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Layered dark structure with a Structuring Field: A $Z_4$-symmetric Inert Doublet-Singlet realization and implications for the $S_8$ tension

Marriam Naeem, Mohid Farhan

Abstract

We introduce the Layered Dark Sectors with a Structuring Field (LDS-SF), a structured cosmological framework where the internal architecture of a multi-component dark sector naturally generates scale-dependent growth of structure. In this framework, the characteristic scale dependence is derived from the dominant eigenvalue, $λ(k)$, of the dark sector's perturbation matrix. This structurally-driven mechanism modifies structure growth while preserving the standard $Λ$CDM background expansion and General Relativity. We provide a minimal realization of this framework within a two-component DM $Z_4$-symmetric Inert Doublet Singlet Model ($Z_4$-IDSM). By integrating out the heavy inert doublet mediator, we derive a contact-interaction Effective Field Theory (EFT) for a 60~GeV singlet dark matter candidate. This interaction manifests macroscopically as an effective sound speed $c_s^2$, which we map to the LDS-SF eigenmode evolution. We implement this system into the CLASS Boltzmann code, employing a late-time activation function that projects virialized halo properties into the linear perturbation framework. We also compute the relic density using micrOMEGAs to further stress-test the relic abundance predictions of viable parameters. Our numerical analysis demonstrates that while the model remains indistinguishable from $Λ$CDM at the era of recombination, it introduces a targeted suppression of the matter power spectrum at late times ($z < 10$) and small scales ($k > 0.1~h/\text{Mpc}$). Confronting the model with Planck CMB, BAO, and growth-rate measurements, we find three instances of couplings that successfully alleviate the $S_8$ tension, bringing the predicted amplitude into $1σ$ agreement with weak-lensing data from KiDS-1000 and DES. This work establishes LDS-SF as a mathematically consistent and observationally viable extension of standard cosmology.

Layered dark structure with a Structuring Field: A $Z_4$-symmetric Inert Doublet-Singlet realization and implications for the $S_8$ tension

Abstract

We introduce the Layered Dark Sectors with a Structuring Field (LDS-SF), a structured cosmological framework where the internal architecture of a multi-component dark sector naturally generates scale-dependent growth of structure. In this framework, the characteristic scale dependence is derived from the dominant eigenvalue, , of the dark sector's perturbation matrix. This structurally-driven mechanism modifies structure growth while preserving the standard CDM background expansion and General Relativity. We provide a minimal realization of this framework within a two-component DM -symmetric Inert Doublet Singlet Model (-IDSM). By integrating out the heavy inert doublet mediator, we derive a contact-interaction Effective Field Theory (EFT) for a 60~GeV singlet dark matter candidate. This interaction manifests macroscopically as an effective sound speed , which we map to the LDS-SF eigenmode evolution. We implement this system into the CLASS Boltzmann code, employing a late-time activation function that projects virialized halo properties into the linear perturbation framework. We also compute the relic density using micrOMEGAs to further stress-test the relic abundance predictions of viable parameters. Our numerical analysis demonstrates that while the model remains indistinguishable from CDM at the era of recombination, it introduces a targeted suppression of the matter power spectrum at late times () and small scales (). Confronting the model with Planck CMB, BAO, and growth-rate measurements, we find three instances of couplings that successfully alleviate the tension, bringing the predicted amplitude into agreement with weak-lensing data from KiDS-1000 and DES. This work establishes LDS-SF as a mathematically consistent and observationally viable extension of standard cosmology.

Paper Structure

This paper contains 8 sections, 50 equations, 39 figures, 1 table.

Table of Contents

  1. Introduction
  2. The LDS-SF Framework
  3. Particle Physics Realization: The Z$_4$-Inert Doublet + Complex Singlet Model
  4. Methodology
  5. Results and Discussion
  6. Conclusion
  7. Appendix
  8. Renormalization Group Equations and EFT Stability of the Z$_4$-IDSM

Figures (39)

  • Figure 1: Full eigenvalue spectrum of the $6\times6$ perturbation matrix $\mathbf{M}(k,a)$ evaluated at $a=1$. The dominant eigenvalue $\lambda_1(k)$ (blue curve) remains positive and separated from the remaining eigenvalues across the full wavenumber range. All subdominant eigenvalues are significantly more negative, and no eigenvalue crossing or degeneracy is observed. This confirms a stable spectral hierarchy and single-mode dominance in the growth sector.
  • Figure 2: Eigenvalue gap $\Delta\lambda(k)=\lambda_1(k)-\lambda_2(k)$ computed from the full $6\times6$ perturbation matrix at $a=1$. The gap remains finite and satisfies $0.38 \lesssim \Delta\lambda \lesssim 0.50$ across the full $k$-range shown.
  • Figure 3: $D(k)$ at zero redshift plotted as a function of $k$ for $\Lambda$CDM and LDS--SF.
  • Figure 4: The power spectrum displayed in a logarithmic scale, highlighting the observational validity of LDS--SF framework.
  • Figure 5: The ratio only goes up at high scales, and by moderate amounts as shown here. This highlights the suppression mechanism at the low-$k$ regime.
  • ...and 34 more figures