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5.4 Binary pulsars and scalar-tensor gravity

Making the usual assumption that both members of the system are neutron stars, and using the methods summarized in TEGP 10 – 12 [281Jump To The Next Citation Point], one can obtain formulas for the periastron shift, the gravitational redshift/second-order Doppler shift parameter, and the rate of change of orbital period, analogous to Equations (78View Equation). These formulas depend on the masses of the two neutron stars, on their self-gravitational binding energy, represented by “sensitivities” s and κ ∗, and on the Brans–Dicke coupling constant ωBD. First, there is a modification of Kepler’s third law, given by
( )1 ∕2 2πfb = 𝒢m-- . (81 ) a3
Then, the predictions for ⟹ω˙⟩, γ ′ and P˙ b are
2∕3 2− 1 −4∕3 ⟹ω˙⟩ = 6 πfb(2πmfb ) (1 − e ) 𝒫𝒢 , (82 ) γ′ = e (2πf )−1(2πmf )2∕3m2-𝒢 −1∕3(α ∗+ 𝒢 m2- + κ∗η∗), (83 ) b b m 2 m 1 2 192 π 5∕3 ′ 2 P˙b = − --5--(2π ℳfb ) F (e) − 4π(2π μfb)ξ𝒮 G(e), (84 )
where ℳ ≡ χ3∕5𝒢 −4∕5η3∕5m, and, to first order in ξ ≡ (2 + ωBD )− 1, we have
5 ( 1 1 ) F ′(e) = F(e) + ----ξ(Γ + 3Γ ′)2 -e2 + -e4 (1 − e2)−7∕2, (85 ) 144 ( ) 2 8 2 −5∕2 1-2 G (e) = (1 − e ) 1 + 2e , (86 ) 𝒮 = s1 − s2, (87 ) 𝒢 = 1 − ξ (s1 + s2 − 2s1s2), (88 ) [ ] 𝒫 = 𝒢 1 − 2ξ + 1ξ (s1 + s2 − 2s1s2) , (89 ) 3 3 α ∗2 = 1 − ξs2, (90 ) ∗ η 2 = (1 −[ 2s2)ξ, ] (91 ) 2 1- -1- 2 χ = 𝒢 1 − 2ξ + 12 ξΓ , (92 ) Γ = 1 − 2 m1s2-+-m2s1-, (93 ) m Γ ′ = 1 − s1 − s2, (94 )
where F(e) is defined in Equation (72View Equation). The quantities sa and ∗ κ a are defined by
( ) ( ) sa = − ∂(ln-ma)- , κ∗= − ∂(lnIa)- , (95 ) ∂(lnG ) N a ∂(lnG ) N
and measure the “sensitivity” of the mass ma and moment of inertia Ia of each body to changes in the scalar field (reflected in changes in G) for a fixed baryon number N (see TEGP 11, 12 and 14.6 (c) [281Jump To The Next Citation Point] for further details). The quantity s a is related to the gravitational binding energy. These sensitivities will depend on the neutron-star equation of state. Notice how the violation of SEP in Brans–Dicke theory introduces complex structure-dependent effects in everything from the Newtonian limit (modification of the effective coupling constant in Kepler’s third law) to gravitational radiation. In the limit ξ → 0, we recover GR, and all structure dependence disappears. The first term in ˙ Pb (see Equation (84View Equation)) is the combined effect of quadrupole and monopole gravitational radiation, while the second term is the effect of dipole radiation.

Unfortunately, because of the near equality of the neutron star masses in the binary pulsar, dipole radiation is suppressed, and the bounds obtained are not competitive with the Cassini bound on γ [293], except for those generalized scalar-tensor theories, with β0 < 0 [74Jump To The Next Citation Point]. Bounds on the parameters α0 and β0 from solar system, binary pulsar, and gravitational wave observations (see Sections 5.1 and 6.3) are found in [74].

Alternatively, a binary pulsar system with dissimilar objects, such as a white dwarf or black hole companion, would provide potentially more promising tests of dipole radiation. In this regard, the recently discovered binary pulsar J1141+6545, with an apparent white dwarf companion, may play an important role. Here one can treat sWD ∼ 10–4 as negligible. Then, from Equation (84View Equation), it is straightforward to show that, if the timing reaches sufficient accuracy to determine P˙b to an accuracy σ in agreement with the prediction of GR, then the resulting lower bound on ωBD would be

4 s2NS ωBD > 4 × 10 ----. (96 ) σ
Thus, for sNS ∼ 0.2, a 4 percent measurement would already compete with the Cassini bound (for further details, see [118102]).


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