The Effective Field Theory of Large Scale Structure for Mixed Dark Matter Scenarios

Technical Summary: The Effective Field Theory of Large Scale Structure for Mixed Dark Matter Scenarios

Problem Statement
Standard cosmological models (ΛCDM) assume Dark Matter (DM) is a collisionless, cold fluid, leading to scale-free hierarchical structure formation. However, particle physics scenarios often predict a fraction of DM that is "non-cold," such as ultra-light axions (ULAs) or light thermal relics (e.g., massive neutrinos, gravitinos). These species possess a characteristic physical scale (e.g., Jeans scale or free-streaming scale) below which the growth of perturbations is suppressed. While constraints on such scenarios exist, current and upcoming galaxy surveys (DESI, Euclid, Rubin) will probe scales where these effects are significant. A major challenge arises because the standard analytical solutions for Large Scale Structure (LSS) developed for pure Cold Dark Matter (CDM) or massive neutrinos do not generally apply to mixed scenarios where a fraction $f_\chi$ of the DM is non-cold. Existing perturbative approaches often fail to accurately model the coupled dynamics of cold and non-cold fluids in the nonlinear regime, potentially leading to biased cosmological inferences.

Methodology
The authors extend the Effective Field Theory of Large Scale Structure (EFTofLSS) to a two-fluid system comprising a cold component ($c$, representing CDM+baryons) and a warm component ($\chi$, representing the non-cold species).

1. Linear Framework:

   • The non-cold species is modeled as a perfect fluid with a finite sound speed $c_s$ at the linear level. The sound speed is parameterized by a power-law dependence on time and scale: $c_s^2 \propto k^p a^{-\gamma}$.
   • The authors derive new analytical linear solutions for the coupled fluid equations. They perform a perturbative expansion in the small fraction $f_\chi \ll 1$, keeping terms up to $O(f_\chi^0)$ for the warm species and $O(f_\chi)$ for the cold species.
   • They introduce a momentum-dependent time variable $\tilde{\eta}$ to decouple the scale dependence of the sound speed term, allowing for closed-form analytical solutions for specific cases (e.g., $\gamma=2$).
   • Two benchmark models are matched to this framework: Ultra-Light Axions ($p=2, \gamma=2$) and light thermal relics like massive neutrinos ($p=0, \gamma=2$).

2. Nonlinear Dynamics and Kernels:

   • The authors derive analytical expressions for the nonlinear kernels (perturbative coefficients) in specific kinematic regimes:
     • Infrared (IR) limit: They identify exact solutions showing that IR divergences cancel as in standard ΛCDM, ensuring the consistency of the 1-loop power spectrum.
     • Ultraviolet (UV) limit: They analyze the UV behavior, identifying new EFT counterterms required to absorb divergences arising from the two-fluid nature.
     • Intermediate regimes: For configurations where momenta are separated from the characteristic scale $k_s$, they derive new analytical expressions for the kernels, noting the emergence of antisymmetric mode-coupling functions absent in scale-free theories.

3. Computational Prescription:

   • Recognizing that exact numerical solutions for the kernels are computationally expensive, the authors devise a "prescription" for evaluating the 1-loop galaxy power spectrum.
   • This prescription constructs the nonlinear kernels by multiplying standard ΛCDM kernels by ratios of linear transfer functions ($g, h, s$). This approach is motivated by exact IR results and ensures that the 1-loop corrections can be computed rapidly enough for Markov Chain Monte Carlo (MCMC) parameter exploration.
   • The prescription is validated against exact numerical solutions, showing relative errors significantly smaller than current and upcoming survey uncertainties for the total matter power spectrum.

4. Bias and Redshift Space:

   • The galaxy bias expansion is extended to include a linear response to the warm species ($b_\chi$) and velocity bias.
   • Redshift-space distortions are treated by mapping real-space kernels to redshift space, incorporating the new scale-dependent growth rates.

Key Contributions

• Theoretical Extension: A systematic derivation of the EFTofLSS for mixed DM scenarios, providing the first analytical treatment of coupled cold and non-cold fluids beyond linear order.
• Analytical Solutions: New closed-form linear solutions and nonlinear kernel expressions for specific kinematic regimes, including the identification of exact IR limits and the structure of UV divergences.
• Computational Tool: A practical, accurate prescription for computing the 1-loop galaxy power spectrum in mixed DM scenarios, enabling efficient parameter estimation.
• Intrinsic Nonlinearities: An analysis showing that intrinsic nonlinearities arising from the wave nature of ULAs (quantum pressure) and the Vlasov equation for thermal relics have a minor impact on the 1-loop power spectrum, justifying the perfect fluid approximation used in the main framework.

Results

• Accuracy: The proposed prescription reproduces the exact numerical 1-loop matter power spectrum with better than 2% accuracy up to $k \approx 0.4 \, h \, \text{Mpc}^{-1}$ for benchmark scenarios.
• Constraints on ULAs: Applying the framework to Planck CMB data and BOSS galaxy power spectrum data, the authors derive updated constraints on the energy density of ultra-light axions ($\Omega_a h^2$).
  • The refined theoretical modeling, which includes the new bias parameter $b_\chi$ and associated counterterms, leads to weaker bounds compared to previous single-fluid analyses (e.g., Ref. [66]).
  • For axion masses where the characteristic scale falls within the BOSS observational window ($m_a \sim 10^{-27} \, \text{eV}$), the inclusion of the two-fluid EFT parameters reduces the constraining power of LSS data relative to CMB-only constraints, highlighting the necessity of consistent theoretical modeling to avoid biased inferences.
  • The analysis confirms that for $m_a \gtrsim 10^{-25} \, \text{eV}$, the axion behaves indistinguishably from CDM on observable scales, while for $m_a \lesssim 10^{-30} \, \text{eV}$, it mimics a cosmological constant.

Significance and Claims
The paper claims that a consistent description of Beyond-Standard-Model (BSM) effects in the dark sector is essential for the robustness of future discoveries by galaxy surveys. The authors emphasize that neglecting the specific nonlinear dynamics of mixed DM scenarios (or using simplified single-fluid approximations) can lead to biased constraints on fundamental parameters.

The work serves as a foundational step for testing mixed DM scenarios with future high-precision data from DESI, Euclid, and the Vera Rubin Observatory. It provides a "model-independent" framework where constraints can be expressed in terms of the non-cold fraction $f_\chi$ and characteristic scale $k_s$, which can then be projected onto specific microscopic models. The authors modestly note that while their current focus is on perturbative scales and specific benchmarks ($\gamma=2$), the framework opens the door to exploring a broader range of dark sector models and higher-point correlation functions in future work.