Supernova constraint on self-interacting dark sector particles

We consider a dark photon portal DS model wherein the Dirac dark fermion $\chi$ couples to dark photon $A^{\prime}$ and the dark photon kinetically mixes with the SM photon Holdom (1986); Okun (1982); Pospelov et al. (2008). The corresponding Lagrangian of the dark sector is given by

where $\epsilon$ is the mixing parameter, $m_{A^{\prime}}$ is the dark photon mass, $m_{\chi}$ is the mass of the dark fermion, and $g_{D}$ is the DS coupling constant. We define the DS fine structure constant $\alpha_{D}\equiv g_{D}^{2}/4\pi$ analogous to the electromagnetic fine structure constant $\alpha_{e}\equiv e^{2}/4\pi$.

Through the mixing of dark photon with the SM photon, dark photons and dark fermions can be produced via processes analogous to the SM electromagnetic ones. We follow Refs. Chang et al. (2017, 2018) to consider the following production channels for light dark photons and dark fermions inside the hot and dense interior of a PNS with a temperature $\simeq 30$ MeV and a core mass density $\gtrsim 10^{14}$ g cm^-3. For the dark photon, we include the nucleon-nucleon bremsstrahlung $np\rightarrow npA^{\prime}$, Compton-like interaction $\gamma e^{-}\rightarrow e^{-}A^{\prime}$, and electron-positron annihilation $e^{-}e^{+}\rightarrow A^{\prime}$. For the dark fermion, we consider three pair-production channels including nucleon-nucleon bremsstrahlung $np\rightarrow np\chi\bar{\chi}$, electron-positron annihilation $e^{-}e^{+}\rightarrow\chi\bar{\chi}$, and plasmon decay $\gamma^{\ast}\rightarrow\chi\bar{\chi}$.

When the PNS interior is optically thin to dark photons or dark fermions, the rates of the above production channels directly determine the energy luminosity carried away by the dark particles. However, when the interactions between dark particles and the SM medium, as well as the self-interactions in the DS, are strong enough, dark photons and fermions can be trapped in the PNS. These interactions include the inverse processes of the above production channels, the dark photon (fermion) pair-annihilation $A^{\prime}A^{\prime}\leftrightarrow\chi\bar{\chi}$, the DS Compton scattering $A^{\prime}\chi\leftrightarrow\chi A^{\prime}$, the scattering of dark fermions $\chi\chi\rightarrow\chi\chi$ and $\chi\bar{\chi}\rightarrow\chi\bar{\chi}$, as well as the dark fermion scattering with SM charged fermions $\chi p\rightarrow\chi p$ and $\chi e^{-}\rightarrow\chi e^{-}$. These interactions determine the mean free path of the DS particles and thus the energy loss rate through the neutrinosphere in the diffuse regime. Moreover, when $m_{A^{\prime}}>2m_{\chi}$, the (inverse) decay of dark photon $A^{\prime}\leftrightarrow\chi\bar{\chi}$ needs to be considered. In this scenario, the inverse decay process dominates all the dark fermion pair-absorption processes and the DS self-interactions due to the resonances.²²2$\chi\bar{\chi}\rightarrow\chi\bar{\chi}$ has a dark photon resonance and $A^{\prime}\chi\rightarrow A^{\prime}\chi$ has a dark fermion resonance.; see Eq. (37) and text below in the Appendix VI for details. Other DS self-interactions without resonances are suppressed by an extra factor of $\alpha_{D}$ compared to the (inverse) decay rate and thus can be neglected for $m_{A^{\prime}}>2m_{\chi}$. We list all the included interactions in this work in Tables 1 and 2 for dark photon and dark fermion, respectively.

Throughout this work, we will use perturbative calculation for the interaction rates. Since the temperature in the PNS is $\sim 30$ MeV, we only consider DS particles of masses below $\simeq 1$ GeV.

Without the presence of dark particles, long-term SN simulations predicted that the PNS cools by emitting neutrinos of all flavors in $\sim 10$ s. The total energy carried away by neutrinos is $\simeq 3\times 10^{53}$ erg, which is fixed by the available gravitational energy released to form a compact neutron star. The potential emission of exotic particles that may be produced inside the PNS will thus reduce the duration of the neutrino emission and can be constrained by the observed neutrino events from SN1987a. Based on comparisons with simulations including the emission of axions from the PNS, a well-known criterion (Raffelt’s criterion) was formulated to constrain the maximal energy luminosity carried by any exotic particles Raffelt (1990),

where $L_{D}$ denotes the energy luminosity of dark emission, evaluated during the early PNS cooling phase, and $L_{\nu}$ is approximately the time-averaged neutrino energy luminosity in SM.

In this work, we use the PNS density and temperature profile at 1 s post the SN core bounce obtained in Ref. Fischer et al. (2010) to compute the emission of dark photons and dark fermions. This PNS profile was widely used recently for similar purposes (see e.g., Refs. Chang et al. (2017); Sung et al. (2019)). Note that the choice of a particular SN model may introduce uncertainties of a factor of a few for the derived bounds Chang et al. (2017); Mahoney et al. (2017). Figure 1 shows the radial evolution of the density $\rho$, temperature $T$, the electron chemical potential $\mu_{e}$, and the plasma frequency $\omega_{p}$ [see Eq. (45)]. The position of the spectral-averaged neutrino decoupling sphere, i.e., neutrinosphere, is indicated by the vertical dash line at $R_{\nu}\simeq 22$ km. The density profile shows a monotonically decreasing behavior as a function of radius, while the temperature profile exhibits a peak at $r\simeq 11$ km, due to the inefficient compression heating at the densest core region. The electrons are highly degenerate inside the PNS. The plasma frequency $\omega_{p}\simeq 14$ MeV at the PNS center and decreases at larger radii. The plasma effect effectively alters the mixing between the dark photon and the SM photon differently for the transverse and the longitudinal polarizations An et al. (2013); Hardy and Lasenby (2017); Chang et al. (2017). We have included this effect throughout this work and give the details in Appendix VII.

In the nondiffuse regime, we consider the bulk emission rates of DS particles inside the neutrinosphere and the attenuation due to absorption and decay, following Refs. Chang et al. (2017); Sung et al. (2019). The luminosity of the dark photon is given by

where $g_{L}=1$, $g_{T}=2$, and $\Gamma^{L,T}_{A^{\prime},\rm prod}$ and $\tau_{L,T}$ are the production rate and optical depth respectively. The exponential factor accounts for the absorption of dark photons by the medium and their decay. We separate the longitudinal (L) and transverse (T) modes because the medium effect modifies their dispersion relations and leads to different effective mixings of the two modes with the SM photon (see Appendix VII). The production rate $\Gamma^{L,T}_{A^{\prime},\rm prod}$ is determined by the interactions involving the SM particles listed in Table 1. By detailed balance, the rates of each production process and its inverse process are related by $\Gamma^{L,T}_{A^{\prime},\rm prod}=e^{-\omega/T}\Gamma^{L,T}_{A^{\prime},\rm abs}$, where $\Gamma^{L,T}_{A^{\prime},\rm abs}$ is the total absorption rate of the inverse process. Therefore,

For the optical depth, we include the absorption and decay of $A^{\prime}$ by the same processes with SM particle as above and ignore those involving DS particles, to avoid double counting the total DS luminosity (see also later discussions in this subsection).³³3The pair-annihilation rate of $A^{\prime}A^{\prime}\rightarrow\chi\bar{\chi}$ is also ignored since the dark photon abundance inside the PNS is relatively small compared to that of SM particles in the nondiffuse regime. This gives

where $v=|\vec{k}|/\omega$ is the dark photon velocity, and $f(r)$ is a geometric factor used in Chang et al. (2017) that effectively takes into account different path lengths of dark photons emitting locations to the neutrinosphere. The explicit forms of these absorption rates are given in Appendix VIII.

Next, we compute the luminosity of the dark fermion ($\chi$) in the nondiffuse regime as

where $g_{\chi}=2$ is the physical degrees of freedom of $\chi$, and $\Gamma_{\chi,\rm prod}$ is the production rate of $\chi$ by the SM medium (see Table 2). Note that since $\chi$ and $\bar{\chi}$ are symmetric in our model, the total luminosity in the dark fermion pair is $L_{\chi}+L_{\bar{\chi}}=2L_{\chi}$. Here we do not include the attenuation due to pair absorption. This is because in the nondiffuse regime, the dark fermion abundance below the neutrinosphere is very low, which suppresses the pair absorption of dark fermions.

If the dark fermions were in equilibrium with the medium, detailed balance could relate the production rates to the absorption rates by $\Gamma^{\rm eq}_{\chi,\rm prod}=e^{-E/T}\Gamma^{\rm eq}_{\chi,\rm abs}$. We take the equilibrium production rates $\Gamma^{\rm eq}_{\chi,\rm prod}$ as an approximation for the production rates $\Gamma_{\chi,\rm prod}$ used in Eq. (6). That is,

This approximation in principle underestimates a bit the production rates of $\chi$ and $\bar{\chi}$ due to the assumed equilibrium occupation number, which effectively results in Pauli blocking. However, since the $\chi$ and $\bar{\chi}$ are always produced in pairs, the effective Pauli-blocking suppression of the rates is small due to their zero chemical potentials.

We note that when $m_{A^{\prime}}\geq 2m_{\chi}$, the dark fermion pair production is in fact dominated by the decay of on-shell dark photons. Since in Eq. (3) we do not include the decay of $A^{\prime}$ to $\chi$ and $\bar{\chi}$ when $A^{\prime}$ are in nondiffuse regime, it leads to double counting of the total dark luminosity if we also include the dark fermion production. Thus, we do not consider the contribution of Eq. (6) when dark photons are in nondiffuse regime and when $m_{A^{\prime}}\geq 2m_{\chi}$. We also note here that if dark photons are in the diffuse limit but dark fermions are not, then effectively the non-diffuse dark fermion luminosity can be determined by the decay rate of the trapped dark photons that are in thermal equilibrium with the SM medium (see Appendix VI).⁴⁴4Note that we consider separately the diffuse condition for the longitudinal and the transverse dark photons. Thus, we further multiply the nondiffuse dark fermion luminosity by a factor of $1/3$ ($2/3$) if only the longitudinal (transverse) mode of the dark photons are trapped (see Appendix IX).

Before we discuss the detailed numerical results of the DS luminosity in the nondiffuse regime, let us provide an analytic estimation for the relevant region of the parameter space. First, for most cases, the production rates of longitudinal dark photons and dark fermions are suppressed by a factor of $m_{A^{\prime}}^{2}/\omega^{2}$ (see Appendix VIII) and by the coupling constant $\alpha_{D}$, respectively, compared to the production rate of the transverse dark photons. We can thus approximate the nondiffuse luminosity of DS particles by considering transverse dark photons only. Second, we consider for simplicity a homogeneous PNS with radius $R_{c}\simeq 10\mbox{ km}$, temperature $T\simeq 30\mbox{ MeV}$, density $\rho\simeq 3\times 10^{14}\mbox{ g}/\mbox{cm}^{3}$ and electron fraction $Y_{e}\simeq 0.3$. We also assume that dark photons are relativistic. Taking the nucleon-nucleon bremsstrahlung $A^{\prime}np\rightarrow np$ without the plasma effects (see Appendix VIII.1), the luminosity of DS particles, $L_{D}$, is approximately

for $m_{A^{\prime}}\lesssim 1$ GeV.

In the diffuse regime, we assume that DS particles are in good thermal contact with the SM medium. Due to the temperature gradient, the DS phase space distributions are slightly anisotropic, which induces an outward energy flux. We use the radiative transfer equation for the DS particle energy flux through the neutrinosphere in Appendix X. The energy flux of a particle species $i$ is approximately given by

where the upper (lower) sign is for fermions (bosons), $g_{i}$ is the physical degrees of freedom of particle $i$, $T_{\nu}$ is the temperature at the neutrinosphere, $m_{i}$ is the mass of particle $i$, and $\langle\lambda^{-1}_{i}(r)\rangle$ is the thermally averaged inverse mean free path (IMFP)⁵⁵5The IMFP used here is rescaled by the relative abundances of the DS particles. See Sec. III.3 for the definition. of particle $i$ at radius $r$ defined by

where $f_{i}(E,T(r))$ is the distribution function of particle $i$ at radius $r$. We distinguish between the absorptive and scattering IMFP, $\lambda_{i,abs}^{-1}$ and $\lambda_{i,sca}^{-1}$, and define the total IMFP as

As in the nondiffuse regime, we compute the energy fluxes of the longitudinal and transverse dark photon separately, and assume that the energy fluxes of $\chi$ and $\bar{\chi}$ are equal. Hence, the total energy loss rate in the DS particles is $L_{tot}=L_{A^{\prime},L}+L_{A^{\prime},T}+2L_{\chi}$. The IMFP calculations can be found in Appendix VIII.

We now estimate the relevant parameter space region in the limit where the DS self-interaction is the dominant opacity source. When $m_{A^{\prime}}>2m_{\chi}$, the dominant DS self-interaction is the (inverse) decay $A^{\prime}\leftrightarrow\chi\bar{\chi}$. Given $T_{\nu}\simeq 3.9\mbox{ MeV}$ and the temperature gradient $|dT/dr|_{R_{\nu}}\simeq 6.2\times 10^{-4}\mbox{ MeV}/\mbox{m}$, if we fix $m_{A^{\prime}}/m_{\chi}=3$, the luminosities of the DS particles can be fitted by

for $m_{A^{\prime}}\lesssim 25$ MeV, and

for $m_{\chi}\lesssim 30$ MeV. If we take $m_{A^{\prime}}=9$ MeV and $m_{\chi}=3$ MeV, then the total DS luminosity is

The corresponding $\chi\bar{\chi}\rightarrow A^{\prime}$ cross section is⁶⁶6The estimated value here is the thermally averaged cross section evaluated at the center of PNS $\sigma_{\chi\bar{\chi}\rightarrow A^{\prime}}\equiv\langle\lambda^{-1}_{\chi\bar{\chi}\rightarrow A^{\prime}}(r=0)\rangle/n_{\chi}(r=0)$.

When $m_{A^{\prime}}<2m_{\chi}$, the diffuse luminosities of dark photons and dark fermions depend on the values of $\epsilon$ and $\alpha_{D}$. For regions where $\epsilon$ is small enough such that the dominating opacity source for dark photons is DS self-interaction $A^{\prime}A^{\prime}\rightarrow\chi\bar{\chi}$, one may estimate the DS luminosity by considering the dark photon luminosity only. If we fix $m_{A^{\prime}}/m_{\chi}=1/3$, then the dark photon luminosity can be fitted by

for $m_{A^{\prime}}\lesssim 25$ MeV. Adopting $m_{A^{\prime}}=3\mbox{ MeV}$ and $m_{\chi}=9\mbox{ MeV}$, this gives

The corresponding $\chi\chi\rightarrow\chi\chi$ cross section is⁷⁷7The cross section is evaluated in the same way as Eq. (15).

and the corresponding $A^{\prime}A^{\prime}\rightarrow\chi\bar{\chi}$ cross section is

These analytical formulas Eqs. (14)–(19) already illustrate that DS particles can be self-trapped by interaction cross sections of $\gtrsim\mathcal{O}(10^{-40})$ cm² such that the corresponding diffuse DS luminosity $L_{D}\lesssim L_{\nu}$. Only very weakly interacting DS particle with $\alpha_{D}$ much smaller than those nominal values in Eqs. (14)–(19) can result in $L_{D}>L_{\nu}$.

Whether DS particles are in the nondiffuse or in the diffuse regime sensitively depends on their abundances in the PNS. For example, if the dark fermion abundance is significant, the contribution of their pair-absorption processes (e.g., $\chi\bar{\chi}np\rightarrow np$, $\chi\bar{\chi}\rightarrow e^{-}e^{+}$, and $\chi\bar{\chi}\rightarrow\gamma^{\ast}$) to the dark fermion optical depth cannot be neglected. Furthermore, the DS abundances lead to significant DS self-interactions (e.g. $\chi\chi\rightarrow\chi\chi$, $\chi\bar{\chi}\rightarrow A^{\prime}A^{\prime}$, $\chi A^{\prime}\rightarrow A^{\prime}\chi$,…) that can trap themselves in the PNS. These DS self-interactions delay the escape times for both the dark photon and the dark fermion, which enhances their abundances and optical depths. Therefore, the abundances of DS particles is critical in determining whether self-trapping is important. Below, we formulate the diffuse criteria in terms of the DS abundances and their IMFP at the neutrinosphere.

To obtain the exact DS abundances, detailed transport incorporating their production, scattering, and absorption needs be solved. Here we roughly estimate the abundance of each particle species by their production rate and a relevant timescale. The abundance of a DS particle species $i$ is approximated as

where the quantity $\Delta t_{i}$ is subject to the following considerations. First, the longest possible time for DS particles to accumulate is the cooling time of the PNS. We set this upper bound of $\Delta t_{i}$ as $t_{\rm cool}=1$ s. Second, the shortest possible timescale, i.e., the lower bound of $\Delta t_{i}$, is estimated by the free-escaping timescale $t_{\rm free}\equiv R_{\nu}/v_{i}$. Another characteristic timescale here in the limit where a DS particle is trapped is the diffusion timescale $t_{i,\rm diff}\simeq R_{\nu}^{2}/D_{i}$ where $D_{i}\simeq v_{i}/\lambda^{-1}_{i}$ is the diffusion coefficient (see below for details). For $t_{\rm free}<t_{i,\rm diff}<t_{\rm cool}$, we then take $t_{i,\rm diff}$ as $\Delta t_{i}$. Combining these criteria, the relevant timescale $\Delta t_{i}$ used in Eq. (20) is given by

Since the PNS is not homogeneous, the diffusion of DS particles cannot be described by a constant $D_{i}$. As shown in Tables 1 and 2, the interactions relevant to the IMFP of DS particles can be categorized as SM-type or DS-type by particles involved in the interactions. We choose particular locations in the PNS where the IMFPs of each type are maximal to roughly estimate their contribution to the diffusion timescale.⁸⁸8Overestimating the IMFP makes Eq. (24) easier to satisfy. Because the luminosity in the diffuse regime is generally lower than that in the nondiffuse regime for the same parameters, our estimation leads to a conservative bound. For the SM-type interactions, we estimate the IMFP at the center ($r_{\rm SM}=0~{}\mathrm{km}$) of the PNS where the density of SM particles is the largest. As for the DS-type interactions, we estimate the IMFP at $r_{\rm DS}=11\mathrm{km}$ where the temperature is the highest. This location has the largest DS particle density, if DS particles are in thermal equilibrium with the SM medium. Knowing the IMFPs of both types at these respective locations, we can then compute the diffusion timescale

where $\tilde{\lambda}^{-1}_{i,SM}$ and $\tilde{\lambda}^{-1}_{i,DS}$ are SM-type and DS-type contribution to the IMFP of particle $i$. When computing the IMFP contribution from DS particles, $\tilde{\lambda}^{-1}_{i,DS}$ in Eq. (22), we take the assumption that DS particles are in full thermal equilibrium with the SM medium. Combining Eqs. (20)–(22) then allows us to compute the DS abundance $N_{i}$ for particle $i$.

After deriving $N_{i}$, we then define a scaling factor $\eta_{i}\equiv N_{i}/N_{i}^{\rm eq}$ for each dark particle species $i$ for $N_{i}<N_{i}^{\rm eq}$ ($\eta_{i}=1$ if $N_{i}\geq N_{\rm eq}$), where

is the corresponding total equilibrium abundance of $i$. These scaling factors are then used to better estimate the IMFP, $\lambda^{-1}_{i,DS}$ due to the DS particles self-interaction, used in Eq. (10) for computing the DS diffuse luminosities. We also use $\eta_{i}$ to calculate the thermally averaged total IMFP of $i$ at $R_{\nu}$, $\langle{\lambda}^{-1}_{i}(R_{\nu})\rangle$.

We are now finally ready to write down our diffuse criteria. We say that a DS particle species $i$ can be treated as in diffuse limit if the following conditions are satisfied:

The first condition ensures that a DS particle species $i$ is only considered to be diffusive when their production is efficient enough such that their amount can exceed the equilibrium number within the relevant timescale $\Delta t_{i}$. The second condition requires that the IMFP of $i$ at $R_{\nu}$, where both the density and temperature are the lowest in the PNS, is large enough to trap particle $i$ inside the PNS (see Fig. 1).⁹⁹9Note that in computing the second criterion, we exclude the contribution from the dark photon decay via $A^{\prime}\rightarrow e^{-}e^{+}$ This is because including this decay process leads to an artificially enhancement of the IMFP by several orders of magnitudes for dark photon heavier than $\simeq 30$ MeV, as the electron chemical potential $\mu_{e}\simeq 15$ MeV at $R_{\nu}$. However, this process should be strongly Pauli-blocked for $m_{A^{\prime}}\lesssim 200$ MeV in most region inside the PNS (see Fig. 1) where dark photons are most abundant in the diffuse limit. When Eq. (24) is satisfied, we use Eq. (9) to compute the DS diffuse luminosity for $i$. Otherwise, we take the nondiffuse luminosities, Eqs. (3) and (6) for dark photons and dark fermions, respectively. Note that in the diffuse regime and when $\Delta t_{i}=t_{i,\rm diff}$, the first condition in Eq. (24) is approximately equivalent to having the characteristic thermalization IMFP Keil et al. (2003); Shapiro and Teukolsky (1983) of DS particles with SM medium, $\sqrt{\lambda^{-1}_{i,SM,\rm abs}\times\lambda^{-1}_{i,\rm total}}$ averaged over the phase space and spatial volume inside $R_{\nu}$, being larger than $R_{\nu}^{-1}$. This means that the energy exchange between the DS particles and the SM medium, enhanced by the DS self-interactions, is efficient enough to keep themselves in thermal contact with SM medium.

We provide in Appendix IX a detailed work flow to additionally describe how we use the results in these sections to compute the dark sector luminosity for interested readers.

We now apply formulas derived in previous sections to compute the dark photon and dark fermion luminosities and examine how they depend on the coupling constants $\epsilon$ and $\alpha_{D}$. Figure 2 shows the luminosities of different DS particles as functions of $\alpha_{D}$ with $\epsilon=10^{-8}$ for two different choices of DS masses. In the left panel, the masses are chosen such that the decay process $A^{\prime}\rightarrow\chi\bar{\chi}$ is allowed. In this scenario, dark photons of different polarization modes are trapped diffusively when $\alpha_{D}\gtrsim 5\times 10^{-18}$, while the dark fermions are diffusive when $\alpha_{D}\gtrsim 10^{-17}$. The nondiffuse luminosity of the dark photon for small $\alpha_{D}$ mainly depends on the interaction with the SM particles and thus is independent of $\alpha_{D}$.¹⁰¹⁰10The difference of $\sim 10^{2.5}$ between the luminosities of the longitudinal and transverse dark photon is due to the plasma effects.. The dark fermion luminosity is proportional to $\alpha_{D}$ in the diffuse limit, as analyzed in Appendix VI However, the dark fermion luminosity for $\alpha_{D}\lesssim 5\times 10^{-18}$ is set to zero to avoid double counting because they are predominantly produced through the decay of on-shell dark photons (see discussions in Sec. III.1). In the diffuse regime, the luminosities of the DS particles are proportional to $\alpha_{D}^{-1}$ due to the self-trapping interactions effectively dominated by the (inverse) decay process $A^{\prime}\leftrightarrow\chi\bar{\chi}$.

For the right panel in Fig. 2, we choose DS masses such that the decay process $A^{\prime}\rightarrow\chi\bar{\chi}$ is not allowed. In this case, dark photons of different polarization modes are in diffuse regime when $\alpha_{D}\gtrsim 8\times 10^{-9}$, while dark fermions are always in the nondiffuse regime for the range of $\alpha_{D}$ shown in the plot. The main reason leading to several orders of magnitude larger difference in $\alpha_{D}$ here than the previous case is due to the extra $\alpha_{D}$ dependence in the IMFP of DS self-interactions [see, e.g., Table 1 or Eqs. (14) and (17)]. Similar to the previous scenario, the nondiffuse luminosity of the dark photon is independent of $\alpha_{D}$. However, the luminosities in the diffuse regime scale as $\alpha_{D}^{-2}$. Once again, this is because the dominant interaction is responsible for trapping the dark photon being $A^{\prime}A^{\prime}\rightarrow\chi\bar{\chi}$, whose interaction rate is proportional to $\alpha_{D}^{2}$. The dark fermion luminosity is significantly smaller than the dark photon luminosity for the range of $\alpha_{D}$ in the plot, because the dark fermion production rate is proportional to $\epsilon^{2}\alpha_{D}$, suppressed by an extra factor of $\alpha_{D}$ when compared with the dark photon production rate.

In Fig. 3, we show the luminosities of the DS particles as functions of $\epsilon$ with two choices of $\alpha_{D}$ and same DS masses as those in Fig. 2. For the case with $\alpha_{D}=10^{-17}$ in the left panel where the decay process $A^{\prime}\rightarrow\chi\bar{\chi}$ is again allowed, the longitudinal (transverse) dark photons are trapped when $\epsilon\gtrsim 3\times 10^{-9}$ ($\epsilon\gtrsim 2\times 10^{-9}$). For dark fermions, they just free stream independent of the value of $\epsilon$, because of the small value of $\alpha_{D}$ (cf., Fig. 2). The nondiffuse luminosities of the dark photons of both modes are proportional to $\epsilon^{2}$ as determined by their production rates. The dark fermion luminosity for $\epsilon\lesssim 2\times 10^{-9}$ is set to zero for the same reason discussed above. The diffuse luminosities of dark photons are affected by both the interactions with the SM particles and the DS self-interactions. For $\epsilon\gtrsim 10^{-7}$, dark photon–SM interactions dominate over DS self-interactions, so the dark photon luminosities are proportional to $\epsilon^{-2}$ as determined by the mean free path of the dark photon–SM interactions. Interestingly, the dark fermion luminosity becomes independent of $\epsilon$ and can thus remain larger than $L_{\nu}$ for large $\epsilon$. This is because the branching ratio of the dark photon absorption processes approaches to $1$, such that dark fermions can be produced via the decay of trapped dark photons (see Appendix VI).

For the right panel in Fig. 3, we choose $\alpha_{D}=10^{-9}$ for the case where $A^{\prime}\rightarrow\chi\bar{\chi}$ is not allowed. There, longitudinal (transverse) dark photons are diffusively trapped when $\epsilon\gtrsim 3\times 10^{-7}$ ($\epsilon\gtrsim 2\times 10^{-7}$), while dark fermions are in the diffuse limit when $\epsilon\gtrsim 2\times 10^{-5}$. The dark photon luminosities are proportional to $\epsilon^{2}$ for $\epsilon\lesssim 10^{-9}$, where the optical depths are negligible. The diffuse luminosities of dark photons scale as $\epsilon^{-2}$ because the absorption processes by the SM medium are dominant. The dark fermion luminosity is much smaller than that of the dark photons until $\epsilon\gtrsim 10^{-5}$, because the dark fermion production rate is suppressed by an additional factor of $\alpha_{D}$ when compared with the dark photon production rate. The diffuse luminosity of the dark fermion is proportional to $\epsilon^{-2}$ for $\epsilon\gtrsim 10^{-4}$, where $\chi e^{-}\rightarrow\chi e^{-}$ dominates the IMFP at $R_{\nu}$.