Halo model approach for the 21-cm power spectrum at cosmic dawn

Prior to the epoch of reionization, the 21-cm signal of the cosmic dawn is dominated by the Lyman-$\alpha$ coupling and gas temperature fluctuations caused by the first sources of radiation. While early efforts to model this epoch relied on analytical techniques, the community quickly transitioned to more expensive seminumerical models. Here, we reassess the viability of simpler approaches that allow for rapid explorations of the vast astrophysical parameter space. We propose a new analytical method to calculate the 21-cm power spectrum based on the framework of the halo model. Both the Lyman-$\alpha$ coupling and temperature fluctuations are described by overlapping radiation flux profiles that include spectral redshifting and source attenuation due to look-back (light-cone) effects. The 21-cm halo model is compared to the seminumerical code 21 cmFAST exhibiting generally good agreement, i.e., the power spectra differ by less than a factor of 3 over a large range of $k$-modes and redshifts. We show that the remaining differences between the two methods are comparable to the expected variations from modeling uncertainties associated with the abundance, bias, and accretion rates of haloes. While these current uncertainties must be reduced in the future, our work suggests that inference at acceptable accuracy will become feasible with very efficient halo models of the cosmic dawn.

Figure 1

Halo growth, mass accretion rate, and star-formation rate density (SFRD) for the three different models discussed in Sec. 2b. Left: halo mass as a function of redshift for four different final masses $M$ (green, brown, red, and blue lines). The orange lines correspond to the mean halo growth from the B20 [60] simulations. Center: mass accretion rates for the same models and halo masses. Right: SFRD for the three different mass-accretion rate models [see Eq. (13)].

Figure 2

Normalised Lyman-$\alpha$ coupling (left), x-ray energy deposition (center), and heating (right) profiles. In the top row we vary the redshift at fixed halo mass, and in the bottom row we vary the halo mass at fixed redshift.

Figure 3

Normalized Fourier transforms of the Lyman-$\alpha$ (solid) and the heating profiles (dashed). In the left- and right-hand panels we vary redshift and halo mass while keeping the other one constant.

Figure 4

Leftmost panels: star-formation efficiency (top) and resulting global signal (bottom) for the three benchmark models with suppressed, unchanged, and boosted small-scale behavior (blue, cyan, and magenta). Remaining panels: power spectra as a function of $k$-modes (top) and redshift (bottom) for the same models.

Figure 5

Comparison between the 21-cm halo model developed in this paper (solid lines) and the BLPF method from Refs. [14, 27] (colored bands) assuming astrophysical parameters from the benchmark model B (see Fig 4). Top panels: global signal (left) and the total 21-cm power spectrum with respect to $k$-modes (center) and redshift (right). Middle panels: auto power spectra of the Lyman-$\alpha$ coupling (left), temperature (center), and mass (right) components. Bottom panels: corresponding cross power spectra of the individual components.

Figure 6

Comparison between the halo model and 21 cmFAST. Top left: star-formation efficiency of Eq. (11) fitted to the one from 21 cmFAST (blue line and band, respectively). Top right: global differential brightness temperature obtained by assuming the EXP and EPS accretion rates (dash-dotted and solid lines) and matched to the 21 cmFast results (band) by refitting the total Lyman-$\alpha$ and x-ray photon numbers. Center and Bottom: corresponding power spectra as a function of $k$-modes and redshift. Note that redshifts below $z\sim 10$ should not be trusted since effects related to reionization become important.

Figure 7

Mass accretion modeling and how it affects the 21-cm signal assuming the source parameters of fiducial model B. Top left: halo growth for the three accretion models EPS, EXP, and AM (see Sec. 2b). Top center: star-formation rate density for the same models. Top right: resulting global 21-cm signal, where the flux parameters of the EXP and AM models ($N_{\alpha }$, $f_X$) are either kept the same (solid lines) or where they are modified so that the minimum of the absorption signal lie at exactly the same redshift like the one from the EPS model (dashed lines). Center and Bottom: resulting power spectra as a function of $k$-modes and redshift.

Figure 8

Effects from the mass function and bias modeling on the 21-cm signal, assuming source parameters from the fiducial model B. Top left and center: halo bias and mass functions for different values of $q$ and $p$. The purple and brown lines correspond to the Press-Schechter ($q=1$, $p=0$) and Sheth-Tormen ($q=0.7$, $p=0.3$) mass functions, while the red line shows a case in between ($q=0.85$, $p=0.3$), which is in better agreement with high redshift simulations. Top right: global 21-cm signal resulting from these mass function and bias prescriptions, where the flux parameters ($N_{\alpha }$, $f_X$) are either kept the same (solid lines) or where they are changed so that the absorption troughs lie at the same redshift (dashed lines). Center and Bottom: resulting power spectra as a function of $k$-modes and redshifts.