Primordial black hole evaporation in a thermal bath and gravitational waves

1. Problem Statement

Primordial Black Holes (PBHs) formed in the early Universe are a generic source of stochastic gravitational waves (GWs) via Hawking radiation. Existing literature calculating the GW spectrum from evaporating PBHs almost universally assumes a zero-temperature vacuum environment. In this standard approximation, the mass loss rate is determined solely by the Hawking temperature and greybody factors, ignoring the surrounding cosmological medium.

However, in realistic early-Universe scenarios, PBHs are embedded in a hot, dense thermal plasma of Standard Model particles. When the ambient plasma temperature ($T_b$) is comparable to or exceeds the Hawking temperature ($T_{PBH}$), the interaction between the black hole and the thermal bath significantly modifies the evaporation dynamics. The authors argue that neglecting these thermal effects leads to an inaccurate description of the PBH evaporation history, which in turn distorts predictions for the resulting stochastic gravitational wave background.

2. Methodology

The authors develop a consistent framework to incorporate thermal effects into PBH evaporation and subsequently calculate the resulting GW spectrum.

• Thermal Evaporation Framework:

  • They utilize a thermofield dynamics approach (referencing Refs. [30, 31]) to modify the evaporation function $\epsilon_i$.
  • In the standard vacuum case, the mass loss rate follows Eq. (2.2) with an evaporation function defined in Eq. (2.4).
  • In the thermal bath scenario, the evaporation function is modified to Eq. (2.6). This equation includes an additional term accounting for the suppression or enhancement of emission due to the thermal environment (specifically, the presence of the bath modifies the phase space for particle emission).
  • The authors distinguish two evolutionary phases:
    1. Thermal Phase ($T_b > T_{PBH}$): Occurs early in the PBH's life. The thermal bath enhances the effective decay rate, leading to accelerated mass loss.
    2. Vacuum Phase ($T_{PBH} > T_b$): As the Universe expands and cools, the PBH temperature eventually exceeds the bath temperature, and evaporation proceeds as in the standard vacuum case.

• Cosmological Evolution:

  • The evolution of PBH mass ($M_{PBH}$), radiation energy density ($\rho_R$), and bath temperature ($T_b$) is solved self-consistently using a set of coupled differential equations (Appendix A).
  • The initial conditions are set by the formation temperature ($T_{in}$) and a collapse efficiency factor $\gamma$.
  • The authors use a monochromatic mass distribution of Schwarzschild PBHs with an initial mass $M_{in} = 10^2$ grams and a formation parameter $\beta = 10^{-10}$, ensuring the Universe remains radiation-dominated during evaporation.

• Gravitational Wave Calculation:

  • The stochastic GW energy density $\Omega_{GW}(f)$ is calculated by integrating the Hawking radiation spectrum of gravitons over the PBH lifetime.
  • The standard vacuum GW spectrum is given by Eq. (3.1).
  • The thermally corrected GW spectrum is derived in Eq. (3.2), which incorporates the modified mass evolution $M_{PBH}(a)$ and an additional factor representing the thermal bath's influence on the emission rate.

3. Key Contributions

• First Application of Thermal Corrections to GWs: This is the first study to systematically investigate how thermally corrected PBH evaporation impacts the generation of stochastic gravitational waves. Previous works focused on particle production or dark matter constraints but had not extended these thermal corrections to the GW sector.
• Redistribution of Evaporation History: The paper demonstrates that thermal effects do not merely scale the lifetime but fundamentally redistribute the mass loss profile in time. It creates a "double burst" feature: an early, thermally enhanced mass loss phase followed by the standard final runaway evaporation.
• Spectral Deformation Analysis: The authors quantify how this temporal redistribution translates into specific deformations in the GW frequency spectrum, moving beyond simple amplitude scaling to analyze spectral shape changes.

4. Results

• Mass Evolution: As shown in Fig. 1, the presence of a thermal bath causes a significant enhancement in mass loss during the early phase ($T_b > T_{PBH}$). While the total lifetime of the PBH is only slightly reduced compared to the vacuum case, the timing of the mass loss is shifted. A finite fraction of the mass is radiated away earlier than in the vacuum scenario.
• Gravitational Wave Spectrum:
  • Vacuum Case (Fig. 2, Left): Produces a standard single-peak spectrum characteristic of instantaneous evaporation at high frequencies ($f \sim 10^{17}$ Hz for $10^2$g PBHs).
  • Thermal Case (Fig. 2, Right): The spectrum exhibits a single dominant peak rather than two distinct peaks. However, the shape is measurably distorted:
    • There is a mild enhancement in amplitude.
    • The spectrum shows a redshifted tail at lower frequencies. This low-frequency component arises from the graviton emission during the early, thermally enhanced mass loss phase.
    • The "early burst" is not strong enough to create a second distinct peak resolvable from the main peak with current parameters, but it significantly alters the spectral slope at low frequencies.
• Observational Implications: The peak frequencies for the benchmark mass ($10^2$ g) lie far beyond the sensitivity of current or near-future detectors (LIGO, LISA, etc.). However, the authors note that future proposals for detecting ultra-high-frequency GWs (e.g., via the inverse Gertsenshtein effect) could potentially probe these spectral distortions.

5. Significance

• Theoretical Consistency: The work establishes that a self-consistent treatment of PBH evaporation in the early Universe must include thermal effects. Ignoring the thermal bath leads to an incomplete picture of PBH dynamics.
• New Observational Signature: While the thermal effects do not generate a qualitatively new "multi-peak" structure for the parameters studied, they induce a characteristic deformation in the GW spectrum. This spectral shape distortion serves as a unique signature that could distinguish between vacuum evaporation and thermally regulated evaporation if ultra-high-frequency GW detectors become operational.
• Bridge to Future Physics: The framework developed provides a systematic basis for extending these analyses to other PBH mass ranges and cosmological scenarios, highlighting that thermal corrections propagate directly into GW observables. This is crucial for interpreting future high-frequency GW data as a probe of early Universe thermodynamics.

In conclusion, the paper argues that while thermal effects may not drastically alter the existence of the GW signal, they significantly reshape its spectral profile, offering a potential window into the thermal history of the early Universe through future high-frequency gravitational wave astronomy.