The effect of primordial black holes on 21cm fluctuations

# The effect of primordial black holes on 21cm fluctuations

Hiroyuki Tashiro and Naoshi Sugiyama
Physics Department, Arizona State University, Tempe, AZ 85287, USA
Department of Physics and Astrophysics & Kobayashi-Maskawa Institute for the Origin of Particles and the Universe,
Nagoya University, Chikusa, Nagoya 464-8602, Japan
Institute for the Physics and Mathematics of the Universe (IPMU), The University of Tokyo, Kashiwa, Chiba, 277-8568, Japan
July 31, 2019
###### Abstract

The 21 cm signal produced by non-evaporating primordial black holes (PBHs) is investigated. X-ray photons emitted by accretion of matter onto a PBH ionize and heat the intergalactic medium (IGM) gas near the PBH. Using a simple analytic model, we show that this X-ray heating can produce an observable differential 21 cm brightness temperature. The region of the observable 21 cm brightness temperature can extend to 1–10 Mpc comoving distance from a PBH depending on the PBH mass. The angular power spectrum of 21 cm fluctuations due to PBHs is also calculated. The peak position of the angular spectrum depends on PBH mass, while the amplitude is independent of PBH mass. Comparing this angular power spectrum with the angular power spectrum caused by primordial density fluctuations, it is found that both of them become comparable if at and at for the PBH mass from to . Finally we find that the Square Kilometer Array can detect the signal due to PBHs up to at and at for PBHs with mass from to .

###### keywords:
cosmology: theory – large-scale structure of universe

## 1 Introduction

primordial black holes (PBHs) could have formed in the early Universe (Carr & Hawking, 1974; Carr, 1975). Although there is no direct evidence of their existence of PBHs, PBHs are attracting attention as a way of constraining physics in the early Universe. In particular, one of the main generation mechanisms of PBHs is the gravitational collapse of an overdense region at the horizon scale when the amplitude of the overdensity exceeds a critical threshold. Therefore the resultant mass function and the abundance of PBHs depend on the amplitude of primordial density fluctuations at the horizon-crossing epoch (Green et al., 2004). The PBH abundance is expected to be a probe of primordial density fluctuations on small scales, which cannot be accessed by Cosmic Microwave Background or large-scale-structure observations.

Constraints on the abundance of PBHs have been extensively studied and continue to be updated (Carr et al., 2010). PBHs with mass less than g have evaporated by the present epoch because the evaporation time scale by Hawking radiation is less than the Hubble time scale today (Hawking, 1974). However evaporation of PBHs generates additional entropy in the Universe after inflation (Zel’dovich & Starobinskii, 1976), affects big bang nucleosynthesis (Vainer et al., 1978; Vainer & Naselskii, 1978; Miyama & Sato, 1978; Zeldovich et al., 1977; Lindley, 1980) and distorts the CMB blackbody spectrum (Tashiro & Sugiyama, 2008). PBH evaporation may also produce the observable gamma-ray background (Page & Hawking, 1976; MacGibbon & Carr, 1991). According to measurements of these cosmological phenomena, there are strong constraints on the abundance of PBHs with mass less than g.

PBHs with mass larger than g survive in the present Universe. One of the constraints on such PBHs can be set from the fact that the current density parameter of PBHs, , cannot exceed the cold dark matter density parameter observed at the present epoch, . Conventionally, the constraint on PBH abundance is given by which is the fraction of regions of mass collapsing into PBHs at the formation epoch (Carr, 1975). The constraint on the density parameter of PBHs today, , implies from WMAP 7-year data, i.e., (Komatsu et al., 2011). Microlensing observations also constrain the abundance of non-evaporating PBHs (Alcock et al., 2001). Ricotti et al. (2008) have obtained the constraint on PBHs with mass larger than , by investigating the effects of such PBHs on cosmic reionization and CMB temperature anisotropies. Future gravitational wave observations are also expected to provide a probe of the massive PBH abundance (Ioka et al., 1999; Inoue & Tanaka, 2003).

In this paper, we evaluate the 21 cm brightness temperature produced by PBHs and study the potential of 21 cm observations to give a constraint on the abundance of PBHs. Mack & Wesley (2008) have investigated the signatures of evaporating PBHs in 21 cm brightness temperature. Accordingly, they have concentrated on PBHs whose mass range is . On the contrary, here, we focus on non-evaporating PBHs with mass much larger than g. After the epoch of matter-radiation equality, gas and matter can accrete onto PBHs. It has been shown that PBHs with large mass could produce X-ray and UV photons through the accretion of matter onto PBHs and these photons heat up and ionize intergalactic medium (IGM) (Carr, 1981; Gnedin et al., 1995; Miller & Ostriker, 2001; Ricotti et al., 2008). Therefore, the heated and ionized IGM gas may produce an observable deviation of the 21 cm brightness temperature from the background even before the birth of the first stars and galaxies (). Our aim in this paper is to evaluate this deviation and to discuss the potential of 21 cm observations to constrain the non-evaporating PBH abundance.

The paper is organized as follows. In section 2, using a simple model of X-ray photon flux due to the accretion onto a PBH, we evaluate the ionization and temperature profile near a PBH. In section 3, we calculate the spin temperature and the brightness temperature induced by a PBH. In section 4, the angular power spectrum of 21 cm fluctuations due to PBHs are evaluated. Section 5 is devoted to the conclusion. Throughout this paper, we use parameters for a flat CDM model: , and . These parameters are consistent with WMAP results (Komatsu et al., 2011).

## 2 Ionization and Heating of IGM by a PBH

After the epoch of matter-radiation equality, matter can accrete onto a non-evaporating PBH, whose mass is larger than . Therefore, the accretion disk of a non-evaporating PBH can be a source of X-ray photons before first stars and galaxies form (). For example, the non-evaporating PBHs are one of the candidates for super massive black hole seeds. PBHs whose masses exceed cannot be directly formed through gravitational collapse since the time scale of the collapse becomes longer than cosmological time. Accordingly only the accretion after the formation, which induces X-ray photon emission, makes possible to form such massive black holes (Düchting, 2004; Khlopov et al., 2005). It is difficult to theoretically predict the accurate X-ray photon spectrum from PBHs, because the X-ray spectrum depends on the detailed condition of the accretion and the environment of PBHs such as the amount of neutral hydrogen. Hence, for simplicity, we assume that the PBH accretion powers a miniquasar with a power-law spectrum of X-ray photons, according to Kuhlen & Madau (2005),

 F(E)=AE−1 s−1, (1)

where is set to correspond to the tenth of the Eddington luminosity. Here we consider the range of the photon energy E from 200 eV to 100 keV since we assume that emitted photons whose energy are lower than 200 eV are immediately absorbed by the surrounding gas of the PBH. Following Zaroubi et al. (2007), we evaluate the ionization and heating of IGM due to massive PBHs in this section.

The number density of photons per unit time per unit area at distance from the source is given by

 N(E;r)=e−τ(E;r)A(4πr2)E−1cm−2s−1, (2)
 τ(E;r)=∫r0nHxHσ(E)dr, (3)

where is the hydrogen neutral fraction, is the mean number density of hydrogen at a redshift and is the absorption cross-section per hydrogen atom. In order to take into account the contribution from helium atoms as well as hydrogen atoms, we adopt the fitting formula by Zdziarski & Svensson (1989),

 σ=4.25×10−21(E250 eV)−p cm2,p={2.65for E<250 eV,3.30for E>250 eV. (4)

Using the function , we can write the ionization rate per hydrogen atom at the distance from the source as

 Γ(r)=∫∞E0σ(E)N(E;r)(1+EE0 ϕ(E,xe))dEE, (5)

where the term, , is introduced to consider the secondary ionization due to the photoelectrons produced by energetic photons (). We apply the fitting formula of by Dijkstra et al. (2004) for the low energy region keV and by Shull & van Steenberg (1985) for the high energy region keV.

The neutral fraction of hydrogen is obtained by solving the equation of the ionization-recombination equilibrium

 αHn2H(1−xH)2=Γ(r) nHxH, (6)

where is the recombination cross-section; ,

Solving Eq. (6), we show the neutral fraction of hydrogen for different masses and for different redshifts in the left and right panels of Fig. 1, respectively. Here we assume that the density around a PBH is the same as the mean density of the Universe. The comoving radius of the ionization sphere is made large by a massive PBH. Increasing the mass of the PBH means increasing the number of the ionization photons, because we assume that the flux of the ionization photons is proportional to a tenth of the Eddington luminosity. As the Universe evolves, the comoving radius of the ionization sphere increases slowly.

Next, we evaluate the kinetic temperature of the IGM around a PBH. The heating rate per unit volume per unit time at a distance from the source is obtained by considering the photons absorbed by the IGM at ,

 H(r)=fnHxH(r)∫∞E0σ(E)N(E;r)dE, (7)

where is the fraction of the photon energy absorbed through the collisional excitations of the IGM. Shull & van Steenberg (1985) provided a simple fitting formula , where , and is the ionized fraction .

The kinetic temperature of the IGM at a distance , , is determined by the balance between the heating and the Compton cooling due to CMB photons,

 H(r)=8σT3meT4γ(1−xH)(Tk(r)−Tγ)+2HTk(r), (8)

where is the cross section for the Compton scattering and is CMB temperature. Here we also take into account the cooling by the expansion of the Universe.

Fig. 2 shows the IGM kinetic temperature profiles for different masses in the left panel and for different redshifts in the right panel. Here we add the background kinetic temperature to the kinetic temperature in order to match both temperatures at a large distance from a PBH. Near the source, the temperature is determined by the heating rate and the Compton cooling rate. With increasing the distance from the source, the neutral fraction grows and the optical depth becomes larger. Accordingly, the number density of photons damps as shown in Eq. (2). As a result, the temperature starts to decrease rapidly. Because the Compton cooling depends on the number of free electrons, this cooling becomes ineffective at a distance where the neutral fraction of hydrogen becomes almost unity. For example, this scale corresponds to 0.1 comoving Mpc for a PBH with at . Beyond this point, the temperature mildly decreases due to the cooling of the cosmic expansion. The kinetic temperature at the inner side is independent on a PBH mass. However the region of the high temperature becomes larger as the PBH mass increases. In the right panel of Fig. 2, the redshift dependence of the kinetic temperature is shown. The larger the neutral hydrogen density is, the larger the heating efficiency becomes as in Eq. (7). Therefore the temperature becomes high as the redshift increases.

In this section, we obtained the neutral fraction of hydrogen by solving the equation of the ionization-recombination equilibrium, Eq. (6). However the recombination time scale is much smaller than the ionization time scale at the redshifts we are interested in. Therefore Eq. (6) might be incorrect. In this limit, we can neglect the recombination effect. Without the recombination term, we can write the evolution of the neutral fraction as

 dxHdt=−ΓxH. (9)

The solution of this equation is roughly . Although the radial distance where reaches is same as in the case of the ionization-recombination equilibrium, the resultant ionization profile has a sharper edge. This is because most of the ionization photons are absorbed to ionize the inner region. Accordingly, there are not enough ionization photons to ionize the outer region. This fact also means that the heating efficiency is suppressed as the radial distance increases, compared with the case of the ionization-recombination equilibrium. As a result, the heated region becomes one-fifth times smaller than in the previous case.

## 3 21 cm Brightness temperature due to a PBH

As shown in the previous section, a PBH ionizes and heats the surrounding region by the X-ray emissions. These ionization and heating leave an observable signature as the differential brightness temperature of the 21 cm intensity relative to the CMB temperature.

The 21 cm intensity depends on the ratio between the number density of neutral hydrogen in the excited state and in the ground state of the hyperfine structure. This ratio can be quantified by the spin temperature . The spin temperature is determined by the balance between three processes: absorption of CMB photons, collisional excitation in the kinetic temperature , and Lyman pumping (Wouthuysen, 1952; Field, 1958). In the steady state approximation between these processes, the spin temperature is obtained by (Field, 1958)

 Ts=T∗+Tγ+ykTk+yαTk1+yk+yα, (10)

where is 0.068 K which is the temperature corresponding to the energy difference between the levels in the hyperfine structure, and and are the kinetic and Lyman- coupling efficiencies, respectively. The kinetic efficiency is given by

 yk=T∗A10Tk(CH+Ce+Cp), (11)

where is the Einstein spontaneous emission rate coefficient, . The terms, , and , represent the de-excitation rates due to neutral hydrogen, electrons and protons, respectively. Here we use the fitting formula by Kuhlen et al. (2006),

 CH=nHκ,Ce=neγe,Cp=3.2npκ, (12)

where and are the electron and proton number densities, respectively. In Eq. (12), is the effective single-atom rate coefficient,

 κ=3.1×10−11nHT0.357kexp(−32/Tk) cm3s−1, (13)

and is given by for K and for K.

The Lyman- coupling efficiency is given by Field (1958),

 yα=16π2T∗e2f12J027A10Tkmec, (14)

where is the oscillator strength of the Lyman transition, and and are the electron charge and mass, respectively. In Eq. (14), is the flux of the Lyman photons due to collisional excitations. At the distance from the source, can be written as (Zaroubi et al., 2007)

 J0(r)=ϕαc4πH(z)ναnHxH(r)∫∞E0σ(E)N(E;r)dEhνα, (15)

where is the fraction of the absorbed energy going into the collisional excitation of Lyman . Shull & van Steenberg (1985) gave the following analytical form,

 ϕα≈0.48(1−(1−xH)0.27)1.52. (16)

Now we can calculate the spin temperature for a PBH according to the results in the previous section. We show the results in Fig. 3. The left panel shows the spin temperature for different masses of a PBH at and the right panel represents the spin temperature for different redshifts for .

In the highly ionized region, the profile of the spin temperature is flat and does not depend on the PBH mass as shown in the left panel of Fig. 3. Because the PBH can ionize surrouding gas, the spin temperature is determined by the CMB temperature and the term of the kinetic efficiency. As the neutral fraction goes up with increasing distance, however, the Lyman efficiency becomes larger so that the Lyman coupling begins to be effective. The peaks of at small radius for small PBH masses shown in the left panel are caused due to this effect. It should note that the peak values are also independent of mass since approaches for a large value of . Because the flux of Lyman- quickly decreases as the radius increases, the spin temperature also drops. Finally, the spin temperature settles down to the one determined by the background CMB and kinetic temperatures. The larger the PBH mass is, the larger the region where the spin temperature is higher than the background one becomes.

Because the kinetic temperature strongly depends on the redshift as shown in the right panel of Fig. 2, the amplitude of the spin temperature also has strong dependence on the redshift as shown in the right panel of Fig. 3. The spin temperature is almost same as the kinetic temperature in the ionized region at high redshifts (), since the kinetic efficiency dominates other contributions in Eq. (10). In low redshifts (), we find the region where the spin temperature is lower than the background one at the edge of the heated region. In these redshifts the background spin temperature is almost the CMB temperature, while the background spin temperature is between the CMB and kinetic temperatures in redshifts, . Since the kinetic temperature is lower than the CMB temperature at the edge, the Lyman- coupling is strong and draws the spin temperature toward the kinetic temperature. Therefore, there exists a region where the spin temperature becomes lower than the background one. This tendency appears even in Zaroubi et al. (2007).

The differential brightness temperature from the CMB temperature for a given spin temperature is obtained by (Ciardi & Madau, 2003)

 δTb=(20 mK)(1+δ)(xHh)(1−TγTs)(ΩBh20.0223)[(1+z10)(0.24ΩM)]1/2, (17)

where is the density contrast. Here we assume as mentioned in section 2. In Fig. 4, is shown as a function of the comoving radial distance from a PBH. The left panel is for different masses of a PBH at and the right panel is for different redshifts for . The is almost zero near the source, because hydrogen in such regions are totally ionized. As the neutral fraction increases, the brightness temperature also grows. The peak amplitudes of are independent on the PBH mass for . This is because the spin temperature is much larger than the CMB temperature, . Accordingly, from Eq. (17), becomes independent on and only depends on . This redshift dependence is shown in the right panel of Fig. 4. On the other hand, the peak amplitude becomes smaller for a mass of the PBH higher than in our model.

As in the case of , the region where the differential brightness temperature is below the background value exists near the edge at . Eventually, however, matches the background value at a large distance. The size of the region where we can detect becomes larger with increasing mass, and reaches almost 10 Mpc .

## 4 The angular power spectrum of 21 cm brightness temperature due to PBHs

In order to study the potential of cosmological 21 cm observations to give a constraint on the PBH abundance, we evaluate the angular power spectrum of 21 cm fluctuations due to PBHs with mass .

The observed at the direction is the contribution from the IGM along the line of sight. We can obtain at the observation frequency by the integration over the comoving distance .

 δTobs(^n,ν)=∫dr W(r,ν)δTb(r^n) (18)

where is the window function of a 21 cm observation with the frequency and is the differential brightness temperature at the position , which is produced by PBHs with mass .

In order to calculate the angular power spectrum, we employ the flat sky approximation and the Limber approximation, since we are only interested in the brightness temperature fluctuations on small scales (). In these approximation, the angular power spectrum can be written in terms of the three dimensional power spectrum of the brightness temperature (Dodelson, 2003),

 Cℓ(zobs)=∫drW2(r,ν)r2P(k,r), (19)

where is the redshift by choosing the observation frequency with MHz, and is the power spectrum of the brightness temperature fluctuations due to PBH with mass at .

We obtain the power spectrum of the brightness temperature fluctuations , following the halo formalism (Seljak, 2000). The power spectrum can be divided to two contributions; the Poisson contribution and clustering contribution. On large scales, since PBHs may be biased tracers of the linear matter power spectrum, they can make the clustering contribution. However, there is theoretical uncertainty in PBH bias. To avoid such uncertainty, we focus on only the Poisson contribution. Therefore, the power spectrum due to PBHs is expressed by

 P(k,r)=nPBH(M)|ΔT2b(k)|2, (20)

where is the comoving number density of PBHs with .

The comoving number density of PBHs is described by using the density parameter as

 nPBH(M)=ρcΩPBHM=1.36×10−2(ΩPBH10−9)(M104M⊙)−1 Mpc−3, (21)

where is the critical density at the present epoch. In Eq. (20), is the Fourier transform of the brightness temperature fluctuations from the background value,

 ΔTb(k)=4π∫x2dx(δTb(x)−δTb0)sin(xk)xk, (22)

where is the differential brightness temperature at the comoving radius from a PBH and is the background brightness temperature given by Eqs. (10) and (17) with the background kinetic temperature, .

Using obtained in the previous section as shown in Fig. 4, we calculate the angular power spectrum. The results are shown in Fig. 5. Here we set . For simplicity, we assume that the window function has a strong peak at where corresponds to the comoving radial distance at . We approximate the window function as . Since the promising signal of the cosmological 21 cm fluctuations is one due to the primordial density fluctuations, we also plot the angular power spectrum of those fluctuations obtained through CAMB (Lewis & Challinor, 2007).

In the left panel of Fig. 5, it is shown that the peak amplitude of the angular spectrum is independent on PBH’s mass. The peak locations shift toward large scales for massive PBHs. Moreover, the power law index of the spectrum matches the one due to the primordial density fluctuations on small ’s. The overall amplitude of the spectrum is simply proportional to or the PBH number density. For , it is shown that the angular spectrum of PBHs with mass matches with the one due to the primordial density fluctuations at . It is clear that the spectrum with heavier (lighter) PBHs becomes comparable with primordial density fluctuations’ one if is smaller (larger). To match these two spectra, we find the critical value of as for . The PBH spectrum dominates over the one due to the primordial density fluctuations if exceeds .

The right panel of Fig. 5 shows that, although the peak location does not depend on the redshift, the redshift dependence of the peak amplitude is different from that due to the primordial density fluctuations.

As the result, while the PBH contribution for is subdominant at , even the PBHs with roughly can produce the spectrum comparable with that due to the primordial density fluctuations at .

The brightness temperature profile at lower redshifts, , has both positive and negative peaks as shown in Fig. 4. The reason to have a negative peak is because there exists a region where the brightness temperature is lower than the background, which is rather difficult to see in the figure. Accordingly, the angular power spectrum due to PBHs also has two peaks at as shown in Fig. 5. The peak on a larger scale is due to the negative peak of , while that on a smaller scale is caused by the positive one.

At the last of this section, we discuss the constraint by a future observation such as the square kilometer array (SKA)-like interferometer. Measuring 21 cm anisotropies at high redshifts is a major challenge. The sky at corresponding frequencies is contaminated by foreground due to synchrotron emissions from the Galaxy and extragalactic sources. The noise power spectrum of an observation including the beam effects is given by (Knox, 1995)

 ℓ(ℓ+1)2πN21ℓ=ℓ(ℓ+1)tobsΔν(DλTsysAeff)2exp[ℓ(ℓ+1)ℓ2b], (23)

where is the effective area is the system temperature, is the observation time, is the frequency bandwidth, is the length of the baseline and is given by with the resolution . Here we adopt the current design of SKA; , hour, MHz and km. We set the system temperature to the sky temperature in minimum emission regions at high Galactic latitudes given by . In this design, the noise power spectrum is evaluated as

 ℓ(ℓ+1)2πN21ℓ∼0.9×105 mK2 (ℓ1000)2(1+z30)7.2exp[(ℓ2500z+130)2]. (24)

The noise power spectrum exponentially grows on larger multipoles than . The amplitude of the angular power spectrum due to the PBHs is scaled by . Accordingly we find that, in order to dominate over the noise spectrum, is required for at and is for at . Since the angular spectrum due to PBHs with mass less than has a peak on larger multipoles than in the SKA design, there is no opportunity to measure the anisotropy spectrum due to such small mass PBHs by SKA.

## 5 conclusion

We have investigated the 21 cm signal produced by massive PBHs whose masses are larger than . Assuming a power-law spectrum of X-ray photons from an accretion disk, we have studied the ionization and heating of IGM gas near a PBH and evaluated the differential 21 cm brightness temperature. We have shown that a PBH can induce an observable signal of differential 21  cm brightness temperature. The size of the region where we can find the differential brightness temperature typically reaches 1–10 Mpc for our interested PBH mass range. The exact size depends on the PBH mass.

We have also calculated the angular power spectrum of 21 cm fluctuations due to PBHs. The peak position of the angular spectrum depends on the PBH mass, while the amplitude is independent of the mass. Comparing this spectrum with the angular power spectrum caused by primordial density fluctuations, we have found that both of them become comparable if at and at for PBH’s mass from to . If the density parameter is larger than these values, the angular power spectrum due to PBHs exceeds the one from primordial fluctuations and can be measured. In other words, we cannot set constraints on the PBH density parameter below these values from 21 cm observations. If we consider the sensitivity of the SKA-like observation, for example, we can detect the signal of PBHs up to at and at for PBHs with mass from to .

The ionization of IGM due to PBHs with such density parameters does not affect the global reionization history of the universe since reionization from each PBH only covers a tiny patch of the universe. Unlike reionization from first stars, therefore, such reionization has little impact on CMB temperature anisotropies. On the other hand, PBHs can heat IGM regions whose scale reaches 1-10Mpc and 21 cm fluctuations are sensitive to the heated IGM regions. Accordingly the PBH density parameter constrained from WMAP data, that is (Ricotti et al., 2008), is several order of magnitude larger than the values at which 21 cm fluctuations from PBHs comparable with those from primordial density fluctuations as we mentioned above. In other words, we can conclude that, if observation instruments or foreground removal are more improved than the current SKA design, 21 cm fluctuation observations have a potential to probe the PBH abundance which is impossible to access by CMB observations.

The most theoretical uncertainty in this model is the flux of photons due to the accretion to PBHs. In this paper, we assume that the X-ray photon flux amplitude is a tenth of the Eddington luminosity and has the power law spectrum with for simplicity. We also studied the effect of the power law index by changing to . The temperature profile shifts to a smaller scale due to the decrease of the ionization efficiency. However this shift is small, and the resultant angular power spectrum does not change much.

The amplitude of the luminosity is considered to depend on the matter accretion rate onto a PBH. Ricotti et al. (2008) have studied the luminosity for the Bondi-Hoyle accretion in detail. Although the luminosity depends on the PBH mass and feedback effect on the ionization and temperature, they have shown that the luminosity for a PBH with is roughly a hundredth of the Eddington luminosity at . In our model, the brightness temperature profile near a PBH at a certain redshift depends only on the PBH’s mass and the amplitude of the angular power spectrum is scaled by . Therefore, if we assume that the amplitude of the X-ray photon flux is a hundredth of the Eddington luminosity, the required density parameter for PBHs to dominate the 21 cm fluctuations due to primordial density fluctuations is at and at for PBH’s mass from to .

## Acknowledgments

We thank A. Long for useful comments. H. T. is supported by the DOE at Arizona state university. N. S. is supported by Grand-in-Aid for Scientific Research No. 22340056. This research has also been supported in part by World Premier International Research Center Initiative, MEXT, Japan.

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