Decaying Dark Matter Cosmological Constraints

• Decaying dark matter (DDM) is a theoretical model where a fraction of dark matter decays into relativistic dark radiation, impacting cosmic expansion and structure formation.
• Analyses use modified Boltzmann codes and MCMC techniques to constrain DDM lifetimes and decay fractions via CMB, BAO, Lyman-α, and weak lensing data.
• Observations indicate that while DDM can slightly alleviate H₀ and S₈ tensions, its effects are tightly limited by both linear and nonlinear cosmological probes.

Decaying dark matter (DDM) cosmologies postulate that a fraction of the dark matter sector is unstable and decays into lighter species—typically assumed to be invisible, relativistic “dark radiation.” Such models are motivated by particle physics frameworks and are also explored as possible solutions to small-scale structure anomalies and tensions in ΛCDM, such as the $S_8$ and $H_0$ discrepancies. The decay process alters the expansion history, the growth and distribution of cosmic structures, CMB anisotropies, and leaves potentially observable signatures across a wide array of cosmic data sets. Cosmological probes at both linear and nonlinear scales, together with laboratory and astrophysical limits, allow for stringent constraints on DDM lifetimes, branching fractions, and daughter particle properties.

1. Theoretical Framework and Parameterization

DDM models fundamentally split the dark sector into stable (SDM) and decaying (DCDM) components. The primary parameters are:

• Fractional abundance: $$f \equiv \Omega_{\mathrm{dcdm}}/\Omega_{\mathrm{dm}}$$ is the initial fraction of (cold) DM that is unstable.
• Decay rate (lifetime): $\Gamma$ in Gyr$^{-1}$, $\tau = \Gamma^{-1}$.
• Nature of decay products: Most cosmological analyses consider invisible dark radiation (massless or very light, noninteracting), though models with decays to massive daughters also exist.
• Energy splitting: For two-body decays (parent mass $m_0$ to massive daughter $m_2$ and massless $\phi$), the fraction of rest-mass energy transferred to relativistic products is $\epsilon = \frac{1}{2}\big[1-(m_2/m_0)^2\big]$, with associated kick velocity $H_0$0.

The homogeneous evolution equations (in conformal or cosmic time) for decaying and stable dark matter, and the dark radiation component, are

$H_0$1

with similar equations for massive daughters in nontrivial decay kinematics. The impact on perturbations is captured in a modified Boltzmann hierarchy, with additional source terms determined by the decay kinematics and branching.

2. Observational Signatures and Sensitive Probes

The specific cosmological consequences of DDM depend on the decay rate, the fraction of decaying DM, and the products' properties. Major observable effects include:

• CMB anisotropies: DDM modifies the integrated Sachs-Wolfe effect (late-time potential decay), alters the lensing kernel, and shifts the acoustic peak structure by changing the matter–radiation content. The effect is most visible at low multipoles ($H_0$2) and, for sufficiently early decays, can shift the position of the peaks. The derived constraints primarily come from Planck temperature, polarization, and lensing power spectra (Poulin et al., 2016, Alvi et al., 2022, Clark et al., 2020, Nygaard et al., 2023, Abellán et al., 2021).
• Structure growth and matter power spectrum: The removal of clustered matter and injection of free-streaming (relativistic or semi-relativistic) daughters suppresses the amplitude of matter fluctuations, leading to a reduction in $H_0$3 and a suppression of $H_0$4 at scales $H_0$5 above the daughter free-streaming horizon. The scale and depth of suppression encode both $H_0$6 and $H_0$7 (or $H_0$8) (Aoyama et al., 2014, Cheng et al., 2015, Peter et al., 2010, Peter, 2010, Abellán et al., 2021).
• Lyman-$H_0$9 forest: Decay-generated free-streaming suppresses small-scale power at high redshift ($$f \equiv \Omega_{\mathrm{dcdm}}/\Omega_{\mathrm{dm}}$$0), providing some of the strongest constraints, especially for kick velocities $$f \equiv \Omega_{\mathrm{dcdm}}/\Omega_{\mathrm{dm}}$$1 km/s and lifetimes less than the Hubble time (Wang et al., 2013).
• Halo abundance, internal structure, and substructure: N-body simulations show that sufficient DDM suppresses the abundance of massive halos and subhalos, alters the concentration–mass relation, and can lead to observable features in the cluster mass function and galaxy counts. This directly impacts cluster abundance studies, galaxy clustering, and Milky Way satellite populations (Peter et al., 2010, Peter, 2010, Cheng et al., 2015, Enqvist et al., 2019).
• Redshift-space distortions and kSZ effect: Direct measurement of growth (e.g., $$f \equiv \Omega_{\mathrm{dcdm}}/\Omega_{\mathrm{dm}}$$2), and velocity fields (kSZ) are also suppressed in DDM, with future surveys projected to reach sub-percent level sensitivity to DDM signatures (Xiao et al., 2019).
• Astrophysical gamma rays and cosmic-ray constraints: In scenarios with decays to SM particles, especially photons, electrons, or quarks, one obtains stringent bounds on $$f \equiv \Omega_{\mathrm{dcdm}}/\Omega_{\mathrm{dm}}$$3 (up to $$f \equiv \Omega_{\mathrm{dcdm}}/\Omega_{\mathrm{dm}}$$4 s) from the isotropic gamma-ray background, X-ray and cosmic-ray measurements (Blanco et al., 2018, Garny et al., 2012, Slatyer et al., 2016).

3. Constraints Across Decay Regimes

The constraints are typically segmented by the regime of the decay lifetime:

Regime	Main Bound	Principal Data Sets	Physical Consequence
Short-lived (pre-recomb.)	$$f \equiv \Omega_{\mathrm{dcdm}}/\Omega_{\mathrm{dm}}$$5 (lifetime $$f \equiv \Omega_{\mathrm{dcdm}}/\Omega_{\mathrm{dm}}$$6 yr)	CMB, $$f \equiv \Omega_{\mathrm{dcdm}}/\Omega_{\mathrm{dm}}$$7, BAO	Early energy injection, alters acoustic scales, shifts $$f \equiv \Omega_{\mathrm{dcdm}}/\Omega_{\mathrm{dm}}$$8 up by $$f \equiv \Omega_{\mathrm{dcdm}}/\Omega_{\mathrm{dm}}$$9 km/s/Mpc at best
Intermediate ($\Gamma$0 yr $\Gamma$1)	$\Gamma$2 ($\Gamma$3)	Planck TTTEEE (+lensing+BAO), LSS, clusters	Enhanced ISW, late-time $\Gamma$4 suppression, $\Gamma$5 and $\Gamma$6 remain largely unaffected
Long-lived ($\Gamma$7)	$\Gamma$8 Gyr$\Gamma$9 ($^{-1}$0 Gyr)	Planck TTTEEE+lensing+BAO	Effectively indistinguishable from CDM; tightest bound on instability fraction

These values are robust to moderate extensions (varying $^{-1}$1, curvature, tensor sector) (Alvi et al., 2022, Poulin et al., 2016, Simon et al., 2022). If only a fraction $^{-1}$2 of the DM is unstable, for $^{-1}$3 the bound is effectively on the product $^{-1}$4.

4. Statistical Methodologies and Analysis Pipelines

Modern analyses employ Boltzmann codes (CLASS/CAMB), often with custom implementations of the DDM Boltzmann hierarchy to accommodate the energy/momentum flow between species with arbitrary masses (Poulin et al., 2016, Aoyama et al., 2014, Blackadder et al., 2015, Aoyama et al., 2011). Parameter estimation is carried out via Markov Chain Monte Carlo techniques (e.g., MontePython, CosmoMC), varying standard ΛCDM and DDM-specific parameters under flat or log priors. The principal observational datasets include:

• CMB: Planck (2015 and 2018) TT, TE, EE, lensing
• Large-scale structure: BAO (BOSS, WiggleZ, eBOSS), $^{-1}$5, full-shape galaxy power spectra (EFTofLSS)
• Supernovae: Pantheon, JLA
• Lyman-$^{-1}$6: SDSS, VHS, BOSS 1D flux power
• Weak lensing: KiDS, CFHTLenS, DES
• SZ cluster counts: Planck
• Cosmic shear: KiDS450
• Astrophysical indirect detection: Fermi IGRB, AMS-02, extragalactic X-ray background

Nuisance parameters are marginalized according to the implementations of each survey's likelihood (Alvi et al., 2022, Poulin et al., 2016, Simon et al., 2022, Abellán et al., 2021).

5. Physical Implications and Tensions

$^{-1}$7 and $^{-1}$8 Tensions

Decaying DM generally suppresses $^{-1}$9 and increases $\tau = \Gamma^{-1}$0 compared to ΛCDM, aligning directionally with the observed tensions. However, multiple analyses consistently show that the improvement in tension is modest and statistically insignificant with current data (Poulin et al., 2016, Xiao et al., 2019, Nygaard et al., 2023, Abellán et al., 2021):

• $\tau = \Gamma^{-1}$1 can increase by $\tau = \Gamma^{-1}$2 km/s/Mpc—but is still $\tau = \Gamma^{-1}$3 below local measurements.
• $\tau = \Gamma^{-1}$4 can be shifted down by up to 3\%, but not enough to fully reconcile lensing and CMB-inferred constraints.

The preferred parameter spaces capable of further alleviating these tensions are essentially excluded by CMB and LSS observables due to excessive ISW effects and $\tau = \Gamma^{-1}$5 suppression (see (Clark et al., 2020, Nygaard et al., 2023)).

Small-Scale Structure

For DDM models designed to address the overabundance of small galaxies or too-dense halo cores, Lyman-$\tau = \Gamma^{-1}$6 forest and halo mass function measurements robustly exclude lifetimes $\tau = \Gamma^{-1}$7 Gyr for kick velocities $\tau = \Gamma^{-1}$8 km/s, consistent with the need to avoid substantial suppression of small-scale power (Peter et al., 2010, Cheng et al., 2015, Wang et al., 2013, Aoyama et al., 2014).

Astrophysical Constraints

For models with decays to SM particles, the most stringent bounds arise from extragalactic gamma-ray backgrounds and antiproton fluxes: $\tau = \Gamma^{-1}$9 s for a variety of decay channels and $m_0$0 GeV–EeV (Blanco et al., 2018, Garny et al., 2012, Slatyer et al., 2016), which are far stronger than cosmological limits for such scenarios.

6. Open Questions, Future Prospects, and Model Extensions

• Nonlinear structure: Present limits rely predominantly on linear cosmological observables or simplified halo models. Improved N-body simulations (including baryons and velocity effects of daughter particles) are actively pursued to robustly characterize DDM on nonlinear and small scales (Cheng et al., 2015, Enqvist et al., 2019).
• Complex decay channels: Phenomenology with massive daughters, partial/fractional decay, and time-dependent rates is being systematically explored. Some mixed decay models may permit additional parameter space, but generally only for highly fine-tuned or subdominant fractions (Aoyama et al., 2014, Aoyama et al., 2011, Blackadder et al., 2015, Simon et al., 2022).
• Next-generation data: Upcoming observatories (Euclid, DESI, LSST, CMB-S4) will measure growth, lensing, velocity fields, and the CMB lensing power to unprecedented precision, shrinking the allowed parameter volume for DDM or potentially uncovering signatures in the event of deviations from ΛCDM (Poulin et al., 2016, Alvi et al., 2022, Xiao et al., 2019, Abellán et al., 2021).
• Systematics vs new physics: There is an active effort to disentangle unmodeled systematics in low-$m_0$1 observations (for instance, in cluster mass calibrations or lensing systematics) from true new-physics signatures (Poulin et al., 2016).
• PBH mergers: DDM constraints are particularly restrictive for scenarios where PBHs constitute all DM, as merging events efficiently generate "dark radiation" exceeding permitted fractions (Poulin et al., 2016).

7. Representative Numerical Constraints

Selected robust 95% C.L. constraints derived from recent literature:

Dataset Combination	Regime	Bound	Reference
Planck18 TT,TE,EE + lensing + BAO	$m_0$2	$m_0$3 Gyr	(Alvi et al., 2022)
As above, fraction $m_0$4 decaying	$m_0$5, $m_0$6	$m_0$7 Gyr	(Alvi et al., 2022)
Planck15 + BAO + WiggleZ	long-lived ($m_0$8)	$m_0$9 Gyr$m_2$0, $m_2$1 Gyr	(Poulin et al., 2016)
Planck15 + BAO + RSD	short-lived	$m_2$2	(Xiao et al., 2019)
SDSS Lyman-$m_2$3	$m_2$4 Gyr	$m_2$5 km/s	(Wang et al., 2013)
Lyman-$m_2$6, large $m_2$7	$m_2$8 km/s	$m_2$9 Gyr	(Wang et al., 2013)
Planck15+BAO+SZ+weak lensing	full DDM	$\phi$0 Gyr	(Enqvist et al., 2019)
Fermi IGRB (γ-ray, SM decays)	GeV–EeV DM	$\phi$1 s	(Blanco et al., 2018, Garny et al., 2012)

• (Poulin et al., 2016)
• (Alvi et al., 2022)
• (Clark et al., 2020)
• (Aoyama et al., 2014)
• (Cheng et al., 2015)
• (Peter et al., 2010)
• (Peter, 2010)
• (Simon et al., 2022)
• (Abellán et al., 2021)
• (Xiao et al., 2019)
• (Wang et al., 2013)
• (Aoyama et al., 2011)
• (Blanco et al., 2018)
• (Garny et al., 2012)
• (Enqvist et al., 2019)
• (Nygaard et al., 2023)
• (Slatyer et al., 2016)