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A power spectrum resolves the second-moment strength of a stationary process or random field by temporal frequency or spatial wavenumber. For a real, mean-zero stationary process with integrable autocovariance function $C(\tau )=\mathbb{E}[X_{t+\tau }{X}_t]$, use the angular-frequency conventionThis form of the Wiener-Khinchin theorem makes the interpretation concrete: integrating spectral density over all frequencies recovers the variance. A nonzero mean contributes a spectral atom at zero if one transforms the uncentered correlation instead of the autocovariance function. Some stationary processes have discrete spectral lines, so an ordinary integrable spectral density need not exist.

For an Ornstein-Uhlenbeck process with $C(\tau )=v{e}^{-a|\tau |}$, $a,v>0$, the Fourier transform is $S(\omega )=2av/(a^2+{\omega }^2)$. Its area is $2\pi v$ as required. Small $a$ concentrates power near zero and corresponds to slowly decaying fluctuations. Conventions using cyclic frequency rather than angular frequency move the factors of $2\pi$; stating the convention prevents a normalization error when comparing spectra.

The primordial bispectrum is the Fourier-space connected correlation function of order three for an early-universe perturbation field. For a zero-mean real scalar field $\varphi$, statistical homogeneity impliesA common spatial translation multiplies the left side by $e^{i({\mathbf{k}}_1+{\mathbf{k}}_2+{\mathbf{k}}_3)\cdot \mathbf{a}}$, so translation invariance requires support on closed wavevector triangles. If the field has a nonzero mean, subtract the disconnected mean/two-point products before defining this connected statistic.

Statistical isotropy is the additional assumption that removes orientation dependence, allowing the shorter notation $B(k_1,k_2,k_3)$ with $k_i=|{\mathbf{k}}_i|$. The three positive magnitudes obey the triangle inequality. Statistical homogeneity alone does not justify replacing the vectors by their lengths. The usual primordial convention is illustrated by the Planck definition.^[ref]

A zero-mean Gaussian random field has vanishing three-point function. A measured nonzero connected primordial bispectrum therefore tests primordial non-Gaussianity beyond the power spectrum. The converse fails: a non-Gaussian field whose distribution is invariant under $\varphi \mapsto -\varphi$ has zero odd moments but can have a nonzero connected four-point function. Comparing predictions requires specifying the field, such as a gravitational potential or comoving curvature perturbation, because nonlinear field redefinitions change its primordial bispectrum.

For a scale-invariant primordial bispectrum, the displayed shape function is invariant under common rescaling of all three wavenumbers. It isolates triangle dependence from the overall power spectrum amplitude. It indicates intrinsic shape preference but alone does not determine observational signal-to-noise, which also depends on transfer functions, covariance and survey weights.

An oscillatory feature with finite width in a source's variability power spectrum, rather than a perfectly periodic clock. Orbital, precession and disc-mode interpretations connect different frequencies to the same compact object. Identifying an observed peak with an innermost stable circular orbit orbital frequency is therefore a model assumption, not a direct black-hole mass or spin measurement.

The Rossi X-ray Timing Explorer was a space observatory designed to measure rapid X-ray variability, operating from its 1995 launch until 2012. Its time-resolved observations probed accretion and rotation around compact objects such as neutron stars and black holes.

Fast event timing complements an energy spectrum. A narrow feature in a temporal power spectrum identifies an oscillation or rotation frequency of the source, not the frequency of an individual X-ray photon. Comparing such timing features with changes in spectral shape helps distinguish physical states of an accreting system.^[ref]

For a universal abundance depending on halo peak height, the peak-background split gives a potential-response coefficient $b_{\phi }=2f_{\mathrm{NL}}{\delta }_c{b}_L$, with $b_L$ the linear Lagrangian halo bias. If $\delta =\alpha (k)\phi$, the corresponding deterministic galaxy bias correction is $\Delta b=b_{\phi }/\alpha (k)$, proportional to $k^{-2}$ on sufficiently large scales. For a physical Eulerian tracer, $b_E=1+b_L$, so $b_{\phi }=2f_{\mathrm{NL}}{\delta }_c(b_E-1)$. The deterministic tracer power spectrum is $(b_E+\Delta b{)}^2{P}_m$; it remains nonnegative even if a negative correction changes the sign of the tracer-matter correlation.

A law or statistical object is scale invariant if its form is unchanged by rescaling lengths, with any specified scaling weight for the fields. A dimensionless field's three-dimensional nonzero-mode power spectrum obeys $P(\lambda k)={\lambda }^{-3}P(k)$ when its statistics have this scaling. The meaning of invariance must specify whether zero modes and regulators are being transformed.

For a stationary quenched intrinsic curvature with power spectrum $S_{\kappa }(q)$ and finite second spectral moment, the bulk deficit satisfiesThe moment is $⟨({\kappa }_0^{\prime }{)}^2⟩$ for mean-square differentiable curvature. Infinite-moment disorder and finite endpoint boundary layers can change the exponent, so this is a smooth bulk law rather than a universal law for all random polymers.

A random field is statistically isotropic when all of its joint distributions are invariant under a common spatial rotation of their arguments. A statistically isotropic scalar power spectrum depends only on the magnitude of its wavevector.

The equal-time two-point function of a de Sitter scalar has a contribution $P_{\mathrm{vac}}=1/(2k{a}^2)$, giving the displayed dimensionless power spectrum per logarithmic wavenumber. At fixed comoving wavenumber it decreases as $a^{-2}$; on short physical scales it reproduces the vacuum fluctuations of a massless scalar field in Minkowski spacetime. Its coincident integral is ultraviolet divergent, so the fixed-mode decrease cannot be interchanged with an unregulated integral over all modes.