Testing general relativity and probing the merger history of massive black holes with LISA

TL;DR

This paper extends LISA parameter estimation to include non-precessing spin effects and analyzes tests of gravity theories (scalar-tensor and massive-graviton) using Fisher-matrix and Monte Carlo methods for NS-IMBH and MBH binaries. It shows spin degrades intrinsic mass measurements (notably $\mathcal{M}$ and $\mu$) while leaving angular resolution and $D_L$ relatively robust, and that BD bounds on $\omega_{BD}$ and graviton bounds on $\lambda_g$ weaken by factors of ~10–20 and ~4–5, respectively. It demonstrates that LISA can measure $D_L$ to about 10% out to $z\sim 4$ for $(10^6+10^6)M_\odot$ MBHs and to $z\sim 2$ for $(10^7+10^7)M_\odot$ MBHs, with the chirp mass precisely determined across redshift, while the reduced mass remains challenging for spinning systems. The results imply that most detectable MBH coalescences lie at $z\sim 2$–$6$, enabling reconstruction of MBH merger history when redshifts are known or inferred, provided the low-frequency sensitivity is maintained.

Abstract

Observations of binary inspirals with LISA will allow us to place bounds on alternative theories of gravity and to study the merger history of massive black holes (MBH). These possibilities rely on LISA's parameter estimation accuracy. We update previous studies of parameter estimation including non-precessional spin effects. We work both in Einstein's theory and in alternative theories of gravity of the scalar-tensor and massive-graviton types. Inclusion of non-precessional spin terms in MBH binaries has little effect on the angular resolution or on distance determination accuracy, but it degrades the estimation of the chirp mass and reduced mass by between one and two orders of magnitude. The bound on the coupling parameter of scalar-tensor gravity is significantly reduced by the presence of spin couplings, while the reduction in the graviton-mass bound is milder. LISA will measure the luminosity distance of MBHs to better than ~10% out to z~4 for a (10^6+10^6) Msun binary, and out to z~2 for a (10^7+10^7) Msun binary. The chirp mass of a MBH binary can always be determined with excellent accuracy. Ignoring spin effects, the reduced mass can be measured within ~1% out to z=10 and beyond for a (10^6+10^6) Msun binary, but only out to z~2 for a (10^7+10^7) Msun binary. Present-day MBH coalescence rate calculations indicate that most detectable events should originate at z~2-6: at these redshifts LISA can be used to measure the two black hole masses and their luminosity distance with sufficient accuracy to probe the merger history of MBHs. If the low-frequency LISA noise can only be trusted down to 10^-4 Hz, parameter estimation for MBHs (and LISA's ability to perform reliable cosmological observations) will be significantly degraded.

Figures (4)

• Figure 1: Top: average errors on $\Omega_S$ (dashed lines) and $D_L$ (solid lines). Bottom: average errors on ${\cal M}$ (dashed lines) and $\mu$ (solid lines). Left panel refers to GR, right panel to massive graviton theories; errors are given as a function of the total mass for equal mass MBH binaries. We assume the LISA noise curve can be trusted down to $f_{\rm low}=10^{-5}$ Hz. Black lines are computed omitting spin terms, red lines include spin-orbit terms, blue lines include both spin-orbit and spin-spin terms.
• Figure 2: Left: average bounds on the graviton wavelength as a function of total mass for equal mass MBH binaries. Right: bound on the BD parameter from a NS-IMBH binary as a function of the IMBH mass. The horizontal blue line corresponds to the Cassini bound bertotti. Black lines are computed omitting spin terms; red lines include spin-orbit terms as well. The dashed line shows that the bound on $\lambda_g$ is reduced if the LISA noise curve can only be trusted down to $f_{\rm low}=10^{-4}$ Hz.
• Figure 3: Scatter plot of the correlations $c_{\bar{\phi}_S {\cal M}}(\bar{\phi}_S)$ in GR. From left to right we observe how correlations change when we add spins. The top row corresponds to a $(10^6+10^6)~M_\odot$ MBH binary at $D_L=3$ Gpc; the bottom row refers to a $(1.4+10^3)~M_\odot$ NS-IMBH system observed with single-detector SNR equal to 10.
• Figure 4: Percentage of binaries for which: $\delta \mu = \Delta \mu/\mu<1 \%$ (black circles), $\delta {\cal M}=\Delta {\cal M}/{\cal M}<0.1 \%$ (red squares; $\delta {\cal M }=\Delta {\cal M}/{\cal M}<1 \%$ in all cases considered), $\delta D_L = \Delta D_L/D_L<10 \%$ (blue diamonds), $\delta D_L = \Delta D_L/D_L<5 \%$ (blue diamonds, dashed lines), $\Delta \Omega_S<10^{-4}$ (green triangles). On the left we consider a BH-BH binary of mass $(10^6+10^6)~M_\odot$, on the right a BH-BH binary of mass $(10^7+10^7)~M_\odot$. We assume $f_{\rm low}=10^{-5}$ Hz.