Cosmological Vacuum Decays from Schwinger-Keldysh Formalism

Abstract: In this work, we establish a systematic framework to describe the vacuum decays in a radiation-dominated FLRW universe from the Schwinger-Keldysh formalism. By splitting the phase transition field $Φ$ into the mean field $φ$ and the short-wavelength modes $σ$ and tracing over the latter as the environment, we obtain a classical Langevin-type equation-of-motion for the mean field $φ$, where the quantum effects are encoded in a non-Markov memory kernel and a non-Gaussian noise. As a phenomenological example, we consider a polynomial potential and study the structure of the memory kernel as well as the correlation functions of the noise term. With a less restrictive scale split by allowing $φ$ to carry spatial dependence, further extensions remain possible to describe the whole dynamics of cosmological first-order phase transitions via numerical simulations.

• The paper derives a Schwinger–Keldysh Langevin equation for vacuum decay in a radiation-dominated universe, replacing Euclidean bounce estimates with real-time nonequilibrium dynamics.
• The analysis shows that first-order phase-transition potentials inevitably produce non-Gaussian, colored noise and nonlocal memory, while only linear current terms contribute to deterministic dissipation.
• The paper computes tree-level and one-loop memory kernels and demonstrates that finite-volume coarse-graining suppresses noise for massive environments, leaving Fokker–Planck methods and spatial bubble simulations as open challenges.

This paper develops a systematic open-quantum-system description of vacuum decay during first-order phase transitions (FOPT) in a radiation-dominated (RD) Friedmann–Lemaître–Robertson–Walker (FLRW) universe, using the Schwinger–Keldysh (SK) closed-time-path formalism (2608.12736). Rather than estimating decay rates from Euclidean bounce actions, the author derives a real-time, nonequilibrium Langevin-type equation of motion for the coarse-grained order parameter, with quantum effects encoded in a non-Markovian memory kernel and non-Gaussian noise.

Setup: scale split and reduced dynamics

The starting point is a real scalar field $\Phi$ with a temperature-dependent polynomial potential, $m^2 = M^2(T^2 - T_0^2)$ and $g = -AT$, on a fixed RD background with $T \propto 1/a$. The field is decomposed as $\Phi = \phi + \sigma$, where the mean field $\phi$ is taken to carry no spatial dependence and $\sigma$ denotes short-wavelength modes treated as an environment. The reduced density matrix of $\phi$ is obtained by tracing over $\sigma$ from an initial product vacuum state, yielding an influence functional $F[\phi_+,\phi_-]$ in the SK path integral.

A key technical simplification follows from the RD choice $m^2 = M^2(T^2 - T_0^2)$0: after rescaling $m^2 = M^2(T^2 - T_0^2)$1 and absorbing scale factors into couplings ($m^2 = M^2(T^2 - T_0^2)$2, $m^2 = M^2(T^2 - T_0^2)$3), the environmental theory becomes exactly equivalent to a massive scalar in flat spacetime with conformal time playing the role of physical time. The mode functions are plane waves with dispersion $m^2 = M^2(T^2 - T_0^2)$4. The paper notes that this cancellation relies entirely on $m^2 = M^2(T^2 - T_0^2)$5; for generic FLRW backgrounds an $m^2 = M^2(T^2 - T_0^2)$6 correction to the dispersion relation survives. A more general mode function involving parabolic cylinder functions is provided for the case where the mass term turns tachyonic at late times, though the plane-wave approximation is used throughout.

Influence functional and Keldysh structure

Expanding the influence functional order by order in the system-environment interaction, the linear term vanishes by normal ordering (tadpole removal), so the leading contribution is second order. In the Keldysh basis $m^2 = M^2(T^2 - T_0^2)$7, the result takes the compact form

$m^2 = M^2(T^2 - T_0^2)$8

where $m^2 = M^2(T^2 - T_0^2)$9 are vectors of currents built from powers of $g = -AT$0 and $g = -AT$1 up to cubic order, and the kernel matrices satisfy $g = -AT$2 and $g = -AT$3. All components are given explicitly as combinations of powers of propagators, $g = -AT$4 and $g = -AT$5.

Langevin equation via generalized Hubbard–Stratonovich transformation

Varying the effective action with respect to $g = -AT$6 produces terms linear, quadratic, and cubic in $g = -AT$7. While quadratic terms map to Gaussian noise through the standard Hubbard–Stratonovich transformation, the cubic terms require a generalized construction: the author introduces a random field $g = -AT$8 whose generating functional reproduces the full tower of correlation functions $g = -AT$9 derived from the kernels $T \propto 1/a$0. This yields the stochastic EOM

$T \propto 1/a$1

Two structural results deserve emphasis. First, only the linear parts of the currents contribute to the deterministic dissipative/memory sector; all nonlinear current components feed exclusively into higher-order noise cumulatives. Second, because any FOPT potential must contain vertices beyond the quadratic term, the resulting noise is necessarily non-Gaussian and colored, and the dynamics necessarily non-Markovian — a direct consequence of the two-minimum structure of the potential rather than an artifact of truncation.

Structure of the memory kernel

The spatially averaged kernel $T \propto 1/a$2 is computed to one-loop order using diagrammatic rules translated from flat-space perturbation theory. At tree level, the averaged propagator difference evaluates to $T \propto 1/a$3, while the vertex involving derivatives vanishes identically due to $T \propto 1/a$4 — physically because it reduces to a zero-momentum $T \propto 1/a$5 propagator. At one loop, the bubble diagram involves the integral $T \propto 1/a$6, evaluated analytically in terms of Hankel functions via analytic continuation of $T \propto 1/a$7, giving an imaginary part proportional to $T \propto 1/a$8. The sunrise (two-loop) contribution is neglected.

The kernel is manifestly not proportional to $T \propto 1/a$9, confirming non-Markovianity. A Markov approximation via Taylor expansion of the past trajectory is possible when the random-walk time scale of $\Phi = \phi + \sigma$0 is much shorter than the kernel variation time $\Phi = \phi + \sigma$1, splitting the memory integral into a drift term and a dissipation term proportional to $\Phi = \phi + \sigma$2. The paper is explicit that this approximation fails for heavy fields and is not guaranteed even for light fields, since the kernel retains time dependence as $\Phi = \phi + \sigma$3.

Noise correlations and coarse-graining scale

The statistics of the volume-averaged noise $\Phi = \phi + \sigma$4 depend strongly on the coarse-graining length $\Phi = \phi + \sigma$5. For large volumes, $\Phi = \phi + \sigma$6-point correlators are suppressed by factors of $\Phi = \phi + \sigma$7; for a massive environment this suppression is unavoidable, since the IR divergence that would cancel it can only arise for massless modes — and $\Phi = \phi + \sigma$8 must be massive to stabilize the false vacuum at early times. In the strict $\Phi = \phi + \sigma$9 limit all correlations vanish and the classical trajectory remains trapped in the false vacuum, consistent with the absence of quantum fluctuations.

For finite $\phi$0, the central object is the averaged propagator $\phi$1, where the spherical Bessel factor acts as a UV cutoff. Using a projection identity for $\phi$2 and the appendix's Bessel integral $\phi$3, the real part reduces to a one-dimensional radial integral combining $\phi$4 and $\phi$5 pieces across the light-cone boundary $\phi$6; the pole there is integrable. The local limit $\phi$7 recovers the coincident-point propagator $\phi$8, which diverges as $\phi$9 without a UV regulator. The natural physical choice is $\sigma$0 (the larger of the two Hubble radii entering a correlator), or alternatively a scale between the average nucleated bubble radius and the Hubble radius; the paper acknowledges that fixing $\sigma$1 precisely remains a modeling choice.

Limitations and open questions

Several restrictions bound the applicability of the results. The mean field is restricted to its zero mode, so bubble nucleation and spatial profile evolution are outside the present treatment, though the formalism extends straightforwardly to a low-$\sigma$2 $\sigma$3 with a $\sigma$4 term. Back-reaction of fluctuations on the geometry is neglected, and the thermal-form of the parameters assumes adiabatic expansion maintaining local thermal equilibrium. The high-temperature approximation $\sigma$5 discards the late-time tachyonic regime where the full parabolic-cylinder mode functions would be required. Most significantly, the corresponding Fokker–Planck equation — needed to extract an actual vacuum decay rate — does not exist in closed time-local form for non-Gaussian colored noise, posing substantial analytical and numerical challenges that the paper explicitly leaves unresolved. Whether the semi-classical extension with spatially dependent $\sigma$6 can be simulated efficiently despite non-Markovian memory and colored noise is likewise left open.

Conclusion

The paper provides a controlled derivation connecting the underlying quantum field theory of cosmological FOPTs to an effective Langevin description, with explicit one-loop expressions for the memory kernel and noise correlators in an RD background. Its main qualitative findings — inevitable non-Gaussianity, color, and non-Markovianity of the noise for any multi-minimum potential, and the strong suppression of noise under coarse-graining for massive environments — constrain how stochastic treatments of vacuum decay must be constructed, and identify the Fokker–Planck formulation and numerical bubble simulations as the concrete next steps.