Measuring violations of general relativity from single gravitational wave detection by nonspinning binary systems: Higher-order asymptotic analysis

A frequentist asymptotic expansion method for error estimation is employed for a network of gravitational wave detectors to assess the amount of information that can be extracted from gravitational wave observations. Mathematically we derive lower bounds in the errors that any parameter estimator will have in the absence of prior knowledge to distinguish between the post-Einsteinian (ppE) description of coalescing binary systems and that of general relativity. When such errors are smaller than the parameter value, there is a possibility to detect these violations from general relativity (GR). A parameter space with inclusion of dominant dephasing ppE parameters $(\beta ,b)$ is used for a study of first- and second-order (co)variance expansions, focusing on the inspiral stage of a nonspinning binary system of zero eccentricity detectible through Advanced LIGO and Advanced Virgo. Our procedure is an improvement of the Cramér-Rao lower bound. When Bayesian errors are lower than our bound it means that they depend critically on the priors. The analysis indicates the possibility of constraining deviations from GR in inspiral signal-to-noise ratio (SNR) ($\rho \sim 15–17$) regimes that are achievable in upcoming scientific runs (GW150914 had an inspiral $\mathrm{SNR}\sim 12$). The errors on $\beta$ also increase errors of other parameters such as the chirp mass $\mathcal{M}$ and symmetric mass ratio $\eta$. Application is done to existing alternative theories of gravity, which include modified dispersion relation of the waveform; nonspinning models of quadratic modified gravity; and dipole gravitational radiation (i.e., Brans-Dicke-type) modifications.

Figure 1

Top: Sky-averaged SNR plotted with $\epsilon$ varied for system parameters: $m_1=m_2=10M_{\odot }$, $t_a={\varphi }_a=0$, $\beta =-0.2$, $D_L=1100\mathrm{Mpc}$, and $b=-3$ in the three-detector network. Bottom: Fitting factors (2.1) for a range of $\beta$ with $b$ fixed to produce PN-order 0.0, 1.0, and 1.5 modifications for a system of $m_1=m_2=10M_{\odot }$ and $t_a={\varphi }_a=0$. Advanced LIGO noise is assumed. Since the range of $\beta$-values scales differently at each PN order, each $\beta$-interval is scaled (as labeled in the legend). For example, in the PN-order 0.0 modification the $\beta$ values in the domain are each scaled by ${10}^{-2}$.

Figure 2

Sky-averaged errors as a function of $\beta$ for a two-dimensional ppE parameter space for the BBH $1:1$ system of averaged network SNR $\rho =14.6$. SNR results of $\rho =29.3$ are also shown by setting the distance to $D_L=550\mathrm{Mpc}$. As noted in Ref. [11] error estimates are rescaled as $\sigma [1]({\rho }^{*}/\rho )$ and $\sigma [2]({\rho }^{*}/\rho {)}^2$, where ${\rho }^{*}$ is the SNR that error estimates are originally calculated from. In the top panel the far left column represents each system for a PN-order 0.0 modification ($b=-5$), the center column is a PN-order 0.5 modification ($b=-4$), and the far right column is for PN-order 1.0 modifications ($b=-3$). Similarly, the bottom panels are resulting modifications at PN order 1.5 ($b=-2$), 2.0 ($b=-1$), and 3.0 ($b=+1$). $\beta$ is more tightly constrained at lower PN orders and the inclusion of second-order errors for $(\beta ,b)$ drastically diverges from Fisher estimates as $\beta \to 0$.

Figure 3

Sky-averaged errors, similar to Fig. 2, for a BNS system of averaged SNR $\rho =17.0$.

Figure 4

Sky-averaged uncertainties for the equal-mass BBH $1:1$ system for a PN-order 1.0 modification of the seven-dimensional parameter space (ppE parameters $\{\beta ,b\}$ and physical parameters $\{\eta ,\log \mathcal{M},t_c,\text{lat},\text{long}\}$). In the left column the top panel displays $\mathrm{\Delta }\beta$ percent errors as a function of $\beta$ (the sign of $\beta$ provides different error estimates) and below that are $\mathrm{\Delta }b$ errors as a function of $\beta$ (the sign of $\beta$ does not play a role in these error estimates). In the middle and to the right are the physical parameters’ errors, where the constraint of $\beta$ primarily affects the second-order contributions. Enlarging the parameter space increases error estimates from those computed in Fig. 2 at PN order 1.0, thus weakening constraints on $\beta$. For negative $\beta$, the full-dimensional study states $\mathrm{\Delta }\beta [1]=100\%$ at $\beta =-0.16$ and $\mathrm{\Delta }\beta [1+2]=100\%$ at $\beta =-0.32$.

Figure 5

Sky distribution of error estimates. Color bars represent the range of ppE quantities labeled (a), (b), …, (f) in Table 2. This demonstrates the correlation of the SNR and ppE error estimation over the sky. See text for discussion.

Figure 6

Sky-averaged error estimates for the BBH $1:2$ and BHNS system. The left column represents calculations of the ppE parameter errors ($\mathrm{\Delta }\beta ,\mathrm{\Delta }b$) for negative $\beta$-values, the center column shows the mass errors ($\mathrm{\Delta }\eta ,\mathrm{\Delta }\mathcal{M}$), and the far right column shows the arrival time $\mathrm{\Delta }{t}_a$ and latitude-longitude ($\mathrm{\Delta }\text{lat},\mathrm{\Delta }\text{long}$) error estimates. Here latitude-longitude error estimates are not affected by $\beta$ variation, as was previously presented in the equal-mass system. This study states that $\mathrm{\Delta }{\beta }_{\mathrm{BBH}1:2}[1+2]=95.2\%$ at ${\beta }_{\mathrm{BBH}1:2}=-1.8\times {10}^{-4}$ and $\mathrm{\Delta }{\beta }_{\mathrm{BHNS}}[1+2]=95.3\%$ at ${\beta }_{\mathrm{BHNS}}=-4.5\times {10}^{-5}$.

Figure 7

Sky-map error estimates of ppE parameters $\mathrm{\Delta }\beta$ and $\mathrm{\Delta }b$ and mass parameters $\mathrm{\Delta }\eta$ and $\mathrm{\Delta }\mathcal{M}$ for the unequal mass BBH $1:2$ system. The top color bars for ppE parameters are for ${\beta }_{\mathrm{BBH}1:2}=-1.8\times {10}^{-4}$ and the mass parameters below that are for ${\beta }_{\mathrm{BBH}1:2}=-3.0\times {10}^{-4}$ of results in Fig. 6. The SNR color bar is valid for both error estimates. Sky-average estimates provide $\mathrm{\Delta }{\beta }_{\mathrm{BBH}1:2}[1+2]=95.2\%$ at ${\beta }_{\mathrm{BBH}1:2}=-1.8\times {10}^{-4}$ and $\mathrm{\Delta }{\beta }_{\mathrm{BBH}1:2}[1+2]=47.4\%$ at ${\beta }_{\mathrm{BBH}1:2}=-3.0\times {10}^{-4}$.

Figure 8

First-order errors (left panels) and eigenvalues (center and right panels) of the Fisher matrix when computations are extended to the seven-dimensional parameter space.

Figure 9

Constraints on ppE parameters $(\beta ,b)$. Alongside frequentist mean-squared error $\lesssim 100\%$ estimates are constraints imposed by Bayesian estimates [18], solar system tests [50], binary pulsar measurements [37, 38], and the GW150914 event [6]. Regions below each mark/line are where violations cannot be detected based on each respective study. The GR-limit is $\beta =0$. Our frequentist two-dimensional study considers ppE parameter space $(\beta ,b)$, while seven-dimensional studies include physical parameters (masses, etc.). See text for discussion.