High frequency gravitational waves from axionic black holes

High frequency gravitational waves from axionic black holes S. Carneiro1,2 , J. C. Fabris3,4 and J. A. Freitas Pacheco5 1

Instituto de F´ısica Gleb Wataghin, UNICAMP, 13083-970, Campinas, SP, Brazil Instituto de F´ısica, Universidade Federal da Bahia, 40210-340, Salvador, Bahia, Brazil 3 N´ ucleo Cosmo-ufes & Departamento de F´ısica, UFES, 29075-910, Vit´ oria, ES, Brazil 4 National Research Nuclear University MEPhI, 115409, Moscow, Russia 5 Observatoire de la Cˆ ote d’Azur, 06304, Nice, France (Dated: November 7, 2018)

arXiv:1811.02289v1 [gr-qc] 6 Nov 2018

2

In a recent paper we have shown that a minimally coupled, self-interacting scalar field of mass m can form blackholes of mass M ∼ 1/m (in Planck units). In the case of an axion field, the blackhole has a 1cm gravitational radius and the Earth mass. Here we evaluate the presence of axionic blackholes in the Milk Way halo, which mergers would produce detectable gravitational waves of primary frequency ∼ 1GHz. I.

INTRODUCTION

Still hypothetical particles, axions are up to date the most reliable theoretical explanation for the CP problem of strong interactions [1]. They are very difficult to be directly detected, because of their tiny mass, estimated to be of order m ∼ 10−5 eV, and their small cross section. In a recent paper [2], however, we have shown that scalar fields may in general suffer gravitational collapse to√a singularity. The formed blackhole has a mass M = 2 3π/m (in reduced Planck units), which in the axion case gives M ∼ 10−5 M⊙ , i.e., one Earth mass. In the solution found, an spherically symmetric exact solution of General Relativity and Klein-Gordon equations, the scalar field is minimally coupled to gravity, and a particular, hyperbolic self-interaction potential was adopted, which reduces to a free field potential at the first stages of the collapse. For different potentials the solution, if found, would not be the same. Nevertheless, as an initially free field can start the collapse, we expect that it happens in general, except perhaps for a very particular, strongly repulsive interaction. On the other hand, axions are good candidates for the cold dark matter component of the standard cosmological model [3], because they interact very weakly with baryonic matter and radiation and were non-relativistic at the time of decoupling from the baryon-photon plasma. If dark matter is indeed formed, at least partially, by axions, we should expect a huge number of axionic blackholes in our galaxy halo. Their absence, on the contrary, would be a challenge to the presence of axions in the dark matter component.

II.

FIRST ESTIMATES

With the above mass, we can estimate the primary GW mode emitted in those blackholes coalescence, as well as the horizon distance dH for its detection. Roughly speaking, the emitted frequency f peaks around the orbital frequency when the binary separation approaches the gravitational radius, which leads to f ∝ 1/M . For a blackhole of 1 solar mass, the primary mode has fre-

quency f⊙ ≈ 13 kHz [4], and so for a mass M ∼ 10−5 M⊙ we have f ∼ 1GHz, far beyond the LIGO/Virgo’s range. On the other hand, an extrapolation of the LIGO/Virgo horizon vs. mass relations [5] to M ∼ 10−5 M⊙ gives for the former dH ∼ 10 to 102 kpc, below the distance to the nearest major galaxies, but of the order of the Milk Way’s halo size Rhalo > ∼ 50 kpc. An upper limit to the density of axionic blackholes in the halo is obtained by assuming that they dominate the dark matter component, which 3 gives a number density η ∼ (Mhalo /M )/Rhalo ∼ 103 /pc3 , 12 where Mhalo ∼ 10 M⊙ is the halo mass. On the other hand, microlensing constraints limit BH’s of 10−5 M⊙ to at most 10% of halo’s dark matter [6], which gives the upper bound η ∼ 102 /pc3 . Their detection will depend on the reach and performance of high frequency detectors under development [7], as well as on the merger rate. The blackholes velocity can be estimated from the virial condition v 2 ≈ GMhalo /Rhalo , leading to v ∼ 10−3 c. The interaction cross section is σ ≈ (GM 2 /K)2 , where K ≈ M v 2 is the total kinetic energy in the binary rest frame. In this way, we obtain σ ≈ (M Rhalo /Mhalo )2 , and the collisions rate Γ = ηvσ ∼ 10−33 s−1 . This would, of course, rule out any detection. Nevertheless, this estimation does not consider the merger of binaries formed in the very process of gravitational collapse, which accounts for a half of stars in the galaxy. Although this is more difficult to evaluate, we can assume that the radius of axionic binaries has the same proportion to the their gravitational radius as in a typical neutron stars binary. Taking for the latter a separation of order 106 km and 1 solar mass, for axionic BH’s of 10−5 solar masses we have a binary radius R ∼ 10 km. The corresponding chirp time is tch = 5c5 R4 /(8G3 M 3 ) ∼ 105 yrs [8], leading 3 to an event rate Γ ∼ ηRhalo /tch ∼ 104 /s (assuming that all blackholes are forming binaries). A more conservative choice for the binaries radius, say ∼ 100 km, would give a rate about 1 event per second, with a merging time of the order of the Universe age, which marginally assures the possibility of detection. As our position in the galaxy is not central, the signal would not be isotropic and could be distinguished from a cosmological background of same frequency [9].

2 III.

DETECTION LIMITS

The strength of a given signal is characterised by the signal-to-noise ratio (S/N ), which depends on the source power spectrum and on the noise spectral density Sn (f ) of the detector. For the merging of a binary system constituted by compact objects, the optimum S/N ratio is obtained by the matched-filtering technique, e.g.,  2 Z ∞ ˜2 h (f ) S =4 df. (1) N S n (f ) 0 ˜ 2 (f ) is the sum of the square of In the equation above, h the Fourier transform of both polarization components of the gravitational wave signal. Here we will adopt the approach by Finn [10] and, in this case, equation (1) is reduced to  r   M 5/6 S ∗ 0 = 8Θ ζ(fmax ). (2) N D 1.2M⊙ In this equation the angular function Θ depends on the geometrical projection factors of the detector and on the inclination angle i between the orbital angular momentum of the binary and the line of sight, that is, h i
2 Θ2 = 4 1 + cos2 i F+2 + 4 cos2 i F×2 . (3)

According to Finn & Chernoff [11], if Θ is in the range 0 ≤ Θ ≤ 4 , its probability distribution can be approximated with a quite good accuracy by the relation P (Θ) =

5Θ(4 − Θ)3 . 256

(4)

Using this distribution, the average value of Θ that will be adopted in our computations is 4/3. The other quantities in equation (2) are: M∗ = µ3/5 M 2/5 is the chirp mass (µ and M being respectively the reduced and the total mass of the system), D is the distance to the source, and the parameter r0 (having the dimension of a length) is defined by the relation q r0 = 9.25 × 10−19 I7/3 kpc, (5) where I7/3 =



f0 π

1/3 Z

0

∞

df f 7/3 S

n (f )

,

(6)

with f0 = 203.38 kHz. The factor ζ(fmax ) is defined similarly, e.g., Z df (f0 /π)1/3 2fmax . (7) ζ(fmax ) = 7/3 S (f ) I7/3 f n 0 The maximum frequency appearing in the upper limit of the integral above corresponds to the frequency on the last stable orbit, given by the relation [12]  −1 M fmax = 4397 (1 + 0.316 η) Hz, (8) M⊙

where η = µ/M . As seen previously, the black hole mass is given by √ 2 3πMp2 Mbh = , (9) ma where Mp is the reduced Planck mass, which for an axion mass ma = 10−5 eV corresponds to Mbh = 1.1 × 1028 g = 5.8 × 10−6 M⊙ . With such a black hole mass, the maximum gravitational wave frequency is about 817 MHz and the chirp mass is 9.9 × 1027 g. In order to compute the integrals in equations (6) and (7), we adopt the spectral noise density for the planned Einstein Telescope (ET) [13], version shot-noise limited antenna with a knee-frequency around 1 kHz. In this case the scale parameter results to be r0 = 958 Mpc and the expression for the S/N ratio becomes S 380 . ≈ N Dkpc

(10)

For S/N ∼ 7, the typical threshold for a false alarm rate of about one per year, the maximum distance that can be probed by the ET is about 54 kpc. This means that practically all merger events occurring in the galactic halo could be detected. Nevertheless, most of the power of the inspiral signal will be around 600 Hz, frequency at which the antenna has its maximum sensitivity. This frequency corresponds to a pair separation of about 0.74 km or 0.825 yrs before the plunge, which will not be seen since the detector has a poor sensitivity for frequencies beyond 6 - 7 kHz. The detection rate depends on the number of pairs whose separation is approximately in the range 0.1 up to 10 km, corresponding to the frequency interval in which the detector sees the signal. IV.

FURTHER REMARKS

The origin of binary stars is still one of the central problems of astrophysics since momentum conservation rules out gravitational capture between two interacting bodies. Observations of young stars not yet on the main sequence support the scenario in which binaries are formed during the process of star formation. Presently the fragmentation of molecular clouds during the formation of protostars is an acceptable explanation either for the formation of binaries or multiple star systems. However, this scenario (supported by recent millimeter observations of molecular clouds) is difficult to transpose to the possible existence of a large number of mini-black hole binaries that could constitute partially dark matter particles. In order to speculate about a possible formation of binaries among that putative population of miniblack holes some scenarios will be discussed. Clearly, if one expects to detect some mergers presently, it is necessary that the pair separation be of the order of 250 km. This corresponds to a merger timescale of 1010 yrs. For a halo of mass 1.5 × 1012 M⊙ and scale of ∼ 100

3 kpc (since the rotation curve extends beyond 50 kpc), the mean distance between pairs is about 7.3 pc (it was also assumed that mini-black holes represent only 10% of the total dark matter face to upper limits derived from microlensing [6]). In this case, the ratio between the separation of pairs and the mean distance between them is around 10−12 . This ratio seems to be extremely small when compared with typical values observed among binary stars in our galaxy that are of the order of 10−2 10−3 . This could be an indication that mini-black hole binaries, if exist, must have been formed in a higher density environment. Assuming that at the moment of formation the typical distance between mini-black hole binaries is about 100 times their separation, i.e., 25, 000 km, this implies a mean density of 1.9 × 1027 pc−3 . The present cosmological density of these dark matter particles is n0 = 0.1

3H02 Ωdm ≈ 7.4 × 10−4 pc−3 , 8πGMbh

where H0 and Ωdm are, respectively, the present values of the Hubble-Lemaˆıtre and dark matter density parameters. Therefore, the density at which the mean distance between pairs is about 100 times their separation is reached at z ∼ 1010 , very early in the universe, when triple interactions could have played a role in their formation. Acknowledgements

We are thankful to O. Aguiar and R. Sturani for valuable references and discussions, and to CNPq and FAPES for financial support. SC is grateful to P.C. de Holanda and F. Sobreira for the hospitality at UNICAMP.

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