Non-thermal WIMPs as Dark Radiation
NNon-thermal WIMPs as Dark Radiation
Farinaldo S. Queiroz
Department of Physics and Santa Cruz Institute for Particle Physics University of California, Santa Cruz, CA95064, USA
Abstract.
It has been thought that only light species could behave as radiation and account for the dark radiation observedrecently by Planck, WMAP9, South Pole and ATACAMA telescopes. In this work we will show that GeV scale WIMPs canplausibly account for the dark radiation as well. Heavy WIMPs might mimic the effect of a half neutrino species if someof their fraction were produced non-thermally after the thermal freeze-out. In addition, we will show how BBN, CMB andStructure Formation bounds might be circumvented.
Keywords:
WIMPs,Dark Radiation, Supersymmetry,3-3-1 Model
PACS:
INTRODUCTION
Among the variety of dark matter candidates, WIMPs stand as one of the most compelling candidates. They are oftenthought to be GeV-TeV stable particles which are thermally produced in the early Universe and able to plausiblyaddress many direct and indirect detection signals discussed elsewhere [1, 2]. Recent precise measurements of theangular power spectrum of the cosmic microwave background (CMB) by a variety of telescopes and satellites seemto indicate the existence of an extra component of radiation in the early Universe which is typically parametrized interms of the number of effective neutrinos species N e f f . In summary, there is mild evidence for N e f f > X (cid:48) via X (cid:48) → DM + γ . Most importantly,any particle physics model that has a WIMP in its spectrum and satisfies the following criteria is able to provide analternative and interesting solution to this excess of neutrino species: • M X (cid:48) / M DM ≥ × ∆ N e f f . • freeze − out time < li f etime ( X (cid:48) ) < s. • Just a small fraction ( ∼
1% or smaller) of the WIMPs should be produced via this non-thermal mechanism.The lifetime may have to be much smaller than 10 s depending on the other particle produced in the final state inaddition to the WIMP. For simplicity we restrict ourselves to the pure electromagnetic case (photon), but in principle,this other particle could be anything [9]. Here we will explain from first principles how this mechanism works and inaddition we present both a low scale supersymmetry example and a 3-3-1 model where this framework is realized. Wewill also discuss the constraints imposed by Big Bang Nucleosynthesis, Structure Formation, and the CMB. We beginby deriving the relation between the non-thermal production of WIMPs and the number of effective neutrino species. WIMPS-DARK RADIATION RELATION
To do so, we compute the ratio between an effective neutrino species and dark matter. Since the Cold Dark Matter(CDM) and neutrino densities are given by ρ DM = ρ c Ω DM a − and ρ ν = ρ c Ω ν a − eq N ν /
3, at the Matter-RadiationEquality (MRE), the ratio between their energy density is, ρ ν ρ DM = Ω ν Ω DM N ν a eq = . Ω γ Ω DM N ν a eq , (1)where Ω γ (cid:39) . × − , Ω DM ∼ . N ν is the number of neutrinos, and a EQ (cid:39) × − is the scale factor at MRE. a r X i v : . [ a s t r o - ph . C O ] N ov or N ν =
1, we thus find that the energy density of one neutrino species is ∼
16% of the CDM density. Hence, if allDM particles had a boost factor of γ DM (cid:39) .
16 at MRE, they would mimic the same effect of an extra neutrino speciesin the expansion of the Universe at MRE, as can be seen in the left panel of Fig.1 [4]. In order to understand how thissetup works, we will explain how to relate these two quantities.As aforementioned, we are assuming in this work that only a small fraction f of the DM particles in the Universewere produced via the decay X (cid:48) → W IMP + γ . Therefore, using energy and momentum conservation we get, E X (cid:48) = E γ + M DM γ DM , (2) P X (cid:48) = E γ + P DM . (3)where γ DM is the boost factor of the dark matter particle. Assuming that X (cid:48) decays at rest, we get | P DM | = | E γ | = M X (cid:48) ( M X (cid:48) − M DM ) . (4)Substituting Eq. (4) into Eq. (2) and taking E X (cid:48) = M X (cid:48) we find, γ DM = (cid:18) M X (cid:48) M DM + M DM M (cid:48) X (cid:19) . (5)Now we have to take into account the expansion of the Universe in the radiation dominated era. Assuming that alldecays happen at the lifetime t = τ , we get γ DM = + (cid:16) τ t (cid:17) / (cid:20)(cid:18) M X (cid:48) M DM + M DM M (cid:48) X − (cid:19)(cid:21) . (6)Hence at MRE ( t = t EQ ) we finally obtain, γ DM (cid:39) + . × − (cid:16) τ s (cid:17) / (cid:20)(cid:18) M X (cid:48) M DM + M DM M (cid:48) X − (cid:19)(cid:21) . (7)Notice that Eq. (7) gives us the boost factor of a DM particle as a function of time and of the mother-to-daughtermass ratio, for the case where all decays happened when the Universe was radiation dominated. This non-thermalproduction leads to an extra radiation component given by, ρ extra = f × ρ DM ( γ DM − ) , (8)where f is the fraction of DM particles non-thermally produced in the decays. This extra radiation coming from thedark sector is called dark radiation and can be parametrized in terms of the number of effective neutrinos by the factthat ∆ N e f f = ρ extra / ρ ν , where ρ ν is the number density of one neutrino species at the same epoch. Therefore, usingEq. (1), we find ∆ N e f f = f ( γ DM − ) . . (9)Thus, substituting Eq. (6) into Eq. (9), we obtain ∆ N e f f (cid:39) . × − (cid:16) τ s (cid:17) / × (cid:20)(cid:18) M X (cid:48) M DM + M DM M (cid:48) X − (cid:19)(cid:21) × f . (10)This equation informs us that if some fraction of the DM particles of the Universe is produced non-thermally throughthe decay X (cid:48) → DM + γ , this non-thermal production mechanism might mimic a neutrino species for reasonable valuesof the lifetime and mass ratio ( M X (cid:48) / M DM ) [4]. In summary this scenario possesses three free parameters:(i) the X (cid:48) lifetime for the decay process,(ii) the mass ratio M DM / M X (cid:48) , and(iii) the fraction f of the DM density produced via the decay.It is important to emphasize that in this framework the majority of the DM particles must still be produced thermally,with only a small fraction being produced non-thermally as we shall discuss further. Now that we have shown how thenon-thermal production of DM particles can mimic some degree of additional neutrino species, we will investigate thecosmological bounds which apply to this setup. / H S M a (scale factor) -9 -8 -7 -6 -5 -4 -3 -2 -1 ΔN ν = 1ΔN ν = 0.5ΔN ν = 0.1τ = 10 s , M x' /M x = 2. 10 , f = 0.01 τ = 10 s , M x' /M x = 2. 10 , f = 0.01 τ = 10 s , M x' /M x = 2. 10 , f = 0.01 f . ( M x ' / M D M ) −3 −2 −1 τ x' (sec) Li lowD low D high He low X' → DM + γ
CMB boundΔN eff = 0.2ΔN eff = 0.4ΔN eff = 0.6ΔN eff = 0.8ΔN eff = 1 FIGURE 1.
Left : We exhibit the expansion rate of the Universe for additional neutrino species in dashed lines fromtop to bottom respectively as N ν = . ( pink ) , . ( gray ) , ( black ) . Moreover, we show the expansion rate when we includethis non-thermal production of WIMPs that reproduces N e f f = τ = s ( red ) , s ( blue ) , s ( dashed green ) [4]. Right : The parameter space defined by the mother particle X (cid:48) lifetime ( τ X (cid:48) ) andthe fraction of relativistically produced DM ( f ) times the mother-to-daughter mass ratio M X (cid:48) / M DM , and constraints from BBN andCMB. The shaded regions show BBN bounds on the non-thermal production of DM via the decay X (cid:48) → DM + γ . The green curverepresents the CMB bound (regions to the right of the curve are excluded). The diagonal lines corresponding to an excess relativisticdegrees of freedom ∆ N e f f = . , . , . , . ,
1. We have assumed M X (cid:48) (cid:29) M DM . [5] COSMOLOGICAL BOUNDSStructure Formation
It is a well known fact that all dark matter particles could not have had a large kinetic energy at the matter-radiationequality. DM particles with large kinetic energies would not cluster at sufficiently small scales due to their large free-streaming. Hence, it is critical to check the suppression on the growth of structure caused by the fraction of DMparticles which were non-thermally produced in this scenario. At small scales, the matter fluctuations of cold DMparticles is governed by a linear equation according to [10], δ ( f ) ∝ a α ∞ ( f ) . (11)where α ∞ ( f ) is the growth rate of the cold DM field at the matter-dominated epoch which is given by [10] α ∞ ( f ) = (cid:114) − f (cid:39) − / f , (12)where f is the fraction of the DM density produced non-thermally in a relativistic state. Comparing the matterfluctuation given in Eqs.(11)-(12), we can determine the suppression caused by this non-thermal production of WIMPs.To first order, we find g = δ ( f ) δ ( f = ) = a − / fEQ (cid:39) exp ( − . f ) , (13)which is only valid in the matter-dominated regime and for f (cid:28)
1. Combined measurements of the amplitude of matterfluctuations in the Universe on scales of 8 h − Mpc from the WMAP9 results [11] with Lyman-alpha forest data [12]require g > .
95. This bound implies that f < .
01 from Eq. (13).In summary, structure formation bounds the fraction of DM particles that can be non-thermally produced in arelativistic state and requires that only a small fraction (less than 1% or so) of the DM in the Universe might have beenproduced by the decay process being considered here. ig Bang Nucleosynthesis
As we already pointed out, the lifetime and the energy released by the mother particle X (cid:48) are also constrained byBBN bounds. The energy released at a given time in the history of the Universe may induce electromagnetic showersthat create and/or destroy light elements synthesized in the early universe [13]. Thus we should investigate the possibleimpacts of this scenario on BBN. Given the quite impressive success of BBN we must demand that new physics effectsdo not drastically alter any of the light element abundances such as those of D, He, Li. The energy of photons createdin late decays of X (cid:48) would have been rapidly redistributed through scattering off background photons ( γγ BG → e + e − )as well as through inverse Compton scattering ( e γ → e γ ) [13, 14, 5]. Hence, the bounds we obtain from BBN are, toa good approximation, independent of the initial energy distribution of the injected photons and are only sensitive tothe total energy released in the decay process [13].In order to derive BBN limits on the fraction of relativistic, non-thermally produced DM via the decay X (cid:48) → DM + γ ,we need to calculate the total electromagnetic energy released. Let Y = n / n CMB γ , where n is the number density ofparticles of a particular species and n CMB γ is the number density of CMB photons. As given by [4, 5], we find n CMB γ = ζ ( ) π T . (14)The total electromagnetic energy released from the X (cid:48) decay is thus ε EM = E γ Y DM . If for each X (cid:48) particle we havethe production of a DM particle plus a photon, then Y X (cid:48) = Y γ = Y DM , τ = Y DM , , where Y DM , τ determines the numberdensity of particles at a time equal to the lifetime of X (cid:48) , and Y DM , is the number density of DM particles today.We thus find that the normalized number density of DM particles is given by [4, 5], Y DM = n DM n CMB γ = Ω DM ρ c M DM n CMB γ , . (15)This can be rewritten as, Y DM (cid:39) · − (cid:18) TeV M DM (cid:19) (cid:18) Ω DM . (cid:19) (cid:18) f . (cid:19) . (16)The factor f showed up in Eq. (16) because we assume here that only a fraction of the DM in the Universe isproduced in the decay process, whereas the majority of it is produced non-relativistically by some other mechanismthat does not induce any significant energy injection during BBN (for example, via a standard thermal freeze-outprocess).Since the photon energy produced in the decay is, E γ = M X (cid:48) ( M X (cid:48) − M DM ) , (17)using Eq.(16) we find that the total electromagnetic energy released is given by ε EM = . · − GeV × (cid:18) Ω DM . (cid:19) (cid:18) f . (cid:19) (cid:18) M X (cid:48) M DM − M DM M X (cid:48) (cid:19) . (18)In the limit M X (cid:48) (cid:29) M DM we can straightforwardly connect the total energy release given in Eq. (18) with the quantity f × ( M X (cid:48) / M DM ) as well as with ∆ N e f f , as given in Eq. (10). Hence we can translate the results on the total energyreleased obtained in [13], and express them in terms of the quantity f × ( M X (cid:48) / M DM ) . We exhibit in Fig. 1 the shadedregions ruled out by BBN using the “baryometer” parameter η = n b / n γ = × − . The diagonal lines representcombinations of the quantity f × M X (cid:48) / M DM and τ X (cid:48) producing the ∆ N e f f as in the labels.We conclude from Fig. 1 that the BBN bounds are weaker for early decays because at early times the universe is hotenough and the initial photon spectrum is rapidly thermalized, leaving just a few high-energy photons able to modifythe light element abundances. Although for lifetimes longer than 10 s, BBN excludes most of the relevant parameterspace. osmic Microwave Background Similarly to the BBN constraints, CMB bounds depend mostly only on the total energy released in the decay process.The key effect of the additional energy injection in the form of photons is related to spectral distortions caused in theCMB black-body spectrum [15]. For early times ( t < s) the processes of bremsstrahlung, i.e. eX → eX γ (where Xis an ion), Compton scattering and double Compton scattering e γ → e γγ quickly thermalize the injected photon energy[15]. On the other hand, for t > s, the bremsstrahlung and double Compton processes become inefficient, and thephoton spectrum relaxes to a Bose-Einstein distribution with a chemical potential ( µ ) different from zero. Limits on µ are used to constrain this additional energy injection and consequently bound the set f M X (cid:48) / M D M and the lifetimeas discussed in Ref.[4, 5]. The current limit implies that µ < × − [16, 17]. In Fig. 1 we plotted the CMB boundafter converting this upper limit on the chemical potential into a bound on f M X (cid:48) / M D M for a given lifetime. Noticethat CMB constraint becomes only relevant for lifetimes longer than ∼ s.We have discussed the dark radiation setup so far and now it is time to present some realistic models where thisscenario is plausibly realized while simultaneously obeying the cosmological limits aforementioned. SUPERSYMMETRIC FRAMEWORK
Neutralinos are the mass eigenstates resulting from a mixture of neutral B-ino, W-ino, and Higgs-inos. Here we willassume it to be a pure Bino. In low scale supersymmetry the neutralino might be the next-to-lightest supersymemtricparticle with the lightest supersymmetric particle being the gravitino. Thus Binos decay into a gravitino-photon finalstate via the interaction Lagrangian term [18], L = − i π M (cid:63) ˜ G µ [ γ ν , γ ρ ] γ µ ˜ B µ F νρ , (19)where M (cid:63) = . × GeV is the reduced Planck mass. Because we are in the regime of low-scale supersymmetrybreaking M ˜ B (cid:29) M ˜ G , which is exactly the limit needed to realize the dark radiation setup we are focused on. In thiscase, from Eq. (19) we find a neutralino lifetime of τ ( ˜ B → γ ˜ G ) (cid:39)
750 s (cid:18) M ˜ G (cid:19) (cid:18) M ˜ B (cid:19) . (20)Solving Eq.(20) for M ˜ B and substituting into Eq.(10) we find a lower limit on the DM mass, M G < ( ) (cid:16) τ (cid:17) / (cid:18) f ∆ N e f f (cid:19) / . (21)The lower bound is found because we imposing the lifetime to be shorter than 10 s to obey BBN limits. Setting f = .
01 and τ = s, we show in Fig. 2 that a gravitino with mass in the 2 to 20 keV range mimics the effectof an extra neutrino species while still obeying cosmological bounds. Notice that for a gluino mass of ∼ T R close to the electro-weak scale is required to prevent over-closing the universe, since thethermal production of gravitinos in the early universe [19] implies T R
100 GeV (cid:39) (cid:18) Ω ˜ G h . (cid:19) (cid:18) M ˜ G (cid:19) . (22)Such a low reheating temperature would rule out certain scenarios for the production of the baryon asymmetry in theuniverse, such as Leptogenesis, but is in general not phenomenologically implausible.Finally, we note that the constraints on the lifetime and Eq. (20) imply a lower limit on the bino mass, which mustbe larger than about 1 GeV. This is, of course, perfectly compatible with the supersymmetric models relevant here. SU ( ) c ⊗ SU ( ) L ⊗ U ( ) N . Models based on this 3-3-1 gauge symmetry [20] potentially address important theoretical G r a v i t i n o ( G e V ) −6 −5 −4 −3 τ (sec) F1F2F3F4F5 ΔN eff = 0.2ΔN eff = 0.4ΔN eff = 0.6ΔN eff = 0.8ΔN eff = 1f = 0.01 τ = 10 -6 sτ = 10 -4 sτ = 10 -2 s M Φ ( G e V ) M N1 (GeV) Δ = Δ = Δ = ΔN eff = 0.1ΔN eff = 0.3ΔN eff = 0.5g' =1 FIGURE 2.
Left:
Lower limit on the gravitino mass, for a supersymmetric model where a fraction f of gravitinos are non-thermally produced by the decay of pure Binos. The relevant lifetime is derived from Eq. (19). We may conclude that a ∼
10 KeV Gravitino might reproduce ∆ N e f f ∼ . Right:
Bound on the WIMP ( φ ) mass of the 3-3-1LHN model. Notice that a100 − ∆ N e f f ∼ .
1. We have used f = .
01 in both plots. and phenomenological questions which remain unexplained within the SM, such as the number of particle generations[21], the possible Higgs to diphoton excess [22] and have a rich phenomenology which includes new scalars andgauge bosons, as extensively explored in the literature [23]. In particular, more recent work based on this symmetryhave been proposed to additionally address debatable direct and indirect detection signals of WIMPs in our galaxy[24, 25, 26, 27, 28] . For these and many other reasons, 3-3-1 models stand as compelling alternatives to the SM. Herewe will focus on a version that has heavy neutrinos ( N R ) and a stable scalar (the WIMP of the model) in its spectrumknown as 331LHN described in more details in Ref. [27]. In this model the so called WIMP miracle is realizedand the right abundance is easily achieved. Although the heavy neutrinos can be long lived, in the sense that theymay decay after the WIMP ( φ ) freeze-out, some non-thermal production of φ will occur. In supersymmetric models,the supersymetric particles will always decay into the lightest stable particle due to the R-parity symmetry. Similarly,because of a global symmetry (G), described in [27], the 331 particles will decay into φ as well, inducing a non-thermalproduction of dark matter when the decays happen after the WIMP freezes-out. For simplicity we will tune only oneheavy neutrino ( N R ) to be long lived, but a more general approach could be straightforwardly investigated [27]. Insummary, this model will have dominant thermal production as in the standard WIMP paradigm, but an additional asub-dominant one arises. This non-thermal production component is crucial in our setup because it generates somedegree of dark radiation as shown earlier. Nevertheless, the lifetime of the mother particle N R has to be shorter than10 s due to BBN constraints. The most important parameters which control the lifetime of this neutrino are the scaleof symmetry breaking of this model, its mass and the Yukawa coupling, g (cid:48) . From our perspective, the relevant decaymode is N R → W IMP + ν e which has the following lifetime, τ (cid:39) (cid:16) · − s (cid:17) (cid:18) − g (cid:48) (cid:19) (cid:16) v χ (cid:48) GeV (cid:17) (cid:18) GeV M (cid:19) . (23)where λ = g (cid:48) v / v χ (cid:48) and M is the see-saw scale. From Eq.(23) we notice that there will be a very wide range of Yukawacouplings ( g (cid:48) ) that produces a lifetime allowed by BBN ( τ ≤ s) and with decays that happen after the WIMPfreeze-out ( τ (cid:39) − s for a 100 GeV WIMP). However, when we try to reproduce N e f f (cid:39) . M wimp , the mother particle mass( M N ) and the coupling constant g (cid:48) , which sets the thermal relic densities. In Fig. 2 we present our results in themother-daughter mass parameter space for a relatively large coupling, g (cid:48) =
1. Notice we find a line across the Models based on the 3-4-1 gauge symmetry which embeds the 3-3-1 might also have good dark matter candidates (see Refs.[29]). arameter space where all of the constraints are satisfied, and where ∆ N eff = . N masses between 10 and 100 TeV. Larger N masses requireincreasingly larger entropy suppression factors ∆ , and larger WIMP masses to obtain the desired enhancement to ∆ N eff .An entropy injection episode should be invoked for small values of g (cid:48) because the abundance of the mother particle istypically too large to only produce 1% of the WIMP density as required by structure formation. Therefore we postulatethat an entropy injection occurred between the relatively high temperature at which the N froze out and the time ofdecay (the latter is indicated by vertical lines in the figures) causing a dilution factor ∆ in the abundance. In otherwords, the standard thermal relic density Ω N → Ω N / ∆ as a result of the larger entropy density. ∆ = CONCLUSIONS
Recent measurements of the cosmic microwave background radiation point to a mild evidence for dark radiation, i.e,an excess of relativistic degrees of freedom in the early universe. In this work we have shown that if a fraction of thedark matter particles of the Universe had been produced from the decay of a heavy particle after their freeze-out, thisnon-thermal production will induce some degree of dark radiation. Indeed, these non-thermal WIMPs would be able tomimic the effect of one neutrino species in the early universe. Furthermore, we have shown that this mechanism mustobey strong BBN, CMB and Structure Formation constraints. Nevertheless, they can be circumvented if the lifetime ofthe mother particle is shorter than 10 s and just a fraction ( ∼ ∼
10 KeV Gravitino, in low scale supersymmetry, and a ∼
10 GeVscalar, in the 3-3-1LHN model, are both plausibly able to reproduce ∆ N eff (cid:39) . ACKNOWLEDGMENTS
We are grateful to Stefano Profumo, Chris Kelso, Carlos Pires, Paulo Rodrigues for their collaboration. We thank theorganizers of PPC2013 for a well organized conference. This work is partly supported by the Department of Energyunder contract DE-FG02-04ER41286 (SP), and by the Brazilian National Counsel for Technological and ScientificDevelopment (CNPq) (FQ,CP,PR).
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