[Paper Review] Primordial magnetic fields in theories of gravity with non-minimal coupling between curvature and matter

Orfeu Bertolami, Maria Margarida Lima+1 more|arXiv (Cornell University)|Apr 14, 2022

Cosmology and Gravitation Theories50 references6 citations

TL;DR

This paper investigates primordial magnetic field generation in a non-minimal coupling gravity model, where curvature and matter are coupled via arbitrary functions f₁(R) and f₂(R). By solving generalized Maxwell equations in a flat Robertson-Walker metric during slow-roll inflation, it shows that magnetic fields scale as B ∝ 1/(a²f₂(R)), enabling them to avoid extreme dilution. For low reheating temperatures (TRH ≲ 10¹⁰ GeV), the resulting field strengths are compatible with observed large-scale magnetic fields, particularly under dynamo or compression amplification mechanisms.

ABSTRACT

The existence of magnetic fields in the universe is unmistakable. They are observed at all scales from stars to galaxy clusters. However, the origin of these fields remains enigmatic. It is believed that magnetic field seeds may have emerged from inflation, under certain conditions. This possibility is analised in the context of an alternative theory of gravity with non-minimal coupling between curvature and matter. We find, through the solution of the generalised Maxwell equations in the context of non-minimal models, that for general slow-roll inflationary scenarios with low reheating temperatures, $T_{RH}\simeq10^{10}$GeV, the generated magnetic fields can be made compatible with observations at large scales, $\lambda \sim 1 Mpc$.

Motivation & Objective

• To explore whether non-minimal curvature-matter coupling in alternative gravity theories can generate observable primordial magnetic fields.
• To address the conformal invariance problem in General Relativity, which otherwise leads to magnetic field dilution during inflation.
• To determine whether magnetic field seeds generated during inflation can survive and be amplified to observed galactic scales.
• To assess the viability of such models under observational constraints from large-scale magnetic fields.

Proposed method

• Formulates a modified gravity action with non-minimal coupling via f₁(R) and f₂(R), generalizing Einstein-Hilbert action.
• Derives generalized Maxwell equations in a flat Robertson-Walker metric, incorporating the non-minimal coupling.
• Applies the slow-roll approximation and a reheating phase without specifying the inflaton potential, ensuring generality.
• Solves the magnetic field evolution equation using Fourier transforms and approximates the solution under inflationary conditions.
• Estimates the ratio r = ρB/ργ between magnetic and radiation energy densities at reheating and applies the Turner-Widow relation to infer present-day values.
• Compares results with observational constraints under dynamo and compression amplification mechanisms.

Experimental results

Research questions

• RQ1 Can non-minimal coupling between curvature and matter break electromagnetic conformal invariance, enabling primordial magnetic field generation?
• RQ2 How does the magnetic field evolve during inflation in a non-minimal gravity model, particularly under slow-roll conditions?
• RQ3 What is the range of initial magnetic field strength (relative to radiation) required to produce observed large-scale fields today?
• RQ4 How do low reheating temperatures affect the viability of such models in matching observational data?
• RQ5 Can the resulting magnetic field strength be compatible with both dynamo and compression amplification mechanisms?

Key findings

• The magnetic field evolves as B ∝ 1/(a²f₂(R)), where f₂(R) is a cubic function of Ricci scalar R, preventing extreme dilution during inflation.
• For typical inflationary scales and reheating temperatures TRH ≲ 10¹⁰ GeV, the ratio BRH/Bi ranges from 10⁻⁴¹ to 10⁷ at 1 Mpc scales.
• With dynamo amplification, the initial magnetic-to-radiation energy density ratio r = 10χ requires χ ≥ −62 to match observations.
• With compression amplification, χ ≥ −36 is sufficient, indicating a more lenient constraint.
• The model supports the viability of primordial magnetic fields in non-minimal coupling gravity, especially for low reheating temperatures.
• The results suggest that such models can generate magnetic fields compatible with current astrophysical observations at large cosmological scales.