Invisible gravitons and large-scale magnetism

Technical Summary: Invisible Gravitons and Large-Scale Magnetism

Problem Statement
The paper addresses the interplay between two cosmological phenomena: the spectral energy density of relic gravitons (tensor modes) and the origin of large-scale magnetism (hypercharge fields). In the standard concordance paradigm, the post-inflationary evolution is assumed to be dominated immediately by radiation. However, the expansion rate prior to Big-Bang Nucleosynthesis (BBN) is not directly observable. The author investigates scenarios where the post-inflationary timeline involves decelerated expansion stages that are either faster or slower than the standard radiation-dominated era.

The core problem is that such modified timelines alter the total number of $e$-folds ($N_\nu$) elapsed since the crossing of specific comoving frequencies during inflation. This modification has two competing effects:

1. It can suppress the tensor-to-scalar ratio ($r_T$) in the ultra-low frequency (aHz) range, rendering relic gravitons "invisible" to current large-scale observations (CMB and Pulsar Timing Arrays).
2. It simultaneously enhances the spectral energy density of relic gravitons in high-frequency domains (kHz to THz) and affects the amplification of gauge fields, which are candidates for explaining the observed large-scale magnetic fields in protogalaxies.

The paper seeks to determine if a consistent timeline exists where gravitons are invisible in the aHz range while still satisfying magnetogenesis requirements and remaining compatible with high-frequency constraints.

Methodology
The analysis is based on a single-field inflationary model described by a tree-level effective action. The author employs a model-independent approach to the post-inflationary timeline, parametrizing the expansion rate by a parameter $\delta$ (where $\delta=1$ corresponds to radiation, $\delta < 1$ to slower expansion, and $\delta > 1$ to faster expansion).

The methodology involves:

• Graviton Spectra: Calculating the spectral energy density of relic gravitons, $\Omega_{gw}(\nu, \tau_0)$, for single and double post-inflationary stages. The analysis utilizes the relation between the number of $e$-folds ($N_\nu$), the slow-roll parameters, and the expansion history to derive the suppression of $r_T$ and the potential for high-frequency spikes.
• Gauge Field Evolution: Analyzing the evolution of hypermagnetic fields via a general action with dynamical gauge couplings ($g_y$). The study tracks the amplification of gauge modes during inflation (where the coupling increases) and their subsequent evolution in the decelerated stage (where the coupling flattens). The late-time hypermagnetic spectrum is shown to be determined by the hyperelectric fields at the end of inflation.
• Constraints: The derived spectra are constrained by:
  • Magnetogenesis: The requirement that the physical power spectrum of magnetic fields at the scale of protogalactic collapse ($\sim$ Mpc, corresponding to frequencies $\nu_g \sim$ fHz) is sufficient to seed observed galactic magnetic fields (typically requiring $\sqrt{P_B} \gtrsim 10^{-11}$ nG, or relaxed to $10^{-16}$ nG with efficient dynamo action).
  • Gravitational Wave Detectors: Limits from wide-band interferometers (audio band, 20 Hz – 10 kHz) and Pulsar Timing Arrays (nHz range).
  • BBN: Constraints on the expansion rate prior to nucleosynthesis to ensure the correct synthesis of light nuclei.

Key Contributions and Results

1. Invisible Gravitons and High-Frequency Spikes:
   The paper demonstrates that a post-inflationary stage expanding slower than radiation ($\delta < 1$) increases the total number of $e$-folds ($N_\nu > 60$). This leads to a significant suppression of the tensor-to-scalar ratio $r_T$ in the aHz range (potentially $r_T \ll 0.03$), making gravitons "invisible" to CMB B-mode polarization searches. Conversely, this same timeline enhances the spectral energy density at high frequencies (MHz to THz), potentially creating a spike orders of magnitude larger than the standard concordance prediction.

2. Double-Stage Expansion and Audio Band Maxima:
   The author analyzes a scenario with two successive decelerated stages: one faster than radiation ($\delta_1 > 1$) followed by one slower than radiation ($\delta_2 < 1$). This configuration creates a local maximum in the gravitational wave spectrum at intermediate frequencies. If this maximum falls within the audio band ($\nu \sim 100$ Hz), it is subject to direct constraints from interferometers. The paper finds that such a maximum is compatible with current audio-band limits ($\Omega_{gw} < 10^{-9}$) while still allowing for a significant magnetic field generation.

3. Compatibility of Magnetogenesis and Invisible Gravitons:
   The study explicitly combines constraints from relic gravitons and large-scale magnetism.

   • Single Stage: For a single stage expanding slower than radiation, a large high-frequency gravitational signal is marginally compatible with magnetogenesis requirements. The magnetic field strength at the protogalactic scale depends on the gauge coupling growth rate ($\gamma$) and the duration of the stage ($\xi$). While strict magnetogenesis limits ($\sqrt{P_B} \ge 10^{-11}$ nG) are difficult to satisfy simultaneously with a large high-frequency spike, relaxed limits (assuming efficient dynamo action, $\sqrt{P_B} \ge 10^{-16}$ nG) are satisfied.
   • Double Stage: In the case of a double stage with a maximum in the audio band, the constraints are more robust. The paper shows that a configuration exists where the relic gravitons are invisible in the aHz range ($r_T \ll 0.03$), the audio-band signal is within interferometric limits, and the magnetic field strength satisfies the strict magnetogenesis requirement ($\sqrt{P_B} \ge 10^{-11}$ nG).

4. Parameter Space Analysis:
   Through extensive numerical analysis (Figs. 4–13), the paper maps the allowed regions in the parameter space defined by the expansion rates ($\delta, q$), the duration of the stages ($\xi$), the tensor-to-scalar ratio ($r_T$), and the gauge coupling parameters ($\gamma, g_1$). The results indicate that the "invisible graviton" scenario is not in conflict with successful large-scale magnetogenesis; rather, they are mutually constraining.

Significance and Claims
The paper claims that the "invisible graviton" scenario—where the tensor-to-scalar ratio is suppressed in the ultra-low frequency domain due to a modified post-inflationary timeline—is physically viable and potentially testable. The significance lies in the fact that the suppression of low-frequency signals is intrinsically linked to the enhancement of high-frequency signals and the generation of large-scale magnetic fields.

The author concludes that:

• The evolution of hypercharge fields and relic gravitons are correlated through the post-inflationary timeline.
• A timeline that renders gravitons invisible in the aHz range can simultaneously produce a detectable spike in the MHz–THz range or the audio band.
• These high-frequency or audio-band signals can be large enough to be detected by future or current experiments, while the magnetic fields generated at the protogalactic scale satisfy observational requirements.
• Therefore, the search for relic gravitons in complementary frequency ranges (audio vs. ultra-high frequency) and the study of large-scale magnetism provide a consistent framework to constrain the expansion history of the Universe prior to BBN.

The paper does not propose new experimental setups but rather interprets existing constraints (from interferometers, PTAs, and magnetic field observations) to define the viable parameter space for early-universe cosmology. It emphasizes that the "concordance paradigm" is not the only solution and that modified timelines offer a consistent explanation for both the lack of low-frequency tensor modes and the existence of large-scale cosmic magnetism.