Primordial Gravitational Perturbations

• Primordial gravitational perturbations are early universe metric fluctuations that include scalar, vector, and tensor components.
• Lattice simulations using the BSSN formalism reveal that nonlinear interactions during preheating can enhance gravitational wave energy density by an order of magnitude.
• These studies show that fully accounting for metric nonlinearities is crucial for accurate predictions of gravitational wave spectra and detection prospects.

Primordial gravitational perturbations are deviations from perfect homogeneity and isotropy in the early universe's metric, present as fluctuation modes in the spacetime geometry. These perturbations, typically decomposed into scalar (density), vector (vorticity), and tensor (gravitational wave) components, play a fundamental role in the evolution of the universe, the generation of structure, and the production of relic gravitational wave backgrounds. Their analysis combines both linear and nonlinear (second-order and beyond) effects, as the dynamics of the metric and matter fields in the Einstein equations couple these perturbations during violent post-inflationary processes such as preheating.

1. Nonlinear Evolution During Preheating and Enhancement of Gravitational Waves

During preheating following inflation, violent field dynamics and resonant instabilities (parametric resonance) drive the rapid growth of fluctuations in both matter fields and the spacetime metric. If the metric perturbations are evolved by integrating the full (fully nonlinear) Einstein equations—instead of the tensor (gravitational wave, GW) perturbations alone on an unperturbed Friedmann-Robertson-Walker (FRW) background—scalar, vector, and tensor modes are all sourced and become tightly coupled.

In three-dimensional lattice simulations of generic supersymmetric hybrid inflation models, it is found that while the preheating of the metric fluctuations does not backreact significantly on the field sector, the amplification of scalar (and to a lesser extent vector) metric fluctuations inevitably sources tensor modes via second-order and higher nonlinear terms. This nonlinear enhancement can drive the stochastic gravitational wave background to an energy density an order of magnitude larger than that obtained in conventional analyses (which evolve tensors in a fixed FRW geometry only). The amplification occurs predominantly at and below the main resonance scale, with the GW spectrum peaking at the frequencies of maximal scalar activity.

2. Role of Nonlinear Metric Couplings and Source Structure

The evolution of the gravitational wave amplitude involves competition between direct field-sourced anisotropic stress and nonlinear metric sources. The relevant Einstein equation (in BSSN formalism) for the traceless part of the extrinsic curvature $\mathcal{A}_{ij}$, which encodes the tensor GW sector, is:

$$\frac{d}{dt} \mathcal{A}_{ij} + 2 H\, \mathcal{A}_{ij} = 2 e^{-4\beta} (M_{ij}^{TF} - R_{ij}^{TF}) + \ldots$$

Here $M_{ij}^{TF}$ contains the anisotropic stress of the preheating fields, while $R_{ij}^{TF}$ is the traceless part of the Ricci tensor arising from nonlinear, primarily scalar, metric perturbations. During preheating, $R_{ij}^{TF}$ is amplified by resonant scalar fluctuations and becomes of the same order as $M_{ij}^{TF}$, making their contributions to the tensor sector comparable.

This fundamentally alters the solution structure: the gravitational wave background after preheating cannot be computed reliably from the field stress alone—the full metric nonlinearity, and thus the amplitude and spectrum, are governed by the coupled evolution.

3. Computational Approach and Lattice Implementation

The quantitative evolution of primordial perturbations in this context requires three-dimensional lattice simulations evolving both matter and metric sectors. These are carried out in the BSSN formalism, which maintains stable evolution of the full nonlinear Einstein equations for weak, small-scale metric fluctuations on top of an expanding FRW background. The line element is

$$ds^2 = N^2 dt^2 - \gamma_{ij}(dx^i + N^i dt)(dx^j + N^j dt),$$

$$\gamma_{ij} = e^{-4\beta} \tilde{\gamma}_{ij}, \quad \det \tilde{\gamma}_{ij} = 1,$$

with synchronous gauge ($N=1, N^i=0$). Field and metric initial conditions (for, e.g., the inflaton $\phi$ and waterfall field $\chi$), are chosen to satisfy the Hamiltonian and momentum constraints, typically with initial tensor and vector modes set to zero.

Throughout the evolution, modes of different spatial character (scalar, vector, tensor) are decomposed, and the transverse-traceless component of $\gamma_{ij}$ provides the observable GW amplitude $h_{ij}^{TT}$,

$$\rho_{\text{GW}} = \frac{m_p^2}{4} \langle \dot{h}_{ij}^{TT} \dot{h}_{ij}^{TT} \rangle.$$

The computational grid enforces a UV cutoff; results are converged with respect to grid size and box length, ensuring correct representation of all physically relevant resonance scales.

4. Enhancements Relative to Previous Homogeneous-Background Analyses

Traditional analyses, in which only tensor modes on a homogeneous background are evolved, ignore the nonlinear backreaction from amplified scalar and vector metric perturbations. Such approaches underestimate the gravitational wave amplitude: by ignoring the $R_{ij}^{TF}$ source, the GW background energy density is typically found to be an order of magnitude lower than when the full Einstein equations are solved.

Spectrally, the amplification from nonlinearities is concentrated around the main resonance peak and modifies the shape of the spectrum in both the low- and high-frequency tails. When compared directly, the ratio of GW energy densities (full metric vs. homogeneous FRW) is $\mathcal{O}(10)$ at and below the resonance frequency, with the largest scalar metric fluctuations.

5. Implications for Gravitational Wave Observatories

The nonlinear enhancement increases the predicted present-day, normalized gravitational wave energy density:

$$h^2\Omega_{\text{GW}}^{\text{peak}} \sim 5.5 \times 10^{-9} \left(\frac{\phi_c}{0.005 m_p}\right)^2 \left(\frac{T_{\text{RH}}}{m_p}\right)^{1/3}$$

This signal amplitude is within the projected reach of future observatories, such as Advanced LIGO (for $\phi_c \gtrsim 0.005 m_p$) and BBO (for smaller $\phi_c$). To locate the peak within the sensitivity bands ($1-10^3\,\mathrm{Hz}$ for LIGO, $10^{-3}-10^2\,\mathrm{Hz}$ for BBO), the inflationary model's couplings (encoded in $g$, the field coupling constant) must be tuned $g \lesssim 10^{-14}$. Otherwise, the peak frequency is likely much higher, but nontrivial low-frequency tails (set by the transfer and resonance structure) may still be observable.

The increased GW background also improves the prospects for indirect detection of primordial GWs via B-mode polarization in the CMB, since these nonlinear contributions enhance the stochastic GW background's amplitude at cosmologically relevant scales.

6. Robustness and Model Independence

The observed amplification is robust across a range of inflationary parameters and does not strongly depend on specific couplings or field content, provided the system exhibits sufficient resonance and metric fluctuation growth during preheating. The enhancement mechanism is generic in hybrid inflation scenarios exhibiting violent post-inflation field dynamics; the key requirement is the generation of amplified, nonlinear scalar metric fluctuations during the parametric resonance phase, which is then transferred into the tensor sector via the Einstein equations' nonlinearities.

7. Summary Table: Comparison of Approaches and Outcomes

Method	GW Source Terms Included	Enhancement over Homogeneous	Spectral Modification
Homogeneous FRW (Standard)	Field Anisotropic Stress	Baseline	Standard Peak
Full Metric (Nonlinear, BSSN)	Field + Metric (All)	$\mathcal{O}(10)$	Broadened, Enhanced

This enhancement mechanism underscores the necessity of accounting for the full nonlinear dynamics of both scalar and tensor metric perturbations during preheating for accurate predictions of the relic stochastic gravitational wave background. It also implies that direct and indirect searches for primordial GWs should interpret potential detections within the context of nonlinear-and model-dependent-amplification mechanisms.