Incorporating waveform uncertainty into modeling and inference of GWs
Comparison of marginal posterior distributions for a GW150914-like NR-surrogate signal.Abstract
I discuss how systematic errors in waveform models can affect parameter inference and how these errors can be incorporated into the waveform model construction. For models that include these errors one can marginalize the posterior distribution over these additional degrees of freedom. I demonstrate this for an aligned- spin effective-one-body model where the uncertainties are captured by an efficient Fourier domain GPR model.
Parameter inference marginalizing over model calibration errors
We built a effective-one-body (EOB) based waveform model that also incorporates calibration errors of the base effective-one-body model against numerical relativity. We want to incorporate the uncertainty in the waveform model $h(\boldsymbol\lambda) \to p(h | \boldsymbol\lambda)$.
To achieve this we modeled the additional degrees of freedom as amplitude and phase deviations in the Fourier domain w.r.t. a neutral EOB calibration using Gaussian process regression (GPR). The error model has $2 \times 10$ additional parameters $\epsilon$ and is applied as a correction on top of SEOBNRv4_ROM. It has the following form
$$ \delta \tilde h_{CE}(\boldsymbol\lambda, \boldsymbol\epsilon; f) = (1 + \delta A(\boldsymbol\lambda, \boldsymbol\epsilon; f)) , \exp(i , \delta\phi(\boldsymbol\lambda, \boldsymbol\epsilon; f)) $$
For parameter inference (using a NRSur3dq8 signal in zero noise) we sample SEOBNRv4CE over $\lambda$ and $\epsilon$, and marginalize over $\epsilon$. This leads to posterior distributions which are less precise, but reduce biases compared to SEOBNRv4_ROM or SEOBNRv4CE with $\epsilon=0$.
A mixture density network for calibration posteriors
The SEOBNRv4 effective-one body waveform model used a polynomial fit to the means of the calibration posteriors $p(\boldsymbol\theta | \boldsymbol\lambda)$at the numerical relativity points ${ \boldsymbol\lambda_i }$ to tune free parameters.
We propose to improve on this by a using mixture of Gaussians, i.e. a mixture density network (MDN)
$$ p(\boldsymbol\theta | \boldsymbol\lambda) \approx \sum_{k=1}^K \pi_k(\boldsymbol\lambda) \mathcal{N}(\boldsymbol\theta | \mu_k(\boldsymbol\lambda), \sigma_k^2(\boldsymbol\lambda))) $$
Here, $\boldsymbol\lambda$ are the physical parameters of the binary (the mass-ratio and the spins projected onto the orbital angular momentum vector), while $\boldsymbol\theta$ are the calibration parameters in the EOB waveform model.
