Gauge Dependence of Scalar-Induced Gravitational Waves from Isocurvature Perturbations: Analytical Results

TL;DR

The paper analyzes how scalar-induced gravitational waves (SIGWs) from primordial isocurvature perturbations depend on gauge choice during radiation domination, across nine gauges. It develops an analytic kernel framework in the $(d,s)$ domain and introduces a radiative projection that retains only freely propagating tensor modes, yielding a gauge-invariant late-time spectrum identical to the longitudinal benchmark. The study reveals strong gauge artifacts in several gauges (e.g., CO, TT, UC, UD) before projection, but demonstrates that after projecting to the radiative part the SIGW spectrum is finite and gauge independent, with the standard infrared tail and UV cutoff preserved. These results provide a robust method for connecting higher-order GW predictions with observations in the mHz–Hz range and establish a consistent basis for interpreting SGW signals from isocurvature sources. They also outline future work extending to finite-width isocurvature spectra and the RD-to-MD transition within the same formalism.

Abstract

We analytically study the gauge dependence of scalar-induced gravitational waves (SIGWs) sourced by primordial isocurvature perturbations during radiation domination (RD), working across nine gauges. Through analytical integrations of the kernels supported by graphical comparison we identify a clear dichotomy. We find that in some gauges viz. the uniform-density (UD), total-matter (TM), uniform-curvature (UC), comoving-orthogonal (CO) and transverse-traceless (TT) gauges the energy density grows polynomially in conformal time $η^n$, where $n$ varies from $2$ to $8$. While in rest of the gauges viz. the longitudinal (Long.), uniform-expansion (UE), Newtonian-motion (Nm), and N-body (Nb) gauges the late-time energy spectrum converges, and SIGWs behave as radiation. For subhorizon modes ($ kη\gg 1 $), the divergence becomes severe, showing that SIGWs are gauge-dependent observables in this regime. We resolve it through a kernel projection that isolates the luminal, freely propagating gravitational wave components (oscillating as $\sin(kη)$ and $\cos(kη)$), eliminating spurious contributions. The resulting kernel decays as $ (kη)^{-1} $ and yields a finite, gauge-independent late-time spectrum, confirming that only luminal modes represent physical SIGWs.

Figures (26)

• Figure 1: Source term in longitudinal gauge, $f_{\rm long.}(d,s,x)$, evaluated at $d=0$ and $s=1/\sqrt{3}$ as a function of the dimensionless time variable $x=k\eta$. The growth around horizon entry is followed by an oscillatory decay $f_{\rm long.}\propto x^{-2}$ (up to trigonometric factors), signalling the transition from causal scalar sourcing to free GW propagation at late times.
• Figure 2: Squared Green–function kernels in longitudinal gauge at $(d,s)=(0,1/\sqrt{3})$: $I_c^2(x)$ (solid) and $I_s^2(x)$ (dashed). Both exhibit a decaying evolution, $I_{c/s}^2\propto x^{-2}$. This ensures a convergent tensor solution and a time-independent late-time energy density.
• Figure 3: Scalar–induced spectrum $\Omega_{\rm GW}(k)$ in the longitudinal gauge for a Dirac–delta isocurvature peak at $k_p$. The evaluation follows the delta line in $(d,s)$: $d=0$ and $s=\tfrac{2}{\sqrt{3}}(k_p/k)$. The spectrum has the standard low-$k$ tail $\Omega_{\rm GW}\propto k^{2}\ln^{2}\!k$ for $k\ll k_p$, peaks at $k=2c_s k_p$ with $c_s=1/\sqrt{3}$, and exhibits a sharp cutoff at $k=2k_p$ from momentum conservation. At late times ($x\gg1$) the evolution is constant: the longitudinal baseline is convergent.
• Figure 4: The source $f_{\rm CO}(d,s,x)$ evaluated at $(d,s)=(0,1/\sqrt{3})$ versus $x=k\eta$. For $x\gg1$ the evolution grows steadily, indicating a strong late–time growth in this slicing for isocurvature initial conditions.
• Figure 5: $I_c^2(x)$ (solid) and $I_s^2(x)$ (dashed) in the CO gauge at $(d,s)=(0,1/\sqrt{3})$. The kernels inherit the late-time growth of the source and increase toward large $x$ (here roughly $I^2\!\propto x^{4}$), showing a non–convergent late–time behaviour in this gauge.