Spherical Collapse and Halo Formation in a Cosmology with Decaying Dark Matter and a Semi-Cosmographic Dark Energy

Technical Summary: Spherical Collapse and Halo Formation in a Cosmology with Decaying Dark Matter and a Semi-Cosmographic Dark Energy

Problem Statement
The standard $\Lambda$CDM model faces persistent observational tensions, particularly regarding the expansion history and the growth of cosmic structures (e.g., the $S_8$ and Hubble tensions). While Decaying Dark Matter (DDM) offers a potential resolution by suppressing late-time structure growth, previous constraints on DDM have typically relied on fixing the dark energy sector to a cosmological constant ($\Lambda$) or a simple fluid with fixed parameters. This approach risks masking degeneracies between the decay of dark matter and the dynamics of dark energy. Furthermore, the nature of dark energy remains largely unknown, and assuming a specific equation of state ($w = -1$) a priori may bias the reconstruction of non-standard dark-sector physics. This paper addresses the need for a unified framework that simultaneously constrains a one-body DDM scenario and a dynamical, model-agnostic dark energy sector, propagating these constraints into the nonlinear regime of structure formation.

Methodology
The authors employ a "semi-cosmographic" framework that combines a physically motivated matter sector with a flexible dark energy reconstruction:

1. One-Body DDM Model: The dark matter sector consists of a nonrelativistic "mother" component ($m$) that decays into a relativistic "daughter" radiation component ($dr$) with a constant decay rate $\Gamma = \tau_{ddm}^{-1}$. The evolution of densities is governed by coupled continuity equations, modifying the background expansion and the growth of perturbations.
2. Semi-Cosmographic Dark Energy: Instead of assuming a parametric equation of state, the dark energy sector is reconstructed from the expansion history. The authors use a Padé kinematic expansion for the comoving angular diameter distance ($D_M^P$) to describe the Hubble rate $H_P(z)$. The effective dark energy density ($\Omega_\phi^P$) and its equation of state ($w_\phi^P$) are derived as residuals after subtracting the contributions of baryons, radiation, and the DDM components from the Padé expansion history.
3. Data Constraints: The background evolution and linear growth are constrained using DESI Data Release 1 (DR1) Baryon Acoustic Oscillation (BAO) distance measurements and compressed "ShapeFit" data (providing $f\sigma_8$ and shape parameters $m+n$).
4. Nonlinear Propagation: The posterior distributions from the linear data are propagated into the nonlinear regime using the spherical collapse model. The authors calculate the critical linear collapse threshold ($\delta_c$) and the virial overdensity ($\Delta_{vir}$) for the specific DDM + dynamical DE background.
5. Halo Mass Function (HMF): Using the Despali prescription (based on Sheth-Tormen multiplicity), the authors compute the Halo Mass Function. The linear power spectrum is generated via the CLASS code, and the HMF is compared against observed halo abundance data from the DESI Legacy Imaging Surveys DR9.
6. Joint Analysis: Constraints are derived for two scenarios: (1) DESI DR1 BAO + ShapeFit alone, and (2) the joint combination of DESI DR1 BAO + ShapeFit + HMF data.

Key Contributions

• Unified Framework: The paper establishes a pipeline connecting a reconstructed dark-energy sector and decaying dark matter directly to nonlinear structure formation observables (specifically halo abundances).
• Semi-Cosmographic Approach: It introduces a method that retains physical motivation for the matter sector (DDM) while maximizing flexibility in the dark energy sector via Padé reconstruction, thereby reducing degeneracies inherent in fully model-agnostic reconstructions.
• Nonlinear Probes: The study demonstrates how halo mass function measurements serve as a powerful complementary probe to linear clustering data, specifically for breaking degeneracies between DDM lifetime and dark energy parameters.

Results

• Parameter Constraints: When using only DESI DR1 BAO and ShapeFit data, the DDM lifetime ($\tau_{ddm}$) remains broadly unconstrained ($10 < \log_{10}(\tau_{ddm}/\text{yr}) < 13$) due to degeneracies with the flexible dark energy sector. However, the inclusion of HMF data significantly tightens these constraints, yielding a median $\log_{10}(\tau_{ddm}/\text{yr}) \approx 12.42$ with reduced uncertainties.
• Dark Energy Equation of State: The reconstructed dark energy equation of state ($w_\phi$) deviates from the $\Lambda$CDM value of $-1$. The median reconstruction exhibits a dynamical evolution: it lies in a quintessence-like regime ($w > -1$) at low redshift ($z \lesssim 0.5$), crosses the phantom divide ($w < -1$) near $z \approx 1$, and shows phantom-like behavior at intermediate redshifts before crossing back. The uncertainty bands widen significantly at high redshift, indicating limited sensitivity of current data to early dark energy evolution.
• Collapse Thresholds: The critical linear collapse threshold ($\delta_c$) remains very close to the standard $\Lambda$CDM prediction, showing only sub-percent deviations. The virial overdensity ($\Delta_{vir}$) shows a more pronounced, non-monotonic response, being lower than $\Lambda$CDM at $z=0$ and slightly higher at intermediate redshifts, before converging at high $z$.
• Halo Abundance: The most significant signature appears in the Halo Mass Function. The joint analysis predicts a substantial suppression of massive halos compared to $\Lambda$CDM. At $M = 10^{14} M_\odot/h$ and $z > 0.5$, the median suppression exceeds 50%. This suppression is driven primarily by modifications to the mass variance $\sigma(M, z)$ induced by the DDM decay and dynamical dark energy, rather than changes in $\delta_c$.

Significance and Claims
The paper claims that halo abundances provide a sensitive nonlinear probe of coupled dark-sector physics, offering constraints that are substantially tighter than those derived from geometric observables (BAO) and linear growth (ShapeFit) alone. The results demonstrate that:

1. The combination of DDM and dynamical dark energy can be constrained jointly, with halo data breaking degeneracies that leave the DDM lifetime poorly defined in linear analyses.
2. The reconstructed dark energy sector does not require a cosmological constant, favoring a dynamical evolution that alternates between quintessence and phantom phases, though high-redshift behavior remains weakly constrained.
3. The suppression of the high-mass tail of the halo mass function is a robust nonlinear signature of this non-standard cosmology, implying a delayed formation of the most massive gravitationally bound structures and potentially affecting the timing of cosmic reionization.

The authors conclude that their framework successfully links reconstructed dark energy dynamics and decaying dark matter to the formation of cosmic structures, highlighting the potential of upcoming large-scale structure surveys to test such non-standard models. They note limitations regarding the use of a diagonal covariance matrix for the HMF data and the analytic nature of the virialization prescription, suggesting that future analyses with full covariance matrices will be necessary for more precise statistical quantification.