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Unitarity, Causality, and Solar System Bounds May Significantly Limit the Use of Gravitational Waves to Test General Relativity

Alexander Cassem, Mark P. Hertzberg

Abstract

The prospect of detecting/constraining deviations from general relativity by studying gravitational waves (GWs) from merging black holes has been one of the primary motivations of GW interferometers like LIGO/Virgo. Within pure gravity, the only possible way deviations can arise is from the existence of higher order derivative corrections, namely higher powers of the Riemann curvature tensor, in the effective action. Any observational bounds imply constraints on the corresponding Wilson coefficients. At the level of the action, one can imagine the coefficients are sufficiently large so as to be in principle detectable. However, from the point of view of some fundamental principles, namely causality and unitarity, this is much less clear, as we examine here. We begin by reviewing certain known bounds on these coefficients, which together imply a low cut off on the effective theory. We then consider a possible mechanism to generate such terms, namely in the form of many scalars, minimally coupled to only gravity, that can be integrated out to give these higher order operators. We show that a by product of this is the generation of quantum corrections to Newton's potential, whose observable consequences are already ruled out by solar system tests. We point out that over 7 orders of magnitude of improvement in interferometer sensitivity would be required to avoid such solar system constraints. We also mention further constraints from Hawking radiation from black holes.

Unitarity, Causality, and Solar System Bounds May Significantly Limit the Use of Gravitational Waves to Test General Relativity

Abstract

The prospect of detecting/constraining deviations from general relativity by studying gravitational waves (GWs) from merging black holes has been one of the primary motivations of GW interferometers like LIGO/Virgo. Within pure gravity, the only possible way deviations can arise is from the existence of higher order derivative corrections, namely higher powers of the Riemann curvature tensor, in the effective action. Any observational bounds imply constraints on the corresponding Wilson coefficients. At the level of the action, one can imagine the coefficients are sufficiently large so as to be in principle detectable. However, from the point of view of some fundamental principles, namely causality and unitarity, this is much less clear, as we examine here. We begin by reviewing certain known bounds on these coefficients, which together imply a low cut off on the effective theory. We then consider a possible mechanism to generate such terms, namely in the form of many scalars, minimally coupled to only gravity, that can be integrated out to give these higher order operators. We show that a by product of this is the generation of quantum corrections to Newton's potential, whose observable consequences are already ruled out by solar system tests. We point out that over 7 orders of magnitude of improvement in interferometer sensitivity would be required to avoid such solar system constraints. We also mention further constraints from Hawking radiation from black holes.
Paper Structure (30 sections, 114 equations, 2 figures)

This paper contains 30 sections, 114 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Effective Field Theory and Bounds
  3. Building the Effective Field Theory
  4. Constraints from Observed Gravitational Waves
  5. Constraints from Causality
  6. Higher Derivative Operators from Integrating Out Massive Particles
  7. All Masses Equal
  8. Distribution of Masses
  9. One-Loop Observable Consequences
  10. Massive Scalar Amplitude
  11. One-Loop Diagrams
  12. Cosmological Constant Counter Term
  13. Newton's Constant Counter Term
  14. Total Amplitude
  15. Corrections to the Newtonian Potential
  16. ...and 15 more sections

Figures (2)

  • Figure 1: Correction to the potential $\Delta V$ between point masses due to a scalar ($N=1$) running in a loop versus distance $r$. The black curve is the full numerical result from Fourier transforming Eq. (\ref{['expanded version of Jq2 for total amplitude']}). The dashed blue curve is the leading term of the small $m\,r$ result given in Eq. (\ref{['massless limit for massive scalar correction newton']}). The dashed orange curve is the large $m\,r$ result given in Eq. (\ref{['correct massive limit for massive scalar in a loop correction']}).
  • Figure 2: Experimental bounds on the size of corrections to Newton's potential $\alpha$ as a function of scale $\lambda$; taken from Ref. Adelberger:2003zx and Fischbach:1992fa. Left side focuses on microscopic scales $10^{-6}m<\lambda<10^{-2}m$, while the right side focuses on macroscopic scales $10^{-2}m<\lambda<10^{14}m$.