1 Introduction

Primordial black holes (PBHs) could form via the collapse of large density perturbations exist in the radiation dominated universe. It is well known that more massive black hole lives longer \((\tau \propto M_{\textrm{BH}}^{3}).\) Therefore, taking into account the current age of the Universe, \(t\sim 13.7\,\text {Gyr}\), PBHs with masses of \(M_{\textrm{PBH}}< 10^{15}\,\text {g}\) have evaporated, while larger PBHs should still exist in the present universe [1,2,3,4,5,6]. Recently, LIGO and Virgo detected the gravitational wave signals caused by the emerges of black holes [7]. It has been suggested that some of these signals would be from PBHs, which may constitute a fraction of DM depending on their masses [8,9,10,11,12,13,14,15,16,17].

Although DM has been confirmed by many different astronomical observations, its nature remains a mystery [18, 19]. As a popular DM model, weakly interacting massive particles (WIMPs) has been studied widely, and they can annihilate into such as photons, electrons and positrons (see, e.g., Refs. [20, 21] for a review). In the mixed DM scenarios consisting of PBHs and WIMPs, WIMPs can be accreted onto PBHs forming a kind of DM structure named ultracompact minihalos (UCMHs) with a density profile \(\rho _\textrm{DM}(r)\sim r^{-9/4}\) [22, 23]. Since the annihilation rate of WIMPs is proportional to the square of the number density, \({\textrm{Gamma}} \propto n_{\textrm{DM}}^2\), it is expected that the annihilation rate of DM in UCMHs is larger than that of classical DM halos.Footnote 1 The gamma-ray flux from UCMHs due to DM annihilation would have contributions to the relevant observations, and those observations can be used to investigate the cosmological abundance of PBHs [24,25,26,27,28]. Moreover, the particles emitted from UCMHs due to DM annihilation have interactions with that exist in the Universe, leading to the changes of the thermal history of the Universe. The relevant measurements, such as the cosmic microwave background (CMB) and global 21cm signals, can be used to investigate these effects and derive the upper limits on the cosmological abundance of PBHs [24, 27, 29].

It is known that the extra energy injected into the Universe during the early epoch can cause the deviation of the CMB from blackbody spectrum [2, 29, 30, 30,31,32,33,34,35,36]. In the radiation dominated universe, the mass of a UCMH keeps unchanged until the time of matter-radiation equality (\(z_\textrm{eq}\sim 3387\)) [22]. The energy released from UCMHs due to DM annihilation results in the CMB y-type distortion, and the upper limits on the distortion measured by the Far Infrared Absolute Spectrophotometer experiment can be used to constrain the cosmological abundance of PBHs [37]. In addition to affecting the CMB blackbody spectrum, the extra energy injection has influence on the photodissociation of \(^{4}{\textrm{He}}\) in the early universe [30, 38]. The observed abundance of \(^{3}{\textrm{He}}\) and D, produced through dissociating the \(^{4}{\textrm{He}}\) nuclei, can be used to investigate the cosmological abundance of PBHs. Here we will investigate the influence of the energy released from UCMHs due to DM annihilation on the photodissociation of \(^{4}{\textrm{He}}\). Taking into account the measured upper limit on the ratio \({}(^3{He}+D)/H<1.10 \times 10^{-4}\) [39, 40], we will derive the upper limits on the cosmological abundance of PBHs.

This paper is organized as follows. In Sect. 2 we briefly review the basic properties of UCMHs, and then derive the upper limits on the fraction of DM in PBHs using the measured ratio \({}(^3{He}+D)/H\). The conclusion is given in Sect. 3.

2 The properties of UCMHs and upper limits on the cosmological abundance of PBHs

2.1 The basic properties of UCMHs

It is well known that the cosmological large scale structures are from the density perturbations exist in the early universe with a amplitude of \(\delta \rho /\rho \sim 10^{-5}\). Large density perturbation, e.g., \(\delta \rho /\rho > 0.3\), would result in the formation of PBHs [2]. It has been argued that the density perturbation in the range of \(10^{-4}<\delta \rho /\rho <0.3\) could result in the formation of UCMHs [22]. However, the simulations have shown that the direct collapse of these large density perturbations is not suitable for the formation of UCMHs [41, 42]. On the other hand, it is found that the UCMHs can be formed through the accretion of DM particles onto PBHs after their formation [23, 28, 43, 44]. For our purpose, we will not refer to the detailed formation mechanism of UCMHs and adopt the latter scenario.

After the formation of PBHs, they can accrete DM particles onto them, resulting in the formation of UCMHs. The density profile of DM particles in a UCMH is in the form of [27]

$$\begin{aligned} \rho _{\textrm{DM }}(r,\!z)\!=\!\left\{ \begin{array}{ll} \rho _{\textrm{max}}(z)&{}\quad \!\! {r\!\le \! r_{\textrm{cut}}(z)}\\ \rho _{\textrm{max}}(z)\left( \frac{r}{r_\textrm{cut}(z)}\right) ^{-9/4}&{}\!\!\quad {r_{\textrm{cut}}(z)\!<\! r\le r_{\textrm{ta}}(z_\textrm{eq})} \end{array} \right. \nonumber \\ \end{aligned}$$
(1)

where \(\rho _{\textrm{max}}(z)\) is inner core of UCMH taking into account the DM annihilation [45],

$$\begin{aligned} \rho _{\textrm{max}} = \frac{m_{\chi }}{\left<\sigma v\right>(t-t_{i})}, \end{aligned}$$
(2)

where \(t_i\) is the formation time of UCMHs. \(m_{\chi }\) and \(\left<\sigma v\right>\) are the mass and thermally averaged annihilation cross section of DM particles, respectively. \(r_\textrm{cut}(z)\) is the radius of inner core [27, 43],

$$\begin{aligned} r_{\textrm{cut}}(z)=\left( \frac{\rho _{\textrm{max}}(z)}{{\bar{\rho }}_\textrm{DM}(z_{\textrm{eq}})}\right) ^{-4/9}r_{\textrm{ta}}(z_{\textrm{eq}}) \end{aligned}$$
(3)

where \({\bar{\rho }}_{\textrm{DM}}\) is the mean density of DM, \(r_{\textrm{ta}}\) is the turnaround scale of UCMH [27, 43],

$$\begin{aligned} r_{\textrm{ta}}(z) \approx (2GM_{\textrm{PBH}}t^{2}(z))^{1/3}. \end{aligned}$$
(4)

Following the work of [27], here we neglect the DM kinetic energy compared with the potential energy, therefore, the mass of PBH considered by us is in the range of

$$\begin{aligned} M_{\textrm{PBH}}\ge 2.2\times 10^{-4}\mathrm{M_{\odot }}\left<\sigma v\right>_{26}^{-1/3}m_{\chi ,10}^{-73/24}, \end{aligned}$$
(5)

where \(\left<\sigma v\right>_{26} {=}\left<\sigma v\right>/10^{{-}26}\,\text {cm}^{3}\,\text {s}^{{-}1}\) and \(m_{\chi ,10} {=} m_{\chi }/10\,\text {GeV}\).

2.2 Upper limits on the cosmological abundance of PBHs

The annihilation rate of DM particles within a UCMH can be written as [24, 46],

$$\begin{aligned} \Gamma _{\textrm{anni}}(z)= & {} \frac{1}{2}\int n^{2}_{\textrm{DM}}(r,z)\left<\sigma v\right> 4\pi r^{2}dr\nonumber \\= & {} 2\pi \frac{\left<\sigma v\right>}{m_{\chi }^2}\int \rho _{\textrm{DM}}^{2}(r,z)r^{2}dr. \end{aligned}$$
(6)

The energy injection rate of DM annihilation per unit volume can be written as,

$$\begin{aligned} {\textrm{E}}_{\textrm{inj}}(z)= & {} n_{\textrm{PBH}}(z)2m_{{\chi }}{\Gamma }_{\textrm{anni}}(z) \nonumber \\= & {} \frac{2m_{\chi }}{M_{\textrm{PBH}}}{\rho _{c,0}}{\Omega }_{\textrm{PBH}}(1+z)^{3} {\Gamma }_{\textrm{anni}}(z) \end{aligned}$$
(7)

where \(n_{\textrm{PBH}}(z)=\rho _{\textrm{PBH}}(z)/M_{\textrm{PBH}}\) is the number density of PBH at redshift z. \({\Omega }_{\textrm{PBH}}=\rho _{\textrm{PBH}}/\rho _{c,0}\) and \(\rho _{c,0}\) is the current critical density of the Universe.

The energy injected into the Universe from UCMHs due to DM annihilation will result in the photodissociation of \(^{4}{\textrm{He}}\), producing the \(^{3}{\textrm{He}}\) or the D. Taking this issue into account, the ratio \((^3{\textrm{He}}+\text {D})/\text {H}\) can be written as [30],

$$\begin{aligned} \frac{^3{\textrm{He}}+\textrm{D}}{\textrm{H}}=\int \frac{E_{\textrm{inj}}}{n_{\textrm{H}}}\left[ N_{\textrm{D}}(z)+N_{^3{{\textrm{He}}}}(z)\right] dt \end{aligned}$$
(8)

where \(N_{\textrm{D}}(z)\) \((N_{^3{\textrm{He}}}(z))\) is the number of D \((^3{\textrm{He}})\) for a given amount of injected energy at redshift z, and we will use the results given in Ref. [47]. For the calculations, we will use the relation of \(dt=1/H(z)(1+z)dz\), where \(\textrm{H}(z)=\textrm{H}_{0}\sqrt{{\Omega }_{\textrm{m}}(1+z)^{3}+ {\Omega }_{\mathrm{\Lambda }}+{\Omega }_{\gamma }(1+z)^4}.\)

The ratios \({}^3{\textrm{He}}/H\) and \({\mathrm{D/H}}\) have been measured in many celestial systems [2, 39, 48, 49]. Here we use the upper limit on the ratio \((^3{\text {He}}+\text {D})/\text {H}<1.10 \times 10^{-4}\) for our calculations [39]. By requiring that the ratio calculated with Eq. (8) does not exceed the measured value, one can get the upper limits on the cosmological abundance of PBHs. The constraints on the fraction of DM in PBHs, \(f_{\textrm{PBH}}={\Omega }_{\textrm{PBH}}/{\Omega }_{\textrm{DM}}\), are shown in Fig. 1 (red solid lines). Here we have set the thermally averaged annihilation cross section of DM as \(\left<\sigma v\right>=3\times 10^{-26}~ \text {cm}^{3}\,\text {s}^{-1}\). We found that the upper limit is \(f_{\textrm{PBH}} < 0.35(0.75)\) for DM mass \(m_{\chi }=1(10)~\text {GeV}\). Since the energy injection rate of DM annihilation (Eq. (6)) is lower for larger DM mass, the limit is weaker for larger DM mass. On the other hand, the energy injection rate of DM annihilation per unit volume is independent on the PBH mass, \({\textrm{E}}_{\textrm{inj}}\propto n_{\textrm{PBH}}{\Gamma }_{\textrm{anni}}\propto {\textrm{M}}_{\textrm{PBH}}^{-1} {\textrm{M}}_{\textrm{PBH}}\), therefore, the limits are independent on the PBH mass for our considerations. Note that since we have neglect the DM kinetic energy compared with the potential energy for the formation of UCMHs around PBHs, there is a cut off for the upper limit depending on the DM mass (Eq. (5)).Footnote 2

In the mixed dark matter scenarios, the fraction of DM in PBHs can be constrained by many different astronomical measurements [3, 4, 24, 27, 28, 50, 51]. In Fig. 1, we also plot several other upper limits on \(f_{\textrm{PBH}}\) for comparison.Footnote 3 The extra energy injected into the early universe can affect the distribution of photons, resulting in the deviation of the CMB from blackbody spectrum. For the energy released from UCMHs due to DM annihilation, CMB y-type distortion is caused [29] and the upper limit on the distortion have been measured by the Far Infrared Absolute Spectrophotometer experiment (FIRAS) [37]. In previous work, we have used the measured results to constrain the fraction of PBHs [29], and the upper limit is shown in Fig. 1 for DM mass \(m_{\chi }=1\,\text {GeV}\) (cyan dashed line, labelled ‘CMB distortion’). The upper limit from the ratio \(\mathrm (^3{He}+D)/H\) is weaker by a factor of \(\sim 18\) than that from the CMB y-type distortion.Footnote 4

As mentioned above, the DM annihilation rate in UCMHs is higher than that in classical DM halos. Therefore, it is expected that they have significant contributions to the extragalactic \(\gamma \)-ray background (EGB) [24,25,26, 50, 61]. The cosmological abundance of PBHs can be constrained using the EGB data measured by the Fermi-LAT [62, 63]. The authors of [28] investigated this issue and derived a upper limit of \(f_{\textrm{PBH}}<10^{-11}\) for \(m_{\chi }=10\) GeV. The released energy from UCMHs due to DM annihilation can also inject into the intergalactic medium, affecting the thermal history of the Universe and resulting in the changes of the anisotropy of CMB. By using the Planck-2018 data, the authors of [27] found that the upper limit is \(f_{\textrm{PBH}}<3\times 10^{-10}\) for \(m_{\chi }=10\) GeV. Note that these upper limits are not shown in Fig. 1, where we have set the range of \(f_{\textrm{PBH}}\) as \([10^{-5},1]\).

Note that the constraints on \(f_{\textrm{PBH}}\) here depend on the measured upper limits on the ratios \({}^3{\textrm{He}}/H\) and \({\textrm{D}}/{\textrm{H}}\). Compared with the ratio \({}^3{\textrm{He}}/H\), \({\mathrm{D/H}}\) is less effected by the cosmological evolution and can be determined well basing on, e.g., the observations and analysis of DI and HI lines from damped Lyman-\(\alpha \) systems [64,65,66,67]. The ratio \({}^3{\textrm{He}}/H\) is less constraining due to the influences of stars and few observations in the Galactic disk [39, 64, 68,69,70,71]. For the future determination of the ratio \(\text {D}/\text {H}\), more damped Lyman-\(\alpha \) systems are needed for decreasing the errors, which can be achieved by, e.g., the next generation of 30-m class telescopes [64]. The situation is slightly more complicated for the ratio \({}^3{\textrm{He}}/H\). In the future, an better understanding of stellar nucleosynthesis models and more observations in the Galactic disk are the key to improve the accuracy of the ratio \({}^3{\textrm{He}}/H\) [64, 67].

Fig. 1
figure 1

Upper limits on the fraction of DM in PBHs, \(f_{\textrm{PBH}}={\Omega }_{\textrm{PBH}}/{\Omega }_{\textrm{DM}}\). The limits from the ratio \((^3{\text {He}}+\text {D})/\text {H}\) are shown for the DM mass \(m_{\chi }=1\) and 10 GeV (red solid lines). The constraints from several other measurements are shown for comparison: (1) the limits from the CMB y-type distortion measured by the FIRAS experiment (labelled ‘CMB distortion’) [29]; (2) the limits from the estimation on the merger rate of PBHs in light of the sensitivity of LIGO/Virgo (labelled ‘LIGO/Virgo’) [57]; (3) the limits from the investigation of the dynamical evolution of stars in the dwarf galaxy Segue I (labelled ‘Segue I’) [58]; (4) the constraints from the studies on the anisotropy of CMB caused by the accreting PBHs with Planck data (labelled ‘Planck’) [59]; (5) the constraints from the studies on the gravitational lensing results measured by EROS (labelled ‘EROS’) [60]

Full size image

3 Conclusion

We have derived the upper limits on the fraction of DM in PBHs in the mixed dark matter scenarios consisting of PBHs and WIMPs. In this scenarios, a PBH accrete WIMPs to form a UCMH with a density profile of \(\rho _{\textrm{DM}}(r)\sim r^{-9/4}\). Compared with the classical DM halos, the formation time of UCMHs is earlier and the annihilation rate of DM in UCMHs is larger. It is expected that the energy released from UCMHs due to DM annihilation has significant influences on different astrophysical processes. We have investigated these effects on the ratio \((^3{\text {He}}+\text {D})/\text {H}\) produced through dissociating the \(^{4}{\textrm{He}}\) nuclei. By requiring that the ratio caused by the DM annihilation in UCMHs does not exceed the measured value of \((^3{\text {He}}+\text {D})/\text {H}<1.10 \times 10^{-4}\), we obtained the upper limits on the fraction of DM in PBHs. We found that the upper limit is \(f_{\textrm{PBH}} < 0.35(0.75)\) for DM mass \(m_{\chi }=1(10)~\text {GeV}\). Compared with other limits obtained by different astronomical measurements, although our limit is not the strongest, we provide a different way of constraining the cosmological abundance of PBHs. On the other hand, the limits on \(f_{\textrm{PBH}}\) can be stronger with the accurate measurements of the ratios \({}^3{\textrm{He}}/H\) and \(\text {D}/\text {H}\), which can be achieved by the future more observations of the related systems and better understanding of stellar nucleosynthesis models.