More than the sum of its parts: Combining parametrized tests of extreme gravity

We connect two formalisms that describe deformations away from general relativity, one valid in the strong-field regime of neutrons stars and another valid in the radiative regime of gravitational waves: the post-Tolman-Oppenheimer-Volkoff and the parametrized-post-Einsteinian formalisms respectively. We find that post-Tolman-Oppenheimer-Volkoff deformations of the exterior metric of an isolated neutron star induce deformations in the orbital binding energy of a neutron star binary. Such a modification to the binding energy then percolates into the gravitational waves emitted by such a binary, with the leading-order post-Tolman-Oppenheimer-Volkoff modifications introducing a second post-Newtonian order correction to the gravitational wave phase. The lack of support in gravitational wave data for general relativity deformations at this post-Newtonian order can then be used to place constraints on the post-Tolman-Oppenheimer-Volkoff parameters. As an application, we use the binary neutron star merger event GW170817 to place the constraint $-2.4\le \chi \le 44$ (at 90% credibility) on a combination of post-Tolman-Oppenheimer-Volkoff parameters. We also explore the implications of this result to the possible deformations of the mass-radius relation of neutron stars allowed within this formalism. This work opens the path towards theory-independent tests of gravity, combining astronomical observations of neutron stars and gravitational wave observations.

Figure 1

Marginalized posterior distribution for the $\chi$ post-TOV parameter, obtained from the Markov-chain-Monte-Carlo (MCMC) samples released by the LVC for the GW170817 event. The 90% credible interval bound on $\chi$ is $-2.4\le \chi \le 44$, as indicated by the vertical lines. The lower support at zero is not an evidence of a deviation from GR as explained in the text, but rather it reflects a similarly skewed posterior distribution for $\delta {\varphi }_4$, which peaks away from zero due to degeneracies between the various binary parameters and nonstationarity of the detector noise. The long tail of the distribution is produced by a similar tail in the marginalized posterior for $\delta {\varphi }_4$, the parameter that encodes deformations in the gravitational wave Fourier phase at 2PN order (see Fig. 1 in Ref. [31]).

Figure 2

Corner plot showing the posterior for $({\mu }_1,{\mu }_2,{\pi }_2)$, as well as the allowed 90% credible regions (solid contour lines in the off-diagonal panels). We see that whereas ${\mu }_1$ is essentially unconstrained, values of ${\mu }_2$ (${\pi }_2$) which are smaller (larger) are favored with peaks located at the boundary of our prior ranges. The strong degeneracy between these parameters follows from the fact that the constraints derive from an underconstrained system, only requiring to satisfy Eq. (6).

Figure 3

Allowed modifications to the mass-radius relation of neutron stars under the constraint $-2.4\le \chi \le 44$, for EOS APRb [50] (the case for SLy is qualitatively similar). The vertically hatched regions represent the allowed post-TOV deformations to GR the GW170817 constraint on $\chi$ only, while the solid line represents the GR result. Requiring that additional constraints be satisfied, such as the mass measurement of MSP $\mathrm{J}0740+6620$ [54] ($M=2.17_{-0.10}^{+0.11}{M}_{\odot }$, shaded region) and the radius of canonical neutron stars [58] ($R_{1.4}={10.9}_{-1.5}^{+1.9}\mathrm{km}$, horizontal solid line), the allowed region is reduced to the horizontally hatched region. For reference, we also included in the limit set by Schwarzschild BHs ($R=2M$), Buchdahl’s limit ($R=9M/8$), the limit set by causality ($R=2.9M$) [55, 56] in GR and the cutoff mass (dotted line) $M=2.6M_{\odot }$ inferred from the mass distribution of compact binaries containing neutron stars [59].

Figure 4

Similar to Fig. 3, however only showing the more restrictive regions for both EOSs, APRb and SLy. This figure explicitly shows the degeneracies between EOSs assuming a theory of gravity to be known (see the solid and dashed curves) and theory of gravity assuming that the EOS is known a priori (individual hatched regions). Varying both EOS and theory of gravity increases further the degeneracy between matter and gravity models—a degeneracy due to the fact that neutron stars are relativistic objects.