Inflationary Vacuum GW Spectrum

• The inflationary spectrum of vacuum gravitational waves is defined as the statistical distribution of primordial tensor modes produced by quantum fluctuations during cosmic inflation.
• It employs quantization of metric perturbations and adiabatic regularization to extract a nearly scale-invariant plateau and regulate ultraviolet divergences in high-frequency modes.
• Practical implications include probing early-universe dynamics via CMB B-mode polarization, pulsar timing arrays, and future ultrahigh-frequency GW detectors.

The inflationary spectrum of vacuum gravitational waves (GW) describes the statistical distribution of the primordial tensor modes generated by quantum vacuum fluctuations during the inflationary epoch. This spectrum encodes key information about the dynamics of inflation, the mechanism by which vacuum fluctuations are converted into classical relic GWs, and the subsequent evolution and observability of these signals across cosmic history. The mathematical form, physical regularization, and phenomenological imprints of the inflationary vacuum GW spectrum represent essential components for both theoretical cosmology and gravitational wave astrophysics.

1. Theoretical Foundation: Quantum Vacuum Fluctuations in Inflation

During inflation, transverse-traceless perturbations $h_{ij}$ of the spatial metric are quantized in the background of a rapidly expanding FLRW universe. The quadratic action for tensor modes in conformal time $\tau$ is given by

$$S = \frac{M_{\rm Pl}^2}{8} \int d^4x\, a^2(\tau) \left[ h_{ij}' h_{ij}' - (\nabla h_{ij})^2 \right]$$

where $a(\tau)$ is the scale factor and $M_{\rm Pl}$ the reduced Planck mass. Expanding in Fourier space and polarizations, each canonical mode $v_k = \frac{a M_{\rm Pl}}{2} h_k$ obeys the Mukhanov–Sasaki equation

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

In (quasi-)de Sitter inflation, $a(\tau) \simeq -1/(H_{\rm inf} \tau)$ with $a''/a = 2/\tau^2$. Imposing the Bunch–Davies vacuum selects

$$v_k(\tau) = \frac{\sqrt{-\pi\tau}}{2} H_{3/2}^{(1)}(-k\tau)$$

The late-time ($-k\tau \ll 1$) amplitude for each polarization becomes $|h_k|^2 \simeq H_{\rm inf}^2/(2M_{\rm Pl}^2 k^3)$ (Soman et al., 10 Jul 2024, Guzzetti et al., 2016). The dimensionless tensor power spectrum is then

$$P_T(k) = \frac{2}{\pi^2} \left( \frac{H_{\rm inf}}{M_{\rm Pl}} \right)^2 \left( \frac{k}{k_*} \right)^{n_T}$$

with $n_T \simeq -2\epsilon_H$ the tensor spectral tilt and $k_*$ a pivot scale. This scale invariance is modified only by slow roll corrections and quantum initial state or model-specific effects.

2. Details of the Inflationary Vacuum GW Spectrum and Adiabatic Regularization

The general solution for the spectrum during power-law inflation, $a(\tau) = \ell_0 |\tau|^{1+\beta}$, with $u_k$ the mode function, yields the dimensionless spectrum (Wang et al., 2015)

$$\Delta_t^2(k,\tau) = \frac{k^{2(\beta+2)}}{2\pi \ell_0^2 M_{\rm Pl}^2} x^{-(2\beta+1)} H^{(2)}_{(\beta+1/2)}(x) H^{(1)}_{(\beta+1/2)}(x)$$

where $x = k|\tau|$. In the long-wavelength limit ($k|\tau| \ll 1$),

$$\Delta_{\rm vac}^2(k) = a_t^2 \frac{8}{M_{\rm Pl}^2} \left( \frac{H}{2\pi} \right)^2 k^{2\beta+4}$$

High-frequency modes ($k\gg 1/|\tau|$) exhibit ultraviolet (UV) divergences in the unregulated spectrum, specifically quadratic and logarithmic in $k$. Adiabatic regularization of 2nd order subtracts the reference “adiabatic” solution

$|u_k^{(2)}|^2 = \frac{1}{2k} + \frac{a''/a}{4k^3}$

from the raw two-point function, yielding the regulated spectrum (Wang et al., 2015)

$$\Delta_{\rm vac, reg}^2(k,\tau) = 2\frac{k^3}{2\pi^2 a^2} \left[ |u_k|^2 - |u_k^{(2)}|^2 \right]$$

In this construction, the high-$k$ behavior falls as $k^{-2}$ (i.e., $\Delta^2 \propto f^{-2}$), ensuring convergence and identifying a spectral break. The “vacuum” component dominates at $f \gtrsim 10^{11}\,\mathrm{Hz}$, while the lower part of the spectrum is set by facilitated graviton production at cosmic transitions (see section 4).

3. Post-Inflationary Evolution, Equation of State, and Spectral Features

The primordial spectrum defined outside the horizon is preserved if subsequent evolution is a scale-invariant expansion. However, the present-day spectral energy density,

$$\Omega_{\rm GW}(f) = \frac{1}{12}\left(\frac{k}{a_0 H_0}\right)^2 P_T(k) T^2(k)$$

with $T(k)$ the transfer function for subhorizon and horizon-crossing effects, is altered by the background equation of state (EoS) and additional cosmological events (Soman et al., 10 Jul 2024, Jinno et al., 2013). For a constant post-inflation EoS $w$ ($a \propto \tau^{2/(1+3w)}$),

$n_{\rm GW} = 2\frac{3w - 1}{3w + 1}$

Thus, epochs with $w>1/3$ (stiff) produce a blue-tilted high-frequency tail, $w=1/3$ (radiation) yields a flat plateau, and $w<1/3$ (matter or softer) gives a red tilt. Multiple transitions in $w$ produce a broken power-law, with "knees" at frequencies set by the transition times (Soman et al., 10 Jul 2024).

Early universe phenomena such as cosmic phase transitions, entropy injection, or dark-radiation decoupling imprint distinct spectral features (steps, kinks, plateaux, or tilts) via their effect on $T(k)$ (Jinno et al., 2013, Jinno et al., 2011). For example, a brief vacuum-domination (thermal inflation) induces a $f^{-4}$ suppression above a transition scale, while entropy injection from decaying matter yields a $f^{-2}$ decline.

4. Graviton Production, UV-IR Structure, and Physical Composition

The present-axis relic GW background consists of two distinct contributions: the vacuum part at $f \gtrsim 10^{11}\,\mathrm{Hz}$ and the graviton part (quantum excitations) at $f \lesssim 10^{11}\,\mathrm{Hz}$ (Wang et al., 2015). The latter arises from non-adiabatic evolution at rapid cosmic transitions (e.g., end of inflation, reheating, radiation-matter equality), with the inflation-reheating transition being most significant.

The regularized spectral energy density and pressure are rendered finite by adiabatic subtraction to 4th order. The IR (long-wavelength) side remains unaffected by the regularization, thus preserving the "primordial plateau" amplitude $\Delta^2 \simeq 8(H/2\pi M_{\rm Pl})^2$ and the consistency relation for the tensor-to-scalar ratio $r$ and tensor tilt $n_t$.

Frequency range	Dominant contribution	Spectral slope
$f \lesssim 10^{11}$ Hz	Graviton (quantum excitations)	$\propto k^{2\beta+4}$
$f \gtrsim 10^{11}$ Hz	Regularized vacuum tail	$\propto k^{-2}$ ($f^{-2}$)

5. Model Extensions: Modified Gravity, Quantum States, Pre-inflationary, and Collapse Models

Variants of the inflationary vacuum GW spectrum arise in multiple extensions:

• Running vacuum models (e.g., $\Lambda(H)$, $H^2$- or $H^3$-dependence) primarily affect the spectral cutoff and high-frequency tail, yielding subtle changes at $\nu \gtrsim 10^5$–$10^6\,$Hz (Tamayo et al., 2015, Tamayo et al., 2015).
• Pre-inflationary quantum gravity scenarios (e.g., Loop Quantum Cosmology) suppress the spectrum at the lowest frequencies (below a cutoff $f_*$), encoding information about the pre-inflationary universe in the GW “knee” (Afonso et al., 2010).
• Alternative quantization (Krein space) regularizes UV divergences via subtraction of negative-norm sectors, producing at most a negligible exponential cutoff at large $k$ while leaving the IR spectrum unchanged (Mohsenzadeh et al., 2012).
• Collapse models or non-Bunch–Davies initial states allow additional scale dependence and overall rescaling of the tensor spectrum amplitude, possibly resulting in nontrivial $k$-dependent modulations of $n_t$ (Mariani et al., 2014).
• Non-vacuum quantum states (squeezed vacuum): Squeezing enhances both the amplitude and high-frequency oscillations of the GW background, with the effects of reheating and acceleration epoch parameters entering into the spectrum for all modes (Malsawmtluangi et al., 2015).

6. Observational Signatures and Prospects

The vacuum GW spectrum provides a unique probe of the high-energy inflationary universe. Cosmic Microwave Background (CMB) $B$-mode polarization is sensitive to the primordial (IR) plateau and tensor tilt $n_t$. Pulsar Timing Arrays (PTA) and space-based or ground-based GW interferometers probe distinct spectral regions: $\Omega_{\rm GW}(f)$ plateaux, knees, and high-frequency roll-offs (Guzzetti et al., 2016, Soman et al., 10 Jul 2024).

The potential for observing the regularized vacuum tail at $f \gtrsim 10^{11}\,$Hz has motivated proposals for ultrahigh-frequency GW detectors (e.g., polarization-based laser-beam experiments). Characteristic features, such as sharp steps (from phase transitions or entropy injection), blue-tilted segments (from stiff EoS epochs), or a downturn at the very lowest frequencies (from pre-inflationary evolution), would discriminate between competing models and provide direct empirical access to epochs otherwise hidden from electromagnetic observations (Wang et al., 2015, Jinno et al., 2013, Jinno et al., 2011).

7. Summary Table: Core Expressions for the Vacuum GW Spectrum

Regime/Context	Power Spectrum $P_T(k)$	Spectral Energy Density $\Omega_{\rm GW}(f)$	Notes
Standard slow-roll, IR plateau	$$\displaystyle \frac{2}{\pi^2} \left( \frac{H}{M_{\rm Pl}} \right)^2$$	$$\frac{1}{12} \left( \frac{k}{a_0 H_0} \right)^2 P_T(k) T^2(k)$$	$n_t = -2\epsilon$, $r = 16\epsilon$ (Guzzetti et al., 2016, Xu et al., 13 May 2025)
Power-law inflation, $k\ll1/|\tau|$	$$a_t^2\frac{8}{M_{\rm Pl}^2} \left(\frac{H}{2\pi}\right)^2 k^{2\beta+4}$$	$\propto f^{2\beta+4}$ (for $f\ll 10^{11}$ Hz)	$n_t=2\beta+4$ (Wang et al., 2015)
Regularized vacuum tail, $k\gg1/|\tau|$	$\propto k^{-2}$	$\propto f^{-2}$	$f \gg 10^{11}$ Hz (Wang et al., 2015)
Equation of state $w$ after inflation	—	$\propto f^{2(3w-1)/(3w+1)}$	Blue tilt for $w>1/3$; red for $w<1/3$ (Soman et al., 10 Jul 2024)
Multiple $w$ transitions	—	Broken power law with knees at $f_i$	“Knees” mark EoS transitions (Soman et al., 10 Jul 2024)
Squeezed vacuum quantum state	$$P_T(k)\times[1+2\sinh^2 r_k+\sinh2r_k\,\cos2\varphi_k]$$	$\propto [1+2\sinh^2 r_k+\sinh2r_k\cos2\varphi_k]$	Enhancement, oscillatory features (Malsawmtluangi et al., 2015)

The inflationary spectrum of vacuum gravitational waves is thus characterized by a primordial, nearly scale-invariant plateau at CMB/PTA-accessible frequencies, a regulated high-frequency tail due to adiabatic subtraction, and a structure that is sensitive to both early-universe physics and quantum state selection. These features create a direct connection between fundamental inflationary dynamics, high-energy theoretical models, and current and next-generation gravitational wave observatories.