Inflationary Tensor Backgrounds

Inflation can leave a nearly scale-invariant spectrum of transverse, traceless metric perturbations. What is observed later is not that primordial spectrum directly: modes that enter the Hubble radius pass through a particular expansion era, decay as free waves, and may be damped by free-streaming species, while sufficiently long modes can remain outside the Hubble radius at the epoch of interest. The resulting transfer function connects one primordial normalization to CMB tensor anisotropies and to present-day subhorizon gravitational-wave energy density without conflating those two observables.

Helpful background. The Einstein Equation supplies Einstein Equations for tensor modes; Relativistic Matter and Compact Objects supplies anisotropic stress; Gravitational Radiation and Waveforms supplies polarization and stochastic-spectrum conventions; Relativistic Cosmology supplies the FLRW history and primordial spectrum; and Partial Differential Equations and Green Functions supplies mode propagation and sourced Green functions.

Primordial tensor and transfer conventions

Use $a_0=1$, conformal time $\eta$, and $e^{i\mathbf k\cdot\mathbf x}$. Write the spatial metric as

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

and choose polarization tensors with $e_{ij}^{\lambda}e^{\lambda'ij}=2\delta_{\lambda\lambda'}$. The mode amplitudes are defined by the direct decomposition

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

For the parity-even, unpolarized inflationary state treated here, the two polarizations are uncorrelated and have equal power. At one common primordial reference time $\eta_i$, the equal-time covariance defines the standard dimensionless tensor spectrum normalized by the contraction of the metric perturbation,

$$\begin{aligned} \left\langle h_{\lambda,\mathbf k}^{\rm prim} \left(h_{\lambda',\mathbf k'}^{\rm prim}\right)^*\right\rangle &=(2\pi)^3\delta_{\lambda\lambda'} \delta^{(3)}(\mathbf k-\mathbf k') \\ &\quad\times \frac{2\pi^2}{k^3}\mathcal P_{h_\lambda}^{\rm prim}(k), \\ \mathcal P_{h_+}^{\rm prim}(k) &=\mathcal P_{h_\times}^{\rm prim}(k), \\ \mathcal P_t^{\rm prim}(k) &\equiv2\sum_\lambda\mathcal P_{h_\lambda}^{\rm prim}(k). \end{aligned}$$

For the passive, source-free branch, define the transfer function by $h_{\lambda,\mathbf k}(\eta)=T_h(k,\eta) h_{\lambda,\mathbf k}^{\rm prim}$, with $T_h\to1$ for the regular constant mode at $k/\mathcal H\to0$. The homogeneous equation has real coefficients, so this page chooses the real solution fixed by real regular initial data; otherwise every squared transfer below is read as a modulus squared. At two possibly different times the passive covariance therefore factorizes as

$$\begin{aligned} &\left\langle h_{\lambda,\mathbf k}(\eta) h_{\lambda',\mathbf k'}^*(\eta')\right\rangle \\ &\qquad=T_h(k,\eta)T_h^*(k,\eta') \left\langle h_{\lambda,\mathbf k}^{\rm prim} \left(h_{\lambda',\mathbf k'}^{\rm prim}\right)^*\right\rangle, \\ &\mathcal P_t(k,\eta) =\left|T_h(k,\eta)\right|^2\mathcal P_t^{\rm prim}(k). \end{aligned}$$

This factorization keeps primordial production and later propagation separate. It does not apply while a stochastic source remains active: the power then depends on the source’s unequal-time covariance through a retarded double integral.

For the leading vacuum result in canonical slow-roll inflation, the initial condition in this convention is $\mathcal P_t^{\rm prim}(k)=2H_*^2/(\pi^2M^2)$, where $M=(8\pi G)^{-1/2}$ and the star labels inflationary Hubble-radius exit $k=a_*H_*$. The quantum normalization and its assumptions are derived in Primordial Scalar and Tensor Power Spectra. That inflationary exit is distinct from the later entry discussed below. Other production models supply different initial power while using the same linear transfer calculation.

Free propagation and gravitational-wave energy

With no transverse-traceless stress after production, each polarization obeys

$$h_k''+2\mathcal H h_k'+k^2h_k=0.$$

The friction-removing rescaling $\mu_k\equiv ah_k$, which is proportional to the canonically normalized $u_k=aMh_k/\sqrt2$, where $M\equiv(8\pi G)^{-1/2}$ is the reduced Planck mass, instead satisfies

$$\mu_k''+\left(k^2-\frac{a''}{a}\right)\mu_k=0.$$

For $k\ll\mathcal H$, the regular solution is constant and the second solution is proportional to $\int d\eta/a^2$. For $k\gg\mathcal H$, WKB gives $\mu_k\simeq C_+e^{ik\eta}+C_-e^{-ik\eta}$, so $h_k\propto a^{-1}$. Only in this short-wavelength regime is the averaged wave energy density unambiguous. The direct norm-two decomposition gives

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

This is the high-frequency effective stress-tensor density of Isaacson 1968, Eq. (4.1), printed p. 1274, specialized to the FLRW background and the spectrum convention above.

Dividing by $\rho_c=3\mathcal H^2/(8\pi Ga^2)$ then gives

$$\boxed{ \Omega_{\rm GW}(k,\eta) =\frac{1}{12} \left(\frac{k}{\mathcal H}\right)^2 \mathcal P_t^{\rm prim}(k)\, \overline{\left|T_h(k,\eta)\right|^2} }$$

after replacing $\overline{\left|T_h'\right|^2}$ by $k^2\overline{\left|T_h\right|^2}$ for a freely oscillating mode. The overbar is an oscillation average. Applying this expression to a superhorizon tensor mode would assign a local Isaacson energy where the short-wave approximation does not hold. Here $\mathcal P_t^{\rm prim}$ already contains the norm-two contraction. If it were replaced by the literal sum of component spectra, the coefficient would have to double; changing only one side of that dictionary is the normalization error. A polarized or parity-violating primordial state instead requires the spectral-matrix treatment introduced in the chapter overview.

Caprini and Figueroa derive the stochastic and energy-density conventions in Caprini and Figueroa 2018, §3.2, printed pp. 28–31, eqs. (80)–(92) (Open PDF). Their characteristic-strain normalization differs from some detector conventions by a factor of two, so the polarization sum must be translated as one package.

Radiation and matter entry leave different slopes

For a power-law expansion $a\propto\eta^\alpha$, the regular free solution is a Bessel function. Two limits provide exact checks:

$$T_h^{\rm rad}(x)=j_0(x)=\frac{\sin x}{x}, \qquad T_h^{\rm mat}(x)=\frac{3j_1(x)}{x}, \qquad x=k\eta.$$

Both approach one as $x\to0$ and decay as $a^{-1}$ once well inside the Hubble radius. For a scale-invariant primordial $\mathcal P_t$, a mode that enters during radiation has $a_{\rm ent}\propto k^{-1}$; hence $h_0\propto k^{-1}$ and $\Omega_{{\rm GW},0}\propto k^0$. During matter domination, $a_{\rm ent}\propto k^{-2}$; hence $h_0\propto k^{-2}$ and $\Omega_{{\rm GW},0}\propto k^{-2}$. The turnover therefore locates equality in transfer space. Late acceleration smooths the very-low-frequency end, so an instantaneous matter-to-de Sitter matching is not a precision transfer.

These solutions and the inflationary radiation/matter transfer are developed in Caprini and Figueroa 2018, §§2.4 and 5.2, printed pp. 19–22 and 70–78 (Open PDF). The result assumes the regular primordial mode; a decaying mode or a later active source needs separate initial data or a Green-function integral.

Frequency and thermal-history dictionary

The observed cyclic frequency is

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

This conversion includes $2\pi$: $k$ is an angular wavenumber, while $f$ is a cyclic frequency. A mode produced or entering at scale factor $a_*$ has

$$f_0=\frac{x_*}{2\pi}\frac{a_*}{a_0}H_* , \qquad x_*\equiv\frac{k}{a_*H_*}.$$

For adiabatic radiation domination, $aTg_{*s}^{1/3}=\text{constant}$ and $H^2=(8\pi G/3)(\pi^2/30)g_*T^4$. Consequently a source or entry temperature maps schematically as

$$f_0\propto x_*\,T_*\,g_*^{1/2}g_{*s}^{-1/3},$$

while the present fractional energy inherits the entropy-dilution factor $g_*g_{*s}^{-4/3}$. Setting $g_*=g_{*s}$ is appropriate only away from a threshold in an equilibrated radiation bath. Caprini and Figueroa give the explicit radiation-era mapping in Caprini and Figueroa 2018, §3.3, printed pp. 32–33, eqs. (97)–(101) (Open PDF).

Free-streaming and reheating corrections

After neutrino decoupling, a tensor perturbation generates a neutrino quadrupole whose anisotropic stress feeds back with a damping sign. The evolution becomes an integro-differential equation rather than the free Bessel problem. In the standard radiation-era limit the energy-density spectrum for modes entering between neutrino decoupling and equality is suppressed by about $35\%$; a mode is unaffected while it remains outside the Hubble radius, but damping begins around entry if the species is still free streaming then. Modes already deep inside at decoupling receive little effect. Caprini and Figueroa 2018, §5.2, printed pp. 78–79, eq. (224) (Open PDF) states this regime and its limitations.

A non-instantaneous reheating era changes a different band. During a decelerating stage with constant equation of state $w\gt-1/3$, horizon entry satisfies $a_{\rm ent}\propto k^{-2/(1+3w)}$, so a scale-invariant primordial spectrum acquires

$$\Omega_{{\rm GW},0}(k) \propto k^{2(3w-1)/(3w+1)}$$

for modes entering in that era. Radiation gives zero slope, matter-like reheating gives $-2$, and a stiff phase gives $+1$. Changes in relativistic degrees of freedom and free streaming sit on top of this ideal power law.

Scope and observational interpretation

CMB temperature and polarization respond to tensor perturbations around recombination, including modes not deeply subhorizon. Direct stochastic searches instead use strain or $\Omega_{\rm GW}$ for oscillatory waves today. They are connected by the same $T_h$ but are not interchangeable likelihoods. Instrument response functions, observing-run upper limits, foreground subtraction, and mutable detection claims belong to Gravitational-Wave Astronomy, not this transfer derivation.

Exercises

A scale-invariant tensor mode enters during an early epoch with constant $w=0$, and another enters during $w=1$. Find the induced spectral slopes. Then convert $k=0.05\,\mathrm{Mpc}^{-1}$ to present cyclic frequency and state whether the short-wave energy formula is automatically valid for that mode.

Solution

The transfer exponent is $n=2(3w-1)/(3w+1)$. Therefore

$$n(w=0)=-2, \qquad n(w=1)=1.$$

The frequency is

$$f_0=1.546\times10^{-15}\,\mathrm{Hz}\times0.05 =7.73\times10^{-17}\,\mathrm{Hz}.$$

The conversion itself is kinematic. The averaged energy formula additionally requires $k\gg\mathcal H$ at the epoch where $\Omega_{\rm GW}$ is evaluated. A CMB-pivot tensor was not always subhorizon, so one must propagate it first and apply the short-wave expression only after that condition is satisfied.

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.

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• Primordial Tensor Backgrounds