Cosmic Radio Background from Primordial Black Holes at Cosmic Dawn

Astrophysical black holes usually have strong X-ray signal that is either from the hot accretion flow (ADAF), or from the standard thin disc and the hot corona above the disc. If the black hole has a jet, then it also produces strong radio signal by the synchrotron emission of relativistic electrons inside the jet. Jet can also produce strong X-ray signal and it may dominate the X-ray radiation of the whole black hole when the accretion rate is extremely low Yuan and Cui (2005).

In the observed cases, when the accretion rate is high ($\dot{m}\gtrsim 1$), the thin disc forms, and only a small fraction of black holes have jet Rafter et al. (2009). However, when the accretion rate is lower ($\dot{m}\lesssim 1$), the ADAF forms, and the fraction of black holes with jet increases dramatically Yang et al. (2020). For most of the cases, our PBH has the accretion rate $\dot{m}$ well below $\sim 1$, therefore it is reasonable to assume that a sufficient large fraction of them have jets.

Ref. Merloni et al. (2003) analyzed the selected samples including supermassive black holes and stellar-mass black holes with both X-ray and radio signals. They found a tight correlation between the radio luminosity $L_{R}$ ($L_{R}=\nu_{5.0}L_{\nu_{5.0}}$, $\nu_{5.0}=5.0$ GHz), hard X-ray (2-10 keV) luminosity $L_{X}$ and black hole mass $M_{\bullet}$,

where $L_{R}$ and $L_{X}$ are in the unit of ${\rm erg}\,{\rm s}^{-1}$, and $M_{\bullet}$ is in units of ${\rm M}_{\odot}$. They pointed out that for models where the X-ray signal is mainly from the standard thin disc and corona, or mainly from the jet, the predictions do not agree with the observed scaling relation. However, if the radio signal is mainly from the jet and the X-ray signal is mainly from the ADAF accretion, such scaling relation is well interpreted Heinz and Sunyaev (2003).

We assume the PBH has radio from the jet and X-ray from the ADAF accretion. The radio luminosity and hard X-ray luminosity of the PBH also follow the scaling relation Eq. (18). To derive the radio luminosity of the PBH from Eq. (18), we use the following formula to calculate the hard X-ray luminosity

The first line fits the outputs of theoretical calculations of the ADAF accretion Merloni et al. (2003); the second line is derived from the bolometric correction of observed AGNs with accretion rate $\dot{m}\gtrsim 0.02$, assuming a radiative efficiency 0.1 Lusso et al. (2012).

We assume the radio spectrum has a power-law form with a constant spectrum index

where $\alpha=0.75$ Ghisellini et al. (2017).

In Fig. 2, we show the radio luminosity as a function of redshift for different PBH masses. For naked PBHs, $L_{R}$ is less evolved except at the highest redshifts. For PBHs inside dark matter halos, at high redshifts $L_{R}$ is close to the naked PBHs. However at lower redshifts it increases rapidly until the accretion rate reaching the Eddington limit. At $z\sim 20$, when $M_{\bullet}=1-10^{3}\,{\rm M}_{\odot}$, $L_{R}\sim 5\times 10^{-15}-1\times 10^{-9}L_{\rm Edd}$ for naked PBHs; and $L_{R}\sim 8\times 10^{-14}-9\times 10^{-9}L_{\rm Edd}$ for PBHs inside dark matter halos.

To calculate the contribution of PBHs to the CRB, we need to know their mass distribution and number density. In this paper, for simplicity we assume that all PBHs have the same mass $M_{\bullet}$, then the comoving number density of PBHs is

where $f_{\rm PBH}$ is the ratio of total PBH mass density to total matter density in the Universe, and $\Omega_{\rm m}$ is the fractional matter density.

The CRB from PBHs is then

where

is the frequency in the emitter’s rest frame. $f_{\rm DC}\leq 1$ is the duty-cycle for radio emission, which refers to the fraction of PBHs that is active in the radio band. Since the radio emission for these PBHs is produced by the jet, it would be equal to the ratio of the jet lifetime compared to the Hubble timescale.

In Fig. 3 we show the contribution to the 1.4 GHz CRB at $z\sim 17$ from PBHs as a function of $f_{\rm PBH}$, for naked PBHs and for PBHs inside dark matter halos respectively. However, even if we set $f_{\rm DC}=1$, this contribution is still negligible compared with the CMB. In principle, if $M_{\bullet}$ is higher and/or $f_{\rm PBH}$ is higher, one may obtain higher $T_{\rm CRB}$, but the X-ray heating (see Sec. II.4) can reduce the accretion rate, so $T_{\rm CRB}$ increases more gently for higher $M_{\bullet}$ at high $f_{\rm PBH}$ region. More importantly, the X-ray heating works more efficiently in reducing the 21-cm absorption than the extra CRB in boosting the 21-cm absorption. We have checked that using a higher PBH mass and a higher abundance than in Fig. 3 will not help boosting the 21-cm absorption signal.

PBHs emit not only radio photons but also X-ray photons. They can heat and ionize the IGM, which would reduce the 21-cm absorption signal.

The physical luminosity density of X-ray emission that interacts with the IGM is

where the optical depth $\tau(\nu)=[\sigma_{\rm H}(E)n_{\rm H}(1-x_{\rm HII})+\sigma_{\rm He}(E)n_{\rm He}(1-x_{\rm HeII})](c/H)$, and $\sigma_{\rm H}$ and $\sigma_{\rm He}$ are photon-ionization cross-sections for hydrogen and helium respectively Maselli et al. (2003). Here we assume that the X-ray spectrum follows the form

and adopt $E_{\rm min}=2$ keV and $E_{1}=100$ keV, with the normalization

Ignoring the contribution from UV photons, the time evolution of hydrogen ionized fraction $x_{\rm HII}$, helium ionized fraction $x_{\rm HeII}$, and temperature of the IGM $T_{\rm k}$ follows

where $n_{\rm H}=(1-Y_{\rm He})\Omega_{\rm b}\rho_{\rm cr}(1+z)^{3}/m_{\rm H}$ and $n_{\rm He}=Y_{\rm He}\Omega_{\rm b}\rho_{\rm cr}(1+z)^{3}/m_{\rm He}$ are physical number density of hydrogen and helium elements respectively, and $Y_{\rm He}=0.24$; $x_{\rm e}=x_{\rm HII}+n_{\rm He}/n_{\rm H}x_{\rm HeII}$; the total number density of particles $n=n_{\rm H}(1+x_{\rm e})+n_{\rm He}$; $E_{\rm H}=13.6$ eV and $E_{\rm He}=24.6$ eV are ionization energy for hydrogen and helium; $k_{\rm B}$ is the Boltzmann constant; $\chi_{\rm ion,H}$, $\chi_{\rm ion,He}$ and $\chi_{\rm heat}$ are fractions of X-ray energy deposited into hydrogen ionization, helium ionization and heating Valdés and Ferrara (2008); the net ionization ($I_{\rm H}$ and $I_{\rm He}$) and heating ($H_{T}$) rates in the absence of PBH are:

where $\gamma_{\rm H,He}$ and $\alpha_{\rm H,He}$ are rates of collision ionization and recombination, for hydrogen and helium respectively Maselli et al. (2003); $\Lambda(T)$ are the physical volume cooling rate (including the collisional ionization cooling, collisional excitation, Bremsstrahlung cooling and Compton heating/cooling) Maselli et al. (2003). Since $\dot{m}$ depends on $T_{\rm k}$ and $x_{\rm e}$, we solve the thermal evolution of the IGM and $\dot{m}$ self-consistently.

In Fig. 4, we show the kinetic temperature of the IGM at $z\sim 17$ as a function of $f_{\rm PBH}$, for various PBH masses same as in Fig. 3, for naked PBHs and PBHs inside dark matter halos respectively. In both cases, it is easier to heat the IGM temperature above the CMB than to generate the CRB larger than the CMB.

The brightness temperature of 21-cm global spectrum is Pritchard and Loeb (2012):

where the temperature of the radiation field $T_{\rm R}=T_{\rm CMB}+T_{\rm CRB}$. The spin temperature is

where $x_{\alpha}$ and $x_{\rm c}$ are coupling coefficients due to Ly$\alpha$ scattering and collision respectively. Calculations of them can be found in Ref. Mesinger et al. (2011) and references therein.

The PBH does not only generate radio background and X-ray which heats up the gas, but also contribute to the Ly$\alpha$ background. A fraction $\chi_{\alpha}$Valdés and Ferrara (2008) of the X-ray energy is deposited into Ly$\alpha$ photons, so the specific intensity of photon number is given by

In Fig. 5, we show the 21-cm brightness temperature at $z=17$ as a function of $f_{\rm PBH}$, for various PBH masses. To highlight the role played by the PBH, we ignore the radiation from any normal astrophysical objects, e.g. Pop III stars, first galaxies. It is interesting to note that, although the CRB from the PBH is unimportant in this scenario, for the 21-cm absorption there is an optimal $f_{\rm PBH}$ where the absorption reaches the maximum. If $f_{\rm PBH}$ is lower, it produces less Ly$\alpha$ background, which is not strong enough to couple $T_{\rm S}$ with $T_{\rm k}$, therefore the 21-cm absorption is weaker. If $f_{\rm PBH}$ is higher however, the IGM is heated significantly, which still results in weaker 21-cm absorption. Even in the absence normal astrophysical objects, the PBHs can provide the necessary Ly$\alpha$ photons to make the 21-cm signal be close to the maximum absorption in the standard cosmology. This is not easy if only the Pop III stars are the first ionization sources Yajima and Khochfar (2015); Furlanetto (2006); Schauer et al. (2019).

In addition to the isolated PBHs, there are also PBH binaries. Ref. De Luca et al. (2020a) found that: for two PBHs with equal mass, if they form a binary system, their total accretion rate is the Bondi accretion rate of a single black hole with mass equal to the sum of their mass. Since Bondi accretion rate is generally proportional to $\sim M^{2}_{\bullet}$, if the indirect dependence of $\lambda$ on $M_{\bullet}$ is ignored, the accretion rate of each PBH is then boosted by a factor $\sim$2, because in the binary system, the ambient medium density of each PBH is boosted. The boost factor of radiation (X-ray, radio) is even larger than 2 since the bolometric luminosity is proportional to $\dot{m}^{2}$. Therefore, the PBH binaries can help to increase the influence on the 21 cm signal. Ref. De Luca et al. (2020a) also pointed out that, the binary system will finally reach a stationary state where the two members have equal mass, even if at the beginning their mass is different, because the accretion evolution tends to balance each other.

On the other hand, the fraction of PBHs that are in binary systems is Raidal et al. (2017)

where $\delta_{\rm dc}$ describes the clustering of PBHs. For uniform distribution, $\delta_{\rm dc}=1$. In such case, $f_{\rm binary}$ is small when $f_{\rm PBH}\lesssim 0.1$, and the contribution to X-ray heating and CRB from PBH binaries is small. However, if for some reason the PBHs can be highly clustered, say $\delta_{\rm dc}\gtrsim 10$, then the PBH binaries may have a significant influence on the 21 cm signal, depending on $f_{\rm PBH}$. In the left panels of Fig. 3, 4 & 5 we show the $T_{\rm CRB}$, $T_{\rm k}$ and $\delta T_{\rm b}$ of 21 cm signal if the contribution from PBH binaries is also taken into account by thin lines, assuming $\delta_{\rm dc}=1000$. We see that when $f_{\rm PBH}\gtrsim 10^{-4}-10^{-3}$, the influence of PBH binaries becomes more obvious. The gravitational wave observations put constraints on the joint parameter $\delta_{\rm dc}^{16/37}f_{\rm PBH}^{53/37}$. For $\delta_{\rm dc}=1000$, $f_{\rm PBH}\sim 10^{-4}-10^{-3}$ is still within the limit set by LIGO observations Raidal et al. (2017).

If the PBH binaries are located inside dark matter halos, absolutely they will also boost the influence on the 21 cm absorption signal. We do not show the results in the right panels because we suppose the accretion rate, in this case, can be much more complicated than the binaries of naked PBHs.

When accretion is weak, the radiative efficiency of the accretion is low, but the relativistic electrons in the jet can produce strong synchrotron radiation Yuan and Cui (2005); Paliya et al. (2017). In this case we can derive the radio luminosity from a jet model where all jet emission is considered as coming from a blob of electrons with uniform properties (Ghisellini et al., 2010).

The total synchrotron power of the blob is (Ghisellini, 2013)

where ${\rm d}N_{\rm e}/{\rm d}\gamma$ is the energy distribution of relativistic electrons in the blob; $U_{B}=B^{2}/(8\pi)$ is the energy density of magnetic field with strength $B$; $\gamma_{\rm min}$ and $\gamma_{\rm max}$ are the minimum and maximum Lorentz factors respectively.

On the other hand, since the jet blob is far from the central black hole, in redshift ranges of our interests, the seed photons for inverse Compton scattering (ICS) are mainly the synchrotron photons (the synchrotron self-Compton, SSC) and the CMB photons. The total ICS power is

where $U_{\rm R}=\rho_{\gamma}+U_{\rm syn}$ is the energy density of radiation field, consisting of CMB energy density ($\rho_{\gamma}$) and synchrotron radiation field. The latter is

where $R_{\rm blob}$ is the physical size of the blob.

It is generally assumed that the injected relativistic electrons follows a power-law distribution with an exponential cutoff (e.g. Pepe et al. (2015a)),

where the normalization $Q_{0}$ is set by the relation

$P_{\rm e}=\eta_{\rm e}\eta_{\rm jet}\dot{m}L_{\rm Edd}$ is the injected power of the relativistic electrons, where $\eta_{\rm jet}$ is the jet efficiency and $\eta_{\rm e}$ is the fraction of jet power that is finally converted into the relativistic electrons.

With the energy loss due to synchrotron and ICS, the steady state electron distribution is Ghisellini and Tavecchio (2009); Gardner and Done (2014, 2018)

where

$\gamma_{\rm cool}$ is the point where the cooling time equals the crossing timescale of the blob, with the cooling time given by

and $t_{\rm cross}=R_{\rm blob}/c$. $K$ factor in Eq. (38) is

In practice, $\gamma_{\rm cool}$ is found by numerical iterations. Finally the synchrotron luminosity is

where $l_{\rm syn}$ is the synchrotron emission of a single electron Ghisellini (2013). We then obtain the SSC luminosity from $L_{\rm syn}(\nu)$ and ${\rm d}N_{\rm e}(\gamma)/{\rm d}\gamma$, and the luminosity of the ICS between electrons and CMB photons Lefa (2012) from ${\rm d}N_{\rm e}(\gamma)/{\rm d}\gamma$.

So the jet model has following parameters $\{R_{\rm bolb},B,\eta_{\rm jet},\gamma_{\rm min},\gamma_{\rm max},s\}$ that are directly relevant to the radio signal. We assume that the PBH jet is analogous to the jet of typical stellar-mass black hole Cygnus X-1 Pepe et al. (2015a); Gandhi et al. (2017), in spite of its lower accretion rate. We set the jet parameters according to the following considerations:

• •

  For Cygnus X-1, according to Ref. Zdziarski (2012); Heinz (2006), the bulk of the low frequency radio emission is generated at the height $\mathcal{O}(1)\times 10^{7}~{}R_{\rm S}$ along the jet axis, where $R_{\rm S}=2GM_{\bullet}/c^{2}$ is the Schwarzschild radius. We assume half of the opening angle of the blob is 0.1 radian, therefore take $R_{\rm blob}=10^{6}~{}R_{\rm S}$.

• •

  According to Ref. Pepe et al. (2015b), at the jet base which is $1.1\times 10^{8}$ cm above the back hole, $B=5\times 10^{7}$ G. $B$ can be inversely proportional to the height along jet axis, therefore at the end of acceleration region which is $\sim\mathcal{O}(1)\times 10^{12}$ cm above the black hole, $B$ is serval thousand G. According to Ref. Zdziarski et al. (2014), at the onset of the energy dissipation $B\sim 10^{4}-10^{6}$ G. If most of the radio emission is produced at the height $10^{3}$ times the onset, and $B$ is again inversely proportional to the height, then $B$ can be several tens to several thousand. We take $B=2000$ G as the fiducial value. We will also show the results for much lower $B$.

• •

  The observed total jet power can be comparable or even larger than the total accretion power $\dot{m}L_{\rm Edd}$ Ghisellini et al. (2014), because the jet power is not directly from the accretion Ghisellini (2019), but supplied by the black hole spin. This is supported by the GRMHD simulations which found that the steady-state jet efficiency $\eta_{\rm jet}$ can be $\sim$ 140% Tchekhovskoy et al. (2011) for the ADAF-like accretion mode. The maximum jet efficiency can reach $\sim 300$% McKinney et al. (2012). Here we assume that a large fraction of the jet power is finally converted into the kinetic energy of the relativistic electrons, and take $\eta_{e}\eta_{\rm jet}=0.5$ as the optimistic fiducial value. We will also show the results for a much smaller $\eta_{e}\eta_{\rm jet}$.

• •

  Regarding the parameters of energy distribution of relativistic electrons, we take $\gamma_{\rm min}=1$ and $\gamma_{\rm max}\sim 10^{6}$ as in Ref. Zhang et al. (2014). However we take a steeper slope $s=3.2$ as fiducial value, so that the low frequency signal is larger in the jet radiation. In some blazars the distribution of relativistic electrons with $\gamma\gtrsim 100$ can have such steep slope Ghisellini et al. (2010). We will also show the results for a shallower slope.

In Fig. 6 we plot the different components of the SED for a $10\,{\rm M}_{\odot}$ PBH with such fiducial parameters at $z=17$. We see that the SSC and ICS luminosities are much smaller than the synchrotron. The heating and ionization of the IGM will be dominated by accretion X-ray radiation, if the UV radiation is neglected.

We solve the IGM thermal history and $\dot{m}$ evolution self-consistently, then compute the CRB and the 21-cm signal. The contributions from accretion X-ray radiation, synchrotron, SSC and the ICS are all taken into account.

For naked PBHs with $M_{\bullet}=10\,{\rm M}_{\odot}$, we plot $T_{\rm CRB}$, $T_{\rm k}$ and $\delta T_{\rm b}$ at $z=17$ as a function of $f_{\rm PBH}$ for various $f_{\rm DC}$ in Fig. 7. For $f_{\rm DC}\lesssim 0.01$, the X-ray heating would be the dominant effect, the 21-cm signal is hardly enhanced.

In contrast, for $f_{\rm DC}\gtrsim 0.01$, when $f_{\rm PBH}f_{\rm DC}\gtrsim 10^{-5}$, they can significantly contribute to the CRB and boost the 21-cm absorption signal. Although the IGM is also heated, $T_{\rm k}$ is still smaller than $T_{\rm CMB}+T_{\rm CRB}$, so the 21-cm signal is still in absorption. For more massive PBHs and a higher abundance, the heating is more efficient, $T_{\rm k}$ can be larger than $T_{\rm CMB}+T_{\rm CRB}$, so it turns 21-cm signal to an emission. However, the resultant high ionization fraction from efficient heating makes the optical depth too large to be compatible with current Planck constraints. To see this, we calculate the augumented optical depth of the Thomson scattering of CMB photons as

where $\Delta x_{\rm e}$ is the fraction of extra IGM electrons generated by X-rays from PBHs. To avoid distorting the CMB signal, we require that $\Delta\tau_{\rm e}(z>10)$ is smaller than the $2\sigma$ uncertainty level of the Planck observations, i.e. $\Delta\tau_{\rm e}\lesssim 0.014$ Planck Collaboration et al. (2020).

In Fig. 8 we plot the contour map of the 21-cm signal with the PBH mass and abundance when $f_{\rm DC}=1$. We see that, even in the presence of X-ray heating, there is still a large parameter space for which the 21-cm absorption can be boosted, For example, if $M_{\bullet}\gtrsim 10^{2}\,{\rm M}_{\odot}$, the 21-cm signal is obviously boosted even $f_{\rm PBH}$ is as low as $\sim 10^{-6}$. Even after excluding the parameters for which $\Delta\tau_{\rm e}\geq 0.014$ (above the solid curve), it is still possible to boost the 21-cm absorption signal deeper than $\sim-500$ mK. In Fig. 8 the $\delta T_{b}=-500$ mK curve corresponds to $\log(f_{\rm PBH})=-1.8\log(M_{\bullet}/{\rm M}_{\odot})-3.5$ for $1\,{\rm M}_{\odot}\lesssim M_{\bullet}\lesssim 300\,{\rm M}_{\odot}$. For a PBH distribution with a single mass, if $M_{\bullet}\lesssim 1\,{\rm M}_{\odot}$ and $f_{\rm PBH}\lesssim 10^{-4}$, it will not significantly affect the 21-cm signal.

We also investigate the alternative duty-cycle values, and find that if $f_{\rm DC}\lesssim 0.04$, the boosted 21-cm absorption signal would be $\leq 500$ mK if we are to keep $\Delta\tau_{\rm e}\lesssim 0.014$.

In Fig. 9 we show the ratio of the 21 cm signal with PBH binaries to that without PBH binaries, for the fiducial jet model parameters. When $f_{\rm PBH}\lesssim 10^{-4}$, the PBH binaries have negligible influence on the 21 cm signal. When $f_{\rm PBH}\gtrsim 10^{-4}$, the PBH binaries boost the 21 cm absorption signal for $M_{\bullet}\lesssim 10~{}M_{\odot}$, while suppress it for $M_{\bullet}\gtrsim 10~{}M_{\odot}$. This is because when the PBHs are heavier, the X-ray heating can somewhat reduce the absorption depth and/or the accretion rate, by increasing $T_{k}$, although such parameter region should be ruled out when the reionization constraints are taken into account.

In Fig. 10 we show the cases for which either $\eta_{e}\eta_{\rm jet}$ is lower (top), or $B$ is lower (middle), or the energy distribution of injected relativistic electron is harder (bottom), than our fiducial values respectively. In these cases, we find that if $\eta_{e}\eta_{\rm jet}\lesssim 0.02$, or $B\lesssim 200$ G, or $s\lesssim 2.5$, then it would be unable to boost the 21-cm absorption signal to $\sim-500$ mK, while keeping the reionization history almost unchanged at the same time. For the electron distribution with $s=2.5$, the heating by X-rays from the synchrotron is even more important than that from the accretion radiation.

If PBHs are inside dark matter halos, where the gravity of the dark matter halo is still negligible, the accretion rate decreases as the Univese expands, like the naked PBHs. However, the rate can increase at a critical redshift because the gravity of the dark matter halo dominates the accretion. This critical redshift depends on the PBH mass, as seen from Fig. 1. The 21-cm signal is then more complicated in this case.

In such a model, at very high redshift, when the effects of dark matter halo is still negligible, the 21-cm absorption decreases with decreasing redshift, just as the case for naked PBHs. However, the dark matter halo enhances the PBH gas accretion, so at some point the 21-cm absorption signal starts to increases as redshift decreases, because of the enhancement of CRB. The absorption reaches a maximum at a redshift $z_{\rm trough}$, then starts to decrease because of the X-ray heating and the expansion of the Universe. Therefore in the presence of dark matter halos, an absorption trough is produced. The depth and redshift of the trough depend on the PBH mass and abundance.

In Fig. 11 we plot the $\delta T_{b}$ at $z_{\rm trough}$ as a function of $f_{\rm PBH}$ and $M_{\bullet}$, assuming $f_{\rm DC}=1$. The $\delta T_{b}=-500$ mK curve corresponds to $\log(f_{\rm PBH})={\rm max}[-1.7\log(M_{\bullet}/{\rm M}_{\odot})-3.7,-0.2\log(M_{\bullet}/{\rm M}_{\odot})-5.0]$ for $0.3\,{\rm M}_{\odot}\lesssim M_{\bullet}\lesssim 30\,{\rm M}_{\odot}$. Since $z_{\rm trough}$ is also a function of $f_{\rm PBH}$ and $M_{\bullet}$, the cross point of given $\delta T_{\rm b}$ and given $z_{\rm trough}$ curves uniquely determines the mass and abundance of PBHs.

For example, to produce the maximal 21-cm absorption of $\delta T_{\rm b}=-500$ mK at $z_{\rm trough}=17$, it requires $M_{\bullet}\sim 1.05\,{\rm M}_{\odot}$ and $f_{\rm PBH}\sim 1.5\times 10^{-4}$. In Fig. 12 we show $\delta T_{\rm b}$ as a function of redshift for this PBH mass and abundance. For comparison we also plot the curves from some other PBH masses and abundances.

Similar to the case of naked PBHs, we find that if $f_{\rm DC}\lesssim 0.04$ then it would be hard to boost the 21-cm signal significantly.

A dark matter halo does not always boost the accretion rate of its embedded PBH. If a PBH falls into the gravitational well of a previously-formed dark matter halo, then the gravitational potential energy is converted into kinetic energy. In this case, the PBH and ambient medium can have a higher relative velocity than $v_{L}$. As a result, the accretion is suppressed and the dark matter halo is not helpful for producing the 21 cm absorption trough. Such a scenario is not involved in our model.

The redshifted radiation from PBHs at $>z$ forms part of the Cosmic Electromagnetic Background (CEB) in present-day Universe. The intensity can be written as

where $L_{\bullet}$ is the total PBH luminosity, including the contribution from synchrotron, SSC, ICS of the CMB photons, and the accretion X-ray radiation. For the model to be consistent with observations, we require that the produced background radiation should not exceed what have been observed at different frequency bands.

In Fig. 13 we plot the CEB from radio to gamma-ray, contributed by PBHs inside dark matter halos at $z>10$, for $M_{\bullet}=1.05\,{\rm M}_{\odot}$ and $f_{\rm PBH}=1.5\times 10^{-4}$, $f_{\rm DC}=1$. Compared with the observed CRB excess Fixsen et al. (2011), the cosmic X-ray background excess Cappelluti et al. (2017) and the isotropic gamma-ray background (IRGB) Ackermann et al. (2015), the contribution from PBHs in this model is much smaller, so this model is entirely consistent with the existing observations. Of course, it also means that to detect PBHs by using the CEB is a challenge. Such a background must be very isotropic compared with the background built up by astrophysical sources since the dark matter halos hosting the PBHs are much less clustered than galaxies.