Decaying Dark Matter: Cosmological Extensions

• Decaying Dark Matter (DDM) models introduce instability in cold dark matter, allowing it to decay into dark radiation or warm components that alter cosmic expansion and clustering.
• Two-body decay scenarios yield a massive warm daughter and a relativistic partner, whose free-streaming effects suppress small-scale structures and modify the matter power spectrum.
• Observational probes including the CMB, weak lensing, and tSZ power spectra constrain DDM parameters, offering insights into resolving the S8 and H0 tensions.

Decaying dark matter (DDM) denotes a family of extensions of $\Lambda$CDM in which some or all of the cold dark matter is unstable on cosmological timescales and decays into dark radiation, warm daughters, or other invisible states. The generic effect is to reduce the late-time clustering reservoir and/or to inject free-streaming daughters, thereby modifying $H(z)$, suppressing the matter power spectrum $P(k)$, and shifting derived parameters such as $\sigma_8$ and $S_8$ relative to stable-CDM predictions (Tanimura et al., 2023). In a distinct usage of the same acronym, “DDM” also denotes Dynamical Dark Matter, an ensemble framework in which dark matter is distributed across many states with correlated masses, lifetimes, and abundances rather than a single unstable species (Dienes et al., 2016).

1. Model classes and parameterizations

The contemporary DDM literature is not a single model but a model space. The simplest class is one-body or effectively one-channel decay of cold dark matter into noninteracting dark radiation, described by

$$\dot{\rho}_{\rm dm}+3H\rho_{\rm dm}=-\Gamma \rho_{\rm dm},\qquad \dot{\rho}_{\rm dr}+4H\rho_{\rm dr}=+\Gamma \rho_{\rm dm},$$

with constant decay rate $\Gamma$ and lifetime $\tau=\Gamma^{-1}$. A second broad class is two-body decay, in which the parent produces one massive daughter plus one relativistic daughter; the massive daughter behaves as warm dark matter or a stable daughter species with a recoil velocity $v_k$, and the free-streaming induced by that kick suppresses growth on a time- and scale-dependent set of modes (Tanimura et al., 2023).

Two-body parameterizations are often written in terms of either a mass-loss fraction or an energy-fraction parameter. In one common notation, the mass splitting satisfies $f \simeq v_k/c$ for small splittings. In another, used for $H(z)$0,

$H(z)$1

and for non-relativistic daughters $H(z)$2. The time-dependent decayed fraction is

$H(z)$3

A separate line of work considers only a subdominant unstable fraction $H(z)$4 of the total dark matter that decays after recombination into dark radiation, leaving the rest stable (Chudaykin et al., 2017).

A further phenomenological variant takes the decay rate to be proportional to the Hubble rate, $H(z)$5, so that the dark-radiation-to-dark-matter ratio is approximately $H(z)$6 after equality. In that construction, the model deviates most strongly from $H(z)$7CDM in the early-to-intermediate regime relevant for the sound horizon and late-time clustering (Pandey et al., 2019).

Scenario	Decay products	Dominant effect
$H(z)$8	massless dark radiation	lowers clustering matter, modifies $H(z)$9
$P(k)$0	warm massive daughter plus radiation	free-streaming suppression of small-scale $P(k)$1
Fractional post-recombination DDM	unstable subcomponent plus stable CDM	few-percent late-time change in distances and growth
$P(k)$2 DDM	dark radiation sourced continuously	reduced $P(k)$3, lower $P(k)$4, mild $P(k)$5 shift

2. Cosmological dynamics, perturbations, and nonlinear suppression

At the background level, DDM modifies the matter and radiation budgets directly. In $P(k)$6DDM1-type models ($P(k)$7), the late-time matter density decreases relative to $P(k)$8CDM, reducing $P(k)$9 and the linear growth factor $\sigma_8$0. In $\sigma_8$1DDM2-type models ($\sigma_8$2), the total matter budget can remain close to $\sigma_8$3CDM, but the daughter velocity dispersion suppresses clustering below a free-streaming scale. A standard diagnostic is

$\sigma_8$4

with $\sigma_8$5 the effective sound speed of the daughter population. In the two-body scenario studied for weak lensing forecasts, $\sigma_8$6 evolves differently from the massive-neutrino case: while decays are ongoing, it tracks the horizon more closely, whereas after decay injection effectively ceases the daughter momenta redshift and the free-streaming scale shrinks (Wang et al., 2012).

At the perturbation level, the model must be treated with modified Boltzmann hierarchies. This has been implemented in several ways: modified CMBFAST for late two-body decay with a stable daughter and light relativistic particle, modified CAMB for arbitrary daughter masses and lifetimes, and CLASS-based implementations such as CLASS-decays and other DDM modules that treat injected warm daughters either with a viscous-fluid approximation or with more detailed hierarchy solvers (Aoyama et al., 2014). For weak-lensing applications, the fully nonlinear two-body $\sigma_8$7DDM matter spectrum has been modeled as

$\sigma_8$8

with $\sigma_8$9 emulated from gravity-only simulations and then combined with baryonic-feedback suppression factors (Bucko et al., 2023).

Nonlinear evolution is not a small correction in DDM. Dedicated simulations of $S_8$0 show that nonlinear mode coupling enhances the small-scale suppression relative to linear theory, and fitting formulas have been constructed for both the nonlinear power spectrum and halo statistics. One such analysis found that the combined CMB–weak-lensing tension is better described in DDM than in stable CDM, while still yielding only a lower bound on the lifetime, $S_8$1 Gyr at 95% C.L. from CMB+WL data (Enqvist et al., 2015).

3. Observational probes

The observational reach of DDM is broad because the decay affects both geometry and growth. CMB primary anisotropies are sensitive mainly through late integrated Sachs–Wolfe contributions, acoustic-scale shifts if the decay changes pre-recombination expansion, and reduced CMB lensing. In the one-body $S_8$2 case constrained with Planck and KiDS, the dominant information does not come from nonlinear weak lensing but from low-$S_8$3 CMB TT via the ISW effect, which sharply limits how much late-time decay can occur without overproducing potential decay at large angular scales (Bucko et al., 2022).

Weak lensing probes the redshift- and scale-dependent suppression in $S_8$4 through tomographic shear spectra,

$S_8$5

or equivalent Limber-projected expressions. This makes it directly sensitive to daughter free-streaming scales in two-body DDM. Linear-theory forecasts showed that Euclid- or LSST-like surveys using $S_8$6 can reach $S_8$7 km s$S_8$8 for $S_8$9–5 Gyr, while ambitious nonlinear forecasts to $$\dot{\rho}_{\rm dm}+3H\rho_{\rm dm}=-\Gamma \rho_{\rm dm},\qquad \dot{\rho}_{\rm dr}+4H\rho_{\rm dr}=+\Gamma \rho_{\rm dm},$$0 can reach $$\dot{\rho}_{\rm dm}+3H\rho_{\rm dm}=-\Gamma \rho_{\rm dm},\qquad \dot{\rho}_{\rm dr}+4H\rho_{\rm dr}=+\Gamma \rho_{\rm dm},$$1 km s$$\dot{\rho}_{\rm dm}+3H\rho_{\rm dm}=-\Gamma \rho_{\rm dm},\qquad \dot{\rho}_{\rm dr}+4H\rho_{\rm dr}=+\Gamma \rho_{\rm dm},$$2 for $$\dot{\rho}_{\rm dm}+3H\rho_{\rm dm}=-\Gamma \rho_{\rm dm},\qquad \dot{\rho}_{\rm dr}+4H\rho_{\rm dr}=+\Gamma \rho_{\rm dm},$$3 Gyr, contingent on calibrated nonlinear theory (Wang et al., 2012).

Thermal Sunyaev–Zel’dovich measurements provide a complementary route because the tSZ power spectrum is sensitive to the halo mass function, the pressure profile normalization, and the background expansion. For Planck PR4 Compton-$$\dot{\rho}_{\rm dm}+3H\rho_{\rm dm}=-\Gamma \rho_{\rm dm},\qquad \dot{\rho}_{\rm dr}+4H\rho_{\rm dr}=+\Gamma \rho_{\rm dm},$$4 maps, the one-halo tSZ power spectrum was modeled as

$$\dot{\rho}_{\rm dm}+3H\rho_{\rm dm}=-\Gamma \rho_{\rm dm},\qquad \dot{\rho}_{\rm dr}+4H\rho_{\rm dr}=+\Gamma \rho_{\rm dm},$$5

with DDM entering through $$\dot{\rho}_{\rm dm}+3H\rho_{\rm dm}=-\Gamma \rho_{\rm dm},\qquad \dot{\rho}_{\rm dr}+4H\rho_{\rm dr}=+\Gamma \rho_{\rm dm},$$6, the halo mass function, and the pressure normalization $$\dot{\rho}_{\rm dm}+3H\rho_{\rm dm}=-\Gamma \rho_{\rm dm},\qquad \dot{\rho}_{\rm dr}+4H\rho_{\rm dr}=+\Gamma \rho_{\rm dm},$$7. This gives direct leverage on the late-time suppression of clustering that is relevant to $$\dot{\rho}_{\rm dm}+3H\rho_{\rm dm}=-\Gamma \rho_{\rm dm},\qquad \dot{\rho}_{\rm dr}+4H\rho_{\rm dr}=+\Gamma \rho_{\rm dm},$$8 (Tanimura et al., 2023).

Additional probes cover other mass and lifetime ranges. BAO and RSD constrain geometric distances and $$\dot{\rho}_{\rm dm}+3H\rho_{\rm dm}=-\Gamma \rho_{\rm dm},\qquad \dot{\rho}_{\rm dr}+4H\rho_{\rm dr}=+\Gamma \rho_{\rm dm},$$9 for decays after recombination. Milky Way satellites, stellar streams, strong-lensing substructure, and Lyman-$\Gamma$0 constrain kick velocities and lifetimes on halo scales. At much lower masses, eV-scale DDM that decays to monochromatic photons can be sought through UV extragalactic-background tomography: GALEX or ULTRASAT diffuse maps cross-correlated with DESI spectroscopy forecast sensitivity to decay rates of order $\Gamma$1 and to axion-like-particle couplings $\Gamma$2 in the relevant mass window (Libanore et al., 2024).

4. Cosmological tensions and current inference landscape

DDM has been studied intensely as a possible explanation of the $\Gamma$3 discrepancy between Planck-inferred clustering and low-redshift probes. In a joint analysis of Planck high-redshift CMB, BAO, Pantheon SNIa, and the tSZ power spectrum, the $\Gamma$4 model preferred

$\Gamma$5

with a 95% lower bound $\Gamma$6 Gyr. The $\Gamma$7 model preferred

$\Gamma$8

with $\Gamma$9 Gyr and a slight goodness-of-fit preference over the pure-DR model, $\tau=\Gamma^{-1}$0 in its favor (Tanimura et al., 2023).

A fully nonlinear weak-lensing analysis of the two-body $\tau=\Gamma^{-1}$1DDM model with KiDS-1000 cosmic shear and Planck reached a less optimistic conclusion. Weak lensing alone gave the tightest direct DDM limits in that setup, $\tau=\Gamma^{-1}$2 Gyr and $\tau=\Gamma^{-1}$3 km s$\tau=\Gamma^{-1}$4 at 95% credible level, while the combined WL+CMB fit reduced the $\tau=\Gamma^{-1}$5 tension from about $\tau=\Gamma^{-1}$6 to about $\tau=\Gamma^{-1}$7, depending on the tension metric, but did not convincingly solve it (Bucko et al., 2023).

For one-body $\tau=\Gamma^{-1}$8, the CMB ISW effect is more restrictive still. A Planck+KiDS analysis found that this model cannot alleviate the $\tau=\Gamma^{-1}$9 difference: CMB alone required $v_k$0 Gyr if all DM is unstable, and for half-lives comparable to or shorter than one Hubble time allowed at most $v_k$1 of the dark matter to decay. KiDS-1000 weak lensing alone was much weaker, $v_k$2 Gyr and $v_k$3, and adding WL to CMB only improved the short-lifetime bound to $v_k$4 (Bucko et al., 2022).

The literature on the joint $v_k$5 and $v_k$6 tensions is similarly mixed. A general Planck-only analysis of two-body channels $v_k$7, $v_k$8, $v_k$9, and $f \simeq v_k/c$0 found no channel that simultaneously raises $f \simeq v_k/c$1 and lowers $f \simeq v_k/c$2; the best-fit $f \simeq v_k/c$3 values remained essentially equal to $f \simeq v_k/c$4CDM and the CMB-inferred $f \simeq v_k/c$5 stayed near $f \simeq v_k/c$6–$f \simeq v_k/c$7 km s$f \simeq v_k/c$8 Mpc$f \simeq v_k/c$9 (Davari et al., 2022). A different phenomenological model with $H(z)$00 did better when using only Planck+SH0ES, giving $H(z)$01, increasing the best-fit $H(z)$02 by $H(z)$03 km s$H(z)$04 Mpc$H(z)$05, and lowering the $H(z)$06 tension to $H(z)$07; however, adding JLA and BAO reduced the effect to $H(z)$08 km s$H(z)$09 Mpc$H(z)$10 and restored an $H(z)$11 $H(z)$12 discrepancy (Pandey et al., 2019).

A related post-recombination model with only a few-percent unstable fraction $H(z)$13 is strongly constrained by BAO/RSD. Depending on the DR12 pipeline, the 95% limits are $H(z)$14 or $H(z)$15, while Ly$H(z)$16 gives $H(z)$17; allowing the lensing amplitude $H(z)$18 to float produces a modest preference for $H(z)$19–0.07 (Chudaykin et al., 2017). More recently, DDM has also been reconsidered as a way to hide large neutrino masses in background data. BAO+SNIa+CMB-distance-prior analyses can indeed make $H(z)$20 effectively unconstrained in a $H(z)$21DDM background fit, but adding full Planck perturbation data, especially CMB lensing, breaks the degeneracy and restores $H(z)$22 eV together with $H(z)$23 (Montandon et al., 3 Mar 2026).

Taken together, the cited analyses indicate that DDM can reduce late-time clustering and sometimes improve dataset overlap, but the degree of improvement depends strongly on the decay channel, the fraction that decays, and whether the constraining power of ISW, CMB lensing, BAO, and nonlinear growth is fully included.

5. Halo structure, subhalos, and small-scale constraints

Small-scale structure is one of the oldest motivations for DDM. Cosmological DDM simulations with kick velocities $H(z)$24–40 km s$H(z)$25 and half-lives $H(z)$26–14 Gyr show that dark matter decays flatten central densities and reduce concentrations in dwarf halos, producing slowly rising scaled rotation curves closer to SPARC dwarf-galaxy data than standard NFW expectations. In the strongest cases, such as $H(z)$27 km s$H(z)$28 and $H(z)$29 Gyr, the model yields large increases in the characteristic inner scale $H(z)$30 and corresponding decreases in $H(z)$31, consistent with substantial core formation (Chen et al., 2020).

On larger scales, N-body studies of $H(z)$32 with cosmological lifetimes built fitting functions for both the halo mass function and the mass–concentration relation. In the regime $H(z)$33 km s$H(z)$34 and $H(z)$35, the suppression scale is set by the daughter free-streaming length while the amplitude is controlled by the global decayed fraction. Those analyses concluded that DDM with $H(z)$36 is unlikely to resolve the Planck SZ–CMB cluster discrepancy: the recoil velocities needed to affect cluster counts strongly are already incompatible with Lyman-$H(z)$37 limits (Cheng et al., 2015).

A semianalytic realization of two-body DDM has now been incorporated into the open-source code $H(z)$38. In that framework, decays inject kinetic energy and direct mass loss into halos through a recoil $H(z)$39 and a mass splitting $H(z)$40, and adiabatic shell expansion is used to map pre-decay to post-decay profiles. For subhalos with $H(z)$41, the predicted surviving fractions relative to CDM are 34$H(z)$4212%, 49$H(z)$4314%, and 76$H(z)$4422% for $H(z)$45 km s$H(z)$46 with $H(z)$47, 20, and 40 Gyr, and 34$H(z)$4811%, 56$H(z)$4921%, and 87$H(z)$5031% for $H(z)$51 km s$H(z)$52 with $H(z)$53, 40, and 80 Gyr. The same work reports good agreement with isolated-halo and cosmological DDM simulations for density profiles, velocity dispersions, and subhalo populations (Nadler et al., 22 Jan 2025).

Milky Way satellite counts already impose strong direct constraints on this regime. Using DES and PS1 satellite catalogs in a forward model that includes the galaxy–halo connection, spatial selection functions, and baryonic disruption, two-body DDM is conservatively excluded at 95% confidence for $H(z)$54 Gyr when $H(z)$55 km s$H(z)$56 and for $H(z)$57 Gyr when $H(z)$58 km s$H(z)$59 (Mau et al., 2022). Much earlier-decaying models are constrained in a different language: for nonrelativistic daughters the free-streaming scale must satisfy $H(z)$60 Mpc, while in the relativistic regime the bound $H(z)$61 was obtained from CMB and large-scale-structure data (Huo, 2011).

6. Related usages, statistical issues, and outlook

The acronym DDM also denotes Dynamical Dark Matter, which is conceptually different from single-species decay. In that framework the dark sector is an ensemble with total abundance

$H(z)$62

tower fraction

$H(z)$63

and a collective effective equation of state $H(z)$64 that evolves because different states decay at different times. Strongly coupled dark sectors with Hagedorn-like density of states provide explicit realizations in which abundances and lifetimes are balanced across the ensemble (Dienes et al., 2016). The same ensemble logic has also been applied to indirect detection, where many constituents with masses in the $H(z)$65–$H(z)$66 GeV range can reproduce the AMS-02 positron excess while predicting a positron-fraction plateau out to $H(z)$67 TeV (Dienes et al., 2013).

Current inference is also sensitive to statistical methodology. A frequentist profile-likelihood analysis of two-body DDM using Planck+BAO found a best fit at $H(z)$68 Gyr$H(z)$69, $H(z)$70 Gyr, and $H(z)$71 km s$H(z)$72, corresponding to $H(z)$73, whereas Bayesian posteriors had previously stayed close to $H(z)$74CDM-like $H(z)$75 values. The same work argued that some KiDS-1000 exclusions of the Planck-favored DDM region were driven largely by priors on $H(z)$76 and $H(z)$77, and that the scales most tightly measured by KiDS-1000 are centered around $H(z)$78, not exactly the canonical $H(z)$79 kernel (Montandon et al., 26 May 2025). This suggests that full-likelihood analyses, rather than compressed $H(z)$80-only comparisons, are especially important for DDM.

The empirical outlook is correspondingly diverse. Future tSZ measurements from AdvACT and SPT-3G, and upcoming CMB surveys such as the Simons Observatory and CMB-S4, are expected to reduce tSZ power-spectrum uncertainties and sharpen sensitivity to the suppression scale associated with warm daughters (Tanimura et al., 2023). Stage-IV weak-lensing surveys such as Euclid, Rubin/LSST, and Roman should improve constraints on $H(z)$81, $H(z)$82, and baryonic degeneracies by extending tomographic measurements deeper into the nonlinear regime (Wang et al., 2012). On small scales, strong-lensing substructure, stellar streams, and improved satellite censuses are natural next tests of the halo-heating predictions now available in semianalytic form (Nadler et al., 22 Jan 2025). At the opposite end of the mass range, ULTRASAT ultraviolet maps cross-correlated with DESI offer a qualitatively different probe of eV-scale radiative decays (Libanore et al., 2024).

The cumulative picture is therefore not one of a single favored DDM model, but of a tightly structured phenomenological landscape. Some DDM realizations still provide partial relief of late-time clustering discrepancies; others are already disfavored by ISW, CMB lensing, BAO/RSD, or small-scale structure. What remains robust across the literature is the central physical point: any decay channel that converts clustering matter into radiation or warm free-streaming daughters imprints a correlated set of signatures in background expansion, growth, halo structure, and line-of-sight observables, and those signatures are now measurable over an exceptionally wide range of masses and lifetimes.