Separating Cosmological and Astrophysical Backgrounds

A measured stochastic background would not arrive labeled “primordial” or “astrophysical.” Its origin must be inferred from a mixture of unresolved sources after detector response, selection, subtraction, and calibration. Spectral shape is only one coordinate in that inference. Angular structure, duty cycle, higher-order statistics, polarization, resolved-population information, and cross-correlations become useful only when analyzed in one consistent likelihood.

Helpful background. The Einstein Equation supplies tensor propagation; Relativistic Matter and Compact Objects supplies source populations; Gravitational Radiation and Waveforms supplies stochastic power, polarization, and response conventions; Relativistic Cosmology supplies primordial production and transfer; and Partial Differential Equations and Green Functions supplies response kernels.

Two source histories enter one likelihood

For transient astrophysical sources, let $R(z)$ be the total event-rate density per unit source-frame time and comoving volume, and let $p(\vartheta\mid z)$ be the conditional density of source parameters, normalized by $\int d\vartheta\,p(\vartheta\mid z)=1$. A schematic population background is

$$\Omega_{\rm a}(f) =\frac{f}{\rho_{c,0}H_0} \int dz\,d\vartheta\, \frac{R(z)p(\vartheta\mid z)} {(1+z)E(z)} \left. \frac{dE_{\rm GW}(f_s;\vartheta)}{df_s} \right|_{f_s=(1+z)f}.$$

Here $E(z)\equiv H(z)/H_0$. Equivalently, one may introduce the differential rate

$$\mathcal R(z,\vartheta) =\frac{dR(z)}{d\vartheta} =R(z)p(\vartheta\mid z)$$

and use $\mathcal R$ in the integral without multiplying by $p$ again. The exact observer/source-frame bookkeeping and the canonical source-class comparison are developed in Astrophysical and Cosmological Backgrounds. Rate evolution, masses, environments, waveforms, and the resolved-source threshold all change this integral. A passive primordial tensor spectrum has a different, sharply defined transfer problem. Assume linear homogeneous propagation after production, no continuing transverse-traceless source, and one regular primordial mode. If $h_{\lambda,\mathbf k}=T_hh_{\lambda,\mathbf k}^{\rm prim}$, its equal-time power is $\mathcal P_t(k,\eta)=\left|T_h(k,\eta)\right|^2 \mathcal P_t^{\rm prim}(k)$. After the mode is freely oscillating well inside the Hubble radius, use the chapter’s direct decomposition $h_{ij}=\sum_\lambda e_{ij}^{\lambda}h_\lambda$ with $e_{ij}^{\lambda}e^{\lambda'ij}=2\delta_{\lambda\lambda'}$ and $\mathcal P_t=2\sum_\lambda\mathcal P_{h_\lambda}$. Denote the passive cosmological contribution by $\Omega_{\rm GW}^{\rm cos}$. These choices give

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

Here $T_h$ transfers strain amplitude; the factor $k^2/\mathcal H^2$ and the short-wave condition are not optional. Inflation supplies this passive initial-value problem. Phase transitions, cosmic strings, and scalar-induced waves are actively sourced during production and propagate passively only after the relevant source has shut off; a long-lived string network remains active over any interval in which it continues radiating. Define $\mathcal L_k\equiv\partial_\eta^2+2\mathcal H\partial_\eta+k^2$ and its retarded Green function by

$$\mathcal L_k^{(\eta)}G_k(\eta,\widetilde\eta) =\delta(\eta-\widetilde\eta), \qquad G_k(\eta,\widetilde\eta)=0\quad\text{for }\eta<\widetilde\eta.$$

Their transverse-traceless stress is resolved as

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

For vanishing initial tensor, $h_{\lambda,\mathbf k}(\eta_i)=h'_{\lambda,\mathbf k}(\eta_i)=0$, the causal solution is

$$h_{\lambda,\mathbf k}(\eta) =16\pi G\int_{\eta_i}^{\eta}d\widetilde\eta\, G_k(\eta,\widetilde\eta)a^2(\widetilde\eta) \Pi_{\lambda,\mathbf k}(\widetilde\eta).$$

The factor $1/2$ is the inverse projection for the norm-two basis. If the source symbol denoted the raw contraction $e_\lambda^{ij}\Pi_{ij}^{\rm TT}$, the integral would instead carry $8\pi G$. If primordial homogeneous data and a later source coexist, write $h=h^{\rm hom}+h^{\rm src}$ and retain every covariance term,

$$\begin{aligned} \left\langle hh^*\right\rangle ={}&\left\langle h^{\rm hom}h^{{\rm hom}*}\right\rangle +\left\langle h^{\rm src}h^{{\rm src}*}\right\rangle \\ &+\left\langle h^{\rm hom}h^{{\rm src}*}\right\rangle +\left\langle h^{\rm src}h^{{\rm hom}*}\right\rangle . \end{aligned}$$

The cross terms vanish only under an explicit statistical-independence assumption. A continuing source requires the full source-history integral and its unequal-time stress covariance; its energy spectrum must be constructed from that solution in the valid short-wave regime. If $\eta_s$ is after source shutoff, a calculation may define a post-source energy transfer by $\Omega(\eta)=T_\Omega(\eta,\eta_s)\Omega(\eta_s)$. In general $T_\Omega\ne|T_h|^2$: energy depends on derivatives as well as strain, and its fractional normalization also follows the Hubble and entropy history. Source duration shapes the initial post-source spectrum $\Omega(\eta_s)$; it is not an extra part of the later passive $T_\Omega$. If the source is still active at $\eta_s$, this post-source factorization is not available. Caprini and Figueroa review these passive and active branches in Caprini and Figueroa 2018, §2.4, printed pp. 19–22, eqs. (53)–(68), and §§3.2–3.4, printed pp. 28–36, eqs. (80)–(111) (Open PDF).

Let $\mathbf d$ collect all summaries actually analyzed: frequency-bin cross powers, angular multipoles, polarization combinations, higher moments, and cross-tracer statistics. A minimal joint model is

$$p(\mathbf d\mid\theta_{\rm c},\theta_{\rm a},\phi) =\mathcal N\!\left[ \mathbf d;\, \mathbf R(\phi) \bigl(\mathbf s_{\rm c}(\theta_{\rm c}) +\mathbf s_{\rm a}(\theta_{\rm a})\bigr), \mathbf C(\phi) \right],$$

where $\mathbf R$ is the complete response and $\phi$ contains calibration, noise, selection, and subtraction nuisances. An additive spectrum before the response need not remain a trivially separable pair after masking, frequency-dependent selection, or nonlinear source subtraction. Romano and Cornish 2017, §3.6, §§4–5, §7.3, and §§8.1–8.4, printed pp. 41–150 develop the response, cross-correlation, anisotropic inverse problem, and polarization framework summarized by this response-first likelihood. The cosmological-versus- astrophysical mixture is the model being tested, not a separation guaranteed by that general framework.

A local spectral slope is not an origin tag

Across a narrow band, any smooth component has

$$\ln\Omega(f) =\ln\Omega(f_0) +\alpha\ln\frac{f}{f_0} +\frac12\beta \left(\ln\frac{f}{f_0}\right)^2+\cdots.$$

Two mechanisms can share amplitude and slope while differing only in curvature outside the measured range. An inspiral-dominated compact-binary population, one flank of a phase-transition peak, a broad string spectrum, and a transferred primordial power law can therefore be locally degenerate. Multi-band leverage helps only when inter-band calibration, time evolution, and population selection are jointly modeled.

Even a spectral knee is conditional. Relativistic thresholds and nonstandard expansion can create primordial steps; mass cutoffs, environments, and subtraction can create population steps. The relevant formulation axis is the frequency range together with the source population and transfer history, not the visual resemblance of a best-fit curve.

Discriminants and their counterexamples

Discriminant	Primordial expectation often used	Counterexample or nuisance
Spectral curvature	A causal peak, transfer step, or broad inflationary/string shape	A population mass scale or one cosmological flank can produce the same local curvature
Angular anisotropy	Many early sources begin nearly statistically isotropic	Propagation and primordial modulation create anisotropy; a high-duty-cycle distant population can be smooth
Duty cycle and non-Gaussianity	A diffuse early background may approach Gaussianity	Cosmic-string bursts are intermittent; many overlapping binaries can be nearly Gaussian
Polarization and chirality	Parity violation can generate a primordial circular component	Response leakage and source-orientation asymmetries require explicit controls
Matter cross-correlation	Low-redshift populations trace galaxies and lensing matter	Magnification, source bias, propagation, and primordial–matter correlations complicate a null
Resolved-source subtraction	Removing cataloged events should reduce a population component	Selection uncertainty and subthreshold extrapolation can bias the residual

No row is a binary classifier. The canonical source-level comparison develops these mechanisms; this page concentrates on the cosmological inference needed to combine them.

Identifiability is a rank condition

In a linearized summary model,

$$\mathbf d=\mathbf A\boldsymbol a+\mathbf n, \qquad \boldsymbol a=(A_{\rm c},A_{\rm a})^{\mathsf T}, \qquad \operatorname{Cov}(\widehat{\boldsymbol a}) =\left(\mathbf A^{\mathsf T}\mathbf C^{-1}\mathbf A\right)^{-1}.$$

This covariance assumes known templates and noise covariance, no informative amplitude prior, and fixed nuisance parameters. It follows by minimizing $(\mathbf d-\mathbf A\boldsymbol a)^{\mathsf T}\mathbf C^{-1} (\mathbf d-\mathbf A\boldsymbol a)$ and solving the resulting normal equations. It therefore measures the information supplied by the stated response, not by a prior that separates otherwise degenerate amplitudes.

If the two response-folded template columns are proportional, $\mathbf A^{\mathsf T}\mathbf C^{-1}\mathbf A$ is singular. More signal-to-noise in the same summary cannot identify two amplitudes. A new observable helps only if it adds a linearly independent response: for example, an angular cross-correlation may distinguish two spectra that are identical in frequency bins.

This rank test is local. Nonlinear hierarchical models can remain globally degenerate, and a formally invertible matrix can be so ill-conditioned that small calibration changes dominate the answer. Report singular values, posterior correlations, and prior sensitivity rather than only marginal error bars.

Nuisance freedom can also remove apparent rank. If an added angular summary distinguishes the source columns only through a signal pattern identical to an unconstrained calibration template, marginalizing that calibration loses the new information. In a local Gaussian model with joint Fisher blocks for amplitudes $\boldsymbol a$ and nuisances $\boldsymbol b$, the retained amplitude information is

$$\mathbf F_{a,\rm marg} =\mathbf F_{aa}-\mathbf F_{ab}\mathbf F_{bb}^{-1}\mathbf F_{ba},$$

when $\mathbf F_{bb}$ is invertible. Include declared nuisance-prior precision in $\mathbf F_{bb}$ if it is used. The rank and conditioning of this marginalized matrix, rather than of $\mathbf A$ alone, then determine what the data and calibration assumptions can separate.

Injection, recovery, and model comparison

A credible separation benchmark uses one forward model for every candidate:

1. draw astrophysical catalogs and a cosmological field, including their anisotropy, polarization, duty cycle, and cross-correlations;
2. inject both through the same detector network, cadence, mask, calibration, noise, and source-selection pipeline;
3. apply the same resolved-source subtraction used on the data;
4. fit astrophysical-only, cosmological-only, and mixed models with declared priors and shared nuisances;
5. repeat over independent realizations to test bias and 68%/95% interval coverage;
6. calibrate Bayes-factor or likelihood-ratio distributions under every generating model, including misspecified populations.

An injection at one convenient parameter point tests recovery, not coverage. A Bayes factor computed after choosing a favorable spectral basis is not calibrated model selection. Null injections are equally important: an astrophysical-only realization must not repeatedly create a primordial component when the population or calibration is varied within its allowed domain.

Cross-validation by frequency band, time block, sky region, and source-removal threshold can expose a component that follows the instrument or selection rather than the sky. Posterior predictive residuals should be evaluated in summaries not used to tune the model, especially angular and non-Gaussian statistics.

Results and limitations through 2026-08-19

Formulation or evidence	Result	Scope or limitation
Caprini–Figueroa 2018 production review	Establishes that early-universe mechanisms span smooth, peaked, broad, anisotropic, and non-Gaussian possibilities	Mechanism diversity defeats a single universal primordial template
Romano–Cornish stochastic framework	Unifies detector response, cross-correlation, anisotropy, and polarization estimators	A correct estimator does not by itself make the source mixture identifiable
Population synthesis and resolved catalogs	Tie astrophysical spectra to rates, masses, redshift evolution, and selection	Subthreshold populations and environment-dependent evolution remain uncertain
NANOGrav 15-year new-physics comparison	Shows that astrophysical and several cosmological interpretations can compete in one low-frequency data set	Bayes factors depend strongly on the SMBHB model and priors; the collaboration does not claim evidence for new physics
LVK O1–O4a cosmological-background search	Finds no evidence for the tested cosmological components while including a compact-binary background	The null applies to specified frequency ranges, responses, and source templates, not every cosmological model
Joint anisotropy, polarization, and cross-tracer proposals	Add observables not reducible to the isotropic spectrum	Public end-to-end joint likelihoods and coverage tests remain incomplete or analysis-specific

The current observational examples are Afzal et al. 2023, §§3–5 and §7, printed pp. 6–32 and 40–43 (Open PDF) and LIGO–Virgo–KAGRA 2025, §I, printed pp. 1–3, and §X, printed pp. 22–23 (Open PDF). The unresolved status is substantive: evidence does not establish a source-independent discriminator from an isotropic two-point spectrum alone.

What could change the conclusion. The interpretation could change after a multi-band detection, a major resolved-population update, a calibrated anisotropy or cross-correlation measurement, or a public joint likelihood changes mixture identifiability, Bayes-factor coverage, or the accepted population-subtraction uncertainty.

Observational boundary

This page defines the inference architecture and failure tests. Exact cross-correlation estimators, instrument calibration, observing-run products, and mutable detection or upper-limit verdicts belong to Stochastic Cross-Correlation in Gravitational-Wave Astronomy. A source classification must cite that current layer rather than copying a numerical verdict into this theory chapter.

Exercises

Suppose two summaries have response columns $\mathbf t_{\rm c}=(1,0)$ and $\mathbf t_{\rm a}=(1,0.6)$. A noiseless injection has amplitudes $A_{\rm c}=2$ and $A_{\rm a}=3$. Find $\mathbf d$. Can the first summary alone separate the components? Recover the amplitudes using both summaries, and explain what a shared unmodeled 1% calibration error would do.

Solution

The injected summary is

$$\mathbf d =2\begin{pmatrix}1\\0\end{pmatrix} +3\begin{pmatrix}1\\0.6\end{pmatrix} =\begin{pmatrix}5\\1.8\end{pmatrix}.$$

The first row measures only $A_{\rm c}+A_{\rm a}=5$, so it has rank one and cannot separate the amplitudes. The second row gives $0.6A_{\rm a}=1.8$, hence $A_{\rm a}=3$ and $A_{\rm c}=2$. If both summaries are multiplied by an unmodeled gain 1.01, both recovered amplitudes are biased high by 1%. A shared calibration nuisance can absorb that mode, but it also broadens and correlates the component posteriors. The exercise demonstrates why an independent statistic fixes rank while calibration still controls accuracy.

References

• Afzal, Adeela, et al. 2023. “The NANOGrav 15-Year Data Set: Search for Signals from New Physics.” Astrophysical Journal Letters 951, L11. DOI.
• Caprini, Chiara, and Daniel G. Figueroa. 2018. “Cosmological Backgrounds of Gravitational Waves.” Classical and Quantum Gravity 35, 163001. DOI.
• LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration. 2025. “Cosmological and High Energy Physics Implications from Gravitational-Wave Background Searches in LIGO–Virgo–KAGRA’s O1–O4a Runs.” arXiv:2510.26848v2. Open preprint.
• 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.