aa r X i v : . [ g r- q c ] J u l Collider Physics and Cosmology
Jonathan L. Feng
Department of Physics and Astronomy, University of California, Irvine, CA 92697,USA
Abstract.
In the coming year, the Large Hadron Collider will begin colliding protonsat energies nearly an order of magnitude beyond the current frontier. The LHC will,of course, provide unprecedented opportunities to discover new particle physics. Lesswell-known, however, is that the LHC may also provide insights about gravity andthe early universe. I review some of these connections, focusing on the topics of darkmatter and dark energy, and highlight outstanding prospects for breakthroughs at theinterface of particle physics and cosmology.PACS numbers: 13.85.-t, 95.35.+d, 95.36.+x
1. Introduction
The Large Hadron Collider (LHC) is scheduled to begin running in the summer of 2008.Conceived around 1984 and approved in 1994, the LHC will provide the first detailedlook at the weak energy scale M weak ∼
100 GeV − E CM = 14 TeV and ultimate luminosity L = 100 fb − / yr.This is far beyond the current energy frontier, where the Tevatron collides protons andanti-protons with E CM = 2 TeV and L ∼ − / yr. As an illustration of the power ofthe LHC, top quarks, discovered in 1994 with a handful of events and currently producedat the Tevatron at the rate of ∼ raison d’etre for the LHC is the discovery of the Higgs boson and associatedmicrophysics, including supersymmetric and other postulated particles. In recent years,however, the LHC’s potential for providing insights into gravity and cosmology havetaken on increasing importance. My goal here is to review some recent developments andto highlight a few scenarios in which the implications of the LHC for our understandingof gravity and the early universe may, in fact, be profound.
2. Cosmology Now
A wealth of recent cosmological observations now constrain the total energy densitiesof non-baryonic dark matter, dark energy, and baryons to be [1, 2]Ω DM = 0 . ± .
044 (1) ollider Physics and Cosmology No Big Bang e x p a n d s f o r e v e r -1012323 c l o s e d r ec o l l a p s e s e v e n t u a l l y SupernovaeCMBClusters op e n f l a t Knop et al. (2003)Spergel et al. (2003)Allen et al. (2002) ΩΩ Λ M Figure 1.
Constraints on Ω M and Ω Λ from observations of supernovae, the CMB,and galaxy clusters. Boundaries between regions with open and closed universes,between universes that expand forever and those that recollapse, and regions thatdo not extrapolate back to a Big Bang singlularity, are also shown [3]. Ω Λ = 0 . ± .
04 (2)Ω B = 0 . ± . . (3)The constraints are summarized in Fig. 1 and both the central values and the error barsare remarkable. Given that just a decade ago the range 0 . ∼ < Ω DM ∼ < . Λ = 0 was often assumed, this represents spectacular progress.At the same time, this progress highlights many outstanding questions. Theseinclude: • Dark matter: What is its mass? What are its spin and other quantum numbers?Is it absolutely stable? What is the symmetry origin of the dark matter particle?Is dark matter composed of one particle species or many? How and when was itproduced? Why does Ω DM have the observed value? What was its role in structureformation? How is dark matter distributed now? • Dark energy: What is it? Why is Ω Λ not much larger than observed? Why notΩ Λ = 0? Does it evolve? • Baryons: Why not Ω B = 0? Is this related to neutrinos and leptonic CP violation?Where are all the baryons?Although colliders may also shed light on baryogenesis, I will focus here on darkmatter and dark energy. From a microphysical viewpoint, these problems appear, atleast at first, to be completely different. In the case of dark matter, no known particlescontribute, there are reasons to believe that the problem is tied to the weak energyscale M weak , and there are several compelling candidates. In contrast, for dark energy, ollider Physics and Cosmology M Pl ∼ GeV, and there are no compellingsolutions.In the following sections, I will discuss dark matter and dark energy in turn, focusingprimarily on dark matter, where the collider connections to cosmology and gravity areespecially concrete and compelling.
3. Dark Matter
The particle or particles that make up most of dark matter must be • stable, or at least long-lived on cosmological time scales, • cold or warm to properly seed structure formation, and • non-baryonic, to preserve the successes of Big Bang nucleosynthesis (BBN).Unfortunately, these constraints are no match for the creativity of theorists, who haveproposed scores of viable candidates with masses and interaction cross sections varyingover tens of orders of magnitude.Candidates with masses at the weak scale M weak ∼
100 GeV − M weak . This conflicts with precision measurements, which constrain the Higgsmass to m h ∼ M weak . This puzzle is the gauge hierarchy problem. New ideas arerequired to resolve this problem, and these ideas invariably predict new particle stateswith masses around M weak .In addition, although we have not seen any of these new particles, there are alreadyindications that if these particles exist, they are stable. This is the cosmological legacyof LEP, the Large Electron-Positron Collider that ran from 1989-2000. Generically, thenew particles introduced to solve the gauge hierarchy problem induce new interactions(SM)(SM) → NP → (SM)(SM), where SM and NP denote standard model and newparticles, respectively. LEP, along with the Stanford Linear Collider, looked for theeffects of these interactions and found none. At the same time, the new particles cannotbe decoupled completely; to solve the gauge hierarchy problem, they must interact withthe Higgs boson through couplings h ↔ (NP)(NP), and they cannot be too heavy. Asimple and elegant solution is to require a conserved discrete parity that requires allinteractions to involve an even number of new particles [4, 5]. As a side effect, thisdiscrete parity implies that the lightest new particle cannot decay — it is stable, asrequired for dark matter.Finally, if these new particles exist and are stable, they are naturally produced withthe cosmological densities required of dark matter. This fact is sometimes called the ollider Physics and Cosmology T below the dark matter particle’s mass m χ , and thenumber of dark matter particles becomes Boltzmann suppressed, dropping exponentiallyas e − m χ /T . In stage (3), the Universe becomes so cool and dilute that the dark matterannihilation rate is too low to maintain equilibrium. The dark matter particles then“freeze out,” with their number asymptotically approaching a constant, their thermalrelic density.More detailed analysis shows that the thermal relic density is rather insensitive to m χ and inversely proportional to the annihilation cross section:Ω DM ∼ h σ A v i − , (4)where v is the relative velocity of the annihilating particles, and the brackets indicatea thermal average. The constant of proportionality depends on the details of themicrophysics, but we may derive a rough estimate. On dimensional grounds, the crosssection can be written σ A v = k πα m χ (1 or v ) , (5)where the factor v is absent or present for S - or P -wave annihilation, respectively,and terms higher-order in v have been neglected. The constant α is the hyperchargefine structure constant, related to the weak interactions of the standard model, and k parameterizes deviations from this estimate.With this parametrization, given a choice of k , the relic density is determined asa function of m χ . The results are shown in Fig. 2. The width of the band comes fromconsidering both S - and P -wave annihilation, and from letting k vary from to 2. Wesee that a particle that makes up all of dark matter is predicted to have mass in therange m χ ∼
100 GeV − m χ ∼
30 GeV −
300 GeV. There are models in which the effective k is outside ourillustrative range. In fact, values of k smaller than we have assumed, predicting smaller m χ , are not uncommon, as the masses of virtual particles in annihilation diagrams canbe significantly higher than m χ . However, the general conclusion remains: particlesthat interact through weak interactions and have mass at the weak scale naturally havesignificant thermal relic densities.To summarize, viable particle physics theories designed to address the gaugehierarchy problem naturally (1) predict new weakly-interacting particles with mass ∼ M weak that (2) are stable and (3) have the thermal relic densities required to bedark matter. The convergence of particle physics and cosmological requirements fornew states of matter has motivated many new proposals for dark matter. These may ollider Physics and Cosmology Figure 2.
Left: The cosmological evolution of a thermal relic’s comoving numberdensity [6]. Right: A band of natural values in the ( m χ , Ω χ ) plane for a thermalrelic [7]. be grouped into two classes: WIMPs and superWIMPs. In the following subsections,we consider what insights colliders may provide in each of these two cases. WIMPs, weakly-interacting massive particles, interact through the weak force and havemasses near the weak scale M weak . For the reasons given above, the WIMP paradigmis now thriving, and recent years have seen a proliferation of WIMP candidates.These include the traditional prototype, neutralinos in supersymmetry with R -parityconservation [8, 9], but also more recent candidates, including Kaluza-Klein photons inuniversal extra dimensions with KK-parity [10, 11], branons in brane world scenarioswith branon parity [12, 13], and T -odd dark matter in little Higgs models with T -parity [5].If WIMPs are the dark matter, what can colliders tell us? This has been investigatedin numerous studies [14]. Given the energy of the LHC and the requirement that WIMPsinteract through the weak force, WIMPs will almost certainly be produced in largenumbers at the LHC, but their detection will be somewhat indirect. For example, insupersymmetry, the LHC will typically produce pairs of squarks and gluinos. These willthen decay through some cascade chain, eventually ending up in neutralino WIMPs,which escape the detector. Their existence is registered only through the signatureof missing energy and momentum. Although the observation of missing particles isconsistent with the production of dark matter in the lab, it is far from compellingevidence. In particular, colliders can only establish that the neutralino was stable enoughto exit the detector, typically implying that the neutralino’s lifetime was τ > − s, afar cry from the criterion τ ∼ > s required for dark matter. Supersymmetric scenariosare not special in this regard — although not examined in as much detail to date, otherWIMP models, such as those derived from universal extra dimensions [15], share all of ollider Physics and Cosmology σ A of Eq. (4). In the case of supersymmetry, for example, the relevant processes aregiven in Fig. 3. The task at a collider is to determine the masses and couplings of allthe new particles entering these processes, or to bound them sufficiently to ensure thattheir contributions are negligible.How well can the LHC do? The answer depends sensitively on the underlying darkmatter scenario, but several qualitatively different cases have now been studied [16, 17,18, 19]. The results of one (admittedly rather exemplary) supersymmetric case studyare given in Fig. 4. In conjunction with other cosmological observations, the WMAPsatellite constrains the dark matter relic density Ω χ to a fractional uncertainty of ± ± ± e + e − collider, could improve this to ± ollider Physics and Cosmology Figure 3.
Particle physics processes that contribute to the annihilation cross sectionfor χχ → anything, where χ is the neutralino dark matter of supersymmetry [6]. Figure 4.
Constraints in the ( m χ , ∆(Ω χ h ) / Ω χ h ) plane from the LHC and ILC,and from the WMAP and Planck satellite experiments [20]. The satellite experimentsmeasure Ω χ , but are insensitive to the dark matter mass m χ ; the collider experimentsbound both. the question of the microscopic identity of dark matter. Note also that, just as BBNgives us confidence that we understand the universe back to times of 1 second after theBig Bang and temperatures of 1 MeV, such studies also provide a window on the eraof dark matter freezeout, or roughly times of 1 nanosecond after the Big Bang, andtemperatures of ∼
10 GeV.Of course, the thermal relic density prediction from colliders and the cosmological ollider Physics and Cosmology IDENTIFYING DARK MATTER
Are (cid:58) hep and (cid:58) cosmo identical?Congratulations!You’vediscovered the identity of dark matter and extended our understanding of the Universe to T = 10 GeV, t = 1 ns (Cf. BBN at T = 1 MeV, t = 1 s) YesYesYes Calculatethe new (cid:58) hep Can you discover another particle that contributes to DM? Which is bigger?No (cid:58) hep (cid:58) cosmo
Does it account for the rest of DM? YesNoDid you make a mistake? Does itdecay?Can you identify a source of entropy production?No YesNo NoYesCan this be resolved with somewacky cosmology? YesNoNoAre you sure? YesThink about the cosmologicalconstant problemNo
Figure 5.
Flowchart illustrating the possible implications of comparing Ω hep , thepredicted dark matter thermal relic density determined from high energy physics, andΩ cosmo , the actual dark matter relic density determined by cosmological observations. observations need not be consistent. In this case, there are many possible lines of inquiry,depending on which is larger. A flowchart of possibilities is given in Fig. 5.
Strictly speaking, dark matter need only be gravitationally interacting — there is as yetno evidence of any other sort of interaction. However, the “WIMP miracle” describedin Sec. 3.1 might appear to require that dark matter have weak interactions if its relicdensity is naturally to be in the right range. This is not true, however — dark mattermay be composed of superweakly-interacting massive particles, superWIMPs, whichhave interactions weaker than weak, but still naturally have the required relic density.In superWIMP scenarios [21], a WIMP freezes out as usual, but then decays to asuperWIMP, as shown in Fig. 6. As with WIMPs, there has recently been a proliferationof superWIMP candidates. The prototypical example of a superWIMP is a weak-scalegravitino produced non-thermally in the late decays of a supersymmetric WIMP, such asa neutralino, charged slepton, or sneutrino [21, 22, 23, 24]. Additional examples includeaxinos [25] and quintessinos [26] in supersymmetry, Kaluza-Klein graviton and axionstates in models with universal extra dimensions [27], and stable particles in modelsthat simultaneously address the problem of baryon asymmetry [28]. SuperWIMPs haveall of the virtues of WIMPs. They exist in the same well-motivated frameworks and arestable for the same reasons. In addition, in the natural case that the decaying WIMPand superWIMP have comparable masses, superWIMPs also are naturally producedwith relic densities of the desired order of magnitude.Collider evidence for superWIMPs may come in one of two forms. Collider ollider Physics and Cosmology Figure 6.
In superWIMP scenarios, a WIMP freezes out as usual, but then decays toa superWIMP, a superweakly-interacting particle that forms dark matter. experiments may find evidence for charged, long-lived particles. Given the stringentbounds on charged dark matter, such particles presumably decay, and their decayproducts may be superWIMPs. Alternatively, colliders may find seemingly stableWIMPs, but the WIMP relic density studies described in Sec. 3.2 may favor a relicdensity that is too large, providing evidence that WIMPs decay. These two possibilitiesare not mutually exclusive. In fact, the discovery of charged long-lived particles with too-large predicted relic density is a distinct possibility and would provide strong motivationfor superWIMP dark matter.Because superWIMPs are produced in the late decays of WIMPs, their numberdensity is therefore identical to the WIMP number density at freeze out, and thesuperWIMP relic density isΩ sWIMP = m sWIMP m WIMP Ω WIMP . (6)To determine the superWIMP relic density, we must therefore determine thesuperWIMP’s mass. This is not easy, since the WIMP lifetime may be very large,implying that superWIMPs are typically produced long after the WIMPs have escapedcollider detectors. As an example, consider the case of supersymmetry with a staunext-to-lightest supersymmetric particle (NLSP) decaying to a gravitino superWIMP.Gravitinos interact only gravitationally, and so this decay is suppressed by Newton’sconstant G N . On dimensional grounds, we therefore expect the stau lifetime to be1 / ( G N M ). More precisely, we find τ (˜ τ → τ ˜ G ) = 6 G N m G m τ " − m G m τ − ∼ − s . (7)This is outlandishly long by particle physics standards. This gravitino superWIMPscenario therefore implies that the signal of supersymmetry at colliders will be meta-stable sleptons with lifetimes of days to months. Such particles will produce slowly-moving particles that should be obvious at the LHC [29, 30, 31, 32]. ollider Physics and Cosmology SleptontrapReservoir
Figure 7.
Configuration for slepton trapping in gravitino superWIMP scenarios [33].
At the same time, because some sleptons will be slowly moving and highly-ionizing,they may be trapped and studied [33, 34, 35, 36]. As an example, sleptons may betrapped in water tanks placed outside collider detectors. These water tanks may thenbe drained periodically to underground reservoirs where slepton decays can be observedin quiet environments. This possibility has been studied in Ref. [33] and is illustratedin Fig. 7. The number of sleptons that may be trapped is model-dependent, but maybe as large as thousands per year.If thousands of sleptons are trapped, the slepton lifetime may be determined tothe few percent level simply by counting the number of slepton decays as a function oftime. The slepton mass will be constrained by analysis of the collider event kinematics.Furthermore, the outgoing lepton energy can be measured, and this provides a highprecision measurement of the gravitino mass, and therefore a determination of thegravitino relic density through Eq. (6). As with the case of WIMPs, consistency at thepercent level with the observed dark matter relic density will provide strong evidencethat dark matter is indeed composed of gravitino superWIMPs.Perhaps as interesting, the determination of τ , m ˜ G , and m ˜ τ in Eq. (7) implies thatone can determine Newton’s constant on the scale of fundamental particles [37, 38].According to conventional wisdom, particle colliders are insensitive to gravity, since itis such a weak force. We see that this is not true — if G N enters in a decay time,one can achieve the desired sensitivity simply by waiting a long time. In this case, onecan measure the force of gravity between two test particles with masses ∼ − kg, aregime that has never before been probed. If this force is consistent with gravity, thesestudies will show that the newly discovered particle is indeed interacting gravitationally,as is required for the gravitino to be the graviton’s superpartner, and demonstrate thatgravity is in fact extended to supergravity in nature. ollider Physics and Cosmology
4. Dark Energy
Recent observations of dark energy provide profound problems for particle physics. Inquantum mechanics, an oscillator has zero-point energy ¯ hω . In quantum field theory,the vacuum energy receives contributions of this size from each mode, and so is expectedto be ρ Λ ∼ R E d k ¯ hω ∼ E , where E is the energy scale up to which the theory isvalid. Typical expectations for ρ / are therefore the weak scale or higher, whereas theobserved value is ρ / ∼ meV. This discrepancy is the cosmological constant problem.Its difficulty stems from the fact that the natural energy scale for solutions is not at highenergies yet to be explored, but at low energies that one would otherwise have assumedare well-understood.Can upcoming colliders provide any insights? It would be pure fancy at this stageto propose an experimental program to solve the cosmological constant problem. On theother hand, the possibility of probing very early times in the Universe’s history impliesthat one may be sensitive to an era when the effects of dark energy were amplifiedrelative to the present.As an example, assume that the Friedmann equation takes the form H = 8 πG N ρ + ∆ ρ ) , (8)where ∆ ρ is a new, exotic contribution to the energy density. This modification to theHubble parameter directly impacts the evolution of the dark matter density n throughits presence in the Boltzmann equation dndt = − Hn − h σ A v i (cid:16) n − n (cid:17) . (9)If one determines the dark matter thermal relic density as outlined in Sec. 3.2, onetherefore simultaneously bounds contributions ∆ ρ to H .This approach has been explored in a few recent studies [44, 45, 46]. In Ref. [46], theexotic energy density contributions are assumed to be of the form ∆ ρ ∝ T n . For variousvalues of n between 0 and 8, this parametrization can accommodate a wide variety ollider Physics and Cosmology Figure 8.
The fractional change in thermal relic density ∆(Ω χ h ) / (Ω χ h ) for exoticcontributions ∆ ρ ∝ T n , as a function of n and ∆ ρ/ρ at T = 10 GeV [46]. of possibilities, including, for example, a cosmological constant, quintessence, trackingdark energy, and variations in G N . The results are given in Fig. 8. Not surprisingly, thethermal relic density is a sensitive probe of new contributions to dark energy, providedthat they are significant at the time of freeze out. For example, for n = 4, a collidermeasurement that bounds the particle physics prediction for the thermal relic densityΩ χ h with a fractional uncertainty of 10% also bounds variations in the energy densityof the order of 10% at temperatures ∼
10 GeV. This provides a constraint on variationsin G N in the very early universe. Alternatively, these results could favor some proposalsfor dark energy and exclude others.
5. Conclusions
In the coming year, the LHC will probe the weak scale M weak ∼
100 GeV − ollider Physics and Cosmology ∼ Acknowledgments
I thank the organizers of GRG18/Amaldi7 for the invitation to participate in thisstimulating conference and my collaborators for their many insights regarding the workdiscussed here. This work was supported in part by NSF Grants PHY–0239817 andPHY–0653656, NASA Grant NNG05GG44G, and the Alfred P. Sloan Foundation.
References [1] D. N. Spergel et al. [WMAP Collaboration], Astrophys. J. Suppl. , 377 (2007)[arXiv:astro-ph/0603449].[2] M. Tegmark et al. , Phys. Rev. D , 123507 (2006) [arXiv:astro-ph/0608632].[3] R. A. Knop et al. [The Supernova Cosmology Project Collaboration], Astrophys. J. , 102 (2003)[arXiv:astro-ph/0309368].[4] J. Wudka, arXiv:hep-ph/0307339.[5] H. C. Cheng and I. Low, JHEP , 051 (2003) [arXiv:hep-ph/0308199].[6] G. Jungman, M. Kamionkowski and K. Griest, Phys. Rept. , 1419 (1983).[9] J. R. Ellis, J. S. Hagelin, D. V. Nanopoulos, K. A. Olive and M. Srednicki, Nucl. Phys. B ,453 (1984).[10] G. Servant and T. M. P. Tait, Nucl. Phys. B , 391 (2003) [arXiv:hep-ph/0206071].[11] H. C. P. Cheng, J. L. Feng and K. T. Matchev, Phys. Rev. Lett. , 211301 (2002)[arXiv:hep-ph/0207125].[12] J. A. R. Cembranos, A. Dobado and A. L. Maroto, Phys. Rev. Lett. , 241301 (2003)[arXiv:hep-ph/0302041].[13] J. A. R. Cembranos, A. Dobado and A. L. Maroto, Phys. Rev. D , 103505 (2003)[arXiv:hep-ph/0307062].[14] See, e.g. , ATLAS Detector and Physics Performance Technical Design Report,http://atlas.web.cern.ch/Atlas/GROUPS/PHYSICS/TDR/access.html, CMS Physics Techni-cal Design Report, http://cmsdoc.cern.ch/cms/cpt/tdr, and references therein.[15] T. Appelquist, H. C. Cheng and B. A. Dobrescu, Phys. Rev. D , 035002 (2001)[arXiv:hep-ph/0012100]. ollider Physics and Cosmology [16] B. C. Allanach, G. Belanger, F. Boudjema and A. Pukhov, JHEP , 020 (2004)[arXiv:hep-ph/0410091].[17] T. Moroi, Y. Shimizu and A. Yotsuyanagi, Phys. Lett. B , 79 (2005) [arXiv:hep-ph/0505252].[18] A. Birkedal et al. , arXiv:hep-ph/0507214.[19] E. A. Baltz, M. Battaglia, M. E. Peskin and T. Wizansky, Phys. Rev. D , 103521 (2006)[arXiv:hep-ph/0602187].[20] Report of the Cosmology Subgroup, American Linear Collider Physics Group, summarized inJ. L. Feng, In the Proceedings of 2005 International Linear Collider Workshop (LCWS 2005),Stanford, California, 18-22 Mar 2005, pp 0013 [arXiv:hep-ph/0509309]; J. Phys. G , R1(2006) [arXiv:astro-ph/0511043].[21] J. L. Feng, A. Rajaraman and F. Takayama, Phys. Rev. Lett. , 011302 (2003)[arXiv:hep-ph/0302215]; Phys. Rev. D , 063504 (2003) [arXiv:hep-ph/0306024].[22] J. R. Ellis, K. A. Olive, Y. Santoso and V. C. Spanos, Phys. Lett. B , 7 (2004)[arXiv:hep-ph/0312262].[23] J. L. Feng, S. Su and F. Takayama, Phys. Rev. D , 063514 (2004) [arXiv:hep-ph/0404198]; Phys.Rev. D , 075019 (2004) [arXiv:hep-ph/0404231].[24] L. Roszkowski and R. Ruiz de Austri, JHEP , 080 (2005) [arXiv:hep-ph/0408227].[25] L. Covi, J. E. Kim and L. Roszkowski, Phys. Rev. Lett. , 4180 (1999) [arXiv:hep-ph/9905212];L. Covi, H. B. Kim, J. E. Kim and L. Roszkowski, JHEP , 033 (2001)[arXiv:hep-ph/0101009].[26] X. J. Bi, M. z. Li and X. m. Zhang, Phys. Rev. D , 123521 (2004) [arXiv:hep-ph/0308218].[27] J. L. Feng, A. Rajaraman and F. Takayama, Phys. Rev. D , 085018 (2003)[arXiv:hep-ph/0307375].[28] R. Kitano and I. Low, arXiv:hep-ph/0503112.[29] M. Drees and X. Tata, Phys. Lett. B , 695 (1990).[30] J. L. Goity, W. J. Kossler and M. Sher, Phys. Rev. D , 5437 (1993) [arXiv:hep-ph/9305244].[31] A. Nisati, S. Petrarca and G. Salvini, Mod. Phys. Lett. A , 2213 (1997) [arXiv:hep-ph/9707376].[32] J. L. Feng and T. Moroi, Phys. Rev. D , 035001 (1998) [arXiv:hep-ph/9712499].[33] J. L. Feng and B. T. Smith, Phys. Rev. D , 015004 (2005) [arXiv:hep-ph/0409278].[34] K. Hamaguchi, Y. Kuno, T. Nakaya and M. M. Nojiri, Phys. Rev. D , 115007 (2004)[arXiv:hep-ph/0409248].[35] A. Brandenburg, L. Covi, K. Hamaguchi, L. Roszkowski and F. D. Steffen, Phys. Lett. B , 99(2005) [arXiv:hep-ph/0501287].[36] A. De Roeck, J. R. Ellis, F. Gianotti, F. Moortgat, K. A. Olive and L. Pape, Eur. Phys. J. C ,1041 (2007) [arXiv:hep-ph/0508198].[37] W. Buchmuller, K. Hamaguchi, M. Ratz and T. Yanagida, Phys. Lett. B , 90 (2004)[arXiv:hep-ph/0402179].[38] J. L. Feng, A. Rajaraman and F. Takayama, Int. J. Mod. Phys. D , 2355 (2004)[arXiv:hep-th/0405248].[39] K. Sigurdson and M. Kamionkowski, Phys. Rev. Lett. , 171302 (2004) [arXiv:astro-ph/0311486].[40] S. Profumo, K. Sigurdson, P. Ullio and M. Kamionkowski, Phys. Rev. D , 023518 (2005)[arXiv:astro-ph/0410714].[41] M. Kaplinghat, Phys. Rev. D , 063510 (2005) [arXiv:astro-ph/0507300].[42] J. A. R. Cembranos, J. L. Feng, A. Rajaraman and F. Takayama, Phys. Rev. Lett. , 181301(2005) [arXiv:hep-ph/0507150].[43] K. Jedamzik, M. Lemoine and G. Moultaka, JCAP , 010 (2006) [arXiv:astro-ph/0508141].[44] M. Drees, H. Iminniyaz and M. Kakizaki, Phys. Rev. D , 103524 (2007) [arXiv:0704.1590 [hep-ph]].[45] D. J. H. Chung, L. L. Everett, K. Kong and K. T. Matchev, JHEP0710