The bispectrum of redshifted 21-cm fluctuations from the dark ages

TL;DR

This work derives the general second-order expansion of brightness-temperature fluctuations in redshifted 21-cm radiation from the dark ages and computes the corresponding angular bispectrum to assess non-Gaussian signatures. It separates contributions from gravitational instability and primordial non-Gaussianity parameterized by $f_{\rm NL}$, providing analytic and numerical forms for the gravity-driven and primordial bispectra and quantifying their redshift and scale dependences. The results show gravity-induced non-Gaussianity dominates across accessible scales (with $\text{S/N}_{\rm grav} \sim 100$ for an ideal survey), while primordial NG with $f_{\rm NL} \sim 1$ could be detectable with a cosmic-variance-limited, high-resolution experiment and tilted by tomographic depth and optimal estimators. Foregrounds, weak lensing, and systematic effects could hamper primordial NG detection, motivating future work on optimal estimators and comprehensive foreground mitigation to exploit the 21-cm bispectrum as a probe of both gravitational instability and the inflationary origin of fluctuations.

Abstract

Brightness-temperature fluctuations in the redshifted 21-cm background from the cosmic dark ages are generated by irregularities in the gas-density distribution and can then be used to determine the statistical properties of density fluctuations in the early Universe. We first derive the most general expansion of brightness-temperature fluctuations up to second order in terms of all the possible sources of spatial fluctuations. We then focus on the three-point statistics and compute the angular bispectrum of brightness-temperature fluctuations generated prior to the epoch of hydrogen reionization. For simplicity, we neglect redshift-space distortions. We find that low-frequency radio experiments with arcmin angular resolution can easily detect non-Gaussianity produced by non-linear gravity with high signal-to-noise ratio. The bispectrum thus provides a unique test of the gravitational instability scenario for structure formation, and can be used to measure the cosmological parameters. Detecting the signature of primordial non-Gaussianity produced during or right after an inflationary period is more challenging but still possible. An ideal experiment limited by cosmic variance only and with an angular resolution of a few arcsec has the potential to detect primordial non-Gaussianity with a non-linearity parameter of f_NL ~ 1. Additional sources of error as weak lensing and an imperfect foreground subtraction could severely hamper the detection of primordial non-Gaussianity which will benefit from the use of optimal estimators combined with tomographic techniques.

Figures (4)

• Figure 1: Left: Redshift evolution of the coefficients that couple the brightness-temperature fluctuations to perturbations in the gas temperature and density (see the main text and both the Appendices for details). Right: Evolution of the "effective" coupling coefficients $f_1$, $f_2$ and $f_3=f^{\rm{T}}_1$ introduced in equation (\ref{['T21']}).
• Figure 2: The bispectrum of 21-cm anisotropies $B_{\ell_1,\ell_2,\ell_3}$ measured by an ideal experiment with 0.1 MHz bandwidth centered around $z=50$. From top to bottom, we consider equilateral ($\ell_1= \ell_2= \ell_3$), quasi-equilateral ($\ell_2=\ell_1+1,\ell_2=\ell_1-1$) and "squeezed" ($\ell_2=\ell_1, \ell_3=2$) configurations, respectively. In the left panels, the contributions of non-linear gravity (solid) and of primordial non-Gaussianity (with $f_{\rm NL}=1$; dashed) are compared. The right panels show the total bispectrum for different values of the non-linearity parameter. Note that in the equilateral and quasi-equilateral configurations the bispectrum vanishes when $\ell$ is odd (not plotted), is positive for $\ell=2j$ with $j$ an odd integer and negative for $\ell=2j$ with $j$ an even integer.
• Figure 3: Cumulative signal-to-noise ratio for the measurement of the bispectrum of 21-cm anisotropies using modes up to a maximum value of $\ell$. Solid lines refer to non-Gaussianity generated by gravity while dashed lines indicate the primordial signal with $f_{\rm NL}=1$. These quantities have been computed using the signal-to-noise ratio per mode of the quasi-equilateral configuration (left) and of the squeezed configuration (right).
• Figure 4: Redshift evolution and Fourier-space dependence of the kernels $\tilde{\mathcal{G}_1}$ and $\tilde{\mathcal{G}_2}$ relating temperature and density perturbations (see §\ref{['FLUCT']} and Appendix \ref{['PGT']} for further details). Top left: redshift evolution of the first-order function $g_1(z)$. Top right: contour levels of the kernel $\tilde{\mathcal{G}_2}(\mathbf{q_1},\mathbf{q_2},z)$ at $z=50$ with $f_{\rm NL}=1$ and ${\mathbf q_1} \parallel {\mathbf q_2}$; consecutive levels increase by a factor of 10 starting from 0.1 at the centre. Bottom left: redshift evolution of the functions $g_{\rm 2a},g_{\rm 2b}$ and $g_{\rm 2c}$ introduced in equation (\ref{['GDUESOLUTION']}). Bottom right: redshift evolution of the function $g_{\rm 2d}$ defined in equation (\ref{['GDUESOLUTION']}).