Second Order Gravitational Waves

• Second order gravitational waves are tensor perturbations generated by quadratic interactions of first order metric fluctuations, capturing nonlinear effects in cosmology.
• Their evolution requires nonlinear analysis and numerical simulations once density perturbations become significant, marking the breakdown of linear theory.
• Enhancement during extended early matter eras can amplify the gravitational wave spectrum, potentially making these signals detectable by current and future observatories.

Second order gravitational waves refer to the tensor perturbations of spacetime arising from nonlinear quadratic interactions of first order perturbations in Einstein’s equations. In contrast to first order gravitational waves—solutions of linear theory associated with freely propagating, transverse-traceless oscillations—second order gravitational waves emerge from mode coupling, including scalar-scalar, scalar-tensor, and tensor-tensor sources. This nonlinear regime is essential both for fundamental general relativity (capturing the self-interaction of gravitational waves and the feedback of density perturbations) and for the consistency of cosmological predictions in the current era of gravitational wave observatories.

1. Mathematical Formulation of Second Order Tensor Generation

At second order, the metric is expanded as

$$g_{\mu\nu} = \bar{g}_{\mu\nu} + \lambda h_{\mu\nu}^{(1)} + \frac{\lambda^2}{2} h_{\mu\nu}^{(2)} + \cdots$$

with perturbations decomposed into scalar, vector, and tensor parts relative to the background. The key distinction from linear theory is that, under second order Einstein equations, the evolution of $h_{ij}^{(2)}$ includes source terms which are quadratic in first order perturbations: $h''_k + 2\mathcal{H} h'_k + k^2 h_k = S_k^{(TT)}$ where $S_k^{(TT)}$ is the transverse-traceless projection of combinations such as $(\partial\Phi)^2$, $(\Phi \partial^2\Phi)$, $(h^{(1)} h^{(1)})$, and similar. This applies in both inflationary and post-inflationary (e.g., early matter or radiation dominated) eras.

During a pressureless early matter era ($w=0$), the first order Newtonian potential $\Phi$ is constant but the comoving density contrast grows. The solution for $h_k$ in this scenario, initiated at $h_k = h'_k = 0$, exhibits the characteristic behavior: $$h_k(\eta) = \frac{S_k}{k^2}\left[1 + 3\frac{k\eta\cos(k\eta)-\sin(k\eta)}{(k\eta)^3}\right]$$ yielding $h_k \propto \eta^2$ on superhorizon scales and $h_k\to S_k/k^2$ (“frozen”) on subhorizon scales, in contrast to free waves decaying as $a^{-1}$ after horizon entry (0901.0989).

2. Enhancement Mechanisms and the Role of Background Evolution

A crucial feature is the amplification of second order gravitational waves when the source scalar power spectrum (e.g., for $\Phi$) extends to small, sub-horizon scales throughout an extended early matter era. The enhancement factor depends on the ratio of comoving Hubble scales at the beginning (dom) and end (dec) of the matter era: $$F^2 = \left(\frac{k_{\text{dom}}}{k_{\text{dec}}}\right)^2 \sim \left(\frac{H_{\text{dom}}}{H_{\text{dec}}}\right)^{2/3}$$ leading to a significant amplification if the matter era is long. The spectrum at the end of the matter era is: $$\mathcal{P}_h(k,\eta_{\text{dec}}) \simeq 46\left(\frac{k_{\text{dom}}}{k}\right) I_1\left(\frac{k}{k_{\text{dom}}}\right)$$ and the present-day energy density per logarithmic interval is then

$$\Omega_{\mathrm{GW},0}(k) \simeq \Omega_{\gamma,0}\ \frac{1}{12}\left(\frac{k}{k_{\text{dec}}}\right)^2\mathcal{P}_h(k,\eta_{\text{dec}})$$

with $\Omega_{\gamma,0}$ the photon energy density fraction today.

Such enhancement brings the possibility of a detectable stochastic GW background from the early universe. However, extremely large enhancement factors coincide with modes where the linear approximation for density perturbations breaks down (0901.0989, Misyura, 3 May 2025).

3. Nonlinear Regime and Breakdown of Linear Theory

Linear analysis is only valid as long as $(\delta \rho/\rho) \ll 1$, roughly corresponding to

$$k^2 \lesssim k_{NL}^2 \sim \mathcal{P}_\Phi^{-1/2} \mathcal{H}^2$$

Once $k > k_{NL}$, scalar fluctuations become nonlinear and naive extrapolation of the linear solution—which would predict enormous GW backgrounds—fails. In this regime, the scalar power must be truncated at the nonlinear scale, $$k_{\text{cut}} = \min\{k_{\text{dom}},k_{NL}(\eta_{\text{dec}})\}$$, and the tensor power spectrum correspondingly modified: $$\mathcal{P}_h(k, \eta_{\text{dec}}) \simeq 46\left(\frac{k_{\text{cut}}}{k}\right) I_1\left(\frac{k}{k_{\text{cut}}}\right)$$ with additional suppression on smaller scales. Predicting the spectrum then requires numerical simulations (e.g., lattice or N-body) that incorporate the fully nonlinear dynamics (0901.0989).

4. Observational Consequences and Detectability

The frequency of the GW signal today is set by the comoving wavenumber as

$$f \simeq 1.2\times 10^{-8}\ g_*^{1/6} (T/\mathrm{GeV})~\mathrm{Hz}$$

where $T$ is the reheat temperature after the early matter era. For LIGO to detect these signals, $T_{\text{dec}} \lesssim 10^{10}$ GeV is required, and substantial enhancement in the amplitude is necessary. For the amplitude to be within detectable limits via linear mechanisms, the primordial curvature power at small scales must be large, specifically, $\Delta_\mathcal{R}^2 \sim 0.02$ (Advanced LIGO) or $0.005$ (LISA), much above the value inferred at CMB scales, $\Delta_\mathcal{R}^2 \sim 2\times 10^{-9}$ (0901.0989). Otherwise, either large nonlinearities or additional mechanisms must be invoked to generate observable GW backgrounds.

5. Numerical Requirements and Theoretical Uncertainties

When the density perturbations reach the nonlinear regime, the scalar sector can no longer be modeled perturbatively. Full nonlinear evolution—using lattice simulations or N-body techniques in the Newtonian limit—becomes necessary to determine the actual amplitude, cutoff, and scaling of the induced GW background (0901.0989). The analytic estimates provide both an extrapolation (from the cutoff scale) and a conservative lower bound, but not a precise prediction. These uncertainties impact the reliability of predictions for upcoming GW observatories targeting cosmological GW backgrounds.

6. Impact on Cosmological Studies and Structure Formation

Second order gravitational waves generated via these mechanisms not only serve as indirect probes of primordial small scale power but also illustrate the nonlinear feedback loop between structure formation and the tensor background. In addition, they may have relevance for understanding the dynamics of density perturbations during nonstandard expansion histories and transitions between matter and radiation-dominated epochs, as well as for the paper of secondary sources such as phase transitions and scalar field resonances (Misyura, 3 May 2025, Kasuya et al., 2022).

7. Principal Results and Key Formulas

The following summarizes the main mathematical results:

Quantity	Formula/Expression	Context/Comments
Second order tensor evolution	$h''_k + (4/\eta) h'_k + k^2 h_k = S_k$ (matter era)	Sourced by quadratic scalars
Frozen amplitude (sub-horizon)	$h_k \to S_k / k^2$	Enhancement vs. decaying modes
Enhancement factor	$$F^2 = (k_{\text{dom}}/k_{\text{dec}})^2 = (H_{\text{dom}}/H_{\text{dec}})^{2/3}$$	Quantifies amplification
Energy density spectrum	$$\Omega_{\mathrm{GW},0}(k) \simeq \Omega_{\gamma,0}\ \frac{1}{12}\left(\frac{k}{k_{\text{dec}}}\right)^2\mathcal{P}_h(k,\eta_{\text{dec}})$$	Detectability metric
Linear breakdown threshold	$$k^2 \gtrsim k_{NL}^2 \sim \mathcal{P}_\Phi^{-1/2} \mathcal{H}^2$$	Onset of nonlinearity
Modified tensor spectrum (nonlin)	$$\mathcal{P}_h(k, \eta_{\text{dec}}) \sim 46\left(\frac{k_{\text{cut}}}{k}\right)I_1\left(\frac{k}{k_{\text{cut}}}\right)$$	Truncated at $k_{NL}$ or $k_{\text{dom}}$

These relations serve as the backbone for both the analytic interpretation and numerical modeling of second order gravitational wave backgrounds arising from primordial density perturbations during early matter dominated phases.

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Second order gravitational waves provide a window into otherwise inaccessible primordial dynamics, with their amplitude and spectral characteristics encoding the detailed interplay between the growth of scalar perturbations and the expansion history of the universe. Their robust physical interpretation, however, depends critically on controlling nonlinear corrections and making precise, gauge-invariant theoretical predictions and simulations. Detection prospects hinge on both the primordial small-scale power and the duration of any early matter dominated phase, with very large enhancement factors requiring full nonlinear analysis to make reliable cosmological inferences (0901.0989).