The cosmology of long range Yukawa interactions in general backgrounds

Technical Summary: The Cosmology of Long-Range Yukawa Interactions in General Backgrounds

Problem Statement
Primordial black holes (PBHs) are promising dark matter candidates, yet their formation in the standard radiation-dominated scenario requires extreme fine-tuning of the primordial density field to generate sufficient overdensities. Alternative mechanisms involving long-range forces stronger than gravity, mediated by light scalar fields coupled to dark matter, have been proposed to enhance early structure formation. Previous studies have largely focused on linear Yukawa couplings or dilatonic couplings within specific cosmological epochs (e.g., radiation domination) and often assumed non-relativistic fermions. However, the interplay between general scalar-fermion couplings (including higher-order monomials), relativistic effects, and arbitrary cosmological backgrounds (characterized by a constant equation of state $w$) remains underexplored. Furthermore, the role of a bare scalar potential in competing with the fermion-induced effective potential has not been systematically analyzed.

Methodology
The authors analyze the homogeneous dynamics of a coupled scalar-fermion system in a fixed Friedmann-Lemaître-Robertson-Walker (FLRW) background with a constant equation-of-state parameter $w$. The system is treated as a spectator sector, decoupled from the dominant cosmic fluid, allowing for energy exchange between the scalar field $\phi$ and a degenerate Fermi gas of fermions $\psi$ via a Yukawa-type interaction.

The study employs the following methodological framework:

1. Action and Couplings: The authors consider a general action where the fermion mass depends on the scalar field, $m_{\text{eff}}(\phi) = |m_\psi + y f(\phi)|$. They investigate two primary classes of coupling functions $f(\phi)$:
   • Monomial couplings: $f(\phi) \propto \phi^p$ (where $p \geq 1$ is an integer), covering both linear ($p=1$) and higher-order effective field theory (EFT) extensions.
   • Dilatonic couplings: $f(\phi) \propto e^{\phi}$ (specifically the runaway mass case).
2. Fluid Approximation: Fermions are modeled as a zero-temperature degenerate Fermi gas. The effective mass $m_{\text{eff}}$ determines whether fermions are relativistic ($x \ll 1$) or non-relativistic ($x \gg 1$), where $x$ is the ratio of effective mass to Fermi momentum.
3. Dimensionless Formulation: The equations of motion are rewritten using dimensionless variables ($\phi \to \varphi$, time $\to$ scale factor $a$) to facilitate numerical and analytical analysis across different regimes.
4. Scaling Arguments: The authors utilize conformal transformations and approximate scale invariance arguments to derive asymptotic behaviors, particularly in the limit where $m_{\text{eff}} \to 0$.
5. Bare Potential Inclusion: The analysis is extended to include a bare scalar potential $V_{\text{bare}}(\phi) \propto \phi^q$, examining how it competes with the fermion-induced effective potential.

Key Contributions and Results

1. Identification of Two Broad Regimes
The system exhibits two distinct evolutionary regimes:

• Scaling Regime: The scalar field oscillates around a "saturation point" where the effective fermion mass vanishes ($m_{\text{eff}} \to 0$). This regime arises from an approximate scale invariance in the scalar-fermion action (exact for linear couplings, approximate for higher-order monomials near $m_{\text{eff}}=0$).
• Asymptotic Regime: The field evolves toward configurations where fermions recover their bare mass ($m_{\text{eff}} \to m_\psi$). This occurs when the field is trapped in a flat region of the effective potential or when the bare potential dominates.

2. Scaling Solutions and Energy Density
For both monomial and dilatonic couplings admitting a saturation point, the system admits scaling solutions where the scalar field oscillates around the saturation point with an envelope scaling as $|\varphi - \varphi_s| \propto a^{-1}$.

• Energy Density Scaling: In this scaling regime, both the scalar energy density ($\rho_\phi$) and the fermion energy density ($\rho_\psi$) redshift as radiation, i.e., $\rho \propto a^{-4}$, regardless of whether the background is radiation-dominated, matter-dominated, or stiff-fluid dominated ($w=1$).
• Constant Ratio: Consequently, the ratio $\rho_\phi / \rho_\psi$ remains approximately constant.
• Relativistic Stability: Relativistic fermions entering this regime remain relativistic. Non-relativistic fermions remain non-relativistic on average, despite brief transitions through the relativistic regime near mass-zero crossings.

3. Dependence on Coupling Order and Background

• Linear Coupling ($p=1$): The field is driven to the saturation point ($m_{\text{eff}}=0$) in all backgrounds.
• Higher-Order Monomials ($p > 1$):
  • Odd $p$: A saturation point exists. For $p=1, 3, 5$, the field typically reaches the saturation point. For sufficiently large $p$ (e.g., $p \geq 7$ in radiation domination), the field may enter an asymptotic branch where it decays as $\varphi \propto a^{-3w+1/(p-2)}$ without reaching saturation, causing fermions to recover their bare mass.
  • Even $p$: No saturation point exists at $m_{\text{eff}}=0$ for positive couplings; the field oscillates around $\varphi=0$. However, for negative couplings ($f \propto -\phi^p$), two degenerate saturation points ($\varphi = \pm 1$) emerge, potentially leading to domain wall formation.
• Background Influence: Stiff-fluid backgrounds ($w=1$) drive the system toward the scaling solution most efficiently due to the slower redshifting of the background energy density compared to the scalar-fermion sector.

4. Impact of Bare Scalar Potential
When a bare potential $V_{\text{bare}} \propto \phi^q$ is included:

• Competition: The bare potential eventually dominates the dynamics as the universe expands, pulling the field away from the fermion-induced saturation point toward the minimum of the bare potential (typically $\varphi=0$).
• Transition Dynamics:
  • If $q > p$, the off-origin minimum drifts asymptotically toward zero.
  • If $q \leq p$, the off-origin minimum may merge with a local maximum and annihilate at a finite scale factor $a_c$. Beyond this point, the field oscillates around the origin.
• Energy Density Evolution: Once the field settles into oscillations around the bare potential minimum (e.g., quadratic $q=2$), the energy density scaling transitions from radiation-like ($a^{-4}$) to matter-like ($a^{-3}$) for the scalar condensate, while fermions recover their bare mass and behave as standard matter.

Significance and Claims
The authors state that this work provides a "first step toward a general study of the growth of perturbations" in systems with long-range Yukawa interactions. By generalizing previous studies to arbitrary couplings and cosmological backgrounds, they establish the existence of robust scaling regimes where the scalar and fermion sectors evolve in lockstep with radiation-like energy densities.

The paper claims that these scaling solutions arise from an approximate scale invariance (becoming conformal at late times) and that the specific behavior (oscillatory vs. asymptotic) depends critically on the coupling power $p$, the presence of a bare potential, and the background equation of state $w$. The authors modestly note that while their results suggest long-range forces could provide a "natural" mechanism for generating large small-scale overdensities (potentially relevant for PBH formation), definitive claims require further investigation into perturbations and non-linear dynamics, which are left for future work. They also highlight the potential for domain wall formation in even-p negative coupling scenarios as a source of gravitational waves.