Faithfulness and Effectualness

Faithfulness asks whether a model reproduces a reference waveform at the same intrinsic physical parameters. Effectualness asks whether some point anywhere in the allowed model family can reproduce it after intrinsic parameters are also optimized. A model can therefore recover signal power extremely well by moving to the wrong masses, spins, or other parameters. This page compares the two geometric diagnostics without turning either into an event-specific inference or using a current detector curve.

Required background. Probability and Statistical Inference supplies nuisance optimization, identifiability, and the distinction between best fit and unbiased interpretation.

Helpful background. Einstein Equations supplies the waveform family. Waves, Green Functions, and Geometric Optics supplies Fourier-domain alignment. Gravitational Radiation and Waveforms supplies physical parameters, modes, and frames. Computational Relativity supplies reference waveforms and validation data.

Common setup and comparison axes

Let $h_*(\lambda)$ be a reference waveform at intrinsic parameters $\lambda$, and let $h_m(\lambda')$ be a model on domain $\mathcal D$. Fix the detector-frame scaling, mode content, response, one-sided PSD, frequency band, and normalized time-phase match $\mathcal M$.

The comparison axes are: whether intrinsic parameters are held fixed or optimized; which extrinsic transformations are removed; the allowed calibration domain $\mathcal D$; the weighting and band; and whether the task is parameter-preserving representation or signal recovery. Changing any axis changes the numerical result.

The terminology was sharpened in historical approximant comparisons such as Damour, Iyer, and Sathyaprakash 1998, §§II–IV, PDF.

Faithfulness fixes intrinsic parameters

Define faithfulness by

$$\mathcal F(\lambda)= \max_{\text{declared extrinsic shifts}} \mathcal O\!\left[ h_*(\lambda),h_m(\lambda) \right],$$

where $\mathcal O$ is the normalized overlap before those shifts. Common extrinsic maximizations include one time and one phase. Distance normalization may also be removed, but sky orientation, polarization, or mode rotation must be listed if optimized.

Low faithfulness at fixed $\lambda$ exposes shape disagreement that cannot be removed by the declared nuisance transformations. High faithfulness is parameter preserving only within the chosen physical parameterization and reference convention. A reparameterized coordinate label can make “same numerical parameters” meaningless unless both families map to the same masses, spins, epoch, and frame.

Effectualness searches intrinsic space

Define effectualness, often called the fitting factor, by

$$\mathcal E(\lambda)= \max_{\lambda'\in\mathcal D} \max_{\text{declared extrinsic shifts}} \mathcal O\!\left[ h_*(\lambda),h_m(\lambda') \right].$$

If the physical point $\lambda$ lies in $\mathcal D$, the intrinsic optimization includes the faithful comparison as one candidate. Therefore

$$\mathcal E(\lambda)\ge\mathcal F(\lambda).$$

The best-fit displacement $\Delta\lambda=\lambda'_{\rm best}-\lambda$ records how the model absorbs waveform error. A boundary, discrete bank, prior restriction, or excluded physical point can invalidate the simple inequality unless the domains are reconciled.

A model can be effectual but unfaithful

Take orthonormal waveform directions $e_0,e_1$ in a local comparison where the declared extrinsic alignment is already fixed and cannot remove their relative rotation. Let the true signal at physical parameter $\theta=0$ be $h_*=e_0$, while the normalized model family is

$$h_m(\theta)= \cos(\theta+\epsilon)e_0 +\sin(\theta+\epsilon)e_1.$$

Restrict to a neighborhood with $\lvert\theta+\epsilon\rvert\lt\pi/2$; the overlap is positive and the shifted optimum is unambiguous. At the same physical parameter,

$$\mathcal F=\cos\epsilon =1-\frac12\epsilon^2+O(\epsilon^4).$$

If $\theta=-\epsilon$ is allowed, intrinsic optimization gives $h_m=e_0$ exactly, so $\mathcal E=1$. The waveform family is perfectly effectual in this toy problem but assigns the recovered signal to the shifted parameter $\theta'=-\epsilon$. The absorbed error is tangent to the model manifold.

An orthogonal model error cannot be removed this way and lowers both diagnostics. Equal error norms can thus have very different parameter consequences depending on their direction relative to waveform tangents.

What each diagnostic can and cannot support

Faithfulness is the more direct test of parameter-preserving waveform agreement. It still does not calculate a posterior bias, because priors, correlations, signal strength, calibration, and nonlinearity matter. Effectualness tests whether the family contains a nearby shape, which is useful for recovery geometry but can conceal physical displacement.

Neither number is a universal percentage accuracy. Results depend on total detector-frame mass, redshift, orientation, harmonics, PSD, band, and optimization. A model can be effectual in a dominant-mode comparison and fail after higher modes are added. Conversely, an apparently unfaithful source-frame mode can become less consequential after a particular response projection.

Domain and reporting requirements

Report the reference and model versions, intrinsic coordinate definitions, parameter domain, extrinsic and intrinsic maximizations, optimizer/tolerance, PSD and band, mode set, frame, response, and the best-fit parameter displacement. State whether the optimizer reached a boundary or competing maximum. Current noise curves and event-specific biases remain with Gravitational-Wave Astronomy.

Exercises

1. Verify the toy inequality. For $\epsilon=0.1$, compute faithfulness, effectualness, and the best-fit shift when the allowed local domain includes $\theta=-0.1$.

Solution

At fixed $\theta=0$,

$$\mathcal F=\cos(0.1)\simeq0.995004.$$

Choosing $\theta'=-0.1$ makes the model exactly $e_0$, so $\mathcal E=1$ and $\Delta\theta=-0.1$. The model loses only about $0.5\%$ normalized overlap at the physical point yet hides the entire waveform rotation by shifting the intrinsic coordinate.

2. Impose a domain boundary. Repeat with $\mathcal D=[0,\theta_{\max}]$, where $\epsilon\gt0$ and $\epsilon+\theta_{\max}\lt\pi/2$.

Solution

The unconstrained optimum $\theta'=-\epsilon$ is forbidden. On the stated interval, $\cos(\theta'+\epsilon)$ is positive and strictly decreasing, so the best allowed point is the boundary $\theta'=0$:

$$\mathcal E=\mathcal F=\cos\epsilon.$$

The equality now reflects the restricted domain, not absence of an absorbable tangent error. Reporting the boundary status is essential; extending the model domain to include $-\epsilon$ would restore effectualness one and the shifted best fit.

References

• Damour, Thibault, Bala R. Iyer, and B. S. Sathyaprakash. “Improved Filters for Gravitational Waves from Inspiralling Compact Binaries.” Physical Review D 57 (1998): 885–907. DOI. Open PDF.