Extreme gravity tests with gravitational waves from compact binary coalescences: (I) inspiral–merger

Abstract

The observation of the inspiral and merger of compact binaries by the LIGO/Virgo collaboration ushered in a new era in the study of strong-field gravity. We review current and future tests of strong gravity and of the Kerr paradigm with gravitational-wave interferometers, both within a theory-agnostic framework (the parametrized post-Einsteinian formalism) and in the context of specific modified theories of gravity (scalar–tensor, Einstein–dilaton–Gauss–Bonnet, dynamical Chern–Simons, Lorentz-violating, and extra dimensional theories). In this contribution we focus on (i) the information carried by the inspiral radiation, and (ii) recent progress in numerical simulations of compact binary mergers in modified gravity.

Appendices

A: Derivation of the black hole scalar charge in decoupled dynamical Gauss–Bonnet gravity

The goal of this appendix is to derive the scalar charge in D$^2$GB gravity for a stationary black hole, valid to arbitrary order in spin. We closely follow the calculation in dCS gravity in [181]. The scalar charge $\mu$ can be read off from the 1 / r coefficient in the asymptotic behavior of the scalar field at spatial infinity as $\phi = \mu \, M/r + {\mathcal {O}}(M^2/r^2)$, where M is the black hole mass. Since we work within the small-coupling approximation, we can take the background metric to be Kerr, and the above equation becomes

$$\begin{aligned} \frac{\partial }{\partial {{\tilde{r}}}} \left( \varDelta \frac{\partial \phi }{\partial {{\tilde{r}}}} \right) + \frac{1}{\sin \theta } \frac{\partial }{\partial \theta } \left( \sin \theta \frac{\partial \phi }{\partial \theta } \right) =T\,, \end{aligned}$$

(23)

where we work in the rescaled radial coordinate ${{\tilde{r}}} \equiv r/M$ and $\varDelta \equiv {{\tilde{r}}}^2 - 2M {{\tilde{r}}} + \chi ^2$ with $\chi$ representing the dimensionless Kerr parameter, and

$$\begin{aligned} T\equiv & {} - 48 \frac{\alpha _\mathrm{GB}\, M^2}{\Sigma ^5} \left[ {{\tilde{r}}}^6 -15 {{\tilde{r}}}^4 \chi ^2 \cos ^2 \theta +15 {{\tilde{r}}}^2 \chi ^4 \cos ^4 \theta - \chi ^6 \cos ^6 \theta \right] \,, \end{aligned}$$

(24)

with $\Sigma \equiv {{\tilde{r}}}^2 + \chi ^2 \cos ^2\theta$.

In order to solve the above field equation using Green’s functions, we decompose the scalar field $\phi$ and the source term T as [312]

$$\begin{aligned} \phi= & {} \frac{\alpha _\mathrm{GB}}{M^2} \sum _{\ell } R_{\ell }({{\tilde{r}}})\, S_{\ell }(\theta )\,, \end{aligned}$$

(25)

$$\begin{aligned} T= & {} \frac{\alpha _\mathrm{GB}}{M^2} \sum _{\ell } T_{\ell }({{\tilde{r}}})\, S_{\ell }(\theta )\,, \end{aligned}$$

(26)

where $S_\ell (\theta )$ is normalized as

$$\begin{aligned} 2 \pi \int _0^\pi S_\ell ^2\, \sin \theta \, d\theta = 1\,. \end{aligned}$$

(27)

Inverting Eq. (26), one obtains

$$\begin{aligned} T_\ell = 2 \pi \frac{M^2}{\alpha _\mathrm{GB}} \int _{0}^{\pi } T\, S_{\ell }\, \sin \theta \, d\theta \,. \end{aligned}$$

(28)

Eq. (23) can be split into radial and angular parts as

$$\begin{aligned} \frac{\partial }{\partial {{\tilde{r}}}} \left( \varDelta \frac{\partial R_{\ell }}{\partial {{\tilde{r}}}} \right) -\ell (\ell +1) R_{\ell }= & {} T_{\ell }\,, \end{aligned}$$

(29)

$$\begin{aligned} \frac{1}{\sin \theta } \frac{\partial }{\partial \theta } \left( \sin \theta \frac{\partial S_{\ell }}{\partial \theta } \right) + \ell (\ell +1) S_{\ell }= & {} 0\,. \end{aligned}$$

(30)

The solution to the second equation is nothing but the $m=0$ mode of the spherical harmonics $S_{\ell } = Y_{\ell 0}$.

Let us first derive the scalar monopole charge by concentrating on the $\ell = 0$ mode. The solution to Eq. (29) consists of homogeneous and particular solutions. Let us first study the former. Modulo overall integration constants, homogeneous solutions for the $\ell = 0$ mode of Eq. (29) are given by

$$\begin{aligned} R_0^{(\mathrm {hom},1)} ({{\tilde{r}}})= & {} 1, \end{aligned}$$

(31)

$$\begin{aligned} R_0^{(\mathrm {hom},2)} ({{\tilde{r}}})= & {} \frac{1}{2 \sqrt{1-\chi ^2}} \log \left( \frac{{{\tilde{r}}}-1-\sqrt{1-\chi ^2}}{{{\tilde{r}}}-1 + \sqrt{1-\chi ^2}}\right) . \end{aligned}$$

(32)

The asymptotic behavior of $R_0^{(\mathrm {hom},2)}$ at spatial infinity and at the Kerr horizon ${\tilde{r}}_\mathrm {hor} \equiv 1 + \sqrt{1-\chi ^2}$ is given by

$$\begin{aligned} R_0^{(\mathrm {hom},2,\mathrm {inf})} ({{\tilde{r}}})= & {} \frac{1}{{\tilde{r}}} + {\mathcal {O}}\left( \frac{1}{{\tilde{r}}^{2}} \right) , \end{aligned}$$

(33)

$$\begin{aligned} R_0^{(\mathrm {hom},2,\mathrm {hor})} ({{\tilde{r}}})= & {} \frac{1}{2 \sqrt{1-\chi ^2}} \log \left( \frac{{\tilde{r}}-{\tilde{r}}_\mathrm {hor}}{2\sqrt{1-\chi ^2}} \right) + {\mathcal {O}}[({\tilde{r}}-{\tilde{r}}_\mathrm {hor})]. \end{aligned}$$

(34)

Since EdGB gravity is a shift-symmetric theory, one can set $\phi (\infty ) = 0$ without loss of generality (namely no contribution from $R_0^{(\mathrm {hom},1)}$). Imposing further regularity at the horizon, one finds that the homogeneous solution is absent.

Let us now turn our attention to the particular solution $R_0^{(\mathrm {p})} ({{\tilde{r}}})$. Such a solution is obtained by using the Green’s function constructed from the two independent homogeneous solutions above [313,314,315]:

$$\begin{aligned} R_0^{(\mathrm {p})} ({{\tilde{r}}})= & {} \frac{1}{\varDelta \; W} \left[ R_0^{(\mathrm {hom},2)} ({{\tilde{r}}}) \int _{{{\tilde{r}}}_\mathrm {hor}}^{{{\tilde{r}}}} T_0({{\tilde{r}}}')\, R_0^{(\mathrm {hom},1)} ({{\tilde{r}}}')\, d{{\tilde{r}}}' \right. \nonumber \\&\left. - R_0^{(\mathrm {hom},1)} ({{\tilde{r}}}) \int _\infty ^{{{\tilde{r}}}} T_0 ({{\tilde{r}}}') \,R_0^{(\mathrm {hom},2)} ({{\tilde{r}}}') \, d{{\tilde{r}}}' \right] \,, \end{aligned}$$

(35)

where W is the Wronskian:

$$\begin{aligned} W \equiv R_0^{(\mathrm {hom},1)}\, \frac{d}{d{{\tilde{r}}}} R_0^{(\mathrm {hom},2)} - R_0^{(\mathrm {hom},2)}\, \frac{d}{d{{\tilde{r}}}} R_0^{(\mathrm {hom},1)} = \frac{1}{\varDelta }\,. \end{aligned}$$

(36)

The lower bound of the integral in Eq. (35) is determined such that the solution is regular at the horizon and satisfies $\phi (\infty ) = 0$. For the purpose of studying the leading asymptotic behavior at infinity, one only needs to consider the first term in Eq. (35).

Combining Eqs. (25) and (35) and performing the integral in the latter, one reads off the monopole scalar charge as

$$\begin{aligned} \mu ^\mathrm{GB}= & {} \frac{\alpha _\mathrm{GB}}{M^2} \frac{Y_{00}}{\varDelta \; W} \int _{{{\tilde{r}}}_\mathrm {hor}}^\infty T_0({{\tilde{r}}}')\, R_0^{(\mathrm {hom},1)} ({{\tilde{r}}}')\, d{{\tilde{r}}}' \nonumber \\= & {} 4 \frac{\alpha _\mathrm{GB}}{M^2} \frac{\sqrt{1-\chi ^2}-1 + \chi ^2}{\chi ^2}\,. \end{aligned}$$

(37)

We checked that when we expand the above scalar charge around $\chi = 0$, the expression agrees with that in [81] to ${\mathcal {O}}(\chi ^8)$.

In a similar manner, one can calculate the quadrupolar scalar charge $q^\mathrm{GB}$ by extracting the coefficient of $P_2(\cos \theta ) M^3/r^3$ in the asymptotic behavior of the scalar field at spatial infinity. One finds

$$\begin{aligned} q^\mathrm{GB}= & {} -\frac{4}{3 \chi ^3} \frac{\alpha }{M^2} \left\{ \chi \left[ 2 \chi ^2 \left( \chi ^2+\sqrt{1-\chi ^2}-2\right) -5 \sqrt{1-\chi ^2}+8\right] \right. \nonumber \\&\left. \quad +6 \tan ^{-1}\left( \frac{\sqrt{1-\chi ^2}-1}{\chi }\right) \right\} . \end{aligned}$$

(38)

Again, we checked that an expansion of this expression about $\chi =0$ agrees with that in [81] to ${\mathcal {O}}(\chi ^8)$.

B: A “dynamical no-hair theorem” for black holes in scalar–tensor gravity

The goal of this appendix is to show and explain how black hole binaries do not develop scalar hair upon dynamical evolution. That is, we will explain how the dynamics of a black hole binary system in Bergmann–Wagoner theory with vanishing potential in asymptotically flat spacetimes are the same as in GR, focusing first on the inspiral phase of coalescence.

In the inspiral phase of the binary’s evolution it is appropriate to use the PN approximation, an expansion in powers of $v/c \sim (Gm/rc^2)^{1/2}$. It is convenient to introduce a rescaled version of the scalar field $\phi$: $\varphi \equiv \phi /\phi _0$, where $\phi _0$ is the value of $\phi$ at infinity (assumed to be constant). Mirshekari and Will [191] found the equations of motion for the bodies up to 2.5PN order. Schematically, the relative acceleration ${\mathbf {a}} \equiv {\mathbf {a}}_1-{\mathbf {a}}_2$ takes the form

$$\begin{aligned} a^i =&-\frac{G\alpha m}{r^2}{\hat{n}}^i+\frac{G\alpha m}{r^2}(A_\text {PN}{\hat{n}}^i+B_\text {PN}{\dot{r}}v^i)+\frac{8}{5}\eta \frac{(G\alpha m)^2}{r^3}(A_\text {1.5PN}{\dot{r}}{\hat{n}}^i-B_\text {1.5PN}v^i) \nonumber \\&{}+\frac{G\alpha m}{r^2}(A_\text {2PN}{\hat{n}}^i+B_\text {2PN}{\dot{r}}v^i) \, , \end{aligned}$$

(39)

where $m \equiv m_1+m_2$, $\eta \equiv m_1m_2/m^2$, r is the orbital separation, ${\hat{\mathbf {n}}}$ is a unit vector pointing from body 2 to body 1, and ${\mathbf {v}} \equiv {\mathbf {v}}_1-{\mathbf {v}}_2$ is the relative velocity. The coefficients $A_\text {PN}$, $B_\text {PN}$, $A_\text {1.5PN}$, $B_\text {1.5PN}$, $A_\text {2PN}$, and $B_\text {2PN}$ (which are typically time-dependent) are given in [191]. The symbol G represents the combination $(4+2\omega _0)/[\phi _0(3+2\omega _0)]$ [with $\omega _0 \equiv \omega (\phi _0)$], which appears in the metric component $g_{00}$ in the same manner as the gravitational constant G in GR. However, the coupling in the Newtonian piece of the equations of motion is not simply G but $G\alpha$, where

$$\begin{aligned} \alpha \equiv \frac{3+2\omega _0}{4+2\omega _0}+\frac{(1-2s_1)(1-2s_2)}{4+2\omega _0} \, \end{aligned}$$

(40)

and $s_i$ ($i=1\,,2$) are the sensitivities of the two objects:

$$\begin{aligned} s_A \equiv \left( \frac{d\ln M_{\scriptscriptstyle A}(\phi )}{d\ln \phi }\right) _{\phi =\phi _0} \,. \end{aligned}$$

(41)

Higher-order derivatives of $M_{\scriptscriptstyle A}(\phi )$ are used to define higher-order sensitivities, e.g. $s'_{\scriptscriptstyle A}$ and $s''_{\scriptscriptstyle A}$. Note that in GR radiation reaction begins at 2.5PN order (quadrupole radiation), while in scalar–tensor gravity radiation reaction begins at 1.5PN order, due to the presence of dipole radiation.

All deviations from GR can be characterized using a fairly small number of parameters, all combinations of $\phi _0$, the Taylor coefficients of $\omega (\phi )$, and the sensitivities $s_A$, $s_A'$, and $s_A''$. If one object in the system is a black hole (with the other being a neutron star), the motion of the system is indistinguishable from GR up to 1PN order. All deviations beyond 1PN order depend only on a single parameter, which is a function of $\omega _0$ and the neutron star sensitivity. Unfortunately, this parameter alone (if measured) could not be used to distinguish between Brans–Dicke theory and a more general scalar–tensor theory.

Going beyond the equations of motion, the next step is the calculation of gravitational radiation. The tensor part of the radiation, encoded in ${\tilde{h}}^{ij}$, was computed up to 2PN order by Lang [192]. All deviations depend on the same (small) number of parameters that characterize the equations of motion. For black hole-neutron star systems, the waveform is indistinguishable from GR up to 1PN order, and deviations at higher order depend only on the single parameter described above. For binary black hole systems, the waveform is completely indistinguishable from GR. Scalar radiation has recently been computed by Lang [193] using a very similar procedure. The dipole moment generates the lowest-order scalar waves, which are of $-0.5$PN order:

$$\begin{aligned} \varphi = \frac{4G\mu \alpha ^{1/2}}{R}\zeta {\mathcal {S}}_-({\hat{\mathbf {N}}}\cdot {\mathbf {v}}), \end{aligned}$$

(42)

where $\mu \equiv m_1 m_2/m$ is the reduced mass, ${\hat{\mathbf {N}}} \equiv {\mathbf {x}}/R$ is the direction from the source to the detector, $\zeta \equiv 1/(4+2\omega _0)$, and

$$\begin{aligned} {\mathcal {S}}_- \equiv \alpha ^{-1/2}(s_2-s_1). \end{aligned}$$

(43)

Because computing the radiation up to 2PN order requires knowledge of the monopole moment to 3PN order (relative to itself) and knowledge of the dipole moment to 2.5PN order, Lang [193] computed the scalar waveform only to 1.5PN order. The 1.5PN waveform is described by the same set of parameters that describes the 2.5PN equations of motion and the 2PN tensor waveform. Again, the scalar waveform vanishes for binary black hole systems (so that the GW signal is indistinguishable from GR).

Lang [193] used the tensor and scalar waveforms to compute the total energy flux carried off to infinity to 1PN order. A derivation of the quadrupole-order flux in tensor-multiscalar theories (that agrees with Lang’s results in the single-scalar limit) can be found in [12]. A similar calculation for compact binaries in the massive Brans–Dicke theory was performed by Alsing et al. [52] (see also [316, 317]). In the notation used above, and correcting a mistake in [52], they found that the lowest-order flux is given by

$$\begin{aligned} {\dot{E}} = \frac{4}{3}\frac{\mu \eta }{r}\left( \frac{G\alpha m}{r}\right) ^3\zeta {\mathcal {S}}_-^2\left[ \frac{\omega ^2-m_s^2}{\omega ^2}\right] ^{3/2}\varTheta (\omega -m_s), \end{aligned}$$

(44)

where $\omega$ is the binary’s orbital frequency, $m_s$ is the mass of the scalar field, and $\varTheta$ is the Heaviside function (i.e., in massive Brans–Dicke theory, scalar dipole radiation is emitted only when $\omega > m_s$).

The emitted radiation has very special features for a binary black hole system: from Eq. (40) and (42) with $s_1=s_2=1/2$ we see that the dominant terms are identical to the equations of motion in GR, except for an unobservable mass rescaling. This result is a generalization to binary systems of “no-scalar-hair” theorems that apply to single black holes [318]. For generic mass ratio, Mirshekari and Will proved this “generalized no-hair theorem” up to 2.5PN order, but they conjectured that it should hold at all PN orders. Indeed, Ref. [277] has shown that the equations of motion are the same as in GR at any PN order if one considers an extreme mass-ratio system and works to lowest order in the mass ratio, and the conjecture is also supported by numerical relativity studies [287, 288]. This “generalized no-hair theorem” for binary black holes depends on some crucial assumptions: vanishing scalar potential, asymptotically constant value of the scalar field, and vanishing matter content. If any one of these assumptions breaks down, the black hole binary’s behavior will differ from GR.