[Paper Review] Post-Newtonian, Quasi-Circular Binary Inspirals in Quadratic Modified Gravity

Kent Yagi, Leo C. Stein+2 more|arXiv (Cornell University)|Oct 27, 2011

Cosmology and Gravitation Theories4 references3 citations

TL;DR

This paper derives post-Newtonian corrections for quasi-circular binary inspirals in quadratic modified gravity theories, including Einstein-Dilaton-Gauss-Bonnet and Chern-Simons gravity. It shows that black holes can acquire scalar hair while neutron stars do not, and finds that scalar field emission dominates energy loss via electric-type dipole and magnetic-type quadrupole radiation, correcting GR at -1 PN and +2 PN orders, respectively.

ABSTRACT

We consider a general class of quantum gravity-inspired, modified gravity theories, where the Einstein-Hilbert action is extended through the addition of all terms quadratic in the curvature tensor coupled to scalar fields with standard kinetic energy. This class of theories includes Einstein-Dilaton-Gauss-Bonnet and Chern-Simons modified gravity as special cases. We analytically derive and solve the coupled field equations in the post-Newtonian approximation, assuming a comparable-mass, spinning black hole binary source in a quasi-circular, weak-field/slow-motion orbit. We find that a naive subtraction of divergent piece associated with the point-particle approximation is ill-suited to represent compact objects in these theories. Instead, we model them by appropriate effective sources built so that known strong-field solutions are reproduced in the far-field limit. In doing so, we prove that black holes in Einstein-Dilaton-Gauss-Bonnet and Chern-Simons theory can have hair, while neutron stars have no scalar monopole charge, in diametrical opposition to results in scalar-tensor theories. We then employ techniques similar to the direct integration of the relaxed Einstein equations to obtain analytic expressions for the scalar field, metric perturbation, and the associated gravitational wave luminosity measured at infinity. We find that scalar field emission mainly dominates the energy flux budget, sourcing electric-type (even-parity) dipole scalar radiation and magnetic-type (odd-parity) quadrupole scalar radiation, correcting the General Relativistic prediction at relative -1PN and 2PN orders. Such modifications lead to corrections in the emitted gravitational waves that can be mapped to the parameterized post-Einsteinian framework. Such modifications could be strongly constrained with gravitational wave observations.

Motivation & Objective

• To analyze gravitational wave emission from comparable-mass, spinning black hole binaries in quadratic modified gravity theories.
• To resolve inconsistencies in modeling compact objects via point-particle approximations in higher-curvature gravity.
• To derive analytic expressions for scalar fields, metric perturbations, and gravitational wave luminosity in the post-Newtonian regime.
• To determine whether black holes and neutron stars can support scalar hair in these modified gravity frameworks.
• To map deviations from general relativity to the parameterized post-Einsteinian framework for observational constraints.

Proposed method

• Uses the post-Newtonian approximation to solve coupled field equations for metric and scalar fields in weak-field, slow-motion binaries.
• Constructs effective sources for compact objects to reproduce known strong-field solutions in the far-field limit, avoiding divergent point-particle artifacts.
• Applies direct integration of the relaxed Einstein equations (DIRE) techniques to compute metric and scalar perturbations.
• Performs Hadamard regularization to handle divergences in integrals involving multiple point particles and angular momentum structures.
• Evaluates $J$-tensors via path-averaged limits to define finite parts of singular integrals in the field equations.
• Computes gravitational wave luminosity at infinity, including contributions from scalar field emission and metric deformation.

Experimental results

Research questions

• RQ1 Can black holes in quadratic modified gravity theories such as Einstein-Dilaton-Gauss-Bonnet and Chern-Simons gravity support scalar hair?
• RQ2 Do neutron stars in these theories exhibit scalar monopole charge, in contrast to predictions in scalar-tensor theories?
• RQ3 How does scalar field emission modify the energy flux and gravitational wave luminosity in binary inspirals?
• RQ4 What are the leading-order post-Newtonian corrections to the gravitational wave energy flux from scalar and tensor modes?
• RQ5 To what extent can deviations from general relativity in these theories be mapped into the parameterized post-Einsteinian framework?

Key findings

• Black holes in Einstein-Dilaton-Gauss-Bonnet and Chern-Simons gravity can possess scalar hair, while neutron stars do not support scalar monopole charge.
• Scalar field emission dominates the energy flux budget, with electric-type (even-parity) dipole radiation correcting GR at -1 PN order.
• Magnetic-type (odd-parity) quadrupole scalar radiation contributes at +2 PN order, modifying the gravitational wave luminosity.
• The leading-order correction to the energy flux from metric deformation appears at 7PN order in the odd-parity, non-spinning case.
• The total energy flux correction from scalar fields is of order $\mathcal{O}(v^{-1})$ relative to GR, indicating a significant deviation in the waveform evolution.
• These modifications can be systematically mapped into the parameterized post-Einsteinian framework, enabling strong constraints via gravitational wave observations.