\Theta_2^{(1)}|$ as a function of $\eta$ for wavenumber $k=0.01\space\rm{Mpc}^{-1}$. The key is the same as in Figure \ref{['c1']}.
• Figure 5: $\Theta_0^{(1)}$, $3\Theta_1^{(1)}-iv^{(1)}$ and $\Theta_2^{(1)}$ as a function of $\eta$ for wavenumber $k=0.1\space\rm{Mpc}^{-1}$. Note that at small scales $3\Theta_1^{(1)}-iv^{(1)}$ becomes comparable to $\Theta_0^{(1)}$, but its contribution to the bispectrum is suppressed because it is weighted by the derivative of spherical Bessel function. See also Equation 10 and Figure 3 in KW09.