Warm dark matter primordial spectra and the onset of structure formation at redshift $z$

Analytic formulas reproducing the warm dark matter (WDM) cosmological spectra are obtained for WDM particles decoupling in and out of thermal equilibrium; these formulas provide the initial data for WDM nonlinear structure formation. We compute and analyze the corresponding WDM overdensities and compare them to the cold dark matter (CDM) case. We consider the ratio of the WDM to CDM spectrum and the ratio of the WDM to CDM overdensities: They turn out to be self-similar functions of $k/k_{1/2}$ and $R/R_{1/2}$, respectively, with $k_{1/2}$ and $R_{1/2}$ being the wavenumber and length where the WDM spectrum and overdensity are one-half of the respective CDM magnitudes. Both $k_{1/2}$ and $R_{1/2}$ show scaling as powers of the WDM particle mass $m$, while the self-similar functions are independent of $m$. The WDM spectrum sharply decreases around $k_{1/2}$ with respect to the CDM spectrum, while the WDM overdensity slowly decreases around $R_{1/2}$ for decreasing scales with respect to the CDM one. The nonlinear regions where WDM structure formation takes place are shown and compared to those in CDM: The WDM nonlinear structures start to form later than in CDM, and as a general trend, decreasing the DM particle mass delays the onset of the nonlinear regime. The nonlinear regime starts earlier for smaller objects than for larger ones; smaller objects can form earlier both in WDM and CDM. We compute and analyze the differential mass function $dN/dM$ for WDM at redshift $z$ in the Press-Schechter approach. The WDM suppression effect of small scale structure increases with the redshift $z$. Our results for $dN/dM$ are useful to be contrasted with observations, in particular, for $4\lesssim z\lesssim 12$. We perform all of these studies for the most popular WDM particle physics models. Contrasting them to observations should give the value of the WDM particle mass within the keV scale.

Figure 1

The warm dark matter power spectrum ${\Delta }_{\mathrm{wdm}}^2(k)$ for a physical mass of 2.5 keV in four different WDM models, and the CDM power spectrum ${\Delta }_{\mathrm{cdm}}^2(k)$. We plot the ordinary logarithm of the power spectrum ${\Delta }^2(k)$ vs the ordinary logarithm of $k\mathrm{Mpc}/h$.

Figure 2

CAMB values for $T^2(k)$ vs $k\mathrm{Mpc}/h$ and those from the fitting formulas (2.7, 2.8) for fermionic WDM decoupling at thermal equilibrium with masses $m=1$, 2 and 4 keV. The agreement between the fitting formulas with the data is excellent.

Figure 3

The ordinary logarithm of the expected overdensity ${\sigma }^2(R,z=0)$ vs ${log}_{10}[Rh/\mathrm{Mpc}]$ for an $m=2.5\mathrm{keV}$ WDM particle in four different WDM models and in CDM. At small scales, ${\sigma }^2(R)$ is almost constant for WDM. WDM flattens and reduces ${\sigma }^2(R)$ for small scales compared with CDM.

Figure 4

Lines where ${\sigma }^2(M,z)=1$ in the $z$, $\log [hM/M_{\odot }]$ plane for $m=2.5\mathrm{keV}$ in four different WDM models and in CDM. The linear regime [${\sigma }^2(M,z)<1$] applies above these lines, and the nonlinear [${\sigma }^2(M,z)>1$] regime applies below these lines.

Figure 5

The ordinary logarithm of the relative overdensity $D(R)={\sigma }_{\mathrm{WDM}}^2(R,z)/{\sigma }_{\mathrm{CDM}}^2(R,z)$ as a function of ${log}_{10}[Rh/\mathrm{Mpc}]$ for $m=2.5\mathrm{keV}$ in four different WDM models. We see in these curves the small and large scale behavior of the relative overdensity $D(R)$ as described by Eq. (3.4).

Figure 6

The ordinary logarithm of the dimensionless mass function ${log}_{10}S(M,z)$ vs ${log}_{10}[hM/M_{\odot }]$ at redshift $z=0$, 2, 5 and 10 for a WDM particle of $m=2.5\mathrm{keV}$ in four different WDM models and for CDM.

Figure 7

The ordinary logarithm of the dimensionless mass function ${log}_{10}S(M,z)$ vs ${log}_{10}[hM/M_{\odot }]$ for a WDM particle of $m=2.5\mathrm{keV}$ in the $\nu \mathrm{MSM}$ model and in CDM for redshifts $z=0$, 2, 5 and 10.

Figure 8

CAMB values for ${\Delta }_{\mathrm{cdm}}^2(k)$ and those from the fitting formula (a1) as a function of $k\mathrm{Mpc}/h$.