[Paper Review] Nonlinear Perturbations in a Variable Speed of Light Cosmology

J. W. Moffat|arXiv (Cornell University)|Jan 7, 2015

Cosmology and Gravitation Theories23 references3 citations

TL;DR

This paper proposes a variable speed of light (VSL) cosmology as an alternative to inflation for explaining primordial perturbations in the early universe. Using the δN formalism, it calculates nonlinear curvature fluctuations and finds f_NL = 5, a constant value consistent with Planck 2015 data, while g_NL exhibits running behavior, distinguishing VSL from inflationary models.

ABSTRACT

A variable speed of light (VSL) cosmology is described in which the causal mechanism of generating primordial perturbations is achieved by varying the speed of light in a primordial epoch. This yields an alternative to inflation for explaining the formation of the cosmic microwave background (CMB) and the large scale structure (LSS) of the universe. We make use of the $δ{\cal N}$ formalism to identify signatures of primordial nonlinear fluctuations, and this allows the VSL model to be distinguished from inflationary models. In particular, we find that the parameter $f_{ m NL}=5$ in the variable speed of light cosmology. The value of the parameter $g_{ m NL}$ evolves during the primordial era and shows a running behavior.

Motivation & Objective

• To provide a non-inflationary mechanism for generating primordial density fluctuations in the early universe.
• To identify observational signatures—specifically nonlinear perturbations—that can distinguish VSL cosmology from inflation and other early-universe models.
• To calculate the non-Gaussianity parameters f_NL and g_NL in a VSL framework using the δN formalism.
• To test the model against Planck 2015 constraints on f_NL and assess the viability of VSL as an alternative to inflation.

Proposed method

• Formulates a VSL cosmology using a Friedmann-Lemaître-Robertson-Walker metric with c(x) as a dynamical field.
• Applies the δN formalism to compute curvature perturbations up to third order in field fluctuations.
• Derives the effective e-folding number 𝒩(ϕ, γ) in terms of scalar field ϕ and integration constant γ, with γ related to the field evolution via the Lambert W function.
• Computes first-, second-, and third-order curvature perturbations ζ₁, ζ₂, ζ₃ using partial derivatives of 𝒩 with respect to ϕ and γ.
• Identifies f_NL and g_NL from the perturbation expansion, with f_NL derived from ζ₂ and g_NL from ζ₃.
• Uses the background solution φ(t) = φ_f + γ ln(t/t_f) to express γ as a function of φ and φ_f, enabling analytical evaluation of the nonlinearity parameters.

Experimental results

Research questions

• RQ1 Can a variable speed of light cosmology generate primordial perturbations without inflation?
• RQ2 What are the specific values and scale dependence of the non-Gaussianity parameters f_NL and g_NL in a VSL model?
• RQ3 How do the nonlinear perturbations in VSL cosmology compare quantitatively with Planck 2015 observational constraints?
• RQ4 Can the running behavior of g_NL serve as a distinguishing signature between VSL and inflationary models?

Key findings

• The non-Gaussianity parameter f_NL is found to be exactly 5 in the VSL model, a constant value independent of scale.
• This value of f_NL = 5 is consistent with the Planck 2015 constraint of f_NL = 2.7 ± 5.8, providing strong observational compatibility.
• The third-order nonlinearity parameter g_NL is not constant but evolves during the primordial era, exhibiting a running behavior dependent on the scalar field and γ.
• The expression g_NL = 25(2γ - φ + φ_f)/(3γ) shows strong scale dependence, making it a distinctive observational signature of the VSL model.
• The δN formalism successfully captures the nonlinear structure of curvature perturbations in a VSL context, enabling direct comparison with inflationary models.
• The model provides a viable alternative to inflation by generating scale-invariant, nearly Gaussian primordial fluctuations through a superluminal phase of c.