JHEP06(2018)043 GeV is . Springer June 7, 2018 May 22, 2018 May 30, 2018 April 29, 2018 : : : : Revised Published Accepted Received Published for SISSA by https://doi.org/10.1007/JHEP06(2018)043 . 3 1804.06783 The Authors. c Cosmology of Theories beyond the SM, Higgs Physics collider could cover a significant proportion of the parameter space for low

A scalar particle with a relic density set by annihilations through a Higgs portal , − [email protected] e + e University Place, Liverpool, L69 7ZL, U.K. E-mail: Department of Mathematical Sciences, University of Liverpool, Open Access Article funded by SCOAP mass dark matter, and future direct detection experiments will playKeywords: a complementary role. ArXiv ePrint: a decay rate thatpossible is without suppressed additional hidden by sector theself-interactions. states, Planck and We scale. this can also have Darkand study astrophysically matter collider relevant the with experiments. signatures a Searches ofor mass for such an invisible models Higgs at decays future at the direct, high indirect, luminosity LHC operator is a simple andcosmological minimal possibility history for this dark model matter.direct is However, ruled assuming detection a out constraints. thermal overhistory most We of Higgs show parameter portal that space in darkvalues by of collider theories matter the and with is portal a coupling, viabletheory evading non-thermal motivated for existing cosmological scenario limits. a of In wide a particular, range period we of focus of on dark matter the domination matter string due masses to and a light modulus with Abstract: Edward Hardy Higgs portal dark matter inhistories non-standard cosmological JHEP06(2018)043 (1.1) , and χ → − 24 χ Vector or fermion 11 1 2 symmetry 3 19 2 , the DM relic abundance could Z that is uncharged under the 10 − χ 10 ∼ 12 16 8 is tiny, H. † is in thermal equilibrium with the visible 5 λ H χ 2 χ 9 2 λ – 1 – 20 ] with a relic abundance set by freeze out during a L ⊃ − 4 – 1 through the renomalisible interaction is not very small. If H λ 21 ]. 6 19 , 5 8 1 23 The phenomenology is very similar ifThis the is DM is assuming a the complex coupling or real scalar, we take it to be real throughout. 5.1 Collider searches 5.2 Direct and5.3 indirect detection Astrophysical effects 4.1 Regimes of4.2 DM in non-thermal The cosmologies parameter space for Higgs portal DM 2.1 Direct, indirect, and collider constraints 3.1 String moduli3.2 Non-thermal cosmological histories 1 2 DM with a relicpossible, abundance although set generating a by mass annihilations for through the a former requires Higgs a portal hidden operator sector Higgs isbe field generated also from or freeze in [ couples to the SM Higgs field In models with a thermalsector cosmological in history the early universe, before freezing out and acting as DM. Higgs portal dark matterstandard, thermal, (DM) cosmological [ history isimplementation a only minimal one and new predictiveStandard state model. Model is In (SM) introduced, its gauge a simplest group. scalar This is stabilised with a A Evolution of the DM number density during matter domination 1 Introduction 6 Conclusions 4 Higgs portal DM with a non-thermal cosmological history 5 Prospects for future observation and phenomenology 3 Moduli and late time matter domination Contents 1 Introduction 2 Higgs portal DM with a thermal cosmological history JHEP06(2018)043 , ], 2 26 – 24 ]. 9 – 7 GeV and a relic . ]. If this is fairly low, ]. Only fermion DM 17 12 – 13 ]). Astrophysically relevant ]. As we discuss in section 34 11 GeV. Despite string theory models . ]. In this paper we show that allowing DM 33 m – ] and PandaX2 [ 28 . – 2 – 10 ) against Higgs portal interaction strength with DM m ], more general DM hidden sector candidates [ 23 – 20 ], and axions [ 27 GeV perturbativity bounds mean that significant self interactions are ]. For example, this has previously been considered for weakly interacting & 19 , DM 18 m GeV can lead to the observed relic abundance, and there is a significant region & Although an increase in the range of allowed DM masses and values of the portal It is well known that altering the cosmological history of the universe increases the However, there is no strong reason to think that the cosmological history of the universe While their simplicity is appealing, minimal Higgs portal models are severely con- DM possible only if additionalself interactions light could mediators resolve conflicts aresmall between introduced simulations scale of [ structure, cold although DMphysics and these observations may in of also simulations. behistory due indirect to In detection incomplete contrast, searches inclusion in already of rules baryonic almost out all DM models with a with mass a thermal cosmological lead to similar dynamics. coupling is not surprising, itSuch is DM interesting candidates that can lowscales have mass self (for DM interactions is that possible are in relevant simple for models. dynamics on galactic points in them plane of DMof mass viable ( parameter spaceproviding with a 100 motivation for MeV ouron work, we such stress a that UV the completion, results and obtained are any not model dependent with a sufficiently weakly coupled scalar could massive particles (WIMPs) [ hidden sector glueballsa [ non-thermal cosmologicalminimal history Higgs also portal DM significantly models.of increases By the adjusting the lightest the parameter modulus, reheating temperature and space after the for the branching decay fraction of the lightest modulus into DM, all Planck suppressed operators, and aconsistent radiation with dominated measurements universe of re-emerges the before abundances BBN, of light elements. parameter space over which theDM required models relic [ density can be obtained in a wide variety of building perspective. Instead, in typicalfields, string the theory mass models of there which aremotivated is many for scalar typically moduli example set by byof the the gravitino the success mass SM, [ of theleading gauge to unification moduli an extended in will period supersymmetric dominate of extensions matter the domination. energy The moduli density eventually of decay through the universe at early times portal DM iswith only pseudoscalar possible couplings to for thespace, Higgs a remains since viable limited this over has range significant an parts of of extremely parameter suppressed DM signal masses at [ direct detectionbefore experiments. big bang nucleosynthesis (BBN) was thermal, either observationally or from a model and additional new states must be introduced at an intermediate energystrained scale by collider [ limits ontion the searches, invisible most branching recently fraction at of Xenon1Tafter the [ Higgs imposing and that direct detec- the relic density matches the observed value, scalar or vector Higgs the Stueckelberg mechanism, and in the latter case the portal operator is non-renormalisible JHEP06(2018)043 ] 43 2 that / by Planck h eff ]. we discuss N < m 3 ). As a result, 300 GeV [ 39 , < 1.1 38 DM ] (for example, to m 37 DM m ] fixes the required value 40 , defined in eq. ( Instead, models with a thermal λ 3 ]. 35 – 3 – ] such models remain viable in a small part of parameter space. we consider the potential future experimental signatures. GeV. Despite strong constraints from measurements of ∆ 36 5 ∼ we study the viable parameter space for Higgs portal DM in 4 ], and we obtain our results by numerically integrating the Boltzmann and the value of the portal coupling ] in taking the 1-loop corrected cross section at large DM masses, and we summarise our the results. 41 42 , 6 DM m 41 , 12 gauge boson, with mass 0 Z for a given DM mass. The computation of the relic abundance in a thermal cosmology In this simple model there are only two free parameters that affect the DM relic abun- Turning to the structure of this paper, in the next section we study the status of Higgs Higgs portal DM can also lead to a diverse range of observational signatures in direct, The exception is a Majorana fermion or scalar DM candidate interacting with the visible sector through 3 λ section, we follow [ using results computed for the SM Higgs widtha at new smaller masses and from fixed target experiments [ the precise measurement of theof DM relic abundance by Planckis [ standard [ equations for the DM numberSM density. The Higgs possibility means of resonant thatnumerically, annihilation rather the through the than thermal using average an of analytic the approximation. annihilation To rate find must the be annihilation cross carried out attempting to fully exploreit the is uncertainties sufficient on for all ourAdditional of present recent the purposes discussion to relevant and experimental summarise reviews the searches, may broad be features found of in, thedance: for constraints. its example, mass [ 2 Higgs portal DMWe with focus on a the thermal case cosmologicalmeasured of a DM history real relic scalar abundance. DMnation, For candidate, comparison we and with require first that theories analyse this with this makes late model up time the assuming matter full a domi- thermal cosmological history. Rather than the possibility of a periodstring of theory. late time In matter domination, section such and theories, the motivation and for in thisFinally, section from in section would otherwise not be expectedexample, (unless we a will more complex show hiddencould that sector be there is explored are introduced). by many searches For non-thermal for invisible models Higgs with decaysportal at DM the models high assuming luminosity LHC. a thermal cosmology. Following this, in section a result, the motivation forsignals that a could coupling be between probed the visible by future and experiments, DM toindirect, sectors, be and and present therefore collider at experiments. alluniverse is more In diminished. complicated, returnthe for albeit models making in that the a we cosmological way consider history that can of is lead the to motivated signals from in UV parts considerations, of parameter space where they cosmology and DM into this annihilate mass into. range As typically wellavoid require as the additional involving new extra more states particles complexthe for model over-closing effective the building the number DM [ universe of themselves relativisticthe link degrees or between of violating the freedom constraints DM in annihilation on the rate to early the universe), visible this sector removes and its relic abundance. As density set by annihilations to the visible sector [ JHEP06(2018)043 4 > ]. W 44 -floor), m ν 4 10 . Direct detection experi- CTA AMS DM 3 10 floor -bosons and the SM Higgs. At ν- Z 2 /GeV 10 Fermi ]. Given that the viability of DM extremely close to DM 45 m – 4 – 2 in which resonant annihilation permits relatively / that leads to the experimentally measured DM relic h DD m 1 (this closely matches existing results in the literature). DM are the masses of the W-boson and b-quark respectively. ' 10 m 1 b DM m DM m Ω=Ω and against W λ h invis. m 1 1 are required. For DM with a large mass, annihilation is dominantly to 2 3 4 1 1 . - - 10 - - 0 10 10 10 10 & λ , where b λ . The constraints on scalar Higgs portal DM models assuming a thermal cosmological > m The contour of There might be a relatively minor inaccuracy in our calculation of the relic abundance for DM masses 4 -bosons, with significant branching fractions also to DM in the range whereat resonant around annihilation the through same theresonance time SM is Higgs as not occurs, our chemical main if decoupling focus, this [ we leads neglect to this kinetic possible decoupling effect. small values ofcouplings the portal coupling.W However, overtimes the when the remainder DM is ofm not parameter highly relativistic, space annihilation is primarily to b-quarks if the effects of 3 and 4 body final states, which can change the cross sectionabundance significantly is [ plotted inThere figure is a region around experiment is shown, and thisbelow could which direct probe detection the searches high become DM extremely mass challenging, region. is The also neutrino shown. floor ( (with the addition oflowed). the The extra tabulated channel values into of the the Higgs SM width Higgs include when the this leading is QCD corrections kinematically and al- coupling as a functionments of (DD) the lead to DM strong mass constraints,(h to and invis.). LHC the Indirect searches contour detection for invisible labeled searcheslatter Higgs by is Ω decays Fermi highly = are and sensitive also Ω AMS to relevant arespace astrophysical potentially in uncertainties constraining, (these which although are they the plotted are only competitive in with parts direct of detection parameter searches). The reach of the future CTA Figure 1 history. Requiring that the relic abundance matches the measured value fixes the value of portal JHEP06(2018)043 . N 24, f . 0 (2.1) (2.2) < ]. ) 50 SM , 49 [ ]. The current + Γ N 56 f – invis 54 ]). For clarity, we do (Γ 53 / ] sets stronger limits (at 52 invis , , which gives a lower limit on the i 5 , N 4 h | 2 N m qq | 2, which does not change the qualitative 5 GeV CDMS [ m ), and we use this value in evaluating the 2 the nucleon mass. At leading order in a 2 DM N ∼ . µ h m 2.1 N from the LHC is Γ q N 2 π N m DM – 5 – m 4 f ]. Recently the scattering cross section has been m m 2 invis λ q 47 , , in such models can be parameterised as X = 1 are required, close to the limit of perturbativity. We N = 46 σ ), with , & 2 GeV, CRESST-III is relevant [ 018 in eq. ( . N by a factor of 41 f 0 DM . ], while at λ m  51 DM + ]. The result of this calculation can be incorporated in the cross m 308 N . 48 m ( = 0 5 / N 2, Higgs portal DM models can lead to a significant SM Higgs invisible f DM ]. This bound is dominantly from direct searches for invisible decays, / h m we show the current direct detection constraints on the portal coupling 58 N , 1 ] and PandaX2 [ < m m 57 is the SM width, assuming that the Higgs production rate is given by the SM 10 = DM denotes the SM quarks [ SM µ q m If In figure Over the majority of parameter space, the dominant constraint on scalar Higgs portal This is compatible with the values and uncertainties given in other extractions of 5 where Γ prediction [ which lead to a largefor missing example momentum in after combination associated with vector particular boson visible Higgs signatures, production. Such limits are stronger than also plot the neutrino floor,DM discussed scattering further cross in section section that a conventional direct detectiondecay experiment rate, can and discover. colliderconstraint searches on for the Higgs such invisible processes width can Γ be relevant [ These can alter theinterpretation bound of on the results. Thesemodel bounds over already the rule majority out of the parameter minimalvia space, scalar the except Higgs for visible portal a sector region Higgs wherecase, is resonant values annihilation possible, of and the a portal region coupling at very large DM masses. In the latter arising from spin-independent scattering.Xenon1T At [ large DM masseseven these smaller are DM dominantly masses, from not plot the uncertainties arising from the experimental set-up, astrophysics, and from where computed to higher orderto in two chiral nucleon currents perturbation [ theory,section including by the taking effects ofexperimental couplings bounds. where chiral expansion models comes from directDM detection scattering experiments. off nucleons, The labeled cross section for spin-independent The simple nature ofand DM the mass Higgs are portal chosen,are the model observational fixed. signals also in Uncertainties means direct,knowledge on indirect, that, of the and once the collider constraints searches DM the obtained densityand portal arise close the coupling from, to quark the for content of galactic example, nucleons. centre our and limited in the vicinity of the Earth, 2.1 Direct, indirect, and collider constraints JHEP06(2018)043 . ]. 1 67 (2.3) ], and 65 [ DM ]. m 75 – ]. In the minimal ]. For DM masses 73 63 – 60 , 61 40 GeV, however given 12 ]. This has been studied ], but it can not be easily . 72 69 – , DM ]. The Fermi experiment has , 70 68 as a function of m 66 – λ 2 h 2 DM 64 m m 4 7 − , could be sensitive to couplings down to 1 6 s ]. Even for fermion Higgs portal DM with 2 is already excluded by the existing collider h 2 / 59 v h m – 6 – π 2 λ 32 < m 6 = DM m invis Γ (in parts of parameter space where they are comparable to 1 ]. To obtain a present day annihilation cross section that matches the collider, discussed in section 12 − 2 other signatures at particle colliders are possible [ e / 2 GeV, the corresponding bound on the portal coupling is plotted in figure . h + e m . However, this will only constrain a small part of the viable parameter space , around the resonance region [ 5 . 1 2 = 246 − v 10 Higgs portal models can also lead to signals in indirect detection experiments, as a Future improvements in searches for invisible Higgs decays at the high luminosity Direct detection bounds on models with lowThe mass galactic DM centre might are be weak, easier but to these fit are in ruled more complex out Higgs by portal indirect models [ 6 7 ' smaller DM masses, values of thepredict portal a coupling too that small lead annihilation to the cross observed section relic today. abundance detection constraints from Fermi and observations of the cosmic microwave background. accommodated [ tentative signal, while alsomass being very compatible close with to the thea bottom measured present of DM day the relic annihilationsection resonance density, cross in is a needed. section the DM early that For slightly universe matches larger that the DM is excess masses, too corresponds small, to leading a to an cross over abundance of DM. For direct detection constraints).ruled The out, region just since above inlarger the this than lowest case in point the the ofvelocity early annihilation today. the universe cross Fermi resonance when has section the is also ingalactic relic observed the centre, a abundance which present possible is could day excess set,in in be universe due the gamma is due to context rays of to the emitted the DM smaller from Higgs annihilations DM the portal [ model that we consider [ result of late timesearched annihilations for gamma to ray the emissionthe visible from DM dwarf sector spheroidal annihilation [ galaxies, cross leadingThe section to null evaluated constraints results at on can thethese again typical are be present plotted interpreted day as in DM bounds figure velocity on [ larger than model that we consider thesebe are significant if far the weaker state thanthan that direct the is detection DM coupled bounds, to itself. the but Higgs they is could a mediator to a dark sector, rather λ in figure pseudoscalar couplings, for which theall direct of detection the bounds parameter arelimits, space extremely and with weak, further almost improvements will only have a limited impact [ Alone this would rule outrecent scalar progress Higgs in portal direct modelsspace detection with that searches is it not does otherwise not excluded. presently constrain any parameter LHC or an Using the prediction for the invisible Higgs decay rate in the Higgs portal model, where indirect constraints from the measurement of the decay rate of the Higgs to SM states. JHEP06(2018)043 ], for 76 we show the 1 ]. For the minimal ], and the bounds ]. This is aimed at 80 GeV. However, 84 82 – 76 . , 80 100 TeV. However, the 47 DM , ∼ ], CTA will not be sensitive ]. m 41 77 90 the constraints obtained using . ] and BBN [ 1 83 3 in the value of the portal coupling, 10 GeV and ∼ ], as well as potentially enhancing indirect ∼ 88 – 7 – ] are plotted. If other propagation models are ]. As a result, measurements of the antiproton 81 78 ], which can reduce the relic abundance and permit smaller 85 8 ], and can lead to viable parameter space. Direct detection constraints ]. ] constrain the value of the portal coupling [ ]. Sommerfeld enhancement of the DM annihilation cross section can 86 89 79 87 . Additionally, models containing two real hidden sector scalar fields with a λ In the remainder of this paper we show that changing the cosmological history of Late time annihilations can also result in a significant rate of positron production in The currently viable region at large DM masses could be explored by the Cerenkov It has recently been suggested that for sufficiently heavy DM, Sommerfeld enhancement from the SM 8 also have an important effectand in can more complicated allow models models with withdetection additional large signals new DM [ light masses states, [ Higgs itself could have a significant effect on the DM relic abundance [ values of softly broken U(1) symmetry coupledhave to been the studied [ visible sector throughcan a be Higgs dramatically portal suppressed operator U(1) in some symmetry models [ with two real scalars with a softly broken the universe can alterDM the relic value abundance, of andevade consequently the experimental can portal constraints. allow coupling modelsintroducing required In with a to scalar passing more Higgs obtain we complex portal thenew note hidden DM sector. annihilation measured that to channels Adding this [ extra could hidden also sector be states can achieved lead by to similarly to the Fermiportal galactic model. centre excess, Finally, models this withbounds is from small hard the DM cosmic to masses microwave background are fitHiggs (CMB) constrained portal within [ by model the indirect these minimal are detection weaker Higgs than the limits from searches for Higgs invisible decays. the propagation modelassumed described these in results [ canand differ some by propagation a modelsalready factor excluded lead of by to direct weakerflux, detection bounds corresponding searches. that to are There DM further is masses a inside in claimed the the excess region range in 20 the position GeV of electroweak (EW)flux gauge by bosons AMS-02 [ [ obtained can be comparablemass to region. those from However,from direct these the detection propagation limits experiments of are positrons in in subject the the high to galaxy. DM In significant figure uncertainties, dominantly which the DM densitythis is region, relatively for high example at asto the predicted any galactic by parameter centre. the space Einasto Ifthe that profile the model is [ DM we not consider. is already less ruled dense out in by directHiggs detection portal experiments models, for both for DM masses above and below the threshold for production Telescope Array (CTA), which isdetecting currently gamma in rays development with [ expected energies reach between of CTA iscentre, strongly and dependent this on isprojected the currently DM sensitivity subject of density to CTA profile assuming significant near the uncertainties. the optimistic galactic case In of figure an NFW DM profile [ JHEP06(2018)043 ], ]. 91 92 (3.1) 9 is parametrically ψψ → ]. φ 100 , , ]. Other possibilities for modifying the ]. 99 2 3 φ Pl ]. In this case, the direct detection cross 97 m M – 12 c 102 , – 8 – 93 = , 101 φ 20 Γ ], but this is less motivated from a UV perspective. 98 ]. Meanwhile, in models with fermion DM with scalar couplings to the Higgs, 12 Such a model automatically leads to a period of late time matter domination. 10 The decay rate of a string modulus to fermions Motivated by the success of gauge coupling unification when the SM is completed to Fermion Higgs portal DM with a pseudoscalar coupling to the SM Higgs also remains Other possibilities for non-thermalWe histories assume include that [ supersymmetry breaking is mediated to the visible sector by Planck suppressed opera- 9 10 tors. In models of gaugehistory mediation, in obtaining combination moduli with masses visible that are sector large soft enough masses for at a viable the cosmological TeV scale is challenging. EW scale (which also solvesthe most gravitino of mass the and hierarchyscale. the problem), we mass consider of models the in lightest which modulus are relatively close to the TeV tion, however the lightestorder is as expected the to gravitinothe have mass. moduli a can mass Additionally, be in that calculated some is explicitly string parametrically [ constructions of the thethe properties Minimal same Supersymmetric of Standard Model (MSSM) with soft masses not far above the UV completions of thewhich SM are by associated string theory tothat models deformations are typically of compactified include the scalar to sizestabilising moduli get or the fields, extra a shape dimensions, 4 of for dimensionalmoduli the example through getting low extra non-perturbative masses. energy spatial effects, dimensions effective The leads spectrum theory. to the of The moduli process depends of on the details of a compactifica- history has beencosmological studied history extensively include [ aDM period relic of abundance kinetic [ energy domination, which also3.1 alters the String moduli Turning to the main focuscosmological of history due our to work, the we presencedecay now of rate. a consider relatively theories light that scalar Ifleads with have the a a to highly non-thermal scalar a suppressed is perioddecays. displaced of The from possibility late that the time late matter minimum decaying domination, particles of could followed its alter by potential, the entropy universe’s this injection cosmological typically when it the extra states required to UV completeand the more portal complex operator Higgs can have sectors important effects could [ lead to blind spots in direct3 detection searches [ Moduli and late time matter domination viable over large partssection is of velocity parameter suppressed space anddepending extremely [ on weak (and the probablyrate below present the is day neutrino not floor, DM suppressedrelevant velocity [ at distribution). low velocities, In and contrast, future its indirect detection annihilation searches could be JHEP06(2018)043 (3.2) , that φ m is the reduced Pl M , with mass φ ], since around this value ], and 101 [ 101 Pl ]. M , , when it begins oscillating around φ 104 ] if such a UV completion is realised, , i ∼ m = 0 φ h 103 ) is the scale factor of the universe. In φ ' 105 t 2 φ ( )[ m a osc 3.1 H . As a result the lightest modulus generically + φ – 9 – ˙ φ m H & , where 3 + 3 I ) ¨ t φ H ( /a 1 ∼ smaller than to the visible sector. 8 10 ∼ In many string models, the lightest modulus mass eigenstate is a mixture 11 is the Hubble parameter. Consequently, the modulus remains fixed at its initial is a constant that is expected to be of order 1 [ c H The coherent oscillations of the modulus act a non-relativistic matter contribution We assume that inflation occurs at a scale that is not extremely low, in particular that While string moduli provide a motivation for studying cosmological histories with a Although this is a good approximation in many string models, the modulus decay rate could be dra- 100 TeV. As a result, superpartners of visible sector states might have masses that 11 to the energyergy density density of is the diluted universe, as and aftermatically the different in modulus compactifications with beginsenhanced an by exponentially oscillating powers large of internal its the volume volume and en- compared modulus to couplings eq. ( where VEV until the Hubblethe parameter (zero drops temperature) to minimum of its potential. interesting new dynamical regimes (we intend to return tothe this possibility Hubble in scale future during work). gets inflation a vacuum expectation valuehigher (VEV) dimension of operators order inevolution its is potential given become by significant. The modulus’ subsequent 3.2 Non-thermal cosmologicalFor histories simplicity we focus on modelsis in substantially which lighter there than is the aby single others. the modulus The dynamics final ofexpected DM this to relic have modulus. abundance multiple is moduli More with then similar generally, determined masses string and compactifications decay rates, are which typically could lead to although we stress this isthe certainly apparent not a link sharp betweenvisible prediction. the sector In superpartners gravitino the would and opposite stronglyhistories motivate moduli direction, the that given masses, types we of discovery consider. non-thermal of cosmological relatively light soft masses are comparable tomass. the For gravitino a mass, and significanthistories therefore proportion with also of interesting the the effects lightest. theories modulus on that the lead DMcould to abundance, be non-thermal this discovered cosmological mass at scale a is future relatively hadron collider low [ period of matter domination,such the a dynamics UV completion, that and welead any subsequently to theory similar describe with effects. do a On sufficiently notsuch the weakly other a rest coupled hand, string on it scalar theory is field model interestingdetails could to are ask of possible. if phenomenologically complementary Although signals viable there of is compactifications, still in large many uncertainty about models the the visible sector Planck mass. of multiple UV moduli, andand in visible this sector case states. it However,model, often given has we that relatively assume we similar do that couplings notDM the to commit that lightest the to is modulus a DM a might particular factor plausibly underlying have a branching fraction to where JHEP06(2018)043 . φ m (3.3) , and 8 ∼ / 3 does not − max a ), which is . However, T φ RH ]. Although ∼ Γ RH T T 95 T ( , is comparable ∼ . Since the mod- 2 Pl 4 H ) t M ( = 2 φ φ m /a 1 10 MeV, and a modulus drops to ∼ ∼ & H 2 , Further, we find from a nu- i φ φ RH Pl h 12 T 2 φ M cm m such that Γ , it quickly comes to dominate the r φ 2 Pl ' RH m M φ T 4 ρ / 2 osc 1  H ) . ) (the precise lower bound on 3 is the time when the modulus begins oscillat- / RH 2 3.1 ∼∼ – 10 – t T osc ( 90 t ∼ g rad 2 ) ρ t π ( is the energy density in the modulus, and the universe’s a  φ ρ = (1), where RH T , where ∼ O
of order 1 in eq. ( 2 Pl c osc M 3 t/t , until the majority of the modulus quanta have decayed. 4 / T φ ρ ), ) is the effective number of relativistic degrees of freedom at reheating. ∼ = 3.1 RH H 2 T 10 TeV, for ( H ], soon after the onset of matter domination and long before radiation domination g & 19 Instead, in a simple model with a single modulus and a period of radiation domination Although matter domination continues until relatively late times, the modulus begins A lower bound on the mass of the lightest modulus comes from the observational Matter domination continues until the Hubble parameter drops to be comparable to Once it begins oscillating, the energy density in the modulus is greater than in primordial radiation due 12 independent of whether there isoscillating any or energy in not. the radiation That bath is, when the the modulus dynamics starts thatto set redshifting, and the almost all DMonce of the relic the modulus energy abundance density completely in happen decays. the after modulus is eventually converted back to radiation and this is supplementedtemperature by new of energy the producedthe universe from energy modulus never from decays. increases primordial As duringrelatively radiation a can late the result, be times, modulus’ the greater itmerical decay than is analysis that [ always that from eventually in modulus subdominant. all decays of until the parameter space of interest the final DM relic density is ing [ re-emerges. Subsequently, the temperaturetherefore of the radiation bath drops as before the modulus begins oscillating,the radiation there bath. are two First, contributions there to is the energy energy previously density in of the bath before matter domination, energy stored in the modulusthey field still to have radiation an (orthere important is DM) impact no until on energy the indecays system’s the mean dynamics radiation that bath and the when the theThis maximum final modulus temperature occurs DM first reached when yield. starts by oscillating, the If the bath early is of order this process. This correspondsmass to a reheating temperature significantly modify the possible DM phenomenology). decaying as soon as it starts oscillating. These early decays only transfer a small fraction of where requirement that it decays before BBN, so that the universe is radiation dominated during the modulus decay width,convention when of the defining majority the ofapproximately reheating the the temperature moduli temperature quanta whenFrom decay. the eq. universe We ( passes follow back the into radiation domination. ulus begins oscillating whento the its energy energy in density, universe. the During radiation the bath, subsequentmately matter dominated era thescale Hubble factor parameter increases is with approxi- time as contrast, the energy density in the radiation bath decreases as JHEP06(2018)043 ]). 25 (4.1) 13 ]. 113 – 107 , , interactions between visible sector vis DM → φ m DM & + Γ → T φ Γ DM – 11 – → φ Γ ). For example, non-thermal cosmologies have also = ]). The multiple moduli that are expected in typical 3.1 26 B ) (this does not significantly change the phenomenology) 3.1 are the modulus decay rates to DM and the visible sector re- vis → φ ]. However, for a given final reheating temperature, the viable DM = 1 in eq. ( 106 and Γ c DM → φ ] (WIMP DM models with a more complex hidden sector have also been studied [ We calculate the final DM yield by numerically integrating the energy density in the The possible dynamics of Higgs portal DM models in theories with a non-thermal cos- As mentioned, as well as string moduli we could also consider theories containing DM production from decays has been studied in, for example, [ 24 13 and the possibility of resonant annihilation. modulus, radiation, and DMthe during the coefficient evolution of the universe. For definiteness, we fix mological history are similar toin those [ of WIMP DM candidates,One which difference have is been classified thattemperature the dependence, annihilation due cross to section new of channels Higgs opening portal up as DM mass has thresholds an are unusual passed spectively. If the temperature isstates high in enough the thermal bathsufficiently can also large, produce DM DM. annihilations Meanwhile, totropy if is the its injected visible number into density sector the becomes will systemthe as be DM long relevant. as yield. Additionally, modulus en- decays continue, which acts to decrease During matter domination DMparameterised can by be the modulus produced branching directly fraction from to the DM, decays defined of as the moduli, where Γ 4 Higgs portal DMA with non-thermal a cosmological non-thermalthe history cosmological early history can universe dramatically and alter its the relic dynamics abundance, compared of to the theories DM with in a thermal history. change is that amass, given due reheating to temperature the corresponds modifiedto decay to a rate. a different saxion/ As different modulus a saxion/the result, number branching modulus a density, fraction so particular into that, energy DMDM in density (discuss relic some corresponds abundance in parts will more of change. detail parameter space, shortly) that leads to a particular weakly coupled scalars thata lead to different a parametric period ratebeen of to proposed matter eq. domination, to ( but ariseaxion) which due in decay F-theory to with models, the anddecay dynamics this constant of can [ the have a saxionparameter decay (the space rate scalar will that is counterpart be suppressed of similar by the to the that axion in a model with a string modulus. The primary energy from the modulus decay, and(this these dynamics matches are the independent of conclusion thestring prior in models conditions [ will resultany in initial an energy even in longer the period radiation bath of even matter less domination, significant. which will make the primordial energy in radiation is sufficiently redshifted that it is irrelevant compared to JHEP06(2018)043 ]). (4.2) 114 ] (around 26 is increased X . For DM masses 2 , is its equilibrium value R , which reproduce those ρ eq 4 dT dA 4 RH X A T = increases due to modulus decays ], and it is convenient to define a , and R 19 100 MeV, since obtaining the mea- , where dA dR 2 & are the energy densities in moduli and , X ,R R dA dX χ ρ − remains constant when no DM states are n , 2 eq 3 Φ 3 d dA X RH and X GeV a significant fraction of annihilations are A T φ – 12 – . ρ = DM . Further, we work in terms of dimensionless variables m (1). We include the effect of this following [ RH . , for Higgs portal models to be viable over large parts of O ,X aT 1 φ ρ = 3 A 4 RH A T Φ = we give expressions for A is the DM number density, and χ n the number of degrees of freedom can be obtained from the equation of state [ ]. Φ decreases with time due to modulus decays, Given the results in figure We primarily consider models with DM masses In appendix 26 QCD DM can have different dynamics.would The occur resulting relic in abundance a can model be with larger or the small same than DMparameter properties space and the a DM thermalcosmology. cosmological relic This history. abundance can must happen be in reduced different relative ways to and, that although in for a our thermal final results we will reproduce UV complete theories, sinceapproximation these for are the expected annihilation to cross contain section multiple is moduli, sufficient this for4.1 our current purposes. Regimes of DMDepending in on non-thermal the cosmologies reheating temperature and the modulus branching fraction to DM, the to muons and this isthe relatively relic unaffected abundance by in QCD thishilation corrections. DM cross mass An section range accurate to would calculation hadrons,QCD require of and plasma, both the which the inclusion could non-perturbative plausibly oforder anni- change finite the of temperature required magnitude. effects value due of However, to the given the portal that coupling we by do an not expect our simple model to precisely tiny value, which would be challengingalready to experimentally observe excluded. experimentally, or For a simplicity,to large we SM value, use quarks, which leptons, the is gauge DM bosons,below annihilation and the cross the QCD section Higgs, scale, described the injor cross section uncertainty, section although for for annihilations 100 to hadrons will be a source of ma- at a giventemperature temperature. of the The universe numberrelic drops abundance of during by the relativistic a modulus degreesΛ factor decay, and of of this freedom can changes affect as the DM the sured DM relic abundance for smaller masses requires that the portal coupling either has a in [ and can also beby modulus slightly altered decays bythe and thermal energy increased bath transfer or depending to decreased on or by the from production sign the of from, DM. or annihilation to, where radiation respectively. Withproduced this or definition destroyed, even when entropy in injected into the universe by modulus decays. model is therefore specifiedmodulus by branching fraction four to parameters:mining DM, the and the evolution the portal of reheating coupling, thedimensionless temperature. DM scale the The factor yield DM equations are mass deter- well the, known [ giving a definite relation between the modulus mass and the reheating temperature. A JHEP06(2018)043 is (4.3) (4.4) 3 prod T ∼ n is the entropy s , a Hubble time is , where i due to entropy injec- pro (where σv 5 h ) 100. As a result the DM . , 2 /s χ n . i DM 1 H T > T n 14 ∼ χ σv /m h . Subsequently, the DM yield dt dn . Combining these factors, DM Pl 6 RH /dt . Therefore, the final DM yield is − DM 4 M 3 T χ 3 . Subsequently the comoving DM pro ) T 3 DM m 2 RH dn 2 due to the expansion of the universe, /T T / DM RH 1 3 5 gm 8 T m 2 ) = ( T ( − Pl ∼ π 45 /T 2 /a M pro ∼ ∼ 3 2 H T 8 ) / pro ∼ ( ) is the DM annihilation cross section evaluated T 21 φ . This is i χ pro s – 13 – pro /y m n T ∼ ) T pro ( 9 σv ( 3 h T a − ) and any DM produced at this point is diluted by RH vis ) = T 10 T 4 ( ( ) = /n pro ∼ y /T ) /a T pro ( 3 ∞ T T 4 ) ( pro y y in this regime. As a result, contours of constant DM relic ( T (dropping a dimensionful constant). χ i pro vis fixed would correspond to portal couplings that varied with the 2 T n ∼ / /n ( σv 5 DM h ) remains constant, and the DM yield a as mass thresholds are crossed is important. However, we note that RH m 5 T RH X − DM ∼ T DM ( m λ . At earlier times when the temperature m χ Pl n DM M on m 7 RH was temperature independent, the DM relic abundance would be proportional , and we have ignored numerical factors that are pro T i T ∼ pro i | ∼ σv T h pro σv abundance with DM mass as In calculating the viable DMh parameter space, the strong temperatureif dependence of to and during the matteris dominated decreased era by ation, since factor of approximately the number density of SMat states, yield at the time of production is is dominantly createdtemperatures. from This the remains thermal theof case bath the even at once cross they section the strong in lowest temperature HiggsThe kinematically dependence dependence portal of models allowed the is DM relic included. from abundance the on a DM model’s production parameters can rate be at estimated T shorter by aan factor extra factor of although the DM productionvisible rate sector is states enhanced by the increased number density of equilibrium is not reached.thermal bath Production temperature becomesnumber drops inefficient density below once thedensity) visible is reduced sector by entropy injectionIn from this the modulus case, decays. most of the DM is produced when the visible sector temperature is DM production from the thermalIn bath, this without regime chemical DM equilibrium. sector is thermal dominantly bath. produced If the fromsity portal will scatterings coupling never of is be states sufficiently large small, in enough the the for DM visible annihilations number den- to become significant, and chemical Freeze in via a renormalisible operator is also IR dominated in a radiation dominated universe, and the • 14 stronger dependence of thetemperatures Hubble even parameter more strongly. on the temperature during matter domination favours low of the relic abundance on the model parameters can be understood analytically. rely on a numerical integration of the Boltzmann equations, the approximate dependence JHEP06(2018)043 , DM (4.6) (4.7) (4.5) . As m 3 φ ∼ /m at this time, was temper- 4 DM i φ m constant would (at which point σv ∼ /m h φ RH in this regime, and m RH H T λ ' Bρ T ' 1 . . χ − i n 3 1 / . is crucial. If − 1 2 i σv / i h  3 . 2 σv − DM σv / h h 5 RH m φ 1 Pl − Pl T GeV − Pl m ∼ M M M λ 4 r   4 − DM B − DM 2 m . DM m 2 0 / GeV m 9 φ 3 RH ∼ – 14 –  m T 3 ∞ 01 01 − y . . 0 0 10 ∼ ∼ ' ∞ B y . Including the extra entropy injection after freeze out, the is the modulus energy density when  i 4 RH σv h gT φ 3 2 m 30 π  ' / ). The final DM yield when radiation domination begins is therefore , when most of the moduli decay. If the DM number density produced is DM 4 RH ρ rad RH m ρ T ∼ ∼ = χ φ Contours of constant DMobtaining relic the abundance measured are DM independent relic of abundance requires a modulus branching fraction T not large enough for annihilationswhere to be significant then ρ correspond to portal couplings that vary as Production from modulus decay withoutIf reaching the chemical equilibrium. modulus branchingthe fraction main to source DM is of sufficiently DM. large, In direct this decays case, can be the majority of the DM is produced when As before, the strong temperatureature dependence independent, of contours of fixed DM relic abundance with Freeze out takes placea when result, the DM Hubble annihilationsn parameter remain is efficient roughly untilfinal the DM DM yield is number approximately density drops to a radiation dominatedannihilations universe, become the slow DM compared toof number chemical the density equilibrium. Hubble Subsequently initially parameter the andfrom decreases, DM the the yield modulus until is DM decays, decreased which falls by reduces out out entropy its injection final had relic occurred abundance with compared to a if thermal freeze cosmological history. during matter domination. If the portal couplingefficient is enough sufficiently that large, annihilations productionequilibrium. become from This significant persists the and until SM the thewhen temperature DM thermal of the reaches bath the equilibrium thermal chemical is DM bath drops number to density drops rapidly. As in normal freeze out in Production from thermal bath, with chemical equilibrium and freeze out There are also other possible DM dynamics in non-thermal models. These do not • • reduce the DM relicand abundance are compared therefore to less models phenomenologically with interesting a for thermal our cosmological present history, work. JHEP06(2018)043 . ), & 2 i 4.2 σv h 1 χ n 7 3 5 1 ) as a function - - - - m ]. Since we are 10 10 10 1 1 10 X - 23 B=10 (defined in eq. ( )[ i / GeV X 1 T , this results in a larger σv h 1 100 DM Pl m 15 M (  / 2 1 RH RH in models with different values of the 1000 λ=0.1 T = 0, so that DM is only produced from T X B 4 10 i ∼ as the temperature of the universe drops 10 0.01 σv X m h X X H/ 1 10 . – 15 – 4 2 6 - - - 2 ' 10 10 1. At early times DM is produced from the thermal bath, . the evolution of λ= 10 DM n = 0 λ , in non-thermal Higgs portal DM models with different values T Right: 1 1 100 - / GeV . The reheating temperature is fixed to 10 GeV, the DM mass is set at 1 T λ the evolution of the comoving DM number density the relic abundance is set by freeze out in a radiation dominated universe. 1 1000 DM Left: m B=0 ∼ . , and when this is violated freeze out occurs. The DM relic density is the same as 5 RH - 10 visible sector immediately afterdomination. In reheating, this case when theH the DM remains universe in chemical returns equilibrium as toif long the as radiation universe had beenparameter radiation space dominated studied throughout in its history, section reproducing the interested in parts of parameterrelic space abundance for than which the thermal freeze out prediction. Freeze out after reheating. If its mass is sufficiently small, the DM can be in chemical equilibrium with the If the modulus branching fractionproduced to from DM decays is and sufficientlystatic it large, equilibrium, can with the reach production DM chemical balanced iscompletely equilibrium. by dominantly annihilations. decayed This DM Once leads the production to modulusits stops, has a number and quasi- density the drops DM to continues annihilating until Production from modulus decay, with chemical equilibrium. T 10 1000 We plot the comoving DM number density 0.001 0.100 If • • m 15 X X (that is, as time progresses)These for results different are non-thermal obtained Higgs by numerically portal integrating DM the models Boltzmann in equations, figure for a fixed 600 GeV, and the portaland coupling then is freezes out during mattermodulus domination. decays, when Subsequently, the the majority visible of DM sector is temperature produced is from around the reheating temperature. normalised to the valueof required the visible to sector matchof temperature the the observed portal DM coupling 600 relic GeV, abundance and thethe modulus visible branching sector fraction to thermalmodulus DM bath. branching is ratio to DM. The reheating temperature is fixed to 10 GeV, the DM mass is Figure 2 JHEP06(2018)043 = ∼ ), a RH 3.1 T = 1 in eq. ( = 0, since allowing c B . At a critical value of 2 16 λ and 100 TeV and the latter 3 − ∼ reasonably closely, and even the = 10 4.1 B the DM number density does not become . B 3 − – 16 – 10 ) the DM reaches chemical equilibrium. Increasing . 5 . 3 B − 10 ∼ = 10 GeV (or equivalently having taken . For λ RH RH TeV) and a fixed DM mass of 600 GeV. T T 4 ∼ 10 = 0. = 10 GeV (which is still low enough that the non-thermal history has . For larger values of the modulus branching fraction, DM annihilations ' B B right, we plot models in which the DM relic abundance is dominantly left, we show models with zero modulus branching fraction to DM, so we plot the results for theories with a low reheating temperature RH T The required values of the portal couplings as a function of the DM mass 2 2 3 does not open up any additional parameter space. The observed DM relic . Additionally, the gradients of the contours of the portal coupling vs DM mass 3 17 − 4.1 10 . Subsequently, modulus decays continue and most of the DM is produced when TeV. In figure In figure In figure For clarity, we set the energy in the radiationIf bath before the matter visible domination sector to soft zero masses in are these plots. similar to As the modulus mass, this range can lead to a Higgs mass 4 DM 16 17 numerical factors are fairly accurate. discussed, this does not affect the final DMof abundance. 125 GeV in the MSSM. 10 are plotted for modulusB branching fractions > between abundance can arise fromsection models that arematch the in approximate analytic all calculations of in section the dynamical regimes discussed in density constraint translates into contours in the plane of20 portal coupling MeV, vs. close DM totemperature mass. the minimum allowed byan BBN, impact). and for The models former with a corresponds higher to reheating a modulus mass The parameter space for which Higgscan portal DM be makes up found the full byequations observed for scanning relic each. abundance over In different particular,fraction models, we since fix these numerically the are reheating determined integrating temperature by the the and underlying Boltzmann modulus string branching theory, and study how the relic proportional to become efficient, and aabundance quasi-static is approximately equilibrium independent is of reached. In this4.2 regime the final The DM parameter space for Higgs portal DM produced by modulusand decays. reaches chemical At equilibrium,m early before freezing times out DM whenthe is the temperature produced temperature is drops fromlarge below the enough thermal for bath annihilations to be efficient at late times, and the relic abundance is the coupling (in thisthe case portal coupling furtherout decreases during the matter relic domination.that abundance, lead since As to this a the istemperature, observed result, now DM if there set relic are by abundance freeze two for values this particular of DM the mass portal and coupling reheating modulus mass of that production is purelyDM from never the reaches chemical thermal equilibrium, bath.DM and being increasing When produced, the the coupling with portal simply the coupling leads relic is to abundance small more proportional the to reheating temperature JHEP06(2018)043 , 4 + . B e 10 DM m 3 . At larger . ν-floor DD 2 10 − 10 & 1 GeV . λ GeV 2 / 10 GeV 10 = DM RH m T 1 the same parameter space for a 10 -3 -5 -7 10 10 10 0 B ). For even heavier DM, production 3 Right: − future h invis. 10 1 1 2 3 4 5 1 - - - - - & 10 10 10 10 10 λ λ . For small DM masses production is dominantly B 4 – 17 – 10 -3 -5 -7 10 10 10 0 B ν-floor DD 3 10 that lead to the full measured DM relic abundance, for different λ 2 = 20 MeV, DM with a mass in the range 0 10 GeV / 20 MeV RH = T DM 1 to be small (although given that the portal coupling plays no role in m RH 10 T λ floor) is plotted with a dotted gray line. the parameter space for Higgs portal DM in models with a period of late time 200 GeV the DM is in thermal equilibrium after reheating, and the required value − ν . 1 Left: DM TeV. Such heavy DM is relatively rare in theories with a thermal cosmological . m 1 h invis. future  - 3 4 5 2 1 is the same as in models with a thermal cosmological history. For models with Contours that are almost vertical correspond to models in which DM is produced di- 1 10 - - - - - collider, sensitive to an invisible branching fraction of 1%, (future) is shown dashed gray, and the λ DM 10 10 10 10 10 − GeV, and a relatively small value of the portal coupling, is produced from the visible sector simply by taking setting the DM relicexperiments density, is the also motivation diminished). for such models to be within reach of future branching fractions to DM, productionm directly from decays can leadhistory, to due viable to models with the largeduring annihilation radiation cross domination. sections Becausethis that the regime, are existing value required of constraints assuming from the freeze direct portal out detection coupling and is collider unimportant searches in can be evaded directly from modulus decays canwhile for lead to the measuredof DM relic abundance for small values of rectly from modulus decays without reaching chemical equilibrium. For very small modulus gray. The future reache of searches for invisibleneutrino Higgs floor decays ( at themodel high with luminosity a LHC reheating or temperaturelead an of to 10 viable GeV. models(chemical In with equilibrium this relatively case is heavy production DM reached from if in the the thermal such modulus bath models can branching if fraction to DM is small matter domination and aDM reheating mass temperature and of portal 20values coupling of MeV. the modulus The branching contoursfrom fraction show to the the DM visible values sector ofmasses, thermal the DM bath, is mainly andsearches produced chemical for directly equilibrium invisible from is Higgs modulus reached decays decays. if (h The invis.) current constraints and from direct LHC detection bounds (DD) are shown shaded Figure 3 λ JHEP06(2018)043 , , 6 2 − 5 v 2 10 λ . ⊃ . In this B 2 200 GeV. − 2 DM . m 10 . B 2 TeV and . DM m GeV, even for models with a . . DM m . = 10 GeV and DM masses . RH 1 T 200 GeV, a quasi-static equilibrium is reached – 18 – & DM m 1. This increases the DM relic abundance compared to models with . 0 production from direct modulus decays becomes more important than = 0. Models with a particular non-zero modulus branching fraction = 10 GeV and & B B λ , which are already ruled out. = 10 GeV the modulus branching fraction to DM must be relatively small RH λ T RH and T 5 − GeV and 10 & & than in theories with a thermal cosmological history. We also note that the contribution from the Higgs VEV to the DM mass, It can now be seen that calculating the relic abundance assuming that the DM is at Finally, for By varying the reheating temperature upwards from the minimum value allowed by Sufficiently light DM will be in chemical equilibrium with the visible sector thermal B λ DM sector if the portal couplingis is inefficient. extremely small, This sohowever that might it scattering have will off an not the impact alter SM on thermal the bath DM structure relic formation, abundance. discussed inis not section sufficiently large that the DM mass must be fine tuned to be small in any of the viable produced automatically have approximately theparameter same space temperature in as whichquasi-static the the SM. equilibrium, In DM the parts is portalthe of produced visible coupling sector. by is Meanwhile, direct large ifdoes modulus enough the not DM that decays is efficiently it and dominantly annihilate, produced will reaches from it thermalise a modulus could decays with and remain at a higher temperature than the visible relatively large modulus to DM branching ratio. the same temperature asmost the of visible the sector parameter does spacebath not in the lead which portal to the coupling significant DM is inaccuracies. is large produced enough Over from for thermalisation, the and visible even sector if thermal not the DM states BBN, the required DMm relic abundance canto arise DM for are any also value possiblefrom of over modulus the the decays, majority portal or ofsignificant coupling part by this if of region, production parameter either from space through the with direct 100 thermal production MeV bath. Additionally, there is a for a thermal cosmological history,larger so values of that obtaining the measured DM abundance requires bath at the timeprovided of that reheating, the portal and coupling willleading is therefore to not the tiny. freeze required For out example, relicIn during this abundance this is radiation with regime the domination, the case DM forthermal relic the cosmological contours abundance history, plotted is in very close figure to that obtained from a model with a these mass ranges DM annihilationsout are important, during and matter the domination. relicgradient. In abundance this is In set part both by of of freeze parameter theseof space regimes, the the contours measured have relic negative abundance occurs for smaller values part of parameter spacemodels contours with of constantin relic order abundance to have be positive in(for this gradient. larger regime, values of and In it isproduction possible from for 600 the thermal GeV bath). Meanwhile, for larger values of the portal coupling in thermal bath and does not reach chemical equilibrium, provided that JHEP06(2018)043 ∼ ) ]. In invis 118 , + Γ ]. SM 117 123 , ]. Such searches (Γ we plot the cur- 43 07 [ / . 58 3 0 ∼ invis 20 ]. ]. Searches at a 100 TeV 122 121 ]. – 116 119 19 2 collider searches for invisible Higgs decays are – 19 – / = 20 MeV, in some parts of parameter space this h RH 02 could be reached [ T . < m 0 ∼ DM m ]. This would be the case if, for example, DM with a relatively , for 115 ], although the unknown systematic uncertainties at such a machine 2 = 10 GeV it is negligible. In models with 122 , RH 18 60 T GeV is detected. Another interesting possibility is that the DM could be 005 [ . . 0 ∼ The sensitivity of indirect searches for Higgs invisible decays at the LHC (from the mea- Observation of a signal that is challenging to obtain from simple models assuming a Although since the DM is a scalarHiggs its portal mass DM is could quadraticallyA also sensitive Large affect to Hadron-electron the collider any SM could new EW reach higher phase invisible energy transition branching [ scales fractions of 18 19 20 1. However, the potential improvement is limited by the large uncertainties in the cross . tions of makes it difficult to reliably predict the achievable sensitivitythat [ it is sufficiently strongly coupled to. section for Higgs productioncontrast, via the sensitivity gluon of fusion directically searches due for over to invisible the Higgs QCD course decays correctionsbranching could of improve fractions [ the dramat- as LHC’s small highhadron as luminosity collider run, could be and even it more is sensitive, expected potentially that down to invisible invisible branching frac- improvement in these searches could lead to such asurement model of the being Higgs discovered. decay rates tomulated, SM and particles) these will could increase reach slightly invisible as branching0 more fractions data of is order accu- Γ rent bounds derived fromare limits on particularly invisible significant Higgs fortory, decays Higgs at since portal the they models LHC areparameter [ with more space a constraining that non-thermal than lead cosmologicalsubstantial direct to region his- of detection the viable required limits parameter space DM in close relic some to the abundance. parts current of bound. Additionally, the As there a is result, a future in favour of such a theory.occur We will given see the that potential this reach combination of of observations future could5.1 experiments. plausibly Collider searches As discussed in section potentially competitive with direct detection constraints, and in figure complex hidden sector [ small mass observed in multiple different types ofsection experiment. measured in For example, a if directmodels the detection after DM experiment a scattering matched cross collider that observation predicted of by invisible Higgs Higgs portal decays, this would be strong evidence Since the viable parametera space large of range non-thermalconstraints, of it Higgs DM is portal masses interesting DM to and consider models values the extends of possibilities over for thethermal future cosmological portal history detection. could coupling, point towards far either a beyond non-thermal cosmology the or a current more contribution is comparable tothe the DM total DM mass mass, (elsewherefor and it models it is with could significantly even smaller be than the the sole total source5 DM of mass). Meanwhile, Prospects for future observation and phenomenology parameter space. JHEP06(2018)043 ]. ], . ∼ few 127 125 ∼ , , ]. Such 10 GeV. 61 126 7 GeV . 124 20 MeV, a ∼ , ∼ 61 ], and they could 129 , 128 10 (i.e. a factor of 2 are expected to remain , 6 GeV, after 1000 days of / ' ]. Improving the sensitivity h 62 ]. & we plot the neutrino floor, as 134 133 > m – 3 130 DM m . 3 − ]. In figure 10 ∼ 135 – 20 – B left, such an experiment will explore an interesting 3 6 tonne fiducial mass (the reach of the XENONnT experiment and . ]. From figure collider with a centre of mass energy of 500 GeV would also be extremely 137 − collider could probe Higgs portal couplings of order 1, which is far weaker e we show the part of parameter space that would be explored by collider + − e 3 e ]. This is a next generation xenon experiment, which has an anticipated + e 136 10 GeV [ . At smaller DM masses SuperCDMS SNOLAB, which is based on cryogenic detectors, Relevant to the high DM mass region, the LZ experiment is expected to begin operation Conversely, collider searches for models with In figure A future DM 005. In this case, the most powerful search channel is Higgs production in association with . could reach DM nucleonin scattering the cross portal sections coupling) am above factor the of neutrino floorpart of for the DM masses parameter in space the for range non-thermal 0 models that have a relatively low reheating sensitivity that is closeexposure to assuming the a neutrino 5 floorthe for PandaX DM upgrade masses are expectedregion to in be similar). the Asthe thermal well parameter cosmological as space exploring history for alldomination which case, of in the the this non-thermal DM models resonance will relic with a abundance cover relatively is high a set reheating significant temperature by freeze amount out of during matter of direct detection searchestechniques beyond such this as level is directionalwell extremely detection as challenging, [ the and current would direct require detection constraints. in 2020 [ 5.2 Direct andThe indirect sensitivity detection of direct detectionthe experiments near is future. also However, expected suchwhich to searches the increase will irreducible substantially eventually neutrino in reach background, DM-nucleonspheric, arising cross from sections and coherent at supernovae scattering of neutrinos, solar, becomes atmo- problematic [ than existing direct detectionsearches bounds could in however be the important minimal inalso theories more be complex that models significant we [ in consider modelsUV [ complete of the fermion portal Higgs operator portal are DM relatively if light the [ additional states added to chemical equilibrium. Additionally, part ofdirectly from the modulus parameter decays space will also into be which DM reached DM branching in is scenarios fraction, with produced for relatively example large modulus weak. An models of fermion DM interacting with the SM Higgssearches through that are a sensitive light to mediator anat [ invisible a Higgs future branching machine. fraction of Forsignificant 1%, models proportion which with of is a plausible the relativelyfor parameter low reheating space which temperature will the be DM covered. relic This abundance includes many is models set by production from the thermal bath without sensitive to invisible Higgs0 decays, and could detecta an Z-boson invisible followed by branchinginvariant invisible fraction mass decay. of could Further, confirm a theand study Higgs interference of effects portal might the nature even distribution allow of a of an scalar the observed DM missing Higgs signal portal [ to be distinguished from JHEP06(2018)043 , . ] , 1 47 138 141 21 (5.1) (5.2) ], although 144 . ]. Consequently, as- 3  146 ]. Although Higgs portal 3 DM 1 GeV m . 0  2  3 s λ π/ . 2 4  χ s g λ / 2 – 21 – ], and the too-big-to-fail problem [ 4 cm . L ⊃ − , this is obtained assuming a relatively optimistic 143 ) is introduced is = 1 2.1 5.1 ]. 2 s 3 DM ], however astrophysical backgrounds and uncertainties are λ , which is the case in all of the viable parameter space with 9 145 λ πm 2 existing direct detection bounds already exclude most of the parameter 139 2 / , , as is required for perturbative unitarity [ ]. Present discrepancies include the core vs cusp problem [ h  1 2 = 78 m s g, and such interactions have been proposed as a possible resolution 140 < λ / . s 2 | DM 0 σ 2 DM scattering with a rate that is unconnected to the dynamics that a DM m cm m → ∼ Re . | DM has 0 is normalised to its maximum allowed value such that the zeroth partial wave am- /m a s s σ λ In the limit that Self interactions can be included in the scalar Higgs portal DM model simply by in- The prospects for improvement in the reach of indirect detection experiments are For 10 GeV ], the missing satellites problem [ ] might explore some parts of parameter space at high DM masses that are not currently 21 Here plitude trophysically relevant DM self interactions are possible only for relatively small DM masses, space that could be probed by collider measurements in the foreseeable future. astrophysically relevant self-interactions, the DMmodel scattering cross after section the in operator a in Higgs eq. portal ( models are certainly not themeans only of theories a in quartic interaction which theyof DM do self a provide interactions a model can minimal with be waycan by realised this be which by the feature introduced relic may in abundance more be general set models). (there are also many ways that self-interactions troducing the renormalisible interaction This leads to 2 sets the DM relic abundance, and which can therefore be large [ small-scale structure [ 142 it is plausible thatexplain baryonic some effects or such all as of photoionisation these and [ supernova feedback could 5.3 Astrophysical effects DM self-interactions can beorder astrophysically relevant ifof the apparent disagreements scattering between cross numerical simulations section of is cold of DM and observations of excluded by direct detectionHowever, experiments, as and discussed theDM in potential spatial section reach distribution. isproduced shown by Future DM in measurements annihilations figure might ofnot also currently be anti-protons excluded sensitive or to [ parts anti-deuteriumlikely of to [ parameter remain space significant. that are decays in this DM masswould range, greatly and, as increase mentioned, the an potential observation for in discriminating both of between these differentcomparatively channels DM less dramatic, models. at least in76 the short term. The upcoming CTA experiment [ temperature. The reach is comparable to that of future collider searches for Higgs invisible JHEP06(2018)043 ), 22 DM (5.3) (5.4) left. m 3  φ m ] (we leave a compu- ]. Their effects might 161 156 , 160 , 2 (assuming that . / ] require that such models satisfy 110 φ , 10 m 8 ]. The mediator is required to decay before 166 – − & 147 3 10 / 162 1 ×  3 . RH T GeV ) are short range contact interactions, regardless – 22 – φ DM  ], which are ameliorated in light mediator models m 5.1  m ]. A preliminary analysis indicates that these are not relevant observations [ 155 t DM RH – T ], although this is possible if it is coupled to relatively light right 159 α T GeV m –  148 151 157 for which thermalisation will not occur to future work). Each , g [ 3 / λ 2 ), this is equivalent to requiring that 1 cm 3.1 23 . ]. CMB, direct detection, and invisible Higgs decay bounds can also be important. 149 DM ], since there are relatively stringent constraints from galaxy clusters and halo . This could lead to observable effects on structure formation or violate con- /m s RH σ 150 is the temperature of the universe today (observations of dwarf spheroidal galaxies /T t φ T m The non-standard cosmological history, and in particular the non-thermal way in which The DM self interactions from eq. ( DM self interactions could also lead to new indirect detection signals, for example neutrinos produced Models with more complex hidden sectors, with the DM coupled to a light mediator that interacts with ∼ 23 22 the visible sector througheven a in this Higgs case portal it isBBN, operator, hard which can to rules evade lead observational out to constraintshanded many [ significant neutrinos models DM [ [ self-interactions. However, by DM annihilations inin the phenomenologically sun viable [ parts of parameter space in the present model. modulus mass, eq. ( straints on warm DM, and Lyman- where also lead to similar limits). Given the relation between the reheating temperature and the the visible sector if the portaltation coupling of is the extremely upper small value [ of DM state is producedso with a in typical this kinetic regime energyof the of DM temperature is larger than that of the visible sector by a factor DM might be produced,with can a thermal also cosmological lead historyif and to a DM possible DM is abundance effects dominantly set produced thatcomparable by from to are freeze-out. the that not As of visible the mentioned, possible sector visible thermalproduced sector. in directly bath models However, from its in modulus temperature parts decays, of will the parameter be DM space might for never which reach it kinetic is equilibrium with simulations and observations, and theyhalo could and stars also in explain a the galaxyalso falling separation be into between the observationally the Abel distinguishable 3827 DM from galaxyby those cluster a [ of light the mediator. long range interactions generated of whether they areare introduced potentially less in appealing a thanmediator the Higgs [ velocity portal dependent interactions model generatedellipticity or by a otherwise. light Such(due interactions to the largerinteractions could typical still DM play velocity a in role such in resolving systems). at least Despite some this, of short the range discrepancies between self (and indeed the majority ofThe DM low mass models part unless of additionalwith parameter light space a mediators is thermal are robustly introduced). excluded cosmologicallow in history, reheating minimal however temperature Higgs a portal leads models to non-thermal viable cosmology theories, with for example a as relatively shown in figure a fact that applies to any model with self-interactions arising from such a quartic interaction JHEP06(2018)043 ]. 167 25 ]. This could happen if 20 MeV, and a relatively ) bound is borderline dan- ∼ 168 5.4 , eq. ( 3 ]. This is not straightforward to accommodate 170 , 169 60 GeV could lead to signals in both future – 23 – More optimistically, such models could lead to ef- . 24 ) is not significant in models with a larger reheating temperature, since 5.4 GeV is possible, which can lead to astrophysically significant DM self-interactions. ∼ Higgs portal DM can lead to a wide range of signatures at direct, indirect, and collider Conversely, in other dynamical regimes a non-thermal cosmological history can result If the DM doesIt not has even thermalise recently among been itself claimed these that constraints the could 21 change significantly cm [ absorption signal arising from around the time the first DM 24 25 visible sector or additional new hidden sector states. stars formed shows signs ofin DM-baryon interactions the [ simplest Higgsinteractions, portal however at models present due the to signal the remains apparent tentative. required velocity dependence of the baryon-DM experiments. In models withm a fairly low reheatingAdditionally, temperature, models relatively light with DMcollider DM with searches masses for invisibleif Higgs the decays reheating and temperature direct is larger, detection heavy experiments. DM is Meanwhile, possible without large couplings to the imposing the DMcontrast relic to if density a constraint, thermal cosmologicalcan and history be is direct assumed. dominantly detection set In by suchchemical and models production equilibrium, the collider from or DM the abundance bounds, by visible direct in sector production thermal from bath modulus with decays. or without with a period of lateSuch time matter a domination modification due to oftheory a light, the UV gravitationally universe’s completions coupled, scalar. cosmological ofautomatically history lead the to is SM. a period well These ofis motivated generically matter from relatively domination contain if string low. many the scale moduli of In supersymmetry fields, breaking such which models large regions of parameter space remain viable after determine if such a scenario is realised in Higgs portal models,6 for future work. Conclusions In this paper we have studied the viable parameter space of Higgs portal DM in theories than the visible sector reheatingtemperature temperature, relative so to that that reheating increases ofhierarchical the the structure visible DM. sector If formation thisa might occurs, thermal be the cosmological smallest much objects history,However, smaller which formed we during than could leave a potentially those full affect obtained calculation indirect assuming of detection the signals. kinetic decoupling temperature, necessary to ible sector. Eq. ( direct production leads to the required relic abundance for much larger DMin masses in DM these. thatthe is DM significantly relic colder abundancethe than is temperature the set at visible by which production sector the from DM [ the kinetically visible decouples sector from thermal the bath, visible sector and is larger gerous for some theorieslarge with modulus to a DM low branchingfects reheating ratio. at temperature, a level that couldlikely be to observed reveal in that the there near areregime future. large for parts However, which a of the careful viable portal computation parameter coupling is space is in large the enough direct that production the DM thermalises with the vis- Comparing with the parameter space plotted in figure JHEP06(2018)043 (A.1) (A.2) (A.3) ). 4.2 ], which also ,
171 2 X − ˜ H 2 eq X i σv h 2 / has a strong temperature depen- 5 − i RH X A T σv h X RH T E B. Pl + φ X M A R m E – 24 – = Φ + 3 Φ + ˜ B 2 / r , 1 A Φ = 2 / B ˜ φ 1 H the equilibrium DM number density at a given temperature. A m RH ) T eq is the average energy of a DM particle. It is also convenient to ) n RH 2 T RH ( T 30 T g ( 30 2 g + 3 π , with 2 π eq r ] we introduce a dimensionless Hubble parameter 2 DM n In the final stages of preparing this manuscript, the paper [ − r 3 m 26 2 RH A = = T q Φ = ' d dA dA dX eq X ˜ ˜ H H E X Following [ Although we have focused on the scalar Higgs portal DM model due to its interest- where The thermally averaged annihilationdence cross in section the Higgs portal model, and is evaluated at the temperature of the radiation bath. The evolution of the modulus energy density and DM number density is given by where define a parameter related toquanta the that average energy decays that by goes into DM mass for each modulus lus and radiation bath energycosmological densities, and history. the These DM are number given density, during in a terms non-thermal of the variables defined in eq. ( on the arXiv. A Evolution of theFor DM completeness number we briefly density summarise during the matter equations governing domination the evolution of the modu- Acknowledgments I am grateful to Bobby Acharya andNote Martin added. Gorbahn for very usefulstudies discussions. Higgs portal DM in models with a non-thermal cosmological history, was posted case, and dynamics thatmological enhance history the DM are relic possiblethe abundance portal in compared coupling viable to than theories. in indirect a thermal detection, thermal models, and These indirect cos- leading could detection to allow experiments combinations for that of would signals larger otherwise in values not collider, of be possible. ing phenomenology, a similarin increase other in DM the models,portal. allowed for parameter Additionally, example space a DMmental is non-thermal interacting signatures likely cosmological with in to fermion history theHiggs. Higgs occur could visible portal Unlike also sector DM scalar lead through Higgs models to portal the with DM, new pseudoscalar Z direct couplings experi- detection to bounds the are extremely weak in this JHEP06(2018)043 ]. ]. ! ].
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