Testing general relativity with black hole-pulsar binaries

Binary pulsars allow us to carry out precision tests of gravity and have placed stringent bounds on a broad class of theories beyond general relativity. Current and future radio telescopes, such as FAST, SKA, and MeerKAT, may find a new astrophysical system, a pulsar orbiting around a black hole, which will provide us a new source for probing gravity. In this paper, we systematically study the prospects of testing general relativity with such black hole-pulsar binaries. We begin by finding a mapping between generic non-Einsteinian parameters in the orbital decay rate and theoretical constants in various modified theories of gravity and then summarize this mapping with a ready-to-use list. Theories we study here include scalar-tensor theories, varying $G$ theories, massive gravity theories, generic screening gravity and quadratic curvature-corrected theories. We next use simulated measurement accuracy of the orbital decay rate for black hole-pulsar binaries with FAST/SKA and derive projected upper bounds on the above generic non-Einsteinian parameters. We find that such bounds from black hole-pulsars can be stronger than those from neutron star-pulsar and neutron star-white dwarf binaries by a few orders of magnitude when the correction enters at negative post-Newtonian orders. By mapping such bounds on generic parameters to those on various modified theories of gravity, we find that one can constrain the amount of time variation in Newton’s constant $G$ to be comparable to or slightly weaker than the current strongest bound from solar system experiments, though the former bounds are complementary to the latter since they probe different regime of gravity. We also study how well one can probe quadratic gravity from black hole quadrupole moment measurements of black hole-pulsars. We find that bounds on the parity-violating sector of quadratic gravity can be stronger than current bounds by six orders of magnitude. These results suggest that a new discovery of black hole-pulsars in the future will provide powerful ways to probe gravity further.

Figure 1

The ratio in the upper bound on the fractional non-GR correction $\gamma$ to the orbital decay rate between BH-PSR and double PSR systems, as a function of at which PN order the correction enters. This bound uses a millisecond PSR-BH binary with simulations from [29]. This figure shows how much improvement one finds by using the BH-PSR system compared to the double PSR one in terms of testing GR. For example, if the ratio is below unity (horizontal magenta dotted dashed line), the bound from the former is stronger than that from the latter. The ratio is shown for FAST (red dotted) and SKA (black dotted). Notice that FAST (SKA) has a bound that is an order of magnitude stronger at $-4$ ($-3.5$) PN than the double PSR. At $-4\mathrm{PN}$ order which corresponds to corrections due to e.g., time variation in the gravitational constant $G$, the BH-PSR bounds are stronger than the double PSR one by almost two orders of magnitude.

Figure 2

Projected bounds on the time variation of the gravitational constant $G$ over the static gravitational constant as a function of the orbital period. The bound is shown for BH-PSR systems with FAST (red dotted) and SKA (black dashed). We also present two solar system bounds with The NASA Messenger (purple dotted-dashed line) [63] and the Mars ephemeris (blue dotted-dashed line) [1], and the current strongest binary pulsar bound (dot dash orange) [64, 65]. Notice that SKA can produce bounds that are comparable to or weaker than solar system ones, though the former are complementary to the latter as the two bounds probe different regime of gravity.

Figure 3

The measurability of the orbital decay rate $\dot{P}$ as a function of the orbital period $P$ with FAST (red dotted) and SKA (black dashed) [29]. The BH mass and the orbital eccentricity is assumed to be $10M_{\odot }$ and $e=0.1$ respectively. For reference, we also present the measurability for a PSR-WD binary (purple double-dotted-dashed) [19] and the double PSR (brown dotted-dashed) [68].

Figure 4

The upper bound on the fractional non-GR correction to the orbital decay rate $\gamma$ in Eq. (5) at each PN order for various astrophysical systems. We present projected bounds with a BH-PSR system using FAST (red dotted) and SKA (black dashed). We choose a 3 day orbital period for the BH-PSR binary. For comparison, we also show bounds from GW observations with GW150914 (blue circle) and GW151226 (magenta square) [8], together with those from the double PSR system (brown dotted-dashed) [34]. Observe that BH-PSR and double PSR bounds are much stronger for negative PN corrections, while GW bounds have more advantage on probing positive PN corrections.

Figure 5

Bounds on the mass of the graviton in Lorentz-violating massive gravity calculated from Eq. (19). We present projected bounds with BH-PSRs using FAST (red dotted) and SKA (black dashed) as a function of its orbital period. For comparison, we also present the strongest bound from a combination of PSR $\mathrm{B}1913+16$ and $\mathrm{B}1534+12$ (magenta dotted-double-dashed) [37] and GWs (brown dotted-dashed) [12]. Notice that the BH-PSR system can place a stronger bound than the binary PSR one but the former is still weaker than the GW ones. The strongest current bound comes from solar system measurements of the perihelion advance of Mars and Saturn (blue double-dotted-dashed) [78].

Figure 6

Similar to Fig. 5 but for cubic Galileon type massive gravity as calculated from Eq. (26). Observe that BH-PSRs yield stronger bounds than NS-PSR systems (magenta dotted-dashed) [38], but weaker bounds than solar system bounds (blue double-dotted-dashed) [87].

Figure 7

The upper bound on vacuum expectation value of the scalar field per Planck mass in SMG as a function of orbital period by a measurement of orbital decay rate for BH-PSR binaries using FAST (red dotted) and SKA (black dashed) calculated from Eq. (30). We also show the bound obtained from PSR-WD binaries (purple double-dotted-dash) in [39]. Observe that using a PSR-WD has strong advantages over a BH-PSR since the WD compactness is approximately $10^4$ times larger than that of a PSR and the screening effect is less efficient.

Figure 8

The fractional measurability of the BH quadrupole moment with stellar-mass BH-PSRs as a function of the BH mass for two different spin inclination angles ${\theta }_s$ [29]. This assumes a 20 year observation length with SKA, the dimensionless BH spin of $\chi =1$ and the orbital eccentricity of $e=0.5$. The time to coalesce has been fixed to 10 Myr, which corresponds to the orbital period of 0.16 days ($m_{\mathrm{BH}}=30M_{\odot }$) to 0.21 days ($m_{\mathrm{BH}}=80M_{\odot }$). We also plot the simulated measurability of the quadrupole moment of Sgr A* by pulsar timing from [27]. Periapsis only contains measurements by measuring three periastron passages, while full orbit considers the entire orbit in the measurement.

Figure 9

Projected upper bounds on the square root of the coupling constant in dCS (top) and EdGB (bottom) gravity assuming that SKA measures the BH quadrupole moment in stellar-mass BH-PSR binaries, as a function of the companion BH mass. We derive the bounds by truncating the BH quadrupole moment in Eqs. (32) and (37) to $\mathcal{O}({\chi }^2)$ (open circles or squares) and by using the full expression valid to $\mathcal{O}({\chi }^4)$ (filled circles or squares). We consider two different spin inclination angles ${\theta }_s$ of 20° (red circle) and 40° (blue square). Other binary parameters are the same as those for the left panel in Fig. 8. In particular, since the fiducial spin was assumed to be $\chi =1$, the slow-rotation approximation adopted in Eqs. (32) and (37) is not accurate, though the bounds are valid as an order of magnitude estimate. The allowed regions for the small coupling approximation are shaded. The BH-PSR bounds are much stronger than the current strongest bounds from solar system [113] and table-top [61] experiments ($\sqrt{{\alpha }_{\mathrm{dCS}}}=\mathcal{O}({10}^8)\mathrm{km}$) for dCS gravity, while they are much weaker than bounds from BH-LMXBs ($\sqrt{{\alpha }_{\mathrm{EdGB}}}=1.9\mathrm{km}$) [66] for EdGB gravity.

Figure 10

Crude estimate on the upper bound on the PPE parameter $\beta$ for GW150914 as a function of at which PN order the correction enters, for two representative GW frequencies (20 Hz and 100 Hz). For reference we also present the bounds using a numerical Fisher analysis [8]. Observe that the two crude estimates bracket the numerical one at lower PN order.

Figure 11

The GW frequency of GW150914 as a function of PN such that the cruder estimate in Eq. (b1) becomes exactly the same as that obtained from the Fisher analysis in [8]. Observe that such dominant GW frequency becomes lower for lower PN orders corrections.

Figure 12

The upper bound on the fractional non-GR correction to the orbital decay rate $\gamma$ in Eq. (5) at each PN order for various astrophysical systems. This is a replication of Fig. 4 at low PN order with the addition of the speculative bound of GW170817. Note that the GW170817 bound on $\gamma$ is not close to being competitive with those from BH/PSRs and the double pulsar.