Transfer Functions Across Cosmic History

Once a cosmological tensor mode has been produced, its later spectrum is reshaped by horizon entry, the background equation of state, changes in the relativistic plasma, and collisionless anisotropic stress. These effects form a transfer function: they do not alter the primordial source model, but they can create slopes and steps that would otherwise be misread as production features.

Helpful background. The Einstein Equation supplies Einstein Equations for tensor modes; Relativistic Matter and Compact Objects supplies anisotropic stress; Gravitational Radiation and Waveforms supplies wave-energy and polarization conventions; Relativistic Cosmology supplies the thermal expansion history; and Partial Differential Equations and Green Functions supplies mode matching and retarded response.

Constant-w transfer is a Bessel problem

Use the chapter’s direct norm-two decomposition $h_{ij}=\sum_\lambda e_{ij}^{\lambda}h_\lambda$, with $e_{ij}^{\lambda}e^{\lambda'ij}=2\delta_{\lambda\lambda'}$. Resolve the transverse-traceless stress in the same basis,

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

For either tensor polarization on a flat background,

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

The inverse projection supplies the factor $1/2$; using the raw contraction $e_\lambda^{ij}\Pi_{ij}^{\rm TT}$ instead would put $8\pi G$ on the right-hand side. While the source is active, a scalar transfer multiplier is not enough. If $G_k$ is the retarded Green function of the tensor wave operator, the sourced covariance depends on the unequal-time stress correlator,

$$\begin{aligned} &\left\langle h^{\rm src}_{\lambda,\mathbf k}(\eta) \left(h^{\rm src}_{\lambda',\mathbf k'}(\eta')\right)^*\right\rangle \\ &\quad=\left(16\pi G\right)^2 \int_{\eta_i}^{\eta}d\eta_1 \int_{\eta_i}^{\eta'}d\eta_2\, G_k(\eta,\eta_1)G_{k'}^*(\eta',\eta_2) a^2(\eta_1)a^2(\eta_2) \\ &\qquad\quad\times \left\langle\Pi_{\lambda,\mathbf k}(\eta_1) \Pi_{\lambda',\mathbf k'}^*(\eta_2)\right\rangle . \end{aligned}$$

Only after the source shuts off can its resulting $(h,h')$ state be propagated by the homogeneous transfer below. First set the transverse-traceless anisotropic stress to zero. During a decelerating expanding epoch with constant equation of state $w>-1/3$ and the conformal origin at its big-bang boundary,

$$a(\eta)\propto\eta^\alpha, \qquad \alpha=\frac{2}{1+3w}.$$

The primordial passive branch is selected by both $h_k\to1$ and $a^2h_k'\to0$ as $\eta\to0$. The second condition removes the zero-frequency decaying-mode momentum; it is essential when $w>1$, where the second solution can remain finite at the big-bang boundary even though it is not the primordial branch. With these conditions, the real solution is

$$T_w(x) =2^\nu\Gamma(\nu+1)x^{-\nu}J_\nu(x), \qquad \nu=\alpha-\frac12.$$

Here $x=k\eta$ is the dimensionless phase variable. It compares a comoving wavelength with conformal time without mixing physical and comoving units.

The small-$x$ expansion of $J_\nu$ gives $T_w\to1$: the regular superhorizon tensor is constant. At $x\gg1$, $J_\nu(x)\propto x^{-1/2}$, so the envelope is $T_w\propto x^{-\alpha}\propto a^{-1}$. Radiation and matter provide exact checks,

$$T_{\rm rad}(x)=j_0(x)=\frac{\sin x}{x}, \qquad T_{\rm mat}(x)=\frac{3j_1(x)}{x}.$$

Both limits must be reproduced by any numerical transfer code. The homogeneous equation, energy convention, and power-law solutions are reviewed by Caprini and Figueroa 2018, §2.4, printed pp. 19–22, eqs. (53)–(68), and §3.2, printed pp. 28–30, eqs. (80)–(86) (Open PDF). For an accelerating constant-$w$ epoch, the convenient conformal coordinate is shifted toward a finite future endpoint (and is often negative). Initial data are then set on the chosen initial slice or inherited by matching from the preceding era. Choosing that coordinate does not impose a physical boundary condition at the future endpoint, and the $\eta>0$ big-bang normalization above cannot be imported unchanged.

Horizon entry turns expansion into a spectral slope

For constant $w>-1/3$, horizon entry satisfies $k=a_{\rm ent}H_{\rm ent}$. Since $H\propto a^{-3(1+w)/2}$,

$$a_{\rm ent}\propto k^{-2/(1+3w)}.$$

A regular passive mode is constant before entry and then has an $a^{-1}$ subhorizon envelope, so $h_0\propto a_{\rm ent}h_{\rm prim}$. A decaying initial mode, an active source, or mode mixing requires the full state or retarded-source evolution rather than this scalar multiplier. In the convention fixed above, the contracted spectrum is $\mathcal P_t=2\sum_\lambda\mathcal P_{h_\lambda}$. Once the mode is freely oscillating well inside the Hubble radius, this fixes the normalization to

$$\Omega_{\rm GW}(k,\eta) =\frac{\mathcal P_t^{\rm prim}(k)}{12\mathcal H^2} \overline{\left|T_h'(k,\eta)\right|^2} \xrightarrow[\text{free oscillation}]{k\gg\mathcal H} \frac1{12}\left(\frac{k}{\mathcal H}\right)^2 \mathcal P_t^{\rm prim}(k)\, \overline{\left|T_h(k,\eta)\right|^2}.$$

The first form is the short-wave Isaacson result. The arrow additionally uses the WKB relation $\overline{\left|T_h'\right|^2}=k^2 \overline{\left|T_h\right|^2}$ for a freely oscillating mode; it is the second form that makes the horizon-entry slope transparent. The effective stress-tensor step is the high-frequency result of Isaacson 1968, Eq. (4.1), printed p. 1274.

If the primordial tensor spectrum is locally $\mathcal P_t^{\rm prim}(k)\propto k^{n_t}$, the horizon-entry scaling gives

$$\Omega_{{\rm GW},0}(k) \propto k^2\mathcal P_t^{\rm prim}(k)a_{\rm ent}^2 \propto k^{\,n_t+2(3w-1)/(1+3w)}.$$

The expansion history therefore adds $2(3w-1)/(1+3w)$ to the primordial tilt. For $n_t=0$, radiation gives a flat plateau, matter gives $k^{-2}$, and a stiff $w=1$ epoch gives $k^{+1}$; a nonzero tilt shifts each exponent by $n_t$. The result applies only to modes that enter during the stated epoch. It does not describe modes that remain superhorizon, cross a smooth transition while entering, or experience an active source.

At a transition with no singular surface stress, match $h_k$ and $h_k'$. Equivalently, for $\mu_k=ah_k$, match $\mu_k$ and $\mu_k'-\mathcal H\mu_k$. Matching only the subhorizon envelope loses phase and can create a spurious step. A physical transition of finite duration smooths ringing generated by an instantaneous idealization.

Relativistic thresholds change the radiation normalization

During adiabatic radiation domination,

$$aTg_{*s}^{1/3}=\text{constant}, \qquad \rho_r=\frac{\pi^2}{30}g_*T^4.$$

Choose an epoch $T_\star$ after the source and anisotropic stress have become negligible, when the wave is freely oscillating and $k\gg\mathcal H$. From that epoch onward $\rho_{\rm GW}\propto a^{-4}$. Relative to the radiation bath, the present spectrum is

$$\Omega_{{\rm GW},0}(k) =\Omega_{r,0} \left(\frac{g_*(T_\star)}{g_{*,0}}\right) \left(\frac{g_{*s,0}}{g_{*s}(T_\star)}\right)^{4/3} \left[ \frac1{\rho_r(T_\star)} \frac{d\rho_{\rm GW}}{d\ln k} \right]_{T_\star},$$

Here $\Omega_{r,0}$ contains photons and exactly those neutrino species treated as relativistic, while $g_{*,0}$ and $g_{*s,0}$ use that same inventory with neutrino temperatures expressed relative to the photon temperature. A photon-only $\Omega_{\gamma,0}$ convention is equally valid only after the two $g$-ratios are translated consistently. This is not an Isaacson energy evaluation at the exact horizon-entry temperature $T_k$, where $k\sim\mathcal H$. If no particle threshold or entropy-producing event lies between entry and $T_\star$, $T_k$ remains a useful label for the frequency band. Otherwise the mode and plasma history must be evolved together. A threshold in $g_*$ or $g_{*s}$ becomes a smooth step across the frequency band whose modes enter while the species changes regime. Treating $g_*=g_{*s}$ as one constant misses both the step and its width. The frequency–temperature and entropy mapping is given in Caprini and Figueroa 2018, §3.3, printed pp. 32–33, eqs. (97)–(101) (Open PDF).

Free streaming is a nonlocal damping term

A collisionless relativistic species develops a tensor quadrupole. In radiation domination with a constant free-streaming fraction $f_{\rm fs}$, the idealized evolution can be written

$$h_k''+2\mathcal Hh_k'+k^2h_k =-24f_{\rm fs}\mathcal H^2 \int_{\eta_{\rm dec}}^\eta d\widetilde\eta\, K\!\left[k(\eta-\widetilde\eta)\right]h_k'(\widetilde\eta),$$ $$K(u)=\frac{j_2(u)}{u^2}.$$

The memory integral is the anisotropic stress generated by the earlier wave history. Its minus sign transfers wave energy into the collisionless multipoles. Three limits are diagnostic:

• a mode that remains outside the Hubble radius has no local oscillatory energy to damp; if it enters later while the species is still free streaming, damping begins around entry;
• a mode entering while the species is free streaming receives the largest suppression;
• a mode already deep inside at decoupling averages much of the delayed response away.

For the standard neutrino fraction, the idealized radiation-era solution has late strain amplitude $A=0.8026$ relative to the no-stress wave, hence a $35.6\%$ reduction in squared amplitude and short-wave energy over the affected band Weinberg 2004, §III, Eqs. (18), (22)–(24), printed pp. 5–7 (Open PDF). This is not a universal factor at every frequency. Interactions, masses, or additional dark radiation change the lower limit, $f_{\rm fs}(\eta)$, and the kernel. The integro-differential equation and its broader cosmological regimes are summarized in Caprini and Figueroa 2018, §5.2, printed pp. 75–78, eqs. (219)–(224) (Open PDF).

A transfer calculation that can fail visibly

A reliable computation should propagate a state vector $(h,h')$ rather than multiply independent fitting factors without ordering them. It should then pass all of the following checks:

Check	Expected result	Failure diagnosed
$k/\mathcal H\to0$	Regular $h$ remains constant	Incorrect initial or matching mode
Radiation-only history	$T=j_0(k\eta)$	Time-variable or normalization error
Matter-only history	$T=3j_1(k\eta)/(k\eta)$	Wrong friction term
$k/\mathcal H\gg1$	Envelope $a\lvert h\rvert$ is constant without stress	Numerical dissipation or active-source contamination
Free-streaming fraction $f_{\rm fs}\to0$	Nonlocal solution returns to the homogeneous one	Kernel or sign error
Refined transition width and time step	Spectrum stabilizes	Unresolved ringing or stiffness

The repository’s tensor-transfer case implements two independent bounded problems. The first uses the exact smooth radiation-plus-matter background

$$a(\tau)=\tau(\tau+2), \qquad h_{xx}+\frac{4(x+q)}{x(x+2q)}h_x+h=0, \qquad x=q\tau,$$

with the zero-decaying-mode series imposed near $\tau=0$. The second solves the radiation-era Volterra equation above with a composite-trapezoid memory integral, a regular series for $K(u)$ near $u=0$, and Heun predictor–corrector time stepping. The separation is deliberate: it checks background transfer and collisionless damping without pretending to be a full Boltzmann calculation.

The case was rerun on 25 August 2026 with Node.js 24.18.0 on Darwin/arm64, using IEEE-754 binary64 JavaScript Number. It is dependency-free—standard library only. The linked MIT-licensed executable has SHA-256 94ca19a2dc80cb7cede79bfc0e1f548e76f21d7e03e3da4ac2894732c4ab0743.

The numerical diagnostics were:

• Smooth equality transfer. With RK4 steps $\Delta x=0.16,0.08,0.04,0.02$, the last two observed orders were $4.0007$ and $4.0003$.
• Superhorizon constancy. For $q=0.01$ at equality, the maximum relative error was $1.68\times10^{-6}$ against an independent small-$q$ quadrature.
• Subhorizon redshifting. Over $x=100,200,400$, the relative range of $J=\sqrt{(ah)^2+[(ah)_x]^2}$ was $2.54\times10^{-5}$, consistent with $h\propto a^{-1}$.
• Matter-entry and radiation-entry limits. The measured transfer slopes were $-1.999675$ for $q=0.005,0.01$ and $-0.008863$ for $q=128,256$, approaching the expected $-2$ and $0$.
• Free-streaming damping. Steps $\Delta x=0.02,0.01,0.005$ gave amplitudes $0.80256813$, $0.80296056$, and $0.80306607$, with observed order $1.8949$.
• No-stress control. Setting $f_{\rm fs}=0$ at $\Delta x=0.005$ gave late amplitude $0.99995421$.

Richardson extrapolation gives $A=0.80310488$ and a $35.5023\%$ reduction in squared amplitude. The amplitude is within $0.063\%$ of Weinberg’s $A=0.8026$ reference. The comparison checks the standard constant $f_{\rm fs}=0.40523$ radiation-era problem only; it omits finite-width decoupling, changing relativistic degrees of freedom, massive species, dark energy, and late-time sources.

Rerun the isolated case with the standalone download-and-run recipe:

curl -fL https://blackhole.org/code/cosmology/cosmology-benchmarks.mjs -o cosmology-benchmarks.mjs
node cosmology-benchmarks.mjs --case tensor-transfer

A passing run ends with PASS: 1 benchmark case.

Scope and observational interpretation

After production and source shutoff, the passive transfer function maps a specified tensor state to present cosmological strain or energy. A source that continues to operate instead requires its retarded unequal-time correlator; only the field accumulated up to a chosen shutoff time enters the subsequent homogeneous transfer. Detector response, cross-correlation likelihoods, foreground subtraction, and observing-run verdicts belong to Gravitational-Wave Astronomy. Source production belongs to the other pages in this chapter. The invariant spectrum definitions being propagated here are developed in Stochastic Background Spectra.

Exercises

For a scale-invariant primordial spectrum, find the transfer slope for modes entering during $w=0$, $w=1/3$, and $w=1$. Then verify directly from the Bessel index that the radiation solution tends to one at small $x$ and has an $a^{-1}$ envelope at large $x$.

Solution

The slope is

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

so

$$n_{\rm tr}(0)=-2, \qquad n_{\rm tr}(1/3)=0, \qquad n_{\rm tr}(1)=1.$$

For radiation, $\alpha=1$ and $\nu=1/2$. The normalized Bessel expression is $T=\sin x/x$. Its Taylor series is $1-x^2/6+\cdots$, while for $x\gg1$ its envelope is $x^{-1}$. Because $a\propto\eta$ in radiation and $x=k\eta$ at fixed $k$, this is exactly $a^{-1}$. These checks distinguish superhorizon constancy from subhorizon redshifting.

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.
• Weinberg, Steven. 2004. “Damping of Tensor Modes in Cosmology.” Physical Review D 69, 023503. DOI. Open PDF.