Cosmological evolution of warm dark matter fluctuations. II. Solution from small to large scales and keV sterile neutrinos

We solve the cosmological evolution of warm dark matter density fluctuations within the analytic framework of the Volterra integral equations presented in the accompanying paper [H. J. de Vega and N. G. Sanchez, preceding Article, Phys. Rev. D 85, 043516 (2012).]. In the absence of neutrinos, the anisotropic stress vanishes and the Volterra-type equations reduce to a single integral equation. We solve numerically this single Volterra-type equation both for dark matter (DM) fermions decoupling at thermal equilibrium and DM sterile neutrinos decoupling out of thermal equilibrium. We give the exact analytic solution for the density fluctuations and gravitational potential at zero wave number. We compute the density contrast as a function of the scale factor $a$ for a relevant range of wave numbers $k$. At fixed $a$, the density contrast turns to grow with $k$ for $k<k_c$ while it decreases for $k>k_c$, where $k_c\simeq 1.6/\mathrm{Mpc}$. The density contrast depends on $k$ and $a$ mainly through the product $ka$ exhibiting a self-similar behavior. Our numerical density contrast for small $k$ gently approaches our analytic solution for $k=0$. For fixed $k<1/(60\mathrm{kpc})$, the density contrast generically grows with $a$ while for $k>1/(60\mathrm{kpc})$ it exhibits oscillations starting in the radiation dominated era which become stronger as $k$ grows. We compute the transfer function of the density contrast for thermal fermions and for sterile neutrinos decoupling out of equilibrium in two cases: the Dodelson-Widrow model and a model with sterile neutrinos produced by a scalar particle decay. The transfer function grows with $k$ for small $k$ and then decreases after reaching a maximum at $k=k_c$ reflecting the time evolution of the density contrast. The integral kernels in the Volterra equations are nonlocal in time and their falloff determine the memory of the past evolution since decoupling. We find that this falloff is faster when DM decouples at thermal equilibrium than when it decouples out of thermal equilibrium. Although neutrinos and photons can be neglected in the matter dominated matter dominated era, they contribute to the Volterra integral equation in the MD era through their memory from the radiation dominated era.

Figure 1

The free-streaming length in dimensionless variables $l(y,Q)$ divided by $Q$ vs ${log}_{10}y$. The (blue) line of small dots corresponds to $Q=0.1$, the solid (red) line to $Q=1$ and the dotted (green) line to $Q=10$. (Recall that ${\lambda }_{\mathrm{fs}}=({\eta }^{*}/{\xi }_{\mathrm{dm}})Ql(y,Q)$ [1]). We explicitly compute $l(y,Q)$ in Appendix . $l(y,Q)$ is given in the different regimes by Eqs. (b2, b3, b8, b9). Notice that ${log}_{10}y=0$ corresponds to equilibration. We choose here ${\xi }_{\mathrm{dm}}=5000$.

Figure 2

The ordinary logarithm of the normalized density contrast $\breve{\delta }(y,\alpha )$ vs ${log}_{10}y$ from the numerical resolution of the Volterra equation (4.2) for DM fermions in thermal equilibrium with $m=0.6736\mathrm{keV}$ and for sterile neutrinos in the DW model with $m=1.685\mathrm{keV}$. (Both models yield identical density fluctuations for a given value of ${\xi }_{\mathrm{dm}}$). The upper panel displays ${log}_{10}\breve{\delta }(y,\alpha )$ for $0\le \alpha \le 0.1\simeq {\alpha }_c$ and the lower panel for ${\alpha }_c\simeq 0.1\le \alpha \le 10$. We see that for $\alpha <{\alpha }_c$ and fixed $y,\breve{\delta }(y,\alpha )$ increases for increasing $\alpha$ while the opposite happens for $\alpha >{\alpha }_c$. The value $\alpha ={\alpha }_c\simeq 0.1$ determines the transition between these two regimes. In the upper panel the bottom curve corresponds to $\alpha =0$ and then from down to up the curves correspond to $\alpha =0.003$, 0.005, 0.01, 0.03 and 0.1. In the lower panel the uppermost curve corresponds to $\alpha =0.1$ and then from up to down the curves correspond to $\alpha =0.5$, 0.7, 1, 3 and 10. $\breve{\delta }(y,\alpha =0)$ is plotted from the analytic solution Eq. (4.8, 4.9, 4.10, 4.11, 4.12, 4.13, 4.14, 4.15, 4.16, 4.17) for TIC. We see that the different curves have essentially the same shape and are shifted from each other in an almost self-similar manner indicating that $\breve{\delta }(y,\alpha )$ is mainly a function of $\kappa y$. $\breve{\delta }(y,\alpha )$ generically grows with $y$ for fixed $\alpha <1$ while it exhibits oscillations starting in the RD era for $\alpha >1$ which become stronger as $\alpha$ grows. $\breve{\delta }(y,\alpha )$ becomes proportional to $y$ at sufficiently late times. The larger is $\alpha$, the later starts $\delta (y,\alpha )$ to grow proportional to $y$ and the later the oscillations remain.

Figure 3

The transfer function today $T(\gamma )$ vs $\gamma =\sqrt{\frac{I_4^{\mathrm{dm}}}{3}}\alpha$ for ${\xi }_{\mathrm{dm}}=5000$. The (red) solid line curve is for DM fermions in thermal equilibrium with $m=0.6736\mathrm{keV}$ as well as for sterile neutrinos out of equilibrium in the DW model with $m=1.685\mathrm{keV}$. The (blue) dotted line corresponds to sterile neutrinos out of equilibrium in the $\chi$ model with $m=0.7203{\tau }^{-1/4}\mathrm{keV}$ where $\tau$ is a coupling constant [see Eq. (5.1)]. That is, $0.9365<m/\mathrm{keV}<1.665$ [see Eq. (5.4)]. $\gamma$ is defined by Eq. (4.23). The presence here of a single maximum at $\gamma ={\gamma }_c\simeq 0.2$ is consistent with the curves for $\breve{\delta }(y,\alpha )$ and the value of ${\alpha }_c$ in Fig. 2. Notice that the two transfer functions turn out to be very similar although they describe quite different dynamics.

Figure 4

The ordinary logarithm of the normalized distribution functions $f_0^{\mathrm{dm}}(Q)$ vs $Q$. The (red) solid line corresponds to fermions at thermal equilibrium (which turns to be identical to the out of thermal equilibrium DW model). The (green) dotted line corresponds to sterile neutrinos out of thermal equilibrium in the $\chi$ model.

Figure 5

The kernel $\Pi (x)$ defined by Eq. (3.3) as a function of $x$ for fermions in thermal equilibrium (which is identical to the DW model) and for sterile neutrinos out of thermal equilibrium in the $\chi$ model.

Figure 6

The gravitational potential $\breve{\varphi }(y,\alpha )$ and ${\varphi }^0(\zeta )$ vs ${log}_{10}\zeta$ defined by Eqs. (a8, a9). $\breve{\varphi }(y,\alpha )$ is plotted for $\alpha =1$ by a (red) solid line, for $\alpha =10$ by a dotted (green) line, for $\alpha =0.1$ by a (blue) line of small dots. ${\varphi }^0(\zeta )$ is depicted by small (red) dots. We see that ${\varphi }^0(\zeta )$ is a very good approximation to $\breve{\varphi }(y,\alpha )$ in the whole range of $\zeta$.