Constraints on Warm Dark Matter from Cosmological Reionization

TL;DR

The paper develops a semi-analytic framework to constrain warm dark matter (WDM) by linking high-redshift structure formation to reionization. It combines a WDM-modified halo formation model, a calibrated halo mass function, and a semi-analytic reionization treatment to predict the ionizing photon budget as a function of WDM particle mass $m_X$ via the velocity dispersion $v_{\rm rms}$ and the free-streaming cutoff scale. The key results show that reionization by $z\approx 5.8$ sets a robust lower bound $m_X>1.2$ keV under standard assumptions, with weaker bounds depending on the ionizing efficiency and clumping; a quasar at $z=5.8$ provides an independent, though somewhat weaker, constraint ($m_X>0.5$ keV). The work highlights that a ~1 keV WDM particle remains viable, but future observations—especially of the high-$z$ galaxy luminosity function with NGST/JWST or similar—could detect the predicted low-luminosity cutoff and further tighten or challenge WDM scenarios.

Abstract

We study the constraints that high-redshift structure formation in the universe places on warm dark matter (WDM) dominated cosmological models. We modify the extended Press-Schechter formalism to derive the halo mass function in WDM models. We show that our predictions agree with recent numerical simulations at low redshift over the halo masses of interest. Applying our model to galaxy formation at high redshift, we find that the loss of power on small scales, together with the delayed collapse of low-mass objects, results in strong limits on the root-mean-square velocity dispersion v_rms of the WDM particles at z=0. For fermions decoupling while relativistic, these limits are equivalent to constraints on the mass m_X of the particles. The presence of a 4 billion solar mass black hole at z=5.8, believed to power the quasar SDSS 1044-1215, implies m_X > 0.5 keV (or v_rms < 0.10 km/s), assuming that the quasar is unlensed and radiating at or below the Eddington limit. Reionization by z=5.8 also implies a limit on m_X. If high-redshift galaxies produce ionizing photons with an efficiency similar to their z=3 counterparts, we find m_X > 1.2 keV (or v_rms < 0.03 km/s). However, given the uncertainties in current measurements from the proximity effect of the ionizing background at redshift 3, values of m_X as low as 0.75 keV (v_rms = 0.06 km/s) are not ruled out. The limit weakens further if, instead, the ionizing-photon production efficiency is greater at high z, but this limit will tighten considerably if reionization is shown in the future to have occurred at higher redshifts. WDM models with m_X < 1 keV (v_rms > 0.04 km/s) produce a low-luminosity cutoff in the high-redshift galaxy luminosity function which is directly detectable with the Next Generation Space Telescope (abridged).

Figures (10)

• Figure 1: Example of shell trajectories for gas with initial temperature corresponding to $\Lambda$WDM with $m_X=1$ keV ($v_{\rm rms,0}=0.041~$km/s; solid curves) and for gas with initial $T=0$ K (corresponding to $\Lambda$CDM; dashed curves). In each case, four trajectories are shown, in order of increasing enclosed mass (from bottom to top). The curves correspond to enclosed masses of $1/4$, $1/2$, 1 and 2, respectively, in units of $1.6\times 10^8 M_{\odot}$ which is the mass of the halo in this case.
• Figure 2: Halo formation in WDM, at $z=6$. In the top panel, we show the linear extrapolated overdensity $\delta_c(M,z)$ at the time of collapse, as a function of halo mass $M$. The solid curves show the cases of $m_X=1.5$ keV and $m_X=0.75$ keV, respectively from bottom to top. For comparison, we show the $\Lambda$CDM curve which includes the Sheth/Jenkins correction (long-dashed curve), and the mass-independent value given by spherical collapse in $\Lambda$CDM (short-dashed curve). In the bottom panel, we show the mass fluctuation $\sigma(M)$, based on the linearly-extrapolated power spectrum at matter-radiation equality. The solid curves illustrate the effect of the power spectrum cutoff in $\Lambda$WDM, for $m_X=1.5$ keV, and $m_X=0.75$ eV, respectively from top to bottom. Also shown for comparison is $\sigma(M)$ in $\Lambda$CDM (long-dashed curve). In both panels, the vertical dotted line shows the value of the lowest halo mass at $z=6$ in which gas can cool (see § \ref{['sec:reion']}).
• Figure 3: Numerical and semi-analytic halo mass functions at $z=1$ in $\Lambda$CDM and in $\Lambda$WDM. The curves show the comoving number density $n(>M)$ of halos above mass $M$. Solid lines show the semi-analytic calculation, and dashed lines show the results from numerical simulations by Bode. The three cases shown are, from top to bottom, $\Lambda$CDM, $m_X=0.35$ keV, and $m_X=0.175$ keV. The dotted lines show model predictions, for the two $\Lambda$WDM models, which include only the suppression of the power spectrum. The lowest-mass halos, those below $\sim 10^{10} M_{\odot}$ (for $m_X=0.35$ keV) or $\sim 10^{11} M_{\odot}$ (for $m_X=0.175$ keV), form in the simulations through fragmentation and are not accounted for in our models, but these halos do not affect our conclusions about reionization (see text). Note that the models shown in this figure use the cosmological parameters $\Omega_{\rm b}=0$ and $h=0.67$ for consistency with Bode.
• Figure 4: Numerical and semi-analytic halo mass fractions at $z=1$ in $\Lambda$CDM and in $\Lambda$WDM. The curves show the total mass fraction $F(<M)$ in halos up to mass $M$, with $F(<1.5 \times 10^9 M_{\sun})$ subtracted. Solid lines show the semi-analytic calculation, and dashed lines show the results from numerical simulations by Bode. The three cases shown are, from top to bottom, $\Lambda$CDM, $m_X=0.35$ keV, and $m_X=0.175$ keV. Note that the models shown in this figure use the cosmological parameters $\Omega_{\rm b}=0$ and $h=0.67$ for consistency with Bode.
• Figure 5: Semi-analytic halo mass functions at $z=6$ in $\Lambda$CDM and in WDM. The lower panel shows $|dn(>M)/d\log(M)|$, where $n(>M)$ is the comoving number density of halos above mass $M$. The upper panel shows the corresponding total mass fraction in halos above $M$, $F(>M)$. In both panels, solid lines are for $\Lambda$CDM and dashed lines for $\Lambda$WDM with $m_X=1.5$ keV and $m_X=0.75$ keV (top to bottom). Also shown is the minimum mass for cooling at $z=6$ (vertical dotted line).