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Scalar correlation functions in de Sitter space from the stochastic spectral expansion

Tommi Markkanen, Arttu Rajantie, Stephen Stopyra, Tommi Tenkanen

TL;DR

The paper develops a stochastic spectral expansion to compute long-distance two-point correlators of local functions of a light scalar in de Sitter space, applied to V(φ)=½ m^2 φ^2 + (λ/4) φ^4. By solving a Schrödinger-like eigenproblem for the stochastic transfer operator, it yields the eigenvalues Λ_n(α) and coefficients f_n that build the correlators and power spectra, showing how n_f−1 = 2Λ_n/H governs the spectral tilt. For the quartic case it finds a blue tilt with n−1 ≈ 0.579√λ, and demonstrates that common mean-field and Gaussian approximations overestimate power and mis-predict tilt away from the non-interacting limit. The work provides a precise, nonperturbative framework across α that improves predictions for isocurvature perturbations and related inflationary observables, illustrating the necessity of the full stochastic approach.

Abstract

We consider light scalar fields during inflation and show how the stochastic spectral expansion method can be used to calculate two-point correlation functions of an arbitrary local function of the field in de Sitter space. In particular, we use this approach for a massive scalar field with quartic self-interactions to calculate the fluctuation spectrum of the density contrast and compare it to other approximations. We find that neither Gaussian nor linear approximations accurately reproduce the power spectrum, and in fact always overestimate it. For example, for a scalar field with only a quartic term in the potential, $V=λφ^4/4$, we find a blue spectrum with spectral index $n-1=0.579\sqrtλ$.

Scalar correlation functions in de Sitter space from the stochastic spectral expansion

TL;DR

The paper develops a stochastic spectral expansion to compute long-distance two-point correlators of local functions of a light scalar in de Sitter space, applied to V(φ)=½ m^2 φ^2 + (λ/4) φ^4. By solving a Schrödinger-like eigenproblem for the stochastic transfer operator, it yields the eigenvalues Λ_n(α) and coefficients f_n that build the correlators and power spectra, showing how n_f−1 = 2Λ_n/H governs the spectral tilt. For the quartic case it finds a blue tilt with n−1 ≈ 0.579√λ, and demonstrates that common mean-field and Gaussian approximations overestimate power and mis-predict tilt away from the non-interacting limit. The work provides a precise, nonperturbative framework across α that improves predictions for isocurvature perturbations and related inflationary observables, illustrating the necessity of the full stochastic approach.

Abstract

We consider light scalar fields during inflation and show how the stochastic spectral expansion method can be used to calculate two-point correlation functions of an arbitrary local function of the field in de Sitter space. In particular, we use this approach for a massive scalar field with quartic self-interactions to calculate the fluctuation spectrum of the density contrast and compare it to other approximations. We find that neither Gaussian nor linear approximations accurately reproduce the power spectrum, and in fact always overestimate it. For example, for a scalar field with only a quartic term in the potential, , we find a blue spectrum with spectral index .

Paper Structure

This paper contains 19 sections, 102 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The stochastic approach
  3. One-point probability distribution
  4. Temporal correlation functions
  5. Equal-time correlation function, power spectrum and spectral index
  6. Example: a massive self-interacting field
  7. Eigenvalue equation
  8. Eigenvalue equation
  9. Massless limit ($\alpha\ll 1$)
  10. Free limit ($\alpha\gg 1$)
  11. Arbitrary $\alpha$
  12. Validity of common approximations
  13. Full results
  14. Approximation schemes
  15. Linear-MF approximation
  16. ...and 4 more sections

Figures (4)

  • Figure 1: The dimensionless eigenvalues from Eq. (\ref{['scaledlambda']}) for the potential (\ref{['eq:pot2']}) solved from Eq. (\ref{['eq:quar']}) as a function of $\alpha\equiv m^2/(H^2\sqrt{\lambda})$. The interpolation between quadratic (\ref{['eq:quadratic']}) and quartic (\ref{['eq:quartic']}) results is clearly evident. The red curves correspond to the leading perturbative corrections, which can be obtained from Table \ref{['tab:eigenvalues']} with Eq. (\ref{['scaledlambda']}).
  • Figure 2: The scaled eigenfunctions $\tilde{\psi}_n(\alpha;z)$ defined in Eq. (\ref{['eq:scaledefs']}) for the potential (\ref{['eq:pot2']}) solved from (\ref{['eq:quarz']}) by making use of the definitions (\ref{['eq:fsca']}) and (\ref{['scaledlambda']}). The quartic and quadratic limits are reached at $\alpha=0$ and $\alpha=\infty$, respectively.
  • Figure 3: The leading terms in the spectral expansion when $f(\phi)=\phi^j$ with $j\leq4$, scaled according to Eq. (\ref{['eq:fsa']}). The dashed and dotted lines correspond to the quartic $(\alpha\rightarrow0)$ and quadratic $(\alpha\rightarrow\infty)$ limits, respectively, given in Table \ref{['tab:coeffslambda']} and Eqs. (\ref{['equ:coeffsodd']}) and (\ref{['equ:coeffseven']}) for the leading results (left).
  • Figure 4: The full result for the leading spectral coefficient (left) and the spectral index (right) for the density contrast (\ref{['eq:nAgeneral']}) along with the approximations (\ref{['lin-MF']}), (\ref{['eq:lin2']}), (\ref{['lin-Gauss']}) and (\ref{['eq:Ga1']}). Note that $(n^x_\delta-1)/(\sqrt{\lambda}+\frac{{m^2}}{{H^2}})$ is a function of only a single parameter, $\alpha$.