Invisible gravitons and large-scale magnetism

The large-scale limits on the relic signals of gravitational radiation complement the bounds coming from the interferometric detectors (in the audio band) and from the pulsar timing arrays (in the nHz range). Within this inclusive perspective the spectral energy density of the gravitons is sharply suppressed in the aHz region even though the high frequency signal can be comparatively much larger both in the kHz and GHz domains. For there are no direct tests on the expansion rate prior to the formation of the light nuclei, a modified postinflationary timeline affects the total number of $e$-folds and additionally suppresses the tensor to scalar ratio by making the relic signals effectively invisible in the aHz range. The expansion rate prior to nucleosynthesis is further bounded by the evolution of the hypercharge field and the large-scale magnetism also constrains the decelerated expansion rate. The magnetogenesis requirements are compatible with a potentially detectable spectral energy density of the relic gravitons between the MHz and the THz while the tensor to scalar ratio remains suppressed in the aHz region. A maximum of the spectral energy density of the gravitons in the audio domain leads instead to a larger magnetic field when the scale of the gravitational collapse of the protogalaxy (of the order of the Mpc) gets comparable with the Hubble radius before equality. Along a converse viewpoint the results obtained here imply that a long decelerated stage expanding faster than radiation does not affect the high frequency range but reduces the effective number of $e$-folds by so enhancing the tensor to scalar ratio, possibly beyond its observational limit.

💡 Research Summary

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The paper investigates how the post‑inflationary decelerated expansion phase influences both the relic graviton background and the generation of large‑scale magnetic fields. Current observations tightly constrain the tensor‑to‑scalar ratio rₜ at ultra‑low frequencies (aHz ≈ 10⁻¹⁸ Hz) to rₜ ≲ 0.03, implying that the stochastic gravitational‑wave background in this band is essentially invisible. However, the same constraints do not apply to higher frequencies (kHz–GHz), where the spectral energy density Ω_GW could be orders of magnitude larger.

The author emphasizes that the number of e‑folds N between the horizon crossing of the pivot scale (kₚ = 0.002 Mpc⁻¹) and the onset of big‑bang nucleosynthesis (BBN) is not fixed. If the post‑inflationary universe expands faster than radiation (a ∝ t^{β} with β > ½), N is reduced (N < 60). Since rₜ ∝ 1/N² in single‑field slow‑roll inflation, a smaller N raises rₜ, potentially pushing it toward current observational limits. Conversely, if the expansion is slower than radiation (β < ½), N increases (N > 60) and rₜ is further suppressed, making the aHz gravitons “invisible”.

These variations in N also affect the high‑frequency graviton spectrum. In a slower‑than‑radiation phase the scale factor grows more gently, which amplifies Ω_GW at MHz–THz frequencies by a factor roughly proportional to (a₁/a₀)⁴, where a₁ is the scale factor at the end of the decelerated phase. Consequently, while the aHz band remains suppressed, the audio band (few Hz–10 kHz) and even the GHz regime can host a detectable stochastic background, possibly reaching Ω_GW ∼ 10⁻⁸–10⁻⁶.

The second major focus is the quantum amplification of hypercharge (electromagnetic) fields during inflation and the subsequent decelerated era. If the gauge coupling g(τ) evolves with time, the hypermagnetic mode functions acquire a non‑adiabatic growth, leading to a magnetic power spectrum P_B(k) ∝ k^{n_B} with a red tilt (n_B ≈ −2 to −1). The amplitude of this spectrum depends sensitively on the expansion rate of the decelerated phase. A slower‑than‑radiation expansion allows the magnetic field to retain power on large comoving scales (∼ Mpc), which after cosmological evolution can seed μG‑level fields observed in galaxies and clusters. In contrast, a faster‑than‑radiation phase truncates the spectrum at much smaller scales, making it difficult to generate the required large‑scale magnetism.

By jointly analyzing the graviton and hypermagnetic spectra, the author identifies a viable parameter space where three conditions are simultaneously satisfied: (i) rₜ is strongly suppressed at aHz, rendering the low‑frequency graviton background invisible; (ii) Ω_GW is sizable in the MHz–THz window, potentially observable by future high‑frequency gravitational‑wave detectors (e.g., resonant microwave cavities, optomechanical sensors); (iii) the magnetic field generated during the decelerated era reaches strengths of order 10⁻⁶ G on Mpc scales by the epoch of protogalactic collapse, consistent with observed galactic and cluster fields.

A particularly interesting result is that when Ω_GW peaks in the audio band, the characteristic scale of protogalactic collapse (∼ Mpc) becomes comparable to the Hubble radius before matter‑radiation equality, leading to an enhanced magnetic field. Conversely, an extended fast‑expanding decelerated stage does not affect the high‑frequency graviton spectrum but reduces N, thereby increasing rₜ possibly beyond observational bounds.

The paper concludes by outlining observational prospects. High‑frequency stochastic backgrounds could be probed by planned laser‑interferometer extensions, resonant mass detectors, or microwave‑cavity experiments. Large‑scale magnetic fields can be constrained via Faraday rotation of distant radio sources, CMB polarization, and 21 cm tomography. Detecting a combination of a suppressed aHz graviton signal, a detectable MHz–THz Ω_GW, and μG‑level intergalactic magnetic fields would provide a powerful test of the post‑inflationary decelerated expansion history, linking two seemingly unrelated cosmological puzzles.