Modified Gravitation Theory (MOG) and the Aligo GW190521 Gravitational Wave Event

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BH mass 66 = < r ytmwt opnn masses component with system ary o eimcores helium For . bevtos hc aeteupper the have which observations, acia oe fscn n third and second of model rarchical las rdrc olpeproducing collapse direct or ollapse e ofi h aawr nbetter in were data the fit to sed − +21 3,tepi ntblt evsno leaves instability pair the 135, 14 M M ⊙ ⊙ nerdfo h aLIGO the from inferred n eodr component secondary and L25 Canada 2Y5, 2L M tda xiignew exciting an ated f 150 = terloo, m M 1 He 85 = M /M ⊙ vitational and M ⊙ ⊙ > sin is 32, For the binary BH system possible evolutionary channels that could allow for a primary component mass m1 = 85M⊙ has has been reviewed in ref. [6]. An application of MOG to the problem of the gravitational wave binary merger GW190814 [15]. The modified Tolman-Oppenheimer-Volkoff equation allows for a heav- ier neutron star and resolves the problem of the 2.5 5M⊙ mass gap between known neutron stars and BHs [16]. − In scalar-tensor-vector gravity theory (MOG) [17] as well as the spin 2 graviton metric tensor field gµν there is a gravitational spin 1 vector field coupling to matter. The massive vector field φµ is coupled to matter through the gravitational charge Qg = √αGN M, where α is a dimensionless scalar field which in applications of the theory to experiment is treated approximately as a constant parameter and GN is Newton’s gravitational constant. The latter is true for the exact matter-free vacuum Schwarzschild- MOG and Kerr-MOG solutions for which the general coupling strength G constant and G = G (1 + ∼ N α) [18, 19, 20, 21, 22]. It can be shown that for the conservation of the gravitational charge Q˙ g = 0, and the conservation of mass M˙ = 0 there is no monopole gravitational wave radiation. Moreover, because Qg = √αGN M > 0, there is no dipole gravitational wave emission for a massive source. A feature of the motion of particles in MOG is that the weak equivalence principle is satisfied [17, 22]. The final merging of the black holes occurs in a very short time duration and the gravitational wave signal is detected in the frequency range 35 - 300 Hz. The ringdown phase results in a remnant quiescent 2 BH with a total mass Mf and spin parameter a = cJ/GN M , where J is the spin angular momentum. In the MOG gravitational theory, the final BH remnant will be described by the generalized Kerr solution [18]. The high component masses and chirp mass for the binary BH that fit the LIGO GW190521 data can be lowered significantly by MOG to be consistent with standard stellar mass collapse channels.

2 MOG Field Equations and Generalized Kerr Black Hole

The field equations for the case G = GN (1 + α) = constant and Q = √αGN M, ignoring in the present universe the small φ field mass m 10−28 eV, are given by [18, 22]: µ φ ∼ R = 8πGT φ , (1) µν − µν 1 Bµν = ∂ (√ gBµν )=0, (2) ∇ν √ g ν − − B + B + B =0. (3) ∇σ µν ∇µ νσ ∇ν σµ φ ν The energy-momentum tensor T µ is

ν 1 1 T φ = (B Bνα δ ν BαβB ). (4) µ −4π µα − 4 µ αβ The exact Kerr-MOG black hole solution metric is given by [18, 22]:

∆ sin2 θ ρ2 ds2 = (dt a sin2 θdφ)2 [(r2 + a2)dφ adt]2 dr2 ρ2dθ2, (5) ρ2 − − ρ2 − − ∆ − where ∆= r2 2GMr + a2 + α(1 + α)G2 M 2, ρ2 = r2 + a2 cos2 θ. (6) − N 2 Here, a = J/Mc is the Kerr spin parameter where J denotes the angular momentum (a = cJ/GN M in dimensionless units). The spacetime geometry is axially symmetric around the z axis. Horizons are determined by the roots of ∆ = 0:

a2 α r± = G (1 + α)M 1 1 . (7) N ± − G2 (1 + α)2M 2 − 1+ α  s N 

An ergosphere horizon is determined by g00 = 0:

a2 cos2 θ α r = G (1 + α)M 1+ 1 . (8) E N − G2 (1 + α)2M 2 − 1+ α  s N 

2 The solution is fully determined by the Arnowitt-Deser-Misner (ADM) mass M and spin parameter a measured by an asymptotically distant observer. When a = 0 the solution reduces to the generalized Schwarzschild black hole metric solution: −1 2G (1 + α)M α(1 + α)G2 M 2 2G (1 + α)M α(1 + α)G2 M 2 ds2 = 1 N + N dt2 1 N + N dr2 r2dΩ2. − r r2 − − r r2 −     (9) When the parameter α = 0 the generalized solutions reduce to the GR Kerr and Schwarzschild BH solutions. The constant gravitational strength scales as G = GN (1 + α) and the mass M scales as M = MMOG/(1 + α).

3 LIGO GW 1509521 Gravitational Wave detection and Binary Black Holes

There are two independent gravitational wave polarization strains, h+(t) and h×(t) in GR and MOG. During the inspiral of the black holes the polarization strains are given by

2 h+(t)= AGW (t)(1 + cos ι)cos(φGW (t)), (10)

h×(t)= 2A (t)cos ι sin(φ (t)), (11) − GW GW where AGW (t) and φGW (t) denote the amplitude and phase, respectively, and ι is the inclination angle. Post-Newtonian theory is used to compute φGW (t,m1,2,S1,2) where S1 and S2 denote the black hole spins, and the perturbative expansion is in powers of v/c 0.2 0.5. The gravitational wave phase is ∼ −

1 2 φGW(t) 2π ft + ft˙ + φ0, (12) ∼ 2   where f is the gravitational wave frequency. In MOG the strain h+(t) can be expressed as

2 t (R)m1m2 2 ′ ′ h+(t) G (1 + cos ι)cos f(t )dt , (13) ∼ DR(t)c4 Z  where D is the distance to the binary system source, (R) is the effective weak gravitational strength [17]: G (r)= G [1 + α α exp( µr)(1 + µr)], (14) G N − − and R(t) is the radial distance of closest approach during the inspiraling merger. We have

3/8 3 5/8 5 c −3/8 f(t)= (tcoal t) , (15) 8π − GMc  where is the chirp mass: Mc 3/5 3 3/5 (m1m2) c 5 −8/3 −11/3 ˙ c = 1 5 = π f f . (16) M (m1 + m2) / 96 G   For frequency f the characteristic evolution time is

f 8 5 c5 evol coal t = (t t)= 8/3 8/3 5/3 , (17) ≡ f˙ 3 − 96π f ( c) GM and the chirp f˙ is given by 8 3 96 c3f πf / f˙ = . (18) 5 c3 GMc GMc   For well-separated binary components and µ−1 24kpc, where µ−1 is determined by fitting MOG to galaxy rotation curves and stable galaxy cluster dynamics≪ [23, 24, 25], we have G , while for the strong G ∼ N

3 Table 1: Summary of values of α, m1,m2, chirp mass and final mass M for GW190521 Mc f

α m1(M⊙) m2(M⊙) (M⊙) M Mc f 0 85 66 64 150 2 28.3 22 21.3 50 3 21.3 16.5 16 37.5 4 17 13.2 12.8 30

dynamical field merging of binary components G (1+α) [1]. For two orbiting black holes each of which G ∼ N is described by the Kerr-MOG metric (5), the gravitational charges Qg1 = √αGN m1 and Qg2 = √αGN m2 and spins S1 and S2 merge to their final values for the quiescent black hole after the ringdown phase. During this stage the repulsive force exerted on the two black holes, due to the gravitational vector field charges Qg1 and Qg2, decreases to zero and G = GN (1 + α) and Qgf = √αGN Mf where α and Mf are the final values of the quiescent black hole α and mass. The repulsive vector force only partially cancels the attractive force during the rapid coalescing strong gravity phase. We have for G (1 + α) in the final coalescing phase: G ∼ N 2 2 t GN (1 + α) m1m2 2 ′ ′ h+(t) (1 + cos ι)cos f(t )dt , (19) ∼ DR(t)c4 Z  3/5 3 3/5 (m1m2) c 5 −8/3 −11/3 ˙ c = 1 5 = π f f , (20) M (m1 + m2) / G (1 + α) 96 N   and 8 3 96 c3f πf / f˙ = G (1 + α) , (21) 5 G (1 + α) c3 N Mc N Mc   where c and f˙ denote the chirp mass and chirp, respectively. AnM alternative scenario for the merging of compact binary systems is obtained from MOG compared to the scenario based on GR. As the two compact objects coalesce and merge to the final black hole with G GN (1 + α), a range of values of the parameter α can be chosen. The increase of G in the final stage of the∼ merging of the black holes can lead to a fitting of the GW 190521 data for binary BH-BH systems in agreement with the observed aLIGO values of the strain h+ and chirp f˙. We have for the GR component masses m1 = 85M⊙ and m2 = 66M⊙ the chirp mass cGR = 64M⊙ and for GN (1+α) cMOG = GN cGR, we have M M M GR MOG α = Mc −Mc . (22) MOG Mc In Table 1, we show values of α, m1,m2 and for the binary BH GW190521 merging system. Mc The value of GN , obtained from the weak gravitational and slow velocity formula (14) is no longer valid for strong gravitationalG ∼ fields in the final merging stage of the BHs. With α> 0 and G G (1 + α), ∼ N we can fit the audible chirp signal LIGO data with m1 and m2 chosen for the BH-BH binary systems. As the compact objects coalesce and the distance R decreases towards the distance of closest approach, the final quiescent BH will have a total mass Mf , less the amount of mass-energy, MGW 8M⊙, emitted by gravitational wave emission. After the ringdown phase the quiescent black hole will be∼ described by the Kerr-MOG metric (5), and the quasi-normal modes predicted by MOG can be calculated [21].

4 Conclusions

By adopting modified gravity MOG for the gravitational wave BH binary system GW190521, the enhance- ment of the gravitational strength of the binary system by G = GN (1 + α) for α > 0, the primary and secondary component masses can be decreased to allow for a standard stellar mass collapse channel to de- scribe the formation of the BH binary system. The final mass Mf is decreased and removes the need to

4 require the existence of a black hole mass in the intermediary BH mass gap, reinstating the standard pair instability mass gap boundary value MBH < 50M⊙. The detection of many more massive BH binary systems like the GW190521 system by the gravitational wave observatories will increase the necessity of finding a solution to the formation channel needed to explain the massive BH primary and secondary components and the final BH remnant mass detected in the GW190521 system. This would make a modification of GR a viable solution to the problem.

Acknowledgments

I thank Martin Green and Viktor Toth for helpful discussions. Research at the Perimeter Institute for Theoretical Physics is supported by the Government of Canada through industry Canada and by the Province of Ontario through the Ministry of Research and Innovation (MRI).

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