Modified Gravitation Theory (MOG) and the aLIGO GW190521 Gravitational Wave Event
aa r X i v : . [ g r- q c ] S e p Modified Gravitation Theory (MOG) and the aLIGO GW190521Gravitational Wave Event
J. W. MoffatPerimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, CanadaandDepartment of Physics and Astronomy, University of Waterloo, Waterloo,Ontario N2L 3G1, CanadaSeptember 10, 2020
PACS: 04.50.Kd,04.30.Db,04.30Nk,04.30Tv
Abstract
A consequence of adopting a modified gravitational theory (MOG) for the aLIGO GW190521 grav-itational wave detection involving binary black hole sources is to fit the aLIGO strain and chirp datawith lower mass, compact coalescing binary systems such as neutron star-neutron star (NS-NS), blackhole - neutron star (BH-NS), and black hole-black hole (BH-BH) systems. In MOG BH - BH componentmasses can be smaller than the component masses m = 85 M ⊙ and m = 66 M ⊙ inferred from the aLIGOGW190521 gravitational wave event. This reduces the mass of the final remnant mass M f = 150 M ⊙ andallows the primary, secondary and final remnant masses of the black holes to be formed by conventionalstellar collapse models. In a previous paper, we investigated gravitational waves in modified gravity theory (MOG) [1]. The grav-itational wave event sourced by a black hole-black-hole (BH-BH) binary system with component masses m = 10 M ⊙ and m = 8 M ⊙ was fitted to the aLIGO strain and chirp data for the GW 150914 gravitationalwave event [2, 3, 4]. It was argued that the lower black hole masses used to fit the data were in betteragreement with the black hole masses determined from X-ray binary observations, which have the upperbound M BH ∼ M ⊙ . The gravitational wave detection event GW190521 [5, 6] has created an exciting newsituation in astrophysics, because the primary component mass m = 85 +21 − M ⊙ and secondary componentmass m = 66 +17 − M ⊙ cannot be formed in standard stellar collapse scenarios. The mass m = 85 M ⊙ is inthe range where pair instability will suppress BH formation, due to the production of electron-positron pairsin the stellar cores, softening the equation of state by removing pressure support [7]. The core temperatureis increased igniting oxygen or silicon, and the star becomes unstable. For helium cores M He /M ⊙ > < M He /M ⊙ < ∼ M ⊙ [8]. Models have been proposed to avoid the pairinstability in the cores of stars and supernovae explosions [5, 6]. A hierarchical model of second and thirdgeneration BH mergers has been proposed as well as the possibility that heavy black holes can be formed inthe star rich and gaseous accretion disks of supermassive black holes in active galactic nuclei (AGNs). So far,no explanation for the evolution and channel formation of massive intermediate binary BH-BH such as theGW190521 BHs has been conclusively shown to solve the intermediate mass BH problem. Evolution modelshave been proposed to explain how the high-mass binary black hole – black hole BH-BH, neutron star – neu-tron star (NS-NS) and black hole - neutron star (BH-NS) systems could be evolved [3, 4, 9, 10, 11, 12, 13, 14].1or the binary BH system possible evolutionary channels that could allow for a primary component mass m = 85 M ⊙ has has been reviewed in ref. [6]. An application of MOG to the problem of the gravitationalwave binary merger GW190814 [15]. The modified Tolman-Oppenheimer-Volkoff equation allows for a heav-ier neutron star and resolves the problem of the 2 . − M ⊙ mass gap between known neutron stars andBHs [16].In scalar-tensor-vector gravity theory (MOG) [17] as well as the spin 2 graviton metric tensor field g µν there is a gravitational spin 1 vector field coupling to matter. The massive vector field φ µ is coupled tomatter through the gravitational charge Q g = √ αG N M , where α is a dimensionless scalar field whichin applications of the theory to experiment is treated approximately as a constant parameter and G N 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 N (1 + α ) [18, 19, 20, 21, 22]. It can be shown that for the conservation of the gravitational charge ˙ Q g = 0, andthe conservation of mass ˙ M = 0 there is no monopole gravitational wave radiation. Moreover, because Q g = √ αG N M >
0, there is no dipole gravitational wave emission for a massive source. A feature of themotion 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 wavesignal is detected in the frequency range 35 - 300 Hz. The ringdown phase results in a remnant quiescentBH with a total mass M f and spin parameter a = cJ/G N M , where J is the spin angular momentum. Inthe 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 belowered significantly by MOG to be consistent with standard stellar mass collapse channels. The field equations for the case G = G N (1 + α ) = constant and Q = √ αG N M , ignoring in the presentuniverse the small φ µ field mass m φ ∼ − eV, are given by [18, 22]: R µν = − πGT φµν , (1) ∇ ν B µν = 1 √− g ∂ ν ( √− gB µν ) = 0 , (2) ∇ σ B µν + ∇ µ B νσ + ∇ ν B σµ = 0 . (3)The energy-momentum tensor T φµν is T φµν = − π ( B µα B να − δ µν B αβ B αβ ) . (4)The exact Kerr-MOG black hole solution metric is given by [18, 22]: ds = ∆ ρ ( dt − a sin θdφ ) − sin θρ [( r + a ) dφ − adt ] − ρ ∆ dr − ρ dθ , (5)where ∆ = r − GM r + a + α (1 + α ) G N M , ρ = r + a cos θ. (6)Here, a = J/M c is the Kerr spin parameter where J denotes the angular momentum ( a = cJ/G N M in dimensionless units). The spacetime geometry is axially symmetric around the z axis. Horizons aredetermined by the roots of ∆ = 0: r ± = G N (1 + α ) M (cid:20) ± s − a G N (1 + α ) M − α α (cid:21) . (7)An ergosphere horizon is determined by g = 0: r E = G N (1 + α ) M (cid:20) s − a cos θG N (1 + α ) M − α α (cid:21) . (8)2he 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 generalizedSchwarzschild black hole metric solution: ds = (cid:18) − G N (1 + α ) Mr + α (1 + α ) G N M r (cid:19) dt − (cid:18) − G N (1 + α ) Mr + α (1 + α ) G N M r (cid:19) − dr − r d Ω . (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 = G N (1 + α ) and the mass M scales as M = M MOG / (1 + α ). There are two independent gravitational wave polarization strains, h + ( t ) and h × ( t ) in GR and MOG. Duringthe inspiral of the black holes the polarization strains are given by h + ( t ) = A GW ( t )(1 + cos ι ) cos( φ GW ( t )) , (10) h × ( t ) = − A GW ( t ) cos ι sin( φ GW ( t )) , (11)where A GW ( t ) and φ GW ( t ) denote the amplitude and phase, respectively, and ι is the inclination angle.Post-Newtonian theory is used to compute φ GW ( t, m , , S , ) where S and S denote the black hole spins,and the perturbative expansion is in powers of v/c ∼ . − .
5. The gravitational wave phase is φ GW ( t ) ∼ π (cid:18) f t + 12 ˙ f t (cid:19) + φ , (12)where f is the gravitational wave frequency. In MOG the strain h + ( t ) can be expressed as h + ( t ) ∼ G ( R ) m m DR ( t ) c (1 + cos ι ) cos (cid:18)Z t f ( t ′ ) dt ′ (cid:19) , (13)where D is the distance to the binary system source, G ( R ) is the effective weak gravitational strength [17]: G ( r ) = G N [1 + α − α exp( − µr )(1 + µr )] , (14)and R ( t ) is the radial distance of closest approach during the inspiraling merger. We have f ( t ) = 5 / π (cid:18) c GM c (cid:19) / ( t coal − t ) − / , (15)where M c is the chirp mass: M c = ( m m ) / ( m + m ) / = c G (cid:20) π − / f − / ˙ f (cid:21) / . (16)For frequency f the characteristic evolution time is t evol ≡ f ˙ f = 83 ( t coal − t ) = 596 π / c f / ( GM c ) / , (17)and the chirp ˙ f is given by ˙ f = 965 c f GM c (cid:18) πfc GM c (cid:19) / . (18)For well-separated binary components and µ − ≪
24 kpc, where µ − is determined by fitting MOG togalaxy rotation curves and stable galaxy cluster dynamics [23, 24, 25], we have G ∼ G N , while for the strong3able 1: Summary of values of α , m , m , chirp mass M c and final mass M f for GW190521 α m ( M ⊙ ) m ( M ⊙ ) M c ( M ⊙ ) M f G ∼ G N (1 + α ) [1]. For two orbiting black holes each of whichis described by the Kerr-MOG metric (5), the gravitational charges Q g = √ αG N m and Q g = √ αG N m and spins S and S merge to their final values for the quiescent black hole after the ringdown phase. Duringthis stage the repulsive force exerted on the two black holes, due to the gravitational vector field charges Q g and Q g , decreases to zero and G = G N (1 + α ) and Q gf = √ αG N M f where α and M f are the final valuesof the quiescent black hole α and mass. The repulsive vector force only partially cancels the attractive forceduring the rapid coalescing strong gravity phase.We have for G ∼ G N (1 + α ) in the final coalescing phase: h + ( t ) ∼ G N (1 + α ) m m DR ( t ) c (1 + cos ι ) cos (cid:18)Z t f ( t ′ ) dt ′ (cid:19) , (19) M c = ( m m ) / ( m + m ) / = c G N (1 + α ) (cid:20) π − / f − / ˙ f (cid:21) / , (20)and ˙ f = 965 c fG N (1 + α ) M c (cid:18) πfc G N (1 + α ) M c (cid:19) / , (21)where M c and ˙ f denote the chirp mass and chirp, respectively.An alternative scenario for the merging of compact binary systems is obtained from MOG comparedto the scenario based on GR. As the two compact objects coalesce and merge to the final black hole with G ∼ G N (1 + α ), a range of values of the parameter α can be chosen. The increase of G in the final stageof the merging of the black holes can lead to a fitting of the GW 190521 data for binary BH-BH systems inagreement with the observed aLIGO values of the strain h + and chirp ˙ f . We have for the GR componentmasses m = 85 M ⊙ and m = 66 M ⊙ the chirp mass M c GR = 64 M ⊙ and for G N (1+ α ) M c MOG = G N M c GR ,we have α = M c GR − M c MOG M c MOG . (22)In Table 1, we show values of α , m , m and M c for the binary BH GW190521 merging system.The value of G ∼ G N , obtained from the weak gravitational and slow velocity formula (14) is no longervalid for strong gravitational fields in the final merging stage of the BHs. With α > G ∼ G N (1 + α ),we can fit the audible chirp signal LIGO data with m and m chosen for the BH-BH binary systems. Asthe compact objects coalesce and the distance R decreases towards the distance of closest approach, thefinal quiescent BH will have a total mass M f , less the amount of mass-energy, M GW ∼ M ⊙ , emitted bygravitational wave emission. After the ringdown phase the quiescent black hole will be described by theKerr-MOG metric (5), and the quasi-normal modes predicted by MOG can be calculated [21]. 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 = G N (1 + α ) for α >
0, the primary andsecondary 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 M f is decreased and removes the need to4equire the existence of a black hole mass in the intermediary BH mass gap, reinstating the standard pairinstability mass gap boundary value M BH < M ⊙ . The detection of many more massive BH binary systemslike the GW190521 system by the gravitational wave observatories will increase the necessity of finding asolution to the formation channel needed to explain the massive BH primary and secondary components andthe final BH remnant mass detected in the GW190521 system. This would make a modification of GR aviable solution to the problem. Acknowledgments
I thank Martin Green and Viktor Toth for helpful discussions. Research at the Perimeter Institute forTheoretical Physics is supported by the Government of Canada through industry Canada and by the Provinceof Ontario through the Ministry of Research and Innovation (MRI).
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