Creation of Supermassive Kerr Black Holes from the Gravitational Collapse of the Rotating BECs of Ultra-light Scalars Fazlu Rahman P Pa and Patrick Das Guptab a Department
of Physics, Cochin University of Science and Technology; of Physics and Astrophysics,University of Delhi
b Department
ABSTRACT Supermassive Black Holes (SMBHs) have been detected in the nuclei of majority of galaxies. The red-shift observed for quasars and other Active Galactic Nuclei(AGNs) is very high so that the associated SMBHs have formed very early on. Conventional technique of accretion of baryonic matter/dark matter is too slow to be a viable model for the formation of these BHs in the early epoch. In recent years, there has been active research works concerning cosmological structure formation that involve coherent, ultra-light bosons in a dark fluid-like or fuzzy cold DM state. It has been shown recently that SMBHs can be generated at early times on dynamical time scales from collapsing Bose-Einstein Condensates (BECs) of ultra-light scalars that constitute portions of dark matter galactic halos. In this work, we extend the analysis to investigate the possibility of forming Kerr SMBHs from excited states of BECs associated with non-zero angular momenta. GP equation is used to know the evolution of the condensate via time-dependent variational method and different mass constraints are obtained for the identical bosons to form Kerr SMBHs of higher angular momenta. The collapse of the halo BEC clusters can emit Gravitational Waves(GW) and, amplitude, luminosity and frequency of the condensate GW are investigated. Finally, the validity of Hawking Area Theorem for black-hole binaries in the recent four GW events is also analyzed. Keywords: Kerr Black holes, BEC, Dark matter, Gravitational Waves
1. INTRODUCTION Presently, cosmology needs to address three cardinal problems - existence of supermassive black holes (SMBHs), dark matter (DM) and dark energy (DE). Majority of galaxies, including our Milky Way, are observed to harbor SMBHs at their centers.1–3 SMBHs also form the main component of central engines that power the active galactic nuclei (AGNs).4 Several quasars have been detected at high redshifts z ∼ 6, implying formation of SMBHs with mass ∼ 109 M when the universe was barely ∼ 109 yrs of age.5–10 For instance, SDSS J010013.02+280225.8, one of the brightest quasars detected at a redshift of z = 6.3, has a central SMBH of mass ∼ 1.2 × 1010 M .11 Now, dark matter (DM) is needed not only to model the observed flat rotation curves in disc galaxies but also to explain why rich clusters of galaxies are gravitationally bound.12 Evidence for DM also comes from gravitational lensing data associated with galactic clusters as well as from cosmic microwave background anisotropies. Cosmological models corresponding to k=0, cosmological constant dominated, homogeneous and isotropic universe containing cold DM of weakly interacting massive particles (WIMPs) kind, are very successful in explaining cosmic structures on scales larger than galactic scales but fail to describe features at sub-galactic scales.13 Furthermore, searches for WIMP candidates like neutralinos and other SUSY particles predicted by supersymmetric extensions of high energy physics Standard Model have not been very successful.?, 14–17 On the other hand, if DM is constituted by ultra-light scalar/ pseudo-scalar particles like axion or dynamical four-form, one may not only reconcile observed large scale cosmic structures with sub-galactic scale features but Further author information: (Send correspondence to Fazlu Rahman P P ) Fazlu Rahman P P: E-mail:
[email protected] Patrick Das Gupta.: E-mail:
[email protected]
also explain the existence of SMBHs at the centers of most galaxies very early on.?, 13, 18, 19 Other scenarios for SMBH formation like, growth of seed black holes (BHs) through matter accretion or direct collapse of halo matter, face great difficulties in generating SMBHs of mass ≈ 109 M when the universe was only ∼ 109 years old.20–26 In the next section, we employ the Gross-Pitaevskii equation to study the evolution of the ultra-light DM bosons that have undergone Bose-Einstein condensation and show that it is possible to generate Kerr BHs from their collapse. We consider generation of GWs during the gravitational collapse of such DM in the Bose-Einstein condensate (BEC) phase in section III and in the final section, we make use of the estimated parameters from the detected GWs to further strengthen the evidence for BHs.27–30
2. KERR SMBH FROM ROTATING BEC The ultralight scalar/ pseudoscalar particles with m < 2 eV can be in the Bose-Einstein condensate phase as their critical temperature to undergo condensation is always greater than the temperature of the universe . In the T∼ 0o K mean field approximation, the non-linear Schrodinger equation called Gross-Pitaevskii(GP) equation explains the dynamics of the condensate wave function . " # Z 2 3 −¯ h2 2 dψ = ∇ + Vext + N V ~r − ~u |ψ ~u, t | d u ψ ~r, t (1) i¯ h dt 2m where m, Vext ~r , V ~r are the mass of the boson, an external potential energy required to confine the BEC and the interaction potential energy between two bosons, respectively, and 4π¯h2 as 3 V ~r − ~u = δ ~r − ~u + Vg |~r − ~u| (2) m where ~u and ~r are the position vectors of two bosons. First term of the Eq. 2) is the contact interaction characterized by the s-wave scattering length as , while the second term is the gravitational attraction between two bosons.Substituting Eq. 2) in Eq. 1) gives " # Z 2 2 3 dψ −¯ h2 2 = ∇ + Vext + N g|ψ ~r, t | + N Vg ~r − ~u |ψ ~u, t | d u ψ ~r, t (3) i¯h dt 2m where g = 4π¯ h2 as /m It’s an uneasy task to solve the GP equation analytically and the evolution of the condensate is studied using time-dependent variational method. It Rinvolves trial wave functions with time-dependent parameters. It is R obtained by extremizing the action S = dt d3 rL using the following Lagrangian density " # Z ∗ i¯ h ∂ψ ∗ ¯h2 gN 4 N 2 ∗ ∂ψ L= ψ −ψ + ∇ψ∇ψ ∗ + Vext |ψ|2 + |ψ| + |ψ| Vg ~r − ~u |ψ ~u, t |2 d3 u (4) 2 ∂t ∂t 2m 2 2 In this case, we consider the population III stars form at very high red shifts (z > 20), and evolve quickly to give rise to compact objects with mass M0 >150 M . Potential energy of a dark boson due to this one compact object is given by Vext ~r) = −GM0 m/r and gravitational potential between individual dark boson is Vg ~r) = −Gm2 /r The variational method involves choosing a trial wavefunction that can exactly give the information about the condensate. We take the wavefunction, ψ ~r, t) = A t)rl exp(−r/2σ t) exp(−iB t)r)Ylm (θ, φ) (5) where Ylm (θ, φ) is the usual spherical harmonics for different values of l and m, σ(t) and B(t) are the width and the phase of the condensate, respectively.
Here we study the behavior of the condensate for various l and m values via the variational method. For l = 1 and m = 0 ψ ~r, t) = A t)rexp(−r/2σ t) exp(−iB t)r)Y10 (θ, φ) (6) 1 3 2 cos(θ) (7) Y10 (θ, φ) = 4π Normalizing the wave-function to unity gives |A(t)|2 =
1 21 1 ⇒ |A(t)| = exp(iγ(t)) 5 4!σ 4!σ 5
(8)
Time evolution of the trial wave function describing the dynamics of the darkbosons is characterized by the changes in it’s width σ t), and the phase B(t).BEC mass enclosed within a sphere of radius R is given by Z
R
Mbec (< R, t) = N m
|ψ(~r, t)|2 d3 r
(9)
0
so that BEC mass enclosed within the width 4σ(t) is Mbec
Nm = 4!σ 5
Z
4σ(t)
r2 exp(−r/σ(t))d3 r ⇒ .37N m
(10)
0
Using Eq.(4), we can obtain the Lagrangian and this can be applied to the Euler-Lagrange’s equation. Hence, we separately evaluate the terms of the Lagrangian, which is given as Z GM0 m ¯h2 1 0.861N Gm2 2 3 − + B + 2 − 5¯ hσ B˙ (11) L = d rL = h ¯ γ˙ − 4σ 4σ 2m 4σ where we take the s-wave interaction as = 0. Using the above Lagrangian, R the R dynamical evolution of the parameters σ(t) and B(t) can be analyzed. On extremizing the action S = dt d3 rL, we get the Euler-Lagrangian equations d ∂L ∂L − =0 dt ∂ q˙j ∂qj Taking j=1,2 we have q1 = B t) and q2 = σ t), we get the following two equations. G 0.861N m + M0 ]m ¯h2 5¯ hB˙ + = − 4mσ 3 4σ 2 σ˙ = −
(12)
(13)
¯h B 5m
(14)
dVef f dσ
(15)
Eqs.(13) and Eqs.(14) can be combined as m¨ σ=− where Vef f
G 0.867N m + M0 ]m ¯2 h = − 200mσ 2 100σ
(16)
and B = −5mσ/¯ ˙ h Now we take into consideration, the condition for the formation of blackhole. Eq.(16) includes various terms representing the interactions among the darkboson particles. The first term corresponds to the uncertainty quantum repulsion enabling the existence of stable self-gravitating bosonic astrophysical system along with the
second and third terms, the inter-particle and external gravitational interactions, respectively. Solving Eq.(15), we get r 1 ˙2 2 K0 − Vef f (17) mσ + Vef f = Constant ≡ K0 → σ˙ = ± 2 m By setting σ˙ = 0, one can calculate the turning point using Eq.(17) Here we take the initial conditions as σ˙ ≈ 0 and B ≤ 0 with an initial 4σi = 25kpc (Vef f ≈ 0).This gives the constant K0 can be taken as 0 such that s G 0.867N m + M0 ] ¯h2 (18) σ˙ = − − 100σ 200m2 σ 2 provided G 0.867N m + M0 ] ¯h2 >0 − 100σ 200m2 σ 2 From Eq.(18), we can find the time for the evolution s h 32 σ(t) 32 i 1 ¯h2 σi3 ¯h2 ¯h2 t − ti = 1 − − − − 2 2 ¯) ¯ )m σi ¯ )m σi ¯ )m2 15 (2.34GM σi G(2.34M G(2.34M 5G(2.34M h 1−
12 σ(t) 12 i ¯2 h ¯h2 − − ¯ )m2 σi ¯ )m2 σi σi G(2.34M G(2.34M
(19)
(20)
When the width of the condensate evolve in time, at some turning point, σ˙ = 0 ( where Vef f = 0), which gives the min. value of σ ¯h2 σmin = (21) 2G 0.867N m + M0 m2 After the bounce at the turning point (σ˙ = 0), σ(t) starts increasing again, unless a blackhole is formed. Eq.(10) gives the total mass enclosed within the radius 4σ(t).We take ¯ = Mo + .37N m M and the corresponding event horizon for a Kerr Blackhole is given as s ¯ ¯ 2 L 2 GM G2 M − ¯ RBH = 2 + c c4 Mc
(22)
(23)
and the angular momentum L = .37N ¯ h gives RBH
" # r ¯ m2P l GM M0 2 = 2 1+ 1− ¯2 2 1− ¯ c M m M
(24)
As long as the minimum value of σ is greater than RBH , no blackholes are formed. Condition for the formation of rotating blackholes is, 4σmin < RBH (25) ¯ , we have If we take M0 M ¯ 1 − 0.57M0 ⇒ 2.34M ¯ 0.867N m + M0 = 2.34M ¯ M # " r ¯ m2P l 2¯h2 GM M0 2 4σmin → ¯ m 2 < c2 1 + 1 − M ¯ 2 m2 1 − M ¯ 2.34GM
(26)
(27)
which gives finally the condition 2
¯ ≥ q 0.85mP l mM 1.71 − 1 −
M0 ¯ M
2
≈ 1.011m2P l
(28)
By Eq.(20), the time scale for the collapse is 1 t − ti = 15
s
σi3 8 9 ¯ ) ≈ 10 − 10 years (2.34GM
(29)
If we do the same steps for l = 1 and m = 1, we get ¯ ≥q mM
m2P l 1−
1 4
1−
M0 2 ¯ M
≈ 1.15m2P l
(30)
Also for l = 2 and m = 2, ¯ ≥q mM
9.09m2P l 18.2 − 1 −
M0 2 ¯ M
≈ 2.18m2P l
(31)
¯ = 1010 M , we obtain the mass of the darkboson get constrained For the creation of Kerr SMBHs of mass, M as m > 10−20 eV .Also, this inequality tells us that ultralight scalars with m ∼ = 10−23 eV can result in the ¯ > 1012 M , which has no observational validity yet. Hence, we can infer formation of Kerr SMBHs of mass M that dark bosons with mass ∼ 10−20 eV can result in the creation of Kerr SMBHs
3. GRAVITATIONAL RADIATION FROM THE COLLAPSING BEC Gravitational Waves (GW)are the ripples in the curvature of space-time traveling at the speed of light. Any change in the mass quadrupole moment of a system can emit gravitational radiation and these are very weakly interacting and hence, difficult to detect. There are various promising sources for GW: Blackhole binaries,kilonova and supernova, stochastic background radiation etc. Here we analyze the gravitational collapse of condensate dark matter as a source of GW. For a collapsing BEC to form the SMBH, energy released is given as 4E ≈
GN 2 m2 σ0
(32)
and a fraction of the total energy is released as gravitational radiation EGW =
GN 2 m2 ≈ 1064 erg σ0
(33)
EGW = 1048 erg/s τdyn
(34)
within a time-scale τdyn ∼ = 4 × 108 years Gravitational wave luminosity on an average is LGW =
If we take the non-symmetric factor ∼ 1, luminosity obtained is comparable with that of the bright quasars and other AGNs in the electromagnetic regime.
3.1 Amplitude of the Radiation The time-varying quadrupole moment is the reason for the emission of the waves and the amplitude of the wave is, 2G ¨ (35) hij = 4 Q ij rc Quadrupole moment Qij can be computed via, Z 1 Qij = N m ψ ∗ (xi xj − r2 )ψd3 x (36) 3 As the trial wave-function describing the condensate, we have chosen ψ ~r, t) = A t)rl exp(−r/2σ t) exp(−iB t)r)Ylm (θ, φ)
(37)
For l = 1 and m = 0, we have the normalized trial wave-function ψ ~r, t) =
√
3/(4π × 4!σ 5 )
12
exp(iγ(t))rexp(−r/2σ t) exp(−iB t)r) cos(θ)
Qij is symmetric and it can be calculated as −4N mσ 2 0 Qij = 0
0 −4N mσ 2 0
(38)
0 0 = Qji 8N mσ 2
Qij is trace-less and we can make it transverse by using the projection operator P = δij − ηi ηj
(39)
¨ 11 = −8N m(σ˙ 2 + σ¨ Q σ)
(40)
Using Eq.(15), Eq.(16), Eq.(17) and Eq.(40) , Eq.(35) becomes ¯ m1 N 4G2 M σ 25c4 r
(41)
N 1pc 1M pc m 1091 σ(t) r
(42)
h11 = − From the inequality in Eq.(28), h11 ≥ 4.3 × 10−20
3.2 Luminosity of the Radiation GW energy flux is given by c3 ˙ ˙ jk hhjk h i 32πG and the luminosity of emitted radiation for a nearly flat background spacetime( isotropic emission) is FGW =
(43)
G ... ...jk LGW ∼ = 4πr2 FGW = 5 hQjk Q i 2c
(44)
¯m ... 2GN M Q11 = 25σ 2
(45)
Using Eq.(41) and Eq.(35), we get
and the luminosity of radiation is LGW =
N 2 h 4G3 m4P l i σ 4 (t) 625c5
(46)
giving, LGW = 2 × 1046
N 2 0.3 × 1036 erg/s 10182 σ 2 (t)
(47)
The order of magnitude for luminosity and the amplitude of the wave strongly depend on the parameter N , the number of condensate particles. They are time-dependent quantities, and will come in the detectable range only in the later stages of collapse.
3.3 Frequency of the Radiation Now if no blackholes are formed, the system gets bounce back and finally settle down to a width σ0 minimizing the Vef f . dVef f =0 (48) dσ For l=1 and m = 0, G[0.867N m + M0 ]m −¯ h2 + =0 (49) 100mσ03 100σ02 and σ0 =
¯2 h ¯ m2 2.34GM
(50)
d2 Vef f |σ0 dσ 2
(51)
The normal mode frequency ω is given by mω 2 = If we choose m > 10−20 eV, we get ν⇒
¯ 2 m3 ω 5.48G2 M ≈ 1.3 × 10−6 Hz > 2π 10¯h3
(52)
The emitted gravitational radiation is in the lower frequency of the GW spectrum where future space-based detectors can sense the signals.
4. HAWKING AREA THEOREM AND BLACKHOLE MERGERS Einstein’s general relativity entails second law of BH thermodynamics according to which no classical process can decrease the area of a BH event horizon (EH) so that,31 dA ≥0, dt
(53)
where A is the EH area. In particular, if two BHs with EH area A1 and A2 merge then the resulting BH will necessarily have an EH area A greater than A1 + A2 . In what follows, we verify this result for the four GWs events - GW150914, GW151226, GW170104 and GW170814, that were detected via the laser interferometric techniques of LIGO and VIRGO.27–30 For a Kerr BH, area of its EH is given by, ! !2 q 2 G A = 8π 2 M M + M 2 − Lc/GM (54) c where M and L are the mass and spin angular momentum of the BH. From the above equation, it is obvious that for a given mass M , the EH area is maximum when the BH is a Schwarzschild BH (i.e. L = 0). Hence, to make an extra-stringent test of the second law, we assume that the two initial BHs are of Schwarszchild type (in any case, for the four GW events, the angular momenta of the initial BHs are not very well inferred) so that the initial total EH area is given by, !2 i G h 2 Ai = 16π 2 (M1 + M22 (55) c
After the BH-BH merger as a result of the loss of orbital energy energy due to emission of GWs, the final EH area of the ensuing Kerr BH is given by, ! q G2 2 2 Af = 8π 4 Mf 1 + 1 − Lc/GMf2 (56) c The ratio of final EH area to initial EH area is therefore, q h i q h 2 i 2 Mf2 1 + 1 − a2f M 1 + 1 − lc/GMf2 f Af i i h h ⇒ = Ai 2 (M12 + M22 2 (M12 + M22
(57)
+5.0 +4.0 For the event GW150914, parameters are as follows: M1 = 29+4.0 −4.0 M — M2 = 36−4.0 M — Mf = 62−4.0 M +0.05 and af = 0.67−0.07 If we take M1 = 29M |M2 = 36M |Mf = 62M and af = 0.67 and apply in Eq.(57), we get Af /Ai = 1.57 For M1 = 33M |M2 = 41M |Mf = 58M and af = 0.72, Af /Ai min = 1.03 and for M1 = 25M |M2 = 32M |Mf = 66M and af = 0.6, Af /Ai max = 2.38 +0.81 ie. Af /Ai can have the value 1.57−0.54 for the LIGO event GW150914.
The above approach is done for all the four events of GW detection from binary blackhole collisions and is tabulated as shown. Event GW150914 GW151226 GW170104 GW170814
M1
M2
Mf
af
29+4.0 −4.0 M 14.2+8.3 −3.7 31.2+8.4 −6.0 M 30.5+5.7 −3.0 M
36+5.0 −4.0 M 7.5+2.3 −2.3 19.4+5.3 −5.9 M 25.3+2.8 −4.2 M
62+4.0 −4.0 M 20.8+6.1 −1.7 48.7+5.7 −4.6 M 53.2+3.2 −2.5 M
0.67+0.05 −0.07 0.74+0.06 −0.06 0.64+0.09 −0.20 0.70+0.07 −0.05
Af /Ai 1.57+0.81 −0.54 1.40+3.16 −0.92 1.55+1.88 −0.8 1.54+0.78 −0.54
For the events, GW151226 and GW170104, although the maximum values of the ratio are larger than unity as expected from the second law, minimum values are less than unity, which suggests that the initial BHs were not of Schwarzschild type, instead must have been rotating BHs.
5. DISCUSSIONS 10
SuperMassive Kerr Blackholes(M ≈ 10 M ) can be formed via the gravitational collapse of the BEC structures of ultralight scalars(m > 10−20 eV).It is seen that, for higher angular momenta( high values of l and m), the mass constraints also increase. The collapse can result in the emission of gravitational radiation and the order of magnitude calculations for amplitude, frequency and luminosity of the wave are obtained. Also, Hawking Area Theorem is tested for the case of all the four events of LIGO/VIRGO detection of Gravitational Waves, strongly validating the existence of Black holes.
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