Stochastic Background Spectra

A stochastic gravitational-wave background is described statistically rather than by a reproducible phase history. Strain spectral density, characteristic strain, and energy density are equivalent only after the Fourier, sidedness, polarization, sky, and critical-density conventions are fixed. This page derives one internally complete dictionary for a stationary, isotropic, unpolarized tensor background observed today. Search pipelines, upper limits, and current detections are outside its scope.

Helpful background. Einstein Equations supplies gravitational-wave energy. Waves, Green Functions, and Geometric Optics supplies Fourier and stationarity conventions. Gravitational Radiation and Waveforms supplies polarization and flux normalization. Relativistic Matter and Compact Objects supplies astrophysical source populations. Relativistic Cosmology supplies expansion, transfer, and critical density.

One-sided field spectrum

Use

$$h_A(t)=\int_{-\infty}^{\infty} \widetilde h_A(f)e^{-2\pi ift}\,df, \qquad \widetilde h_A(-f)=\widetilde h_A(f)^*.$$

For positive frequencies, define a one-sided polarization spectrum by

$$\left\langle \widetilde h_A(f)\widetilde h_B(f')^* \right\rangle =\frac12\delta_{AB}\delta(f-f')S_A(f).$$

Then

$$\left\langle h_A^2(t)\right\rangle =\int_0^\infty S_A(f)\,df.$$

For the unpolarized tensor background define the total field spectrum $S_h=S_++S_\times$, with $S_+=S_\times=S_h/2$. It has units $\mathrm{Hz}^{-1}$. This is a spacetime-field spectrum before antenna averaging; a detector-channel spectrum contains response factors and noise.

For radiation arriving from the whole sky, the polarization labels are local to each direction. The precise isotropic version of the shorthand above is

$$\left\langle\widetilde h_A(f,\boldsymbol n) \widetilde h_B(f',\boldsymbol n')^*\right\rangle =\frac{S_h(f)}{16\pi}\delta_{AB}\delta(f-f') \delta^2(\boldsymbol n,\boldsymbol n'),$$

where the sphere delta function integrates to one. Each polarization’s one-sided power per solid angle is $S_h/(8\pi)$, so summing both and integrating over $4\pi$ gives $S_h$. The strain-rate variance below is this sky sum of independent plane-wave contributions; it does not require one common plus-cross screen for all directions. This normalization agrees with Romano and Cornish 2017, §2.3.1, eqs. (2.14)–(2.17), p. 20, Open PDF.

The stationary random-field conventions and their detector translation are developed in Allen and Romano 1999, §§II–IV, PDF and Romano and Cornish 2017, §§2–4, PDF.

From strain rate to energy density

In the short-wavelength regime, the tensor-wave energy density in the local cosmological frame is

$$\rho_{\rm gw} =\frac{1}{16\pi G} \left\langle \dot h_+^2+\dot h_\times^2 \right\rangle.$$

Using the one-sided definition,

$$\frac{d\rho_{\rm gw}}{d\ln f} =\frac{\pi}{4G}f^3S_h(f).$$

The factors follow directly: each time derivative gives $2\pi f$, the logarithmic interval gives one more factor of $f$, and the polarization sum is already inside $S_h$. Dimensional analysis gives energy density because $f^3S_h$ has units $\mathrm{s}^{-2}$.

Dimensionless energy spectrum

Let today’s critical density be

$$\rho_{c,0}=\frac{3H_0^2}{8\pi G}.$$

Define

$$\Omega_{\rm gw}(f) =\frac{1}{\rho_{c,0}} \frac{d\rho_{\rm gw}}{d\ln f}.$$

Substitution yields the field-level relation

$$\boxed{ \Omega_{\rm gw}(f) =\frac{2\pi^2}{3H_0^2}f^3S_h(f) }.$$

Both $f$ and $H_0$ must use the same time unit. $\Omega_{\rm gw}(f)$ is dimensionless and is a density per logarithmic observed-frequency interval, not the integral over all frequencies.

Characteristic strain and amplitude spectrum

In this convention define

$$h_c^2(f)=fS_h(f).$$

Then

$$\Omega_{\rm gw}(f) =\frac{2\pi^2}{3H_0^2}f^2h_c^2(f).$$

$h_c$ is dimensionless, while $\sqrt{S_h}$ has units $\mathrm{Hz}^{-1/2}$. If $\Omega_{\rm gw}$ is flat, then $S_h\propto f^{-3}$ and $h_c\propto f^{-1}$. Papers that define $h_c^2=2fS_h$, use a two-sided spectrum, or assign $S_h$ to one polarization instead of the sum will display different factors of two. Translate definitions rather than averaging formulas.

Field spectra versus detector spectra

An interferometer measures

$$\widetilde s_I(f)= \sum_A\int d\Omega_{\boldsymbol n}\, F_I^A(f,\boldsymbol n) \widetilde h_A(f,\boldsymbol n),$$

so its signal cross-spectrum contains angular response products and, for separated detectors, propagation phases. Conventional overlap functions may absorb sky averages, polarization sums, and a normalization chosen to equal unity for a reference geometry. This is why a detector-literature formula can contain a factor such as $10\pi^2$ where the field dictionary above contains $2\pi^2$. The formulas are not inconsistent if their $S_h$ symbols denote different normalized objects.

Noise adds a separate channel spectrum. A measured power spectrum is therefore not automatically $S_h$, and no present empirical result is inferred from these definitions.

For a concrete check, take a right-angle Michelson interferometer in the long-wavelength limit with the usual dimensionless strain response. The screen-independent sky average satisfies $\langle F_+^2+F_\times^2\rangle=2/5$. Equal polarization power then gives a one-sided signal-channel spectrum

$$S_{\rm channel}=\frac15S_h, \qquad \Omega_{\rm gw} =\frac{10\pi^2}{3H_0^2}f^3S_{\rm channel}.$$

This explicitly accounts for the factor five without changing the energy carried by the field. The geometry and low-frequency assumptions matter; a different opening angle or finite-arm response changes the conversion Romano and Cornish 2017, §5.3.3, p. 77, Open PDF.

A normalization checklist

Before translating a published spectrum, ask five questions. Is the Fourier transform written with $f$ or angular frequency $\omega$? Is the spectrum one sided or two sided? Does its strain symbol describe one polarization, the sum of two polarizations, a sky-resolved field, or an antenna-averaged detector channel? Is energy normalized to today’s critical density or to the total radiation density? Is the quoted frequency in the source frame or the observer frame?

These choices generate most apparent disagreements. Replacing $f$ by $\omega=2\pi f$ changes both the integration measure and derivative factor. Converting a two-sided spectrum to a one-sided spectrum doubles its positive-frequency value while leaving the variance unchanged. Summing two equal tensor polarizations doubles a one-polarization spectrum. Antenna averaging introduces geometry factors that should remain with the response rather than be silently called a new gravitational-wave energy density. A reliable conversion first rewrites the covariance and variance definitions, then derives the coefficient instead of matching notation by name.

Assumptions and extensions

Stationarity produces the frequency delta function; isotropy makes the sky covariance direction independent; unpolarized radiation makes the plus-cross matrix proportional to the identity. Dropping these assumptions leads to a sky- and polarization-dependent coherency matrix, treated on the next page. Non-Gaussianity does not change the two-point spectrum, but it means the spectrum is not a complete statistical description.

For cosmological radiation, $f$ is today’s physical frequency after redshift and transfer. A primordial calculation must evolve the mode energy through horizon entry and changes in relativistic degrees of freedom before comparing with $\rho_{c,0}$. An unresolved astrophysical population instead integrates source-frame emission over redshift and rate. The spectral dictionary is common; the production kernel is not.

Exercises

1. Numerical conversion. Take $H_0=2.2\times10^{-18}\,\mathrm{s}^{-1}$, $f=25\,\mathrm{Hz}$, and $\Omega_{\rm gw}=10^{-9}$. Find the total field $S_h$ and $h_c$ in this convention.

Solution

Inverting the boxed relation gives

$$S_h=\frac{3H_0^2}{2\pi^2} \frac{\Omega_{\rm gw}}{f^3} \simeq4.7\times10^{-50}\,\mathrm{Hz}^{-1}.$$

Therefore

$$h_c=\sqrt{fS_h} \simeq1.1\times10^{-24}.$$

The corresponding amplitude spectral density is $\sqrt{S_h}\simeq2.2\times10^{-25}\,\mathrm{Hz}^{-1/2}$. None of these numbers includes an antenna response or detector noise.

As a dimensional check, $H_0^2f^{-3}$ has units of seconds, which equals $\mathrm{Hz}^{-1}$. Multiplying the result by $f$ before taking the square root therefore produces a dimensionless characteristic strain.

2. Recover the spectral slope. If $S_h\propto f^\alpha$, what are the slopes of $h_c$ and $\Omega_{\rm gw}$?

Solution

Since $h_c^2=fS_h$, one has $h_c\propto f^{(1+\alpha)/2}$. Since $\Omega_{\rm gw}\propto f^3S_h$, one has $\Omega_{\rm gw}\propto f^{3+\alpha}$. Thus $\alpha=-3$ produces a flat energy-density spectrum and $h_c\propto f^{-1}$, providing a quick convention check.

For example, white strain power has $\alpha=0$: its characteristic strain rises as $f^{1/2}$ and its energy density rises as $f^3$. “White” therefore depends on which spectral quantity is being described.

References

• Allen, Bruce, and Joseph D. Romano. “Detecting a Stochastic Background of Gravitational Radiation: Signal Processing Strategies and Sensitivities.” Physical Review D 59 (1999): 102001. DOI. Open PDF.
• Romano, Joseph D., and Neil J. Cornish. “Detection Methods for Stochastic Gravitational-Wave Backgrounds: A Unified Treatment.” Living Reviews in Relativity 20 (2017): 2. DOI. Open PDF.