Match, Mismatch, and Alignment

Match compares the shapes of two weighted waveforms after removing only the transformations explicitly declared irrelevant. The usual time-and-phase maximization discards a common arrival-time shift and one overall phase; it does not erase intrinsic phasing, mode-dependent rotations, precession, eccentric structure, or frame errors. Mismatch is one minus that optimized overlap. This page derives the construction without adopting a current detector curve or an event-specific bias conclusion.

Required background. Probability and Statistical Inference supplies covariance-weighted geometry, nuisance maximization, and the distinction between a distance and an inferential verdict.

Helpful background. Einstein Equations supplies radiative solutions. Waves, Green Functions, and Geometric Optics supplies Fourier shifts. Gravitational Radiation and Waveforms supplies modes, frames, and strain. Computational Relativity supplies numerical comparisons.

Normalize before comparing shape

For the declared one-sided PSD and band, write

$$(a\mid b)=4\operatorname{Re}\int \frac{\widetilde a(f)\widetilde b(f)^*}{S_n(f)}\,df, \qquad \widehat a=\frac{a}{\lVert a\rVert}.$$

The unmaximized real overlap is $(\widehat a\mid\widehat b)$. Cauchy–Schwarz bounds its absolute value by one. Normalization removes an overall positive amplitude scale, which may represent distance in a restricted model but can also hide a genuine amplitude error. State whether amplitude is meant to be a nuisance before interpreting the result.

The waveform-space construction and its local metric are developed in Owen 1996, §§II–III, PDF.

Time and phase transformations

A time shift $\Delta t$ and constant phase $\Delta\phi$ act in frequency space as

$$\widetilde b_{\Delta t,\Delta\phi}(f) =e^{-2\pi if\Delta t}e^{i\Delta\phi} \widetilde b(f).$$

This equation acts on positive frequencies. For a real strain time series, set the negative-frequency part by complex conjugation; multiplying the entire real time series by $e^{i\Delta\phi}$ would generally make it complex. Equivalently, define the complex positive-frequency representation

$$b_+(t)=2\int_0^\infty\widetilde b(f)e^{-2\pi ift}\,df, \qquad b(t)=\operatorname{Re}b_+(t).$$

Then the transformed real signal is

$$b_{\Delta t,\Delta\phi}(t) =\operatorname{Re}\!\left[e^{i\Delta\phi}b_+(t+\Delta t)\right].$$

Thus positive $\Delta t$ is a time-coordinate advance. An active delay $b(t-\Delta t)$ instead carries $e^{+2\pi if\Delta t}$ at positive frequency. Maximizing over all real shifts makes this sign convention immaterial to the match value, but it matters when interpreting the fitted shift.

This phase operation is a rotation between a real signal and its quadrature. It is a physical orbital-phase change only for a model in which that change has this action. A multimode source generally acquires different factors for different azimuthal indices $m$; its physical phase must be optimized through the complete model.

Define the complex correlation

$$Z(\Delta t)=4\int_{f_{\min}}^{f_{\max}} \frac{\widetilde a(f)\widetilde b(f)^*}{S_n(f)} e^{2\pi if\Delta t}\,df.$$

Maximizing the real overlap over $\Delta\phi$ gives $\lvert Z(\Delta t)\rvert$. The time-and-phase match is therefore

$$\mathcal M(a,b)= \max_{\Delta t} \frac{\lvert Z(\Delta t)\rvert} {\lVert a\rVert\lVert b\rVert}, \qquad 0\le\mathcal M\le1.$$

The mismatch is $\mathfrak m=1-\mathcal M$. Identical waveforms have zero mismatch, as do waveforms differing only by the maximized time, phase, and normalization transformations.

What alignment removes

At a best fit, the residual is orthogonal—under the chosen inner product—to the tangent directions of the optimized transformations. For time and phase those directions are approximately $\partial_t h$ and $ih$ in a complex representation. Normalization removes the radial direction $h$. The remaining residual contains differences that cannot be mimicked by those shifts.

This projection is useful but lossy. A physical reference time, absolute phase tied to another messenger, or calibrated amplitude can be scientifically relevant even when the conventional match removes it. Maximizing a polarization rotation, sky position, mass, or spin would discard still more information and must not be called “time-and-phase alignment.”

The small-error limit

Let $\widehat h$ have unit norm and write a nearby waveform before renormalization as

$$h'=\widehat h+\delta h.$$

Decompose $\delta h$ into components parallel to normalization and the optimized tangent space, plus an orthogonal residual $\delta h_\perp$. After normalization and alignment,

$$1-\mathcal M =\frac12\lVert\delta h_\perp\rVert^2 +O(\delta h^3).$$

To see the factor $1/2$, write $\delta h=a\widehat h+v$ with $(\widehat h\mid v)=0$. The normalized overlap before alignment is

$$\frac{1+a}{\sqrt{(1+a)^2+\lVert v\rVert^2}} =1-\frac12\lVert v\rVert^2+O(\delta h^3).$$

Minimizing the remaining quadratic distance over infinitesimal time and phase shifts removes the projections of $v$ along their tangent directions, leaving $\delta h_\perp$. This is a local least-squares projection, not an assumption that an arbitrary physical difference is unimportant. The approximation fails for large errors, multiple competing alignment maxima, discontinuous mode conventions, or a parameter boundary.

Alignment windows and multimode waveforms

A time-domain alignment over an interval solves a different optimization problem from a full-band noise-weighted match. Its result can depend on window endpoints, taper, junk removal, and whether phase or frequency is fitted. Hybrid construction must vary the window and PN family; a small residual in the chosen interval does not prove agreement elsewhere. These sensitivities are demonstrated for PN–NR hybrids in MacDonald, Nissanke, and Pfeiffer 2011, §§II–IV, PDF.

For multimode radiation, one global time and rotation acts coherently on all modes. Aligning each $(\ell,m)$ mode with an independent phase can manufacture agreement that no single source-frame transformation realizes. Precessing comparisons must also state inertial or coprecessing frames and the rotation convention.

Interpretation boundaries

Mismatch is conditional on PSD, band, total mass/redshift map, included modes, response, and maximized variables. It is not a universal accuracy percentage. Current detector noise curves and event-specific parameter biases belong to Gravitational-Wave Astronomy; this chapter uses declared reference weights to study model geometry.

For a reproducible comparison, publish the two waveform identifiers and versions, detector-frame scaling, mode set, taper and sampling, band, analytic or tabulated PSD, alignment search range, interpolation method, and whether the reported optimum is global. Reporting only the final mismatch prevents a reader from distinguishing physical disagreement from preprocessing and optimization choices.

Exercises

1. A removable difference. Suppose

$$\widetilde b(f)=Ce^{-2\pi ift_0}e^{i\phi_0} \widetilde a(f)$$

with $C\gt0$. Find the match after amplitude normalization and time-phase maximization.

Solution

Normalization removes $C$. Choosing $\Delta t=-t_0$ and $\Delta\phi=-\phi_0$ makes the transformed normalized waveform identical to $\widehat a$. Its overlap is one, so $\mathcal M=1$ and $\mathfrak m=0$. The original waveforms still differ in amplitude, arrival time, and phase; zero mismatch means only that those declared differences were quotiented out.

2. Quadratic mismatch. Let $\widehat h$ and $e$ be orthonormal and set

$$\widehat h'=\frac{\widehat h+\epsilon e} {\sqrt{1+\epsilon^2}}.$$

Compute the mismatch without further maximization for small $\epsilon$.

Solution

The overlap is

$$(\widehat h\mid\widehat h') =\frac{1}{\sqrt{1+\epsilon^2}} =1-\frac12\epsilon^2+O(\epsilon^4).$$

Therefore $\mathfrak m=\epsilon^2/2+O(\epsilon^4)$. The absence of a linear term reflects normalization and orthogonality; if $e$ were an optimized tangent direction, alignment would project it away as well.

References

• MacDonald, Ilana, Samaya Nissanke, and Harald P. Pfeiffer. “Suitability of Post-Newtonian/Numerical-Relativity Hybrid Waveforms for Gravitational Wave Detectors.” Classical and Quantum Gravity 28 (2011): 134002. DOI. Open PDF.
• Owen, Benjamin J. “Search Templates for Gravitational Waves from Inspiraling Binaries: Choice of Template Spacing.” Physical Review D 53 (1996): 6749–6761. DOI. Open PDF.