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Primordial Black Holes
Teruaki Suyama
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DOI: 10.22661/AAPPSBL.2018.28.6.02

Primordial Black Holes

TERUAKI SUYAMA
DEPARTMENT OF PHYSICS, TOKYO INSTITUTE OF TECHNOLOGY

ABSTRACT

Primordial black holes are hypothetical black holes that might have formed right after the Big Bang of the Universe. Tests of their existence in nature bring us valuable information about the origin of our Universe.

In this article, I will give an introductory presentation of the primordial black holes that covers their formation, observational constraints, recent proposals of the primordial black holes as the sources of the LIGO events, and various ideas to test the primordial black holes in the era of the gravitational-wave astronomy.

INTRODUCTION

In 1971, Steven Hawking proposed the idea that when the age of the Universe was less than a second, highly overdense regions in the Universe underwent gravitational collapse and directly formed black holes [1]. Such black holes, known as primordial black holes (PBHs), do not originate from the end stage of the stars but are formed by the direct collapse of radiation (photons, neutrinos, leptons, quarks etc.) filling the Universe at that epoch. Despite the non-detection of PBHs after various observational efforts for over 40 years since Hawking's original proposal, PBHs are still important in cosmology. There are a couple of reasons that explain why this is so.

First, in modern cosmology, the primordial density perturbations, which are the seeds of the structures in the present Universe, are thought to have been created during the inflationary era. The inflationary era was a period of rapid accelerated expansion of space preceding the radiation dominated era. Currently, there are many inflation models. As will be explained later, if the primordial density perturbations are inhomogeneous enough, PBHs are formed out of the primordial density perturbations. Therefore, if PBHs are discovered by observations in the future, then inflation models will be narrowed down to those that form PBHs. In this way, PBHs provide a probe of the early Universe which is independent of the observations of the temperature fluctuations of the cosmic microwave background radiation. Conversely, if the existence of PBHs is ruled out by the observations, then the inflation models predicting PBHs would be excluded; therefore, evidence excluding the existence of PBHs in the Universe would also be useful in cosmology.

The second reason that PBHs are still important in cosmology is that PBHs behave as cold dark matter. The non-detection of new particles and new physics beyond the standard model of particle physics by terrestrial experiments, in addition to no astrophysical hints of dark matter in the form of elementary particles, have continuously motivated us to consider PBHs as the true identity of cold dark matter. An advantage of the idea of PBHs being dark matter is that there is then no need to introduce a new physics.

The third reason for why PBHs are still important in cosmology is that gravitational-wave (GW) astronomy, which was initiated spectacularly with the detection of GWs from the merger of a BH binary by LIGO [2], would definitely provide novel methods to touch parameter space of PBHs, which has been impossible to explore by means of existing electromagnetic observations. Quite interestingly, it is even possible that the BH binaries detected by LIGO are PBHs, meaning that we might have already detected PBHs for the first time!

The fourth reason why PBHs are still important is due to the observations of supermassive BHs. It is known that almost all galaxies contain BHs heavier than ~105 solar mass (M) in the center. Furthermore, a BH heavier than ten billion solar mass is known to have already existed when the Universe was younger than one tenth of the current age. There are no established astrophysical explanations about their origins. A natural question is then that if it is possible that those supermassive BHs are PBHs.

In what follows, I will explain the basic properties of PBHs and the current constraints on PBH abundance and introduce recent studies on PBHs that appeared after LIGO's detection of the binary BHs. Because of the limitation of pages, this article is not exhaustive. Readers who want to know in more detail are recommended to read a recent review article [3] written by the author of this article and his collaborators. Since reference papers are enormous, I decided to cite only papers that are either extremely fundamental or appeared after the review article [3] was written. Relevant references not cited in this article can be found in [3].

FORMATION OF PBHS

Among the various formation mechanisms of PBHs proposed in the literature, the direct gravitational collapse of primordial density perturbation is the most popular scenario, and I will focus on this. If the relative amplitude of the density perturbations of a particular overdense region exceeds a threshold value, which is about unity, such a region satisfies the Jeans criterion for the gravitational collapse and starts to contract by its own gravity against the radiation pressure at around a time when its size becomes equal to the Hubble horizon and directly turns into a BH (PBH) within the free fall time [4]. Contrary to the astrophysical BHs, which are heavier than the mass of the Sun, PBH mass could be much smaller. The mass of the resultant PBH is simply proportional to the time when the PBH formed. For instance, ten solar mass PBHs are formed when the age of the Universe is merely about millisecond. Depending on the length scale of the overdense region, the PBH mass could be as low as 1 grms.

It may sound like a stupid story that BHs, which could be even heavier than the Sun, are formed when the age of the Universe is less than a second. But, as long as general relativity is correct, PBH formation necessarily occurs whenever there is an overdense region having order unity fluctuations. The non-trivial point of PBH formation is how to realize such overdense regions. According to the theory of inflation, the primordial density perturbations are instantaneously generated by inflation over a wide length scale ranging from the ten billion light years, the current Hubble horizon, down to meter scales (Fig.1). Using cosmological observations such as cosmic microwave background radiation, their amplitudes have been well measured over scales ranging from the current Hubble horizon down to a million light years. Those observations suggest that the amplitudes are too small to produce PBHs corresponding to those scales. Yet, cosmological observations cannot explore the primordial density perturbations whose length scales are below a million light years. In terms of the PBH mass, PBHs that are lighter than a trillion solar mass are not excluded by the cosmic microwave background radiation. Thus, detections/non-detections of PBHs give us independent constraints on inflation models.

 

Fig. 1: Amplitude of the primordial density perturbations at various scales. Top horizontal axis is the corresponding PBH mass. Amplitude on large scales (red) are known by the cosmic observations such as the cosmic microwave background and the large-scale structure. Density perturbations relevant to PBH formation are on short scales, which are not constrained by the cosmic observations.

If the mean amplitude of the density perturbations at certain length scales is order unity, then PBHs were efficiently produced and quickly dominated the Universe substantially before the observationally determined time of matter-radiation equality, where radiation becomes a subdominant component. Thus, in order to be compatible with our Universe, PBH formation must be rare. In other words, PBHs are formed only at the extremely high-σ peaks (typically ~10σ) of the density perturbations.

OBSERVATIONAL CONSTRAINTS ON PBHS

If a PBH is lighter than 1015 grms, which is approximately equivalent to the total mass of all mankind, then the PBH evaporates due to the Hawking radiation by the present age of the Universe. Thus, such tiny PBHs do not exist in the present Universe. Yet, they leave characteristic traces that can be used to investigate how many such PBHs could have existed in the early Universe. For instance, PBHs in the mass range 109-1012 grms change an abundance of light elements produced by Big Bang nucleosynthesis due to high energy particles emitted by the evaporating PBHs. Comparison between the observed light elements and the theoretical prediction tightly constrains the abundance of such PBHs [5].

PBHs heavier than 1015grms do not evaporate and remain in the present Universe. Since PBHs move only at a non-relativistic speed and interact only gravitationally, they behave as cold dark matter. At the level of this argument, PBHs could be considered as a candidate for dark matter. However, non-evaporating PBHs also produce various peculiar phenomena such as gravitational lensing, disruption of astrophysical systems, and accretion of baryonic gas. The answer to the question of which physical process becomes the most prominent depends on the PBH mass. Non-observations of such phenomena have placed upper limits on the PBH abundance for various PBH mass ranges. In the rest of this article, I will focus on the non-evaporating PBHs.

Gravitational lensing
Gravitational lensing is a very powerful method to constrain/detect PBHs. PBHs act as lenses and magnify the luminosity of background objects such as stars. An excellent aspect of the gravitational lensing is that the individual lensing signal is solely based on gravitational physics and does not suffer from the uncertainties that exist in the studies of electromagnetic signals, which arise due to the interaction between the PBHs and the surrounding matter. Characteristics of the observational signals depend on the PBH mass, and different experiments are sensitive to different PBH masses.

 

Fig. 2: Upper limits on the fraction of PBHs in dark matter from various observations. All PBHs are assumed to have the same mass.

Fig. 2 shows the upper limits on the fraction of PBHs in dark matter as a function of the PBH mass, using the assumption that all PBHs have the same mass. Curves labeled as Femtolensing, HSC, Kepler, Caustic, EROS, OGLE, and Millilensing are the limits derived by gravitational lensing. It is clear that the gravitational lensing constrains PBH over a wide mass range from 1017 grms to billion solar mass. Beside the Femtolensing constraint, which uses an interference pattern of gamma-rays emitted by gamma-ray bursts, all the other gravitational-lensing constraints are obtained under the geometrical optics approximation. However, the wave nature of light becomes effective for the left half of the HSC constraint, and it is currently believed that the HSC constraint in the mass range 1019-1023 grms no longer holds. A recent study [6] shows that the previous Femtolensing constraint is removed after taking into account the wave nature of gamma-rays and the finite size effects of the gamma-ray bursts. To summarize, non-evaporating PBHs lighter than 1023 grms are still allowed to comprise all dark matter.

Dynamical constraints
PBHs dynamically affect any astrophysical system by their gravitational interactions. If there are abundant PBHs with certain mass, they destroy the known astrophysical systems efficiently. The basic idea behind dynamical constraints is that existence of astrophysical systems is incompatible with the existence of PBHs and this can be used to place an upper limit on PBH abundance. Until recently, various astrophysical systems have been considered in this context. They are shown in Fig. 2 as WD (disruption of white dwarfs), NS (disruption of neutron stars), UFDs (disruption of ultra-faint galaxies), Eri-II (disruption of star cluster), WB (disruption of wide binaries), and DF (dynamical friction). While the figure demonstrates the usefulness of the dynamical constraints, it is important to keep in mind that each constraint suffers from different underlying assumptions, some of which may not hold true. For instance, the NS constraint assumes there are PBHs as dark matter at the cores of globular clusters and it is not known observationally whether dark matter exists in such regions.

Accretion constraints
Accretion of gas onto the PBHs and its impact on the constraint of the PBH abundance has also been a subject of research. Radiation emanating from the accreting gas onto the PBHs in the early Universe leads to the distortion of the spectrum in the cosmic microwave background (CMB). PBHs in the present Universe also produce strong radiation due to the accretion. Non-observations of the effects caused by the radiation place upper limits on the PBH abundance. A characteristic point of the accretion constraints is that the corresponding PBH mass range is around the stellar mass. This means that the accretion provides a one possible method to test if the BHs detected by LIGO are PBHs. The accretion constraints are shown as red curves in Fig. 2. Curves labeled by CMB and CMB-II are derived from the accretion effects on CMB under different models for the accretion processes. Other accretion constraints are derived for the accretion in the present Universe. Compared to the lensing and the dynamical constraints discussed above, physics is much more involved in the case of the accretion constraints. Currently, it is impossible to derive the accretion constraints from a first-principles calculation, and all the accretion constraints make some assumptions or rely on the observationally established empirical rules.

To summarize the constraints on the PBH abundance, taking into account the uncertainties of the underlying assumptions for the accretion and the dynamical constraints, it would be conservative to say that two mass ranges, 1016-1023 grms and 10-100M, remain as potential windows for PBHs as a whole dark matter.

Indirect constraints
The constraints explained above are direct in the sense that all the effects are directly triggered by the PBHs. One advantage of the direct constraints is that they do not resort to the formation mechanism of PBHs. Any model predicting the PBH formation must satisfy the direct constraints. In addition to the direct constraints, there are indirect ones. As already mentioned, the most popular scenario for PBH formation is direct gravitational collapse of the rare high-σ peaks of primordial density perturbations. Although other regions are not inhomogeneous enough to produce PBHs, they are still inhomogeneous enough to induce effects that yield the observable signals. Those effects are not sourced by the PBHs but by the density perturbations that seed the PBHs, and hence the resultant constraints on the PBHs are indirect.

The first such effect is the emission of incoherent GWs from the acoustic oscillations of the primordial density perturbations. Such GWs form the stochastic GW background and are filling the present Universe. The peak frequency of the GW spectrum is related to the PBH mass as

In particular, stellar mass PBHs accompany the stochastic GW background in the nano-Hertz band. Quite interestingly, such low-frequency GWs are severely constrained by pulsar timing. Pulsars, which are believed to be rotating neutron stars, are extraordinarily precise clocks. Namely, consecutive pulses from the pulsar arrive at the Earth with the same interval. If GWs are present between the pulsars and the Earth, modulation of the distance to the pulsar results in the slight distortion of the pulse intervals. Conversely, monitoring the pulse intervals enables to place an upper limit on the amplitude of GWs in the nano-Hertz band, which is translated into the upper limit on the PBH abundance in the stellar mass range. Although the constraint depends on the shape of the power spectrum and the statistical properties of the primordial density perturbations, pulsar timing has the strong potential to place stringent constraints on inflation models yielding PBHs. Recently, an interesting idea has been proposed [7,8,9] that PBHs in the mass range 1020-1022 grms can be tested by searching the stochastic GWs by LISA, a planned space laser interferometer.

The second indirect constraint can be obtained from generation of the CMB spectral distortion of the primordial density perturbations. Acoustic oscillations of the primordial density perturbations eventually dissipate its energy into the background radiation through photon diffusion and produce deviation of the CMB spectrum from the Planckian distribution. This occurs for the PBHs in the mass range between 104-1013M. Non-observations of such a derivation by the COBE (Cosmic Background Explorer)/FIRAS(Far-InfraRed Absolute Spectrophotometer) experiment exclude the presence of (even a single) PBHs in the above mass range if the primordial density perturbations obey Gaussian statistics. This constraint becomes weakened for the non-Gaussian primordial perturbations.

PBHS AS SUPERMASSIVE BHS

As mentioned in the introduction, the origin of supermassive BHs remains an unresolved problem. One could then ask if PBHs can explain the observed supermassive BHs. One good point of the PBH scenario is that we only need very simple physics to form such heavy PBHs, that is, the direct collapse of the primordial density perturbation. As long as there are high-amplitude primordial perturbations at a corresponding scale, PBH formation is inevitable. It's so simple!

Yet, things never work out as one would expect. As explained in the last section, formation of supermassive PBHs is necessarily associated with the spectral distortion of CMB, which largely contradicts with the measurements. One can, in principle, circumvent the CMB constraint by considering highly non-Gaussian primordial density perturbations. Realizing such perturbations in inflation requires contrived inflation models.

PBHS AS SOURCES OF GWS

Direct detection of GWs by LIGO opened a completely novel arena for PBH research. As of October 2018, five mergers of the BH binaries have been found by LIGO/Virgo. We now know that BH binaries that merge within the age of the Universe are common in the Universe. After LIGO, elucidating when and where such BHs were born and how they formed binaries has become a hot topic in astrophysics and cosmology.

The mass and spin of BHs can be estimated from the GW waveforms. The observed data indicate at least the following two interesting features. First, the BHs tend to be heavier than those indirectly suggested by electromagnetic wave observations. Second, the orbital angular momentum of the binary appears not to be aligned with the spin of the individual BH, and the spin of each BH seem to be small. These observations may provide useful hints to clarify the origin of the LIGO BHs.

Soon after LIGO announced the first detection of a BH merger in February 2016, several groups pointed out the possibility that the observed BHs were PBHs. Thus, an exciting scenario has arisen where we might have discovered PBHs for the first time by the direct observation of GWs.

Explaining the LIGO events by the PBHs is not trivial in two aspects. First, since the GWs were emitted from the BH binaries, the formation mechanism of the PBH binary must be identified in order to test the PBH scenario with GW observations. Second, as discussed in the previous section, there are existing constraints on the PBH abundance for the mass around the observed BH mass. It needs to be checked if the PBH scenario does not conflict with those constraints. In what follows, I will explain how these issues are resolved in the PBH scenario.

Formation of the PBH binaries
The physical mechanism of the formation of the PBH binaries was proposed in 1990s. At that time, compact objects called MACHOs with a mass of about 0.5M were discovered by the microlens phenomenon, and PBHs were considered as one possibility to explain the phenomenon. Then, GWs from the PBH binaries were investigated as one method to test the PBH scenario for the MACHOs.

At the time of their formation, PBHs are rare and distributed sparsely. In other words, the PBH mean distance is initially much larger than the Hubble horizon. Given that PBHs are formed out of random fluctuations, it is natural to assume that initial spatial distribution of PBHs obeys the Poisson distribution. All PBHs are initially on the flow of cosmic expansion. However, for any close pair of PBHs whose distance is below a certain threshold, the gravity acting between them eventually defeats the cosmic expansion and the two PBHs begin to approach each other. This approach occurs at the free-fall time. During this time, the surrounding PBHs, especially the nearest one, exert torques on the bound system. As a result, the two PBHs avoid a head-on collision and form typically a highly eccentric binary. By assuming the Poisson distribution of PBHs, it is possible to derive the probability distribution of the major axis and the eccentricity of the binaries at the formation time.

Merger rate of the PBH binaries
Given the probability distribution of the initial binary parameters, using the formula for the life time of the binaries due to GW emission, we can derive the merger rate of the PBH binaries per unit time and unit volume, which can be directly compared with the merger rate given by LIGO. A red curve in Fig. 3 shows the expected merger rate in the PBH scenario as a function of the fraction of PBHs in dark matter, under the assumption that all PBHs are 30M, which is approximately the mass of the first BHs discovered by LIGO. We find that the merger rate largely exceeds LIGO's observations when PBHs dominate the total dark matter and lie in the band estimated by LIGO if a PBH fraction is about a thousandth part. This constraint is severer than the existing constraints and vividly demonstrates the power of GW astronomy for PBH research.

 

Fig. 3: Predicted merger rate of the PBH binaries as a function of the fraction of PBHs in dark matter. Gray band is the merger rate estimated by LIGO. It is assumed that all PBHs have 30M.

TESTING THE PBH SCENARIO

As explained above, the PBH scenario can explain the LIGO observations if PBHs responsible for the LIGO events constitute only a fraction of whole dark matter. However, the PBH scenario is not the only explanation of the LIGO events, and there are several astrophysical scenarios that have been proposed as the origin of the LIGO events.

The next obvious task is to clarify how we can test the PBH scenario and discriminate it from the others by using future observations that would bring us much more information. Several ideas have been already proposed. In the sections that follow, I will briefly explain them.

Search for the sub-solar BHs
The first idea is to search for the BHs that are lighter than the Sun. Since only BHs heavier than the Sun are produced by the astrophysical processes, detection of sub-solar BHs is definitely evidence of PBHs. Recently, the LIGO-Virgo collaboration conducted a search for the GW signal from the merger of sub-solar PBHs in the mass range 0.2-1M [10]. No viable GW candidates were found, and the upper limit on the PBH fraction in dark matter was derived.

Stochastic GW background from the PBH binaries
In addition to the loud merger events, which are loud enough to be detected by GW data analysis as individual events, there are other merger events that occur more frequently and at more distant places and where the GWs from such mergers could not be detected as the individual events. Yet, those events may be detected as a whole by taking correlations of the GW signals among different detectors and integrating it over some time. If such tiny GWs exist, after the time integration, the GW signal will emerge. Such GWs are referred to as stochastic GW background.

Recent studies suggest it is very challenging to test the PBH scenario by using the stochastic GWs. The reason is that other astrophysical scenarios predict a quite similar shape of the GW spectrum with a sizable uncertainty in its amplitude. Because of this, disentangling the degeneracy between the two scenarios is a difficult, if not impossible, problem that must be overcome.

Cosmic evolution of the merger rate
The third idea is very simple. While the PBHs exist from almost the beginning of the Universe, the astrophysical BHs exist only in the low-redshift Universe. Thus, simply measuring the redshift evolution of the mergers provides a unique method to probe PBHs. If the merger rate does not diminish even at very high redshift, it is a smoking gun for the PBH scenario. A recent study shows that future GW detectors such as DECIGO (DECihertz laser Interferometer Gravitational wave Observatory) can measure the mergers up to redshift around 40 and can be used to test the PBH scenario. Although this is a very clean test, one drawback is that such a powerful detector will only be accessible in the distant future.

Mass distribution
The mass distribution function of PBHs strongly depends on the inflation models. Thus, the resultant merger rate of PBH binaries also depends on the inflation models. However, a recent study revealed an intriguing feature; a special quantity constructed out of the merger rate distribution takes a value close to 1 irrespective of the PBH mass distribution function. This dimensionless quantity is predicted to take different values for different scenarios. For example, dynamical formation of BH binaries in the globular clusters predicts a value close to 4. This indicates that measurement of this quantity enables us to test the PBH scenario irrespective of the unknown PBH mass distribution function.

Spin distribution
GWs from the BH binary contain information about the spins of the individual BHs in the binary. Spin magnitude of a BH is commonly represented by a dimensionless quantity χ that takes a value between 0 (no spin) and 1 (maximally spinning). GWs from BH binary consisting of BHs with masses m1 and m2 are primarily sensitive to a particular combination given by

where θ1 / θ2 is angle between the spin of the first/second BH and the orbital angular momentum.

Distribution of χeff is useful to constrain the formation scenarios of the BH-BH binaries. For instance, in the isolated field binary scenario, in which stars are formed as binary and later individual stars collapse to BHs (heavier ones first), BH spins are likely to be aligned with the orbital angular momentum, and χeff > 0 is a natural consequence in this scenario. The dynamical formation scenario, in which BH binaries are formed by dynamical interactions among BHs in dense stellar environments such as globular clusters, predicts isotropic distribution of the individual BH spins. Thus, positive and negative χeff are equally probable in this case.

Since PBH binaries are formed dynamically, the statistical distribution of χeff in the PBH scenario is isotropic. What is non-trivial is the distribution of the spin magnitude χ of the individual PBHs. So far, there is only one paper that has addressed this issue. According to this paper, the probability distribution of χ is suppressed at higher values of χ. Thus, PBHs with lower spin are more favored than the higher spin. This result is physically natural since formation of a rapidly spinning PBH is possible only when the gravity defeats the centrifugal force and hence requires higher (i.e. rarer) amplitude of the overdensity than that for the non-spinning PBH.

SUMMARY

Since its original proposal around 1970, PBH has been an important topic in cosmology. Whether they exist or not in our Universe is a valuable information to probe the extremely early stage of the Universe such as inflation. Furthermore, PBHs are a candidate for cold dark matter; consequently, testing for the existence of PBHs is also important in this respect.

Detections of GWs by LIGO led to the interesting possibility that the sources of the GWs are PBHs. Associated with the advent of GW astronomy, PBH research entered a qualitatively new phase. Detections of much more BH binaries in the future will enable us to gain more knowledge about the PBHs and the early Universe. PBHs are dark but the future of PBH research is bright.

References

[1] S. Hawking, Mon.Not.Roy.Astron.Soc. 152,75 (1971).
[2] LIGO Scientific and Virgo Collaborations, B.P.Abbott et al., Phys.Rev. Lett.116, 061102 (2016).
[3] M.Sasaki, T.Suyama, T.Tanaka, S.Yokoyama, Class.Quant.Grav.35, 063001, (2018).
[4] B.J.Carr, Astrophys.J. 201, 1 (1975).
[5] B.J.Carr, K.Kohri, Y.Sendouda, J.Yokoyama, Phys.Rev.D81, 104019 (2010).
[6] A.Katz, J.Kopp, S.Sibiryakov, arXiv:1807.11495.
[7] K.Ando, M.Kawasaki, H.Nakatsuka, Phys.Rev.D 98, 083508 (2018).
[8] R.Cai, S.Pi, M.Sasaki, arXiv:1810.11000.
[9] N.Bartolo et al., arXiv:1810.12224.
[10] LIGO Scientific and Virgo Collaborations, B.P.Abbott et al., arXiv:1808.04771.

 

Teruaki Suyama is an associate professor of physics at Tokyo Institute of Technology. After receiving a PhD from Kyoto University, he worked at the Institute for Cosmic Ray Research at the University of Tokyo, Louvain Catholic University in Belgium, and the Research Center for the Early Universe (RESCEU) at the University of Tokyo, before joining Tokyo Institute of Technology in 2018. His research field is cosmology and gravitational waves.

 
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