Dissipative dark matter halos: The steady state solution

Dissipative dark matter, where dark matter particle properties closely resemble familiar baryonic matter, is considered. Mirror dark matter, which arises from an isomorphic hidden sector, is a specific and theoretically constrained scenario. Other possibilities include models with more generic hidden sectors that contain massless dark photons [unbroken $U(1)$ gauge interactions]. Such dark matter not only features dissipative cooling processes but also is assumed to have nontrivial heating sourced by ordinary supernovae (facilitated by the kinetic mixing interaction). The dynamics of dissipative dark matter halos around rotationally supported galaxies, influenced by heating as well as cooling processes, can be modeled by fluid equations. For a sufficiently isolated galaxy with a stable star formation rate, the dissipative dark matter halos are expected to evolve to a steady state configuration which is in hydrostatic equilibrium and where heating and cooling rates locally balance. Here, we take into account the major cooling and heating processes, and numerically solve for the steady state solution under the assumptions of spherical symmetry, negligible dark magnetic fields, and that supernova sourced energy is transported to the halo via dark radiation. For the parameters considered, and assumptions made, we were unable to find a physically realistic solution for the constrained case of mirror dark matter halos. Halo cooling generally exceeds heating at realistic halo mass densities. This problem can be rectified in more generic dissipative dark matter models, and we discuss a specific example in some detail.

Figure 1

Optically thin cooling function for solar abundances. The solid line is the result of our code, while the dashed line is the result found by Dopita and Sutherland [64].

Figure 2

Halo heating (thick solid line) and cooling rates (dashed line) for a Milky Way scale galaxy. The lower thin solid line is the heating contributed by SN sourced dark photons. The $\lambda$-density profile was used with the halo temperature and ionization state numerically determined from steady state equations. (a) and (b) give the rates in terms of the mirror metal abundance parameter, $\zeta$, for $\kappa ={10}^{45}\mathrm{erg}/\mathrm{s}$ and (a) $T_{\mathrm{SN}}=25\mathrm{keV}$, (b) $T_{\mathrm{SN}}=1\mathrm{keV}$. While (c) and (d) give the rates in terms of the SN sourced dark photon luminosity, $\kappa$ [$\mathrm{erg}/\mathrm{s}$], for $\zeta =2$ and (c) $T_{\mathrm{SN}}=25\mathrm{keV}$, (d) $T_{\mathrm{SN}}=1\mathrm{keV}$.

Figure 3

The optical depth for a dark photon originating near the Galactic center and escaping a Milky Way scale galaxy, with halo properties as per Fig. 2. The solid (dashed) line shows results for $\zeta =0$ ($\zeta =2$), with $\kappa ={10}^{45}\mathrm{erg}/\mathrm{s}$ and (a) $T_{\mathrm{SN}}=25\mathrm{keV}$, (b) $T_{\mathrm{SN}}=1\mathrm{keV}$.

Figure 4

Properties of the steady state solution obtained for a Milk Way scale galaxy ($m_{\text{baryons}}={10}^{11}{m}_{\odot }$, $f_s=0.8$, $r_D=3.95\mathrm{kpc}$, $M_{\mathrm{FUV}}=-18.4$). Shown are (a) the halo density, (b) the halo temperature, (c) the mean mass, $\overline{m}$, and (d) the heating, cooling rates [$\mathcal{H}$, $\mathcal{C}$] (solid, dashed lines).

Figure 5

(a) The rotation curves ($\mathrm{halo}+\text{baryons}$) derived from the computed steady state solutions. The baryon mass ranges from $m_{\text{baryons}}={10}^8{m}_{\odot }$ (bottom curve) to $m_{\text{baryons}}={10}^{11}{m}_{\odot }$ (top curve). See Table 2 for other baryonic parameters chosen. (b) The corresponding halo rotation curves (halo contribution only).

Figure 6

Halo rotation curves derived from the steady state solutions for the galaxy set of Table 2 (solid lines), and with a factor of $\times 2.5$ increase in the baryonic disk scale length (dashed lines). (a) The results are given for the five smallest galaxies, with $m_{\text{baryons}}={10}^8{m}_{\odot }$ (bottom curve) to $m_{\text{baryons}}=10^{10.5}{m}_{\odot }$ (top curve), while (b) is for $m_{\text{baryons}}={10}^{11}{m}_{\odot }$.

Figure 7

(a) Normalized halo rotational velocity, $v_{\mathrm{halo}}(r)/v_{\mathrm{halo}}(r_{\mathrm{opt}})$, for all 18 modeled galaxies calculated from the steady state solution. (b) Comparison of the spherically symmetric $\lambda$-density profile [Eq. (40)] (solid line) with the corresponding profile for a disk geometry [Eq. (51)] (dash-dotted line). The triangles in the figures are the synthetic rotation curve obtained from dwarf disk galaxies [76].

Figure 8

The quantity $\tilde{\lambda }$ [Eq. (52)] versus $v_{\mathrm{rot}}^{\max }$ computed from the steady state solutions found. (a) The results are given for the 18 modeled galaxies along with an extrapolation (circles for the six canonical baryonic parameters of Table 2, squares for the $r_D\to 2.5r_D$ parameter variation, and triangles for the $M_{\mathrm{FUV}}\to M_{\mathrm{FUV}}-0.8$ variation). (b) The extrapolated results together with the $\tilde{\lambda }$ values from THINGS spirals (stars) [5] and LITTLE THINGS dwarfs (squares) [8].

Figure 9

(a) $L_{\mathrm{FUV}}/L_{\mathrm{FUV}}^{\mathrm{MW}}$ versus $v_{\mathrm{rot}}^{\max }$ and (b) $m_{\text{baryons}}$ versus $v_{\mathrm{rot}}^{\max }$, for each of the steady state solutions found. Circles are the results for the canonical galaxy set of Table 2, squares for the $r_D\to 2.5r_D$ parameter variation, and triangles for the $M_{\mathrm{FUV}}\to M_{\mathrm{FUV}}-0.8$ variation. The dashed lines are the power laws: (a) $L_{\mathrm{FUV}}\propto [v_{\mathrm{rot}}^{\max }{]}^{2.8}$ and (b) $m_{\text{baryons}}\propto [v_{\mathrm{rot}}^{\max }{]}^{3.7}$.

Figure 10

(a) The effect of varying the effective temperature parameter, $T_{\mathrm{eff}}$ [Eq. (35)], on $\tilde{\lambda }$, Eq. (52). Notation as in Fig. 8 but with small (large) symbols for $T_{\mathrm{eff}}=10\mathrm{keV}$ ($T_{\mathrm{eff}}=25\mathrm{keV}$). (b) The extrapolated results for $T_{\mathrm{eff}}=10\mathrm{keV}$ (thin solid line) and $T_{\mathrm{eff}}=25\mathrm{keV}$ (thick solid line). Also shown are the $\tilde{\lambda }$ values from THINGS spirals (stars) [5] and LITTLE THINGS dwarfs (squares) [8].