Theoretical physics implications of the binary black-hole mergers GW150914 and GW151226

The gravitational wave observations GW150914 and GW151226 by Advanced LIGO provide the first opportunity to learn about physics in the extreme gravity environment of coalescing binary black holes. The LIGO Scientific Collaboration and the Virgo Collaboration have verified that this observation is consistent with Einstein’s theory of general relativity, constraining the presence of certain parametric anomalies in the signal. This paper expands their analysis to a larger class of anomalies, highlighting the inferences that can be drawn on nonstandard theoretical physics mechanisms that could otherwise have affected the observed signals. We find that these gravitational wave events constrain a plethora of mechanisms associated with the generation and propagation of gravitational waves, including the activation of scalar fields, gravitational leakage into large extra dimensions, the variability of Newton’s constant, the speed of gravity, a modified dispersion relation, gravitational Lorentz violation and the strong equivalence principle. Though other observations limit many of these mechanisms already, GW150914 and GW151226 are unique in that they are direct probes of dynamical strong-field gravity and of gravitational wave propagation. We also show that GW150914 constrains inferred properties of exotic compact object alternatives to Kerr black holes. We argue, however, that the true potential for GW150914 to both rule out exotic objects and constrain physics beyond general relativity is severely limited by the lack of understanding of the coalescence regime in almost all relevant modified gravity theories. This event thus significantly raises the bar that these theories have to pass, both in terms of having a sound theoretical underpinning and reaching the minimal level of being able to solve the equations of motion for binary merger events. We conclude with a discussion of the additional inferences that can be drawn if the lower-confidence observation of an electromagnetic counterpart to GW150914 holds true, or such a coincidence is observed with future events; this would provide dramatic constraints on the speed of gravity and gravitational Lorentz violation.

Figure 1

(Left) Third-order PN estimates of the orbital separation (top) and velocity (bottom) as a function of the GW frequency (see also Fig. 2 of [1]). (Right) An estimate of the square root of the spectral noise density curve of aLIGO when GW150914 was detected (as interpolated from the data made publicly available by the LVC [121] as described in Appendix pp3), and two models (PhenomB [90] and PhenomD [92, 93]) of the amplitude of the GW Fourier spectrum of GW150914 (GW151226) multiplied by twice the square root of the frequency and scaled to SNR 24 (13).

Figure 2

Schematic diagram of the curvature-potential phase space sampled by various experiments that test GR. The vertical axis shows the inverse of the characteristic curvature length scale, while the horizontal axis shows the characteristic gravitational potential, based on Table 2. GW150914 and GW151226 sample a regime where the curvature and the potential are both simultaneously large and dynamical, indicated here by the finite range the curves sweep in the figure. The finite area of pulsar timing arrays is due to the range in the GW frequency and the total mass of supermassive BH binaries that such arrays may detect in the future. Figure 3 is a companion plot that illustrates the dynamical aspects of gravity probed by these experiments; the lighter (blue) dots here are to indicate that the Shapiro time delay from binary pulsars and the Cassini satellite do not give information on the dynamical regime.

Figure 3

(Left) Schematic diagram of the curvature-radiation reaction time-scale phase space sampled by relevant experiments shown in Fig. 2. As is evident, GW150914 and GW151226 sample a regime of dynamic gravity where the radiation-reaction time scale is the shortest by many orders of magnitude. (Right) Characteristic curvature and strength of the Newtonian gravitational potential as a function of GW frequency.

Figure 4

90%-confidence constraints on the ppE parameter $|\beta |$ at $n$th PN order. The green crosses represent the bounds reported in [5, 19] through a Bayesian analysis of event GW150914, mapped to constraints on $\beta$. The red (magenta) dots and line represent bounds from GW150914 (GW151226) estimated with a Fisher analysis, using the IMRPhenom waveform (without spin precession) and a fit to the aLIGO spectral noise density. The constraints obtained with a Fisher analysis agree very well with the Bayesian constraint reported in [5, 19]. The blue dotted line shows projected constraints predicted in 2011 by [142] for a system similar to GW151226. The dashed black line is a rough estimate on the constraints that the double binary pulsar PSR J0737-3039 [127, 128, 129] can place on the ppE $\beta$ parameter [188], while the cyan star refers to the bound on $\beta$ at 1PN from the perihelion precession of Mercury [150]. Binary pulsar observations can constrain negative-PN-order deviations better than aLIGO, while aLIGO does better than binary pulsar observations at higher PN order, as first calculated in [188]. However, note also that binary pulsar and Solar System bounds cannot be directly compared to GW ones as the binary pulsar (Solar System) one corresponds to the extreme case of no conservative (no dissipative) corrections. Moreover, stronger constraints on $\beta$ for these latter tests do not necessarily mean stronger constraints on modifications to GR for BH mergers, as $\beta$ depends not only on theoretical coupling parameters but also on system parameters, and in certain theories (like EdGB gravity), non-GR corrections are suppressed in stars compared to BHs.

Figure 5

Upper bound on corrections to the binding energy $|A|$, energy flux $|B|$ [see Eq. (28)] and a combination of these two $|C|$ [see Eq. (29)] as a function of the PN order that they enter for GW150914 (red) and GW151226 (blue).

Figure 6

90%-confidence upper bound on the ppE parameter $|\beta |$ at $-1\mathrm{PN}$ as a function of the transition frequency $f_{*}$ for GW150914 (top) and GW151226 (bottom), for theories in which the scalar field only activates when $f>f_{*}$ (red solid curve) and $f<f_{*}$ (blue dashed curve). The vertical dotted-dashed line corresponds to the transition frequency $f_{\mathrm{Int}}$ between the inspiral and intermediate phase in the IMRPhenom waveform.

Figure 7

Upper bound on the amplitude of the correction to the graviton dispersion relation $\mathbb{A}$ in Eq. (22) from GW150914 (the GW151226 bound is almost indistinguishable) as a function of $\alpha$ for $\mathbb{A}>0$ (top) and $\mathbb{A}<0$ (bottom) obtained from 90%-confidence constraints on the ppE parameter $\beta$. The top axis shows the corresponding PN order at which the correction enters. The red circles are the Fisher estimates derived in this paper, while the green crosses are a mapping of the Bayesian bound in [19] on the graviton mass through $m_g=\sqrt{\mathbb{A}}$ at $\alpha =0$. Cyan stars are rough bounds on $\mathbb{A}$ in Eq. (30) based on the Bayesian bound at $\alpha =0$. The magenta squares correspond to a bound derived from the time of arrival of GW150914 at Hanford and Livingston [87]. Blue pluses present the bound on $\mathbb{A}$ from the absence of gravitational Cherenkov radiation in cosmic ray observations [84], assuming that cosmic ray particles of $p=10^{11}\mathrm{GeV}$ arrive from a distance of 100 Mpc. The GW150914 observation constrains $|\mathbb{A}|$, while cosmic ray observations only constrain the negative sector of $\mathbb{A}$. The former places a stronger bound on the $\mathbb{A}<0$ region than the latter for $\alpha \lesssim 2$, while it places a unique bound on the $\mathbb{A}>0$ region.

Figure 8

90%-confidence upper bound on the amplitude of an additional ringdown mode ${\overline{\gamma }}_1$ relative to the amplitude ${\gamma }_1$ of the primary $\ell =m=2$ mode as a function of the former’s ringdown frequency ${\overline{f}}_{\mathrm{RD}}$ and damping time $\overline{\tau }$. ${\overline{\gamma }}_1$ can be constrained to be less than $\sim 10\%$ of the primary mode’s amplitude if the damping time is larger than 100 ms, which is typical for boson star QNMs. The constraint becomes relatively weak around the white dot corresponding to the frequency and damping time of the primary $\ell =m=2$ mode, due to degeneracies between the primary mode and the additional mode. The black dot represents the subleading $\ell =m=3$ mode of a BH [36], whose amplitude is smaller than 10% [37].

Figure 9

GW150914 constraints on the fractional deviation in the propagation speed of GWs away from the speed of light for the ${\delta }_g>0$ (top) and ${\delta }_g<0$ (bottom) region, assuming that the event Fermi observed was associated with GW150914. We also show the cosmic ray constraints from the absence of the gravitational Cherenkov radiation in Eq. (46) in the top panel. The vertical dotted-dashed line in the bottom panel corresponds to $\mathrm{\Delta }{\tau }_{\mathrm{int}}=0.4\mathrm{s}$ assumed in [108, 109]. We do not show the region with $\mathrm{\Delta }{\tau }_{\mathrm{int}}<0.256\mathrm{s}$, since then the binning of the Fermi observation in the time dominates the error budget over $\mathrm{\Delta }{\tau }_{\mathrm{int}}$.

Figure 10

Comparison of 90%-confidence GW150914 constraints on $|\beta |$ with PhenomD and PhenomB waveforms for modified generation (left) and propagation (right) effects on GWs. We also show the PhenomD result with a sudden transition of the non-GR effect at $f=f_{\mathrm{Int}}$ like PhenomB, which corresponds to the sudden deactivation of the scalar field with a transition at $f_{*}=f_{\mathrm{Int}}$ in Sec. 3a2 (but for arbitrary $b$). Green dotted-dashed curves present a rough estimate of the impact of mismodeling error in the PhenomD waveform on constraints on $|\beta |$, which serve as the threshold on future $|\beta |$ constraints.

Figure 11

(Left) Probability distribution of the PhenomD waveform dephasing between the injected parameters and parameters within statistical errors at $f=50\mathrm{Hz}$ with a ppE modification at $-1\mathrm{PN}$ order. The asymmetry in the distribution arises from the condition $\eta \le 0.25$. (Right) The mean of the probability distribution of the dephasing as a function of the PN order of the ppE modification, together with $1\sigma$ error bars. The mean and $1\sigma$ error of the left panel (at $-1\mathrm{PN}$ order) are shown in blue. The absolute dephasing within $1\sigma$ errors of the distribution can be as large as 10–50 rad, which is much larger than the maximum mismodeling dephasing of 0.015 rad. This suggests that the latter is negligible in constraining $\beta$.

Figure 12

(Top) Fisher estimates of the accuracy one can constrain the ppE parameter ${\beta }_{\mathrm{BD}}$ at $-1\mathrm{PN}$ with GW150914 and GW151226, as a function of the highest PN order included in the modified waveform phase. The bound at $-1\mathrm{PN}$ order is the same as that in Fig. 4. For reference, we also show the bound from a NS-BH and NS-NS binary with $\mathrm{SNR}=24$ and with masses $(10,1.4)M_{\odot }$ and $(1.5,1.3)M_{\odot }$, respectively. (Bottom) The absolute fractional difference of the bound on $|{\beta }_{\mathrm{BD}}|$ as a function of PN order. Including higher-order corrections only affects the bound obtained with only the leading-PN-order phase correction by at most $\mathcal{O}(10\%)$ for GW150914. The fractional difference for GW151226, NS-BH and NS-NS binaries is even smaller.

Figure 13

(Top) Upper bound on the leading ppE parameter $|\beta |$ for GW150914 with only the leading PN correction at $n\mathrm{PN}$ order added (red solid line) and with relative corrections the same as BD theory added up to $(n+2.5)\mathrm{PN}$ order (blue dashed line). The vertical dotted-dashed line shows $n=-1$, which corresponds to the BD case in Fig. 12. (Bottom) The fractional difference between the two curves in the top panel. Such a difference generally becomes larger for high PN terms but is smaller than $\sim 20\%$ in most cases.

Figure 14

(Top) Square root of the noise spectral density as a function of frequency (in hertz) for the Hanford detector during the O1 (red solid line) and using the fitted function of Eq. (c1) (blue dashed line). (Bottom) The relative fractional difference between the data and the fit. The data contains spikes that are absent from the fit, but we have checked that these spikes do not significantly affect the constraints quoted in this paper, if the SNR is fixed.

Figure 15

GW150914 upper bound on $\sqrt{|{\alpha }_{\mathrm{EdGB}}|}$ in kilometers derived by mapping the constraint on $\beta$ at $-1\mathrm{PN}$ order with each (${\chi }_1$, ${\chi }_2$) allowed from the aLIGO measurement. Orange dashed lines show the allowed region from the effective spin ${\chi }_{\mathrm{eff}}$ measurement, while white curves show spins that shut off dipole radiation completely. However, note that the small-coupling approximation used to derive these bounds is only valid within the region between the yellow curves.