Supermassive black hole tests of general relativity with eLISA

Motivated by the parametrized post-Einsteinian (ppE) scheme devised by Yunes and Pretorius, which introduces corrections to the post-Newtonian coefficients of the frequency domain gravitational waveform in order to emulate alternative theories of gravity, we compute analytical time domain waveforms that, after a numerical Fourier transform, aim to represent (phase corrected only) ppE waveforms. In this formalism, alternative theories manifest themselves via corrections to the phase and frequency, as predicted by general relativity (GR), at different post-Newtonian (PN) orders. To present a generic test of alternative theories of gravity, we assume that the coupling constant of each alternative theory is manifestly positive, allowing corrections to the GR waveforms to be either positive or negative. By exploring the capabilities of massive black hole binary GR waveforms in the detection and parameter estimation of corrected time domain ppE signals, using the current eLISA configuration (as presented for the European Space Agency Cosmic Vision L3 mission), we demonstrate that for corrections arising at higher than 1PN order in phase and frequency GR waveforms are sufficient for both detecting and estimating the parameters of alternative theory signals. However, for theories introducing corrections at the 0 and 0.5PN orders, GR waveforms are not capable of covering the entire parameter space, requiring the use of non-GR waveforms for detection and parameter estimation.

Figure 1

Maximum allowed values for $\beta$ as a function of mass ratio $q$ given the constraint of ${\kappa }_i/{\mathrm{\Phi }}_i\le 0.5$ for different alternative theories. Cell (a) represents non-GR corrections with positive sign, while cell (b) represents non-GR corrections with negative sign, with $b=-1/3$ (red dotted line), $b=-2/3$ (blue dashed line), $b=-1$ (green dot-dashed line), $b=-4/3$ (orange double dot-dashed line) and $b=-5/3$ (magenta double dash-dot line).

Figure 2

Instrumental noise model for a $10^9\mathrm{m}$ arm eLISA configuration.

Figure 3

Power spectra for both GR and non-GR waveforms assuming a SMBHB with individual source-frame masses of $10^7-10^6{M}_{\odot }$ at $z=1$, for an alternative theory with $b=-5/3$, and for differing values of $\beta$. The top figure represents positive non-GR corrections, while the bottom figure displays negative non-GR corrections.

Figure 4

Detection horizons as a function of redshifted total mass for positive (top figure) and negative (bottom figure) non-GR corrected waveforms. In each figure we compare the different theories: GR (solid black line), $b=-1/3$ (red dotted line), $b=-2/3$ (blue dashed line), $b=-1$ (green dot-dashed line), $b=-4/3$ (orange double dot-dashed line) and $b=-5/3$ (magenta double dash-dot line).

Figure 5

Allowed upper limits for $\beta$ (dashed black line), detection limits (yellow curve) and limits where unbiased parameter estimation is still possible with GR templates (red curve). We account for positive (left) and negative (right) corrections and corrections at increasing PN orders (i.e. increasing values of $b$ from top to bottom). Except for the positive correction with $b=-5/3$, GR templates manage to detect all injected non-GR waveforms. With increasing values of $b$, the red curve below which GR templates are sufficient for parameter estimation approaches the detection limit.