Gravitational radiation from compact binary systems in the massive Brans-Dicke theory of gravity

We derive the equations of motion, the periastron shift, and the gravitational radiation damping for quasicircular compact binaries in a massive variant of the Brans-Dicke theory of gravity. We also study the Shapiro time delay and the Nordtvedt effect in this theory. By comparing with recent observational data, we put bounds on the two parameters of the theory: the Brans-Dicke coupling parameter ${\omega }_{\mathrm{BD}}$ and the scalar mass $m_s$. We find that the most stringent bounds come from Cassini measurements of the Shapiro time delay in the Solar System that yield a lower bound ${\omega }_{\mathrm{BD}}>40000$ for scalar masses $m_s<2.5\times {10}^{-20}\mathrm{eV}$ [or Compton wavelengths ${\lambda }_s=h/(m_{s}c)>5\times {10}^{10}\mathrm{km}$], to 95% confidence. In comparison, observations of the Nordtvedt effect using lunar laser ranging experiments yield ${\omega }_{\mathrm{BD}}>1000$ for $m_s<2.5\times {10}^{-20}\mathrm{eV}$. Observations of the orbital period derivative of the quasicircular white dwarf-neutron star binary PSR $\mathrm{J}1012+5307$ yield ${\omega }_{\mathrm{BD}}>1250$ for $m_s<{10}^{-20}\mathrm{eV}$ (${\lambda }_s>1.2\times {10}^{11}\mathrm{km}$). A first estimate suggests that bounds comparable to the Shapiro time delay may come from observations of radiation damping in the eccentric white dwarf-neutron star binary PSR $\mathrm{J}1141-6545$, but a quantitative prediction requires the extension of our work to eccentric orbits.

Figure 1

Left panel: Lower bound on $({\omega }_{\mathrm{BD}}+3/2)$ as a function of the mass of the scalar $m_s$ from the Cassini mission data (solid black line; cf. 18), period derivative observations of PSR $\mathrm{J}1141-6545$ (dashed red line) and PSR $\mathrm{J}1012+5307$ (dot-dashed green line), and lunar laser ranging experiments (dotted blue line). Vertical lines indicate the masses corresponding to the typical radii of the systems: 1 AU (solid black line) and the orbital radii of the two binaries (dashed red and dot-dashed green lines). Right panel: Upper bound on $\xi$ as a function of $m_s$. Line styles are the same as in the left panel. Note that the theoretical bounds on the coupling parameters are $\omega >-3/2$ and $\xi <2$.

Figure 2

Deformed integration contour $C_1+C_2+C_3+C_4$ along lines of steepest descent for $\rho (u)=iR(\omega u-m_s\sqrt{u^2-1})+i3\pi /4$ (corresponding to $n_1=+1$, $n_2=-1$) on the two sheets of the Riemann surface associated with $\sqrt{u^2-1}$. Here only the saddle point on the negative imaginary axis contributes.