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The modeled matched filter search found an arrival difference of about $7\mathrm{ms}$, within the roughly $10\mathrm{ms}$ intersite light travel time. Background events were obtained by shifting one detector's timestamps far beyond that time and repeating the coincidence analysis. The binary search bounded the false alarm rate below one event per $203000$ years. That is an empirically estimated background rate, not a probability that the source interpretation is false.^[ref]

The logic is to destroy a common astronomical arrival while retaining each instrument's real glitches. A Poisson distribution approximation would turn a background rate $r$ into $1-e^{-rT}$ for at least one equally strong noise event during live time $T$, only after accounting for the search's ranking and trials. Repeated time slides reuse data; their accumulated background duration is not an equally long series of independent new observations. Correlated disturbances also require separate controls because shifting timestamps can erase them along with a true signal.

The dedicated noise study inspected auxiliary channels and deliberately applied magnetic, acoustic and mechanical disturbances to measure their coupling into strain. For GW150914, checks included environmental coincidences, instrument state, and exclusion of an intentional hardware injection. It also examined potentially correlated global magnetic disturbances, which time slides alone cannot exclude.^[ref] The point of an injection test is quantitative: a quiet environmental monitor is informative only if its sensitivity and coupling bound the disturbance needed to explain the candidate.

The condition for preserving a spectral signal under linear compression specifies when compression retains the covariance-weighted signal measure of a known shape. Let the mean be $\theta \mathbf{s}$, with fixed positive-definite noise covariance $C$, and compress by a full-row-rank matrix $B$. The measures in noninvertible spectral compression versus a coordinate change obeyThus the rows of the compression must retain the noise-weighted signal direction, not necessarily the unweighted plotted shape.

For a short proof, whiten the shape as $\mathbf{t}=C^{-1/2}\mathbf{s}$ and put $M=B{C}^{1/2}$. The parent's orthogonal projection formula givesEquality holds exactly when $\mathbf{t}$ lies in the row space of $M$. Multiplying that subspace relation by $C^{-1/2}$ yields the stated condition. Linear covariance propagation gives the compressed covariance $BC{B}^T$.^[ref]

One useful construction is the scalar matched filter statistic $z={\mathbf{s}}^T{C}^{-1}\mathbf{y}$. Its mean is $\theta J$ and variance $J$, so $z/J$ has variance $1/J$, the original optimal linear amplitude precision. In the Gaussian known-covariance model, this statistic also retains the likelihood's dependence on that single amplitude.

For an original example, choose $C=\mathrm{diag}(1,4)$ and $\mathbf{s}=(1,2{)}^T$. Then $J=2$ and the retaining row is $B=(1,1/2)$. Its signal response is two and noise variance two, giving $J_B=4/2=2$. An unweighted sum $B=(1,1)$ instead has response three and variance five, giving $J_B=9/5=1.8$: ten percent of this signal measure is lost.

Signal-subspace preservation for several spectral amplitudes requires retaining the corresponding family of weighted mean directions when all those amplitudes are to be preserved. Unknown nuisance shapes can also change the relevant estimation problem. A compression designed for one known shape does not automatically preserve every spectral question or uncertainty component.

GW150914 is the first directly detected gravitational wave, interpreted as a binary black hole inspiral, merger and ringdown. Its discovery paper reports a peak projected gravitational wave strain of approximately $10^{-21}$ and a combined matched filter signal-to-noise ratio of 24. The inferred source-frame component masses were $36_{-4}^{+5}$ and $29_{-4}^{+4}$ solar masses, with 90 percent credible intervals. These are model-dependent estimates from a detector network.^[ref]

The interesting measurement is a chain: curvature changes relative light travel times, wave interference turns those changes into a photodetector signal, calibrated records from separate sites constrain a common source, and statistical inference tests competing explanations. The following original models explain that chain; the plotted time series are stipulated calculations, never measured GW150914 data.

For detector $k$, model the calibrated data as $d_k(t)=h_k(t;\theta )+n_k(t)$, with a shared source parameter vector $\theta$, separate detector responses, propagation delays and noise $n_k$. Under an independent stationary Gaussian process approximation with one-sided power spectral density $S_k(f)$, define the noise-weighted inner product and likelihood functionStrain power spectral density has units ${\mathrm{Hz}}^{-1}$; the displayed inner product is dimensionless when the Fourier transform integrates over seconds. A matched filter rewards agreement over the full evolving wave phase, weighting frequency bands by their noise. One conspicuous peak is insufficient: a candidate should also have the right distribution of signal across frequency bands and compatible projections in the network.

The parameter paper used coherent Bayesian inference, numerical relativity-informed inspiral–merger–ringdown models and explicit calibration uncertainties. It found a strong distance–inclination correlation; source-frame masses additionally require converting the detector-frame masses by a cosmological redshift.^[ref] Conceptually, multiply the likelihood function by a prior distribution to obtain a posterior distribution. Changes in waveform families, residual structure after subtraction, and checks with a generic transient reconstruction test parts of the modeling assumptions. Detection significance and parameter uncertainty answer different questions.