Black hole MACHO and its identification

Physics Reports 307 (1998) 181—190

Black hole MACHO and its identification Takashi Nakamura* Yukawa Institute for Theoretical Physics, Kyoto University, 08190 Kyoto 606, Japan

Abstract If MACHOs are primordial black holes of mass &0.5M (BHMACHO), it is extremely difficult, if not impossible, to > identify BHMACHOs by their accretion-driven emission. However using gravitational wave detectors, BHMACHOs may be identified. There may exist &5;10 BHMACHO binaries in the halo up to &50 kpc whose coalescing time due to the emission of gravitational waves is comparable to the age of the universe. This means that the event rate will be &5;10\ events/yr/galaxy and several events/yr within 15 Mpc are expected. The gravitational waves from such coalescing BHMACHO binaries, if they exist, can be detected by LIGO, VIRGO, TAMA and GEO within next 5 years.  1998 Elsevier Science B.V. All rights reserved. PACS: 97.60.Lf

1. Introduction The analysis of the first 2.1 years of photometry of 8.5 million stars in Large Magellanic Cloud (LMC) by MACHO collaboration [1] suggests that 0.62>  of the halo consists of MACHOs \  of mass of 0.5> M in the standard spherical flat rotation halo model. The preliminary analysis \  > of the four year data suggests the existence of at least five additional microlensing events in the direction of LMC [2]. The estimated mass of MACHOs is just the mass of red dwarfs. However the contribution of the halo red dwarfs to MACHO events should be at most a few percent from the observations of the number density of red dwarfs [3—6]. As for white dwarf MACHOs, the IMF should have a sharp peak around 2M [7—9]. Several times more gas mass than MACHOs is > needed to make white dwarf MACHOs since progenitors of white dwarfs are massive stars with mass (8M . Extreme parameters or models are needed for the case of white dwarf MACHOs, > although future observations of the high velocity white dwarfs in our solar neighborhood might prove the existence of white dwarf MACHOs. * E-mail: [email protected]. 0370-1573/98/$ — see front matter  1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 8 ) 0 0 0 6 1 - 1

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The measurement of the optical depth to other lines of sight including SMC and M31 are needed to confirm that MACHOs exist everywhere in the halo. One microlensing event toward SMC [10,11] is not enough to determine the optical depth toward SMC reliably. At present only the optical depth toward LMC is available so that in principle MACHOs may not exist in the other line of sight. Any objects somewhere between LMC and the sun with the column density larger than 25M pc\ [12] might interpret the data. They include; LMC-LMC self-lensing, spheroids, > the thick disk, the dwarf galaxy, the tidal debris, the warping and flaring of the Galactic disk [13—16]. However none of them are confirmed to explain microlensing events toward LMC. This means that MACHOs may not be stellar objects but absolutely new objects such as black holes of mass &0.5M or boson stars with the mass of the boson &10\ eV. > In this review we consider black hole MACHO (BHMACHO) case. Since it is impossible to make a black hole of mass &0.5M as a product of the stellar evolution, it is necessary to consider > the formation of solar mass size black holes in the very early universe due to large metric perturbations or some unknown mechanism in various possible phase transitions [17—19]. A stand point here, however, is not to review the detailed models of formation but to review what is the outcome of the formation of BHMACHOs. In short the formation of solar mass black holes in the early universe is assumed without asking its theoretical origin and argue how to identify BHMACHOs. Note here that BHMACHOs of mass &M do not contribute much to the > background radiation [20,21]. In Section 2 identification of BHMACHOs by IR, optical and X-ray detectors will be discussed. In Section 3 formation of BHMACHO binary is discussed. In Section 4 gravitational waves from coalescing BHMACHO binary and its detectability will be argued.

2. Identification of BHMACHOs by IR, optical and X-ray detectors Since BHMACHOs are moving in the halo, some of them may exist in dense molecular clouds such as Orion nebula. Such a BHMACHO of mass M moving with velocity » (
(1)

The accretion rate is











 

M  n » \  MQ &nr o »+7.4;10 g s\   M 10 cm\ 10 km s\ > M n » \  , (2) +5.3;10\M Q # M 10 cm\ 10 km s\ > where o and n are, respectively, the mass and number density of the medium, and   M "1.39;10 (M/M ) g s\ is the Eddington accretion rate with unit efficiency. The advec# > tion dominated accretion flow (ADAF) model [22—26] is relevant for systems with low accretion rates compared with the Eddington one. With minimal assumptions and few free parameters,

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Fig. 1. The luminosity spectra of BHMACOs in the ADAF model (solid lines) with M"0.5M and three different > values of M Q (marked on each curve in Eddington units), which correspond to black hole velocities »"5, 10, 20 km s\ for n "10 cm\. To be compared are the Ipser-Price model spectra (dashes lines) for two MQ values. The typical point  source detection limits for various observational facilities assuming a distance of 400 pc (e.g. Orion).

ADAFs self-consistently predict a stable, hot, two temperature structure that generates broad-band spectra from radio to gamma-rays, and have been successfully applied to a variety of lowluminosity objects. For high accretion rate » should be smaller than &20 km/s. Fig. 1 [27] shows the luminosity spectra of BHMACHOs in the ADAF model for »"5, 10, 20 km/s. The fraction of BHMACHOs with »(20 km/s is estimated as 2;10\ for the velocity dispersion 155 km s\ of BHMACHOs. If the density of MACHOs in the solar neighborhood is 0.0079M pc\ [28], the BHAMCHO > number density is 0.016 pc\. Then the number of near-IR observable objects in Orion is only &0.4, while the chances of an X-ray detection are hopelessly small. For BHMACHOs in the general interstellar matter of n "1 cm\ the detection requirements are »(4 km s\ (near-IR)  and »(0.9 km s\ (X-ray) at distances (400 pc; fraction ((4 km s\)"2;10\ then results in &2 near-IR BHMACHOs, with the X-ray numbers again being miniscule. BHMACHO spectra in the Ipser and Price spherical accretion model [29,30] are also shown in Fig. 1. Its greater luminosity at optical wavelengths may raise the number of detections to &10. However discriminating isolated, accreting black holes from other sources based solely on IRoptical observations is akin to searching for needles in a haystack. Therefore the identification of BHMACHOs by their accretion-driven emission is extremely difficult if not impossible [27].

3. Formation of solar mass size black hole MACHO binary Since there are a huge number (&4;10) of black holes in the halo it is natural to expect that some of them are binaries. The fraction and the distribution function with respect to the semimajor

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axis and the eccentricity of binary BHMACHOs are estimated [21,31]. The main purpose of the estimate is the following two points. Firstly if the semimajor axis of a BHMACHO binary in a circular orbit is &10 cm, the binary coalesces in the time scale of the age of the universe so that the gravitational waves in the last three minutes [32] can be detected by LIGO, VIRGO, TAMA and GEO [33—36]. Secondly in the microlensing events one event toward LMC is due to a binary with separation &2;10 cm [37]. Although this may be a binary in the Galactic disk or in LMC, it is important to ask the possibility of it being a BHMACHO binary. The density parameter of BHMACHOs, X , should be comparable to X (or X ) to explain &+
!"+ the number of the observed MACHO events. In order to simplify the discussion it is assumed that BHMACHOs are the only dark matter and dominate the matter energy density, i.e., X"X , &+ although it is possible to consider other dark matter components than BHMACHOs. To determine the mean separation of the BHMACHOs, it is convenient to consider it at the time of matterradiation equality, t"t with the normalization of the scale factor R"1 at t"t . At t"t ,    both energy densities of the radiation and BHMACHOs are the same and given by o "1.4;10\(Xh) g/cm , (3)  where h is the Hubble parameter in units of 100 km/s Mpc. The mean separation of BHMACHOs with mass, M , at this time is given by & M  M  & & "1.1;10 (Xh)\ cm . (4) xN " o M  > Since the scale factor R is unity at t"t , xN is regarded as the comoving mean separation. The  Hubble horizon scale at t"t becomes  3c ¸ & "1.1;10(Xh)\ cm . (5)  8nGo  During the radiation dominated era, the total energy inside the horizon changes as R. Since the Jeans mass in this era is essentially the same as the horizon mass, black holes are formed only at the time when the horizon scale is equal to the Schwarzschild radius of a BHMACHO. Thus the scale factor at the formation epoch becomes














GM M  &"1.2;10\ & (Xh) . (6) c¸ M  > The age of the universe and the temperature at R"R are &10\ s and &GeV, respectively.  Note here that at R"R only the fraction of &R of the total energy density is in the form of black   holes since the ratio of the radiation density to the black hole is in proportion to the scale factor R. Consider a pair of black holes with the same mass M and the initial comoving separation & x;xN at R"R in the radiation dominated universe. At the beginning, the comoving separation  does not change and the physical length of the separation increases in proportion to R. The energy density due to a pair of black holes is given by R" 

o xN  1 , o ,  & x R

(7)

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while that due to the radiation is given by o o "  .  R

(8)

Hence, the energy density of BHMACHOs dominates that of the radiation in the neighborhood of this BHMACHO binary for [21]




x  . R'R ,

xN

(9)

This means that a pair of black holes decouples from the cosmic expansion when R"R to be

a bound system. To confirm this simple argument Newtonian numerical simulations of formation of such a bound system in the expanding radiation dominated universe have been done [31]. Fig. 2 [31] shows simulations for x"0.1xN with zero initial relative velocity at R"R . From Eq. (9), R is &10\, while Fig. 2 shows R is &1.5;10\ irrespective of the starting time R ,

which proves the simple estimate of Eq. (9). It is also confirmed numerically that the formation time is almost independent of the initial velocity at R"R since due to the cosmic expansion before R the peculiar velocity ceases out even if the peculiar velocity is comparable to the light velocity at R"R [31].  In Fig. 2, the pair of black holes has no angular momentum so that they coalesce to be a single black hole in the free fall time scale. However, in reality, there exists tidal force from neighboring black holes so that the pair of black holes obtains the angular momentum and becomes a binary black holes with an eccentric orbit.

Fig. 2. The time evolution of the separation of a pair of black holes with x"0.1xN at R"R . Solid lines show the relative physical separation of the pair of black holes as a function of the scale factorR for three values of initial R . At R"R , the relative velocity of black holes is assumed to be zero. At first the relative separation increases due to the cosmic expansion but eventually due to the gravity between black holes, a pair of black holes decouples from the cosmic expansion. Finally two black holes will collide.

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Define the semimajor axis and the semiminor axis of the binary as a and b, respectively. Suppose that the comoving separation of the nearest neighboring black hole from the center of mass of the binary is y. a is estimated as [21] x a"xR " ,

xN 

(10)

while b is evaluated by (tidal force);(free fall time) as




GM xR (xR ) x  &

" b" a. (yR ) GM y

& Hence, the eccentricity e is given by [21]




e" 1!

x  . y

(11)

(12)

Fig. 3 [31] shows a numerical simulation of such a three body problem in the expanding radiation dominated universe. The dotted and solid lines show the relative orbits of the third and the second black hole to the first one, respectively. The third black hole follows the expansion of the universe and goes away giving the angular momentum to the first and the second one. From 300 simulations with various x and y, it is confirmed that relations (10) and (12) hold with a slight modification [31]. Ioka et al. [31] considered the various effects on the above three body models such as angle dependence of the third body, 3-body collision, effect of mean fluctuation field, initial condition dependence, radiation drag effect and so on. They found that within 50% ambiguity the estimate of

Fig. 3. Formation of a binary black holes. The dotted and solid lines show the relative orbits of the third and the second black hole to the first one in the physical two-dimensional plane, respectively. The third black hole follows the expansion of the universe and goes away giving the angular momentum to the first and the second one. The initial parameters are shown in the figure.

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the distribution function and event rate of coalescence based on a simple estimate of Eqs. (10) and (12) are correct [31]. Now assume that x and y have the uniform probability distribution in the range x(y(xN . Then the probability distribution of the binary parameters, a and e, becomes [21] 18xy dx dy , f (a, e) da de" xN 

(13)

3ea da de . " 2xN (1!e)

(14)

Integrating f (a, e) with respect to e, the distribution of the semimajor axis f (a) is given as ?





 

a  a  da ! . xN xN a

3 f (a) da" ? 2

(15)

From Eq. (15), it is found that the fraction of the binary BHMACHOs with a&2;10 cm is &8% and &0.9% for Xh" 1 and 0.1, respectively. The estimated fraction of the &10 AU size BHMACHO binary is slightly smaller than the observed rate of the binary event (i.e. one binary event in 8 observed MACHOs).

4. Gravitational waves from coalescing BHMACHO binary Short period BHMACHO binaries may coalesce due to the emission of gravitational waves within the age of the universe [21]. The coalescing time is approximately given by [38,39] t"t




a  (1!e) ,  a 

(16)

where t "10 yr and 




M  & a "2;10 cm ,  M >

(17)

is the semimajor axis of the circular orbit BHMACHO binary which coalesces in t . Integrating  Eq. (14) for a fixed t with the aid of Eq. (16), the probability distribution of the coalescing time is obtained as [21]




3 f (t) dt" R 29

t

t




 

 !

t

t



 dt , t

(18)

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where




xN  . (19) "t

  a  If the halo of our galaxy consists of BHMACHOs of mass &0.5M , &10 BHMACHOs exist > up to LMC. The number of coalescing binary BHMACHOs with t&t becomes &5;10 for  Xh"0.1 from Eq. (18) so that the event rate of coalescing BHMACHO binaries becomes &5;10\ events/yr/galaxy. This can be compared with the event rate of coalescing binary neutron stars which is one of the most important sources of gravitational waves. Based on the number of the three known binary neutron stars, the event rate is estimated as 10\&2;10\ events/yr/galaxy [40—42]. The event rate of the coalescing BHMACHO binary is three orders of magnitude larger than this and is comparable to or greater than the upper limit [40]. If, however, BHMACHOs extend up to the half way to M31, the number of coalescing binary BHMACHOs with t&t can be &3;10 and the event rate becomes even higher as  &0.3 events/yr/galaxy. The detectability of these waves by interferometers is most easily discussed in terms of the waves’ “characteristic amplitude” h given by [43]  M  l \ r \   , (20) h "5.3;10\  M 100 Hz 15 Mpc > where l and r are the frequency of the gravitational waves in the last three minutes [32] and the distance to the BHAMCHO binary, respectively, while M "(M M )/(M #M ) is the       “chirp mass” of the binary whose components have individual masses M and M . For the first   LIGO and VIRGO interferometers [33,34], which are expected to be operational in 2001, the sensitivity is expected as h K3;10\. For equal mass BHMACHO binary, M "M "0.5M , 1    > M becomes 0.43M and h is &3;10\ for l"100 Hz and r"15 Mpc. LIGO/VIRGO   >  should be able to detect coalescing BHMACHO binaries, with high confidence, out to about 15 Mpc distance with an event rate of several per year since the number density of the galaxy is &0.01 Mpc\. If BHMACHO binary coalescence occurs in our halo, TAMA300 [35] and GEO600 [36] can see the event by the end of this century. If the gravitational waves from the coalescing BHAMCHO binary is detected, by making a cross correlation of the observational data with the theoretical template of the waves in the last three minutes [32], each mass, the distance and the direction of BHMACHO binary [44] can be determined so that BHAMCHO can be identified. One of the difference between the coalescing BHMACHO binary and the coalescing neutron stars is the mass determined by the gravitational waves. Even if the mass of a BHMACHO is &1M , the gravitational wave forms of coalescing neutron star binaries may be different from > that of BHMACHO binaries in the final merging phase since the radius of the neutron star is larger than the Schwarzschild radius. Moreover if the coalescing binary neutron stars are sources of the gamma-ray bursts [45], the gamma-ray appears &1 s after the amplitude of the gravitational waves becomes zero while in the coalescence of binary BHMACHOs the emission of gamma-ray is not expected. Therefore we may be able to distinguish coalescing binary BHMACHOs from coalescing binary neutron stars. t












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In conclusion, if MACHOs are primordial black holes of mass &0.5M , they will be identified > by gravitaional wave detectors within next 5 years.

Acknowledgements The author would like to thank Thorne, Sasaki, Tanaka, Ioka, Chiba, Fujita and Inoue for useful discussions. This work was supported in part by the Grant-in-Aid for Basic Research of the Ministry of Education, Culture, and Sports No. 08NP0801 and No. 09640351.

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