Combined binary-pulsar tests

4.4 Combined binary-pulsar tests

Because of their different orbital parameters and inclinations, the double-neutron-star systems PSR B1913+16 and B1534+12 provide somewhat different constraints on alternative theories of gravity. Taken together with some of the limits on SEP violation discussed above, and with solar-system experiments, they can be used to disallow certain regions of the parameter space of these alternate theories. This approach was pioneered by Taylor et al. [133

], who combined PK-parameter information from PSRs B1913+16 and B1534+12 and the Damour and Schäfer result on SEP violation by PSR B1855+09 [43

] to set limits on the parameters $β ′$ and $β ′′$ of a class of tensor-biscalar theories discussed in [36 ] (Figure 9

). In this class of theories, gravity is mediated by two scalar fields as well as the standard tensor, but the theories can satisfy the weak-field solar-system tests. Strong-field deviations from GR would be expected for non-zero values of $β′$ and $β′′$ , but the theories approach the limit of GR as the parameters $β′$ and $β ′′$ approach zero.

Figure 9: Portions of the tensor-biscalar $′ ′′ β –β$ plane permitted by timing observations of PSRs B1913+16, B1534+12, and B1855+09 up to 1992. Values lying above the curve labeled “a” are incompatible with the measured $ω˙$ and $γ$ parameters for PSR B1913+16. The curves labeled “b” and “d” give the allowed ranges of $′ β$ and $′′ β$ for PSRs B1913+16 and B1534+12, respectively, fitting for the two neutron-star masses as well as $′ β$ and $′′ β$ , using data available up to 1992. The vertical lines labeled “c” represent limits on $β′$ from the SEP-violation test using PSR B1855+09 [43 ]. The dot at (0,0) corresponds to GR. (Reprinted by permission from Nature [133 ], Ⓒ 1992, Macmillan Publishers Ltd.)

A different class of theories, allowing a non-linear coupling between matter and a scalar field, was later studied by Damour and Esposito-Farèse [38 , 40 ]. The function coupling the scalar field $ϕ$ to matter is given by $A (ϕ) = exp(12β0 ϕ2)$ , and the theories are described by the parameters $β0$ and $α0 = β0ϕ0$ , where $ϕ0$ is the value that $ϕ$ approaches at spatial infinity (cf. Section 3.3). These theories allow significant strong-field effects when $β0$ is negative, even if the weak-field limit is small. They are best tested by combining results from PSRs B1913+16, B1534+12 (which contributes little to this test), B0655+64 (limits on dipolar gravitational radiation), and solar-system experiments (Lunar laser ranging, Shapiro delay measured by Viking [114 ], and the perihelion advance of Mercury [116 ]). The allowed parameter space from the combined tests is shown graphically in Figure 10 [40 ]. Currently, for most neutron-star equations of state, the solar-system tests set a limit on $α0$ ( $2 −3 α 0 < 10$ ) that is a few times more stringent than those set by PSRs B1913+16 and B0655+64, although the pulsar observations do eliminate large negative values of $β0$ . With the limits from the pulsar observations improving only very slowly with time, it appears that solar-system tests will continue to set the strongest limits on $α 0$ in this class of theories, unless a pulsar–black-hole system is discovered. If such a system were found with a $∼ 10 M ⊙$ black hole and an orbital period similar to that of PSR B1913+16 ( $∼$ 8 hours), the limit on $α0$ derived from this system would be about 50 times tighter than that set by current solar-system tests, and 10 times better than is likely to be achieved by the Gravity Probe B experiment [40 ].

Figure 10: The parameter space in the non-linear $α0$ , $β0$ gravitational theory, for neutron stars described by a polytrope equation of state. The regions below the various curves are allowed by various pulsar timing limits, by solar-system tests (“1PN”), and by projected LIGO/VIRGO observations of NS–NS and NS–BH inspiral events. The shaded region is allowed by all tests. The plane and limits are symmetric about $α0 = 0$ . (From [40 ]; used by permission.)