Cosmological Gravitational-Wave Backgrounds

A cosmological gravitational-wave background is a fossil only after three maps have been kept distinct: a source creates tensor perturbations, the expanding universe transfers them across horizon entry and thermal thresholds, and an observing system converts the arriving field into a data vector. This chapter develops that chain for inflation, first-order phase transitions, cosmic defects, and scalar-induced waves. Its goal is not a list of proposed spectra, but a framework for deciding which feature belongs to production, propagation, or inference.

Helpful background. Einstein Equations and relativistic matter supply the tensor field equation and its stress source. Gravitational radiation introduces polarization and stochastic spectra; the cosmology overview locates the expansion and primordial perturbation chapters; and PDEs and Green functions provides propagation and retarded-response methods. The normalization needed for this chapter is developed explicitly below.

Enter the source–transfer–inference chain

The six pages support two complementary routes.

1. For a source-first route, read Inflationary Tensor Backgrounds, First-Order Phase-Transition Backgrounds, Cosmic Strings and Defect Backgrounds, and Scalar-Induced Gravitational Waves. Compare a passive primordial initial condition, a finite-duration plasma source, a scaling network that remains active, and a second-order source built from scalar statistics.
2. For an interpretation-first route, establish the tensor normalization on the inflationary page, then read Transfer Functions Across Cosmic History and Separating Cosmological and Astrophysical Backgrounds. This route asks which spectral features survive propagation and which additional observables are required before an origin can be identified.

The page order is deliberate. A measured slope is not a source parameter until the transfer history has been specified, and a transferred spectrum is not an observational verdict until detector response, foregrounds, selection, and covariance have been included.

One tensor convention across all six pages

For a spatially flat background, write

$$g_{ij}=a^2(\eta)(\delta_{ij}+h_{ij}), \qquad \partial_i h_{ij}=0, \qquad h^i{}_i=0,$$

with $e^{i\mathbf k\cdot\mathbf x}$ and the direct polarization decomposition

$$h_{ij}(\eta,\mathbf x) =\sum_{\lambda=+,\times}\int\frac{d^3k}{(2\pi)^3}\, e_{ij}^{\lambda}(\widehat{\mathbf k}) h_{\lambda,\mathbf k}(\eta)e^{i\mathbf k\cdot\mathbf x},$$

where $e_{ij}^{\lambda}e^{\lambda'ij}=2\delta_{\lambda\lambda'}$ in the real plus/cross basis used below. More generally, a complex polarization basis uses the Hermitian inner product and its inverse projection,

$$e_{ij}^{A}e^{B*ij}=2\delta_{AB}, \qquad h_{A,\mathbf k}=\frac12e_A^{*ij}h_{ij,\mathbf k}.$$

Choose the real plus/cross frame on each Fourier pair so that

$$e_{ij}^{\lambda}(-\widehat{\mathbf k}) =e_{ij}^{\lambda}(\widehat{\mathbf k}), \qquad h_{\lambda,-\mathbf k}=h_{\lambda,\mathbf k}^*.$$

The stars may then be omitted in real-basis contractions. This is a convention on each pair $\{\mathbf k,-\mathbf k\}$, not a claim that one polarization frame can be chosen smoothly over the entire direction sphere. For a zero-mean, statistically homogeneous background, encode the two-time correlation in a polarization spectral kernel,

$$\left\langle h_{\lambda,\mathbf k}(\eta) h^*_{\lambda',\mathbf k'}(\eta')\right\rangle =(2\pi)^3\delta^{(3)}(\mathbf k-\mathbf k') \frac{2\pi^2}{k^3}\mathsf P_{\lambda\lambda'} (\mathbf k;\eta,\eta').$$

At equal time, $\mathsf P(\mathbf k,\eta)\equiv \mathsf P(\mathbf k;\eta,\eta)$ is Hermitian. Statistical isotropy removes physical dependence on the direction $\widehat{\mathbf k}$ and constrains the matrix’s polarization structure; it does not make an evolving spectrum time independent. After fixing a polarization-frame convention, write $\mathcal P_{h_\lambda}(k,\eta)\equiv \mathsf P_{\lambda\lambda}(k,\eta)$ and define the contraction-normalized equal-time tensor spectrum

$$\mathcal P_t(k,\eta) \equiv2\,\operatorname{tr}\mathsf P(k,\eta) =2\sum_\lambda\mathcal P_{h_\lambda}(k,\eta) =2\left[\mathcal P_{h_+}(k,\eta)+\mathcal P_{h_\times}(k,\eta)\right].$$

Thus $\mathcal P_t$ is the dimensionless spectrum of $h_{ij}h^{ij}$, not the unweighted component sum. An unpolarized parity-even background has a diagonal matrix with equal entries. Define the circular basis and its amplitudes by

$$e_{ij}^{R,L}=\frac{e_{ij}^{+}\pm i e_{ij}^{\times}}{\sqrt2}, \qquad h_{R,L}=\frac{h_+\mp i h_\times}{\sqrt2}.$$

With the real even parent frame above, field reality becomes $h_R(-\mathbf k)=h_L(\mathbf k)^*$ and $h_L(-\mathbf k)=h_R(\mathbf k)^*$. Circular polarization or primordial chirality therefore requires the full Hermitian matrix. The transformation is unitary, so its trace—and hence the total energy normalization below—is basis independent.

For a passive primordial background with a common, polarization-independent transfer function, define $h_{\lambda,\mathbf k}(\eta)=T_h(k,\eta) h_{\lambda,\mathbf k}^{\rm prim}$ and choose the regular mode $T_h\to1$ as $k/\mathcal H\to0$. The equal-time and primordial matrices are then related by

$$\mathsf P(k;\eta,\eta') =T_h(k,\eta)T_h^*(k,\eta')\mathsf P^{\rm prim}(k), \qquad \mathcal P_t(k,\eta)=\left|T_h(k,\eta)\right|^2 \mathcal P_t^{\rm prim}(k).$$

This factorization does not apply to a general active source. For one, use the same basis to define

$$\Pi_{ij}^{\rm TT}(\eta,\mathbf k) =\sum_\lambda e_{ij}^{\lambda}(\widehat{\mathbf k}) \Pi_{\lambda,\mathbf k}(\eta), \qquad \Pi_{\lambda,\mathbf k}(\eta) \equiv\frac12e_\lambda^{ij}(\widehat{\mathbf k}) \Pi_{ij}^{\rm TT}(\eta,\mathbf k).$$

The retarded component equation is then

$$h_{\lambda,\mathbf k}''+2\mathcal Hh_{\lambda,\mathbf k}' +k^2h_{\lambda,\mathbf k} =16\pi Ga^2\Pi_{\lambda,\mathbf k}.$$

The factor $1/2$ is the inverse of the norm-two basis contraction. If the source variable were instead the raw contraction $e_\lambda^{ij}\Pi_{ij}^{\rm TT}=2\Pi_\lambda$, its equation would carry $8\pi G$. An active source’s unequal-time stress correlations cannot in general be replaced by one primordial amplitude. Caprini and Figueroa give a common treatment of the source and stochastic conventions in Caprini and Figueroa 2018, §2.4, printed pp. 19–20, eqs. (53)–(61), and §§3.1–3.2, printed pp. 22–30, eqs. (80)–(86) (Open PDF).

For the passive field after it is freely oscillating inside the horizon, let $\mathcal P_{h'_\lambda}$ denote the equal-time spectrum of the derivative mode. The basis contraction and common transfer function give

$$\frac{d}{d\ln k}\left\langle h'_{ij}h'^{ij}\right\rangle =2\sum_\lambda\mathcal P_{h'_\lambda}(k,\eta) =\mathcal P_t^{\rm prim}(k)\left|T_h'(k,\eta)\right|^2.$$

The effective density uses an average over several wave periods, replacing $\left|T_h'\right|^2$ by $\overline{\left|T_h'\right|^2}$. Then $\rho_{\rm GW}=\overline{\langle h'_{ij}h'^{ij}\rangle}/(32\pi Ga^2)$ is the high-frequency result of Isaacson 1968, Eq. (4.1), printed p. 1274, specialized to this FLRW frame. Hence

$$\frac{d\rho_{\rm GW}}{d\ln k} =\frac{1}{32\pi Ga^2}\, \mathcal P_t^{\rm prim}(k)\, \overline{\left|T_h'(k,\eta)\right|^2}.$$

Together with $\rho_c=3\mathcal H^2/(8\pi Ga^2)$, the corresponding energy fraction for the passive background is therefore

$$\Omega_{\rm GW}(k,\eta) =\frac1{12} \left(\frac{k}{\mathcal H}\right)^2 \mathcal P_t^{\rm prim}(k)\, \overline{\left|T_h(k,\eta)\right|^2}.$$

The oscillation average and the short-wave condition are part of this formula. It must not be assigned to a superhorizon tensor merely because $\mathcal P_t$ is defined there. The factor two in the definition of $\mathcal P_t$ and the coefficient $1/12$ are one convention package: using the literal component sum with the same coefficient would lose half the energy. With $a_0=1$, angular wavenumber and observed cyclic frequency are related by

$$f_0=\frac{k}{2\pi a_0} =1.546\times10^{-15}\,\mathrm{Hz} \left(\frac{k}{1\,\mathrm{Mpc}^{-1}}\right).$$

Keeping the $2\pi$ explicit prevents a production scale stated as $k$ from being shifted by an entire logarithmic frequency interval.

At the present epoch, the same stochastic field can instead be described by the total one-sided strain spectrum $S_h(f)=S_+(f)+S_\times(f)$ and characteristic strain $h_c^2(f)=fS_h(f)$. With $a_0=1$, $k=2\pi f$, and $d\ln k=d\ln f$, the exact bridge to the spatial tensor spectrum is

$$\mathcal P_t(k,\eta_0)=2fS_h(f)=2h_c^2(f).$$

It makes the two energy-density formulas identical:

$$\frac1{12}\left(\frac{k}{H_0}\right)^2\mathcal P_t =\frac1{12}\left(\frac{2\pi f}{H_0}\right)^2 \left[2fS_h(f)\right] =\frac{2\pi^2}{3H_0^2}f^3S_h(f).$$

Here $S_h$ is the total field spectrum, not the strain spectrum of one detector channel. Stochastic Background Spectra develops the one-sided convention and detector-response distinction.

What production and transfer each contribute

Inflation supplies an initial tensor spectrum whose later evolution is homogeneous once anisotropic stress is neglected. A first-order transition instead combines bubble nucleation, wall motion, acoustic fluid power, and turbulence; its peak and amplitude depend on duration, efficiencies, and the lifetime of the source. A cosmic-string network continually generates loops, bursts, and an integrated stochastic background, so network and loop distributions are inseparable from the prediction. Scalar-induced waves are second order in enhanced scalar perturbations, and the same scalar statistics must be used when comparing them with primordial-black-hole production.

After production, a regular tensor is constant outside the Hubble radius and has an $a^{-1}$ envelope after entry. For a scale-invariant primordial spectrum, modes entering during a decelerating constant-$w$ epoch ($w\gt-1/3$) acquire the transfer slope

$$n_{\rm tr}=\frac{2(3w-1)}{1+3w}.$$

Radiation gives a flat plateau, matter-like expansion gives $-2$, and a stiff epoch gives $+1$. Changes in $g_*$ and $g_{*s}$ add entropy-transfer steps; collisionless relativistic species add a history-dependent damping kernel. These are propagation features, not new production mechanisms. Conversely, an active source operating through horizon entry cannot always be represented as a production spectrum multiplied by a universal transfer factor.

Page	Physical object	Decisive check
Inflationary tensors	passive primordial spectrum	radiation- and matter-entry asymptotes plus the $k$-to-$f$ conversion
Phase transition	finite-duration plasma stress	energy budget, acoustic lifetime, and redshifted peak
Strings and defects	scaling network and loop population	propagate two loop models through the same history
Scalar-induced waves	scalar convolution source	narrow-spectrum support and resonance with matched scalar statistics
Cosmic transfer	homogeneous and free-streaming propagation	superhorizon constancy, Bessel limits, and refined transition matching
Background separation	response-folded component mixture	rank, injection recovery, and prior-sensitive model comparison

From a spectrum to an origin claim

A frequency spectrum alone rarely identifies its source. Smooth components can share a local amplitude and slope; transfer steps can resemble population cutoffs; and a high-duty-cycle astrophysical population can be nearly Gaussian and isotropic. Useful separation therefore combines frequency curvature with angular structure, polarization, duty cycle, higher moments, resolved-source subtraction, and cross-correlation with matter tracers.

In a local linear approximation, component amplitudes are identifiable only when the response-folded template matrix has full rank,

$$\operatorname{Cov}(\widehat{\boldsymbol a}) =\left(\mathbf A^{\mathsf T}\mathbf C^{-1}\mathbf A\right)^{-1}.$$

Higher signal-to-noise cannot repair proportional columns. The stochastic likelihood, calibrated response, cross-correlation, anisotropic inverse problem, and polarization separation are developed in Romano and Cornish 2017, §3.6, §§4–5, §7.3, and §§8.1–8.4, printed pp. 41–150. Those methods supply the inference machinery; they do not by themselves identify an astrophysical or cosmological production channel. Gravitational-Wave Astronomy explains detector searches and the upper limits they establish. This chapter develops the cosmological source and transfer calculations needed to interpret those results. The research examples use an evidence cutoff of 2026-08-19.

Review the chapter

A passive primordial tensor power has a local spectrum $\mathcal P_t^{\rm prim}\propto k^3$. Its modes enter during an early matter-like epoch with $w=0$. Find the local present energy-density slope and convert $k=10^{12}\,\mathrm{Mpc}^{-1}$ to observed cyclic frequency. Next suppose two whitened frequency summaries have cosmological and astrophysical template columns $\mathbf c=(1,2)^{\mathsf T}$ and $\mathbf a=(2,4)^{\mathsf T}$. Decide whether the two amplitudes are identifiable. Finally add an anisotropy summary for which the extended columns are $\mathbf c=(1,2,0)^{\mathsf T}$ and $\mathbf a=(2,4,1)^{\mathsf T}$, and repeat the rank test.

Solution

For $w=0$, the passive transfer contributes $n_{\rm tr}=-2$. With primordial tensor tilt $n_t=3$, the present energy-density slope is therefore

$$n_{\Omega,0}=n_t+n_{\rm tr}=3-2=1.$$

The frequency is

$$f_0=1.546\times10^{-15}\,\mathrm{Hz}\times10^{12} =1.546\times10^{-3}\,\mathrm{Hz}.$$

With only the two frequency summaries, $\mathbf a=2\mathbf c$, so the two-column response matrix has rank one and its Fisher matrix is singular. The amplitudes cannot be separated regardless of signal-to-noise. After the anisotropy summary is added,

$$\mathbf A^{\mathsf T}\mathbf A =\begin{pmatrix}5&10\\10&21\end{pmatrix}, \qquad \det(\mathbf A^{\mathsf T}\mathbf A)=5.$$

The determinant is nonzero, so the two local amplitudes are identifiable in this idealized model. A real claim would still require calibration, foreground, selection, and template uncertainty rather than treating $\mathbf A$ as exact.

References

• Caprini, Chiara, and Daniel G. Figueroa. 2018. “Cosmological Backgrounds of Gravitational Waves.” Classical and Quantum Gravity 35, 163001. DOI.
• Isaacson, Richard A. 1968. “Gravitational Radiation in the Limit of High Frequency. II. Nonlinear Terms and the Effective Stress Tensor.” Physical Review 166, 1272–1280. DOI.
• Romano, Joseph D., and Neil J. Cornish. 2017. “Detection Methods for Stochastic Gravitational-Wave Backgrounds: A Unified Treatment.” Living Reviews in Relativity 20, 2. DOI.