Revisiting cosmological constraints on supersymmetric SuperWIMPs

Deshpande, Meera; Hamann, Jan; Sengupta, Dipan; White, Martin; Williams, Anthony G.; Wong, Yvonne Y. Y.

doi:10.1140/epjc/s10052-024-12992-3

Revisiting cosmological constraints on supersymmetric SuperWIMPs

Regular Article - Theoretical Physics
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Published: 05 July 2024

Volume 84, article number 667, (2024)
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Revisiting cosmological constraints on supersymmetric SuperWIMPs

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Meera Deshpande¹,
Jan Hamann²,
Dipan Sengupta ORCID: orcid.org/0000-0002-5107-0098^1,2,
Martin White¹,
Anthony G. Williams¹ &
…
Yvonne Y. Y. Wong²

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Abstract

SuperWIMPs are extremely weakly interacting massive particles that inherit their relic abundance from late decays of frozen-out parent particles. Within supersymmetric models, gravitinos and axinos represent two of the most well motivated superWIMPs. In this paper we revisit constraints on these scenarios from a variety of cosmological observations that probe their production mechanisms as well as the superWIMP kinematic properties in the early Universe. We consider in particular observables of Big Bang Nucleosynthesis and the Cosmic Microwave Background (spectral distortion and anisotropies), which limit the fractional energy injection from the late decays, as well as warm and mixed dark matter constraints derived from the Lyman-$\alpha $ forest and other small-scale structure observables. We discuss complementary constraints from collider experiments, and argue that cosmological considerations rule out a significant part of the gravitino and the axino superWIMP parameter space.

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1 Introduction

The pursuit of signatures of beyond-the-Standard-Model (BSM) physics and an explanation for the dark matter of the Universe has been the holy grail for particle physicists for over three decades. To this end, the Large Hadron Collider (LHC) has probed large swathes of parameter space in a variety of well motivated BSM models. These include Supersymmetry (SUSY), the leading BSM theory that not only solves the hierarchy problem but also provides a slew of particle candidates for the dark matter. Within the context of specific SUSY breaking scenarios as well as simplified models, null measurements at the LHC have translated into constraints on a significant chunk of SUSY particles in the GeV-to-TeV mass range [1,2,3].

However, SUSY/BSM searches at the LHC rely primarily on prompt decays or, at best, decays with proper lengths of $\mathcal {O}(10-100)$ m in the so-called long-lived particle (LLP) searches [4, 5]. These searches are further subject to the constraints that the produced particles are within kinematic reach of the LHC, i.e., their masses are at most a few TeV, and that they are produced with a cross-section sufficient to generate a detectable signal over the enormous Standard-Model background. Extremely weakly-interacting particles and/or those with very long lifetimes – many of which also reside in well-motivated SUSY/BSM model and parameter spaces – are thus inherently out of the LHC’s reach, even if their masses lie within the conventional GeV-to-TeV collider window.

Interestingly, when the proper decay lengths/lifetimes of these particles exceed $\mathcal {O}(10)$ m, a second, albeit less conventional, window to explore their properties opens up. Disregarding concerns of naturalness, scenarios of extremely long particle lifetimes could be easily realised in a wide range of BSM theories by particle masses and couplings spanning orders of magnitude (e.g., $m \sim \mathcal {O}(1)~\textrm{MeV}-\mathcal {O}(100)~\textrm{TeV}$). In the context of SUSY, these scenarios fall under the superWIMP class of models [6, 7], wherein quasi-stable particles can be efficiently produced in the early Universe and decay at a very late time, i.e., $t \gg {{\mathcal {O}}}(1)$ s post-Big Bang, during the standard cosmological history. Regardless of whether these quasi-stable particles can account for all of the observed dark matter abundance of the Universe at early times, late-decaying particles leave potentially observable signatures in the cosmic microwave background radiation (CMB), as well as the light element abundances from Big Bang Nucleosynthesis (BBN) and the large-scale matter distribution, particularly the Lyman-$\alpha $ (Ly$\alpha $) forest. Measurements of these observables can in turn be used to probe and constrain regions of SUSY/BSM parameter space inaccessible to collider searches.

As a concrete example, consider the following: in R-parity-conserving SUSY, the conventional lightest SUSY particle (LSP) dark matter candidate is the lightest neutralino, superpartner of the electroweak gauge particles. With masses $m_{\textrm{LSP}} \sim {{\mathcal {O}}}(0.1$–1) TeV and weak-like interactions with the Standard Model (SM), the neutralino easily satisfies the observed relic abundance of the Universe and has a range of signatures at collider physics experiments, as well as at direct and indirect dark matter searches [8, 9]. However, in models of Supergravity or in SUSY models extended to include the axion, the lightest neutralino may not be the LSP but can decay to lighter particles of these theories, such as the gravitino $\tilde{G}$ or the axino $\tilde{a}$. That is, the neutralino is now the next-to-lightest supersymmetric particle (NLSP), while the gravitino or the axino serves as the LSP [6, 7, 10,11,12].

In the latter scenarios, the decay widths of the NLSP neutralino to $\tilde{G}$ and $\tilde{a}$ are generally suppressed, either by the Planck mass $m_{\textrm{Pl}}$ or by the axion decay constant $f_a$, such that the lifetime of the NLSP can be much longer than its freeze-out time scale. In this case, the decay of the NLSP can also generate an axino or gravitino population. Provided that the reheating temperature is low enough to avoid significant production of $\tilde{G}$ or $\tilde{a}$ from thermal scattering in the very early Universe [13], it is the late-time NLSP-to-LSP decay process that dominates the final $\tilde{G}$ or $\tilde{a}$ abundance.

Then, the relic LSP production can be thought of as a two-stage process. First, a neutralino NLSP population is produced by interactions with the SM.^{Footnote 1} Such a neutralino population with the right observed relic density can build up either via the usual thermal freeze-out mechanism through annihilations with SM particles, or via freeze-in if extremely-weakly coupled to the parent SUSY and other SM particles. Irrespective of the details of this first step, at very late times the neutralino NLSP decays into the gravitino or axino LSP, constituting the second step of the LSP production process. The gravitino or axino LSP thus generated – dubbed as SUSY superWIMPs in the literature – can provide part or all of the observed dark matter in the Universe [6, 16,17,18].

Because superWIMPs are extremely weakly coupled – the interactions of the gravitino and axino are $m_{\textrm{Pl}}$- or $f_a$-suppressed – prompt searches at colliders are insensitive to a large part of their parameter space. Only a small sliver can potentially be probed [11, 12, 18,19,20,21,22,23,24,25,26, 26,27,28,29], via searches for long-lived particles by ATLAS, CMS, or future experiments such as FASER [30] and Mathusla [31].^{Footnote 2} In a similar vein, bounds on the superWIMP parameter space from direct and indirect dark matter searches are practically non-existent. Our best prospects for probing and constraining superWIMP scenarios lie in cosmological observations.

A number of early studies have considered how gravitino superWIMPs can be probed cosmologically [6, 7, 22, 34,35,36], based primarily on the premise that electromagnetic and/or hadronic energy released from the NLSP-to-LSP decay has consequences for the light element yields from BBN and the CMB black-body energy spectrum. Using measurements of the Deuterium and Helium-4 abundances, as well as the COBE-FIRAS constraint on $\mu $-type spectral distortions, stringent constraints on the gravitino superWIMP parameter space can be set for NLSP lifetimes in the range $t_{\textrm{NLSP}} \sim 10^4 - 10^8$ s. Similar considerations have also been applied to the axino superWIMP scenario, wherein a frozen-out neutralino or stau decays to an axino accompanied by an SM particle [10, 11, 24,25,26,27, 36, 37]. Independently of whether a particular axion-axino superfield realisation solves the strong CP problem, if the axino is the LSP, cosmological data can be expected to constrain a large part of the parameter space.

In this work, we extend these early analyses to include constraints from the CMB temperature and polarisation anisotropies from the Planck CMB mission [38]. Just as they impact on the light elements and the CMB energy spectrum, electromagnetic energy injections from NLSP decay can likewise have drastic consequences for the reionisation history of the Universe. Energy injection over NLSP lifetimes of $t_{\textrm{NLSP}} \sim 10^{10}$–$10^{24}$ s, in particular, can significantly modify the evolution of the free-electron fraction in the cosmic plasma, altering the CMB anisotropies in ways that are strongly disfavoured by current anisotropy measurements [38, 39]. As we shall demonstrate, this in turn allows us to place stringent constraints on large swathes of superWIMP parameter space previously considered viable, providing a powerful complement to conventional SUSY dark matter searches at colliders as well as at direct and indirect dark matter detection experiments.^{Footnote 3}

The paper is organised as follows. We begin in Sect. 2 with a summary of some well-motivated SUSY superWIMP scenarios amenable to the observational and experimental constraints of this work. In Sect. 3 we describe how these constraints can be applied to derive limits on the superWIMP parameter space, starting with cosmological observations and concluding with collider searches. Sect. 4 summarises the limits thus derived on the gravitino and axino superWIMP parameter space, assuming the initial NLSP abundance matches the observed DM abundance. We conclude in Sect. 5. Appendix Aoutlines the derivation of the LSP momentum distribution expected from NLSP-to-LSP decay, while Appendices B and C discuss scenarios in which the NLSP population is under- or over-produced.

2 SUSY superWIMPs

The general mechanism of superWIMP production in the early Universe is straightforward. Heavier SUSY particles undergo cascade decays to lighter SUSY particles and eventually to the NLSP. The NLSP then freezes out, typically at $x_{\textrm{f}} \equiv m_{\textrm{NLSP}}/T\sim 25$–30, with a yield $Y_{\textrm{NLSP}} \equiv n_{\textrm{NLSP}}/s$, where $n_{\textrm{NLSP}}$ is the NLSP number density and s is the entropy. Long after freeze-out, the NLSP decays to the LSP. Assuming the decay is complete, $Y_{\textrm{LSP}} \simeq Y_{\textrm{NLSP}}$, and the final LSP abundance is simply given by $\Omega _{\textrm{LSP}} h^{2} \simeq (m_{\textrm{LSP}}/m_{\textrm{NLSP}})\, \Omega _{\textrm{NLSP}} h^{2}$, where h is the reduced Hubble parameter. In most superWIMP scenarios, $m_\textrm{NLSP}\simeq m_{\textrm{LSP}}$, such that LSP inherits the same abundance as the NLSP. A large mass difference is however not precluded and can be a means to relax constraints on the LSP parameter space from relic density considerations.

For SUSY superWIMPs, if there exists a thermal production mechanism in the early Universe generating an abundance proportional to the reheating temperature, then the total superWIMP abundance today is simply the sum of the thermal population and the population arising from NLSP decay (“non-thermal”), i.e.,

$$\begin{aligned} \Omega _{\textrm{LSP}}h^{2} = \Omega _{\textrm{LSP}}^\textrm{thermal}h^{2} + \Omega _{\textrm{LSP}}^\mathrm {non-thermal}h^{2}. \end{aligned}$$

(1)

In general, however, to generate by thermal means a GeV-to-TeV-mass SUSY superWIMP population to match the observed dark matter abundance requires a reheating temperature in excess of $T_{\textrm{rh}} \sim 10^{10}$ GeV [13]. Thus, if the reheating temperature turns out to be low, the production of a sizeable population of superWIMP relics must rely entirely on NLSP-to-LSP decay. For axinos in the mass range $\mathcal {O}(\mathrm{MeV-GeV})$, thermal scatterings can account for the correct relic density for reheating temperatures ($T_{rh}$) of about $10^{4}$ GeV or lower [10].^{Footnote 4}

At this stage it is important to emphasise that, within the general Minimal Supersymmetric Standard Model (MSSM), the mechanism of thermal neutralino freeze-out that generates the right relic abundance is quite restricted given collider and electroweak precision observables, as well as constraints on the Higgs and Z-boson invisible widths [14, 15, 48,49,50,51]. The mechanism of thermal freeze-out for a relic neutralino depends on the nature of the gauge composition of the neutralino; for a comprehensive recent summary, see, e.g., [15]. If the neutralino is light, i.e., $\textrm{m}_{\chi _{1}^{0}}\lesssim 100 ~\textrm{GeV}$, limits on the charged components of the neutralino sector demand that the light neutral component $\chi _{1}^{0}$ be predominantly bino.

Then, imposing the Planck-inferred dark matter density, $\Omega _{\textrm{DM}} h^{2}\lesssim 0.12$ [38], on the neutral relic density leads immediately to a lower limit of $m_{\chi _{1}^{0}}\gtrsim 34$ GeV on the neutralino mass [14]. Note however that for a decaying neutralino the lower limit is relaxed due to the dilution of the relic abundance. The precise lower limit will depend on the amount of dilution, along with the underlying mechanism and abundance of the relic neutralino. We use this bound as an indicative limit, but do not strictly enforce it. Thus, on the light neutralino side, assuming a thermal freeze-out mechanism the two places with maximally efficient enhancements in the annihilation cross-section so as not to overclose the Universe are at the Z-funnel and the Higgs funnel regions [14].^{Footnote 5} For a predominantly bino LSP, the cross-section $\sigma $ for annihilation for light neutralinos to fermions f for the Higgs/Z funnel region is given by

$$\begin{aligned} \sigma (s)_{\chi _{1}^{0}\chi _{1}^{0}\rightarrow f \bar{f}}\simeq & {} \frac{N^{2}_{\chi _{1}^{0}-Z/h}}{m_{Z/h}} \sqrt{1- \frac{4m_{\chi _{1}^{0}}^{2}}{s}} \nonumber \\{} & {} \times \frac{s\Gamma _{Z/h}}{(s- m_{Z/h}^{2} )^{2} + (m_{h/Z}\Gamma _{Z/h})^{2} }, \end{aligned}$$

(2)

where s represents the centre-of-mass energy, $m_{Z/h}$ the mass the Z/h boson, $\Gamma _{Z/h}$ the partial width of Z/h boson to fermions, and $N_{\chi _{1}^{0}-Z/h}$ denotes the coupling of neutralinos to Z/h bosons. Direct detection constraints however rule out a significant part of the neutralino parameter space in the 10 GeV-to-1 TeV mass range [9, 14, 33, 53], with limits depending on the specifics of the model parameters. In general spin-independent limits from the Xenon-nT/LZ direct detection experiments [49] are quite constraining in the light dark matter scenario ($\textrm{m}_{\chi _{1}^{0}}\lesssim 200~\textrm{GeV}$), leaving viable the Z/H funnel regions.^{Footnote 6}

At higher masses, depending on the gauge content of the neutralino and the SUSY mass spectrum, a variety of new annihilation mechanisms can open up. Given the strong limits from collider searches, the most promising scenarios proceed through co-annihilations with sleptons and squarks. If light staus $\tilde{\tau }$ or stops $\tilde{t}$ appear in the t-channel, the annihilation cross-section for a predominantly bino neutralino scale as

$$\begin{aligned} \sigma (s)_{(\chi _{1}^{0}\chi _{1}^{0}\rightarrow f\bar{f})}\simeq \frac{g_{W}^{4}s}{64\pi m^{2}_{\tilde{t}(\tilde{\tau )}}}, \end{aligned}$$

(3)

where $g_{W} = (8/\sqrt{2}) G_F m_W^2$, with Fermi constant $G_F$ and W-boson mass $m_W$.) In this case, co-annihilations aided by Sommerfeld enhancements can lead to the correct relic density [9, 54].

We also add that if the neutralino has a sizable Higgsino component, a TeV scale Higgsino can generate the relic abundance of the Universe through co-annihilation with nearly mass-degenerate charginos [55]; this scenario requires the so-called “well-tempered” neutralino, a right admixture of bino and Higgsino for efficient annihilation [55, 56]. Since the charginos are TeV scale electroweak gauginos, collider limits can be evaded if the rest of the SUSY spectrum is decoupled as in the Split SUSY cases [57]. These considerations are generally encoded within the idea of the so-called relic neutralino surface [9]. General phenomenological and simplified MSSM model studies for the electroweakino sector using LHC data have shown that large swathes of parameter space are allowed within the gaugino sector, implying that there is no generic model-independent lower bound on the light neutralino [33]. The situation is relaxed further in non-minimal models like the next-to-Minimal Supersymmetric Standard Model (NMSSM) or non-universal Gaugino Models (NUGM).

Lastly, we note that, in models with over-abundant dark matter (e.g., models involving a light bino-like neutralino), the superWIMP mechanism is a way to dilute the final relic abundance. We emphasise, however, that, unless otherwise specified, our assumption throughout the present analysis is that the NLSP is always produced with the correct relic abundance prior to decay. That is, had the NLSP been stable, its present-day reduced energy density would be equal to $\Omega _{\textrm{DM}}$. This assumption also implies that, in those scenarios where a large mass gap exists between the NLSP and the LSP, the latter would constitute only a fraction of the observed dark matter today; the remainder would have to be explained by some other physics.

In what follows, we briefly describe two well-motivated SUSY superWIMPs, the gravitino $\tilde{G}$ and the axino $\tilde{a}$. As we shall see in Sect. 4, irrespective of the freeze-out/freeze-in mechanism that produces the NLSP neutralino, energy injection constraints of PMSSM and BBN, coupled with free-streaming bounds from the Ly$\alpha $ data will constrain the bulk of their parameter spaces.

2.1 Gravitino superWIMPs

Gravitinos $\tilde{G}$ are spin-$\nicefrac {3}{2}$ superpartners of gravitons. Depending on the SUSY breaking mechanism, the gravitino mass – given approximately by $m_{\tilde{G}} \simeq \langle F \rangle /m_{\textrm{pl}}$, where $\langle F \rangle $ is the SUSY breaking scale – can range from keV to TeV and is thus essentially a free parameter in this study. Because interactions of the gravitino are $m_{\textrm{Pl}}$-suppressed, we do not expect them to be efficiently produced via scattering in the early Universe unless the reheating temperature is high [13].

Production from NLSP decay can proceed via the decay of the lightest neutralino $\chi _{1}^{0}$. Stringent BBN constraints on hadronic energy injection from the decays $\chi _{1}^{0}\rightarrow \tilde{G} h/Z$ essentially rules out a predominantly wino- or Higgsino-like neutralino [6, 16]. Then, what remains is a bino-like neutralino, which decays into a gravitino predominantly via the two-body decay $\chi _{1}^{0}\rightarrow \tilde{G} \gamma $, whose width is given by [6]

$$\begin{aligned} \Gamma (\chi _{1}^{0} \rightarrow \tilde{G} \gamma ) = \frac{m_{\chi _{1}^{0}}^{5} \cos ^{2}\theta _{\textrm{W}}}{48 { \pi } m_{\textrm{Pl}}^{2}m_{\tilde{G}}^{2}}\left( 1 - \frac{m_{\tilde{G}}^{2}}{m_{\chi _{1}^{0}}^{2}} \right) ^{3}\!\! \left( 1 + 3\frac{m_{\tilde{G}}^{2}}{m_{\chi _{1}^{0}}^{2}} \right) ,\nonumber \\ \end{aligned}$$

(4)

where $m_{Pl}$ is the reduced Planck mass, and $\theta _{\textrm{W}}$ is the weak mixing angle.

Assuming decay at rest and that the energy carried by the photon, $E_{\gamma }= (m_{\chi _{1}^{0}}^{2} - m_{\tilde{G}}^{2})/(2m_{\chi _{1}^{0}})$, is injected entirely into the cosmic plasma, it is convenient to recast the width (4) as

$$\begin{aligned} \begin{aligned} \Gamma (\chi _{1}^{0}\rightarrow \tilde{G} \gamma )&= \frac{m_{\chi _{1}^{0}}^{3} \cos ^{2}\theta _{\textrm{W}}}{3 \pi m_{\textrm{Pl}}^{2}} \epsilon _{\textrm{em}}^3 \frac{2-3 \epsilon _{\textrm{em}}}{1- 2 \epsilon _{\textrm{em}}} \\&\simeq 2.2 \times 10^{-14}~{\textrm{s}}^{-1} \, \epsilon _{\textrm{em}}^3 \frac{2-3 \epsilon _{\textrm{em}}}{1- 2 \epsilon _{\textrm{em}}} \left( \frac{m_{\chi _1^0}}{\textrm{GeV}} \right) ^3, \end{aligned}\nonumber \\ \end{aligned}$$

(5)

with

$$\begin{aligned} \epsilon _{\textrm{em}} \equiv \frac{E_{\gamma }}{m_{\chi _{1}^{0}}}=\frac{ m_{\chi _{1}^{0}}^{2} - m_{\tilde{G}}^{2}}{2m^2_{\chi _{1}^{0}}} \end{aligned}$$

(6)

denoting the fraction of the neutralino mass released as electromagnetic energy. Where kinematically allowed, the additional decay channels $\chi _{1}^0 \rightarrow \tilde{G}Z/h$ are also available. But, as said above, these channels are suppressed for a bino-like $\chi ^{0}_{1}$. Note that maximal energy injection is represented by $\epsilon _{\textrm{em}}\rightarrow 0.5$, which occurs as $m_{\tilde{G}}\rightarrow 0$.^{Footnote 7}

2.2 Axino superWIMPs

Axinos $\tilde{a}$ are the supersymmetric partners of the axion – the dynamic field expected to solve the strong CP problem – and appear in the axion supermultiplet after the Peccei-Quinn (PQ) symmetry breaking in the form $A = (s +i a)/\sqrt{2} + \sqrt{2}\theta a + \theta ^{2}F$, where a is the axion, s the saxion,^{Footnote 8}F the auxiliary superfield, and $\theta $ is the Grassmanian coordinate. The axion couples derivatively to quarks and to the gauge bosons with interactions suppressed by the PQ breaking scale $f_{a}$; the accompanying SUSY interactions can be found by simply supersymmetrising the effective SM-axion interactions, i.e., the axion supermultiplet A couples to the vector supermultiplet $V_{a}$. The axion supermultiplet acquires a mass after SUSY is broken. While the saxion mass is roughly set by the the soft SUSY breaking scale, the axino mass depends on the superpotential. For the purposes of this work, we will take the axino mass to be a free parameter, and note that its mass can range from eV to TeV scales.

Like gravitinos, axinos can be produced in the early Universe in abundance via thermal scattering if the reheating temperature is large [59]. However, if the axino is the LSP, production from the decay of a NLSP neutralino population is also possible. Assuming a (pure) bino decay, the decay width is given by [11, 12]

$$\begin{aligned} \begin{aligned} \Gamma \left( \chi _1^0 \rightarrow \tilde{a} \gamma \right)&= {\left( \frac{\alpha ^{2}}{4\pi }\right) } C_{aYY}^2 \frac{m_{\chi _{1}^{0}}^{3}}{4 \pi ^{2} {f'_{a}}^{2} \cos ^{2}\theta _{\textrm{W}}} \epsilon _{\textrm{em}}^{3}\\&\simeq 2.1 \times 10^{-15}~{\textrm{s}}^{-1} \\&\quad \times C_{aYY}^2 \epsilon _{\textrm{em}}^{3} \left( \frac{m_{\chi _1^0}}{\textrm{GeV}} \right) ^3 \left( \frac{\textrm{GeV}}{f'_a/10^{16}} \right) ^2. \end{aligned}\nonumber \\ \end{aligned}$$

(7)

Here, $f'_{a} \equiv f_{a}/N$, where the factor $N=1$ and $N=6$ applies to the KSVZ and DFSZ axion, respectively; the coefficient $C_{aYY}$ is a model-dependent $\mathcal {O}(1)$ number [$\sim 100$ MeV in the case of BBN and $\sim 1$ MeV in the case of CMB. To illustrate the power of these constraints, the black dashed lines indicate predictions for the gravitino superWIMP scenario, based on Eq. (5), for several neutralino mass values, assuming that the neutralino population has been produced at the correct relic abundance prior to decay

Full size image

3.1 Light element abundances

The main effect of electromagnetic energy injection into the plasma on the abundances of the light elements (Deuterium $^2$H, Helium-3 $^3$He, and Helium-4 $^4$He) from BBN is photo-dissociation, provided that the injected energy exceeds the reaction threshold (typically ${{\mathcal {O}}}(2$–30) MeV; see, e.g., Table 1 of Ref. [$\tau \simeq 10^{12}$ s in the figure is simply due to the lack of calculations in this parameter region in the literature; we are otherwise not aware of any fundamental reason why BBN constraints could not be extended to longer decay lifetimes.

To illustrate the power of energy injection constraints, we have also plotted in Fig. 1 predictions for the gravitino superWIMP scenario, based on Eq. (5), for a number of $\chi _{1}^{0}$ masses from $m_{\chi _1^0}=1$ GeV and up to the unitarity limit of $m_{\chi _1^0}=100$ TeV.^{Footnote 10} We note however that the unitarity bound is indicative, since the bound will be relaxed due to the decaying neutralino. We emphasise that for a given mass hierarchy between the neutralino and the gravitino, the amount of energy released and the lifetime are fixed by Eq. (5). That is, there are no further variables that can affect the gravitino superWIMP predictions in Fig. 1. As mentioned earlier, the maximal fractional energy released can never be exactly equal to 0.5, which would correspond to a massless gravitino.

3.2 CMB spectral distortions

Electromagnetic energy injection in the early Universe must also perturb the Planck blackbody energy spectrum of the CMB photons, creating a spectral distortion. If both photon number-changing processes, (e.g., double Compton scattering and Bremsstrahlung) and energy-changing processes (e.g., Compton scattering) occur efficiently, then these distortions are quickly wiped out, leaving no trace of the decay in the CMB energy spectrum besides a temperature shift (which is unobservable by spectral measurements, but may be detectable in the anisotropies via $N_{\textrm{eff}}$). However, should these processes be inefficient at the time of energy injection and remain inefficient until the present time, the spectral distortions they cause may freeze in and become observable.

A detailed review of CMB spectral distortions can be found in, e.g., [69, $t \gg 10^{-13}$ s. Nonetheless, increased ionisation of the intergalactic medium between the epochs of recombination ($z \sim 1100$) and reionisation ($z \sim 10$) raises the optical depth, which manifests itself most prominently in a stronger E-polarisation signal at multipoles $\ell \sim 20$ [$\sim 1$ MeV and that the energy is deposited in the plasma instantly upon injection.

3.4 Lyman-$\alpha $ forest

The production of superWIMPs via NLSP-to-LSP decay is always accompanied by a finite momentum for the LSP, given at the production time $t_{\textrm{prod}}$ by $p_{\textrm{LSP}}(t_{\textrm{prod}})= p_\star = (m^2_{\chi _1^0} - m_{\textrm{LSP}}^2)/(2 m_{\chi _1^0})=\epsilon _{\textrm{em}} m_{\chi _1^0}$, assuming decay at rest. Universal expansion causes $p_{\textrm{LSP}}$ to redshift subsequently as $p_{\textrm{LSP}}(t)= p_\star R_{\textrm{prod}}/R(t)$, where R(t) is the scale factor and $R_{\textrm{prod}}\equiv R(t_{\textrm{prod}})$; at a later time t the corresponding LSP velocity therefore reads

$$\begin{aligned} v_{\textrm{LSP}}(t)= & {} \frac{p_\star }{\sqrt{p_\star ^2+ m^2_{\textrm{LSP}} R^2(t)/R^2_{\textrm{prod}}}}\nonumber \\= & {} \frac{1}{\sqrt{1+R^2(t)/v_0^2}}, \end{aligned}$$

(8)

where

$$\begin{aligned} \begin{aligned} v_0 \equiv \epsilon _{\textrm{em}} \frac{m_{\chi _1^0}}{m_{\textrm{LSP}}} R_{\textrm{prod}} = \frac{\epsilon _{\textrm{em}}}{\sqrt{1-2 \epsilon _{\textrm{em}}}} R_{\textrm{prod}} \end{aligned} \end{aligned}$$

(9)

is the present-day ($R(t_0)=1$) LSP velocity, under the assumption that $m_{\textrm{LSP}} \gg p_{\textrm{LSP}}(t_0)$ holds.

Because the NLSP-to-LSP decay is isotropic, the overall effect of a finite $v_{\textrm{LSP}}$ is that of isotropic LSP free-streaming. Furthermore, because NLSP-to-LSP decay is a continuous process, together with a redshifting $p_{\textrm{LSP}}$ we can expect a present-day comoving LSP number density $n_{\textrm{LSP}}(t_0)$ to be composed of a distribution of particles in momentum space, given approximately by

$$\begin{aligned} \begin{aligned}&n_{\textrm{LSP}}(t_0) \simeq \frac{\Omega _{\textrm{DM}} \rho _{\textrm{crit}}}{m_{\chi _1^0}} \\&\quad \times \left\{ 2\int _0^{p_\star R_{\textrm{eq}}} \textrm{d} \ln p \, \left( \frac{p}{p_\star R_\Gamma } \right) ^2 \exp \left[ -\left( \frac{p}{p_\star R_\Gamma } \right) ^2 \right] \right. \\&\quad \left. +\frac{3}{2}\int _{p_\star R_{\textrm{eq}}}^{p_\star } \textrm{d} \ln p \, \left( \frac{p}{p_\star R_\Gamma } \right) ^{3/2} \exp \left[ -\left( \frac{p}{p_\star R_\Gamma } \right) ^{3/2} \right] \right\} , \end{aligned}\nonumber \\ \end{aligned}$$

(10)

where the prefactor assumes the $\chi _1^0$ population was produced with the correct relic abundance, $R_\Gamma $ is the scale factor corresponding to NLSP lifetime $t=\tau \equiv 1/\Gamma $, and $R_\textrm{eq}$ is the scale factor at matter-radiation equality. See Appendix A for the derivation of Eq. (10). Thus, phenomenologically, the LSP population today is dispersive and akin to a warm dark matter (WDM) with a present-day characteristic velocity given approximately by $v_0 \sim p_\star R_\Gamma /m_{\textrm{LSP}}$. This also means that limits on thermal WDM properties from small-scale fluctuation measurements such as the Lyman-$\alpha $ forest can be reinterpreted to constrain properties of the LSP and the corresponding superWIMP parameter space [74].

Several recent studies have investigated how to map thermal WDM bounds from the Lyman-$\alpha $ forest to constraints on WDM from particle decays, e.g., [75, 76]. Roughly, one would compute the linear transfer function of the WDM given the daughter particle’s phase space distribution, and match it to the thermal WDM transfer function that exhibits free-streaming suppression at approximately the same location in wave number k. Here, however, we adopt the simpler approach of estimating the comoving particle horizon – also called the free-streaming horizon – of the LSP population.

The free-streaming horizon arises naturally in semi-analytical solutions of the Vlasov equation that governs the evolution of the linear transfer function. Its inverse correlates with the location in k-space of power suppression due to free-streaming, and can therefore be seen as a proxy for the transfer function. For a LSP produced at a fixed time $t_{\textrm{prod}}$ and observed later at a time $t_{\textrm{obs}}$, the free-streaming horizon is given by

$$\begin{aligned} \lambda _{\textrm{FS}} (t_{\textrm{obs}},t_{\textrm{prod}})= & {} \int _{t_\textrm{prod}}^{t_{\textrm{obs}}} \frac{\textrm{d}t}{R(t)} v_{\textrm{LSP}}(t)\nonumber \\= & {} \int ^{R_{\textrm{obs}}}_{R_{\textrm{prod}}} \frac{\textrm{d}R}{R^2 H(R)} v_{\textrm{LSP}}(R), \end{aligned}$$

(11)

where $t_{\textrm{obs}}$ corresponds typically to a low redshift of $z \sim 2$, $t_{\textrm{prod}}$ to some time prior to matter-radiation equality, and $H\equiv (1/R)(\textrm{d} R/\textrm{d} t)$ is the Hubble expansion rate. Because NLSP decay is a continuous process, in principle we must calculate $\lambda _{\textrm{FS}}$ for all possible production times $t_{\textrm{prod}}$ and then average it over the momentum distribution of Eq. (10). For simplicity, however, we assume all LSPs to be produced at $t_\textrm{prod}=1/\Gamma $; this is a reasonable approximation given that, as shown in Eq. (10), LSPs produced at $t_{\textrm{prod}}\sim 1/\Gamma $ – whose characteristic momentum is $\sim p_\star R_\Gamma $ – in any case dominate the LSP distribution today. Then, using Eq. (8) and setting $t_{\textrm{prod}} = 1/\Gamma $, Eq. (11) can now be rewritten as

$$\begin{aligned}{} & {} \lambda _{\textrm{FS}}(R_{\textrm{obs}}, R_\Gamma ) \nonumber \\{} & {} \quad = \frac{1}{H_0}\sqrt{\frac{R_{\textrm{eq}}}{\Omega _{\textrm{m}}}} \int _{R_\Gamma /R_{\textrm{eq}}}^{R_{\textrm{obs}}/R_{\textrm{eq}}} \frac{\textrm{d} y}{\sqrt{(1+y)[1+ (R_{\textrm{eq}}/v_0)^2 y^2] }} \nonumber \\{} & {} \quad \simeq 91 \, h^{-1}\textrm{Mpc}\; \sqrt{\left( \frac{3401}{1+z_{\textrm{eq}}} \right) \left( \frac{0.32}{\Omega _{\textrm{m}}} \right) } \nonumber \\{} & {} \qquad \times \int _{R_\Gamma /R_{\textrm{eq}}}^{R_{\textrm{obs}}/R_{\textrm{eq}}} \frac{\textrm{d} y}{\sqrt{(1+y)[1+ (R_{\textrm{eq}}/v_0)^2 y^2] }}, \end{aligned}$$

(12)

valid across the radiation and matter domination epochs, where the subscript “eq” denotes matter-radiation equality, $\Omega _{\textrm{m}}$ is the present-day reduced total matter density, and h is the reduced Hubble parameter defined via $H_0 = 100 \, h~\textrm{km} \, \textrm{s}^{-1} \, \textrm{Mpc}^{-1}$.

The typical Ly$\alpha $ WDM bound in the literature is presented as a lower limit on the WDM mass $m_{\textrm{WDM}}$, assuming that the WDM constitutes the entirety of the dark matter abundance of the Universe and that the WDM population follows a relativistic Fermi-Dirac distribution, with a temperature linked to the dark matter abundance, i.e.,

$$\begin{aligned} \Omega _{\textrm{WDM}} h^2= \left( \frac{T_{\textrm{WDM}}}{T_\nu }\right) ^3 \left( \frac{m_{\textrm{WDM}}}{94~\textrm{eV}} \right) , \end{aligned}$$

(13)

where $T_\nu $ is the temperature of the neutrino background ($T_{\nu ,0} = 1.95$ K). Fixing $\Omega _{\textrm{WDM}} h^2 = 0.12$ and using the current best limit $m_{\textrm{WDM}} \gtrsim 5.3$ keV (95% C.I.) [77], we find an upper bound on the present-day WDM temperature of $T_{\textrm{WDM},0} \lesssim 2.2 \times 10^{-5}$ eV, or equivalently, a bound on the present-day average WDM velocity of $v_{\textrm{WDM,0}} \simeq 3\, T_x/m_x \lesssim 1.2 \times 10^{-8}$. Following the steps outlined above and letting $R_\textrm{prod} \rightarrow 0$, it is straightforward to show that the equivalent upper limit on the comoving WDM free-streaming horizon at $z_\textrm{obs}=2$ is

$$\begin{aligned} \lambda _{\textrm{FS}}^{\textrm{WDM}} (z_{\textrm{obs}}=2) \lesssim 0.045 \, h^{-1} \textrm{Mpc} \; \sqrt{\left( \frac{3401}{1+z_{\textrm{eq}}} \right) \left( \frac{0.32}{\Omega _{\textrm{m}}} \right) }.\nonumber \\ \end{aligned}$$

(14)

This limit can in principle serve as an upper bound on $\lambda _\textrm{FS}(z_{\textrm{obs}}=2,z_\Gamma )$ for those superWIMP scenarios wherein the LSP explains all of the observed dark matter abundance.^{Footnote 12}

Table 1 Upper limit on the free-streaming horizon $\lambda _{\textrm{FS}}^{\textrm{WDM}}$ as a function of the WDM fraction $f_{\textrm{WDM}}$. These limits have been mapped from the two-dimensional 68%-credible contours in $(f_{\textrm{WDM}},m_\textrm{WDM})$-plane in Fig. 12 of Ref. [79], which dates from 2009. In the comparable case of $f_{\textrm{WDM}}=1$, we note that the limit $m_{\textrm{WDM}} \gtrsim 5.3$ keV (95% C.I.) quoted in the text comes from a more recent 2017 analysis [77] and translates to $\lambda _{\textrm{FS}}^{\textrm{WDM}} \lesssim 0.045\, h^{-1}$Mpc (see also Eq. (14)), in contrast to $\lambda _{\textrm{FS}}^{\textrm{WDM}} \lesssim 0.0708\, h^{-1}$Mpc tabulated below. The Ly$\alpha $ limits used in our analysis are therefore quite conservative

Full size table

A more versatile analysis of the Ly$\alpha $ data could however also consider the possibility of a mixed cold+warm dark matter cosmology and vary as part of the fitting procedure the fraction of the total dark matter in the form of WDM, $f_{\textrm{WDM}} \equiv \Omega _\textrm{WDM}/\Omega _{\textrm{DM}}$. Reference [79] has provided such an analysis and presented the outcome as two-dimensional constraints in the $(f_{\textrm{WDM}},m_{\textrm{WDM}})$-plane and in the $(f_{\textrm{WDM}},v_0)$-plane (see Fig. 12 of [79]). We have translated these constraints to constraints in the $(f_{\textrm{WDM}},\lambda _{\textrm{FS}}^{\textrm{WDM}})$-plane. See a representative set in Table 1. Observe that the upper limit on $\lambda _{\textrm{FS}}^{\textrm{WDM}}$ deteriorates as we decrease the WDM fraction $f_{\textrm{WDM}}$; at $f_{\textrm{WDM}} \lesssim 0.15$, no limit can be set on $\lambda _\textrm{FS}^{\textrm{WDM}}$. In our analysis of superWIMPs, we take the WDM energy density to be $\Omega _{\textrm{WDM}} \simeq (m_{\textrm{LSP}}/m_\textrm{NSLP}) \Omega _{\textrm{DM}}$, in accordance with our assumption that the NLSP population is always produced with the correct abundance prior to decay. It then follows simply that $f_{\textrm{WDM}} = m_\textrm{LSP}/m_{\textrm{NLSP}}$, with the proviso that the remaining $1-f_\textrm{WDM}$ of the dark matter is cold and explained by some other physics. We apply the Ly$\alpha $ constraints only to those cases where production takes place before matter-radiation equality, i.e., $1/\Gamma \le t_{\textrm{eq}}$, because using $\lambda _{\textrm{FS}}$ as a proxy for small-scale suppression is likely unreliable if the bulk of the NLSP decay happens deep in matter domination.^{Footnote 13}

3.5 Collider constraints

We assume from the outset that the neutralino is not ruled out by conventional jets/leptons + missing energy searches. In principle, within the scope of specific mass spectra, part of the parameter space can indeed be ruled out; however, this requires a larger global fit within specific simplified or full SUSY models. The GAMBIT collaboration has performed a global fit of the pMSSM within a reduced 7-parameter space – the relevant parameters being the trilinear couplings, Higgsino mass parameters, diagonal sfermion masses, and $\tan \beta $ – taking into account collider, direct detection and relic density observables [33]. They conclude that a large volume of the pMSSM parameter space remains unconstrained, with the neutralino mass ranging from the Z/H funnel region to the mulit-TeV scale. In an extended set-up that includes gravitinos, for a gravitino mass fixed at $m_{\tilde{G}}=1$ eV, best-fit points that take into account collider searches for the rest of the electroweak gaugino sector indicate that both light and heavy neutralinos remain unconstrained within a variety of simplified-model scenarios [80]. Scenarios like split SUSY models [57, 81] also predict mass spectra that are unconstrained by LHC searches. Thus, we re-emphasize that collider bounds on neutralinos are model independent and depends on the specifics of the mass spectrum and neutralino composition.

Bearing the above in mind, we summarise in this subsection collider constraints on the gravitino and the axino LSP originating from neutralino decay. The neutralino proper decay length to gravitino can be expressed following Eq. (5) as a function of the fractional energy $\epsilon _{\textrm{em}}$ and the neutralino mass $m_{\chi _{1}^{0}}$:

$$\begin{aligned} L= c\tau \simeq 1.4 \times 10^{22}~{\textrm{m}}\, \frac{1-2\epsilon _{\textrm{em}} }{\epsilon _{\textrm{em}}^{3} (2-3\epsilon _{\textrm{em}}) } \left( \frac{\textrm{GeV }}{m_{\chi _{1}^{0}}}\right) ^{3}. \end{aligned}$$

(15)

Collider searches – including searches of prompt decays that occur at the interaction vertex and are hence sensitive to photons plus missing energy signatures, and LLP searches sensitive to delayed decays – probe proper decay lengths of about 100 m. Thus, in order for colliders to be sensitive to the gravitino superWIMP scenario, it is a priori clear that a large mass hierarchy must exist between the neutralino and the gravitino.

The LEP experiment has placed a lower bound on the gravitino mass from the process $e^{+}e^{-} \rightarrow \tilde{G}\tilde{G}\gamma $ of $m_{\tilde{G}}\gtrsim 1.09 \times 10^{-5}$ eV [82]. Furthermore, under the assumption that the rest of the SUSY spectrum is decoupled apart from the selectron and the neutralino $\chi _{1}^{0}$, the LEP searches exclude a neutralino mass of up to $m_{\chi _{1}^{0}}\simeq $ 200 GeV for a gravitino mass of $m_{\tilde{G}}\lesssim 10^{-5}$ eV. At the LHC, gravitino searches have been conducted within the context of gauge mediated supersymmetry (GMSB) breaking models [83]. These searches look for displaced photons assuming a SUSY topology that yields the neutralino NLSP. Assuming a decay channel with maximal production cross-section in the $pp \rightarrow \tilde{q}\tilde{q}\rightarrow qq\chi _{1}^{0}\chi _{1}^{0}$ followed by the displaced photon signature of $\chi _{1}^{0}\rightarrow \tilde{G}\gamma $, the ATLAS experiment rules out neutralino masses in the range $\sim $ 100–400 GeV for $c\tau \sim 10$–$10^{4}$ cm [84]. The latest CMS result [85] at 13 TeV with 78 $\mathrm fb^{-1}$ luminosity excludes within these scenarios a neutralino mass in the range $m_{\chi _1^0}\sim 200$–550 GeV, for $c\tau \sim 10$–$10^{4}$ cm. We will use these experimental bounds for illustrative purposes, but emphasise that they are model-dependent.

Similar considerations apply to the axino superWIMP, in which case we have an additional degree of freedom in the neutralino decay width, namely the axion decay constant $f_a'\equiv f_a/N$. The proper decay length for the axino follows simply from Eq. (7):

$$\begin{aligned} L= c\tau \simeq 14.15~{\textrm{m}} ~ \epsilon _{\textrm{em}}^{-3} \left( \frac{f_{a}^{'}}{10^{8}~{\textrm{GeV}}}\right) ^{2}\left( \frac{100~{\textrm{GeV}}}{{\textrm{m}}_{\chi _{1}^{0}}}\right) ^{3}. \end{aligned}$$

(16)

As with the gravitino, collider constraints on the axino superWIMP scenario depend in general on the model specifics. Independently of the specifics, however, in order for colliders to be sensitive to the parameter space, the decay length should be $\lesssim \mathcal {O} (100)$ m. Within the context of specific models, estimates have been made on the capability of the LHC to probe axinos from neutralino decays in prompt and LLP searches [28]. In this work we will reinterpret existing LLP search results to put limits on this decay process.

Finally, since we are dealing with long-lived neutralinos, if they are light (GeV/sub-GeV), there is potentially sensitivity at fixed-target experiments such as CHARM [86, 87], NA62 [88], NOMAD [89], SHiP [90] and SEAQuest [91], as well as at forward physics facility experiments like FASER [30].

The production of neutralinos in fixed-target experiments depend on the specifics of the model. Typically in such experiments, a beam of particles with energies ranging from $\sim 100$ GeV (SEAQuest) to $\sim 450$ GeV (SHiP, NOMAD, NA62) collides with the target, thereby producing hadrons and weakly-interacting particles that could be captured in a far detector. If the rest of the SUSY spectrum is decoupled and heavy, neutralino production proceeds primarily from the decay of pseudo-scalar and vector mesons. These neutralinos then decay to gravitinos/axinos with decay widths given by Eqs. (4) and (7). Thus, the number of long-lived neutralino decays within a fixed-target experiment depends on the decay lifetime, the Lorentz boost factor, and the energy spectrum of the mesons specific to the experiment. While a full analysis for each experiment is beyond the scope of this work, a crude estimate suggests that the NOMAD experiment can exclude up to $m_{\chi _{1}^{0}}\simeq 300$ MeV for a fixed $m_{\tilde{a}}\simeq 20$ MeV, assuming $f_{a}\simeq 10^{3}$ GeV. For the same fixed value of $m_{\tilde{a}}$, a future experiment like SHiP can rule out $f_{a}=10^{4}$ GeV for $m_{\chi _{1}^{0}}\simeq 300$ MeV.^{Footnote 14}

In the case of FASER, a recent assessment of its feasibility to probe light axinos and gravitinos found that, for $m_{\tilde{a}}\simeq 10$ MeV, it is possible to rule out $f_{a} \simeq 10^{2}$–$10^{3}$ GeV for $m_{\chi _{1}^{0}}\simeq 300$ MeV [93].

4 Impact of constraints on the superWIMP parameter space

Having discussed the relevant cosmological and collider probes of superWIMPs, we are now in a position to present the constraints on the gravitino and the axino superWIMP parameter spaces, assuming that the neutralino has been produced with an abundance matching the observed DM density prior to decay. These are shown in Figs. 2 and 3 in the $(m_{\chi _1^0},m_{\textrm{LSP}})$- and $(m_{\chi _1^0},\epsilon _\textrm{em})$-projections, highlighting respectively the hierarchical ($m_{\chi _1^0} \gg m_{\textrm{LSP}}$) and degenerate ($m_{\chi _1^0} \simeq m_{\textrm{LSP}})$ regions (recall that $2\epsilon _{\textrm{em}} = \Delta m^2/m_{\chi _1^0}^2$, where $\Delta m^2 \equiv m_{\chi _1^0}^2 - m_{\textrm{LSP}}^2$ is the NLSP-LSP squared-mass gap). Detailed discussions of these results follow below. The equivalent constraints in the cases of an under- or over-production of neutralinos at 10%, 0.1%, and 10 times the observed DM abundance are presented in Appendices B and C.

4.1 Gravitino

Figure 2 shows the current and projected cosmological and collider constraints on the gravitino superWIMP parameter space in the $(m_{\chi _1^0},m_{\tilde{G}})$-plane (left panel) and the $(m_{\chi _1^0},\epsilon _{\textrm{em}})$-plane (right panel). A quick glance at both reveals that a large parameter region from the hierarchical to the degenerate limit is strongly constrained by a plethora of cosmological observations. In particular, we observe that for a neutralino mass fixed at $m_{\chi _{1}^{0}}=100$ GeV, the totality of cosmological observations rules out gravitino masess lying in the range

$$\begin{aligned} 0.8 \lesssim m_{\tilde{G}}/\textrm{GeV} \lesssim 99.998, \end{aligned}$$

(17)

while for $m_{\chi _{1}^{0}}=1$ TeV the exclusion region lies in the range

$$\begin{aligned} 270 \lesssim m_{\tilde{G}}/{\textrm{GeV}} \lesssim 999.997. \end{aligned}$$

(18)

As a rule of thumb, the lower end of these exclusion regions in the gravitino mass $m_{\tilde{G}}$ is driven by how short a neutralino lifetime $\tau $ a particular cosmological observation can probe with some precision, and is currently dominated by energy injection constraints from observations of the primordial $^2$H and $^3$He abundances. This limit can be seen most clearly in the left panel of Fig. 2 represented by the lower edge of the orange-shaded area at $m_{\chi _1^0} \lesssim 1$ TeV, and in the right panel by the upper edge of the orange-shaded area at $m_{\chi _1^0} \gtrsim 1$ TeV. A future CMB spectral measurement by a PRISM-like instrument will contribute to tightening the lower limit on $m_{\tilde{G}}$ primarily in the degenerate ($m_{\chi _1^0} \simeq m_{\tilde{G}}$) region at $m_{\chi _1^0}\gtrsim 1$ TeV (right panel, upper edge of the brown-shaded area).

On the other hand, the upper end of the exclusion ranges (17) and (18) is mainly determined by how long a decay lifetime an observation can probe, and in this regard, it is the energy injection limits from the Planck CMB anisotropy measurements that dominate the current constraints. The Planck CMB anistropy limit is most evident in the right panel of Fig. 2 (lower edge of the red-shaded area), and extends to lifetimes well in excess of $\tau \sim 10^{20}$ s, i.e., longer than the lifetime of the Universe thus far ($t\sim 4 \times 10^{17}$ s). Improvement from a future CMB anisotropy measurement by LiteBIRD+CMB-S4 in this region will however likely be marginal (right panel, lower edge of the brown-shaded area), as the reach in decay lifetime of these future observations is unlikely to outperform the status quo by more than a factor of a few (see also Fig. 1).

It is also interesting to consider what limits Ly$\alpha $ observations impose on the gravitino superWIMP parameter space; the corresponding exclusion region is shaded pink in both the left and right panels of Fig. 2. At face value the Ly$\alpha $ bounds do not appear to add much to the energy injection constraints already discussed above in either the $(m_{\chi _1^0},m_{\tilde{G}})$- or $(m_{\chi _1^0},\epsilon _\textrm{em})$-plane. However, while BBN and CMB observations probe the electromagnetic energy injected into the cosmic plasma, Ly$\alpha $ is sensitive to the free-streaming properties of the gravitino itself and therefore offers a somewhat different perspective on the superWIMP scenario. It also provides a useful constraint on those scenarios not considered in this work wherein the NLSP does not decay electromagnetically (e.g., decay into neutrinos). We devote several paragraphs below to explain the essential features of the Ly$\alpha $ constraint on the gravitino superWIMP parameter space.

Observe first in the left panel of Fig. 2 that Ly$\alpha $ rules out a substantial region around neutralino masses of ${{\mathcal {O}}}(1)$ GeV to ${{\mathcal {O}}}(1)$ TeV in the $(m_{\chi _1^0},m_{\tilde{G}})$-plane, and, like the energy injection constraints, provides an upper limit on viable values of $m_{\tilde{G}}$, albeit a much weaker one. This may at first glance appear counter-intuitive to the common understanding that Ly$\alpha $ observations limit the particle masses of free-streaming dark matter from below, to $m_{\textrm{WDM}} \gtrsim {{\mathcal {O}}}(1)$ keV. To reconcile these seemingly conflicting results, note first of all that thermal WDM mass constraints in the literature typically carry the assumption that the WDM makes up all of the dark matter content of the Universe. On the other hand, the upper limit on $m_{\tilde{G}}$ in Fig. 2 arises from the fact that once the WDM fraction – defined here as $f_\textrm{WDM}= m_{\tilde{G}}/m_{\chi _1^0}$ (see Sect. 3.4) – drops below $f_{\textrm{WDM}} <0.15$, the free-streaming properties of the gravitino LSP become unconstrainable by current observations (see also Table 1). In fact, this upper limit on $m_{\tilde{G}}$ parallels cosmological bounds on the absolute mass scale of Standard-Model neutrinos, where extremely small masses (and hence very large free-streaming scales) cannot be constrained because the energy density low-mass neutrinos contribute to the total dark matter content is too minute for their free-streaming effects to be observable.

Secondly, free-streaming is a kinematic effect dependent only on the characteristic velocity of the WDM in question. Its use as tool to constrain WDM masses to ${{\mathcal {O}}}(1)$ keV masses and above rests strongly on the assumption that the WDM has been produced via scattering processes with the SM thermal bath, such that the WDM population inherits the characteristic momentum (or temperature) of the bath even in the event that thermalisation is incomplete. In the case of production via NLSP decay, however, the characteristic momentum inherited by the LSP at production, $p_\star = \epsilon _{\textrm{em}} m_{\chi _1^0}$, is unrelated to the properties of the SM thermal bath; rather, it is determined by the NLSP mass – which, in our scenarios, is typically orders of magnitude larger than the SM bath temperature at the time of decay – and the NLSP-LSP mass gap. Consequently, the LSP masses constrained by velocity-based free-streaming arguments also fall in a range of orders of magnitude above naïve expectations.

Turning our attention now to the Ly$\alpha $ limits displayed in the right panel of Fig. 2, we see immediately that there is in fact also a lower limit on viable values of $m_{\tilde{G}}$, manifesting in the $(m_{\chi _1^0},\epsilon _\textrm{em})$-plane as a lower limit on viable values of $\epsilon _{\textrm{em}}$ at $m_{\chi _1^0} \gtrsim 1$ TeV (right edge of the pink-shaded region). To explain this limit, recall that in this plot, the energy injection is generally tiny, i.e., $\epsilon _{\textrm{em}} \ll 0.5$, such that the WDM fraction is saturated at $f_{\textrm{WDM}} = 1$ over the bulk of the displayed parameter region. Thus, the same $f_{\textrm{WDM}} = 1$ free-streaming horizon limit ($\lambda _{\textrm{FS}} \lesssim 0.0708\, h^{-1}$Mpc from Table 1) applies (almost) everywhere. Indeed, the right edge of the Ly$\alpha $ exclusion region aligns closely with the contours of constant free-streaming horizon $\lambda _{\textrm{FS}}$ in the right panel of Fig. 2 as expected. In contrast, the same edge in the $(m_{\chi _1^0},m_{\tilde{G}})$-plane in the left panel shows no such alignment; rather, the edge shifts to higher values of $\lambda _{\textrm{FS}}$ as we decrease $m_{\tilde{G}}$, reflecting the weakening of the Ly$\alpha $ limit on $\lambda _{\textrm{FS}}$ with decreasing $f_{\textrm{WDM}}$ evident in Table 1.^{Footnote 15}

Finally, we remark that collider experiments constrain only a very tiny region of the gravitino superWIMP parameter space, in a spot where an extremely large hierarchy exists between $m_{\chi _1^0}$ and $m_{\tilde{G}}$ so as to produce a detectable LLP signal (left panel of Fig. 2, purple-shaded region). Specifically, reinterpreting the ATLAS and CMS limits from Refs. [84, 85], which are presented on the $(m_{\chi _1^0},\tau )$-plane with GMSB SPS8 benchmarks as the reference model, we find that for neutralino masses up to $m_{\chi _1^0} \simeq 400$ GeV, current LHC data exclude gravitino masses up to $m_{\tilde{G}} \simeq 10^{-5}$ GeV. As discussed in Sect. 3.5, LLP search limits are model-dependent: thus the LHC exclusion region presented in Fig. 2 can move (within small confines) depending on the model specifics of the search analysis.

Note also that our LHC exclusion region has a vertical cut-off at $m_{\chi _1^0} \simeq 200$ GeV; this follows simply from the fact that Refs. [84, 85] had chosen to focus on a specific region of parameter space in their analyses. It is not inconceivable that current LHC data could constrain also some parts of the parameter region to the left of the cut-off; it is however extremely unlikely that the exclusion region extends all the way to $m_{\chi _1^0} =0$, as the search would become background-limited.

For completeness we estimate also the sensitivity of the high-luminosity LHC (HL-LHC) run, by simply rescaling the current LHC constraint to $3000~\textrm{fb}^{-1}$ integrated luminosity (left panel, blue-shaded region). We find that HL-LHC can improve the reach in gravitino masses by at most an order of magnitude to $m_{\tilde{G}}\simeq 10^{-4}$ GeV for neutralino masses up to $m_{\chi _1^0} \simeq 1$ TeV. We note however that the projection to HL-LHC originating from LLP searches depend on whether the neutralino is produced with enough cross-section and is not ruled out by jets + missing energy searches.

4.2 Axino

Analogously to Figs. 2, 3 shows the current and projected cosmological and collider constraints on the axino superWIMP parameter space in the $(m_{\chi _1^0},m_{\tilde{a}})$-plane (left panels) and the $(m_{\chi _1^0},\epsilon _{\textrm{em}})$-plane (right panels), assuming three different values of the axion decay constant $f_a'\equiv f_a/N \in [10^8, 10^{12}, 10^{16}]$ GeV. As in Fig. 2, overclosure excludes in principle $m_{\chi _1^0} \lesssim 34$ GeV if the neutralino is produced via thermal freeze-out; we have however omitted plotting this constraint in Fig. 3 since it can be easily circumvented by other production mechanisms and we in any case do not enforce it.

As can be gleaned from the neutralino-to-axino decay width (7), with $\epsilon _{\textrm{em}}$ held fixed, the decay lifetime scales as $\tau \propto {f_a'}^2/m_{\chi _1^0}^3$. Thus, in both the $(m_{\chi _1^0},m_{\tilde{a}})$- and $(m_{\chi _1^0},\epsilon _\textrm{em})$ planes, we generally expect experimental constraints (cosmological and collider) to apply to larger neutralino mass values as we crank up $f_a'$. Indeed, as is evident in the left panels of Fig. 3, the right edges of the cosmological exclusion regions (BBN, FIRAS+ Planck, and FutureCMB) mostly align with the lifetime reach of the observations at the short end. Where the neutralino and axino masses are strongly hierarchical (so that $\epsilon _{\textrm{em}} \simeq 0.5$), the decay lifetime also becomes independent of $m_{\tilde{a}}$. Consequently, the points at which the edges of the exclusion regions intersect the horizontal $m_{\chi _1^0}$-axis also scale as ${f_a'}^{2/3}$. As in the case of the gravitino superWIMP, the most stringent limit at the short end of $\tau $ comes currently from observations of the light element abundances (shaded orange), although there is also a small region in which BBN constraints are outperformed by Ly$\alpha $ observations (shaded pink).

Interestingly, for a large range of neutralino masses subject to cosmological energy injection constraints – specifically, where the aforementioned scaling of the $m_{\chi _1^0}$ bound applies – the same constraints also rule out a massless axino ($m_{\tilde{a}} =0$). It is only in the degenerate ($m_{\chi _1^0} \simeq m_{\tilde{a}}$ and $\epsilon _{\textrm{em}} \ll 0.5$) region where the excluded parameter space does not necessarily include $m_{\tilde{a}}=0$. This conclusion is independent of the assumed value of $f_a'$ and stands in stark contrast to the gravitino superWIMP case, where the lower end of the exclusion region is always finite in $m_{\tilde{G}}$, whose precise value depends on our choice of $m_{\chi _1^0}$ (see Eqs. (17) and (18)). Cosmological energy injection constraints therefore appear to have more drastic consequences for the axino than for the gravitino.

At the long end of the decay lifetime ($\tau \sim 10^{12}-10^{23}$ s), the constraints again originate primarily from the Planck measurements of the CMB anisotropies (although we remind the reader here that the BBN limits have been cut off in this region only because of the lack of available calculations in the literature, while for the Ly$\alpha $ constraints our estimates of the free-streaming horizon may not suffice to model the observable effects of this parameter region; see Sects. 3.1 and 3.4 for details). The limits imposed by the CMB anisotropies at this end are most easily discernible in the right panels of Fig. 3 showing the $(m_{\chi _1^0},\epsilon _{\textrm{em}})$-plane (lower edge of red-shaded area), and translate generally into an upper limit on the axino mass. Taking all observations into account, for the three values of $f_a'$ considered in this work and various fixed neutralino mass values, we find that cosmology constrains the axino mass $m_{\tilde{a}}$ to lie within the following ranges:

$f_{a}'=10^{16}$ GeV:
$$\begin{aligned} \begin{aligned}&m_{\chi _{1}^{0}}=100~\textrm{GeV}&:&\; 0\lesssim m_{\tilde{a}}/\textrm{GeV} \lesssim 99.992, \\&m_{\chi _{1}^{0}}=1~\textrm{TeV}&:&\; 0 \lesssim m_{\tilde{a}}/\textrm{GeV} \lesssim 999.98,\\&m_{\chi _{1}^{0}}=2.7~\textrm{TeV}&:&\; 0 \lesssim m_{\tilde{a}}/\textrm{GeV} \lesssim 2699.990,\\&m_{\chi _{1}^{0}}= 4~\textrm{TeV}&:&\; 530 \lesssim m_{\tilde{a}}/\textrm{GeV} \lesssim 3999.986, \end{aligned} \end{aligned}$$
(19)
$f_{a}'=10^{12}$ GeV:
$$\begin{aligned} \begin{aligned}&m_{\chi _1^0}=100~\textrm{GeV}&:&\; 98 \lesssim m_{\tilde{a}}/\textrm{GeV} \lesssim 100-5 \times 10^{-5}, \\&m_{\chi _{1}^{0}}=1~\textrm{TeV}&:&\; 998.9 \lesssim m_{\tilde{a}}/\textrm{GeV} \lesssim 1000-8 \times 10^{-5} \end{aligned} \end{aligned}$$
(20)
$f_{a}'=10^{8}$ GeV:
$$\begin{aligned} \begin{aligned}&m_{\chi _{1}^{0}}=10~\textrm{GeV}&:&\; 0.00017 \lesssim m_{\tilde{a}}/\textrm{GeV} \lesssim 0.000019. \end{aligned} \end{aligned}$$
(21)

Thus, in terms of ruling out a large chunk of interesting particle masses in the GeV-to-TeV region conventionally of interest to collider searches, cosmological constraints have the strongest impact for large axion decay constants in the vicinity of $f_{a}'=10^{16}$ GeV, ruling out neutralino masses of up to $\sim 3$ TeV and all possible axino masses up to the degenerate limit. The equivalent constraints in the same mass region are currently significantly weaker for smaller $f_a'$ values (i.e., $f_a'=10^{12}, 10^8$).

Having said the above however, we also observe that only for $f_{a}'\lesssim 10^{8}$ GeV does the neutralino lifetime become short enough to be sensitive to LLP searches at the LHC (bottom left panel of Fig. 3, purple- and blue-shaded areas). Specifically, for $f_a' = 10^8$ GeV, LLP searches at the LHC currently rule out neutralino and axino masses in the $\sim 50$–500 GeV region, which can be extended to $\sim 80$–800 GeV in the future by HL-LHC. As in the gravitino case, reinterpretation of published ATLAS and CMS limits from the model-specific analyses of [94], clockwork [95, 96], long-lived KK gravitons and radions [97], and continuum dark matter [98], etc. The largely model-independent analysis presented in this work can be easily adapted to investigate how precision cosmological observations impact on such BSM scenarios. We leave these studies for future works.

Data Availability Statement

This manuscript has no associated data or the data will not be deposited. [Authors’ comment: All data generated during this study are contained in this published article.]

Code Availability Statement

My manuscript has no associated code/software. [Authors’ comment: The code/software generated during and/or analysed during the current study is available from the corresponding author on reasonable request.]

Notes

The mechanics of this production depends on the nature of the neutralino. For a light neutralino $\chi _{1}^{0} \lesssim 100$ GeV, one requires a bino-like neutralino to have enough annihilations to avoid overclosing the Universe [14, 15].
This statement assumes the neutralino $\chi _{1}^{0}$ has not been ruled out by prompt jets/leptons + missing energy searches at the LHC. Within the general 19-parameter phenomenological MSSM (PMSSM) scenario [32], large regions of the parameter space are unconstrained [33]. This is discussed in more detail in Sect. 2.
We note that, in the recent years, several works have also invoked late-decaying dark matter to explain cosmological anomalies such as the Hubble and the $\sigma _{8}$ tensions [40,41,42]. Most of these scenarios do not generate significant electromagnetic emissions and are thus not subject energy injection constraints. The dominant constraints applicable to these scenarios arise from free-streaming effects and/or radiation excess (i.e., a non-standard $N_{\textrm{eff}}$ relative to the Standard-Model prediction of $N_{\textrm{eff}}^{\textrm{SM}}=3.0440 \pm 0.0002$ [43,44,45,46].
Production of dark matter from the decay of parent particles in thermal equillibrium with the SM bath to out-of-equillibrium LSPs (with negligible initial density) can also be occur within the freeze-in mechanism [47]. For gravitinos, it can be shown that for heavy scalar superpartners, a significant fraction of dark matter can be produced by the freeze-in from superpartners, largely independently of the reheating temperature [29]. A significant part of this parameter space is however constrained by LHC collider searches for superparticles. Axino freeze-in has been considered recently in [28] for Higgsino dark matter and for an axion decay constant $f_{a}\sim 10^{8}$–$10^{12}$ GeV in modified cosmological scenarios, with consequences for collider searches.
This rather stringent condition on the neutralino mass can be relaxed within the next-to-Minimal Supersymmetric Standard Model (NMSSM), where the presence of additional scalars ensure an efficient annihilation [52]. Alternatively, a non-thermal neutralino will also ensure that these limits are significantly weakened [14].
The limits are sensitive to the Higgsino mass parameter $\mu $. While the $\mu > 0$ parameter space is severely restricted, constraints on $\mu < 0$ are not as restrictive [14, 49].
The limit $m_{\tilde{G}} = 0$ is ill-defined within theories of SUSY breaking mechanisms. Since $m_{\tilde{G}}$ is related to the SUSY breaking scale $\langle F\rangle $, the $\langle F\rangle \rightarrow 0$ limit simply means a decoupled massless gravitino. Swampland conjectures relate it to the massless limit of an infinite tower of states and the breakdown of the effective field theory [58].
Although we will ignore the saxion for this work, saxions can also form superWIMPs and be subject to cosmological energy injection constraints.
If the misalignment angle is ${{\mathcal {O}}}(1)$, such as in the post-inflationary scenario, then to explain the observed dark matter abundance of the Universe fixes $f_{a}\sim 10^{12}$ GeV. However, in the “anthropic” or pre-inflationary scenario, the misalignment angle is random; in this case there is no upper limit on $f_a$.
The unitarity limit quoted here comes from Ref. [68], which argues as follows: in order for the neutralino to not exceed the upper limit on the relic abundance, the velocity-averaged cross-section must satisfy $\langle \sigma v\rangle \gtrsim 3\times 10^{-26} \,\mathrm cm^{3}\, \textrm{s}^{-1}$. For an s-wave annihilation with perturbative couplings, this imposes an upper limit on the neutralino mass of $m_{\chi _1^0}\lesssim 30$–100 TeV, depending on the nature of the interaction. We take $m_{\chi _1^0}\lesssim 100$ TeV as a conservative limit.
The analysis of Ref. [39], on which Fig. 1 is based, in fact used the full energy spectrum measurement from FIRAS (i.e., not just derived limits on the $\mu $ and y parameters) to put constraints on the $(\tau ,f_{\textrm{frac}})$-plane.
Constraints on the free-streaming properties of the dark matter can also be derived from the distribution of Milky Way satellite galaxies. Observations currently limit the thermal WDM mass to $m_{\textrm{WDM}} \gtrsim 6.5$ keV (95% C.I.) [78] assuming $f_{\textrm{WDM}}=1$, marginally better than the Ly$\alpha $ bound of Ref. [77]. We therefore do not consider Milky Way constraints here.
Since we must always compute $v_0$, the possibility also exists to constrain superWIMP scenarios using the thermal WDM limit on $v_{\textrm{WDM},0}$, instead of the more complicated $\lambda _\textrm{FS}^{\textrm{WDM}}$. Indeed, the limits in the superWIMP parameter space do turn out to be quite similar if we restrict $t_{\textrm{prod}}$ to the radiation-domination epoch. Conceptually though, we note that $v_0$ describes only the instantaneous free-streaming behaviour of the LSP population, while $\lambda _{\textrm{FS}}$ is able to capture a more complete free-streaming history; the difference between the two measures becomes more marked if LSP production takes place during matter domination. We emphasise however that neither measure is likely accurate if NLSP-to-LSP decay happens deep in the matter domination epoch, as the growth histories of the density perturbations in such scenarios – especially as the growth enters the nonlinear regime – deviate too strongly from the conventional thermal WDM scenario.
Also see detailed estimates provided by [92], with which we agree.
References [75, 76] have also derived Ly$\alpha $ constraints on particle decay scenarios. It is however difficult to make quantitative comparisons of our results with these previous works, because of the different assumptions involved. One major difference, for example, is that we always assume the NLSP to be produced in the correct abundance (or at some fixed abundance as explored in Appendices B and C), while [75, 76] demand that the decay product makes up the entire observed dark matter abundance irrespective of the original NLSP abundance, i.e., they “marginalise” over the NLSP abundance. As demonstrated in Appendix C, qualitatively, over-producing NLSP tends to shift (but not extend) the Ly$\alpha $ sensitivity region to smaller LSP masses. This understanding may explain why these earlier analyses find lower limits on the LSP mass in the keV region.
While it is commonly said that CMB anisotropy measurements constrain the dark matter density $\Omega _{\textrm{DM}} h^2$ to 2%-level uncertainty at 95% [38], this statement is in fact strongly dependent on the background cosmology and free parameters adopted in the parameter estimation analysis. Analyses that allow for a non-standard radiation content (i.e., $N_{\textrm{eff}} \ne 3.044$) and/or non-zero neutrino masses in particular typically yield significantly larger (up to a factor of four) error bars for $\Omega _{\textrm{DM}} h^2$ because of these parameters’ degeneracy with $\Omega _{\textrm{DM}} h^2$. See the Planck Legacy Archive (https://pla.esac.esa.int/) for a thorough exploration of how cosmological constraints vary with respect to model assumptions and/or data used.

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Acknowledgements

We thank Geneviève Bélanger, Daniel R. Green, Paul D. Jackson, and Matteo Lucca for useful discussions. Support for this work has been provided by the University of Adelaide and the Australian Research Council through the Centre of Excellence for Dark Matter Particle Physics (CE200100008). DS is partially supported through the the University of New South Wales, startup grant PS-71474. Y$^3$W is supported in part by the Australian Government through the Australian Research Council’s Future Fellowship (FT180100031) and Discovery Project (DP240103130).

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Department of Physics, ARC Centre of Excellence for Dark Matter Particle Physics, The University of Adelaide, Adelaide, SA, 5005, Australia
Meera Deshpande, Dipan Sengupta, Martin White & Anthony G. Williams
School of Physics, Sydney Consortium for Particle Physics and Cosmology, The University of New South Wales, Sydney, NSW, 2052, Australia
Jan Hamann, Dipan Sengupta & Yvonne Y. Y. Wong

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Meera Deshpande
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Correspondence to Dipan Sengupta.

Appendices

Appendix A: The LSP momentum distribution

Consider the decay process $\chi _1^0 \rightarrow \textrm{LSP} + \gamma $, whose rate in the rest frame of $\chi _1^0$ (which coincides with the cosmic frame) is $\Gamma $. The comoving number density of $\chi _1^0$ is

$$\begin{aligned} n_{\chi _1^0}(t) = \frac{\Omega _{\textrm{DM}} \rho _\textrm{crit}}{m_{\chi _1^0}} \exp \left( - \Gamma t\right) , \end{aligned}$$

(A1)

where $\rho _{\textrm{crit}}$ is the present-day critical density, and we have assumed $\chi _1^0$ to be produced at an abundance that would match the observed reduced dark matter density $\Omega _{\textrm{DM}}$ in the absence of decay. Demanding that the rate of change of the comoving LSP number density $n_{\textrm{LSP}}$ matches the negative rate of change of $n_{\chi _1^0}$, the evolution equation for $n_\textrm{LSP}$ can be written as

$$\begin{aligned} \frac{\textrm{d} n_{\textrm{LSP}}}{\textrm{d}t} = \Gamma n_{\chi _1^0} =\frac{\Omega _{\textrm{DM}} \rho _{\textrm{crit}}}{m_{\chi _1^0}} \Gamma \exp \left( - \Gamma t\right) , \end{aligned}$$

(A2)

or equivalently in terms of the scale factor R,

$$\begin{aligned} \frac{\textrm{d} n_{\textrm{LSP}}}{\textrm{d}\ln R} = \frac{\Omega _{\textrm{DM}} \rho _{\textrm{crit}}}{m_{\chi _1^0}} \left( \frac{R}{R_\Gamma } \right) ^n n \exp \left[ - \left( \frac{R}{R_\Gamma } \right) ^n\right] , \end{aligned}$$

(A3)

where $n=2,\nicefrac {3}{2}$ applies to radiation domination and matter domination respectively, and $R_\Gamma $ is the scale factor at the time $t = 1/\Gamma $.

Observe that, at the moment of production, the LSP always carries a physical momentum $p_\star = \epsilon _{\textrm{em}} m_{\chi _1^0}$, irrespective of the the time of of production. Universal expansion, however, will cause the momentum to redshift, such that an LSP produced at a scale factor R will be observed today uniquely with a physical momentum of $p=p_\star R$. Thus, in the same way that we use R as a proxy for time, we can also use the observed momentum p as a proxy for R, and recast Eq. (A3) as

$$\begin{aligned} \begin{aligned} \frac{\textrm{d} n_{\textrm{LSP}}}{\textrm{d}\ln p} = \frac{\Omega _{\textrm{DM}} \rho _{\textrm{crit}}}{m_{\chi _1^0}} \left( \frac{p}{p_\star R_\Gamma } \right) ^n n \exp \left[ - \left( \frac{p}{p_\star R_\Gamma } \right) ^n\right] . \end{aligned}\nonumber \\ \end{aligned}$$

(A4)

Approximating the transition from radiation to matter domination to be instantaneous at $R_{\textrm{eq}}$, LSP produced during radiation domination ($n=2$) can be observed today with a physical momentum up to $p_\star R_{\textrm{eq}}$, while those arising from decay during matter domination ($n=\nicefrac {3}{2}$) will have momenta in the range $p_\star R_{\textrm{eq}} < p \le p_\star $. Integrating Eq. (A4) under this approximation and assuming no pre-existing LSP population then yields Eq. (10).

Appendix B: Constraints on the superWIMP parameter space for under-production of neutralinos

Figures 4 and 5 show constraints in the $(m_{\chi _1^0},m_{\textrm{LSP}})$- and $(m_{\chi _1^0},\epsilon _{\textrm{em}})$-planes for a gravitino LSP and an $f_a = 10^{16}$ GeV axino LSP, respectively, assuming that the neutralino population is produced at only the 10% and 0.1% of the abundance required to match the observed DM energy density.

Relative to the case in which the neutralino abundance matches the observed DM abundance, under-production cases generally result in a shrinkage in the parameter regions constrained by BBN and CMB energy injection bounds. This shrinkage is primarily due to a reduced sensitivity to (i) the short lifetimes ($\tau \lesssim 10^4$ s) on the $(m_{\chi _1^0},m_{\textrm{LSP}})$-plane, and (ii) the energy injection fraction for extremely long lifetimes ($\tau \gg 10^{12}$ s). Observe also that the Ly$\alpha $ constraints have completely disappeared from Figs. 4 and 5. This is because an under-production of neutralinos also implies an under-production of gravitino and axino warm dark matter. At, e.g., 10% under-production of $\chi _1^0$, the WDM fraction is at most $f_{\textrm{WDM}} =0.1$, which falls below the smallest constrainable WDM fraction by the Ly$\alpha $ data of Table 1.

Appendix C: Constraints on the superWIMP parameter space for over-production of neutralinos

It is also interesting to see how the cosmologically-constrained regions change when the neutralino population is over-produced. Figures 6 and 7 show the corresponding constraints in the $(m_{\chi _1^0},m_{\textrm{LSP}})$-plane for a gravitino LSP and an $f_a = 10^{16}$ GeV axino LSP, respectively, assuming the neutralino population to be over-produced by a factor of 10 relative to the observed DM abundance.

Over-production scenarios are subject to two additional constraints: (i) the maximum mass of the LSP is always limited by $m_\textrm{LSP}/m_{\chi _1^0} \le 1/F_{\textrm{over}}$, where $F_{\textrm{over}}$ is the over-production factor (10 in the case of Figs. 6 and 7), and (ii) the NLSP-to-LSP decay must occur at a sufficiently early time such that no substantial DM and radiation excesses remain at CMB times to interfere with the standard expansion history. Condition (i) is straightforward to apply and immediately rules out the light grey region in Figs. 6 and 7. Furthermore, for an over-production factor of 10, the viable parameter regions under condition (i) – i.e., $m_{\textrm{LSP}}/m_{\chi _1^0} \lesssim 1/10$ – must yield an energy injection fraction no less than $\epsilon _{\textrm{em}} \simeq 0.5$ irrespective of $m_{\chi _1^0}$. In other words, the entire $(m_{\chi _1^0},\epsilon _{\textrm{em}})$-plane up to $\epsilon _{\textrm{em}} \simeq 0.5$ is excluded; for this reason we have chosen not to plot it.

Condition (ii) can be implemented as two separate constraints, one on the DM abundance and one on the radiation energy. For the former, we demand that combined neutralino and LSP abundance, given as a function of time by

$$\begin{aligned} \begin{aligned}&\Omega _{\textrm{LSP}} + \Omega _{\chi _1^0} \\&\quad = F_{\textrm{over}} \Omega _{\textrm{DM}} \left\{ \frac{m_\textrm{LSP}}{m_{\chi _1^0}} + \left[ 1- \frac{m_\textrm{LSP}}{m_{\chi _1^0}}\right] \exp (-\Gamma t) \right\} , \end{aligned}\nonumber \\ \end{aligned}$$

(C1)

not to exceed the observed DM abundance by more than a few percent at $z\lesssim 10^{4}$. That is, we demand that the DM abundance should already be more or less “correct” by the time the smallest scales probed by the CMB anisotropies enter the horizon, so as not create too much deviation in the evolution history of the anisotropies from the standard case. Since Eq. (C1) is a decreasing function of time, it suffices to demand approximate concordance at $z = 10^{4}$, i.e., the time point at which the impact of (C1) on the CMB anisotropies is the largest, and we impose a maximum deviation of 10%, motivated by a conservative 95% C.L. error on the observed DM abundance.^{Footnote 16} Then, the condition becomes a simple upper limit on the neutralino lifetime,

$$\begin{aligned} \tau \lesssim \dfrac{2 \times 10^{11}~\textrm{s}}{\ln \left[ \dfrac{m_{\chi _1^0}-m_{\textrm{LSP}}}{((1+x)/F_{\textrm{over}}) m_{\chi _1^0}-m_{\textrm{LSP}}} \right] }, \end{aligned}$$

(C2)

where the prefactor $2 \times 10^{11}$ s corresponds to the age of the universe at $z=10^4$, and $x=0.1$ for a 10% tolerance. For masses saturating $m_{\textrm{LSP}}/m_{\chi _1^0}=1/F_{\textrm{over}} = 1/10$, Eq. (C2) is equivalent to a constraint of $\tau \lesssim 4 \times 10^{10}$ s on the neutralino lifetime. For mass combinations not saturating the $1/F_{\textrm{over}}$ bound, the constraint on $\tau $ will be weaker, but not significantly so: indeed, in the limit $m_{\textrm{LSP}}/m_{\chi _1^0} \rightarrow 0$, the upper limit on the neutralino lifetime asymptotes to $\tau \lesssim 9 \times 10^{10}$ s. The vertically-hatched regions in Figs. 6 and 7 are excluded by this lifetime constraint consideration. Note that while we have imposed a tolerance $x=0.1$ on the DM abundance (C1), in practice the form of the neutralino lifetime constraint (C2) implies an extremely weak dependence of the excluded superWIMP parameter region on the choice of x, provided $x \ll 1$.

To implement the constraint on the excess radiation density in the CMB window, we note first of all that the time evolution of the excess radiation density from NLSP-to-LSP decay is described by

$$\begin{aligned} \begin{aligned} \frac{\textrm{d} \rho _{\textrm{em}}}{\textrm{d} t} + 4 H(t) \rho _{\textrm{em}}&=m_{\chi _1^0} \epsilon _{\textrm{em}} R^{-3} \frac{\textrm{d} n_{\textrm{LSP}}}{\textrm{d} t} \\&= F_{\textrm{over}} \Omega _{\textrm{DM}} \rho _{\textrm{crit}} \epsilon _{\textrm{em}} R^{-3} \Gamma \exp (-\Gamma t). \end{aligned}\nonumber \\ \end{aligned}$$

(C3)

Defining the comoving radiation energy density to be $\tilde{\rho }_{\textrm{em}} \equiv \rho _{\textrm{em}} R^{4}$ and adopting the scale factor R in place of the time variable t, Eq. (C3) can be recast into

$$\begin{aligned} \frac{\textrm{d} \tilde{\rho }_{\textrm{em}}}{\textrm{d} R} = F_{\textrm{over}} \Omega _{\textrm{DM}} \rho _{\textrm{crit}} \epsilon _{\textrm{em}} \left( \frac{R}{R_\Gamma } \right) ^n n \exp \left[ -\left( \frac{R}{R_\Gamma } \right) ^n\right] , \nonumber \\ \end{aligned}$$

(C4)

where $n =2,\nicefrac {3}{2}$ during radiation domination and matter domination, respectively. Evaluating Eq. (C4) for $n=2$ from 0 to $R_{\textrm{eq}}$ and then for $n=\nicefrac {3}{2}$ from $R_{\textrm{eq}}$ to $R>R_{\textrm{eq}}$, we find the solution

$$\begin{aligned} \begin{aligned} \tilde{\rho }_{\textrm{em}}&= F_{\textrm{over}} \Omega _{\textrm{DM}} \rho _\textrm{crit} \epsilon _{\textrm{em}} \\&\quad \times \left\{ \frac{1}{2} R_\Gamma \sqrt{\pi }\, \textrm{erf} \left( \frac{R_{\textrm{eq}}}{R_\Gamma } \right) -R_{\textrm{eq}} \exp \left[ -\left( \frac{R_{\textrm{eq}}}{R_\Gamma } \right) ^2\right] \right. \\&\quad \left. + \left. R' \left( \frac{R'}{R_\Gamma }\right) ^{3/2} E_{-2/3}\left[ \left( \frac{R'}{R_\Gamma }\right) ^{3/2} \right] \right| ^{R'=R_{\textrm{eq}}}_{R'=R}\right\} , \end{aligned}\nonumber \\ \end{aligned}$$

(C5)

where erf stands for the error function, and $E_n(z)\equiv \int _1^\infty \textrm{d}t\, e^{-z t}/t^n$ is the exponential integral function.

Assuming three SM neutrinos of fixed energy densities, the energy density in electromagnetic radiation during the CMB epoch is measured to a 95% uncertainty of $\sim 10\%$, motivated by the current $\sim 10\%$ uncertainty on the effective number of neutrinos $N_{\textrm{eff}} \equiv (1/0.227) (\rho _\nu /\rho _{\textrm{em}})$, depending on exactly how the parameter estimation analysis is done (see Footnote 16). Then, a simple constraint can be set on $R_{\Gamma }$ by demanding that $\tilde{\rho }_\textrm{em}/\rho _{\textrm{crit}}$ conform within errors to the measured value in the CMB window $z \simeq z_\star \rightarrow 10^4$, where $z_\star \simeq 1100$ is the redshift of recombination. Since $\tilde{\rho }_\textrm{em}$ increases with time, we choose to impose a 10% tolerance on the excess radiation at $z = z_\star $, where the impact of $\tilde{\rho }_{\textrm{em}}$ on the CMB primary anisotropies is the largest. Then, solving Eq. (C5) for $F_\textrm{over}=10$, $\epsilon _{\textrm{em}} = 0.5$, and $\Omega _{\textrm{DM}}=0.244$ [38], and applying the $\Omega _{\textrm{rad}}$ constraint, we find that

$$\begin{aligned} R_\Gamma \simeq 9 \times 10^{-6} \rightarrow 1.5 \times 10^{-2}, \end{aligned}$$

(C6)

or, equivalently,

$$\begin{aligned} \tau \simeq 2 \times 10^9 \rightarrow 9.5 \times 10^{14}~\textrm{s}, \end{aligned}$$

(C7)

can be excluded irrespective of the particle masses. This exclusion region is indicated with horizontal hatching in Figs. 6 and 7. Reducing the tolerance from 10% to, e.g., 5% has but a minor impact on the exclusion region: the boundaries shift by less than a factor of two, practically indiscernible on a log-log plot spanning five to ten orders of magnitude in each direction.

Note that this excess radiation energy constraint applies in principle also to scenarios in which the NLSP is produced at the correct abundance or is under-produced. However, these constraint are uncompetitive: for NLSP produced at the correct abundance, i.e., $F_{\textrm{over}}=1$, the exclusion region corresponds to

$$\begin{aligned} R_\Gamma \simeq 9 \times 10^{-5} \rightarrow 3 \times 10^{-3}, \end{aligned}$$

(C8)

or equivalently,

$$\begin{aligned} \tau \simeq 2 \times 10^{11} \rightarrow 8 \times 10^{13}~\textrm{s}, \end{aligned}$$

(C9)

i.e., a mere sliver of the parameter superWIMP space already excluded by CMB and BBN energy injection bounds depicted in Figs. 2 and 3; in cases of underproduction, e.g., $F_{\textrm{over}} \le 1/10$ in Figs. 4 and 5, equivalent constraints do not even exist.

Other than the above new conditions, the constraints on the $(m_{\chi _1^0},m_{\textrm{LSP}})$-plane from BBN and CMB energy injection considerations are only marginally tighter than those we saw earlier in Figs. 2 and 3, where the neutralino had been assumed to be produced at the correct abundance. On the other hand, the Lyman-$\alpha $ exclusion regions, while looking generally similar in shape as before, have “shifted downwards” by a factor of 10 (corresponding to the over-production factor). In the case of the axino, the exclusion region also protrudes further to the right following the free-streaming horizon $\lambda _{\textrm{FS}}$ contours (see Fig. 3), allowing Ly$\alpha $ to outperform the energy injection constraints by a larger margin than in Figs. 2 and 3. Importantly, however, the Ly$\alpha $ exclusion regions remain bounded from below by $f_{\textrm{WDM}} = 0.15$, or, equivalently in this case, by $m_{\textrm{LSP}}/m_{\chi _1^0} = 0.15/F_{\textrm{over}}$, since once the LSP mass drops too low to make up a substantial fraction of the DM, its free-streaming effects also become unobservable.

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Deshpande, M., Hamann, J., Sengupta, D. et al. Revisiting cosmological constraints on supersymmetric SuperWIMPs. Eur. Phys. J. C 84, 667 (2024). https://doi.org/10.1140/epjc/s10052-024-12992-3

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Received: 19 September 2023
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Revisiting cosmological constraints on supersymmetric SuperWIMPs

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1 Introduction