Testing Chern-Simons modified gravity with gravitational-wave detections of extreme-mass-ratio binaries

The detection of gravitational waves from extreme-mass-ratio inspirals (EMRI) binaries, comprising a stellar-mass compact object orbiting around a massive black hole, is one of the main targets for low-frequency gravitational-wave detectors in space, like the Laser Interferometer Space Antenna (LISA) or evolved LISA/New Gravitational Observatory (eLISA/NGO). The long-duration gravitational-waveforms emitted by such systems encode the structure of the strong field region of the massive black hole, in which the inspiral occurs. The detection and analysis of EMRIs will therefore allow us to study the geometry of massive black holes and determine whether their nature is as predicted by general relativity and even to test whether general relativity is the correct theory to describe the dynamics of these systems. To achieve this, EMRI modeling in alternative theories of gravity is required to describe the generation of gravitational waves. However, up to now, only a restricted class of theories has been investigated. In this paper, we explore to what extent EMRI observations with a space-based gravitational-wave observatory like LISA or eLISA/NGO might be able to distinguish between general relativity and a particular modification of it, known as dynamical Chern-Simons modified gravity. Our analysis is based on a parameter estimation study which uses approximate gravitational waveforms obtained via a radiative-adiabatic method. In this framework, the trajectory of the stellar object is modeled as a sequence of geodesics in the spacetime of the modified-gravity massive black hole. The evolution between geodesics is determined by flux formulae based on general relativistic post-Newtonian and black hole perturbation theory computations. Once the trajectory of the stellar compact object has been obtained, the waveforms are computed using the standard multipole formulae for gravitational radiation applied to this trajectory. Our analysis is restricted to a five-dimensional subspace of the EMRI configuration space, including a Chern-Simons parameter which controls the strength of gravitational deviations from general relativity. We find that, if dynamical Chern-Simons modified gravity is the correct theory, an observatory like LISA or even eLISA/NGO should be able to measure the Chern-Simons parameter with fractional errors below 5%. If general relativity is the true theory, these observatories should put bounds on this parameter at the level ${\xi }^{1/4}<{10}^4\mathrm{km}$, which is four orders of magnitude better than current Solar System bounds.

Figure 1

Depiction of a section of the inspiral orbit for an EMRI (system $A$ in Table ) with parameters $M_{•}=5·{10}^5{M}_{\odot }$, $a=0.25M_{•}$, $e_0=0.25$ and $\zeta =5·{10}^{-2}{M}_{•}^5$.

Figure 2

This 2-dimensional plot shows the symmetric normalized overlap of Eq. (71) for EMRI systems with the following parameters: $M_{•}=5·{10}^5{M}_{\odot }$, $m_{\star }=10M_{\odot }$, $e_0=0.25$, $p_0=11$, ${\theta }_{\mathrm{inc},0}=0.569$, ${\theta }_{\mathrm{S}}=1.57$, ${\varphi }_{\mathrm{S}}=1.57$, ${\theta }_{\mathrm{K}}=0.329$, ${\varphi }_{\mathrm{K}}=0.78$, $D_{\mathrm{L}}/\mu =5·{10}^4\mathrm{Gpc}$, ${\psi }_0=0.78$, ${\chi }_0=0.78$, ${\varphi }_0=0.78$. The parameters $\xi /M_{•}^4$ and $a/M_{•}$ take values in the interval [0,0.5] with a step of 0.05 for a total of 121 points.