[Paper Review] Generation of gravitational waves from symmetry restoration during inflation

Cody Goolsby-Cole, Lorenzo Sorbo|arXiv (Cornell University)|May 10, 2017

Cosmology and Gravitation Theories40 references3 citations

TL;DR

This paper proposes that gravitational waves (GWs) can be generated during inflation when a scalar field χ acquires a time-dependent mass due to symmetry restoration, remaining massless for the remainder of inflation. Unlike prior models where the field's mass recovers after crossing zero, here the field stays massless, leading to enhanced GW production. The key result is that for a single χ species, the tensor-to-scalar ratio is at most r ~ 10⁻⁵ for CMB experiments, but for multiple species, GW amplitudes in the LISA band may exceed 10⁻¹³, making them potentially detectable.

ABSTRACT

We discuss the possibility of a feature in the spectrum of inflationary gravitational waves sourced by a scalar field $χ$ whose vacuum fluctuations are amplified by a rapidly time dependent mass. Unlike previous work which has focused on the case in which the mass of the field $χ$ vanishes only for an instant before becoming massive again, we study a system where the scalar field becomes and remains massless through the end of inflation as the consequence of the restoration of a shift symmetry. After applying appropriate constraints to our parameters, we find, for future CMB experiments, a small contribution to the tensor-to-scalar ratio which can be at most of the order $r \sim 10^{-5}$. At smaller scales probed by gravitational interferometers, on the other hand, the energy density in the gravitational waves produced this way might be above the projected sensitivity of LISA, $Ω_{GW}\,h^2 \sim 10^{-13}$, in a narrow region of parameter space. If there is more than one $χ$ species, then these amplitudes are enhanced by a factor equal to the number of those species.

Motivation & Objective

• To explore the generation of primordial gravitational waves (PGWs) from a scalar field χ whose mass vanishes and remains zero due to symmetry restoration during inflation.
• To investigate whether such a mechanism can produce a detectable GW signal, especially in the frequency band accessible to gravitational wave interferometers like LISA.
• To compare the resulting GW spectrum with standard vacuum fluctuations and assess its phenomenological distinctness.
• To quantify the dependence of GW amplitude on the number of χ species and model parameters such as the energy scale Λχ.

Proposed method

• Model the scalar field χ as acquiring a time-dependent mass via coupling to a secondary field σ, with the mass vanishing and remaining zero due to shift symmetry restoration.
• Use a time-dependent effective mass mχ(t) that remains zero after a critical time, leading to sustained particle production of χ quanta.
• Compute the two-point function of the metric perturbation h_ij using the stress-energy tensor sourced by χ fluctuations, treating χ as a classical source.
• Apply the in-in formalism to compute the GW power spectrum P_T(k), focusing on the contributions from χ self-interactions and cross-terms with vacuum gravitons.
• Perform numerical integration of the resulting power spectrum expressions over momentum space, with UV cutoff at p_UV = 50Λχ.
• Use de Sitter and Minkowski approximations to derive analytical scaling laws and validate numerical results.

Experimental results

Research questions

• RQ1 Can a scalar field χ that remains massless after symmetry restoration during inflation produce a significant gravitational wave signal?
• RQ2 How does the GW amplitude from such a mechanism compare to standard vacuum fluctuations in inflation?
• RQ3 What is the dependence of the GW power spectrum on the number of χ species and the energy scale Λχ?
• RQ4 Can the resulting GW signal be detectable by future experiments like CMB-S4 or LISA?
• RQ5 Does the prolonged masslessness of χ lead to a resonant enhancement in GW production compared to models where mχ recovers after crossing zero?

Key findings

• For a single χ species, the tensor-to-scalar ratio r is bounded by r ~ 10⁻⁵, which is within the sensitivity range of next-generation CMB experiments like CMB-S4.
• At smaller scales probed by gravitational wave interferometers, the GW energy density Ω_GW h² can exceed 10⁻¹³ in a narrow region of parameter space, potentially detectable by LISA.
• The peak amplitude of the GW power spectrum scales as P_T ∝ Λχ⁵ / M_P⁴ for the χχ source term, and as P_T ∝ k Λχ³ / M_P⁴ for the cross-term with vacuum gravitons.
• The GW signal exhibits a characteristic peak in the power spectrum at kt ≈ 2.37 for the cross-term and kt ≈ 2.65 for the self-interaction term.
• The amplitude is enhanced by a factor equal to the number of χ species, making multi-scalar models particularly promising for detectable GW signals.
• The GW spectrum shows oscillatory behavior due to coherent χ particle production, distinguishing it from the featureless spectrum of vacuum fluctuations.