Cancellation of one-parameter graviton gauge dependence in the effective scalar field equation in de Sitter

We investigate gauge dependence of one-graviton-loop corrections to the effective field equation of the massless, minimally coupled scalar in de Sitter, obtained by including source and observer corrections to the effective self-mass correcting the equation. Using the $Δα$ variation of the de Sitter-breaking graviton propagator in a one-parameter family of gauges, we compute the gauge-dependent contributions to the effective self-mass of a massless minimally coupled scalar mediating interactions between heavy scalars. We show that gauge dependence cancels provided the contributions from all diagram classes are collected, including one-loop corrections to external mode functions, which play a qualitatively new role relative to flat space. The resulting cancellation supports the construction of graviton gauge-independent cosmological quantum-gravitational observables from quantum-corrected effective equations.

💡 Research Summary

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The paper addresses the long‑standing issue of gauge dependence in one‑graviton‑loop corrections to the effective field equation of a massless, minimally coupled (MMC) scalar field in de Sitter space. The authors work within a simple two‑scalar model: a heavy scalar Ψ (mass m ≫ H) interacts with the MMC scalar ϕ via a cubic coupling λ ϕ Ψ² on a spatially flat de Sitter background (scale factor a(η)=−1/(Hη)). The quantum fluctuations of the metric are treated perturbatively, with the graviton field h_{\mu\nu} defined by g_{\mu\nu}=a²η_{\mu\nu}+κh_{\mu\nu}.

The central object of interest is the retarded self‑mass M²(x;x′) of ϕ, obtained in the Schwinger‑Keldysh formalism, which corrects the classical equation Dϕ(x)=K a δ³(⃗x) to

 Dϕ(x)−∫d⁴x′ M²(x;x′) ϕ(x′)=K a δ³(⃗x).

In earlier work the self‑mass was computed using the simplest graviton gauge (Δα = 0). However, gauge‑dependent pieces are known to appear in 1PI two‑point functions, and it is not guaranteed that the secularly enhanced logarithmic terms survive gauge‑independent. To test the proposal of Ref.