Gravitational waves from axion inflation in the gradient expansion formalism. Part II. Fermionic axion inflation

This paper extends the gradient expansion formalism to fermionic axion inflation, demonstrating that Schwinger pair creation of charged fermions damps gauge-field production and attenuates the gravitational wave signal, thereby allowing observable signals from LISA and ET to be compatible with $\Delta N_{\rm eff}$ constraints while identifying a new damped oscillatory backreaction regime.

The Gist

The Big Picture: A Cosmic Symphony

Imagine the very early universe as a giant, vibrating drum. When this drum was hit during the "Inflation" era (a time when the universe expanded faster than the speed of light), it created ripples in space-time called Gravitational Waves. Scientists are trying to "hear" these ripples using future detectors like LISA (a space-based microphone) and ET (a ground-based one).

This paper is the second part of a study about a specific type of drumming called Axion Inflation.

• Part I looked at a "Pure" version where the drum only had two players: the Axion (the drummer) and a Gauge Field (the sound waves).
• Part II (This Paper) asks: "What happens if we add a third player to the band? What if there are Fermions (charged particles like electrons) in the mix?"

The Problem with the "Pure" Band (Part I)

In the first study, the authors found a problem. When the Axion drummed hard, it created a massive amount of sound (gauge fields). This sound was so loud that it created a feedback loop:

1. The sound got louder.
2. The louder sound pushed back on the drummer, making them drum even harder and longer.
3. This created an explosion of gravitational waves.

The Catch: This explosion was too loud. It violated a cosmic rule called $\Delta N_{eff}$ (which is like a limit on how much "extra energy" the universe can have without messing up the formation of stars and galaxies). In the "Pure" model, if you could hear the waves, the universe would have broken.

The New Twist: The "Fermionic" Band (Part II)

In this new paper, the authors add Fermions (charged particles) to the mix. Here is the magic analogy:

The Analogy: The Spongy Floor
Imagine the Axion is a dancer spinning on a floor.

• In the Pure Model: The floor is made of ice. The dancer spins fast, creating huge, uncontrollable whirlwinds (gauge fields). The whirlwinds get so big they knock the dancer over (backreaction), creating a chaotic, deafening noise (gravitational waves) that breaks the rules of the universe.
• In the Fermionic Model: The floor is covered in sponges (the fermions). As the dancer spins, the sponges soak up the energy.
  • The Schwinger Effect is the mechanism where the strong electric/magnetic fields created by the dancer create these sponges out of thin air.
  • Once the sponges appear, they act like a dampener. They absorb the energy of the whirlwinds.

The Result: A "Tempered" Performance

Because the fermions act as sponges, the chaotic explosion is stopped.

1. No More Overproduction: The gravitational waves are still generated, but they are "tempered" (soothed). They don't violate the cosmic energy limits ($\Delta N_{eff}$).
2. Detectable but Safe: The signal is now "Goldilocks" sized—not too quiet, not too loud. It falls right into the sweet spot where future detectors like LISA and the Einstein Telescope can hear it without breaking the laws of physics.
3. A New Dance Style: The authors discovered a new regime they call "Fermion-Tempered Backreaction." Instead of the dancer spinning out of control, they start to wobble and oscillate gently around their path. The universe expands a bit longer, but it's a controlled, rhythmic wobble rather than a crash.

The "Fermion Gas" Bonus

There is a second layer to this. The paper suggests that the fermions themselves (the sponges) might also make noise.

• If the sponges are moving in a messy, uneven way (anisotropic), they could create their own gravitational waves.
• This would make the signal even louder at high frequencies, potentially making it easier for detectors to hear, but it also makes the signal "bluer" (higher pitch).
• Interestingly, this extra noise might actually make it harder to explain a recent signal detected by the NANOGrav collaboration (which uses pulsars as clocks), because the signal would be too "high-pitched" compared to what NANOGrav sees.

Why This Matters

• Realism: This model is more realistic because it includes the Standard Model particles (like electrons) that actually exist in our universe, rather than just hypothetical particles.
• Hope for Detection: It opens up a new window for discovery. We might actually be able to detect these waves with LISA or ET in the coming decades.
• The "Gradient Expansion" Tool: The authors used a mathematical shortcut (Gradient Expansion Formalism) to do this. They admit it's an approximation, but they argue that because the fermions "soak up" the chaos, this shortcut is actually very accurate for this specific scenario. They are calling on computer scientists to run full simulations (lattice studies) to double-check their work.

Summary in One Sentence

By adding charged particles (fermions) to the early universe's inflationary model, the authors found that these particles act like a cosmic shock absorber, preventing a catastrophic explosion of gravitational waves and creating a signal that is perfectly tuned to be detected by our future telescopes without breaking the laws of physics.

Technical Summary

1. Problem Statement

The paper addresses the generation of a Stochastic Gravitational Wave Background (SGWB) during Axion Inflation, specifically focusing on the interplay between the inflaton, gauge fields, and charged fermions.

• Context: In "Pure Axion Inflation" (PAI), where an axion inflaton couples to a gauge field without charged matter, the tachyonic instability leads to explosive gauge field production. This results in a strongly blue-tilted gravitational wave (GW) spectrum. However, previous studies (Part I of this series) showed that achieving an observable GW signal in PAI inevitably violates constraints on the effective number of relativistic degrees of freedom ($\Delta N_{\text{eff}}$) due to the overproduction of radiation.
• The Gap: The paper investigates Fermionic Axion Inflation (FAI), a more realistic scenario where the gauge field is the Standard Model (SM) hypercharge field ($U(1)_Y$). In this regime, the strong electric and magnetic fields generated by the axion coupling induce Schwinger pair production of SM fermions. The authors aim to determine if the presence of these fermions alters the GW phenomenology, potentially resolving the tension with $\Delta N_{\text{eff}}$ constraints while maintaining detectability.

2. Methodology

The authors employ the Gradient Expansion Formalism (GEF), a semi-analytical approach that treats the gauge field as a classical stochastic background while retaining quantum fluctuations.

• Model Setup:
  • Lagrangian: Includes a pseudoscalar inflaton $\phi$, a $U(1)$ gauge field $A_\mu$, and charged fermions $\chi_i$. The coupling is via the Chern-Simons term $I(\phi)F\tilde{F}$.
  • Fermion Treatment: Instead of simulating fermions as independent degrees of freedom (which is computationally expensive), they are modeled effectively via an induced current $J$.
  • Schwinger Damping: The induced current is derived from the Schwinger effect formula, relating the current to the electric and magnetic fields via a conductivity $\sigma$. This introduces a damping term in the gauge field evolution equation.
  • Scale Dependence: To avoid unphysical damping of sub-horizon modes, the authors introduce a spectral function $\Theta(t, k)$ that limits the damping to modes with wavelengths larger than the typical particle-antiparticle separation scale ($k_S$). They compare two modeling approaches: $k_S$-damping (physically motivated but complex) and $k_H$-damping (approximate, using the instability scale).
• Dynamics: The system is solved using coupled equations of motion for the inflaton, gauge fields, and fermion energy density. The anisotropic stress from both the gauge field and the fermion gas is calculated to determine the tensor power spectrum.
• Observables: The authors compute the GW energy density $\Omega_{\text{GW}}(f)$ and compare it against:
  • Detectors: LISA, Einstein Telescope (ET), NANOGrav 15-year (NG15), and LIGO-Virgo-O3 (HLVO3).
  • Constraints: Planck CMB data (scalar amplitude $A_s$, tensor-to-scalar ratio $r$, non-Gaussianity), $\Delta N_{\text{eff}}$ limits, and BBN bounds.

3. Key Contributions

1. Fermion-Tempered Backreaction: The paper identifies a novel dynamical regime where the Schwinger effect does not merely suppress gauge fields but fundamentally alters the backreaction mechanism. Unlike PAI, where backreaction causes a violent, explosive instability, FAI exhibits fermion-tempered backreaction. Here, the inflaton velocity oscillates around the slow-roll trajectory, and the gauge field production is damped by fermion conductivity, preventing the runaway energy transfer that leads to $\Delta N_{\text{eff}}$ violations.
2. Resolution of the $\Delta N_{\text{eff}}$ Tension: The study demonstrates that FAI can produce observable GW signals without violating the upper limit on extra relativistic degrees of freedom. The damping of the gauge field by fermions reduces the total energy density in radiation, keeping $\Delta N_{\text{eff}}$ within bounds even for parameters that would be ruled out in PAI.
3. Fermionic GW Contribution: For the first time, the authors estimate the direct contribution of the fermion gas to the SGWB. They find that while the gauge field dominates at lower frequencies, the fermion contribution can become dominant at high frequencies due to the anisotropy of the fermion distribution (specifically, left-right symmetric fermions in high Landau levels).
4. Benchmark for Lattice Simulations: The authors provide specific benchmark points and dynamical regimes (fermion-tempered backreaction) that serve as targets for future full lattice simulations, validating the GEF approach for this specific class of models.

4. Key Results

• GW Spectrum Characteristics:
  • The GW spectrum in FAI is mildly blue-tilted ($h^2\Omega_{\text{GW}} \propto f^{0.1}$), contrasting with the strongly blue-tilted spectrum of PAI.
  • The amplitude is sufficient to be detected by LISA and the Einstein Telescope (ET) for coupling strengths $\beta \gtrsim 45$ and inflaton masses $m \lesssim 3.1 \times 10^{-6} M_P$ (consistent with Planck constraints).
• NANOGrav (NG15) Interpretation:
  • While FAI can produce signals in the nanohertz regime (NG15 band) for very large couplings ($\beta \gtrsim 54$) and masses, these regions are excluded by Planck constraints on the inflaton mass.
  • Furthermore, even if Planck constraints were ignored, the spectral slope of FAI signals ($\sim f^{0.06}$) is too shallow to match the steeper slope favored by NG15 data ($\sim f^{1.5}$).
• Reheating Effects:
  • Lowering the reheating temperature ($T_{\text{reh}}$) redshifts the GW spectrum. This can enhance sensitivity for lower-frequency detectors but causes a cutoff at frequencies corresponding to $T_{\text{reh}}$.
  • For $T_{\text{reh}} \sim 1$ MeV, the sensitivity of ET and LISA is significantly reduced or lost entirely for the relevant parameter space.
• Fermion Contribution Impact:
  • Including the estimated fermionic contribution boosts the GW signal at high frequencies (ET band).
  • This enhancement creates a tension with HLVO3 (LIGO-Virgo) non-detection limits. If the fermion contribution is significant, signals explaining NG15 would likely have been detected by HLVO3, further disfavoring FAI as the source of the NG15 signal.

5. Significance

• Phenomenological Viability: The paper establishes that Fermionic Axion Inflation is a phenomenologically viable source of observable gravitational waves, unlike its "pure" counterpart. It opens a new window for detecting early Universe physics via space-based (LISA) and ground-based (ET) interferometers.
• Theoretical Insight: The discovery of "fermion-tempered backreaction" provides a crucial theoretical insight into how charged matter stabilizes inflationary dynamics, preventing the catastrophic instabilities seen in pure gauge models.
• Standard Model Connection: By explicitly modeling the coupling to the SM hypercharge sector, the work bridges the gap between inflationary model building and the Standard Model, suggesting that the most realistic axion inflation scenarios naturally include these damping effects.
• Future Directions: The results highlight the necessity of computing fermion backreaction from first principles and reassessing the impact of axion inhomogeneities. The paper sets a clear agenda for future lattice simulations to validate the GEF predictions in the fermion-tempered regime.

In summary, this work demonstrates that the inclusion of Standard Model fermions in axion inflation acts as a natural regulator, allowing for observable gravitational wave signals that are consistent with cosmological constraints, thereby making FAI a compelling target for next-generation GW astronomy.