The Statistics of Density Peaks and the Column Density Distribution of the Lyman-Alpha Forest

TL;DR

The paper develops an efficient semi-analytic framework to predict the Lyman-α forest column density distribution by coupling the Zel'dovich approximation with a density-peak statistic (Density-Peak-Ansätz). It demonstrates that peculiar velocities largely shape absorption line profiles but have a limited impact on the distribution of column densities, and derives how the slope and normalization of the distribution depend on the temperature–density relation and the small-scale power spectrum. The authors validate their approach against full hydrodynamic simulations for a test CDM model and extend it to CHDM models, showing CHDM tends to produce steeper distributions due to reduced small-scale power. They also provide an approximate relation for the slope that encapsulates the roles of $\gamma$, $\sigma_0$, and $n_{\rm eff}$, offering a practical way to constrain small-scale power from Lyα forest statistics. The work highlights the need for careful consideration of line-identification biases and resolution when comparing to data and motivates further hydrodynamic tests across models.

Abstract

We develop a method to calculate the column density distribution of the Lyman-alpha forest for column densities in the range $10^{12.5} - 10^{14.5} cm^{-2}$. The Zel'dovich approximation, with appropriate smoothing, is used to compute the density and peculiar velocity fields. The effect of the latter on absorption profiles is discussed and it is shown to have little effect on the column density distribution. An approximation is introduced in which the column density distribution is related to a statistic of density peaks (involving its height and first and second derivatives along the line of sight) in real space. We show that the slope of the column density distribution is determined by the temperature-density relation as well as the power spectrum on scales $2 h Mpc^{-1} < k < 20 h Mpc^{-1}$. An expression relating the three is given. We find very good agreement between the column density distribution obtained by applying the Voigt-profile-fitting technique to the output of a full hydrodynamic simulation and that obtained using our approximate method for a test model. This formalism then is applied to study a group of CDM as well as CHDM models. We show that the amplitude of the column density distribution depends on the combination of parameters $(Ω_b h^2)^2 T_0^{-0.7} J_{HI}^{-1}$, which is not well-constrained by independent observations. The slope of the distribution, on the other hand, can be used to distinguish between different models: those with a smaller amplitude and a steeper slope of the power spectrum on small scales give rise to steeper distributions, for the range of column densities we study. Comparison with high resolution Keck data is made.

Figures (18)

• Figure 1: A line of sight through a $\sigma_8 = 0.7$ CDM simulation (produced using the truncated Zel'dovich approximation) at $\bar{z}=3$, with $h = 0.5$. The transfer function is taken from Ma (1996). Box size is $12.8 \, {\rm Mpc}$ with a grid spacing of $0.032 \, {\rm Mpc}$. The parameters are $\Omega_b h^2 = 0.0125$, $J_{\rm HI} = 0.5$, $T_0 = 10^4 \, {\rm K}$ ,$\gamma = 1.5$ and $k_{\rm S} = 2.3 \, {\rm Mpc^{-1}}$. All distances are comoving. See Sec. \ref{['ingr']} for definitions of the symbols. The abscissas for the lower two panels are the comoving distances along the line of sight in units of ${\rm Mpc}$. The lower of the two panels is the profile of overdensity $\delta_{\rm b}$ (eq. [\ref{['deltadef']}]) and the upper one is the profile of velocity $u$ (eq. [\ref{['ux']}]). The top two panels are both transmission profiles where $\tau$ is the Ly$\alpha$ optical depth and the abscissas represent $u_\circ$, which is related to the observed frequency through equation (\ref{['uuO2']}). The profile with solid line is obtained using the full density and peculiar velocity fields. The profile with dashed line is obtained using the same density field but setting the peculiar velocity to zero everywhere (in which case, $u$ becomes linear in $x$).
• Figure 2: The same as in Fig. \ref{['delvtrans7_21_50PCL1_52']} but a different line of sight.
• Figure 3: The column density distributions for the same model and parameters as those in Fig. \ref{['delvtrans7_21_50PCL1_52']}. The quantity $d^2 N_{{\rm Ly}\alpha}/dN_{\rm HI}/dz$ has units of ${\rm cm^2}$. Crosses represent the distribution obtained by applying the Threshold-Algorithm (threshold at $0.89$, which is the mean transmission) to spectra generated using the density and peculiar velocity fields predicted by the truncated Zel'dovich approximation. Open triangles represent the same except that peculiar velocities are set to zero. Open squares are obtained by applying the Threshold-Deblending-Algorithm at the threshold of $0.89$.
• Figure 4: Same parameters as in Fig. \ref{['delvtrans7_21_50PCL1_52']}. The column densities computed using two different methods are plotted against each other. First, we identify absorption lines by the Threshold-Deblending-Algorithm using a transmission threshold of $0.89$ and assign column densities according to equation (\ref{['tcd']}), which are plotted as the abscissas. We then take each absorption line identified using the Threshold-Deblending-Algorithm and search for the corresponding maximum in $\delta_{\rm b}$ and apply equation (\ref{['npk3']}) to assign a second set of column densities, which are plotted as the ordinates.
• Figure 5: The model CDM1b (Table \ref{['cdmmodels']}) at $\bar{z}=3$, $J_{\rm HI} = 0.325$, $T_0 = 10^4 K$ and $\gamma = 1.45$. Solid triangles represent the distribution obtained by applying the Voigt-profile-fitting-technique to synthetic spectra from a full hydrodynamic simulation (Zhang et al. 1996) with box size of $9.6 \, \,{\rm Mpc}$ comoving and grid spacing of $0.075 \, {\rm Mpc}$. Open triangles and open squares are the predictions of the Density-Peak-Ansätz (DPA) coupled with the truncated ($k_{\rm S} = 2.3 \,{\rm Mpc^{-1}}$) Zel'dovich approximation, the former using the same box size and grid-spacing as the hydrodynamic simulation and the latter using a box size of $12.8$ Mpc and grid spacing of $0.05$ Mpc. Crosses are the results of applying the Density-Peak-Ansätz and the Threshold-Algorithm (TA; transmission threshold at $0.83$, the mean transmission) to the same density field as for the open squares. The short-dashed and long-dashed curves are the predictions of the Density-Peak-Ansätz coupled with the lognormal approximation, the former with $k_{\rm S} = 2.3 \,{\rm Mpc^{-1}}$ and the latter with the smoothing scale chosen so that the final rms density fluctuation matches that of the Zel'dovich approximation ($k_{\rm S} = 3.6 \,{\rm Mpc^{-1}}$).