Saxion Cosmology for Thermalized Gravitino Dark Matter

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Title Saxion cosmology for thermalized gravitino dark matter

Permalink https://escholarship.org/uc/item/6qh0b893

Journal Journal of High Energy Physics, 2017(7)

ISSN 1126-6708

Authors Co, RT D’Eramo, F Hall, LJ et al.

Publication Date 2017-07-01

DOI 10.1007/JHEP07(2017)125

Peer reviewed

eScholarship.org Powered by the California Digital Library University of California JHEP07(2017)125 ) s a,b , m a , f 2 Springer / 3 July 11, 2017 July 26, 2017 April 12, 2017 m : : : Accepted Published Received , and Keisuke Harigaya 100 TeV. Constraints on TeV, axion decay constant, < a,b < s 2 / 3 < m < m Published for SISSA by https://doi.org/10.1007/JHEP07(2017)125 [email protected] , Lawrence J. Hall c,d [email protected] , or a pair of Standard Model fermions. Freeze-in may lead to a . GeV, and saxion mass, 10 MeV 3 Francesco D’Eramo, 16 10 1703.09796 , gg, γγ a,b ) The Authors. < c Beyond Standard Model, Cosmology of Theories beyond the SM, Supersym-

a In all supersymmetric theories, gravitinos, with mass suppressed by the Planck , [email protected] < f W W, ZZ, hh GeV 9 Santa Cruz, CA 95064, U.S.A. Santa Cruz Institute forSanta Particle Cruz, Physics, CA 95064, U.S.A. E-mail: [email protected] Department of Physics, UniversityBerkeley, of California California, 94720, U.S.A. Theoretical Physics Group, Lawrence BerkeleyBerkeley, National California Laboratory, 94720, U.S.A. Department of Physics, University of California Santa Cruz, b c d a Open Access Article funded by SCOAP lead to dark radiation. Keywords: metric Standard Model ArXiv ePrint: alignment and parametric resonanceremain, mechanisms. but Large allowed differ regionsmay for of lead DFSZ ( to and eventsto KSVZ with ( displaced theories. vertices andsub-dominant Superpartner kinks, warm production component and of at may gravitino contain colliders dark saxions matter, and decaying saxion decay to axions may such dilution allows previouslymatter over thermalized very gravitinos wide ranges to10 of account gravitino mass, for keV thethis observed parameter space dark are studiedproduction from BBN, from supersymmetry freeze-in breaking, and gravitino and saxion axino decay, and from axion production from both mis- Abstract: scale, are an obviousequilibrium, candidate such for dark dark mattercluded. matter; is However, apparently but in either ifearly theories too gravitinos with era abundant ever an of or reached axion, cosmological thermal too a history saxion hot, and condensate and its is is late generated ex- decay during dilutes an dark matter. We show that matter Raymond T. Co, Saxion cosmology for thermalized gravitino dark JHEP07(2017)125 11 12 15 ] broken 6 , 5 ], to strongly 4 14 ] for dilution by 11 , 10 15 4 ]. (See [ 9 6 gg hh/ZZ/W W 20 is severely restricted [ → → s s R T ] which is generated by supersymmetry 5 8 , 7 17 7 – 1 – 8 ]. To obtain the dark matter abundance revealed by ]. These pioneering works assumed that gravitinos, like 2 3 2 18 1 18 to solve the strong CP problem, the gravitino abundance can be diluted PQ ]. However, the cosmology of gravitinos has long been viewed as problematic. In V 1 However, in supersymmetric theories with a Peccei-Quinn (PQ) symmetry [ 4.3 Dark radiation 3.3 Further constraints3.4 in DFSZ theories A with limit on the saxion condensate from4.1 axion parametric resonance Displaced vertices,4.2 kinks and saxion A resonances warm at component colliders of dark matter 2.2 Cosmology of2.3 the “Decoupled Saxion” Misalignment axion contribution to dark matter 3.1 The maximal3.2 parameter space Further constraints in KSVZ theories with 2.1 The matter dominated era of the saxion condensate at scale by the latebreaking decay during of an early a era,thermally saxion for produced example saxions.) condensate during Hence, [ inflationWitten, in [ Pagels this and paper Primack we that return to gravitinos the were original in assumption thermal of equilibrium in the very early photons and neutrinos, would behigh temperature, in so thermal that equilibrium theirthen in number the it density very would has be early typicallyreheat given universe been by with temperature assumed thermodynamics. a of that, Since the inlimit universe theories the after gravitino with inflation abundance. weak scale supersymmetry, the Witten [ 1981 Pagels and Primackwere found heavier that than light gravitinos therecent would keV measurements, overclose scale the the [ universe gravitinoto if mass the they must warmness of be the around gravitino 100 [ eV, which is excluded due 1 Introduction If supersymmetry is relevant forby the hierarchy the problem, Planck gravitinos, mass, with a become mass an suppressed interesting candidate for dark matter, as pointed out by 5 Conclusions A Warm dark matter from freeze-in gravitinos 4 Signals 3 Thermalized gravitino dark matter Contents 1 Introduction 2 The cosmological history JHEP07(2017)125 ] ) PQ 20 , (2.1) ,V 19 2 / 3 m ] is frequently 14 – 12 has no potential. In the s is the axion decay constant a f ]. Hence we also explore the 18 – 16 2 s 2 s m (TeV). Thus, to obtain the main results GeV, where 1 2 O 12 = 10 – 2 – V × DW N < ∼ 2 √ / ], arising from decays to gravitinos. below the TeV scale of superpartners. Thus saxion dilution al- DW 15 R N , the masses of the SM superpartners, specifically the masses of the T a f TeV until it decays, the saxion condensate oscillates about the present m = ∼ ] models where it can be a significant component of dark matter. to be arbitrarily high. For comparison, without dilution a light gravitino, ˜ m is the soft supersymmetry breaking mass of the saxion. PQ 22 R s , V scenaria for the interesting case of displaced vertex signals at LHC, and its ∼ T m 21 R T is the domain wall number. However, with dilution from a saxion condensate T MeV, requires DW From vacuum in a quadratic potential where Before reaching the TeV scale, thepresent saxion value field and acquired there a was large an displacement era from where its gravitinos were in thermal equilibrium. < N In the absence of supersymmetry breaking, the saxion field Cosmological axion production from the misalignment mechanism [ Dilution from the saxion condensate allows a much wider set of cosmologies, in partic- 2 • • / 3 above the TeV scale is extremely mild: Furthermore, for later evolution of the saxion field we assume the saxion condensate. Rather thanrelevant studying for a gravitino particular dark model,less matter we than show depends that or on the of physics the orderhiggsino saxion and ˜ evolution the only LOSP, atof which this temperatures we paper take the to assumption be on the cosmological history of the universe at temperatures and the thermal bath, and we give results forearly the universe, at axion any era abundance. the non-zeroand energy hence density of the the universe form breaksuniverse of supersymmetry, evolves the through inflation, saxion post-inflation, potentialpotential reheating is and and a its subsequent minimum highly eras changes the model-dependent leading, saxion in question. general, to As a the highly complicated evolution of and KSVZ [ 2 The cosmological history In this section we provide an overview of the cosmological evolution of the saxion condensate taken to limit and this limit isabundance weakened of by axion 3-4 dark matter, orders finding regions of of magnitude parameter space [ in both DFSZ [ particle (LOSP). ular allowing m lows high MATHUSLA extension [ study constraints on such gravitino darkplane matter, are exploring preferred which and regions hence of relatingconstraints the are the ( independent scales of of the supersymmetry very anddepending early PQ only cosmological breaking. on history the Most of the saxionof saxion evolution oscillations, superpartners, at temperatures especially less those than of or of the order higgsino of and the lightest masses observable supersymmetric universe. We take the gravitino to be the Lightest Supersymmetric Partner (LSP) and JHEP07(2017)125 β W . In (2.4) (2.5) (2.6) (2.2) (2.3) DW . /N h m PQ V 2 s √ m = , even misalignment a PQ f V , , 2 f 2 m PQ 4 f V µ s N 2 µ 2 q , . / is normalized such that all charges s πm 3 s 2 a 4 µν limits as well as µ f m ˜ quarks and leptons, respectively). ,ZZ 4 h p 2 s π m − ) = f – 3 – m g PQ ' 32 2 W µ gg π V s q 4 + , the saxion mainly decays into a pair of Higgs, = R → T ˜ W G s ) = ( m ¯ is defined by the axion coupling with the gluon field as s aG f , is dependent on the saxion mass and on whether the 2 hh, W s Γ f L a f > → → s s s ( m ( s s of the Higgs mass parameter Γ Γ µ and the saxion decays become significant, at temperature q H , the decay of the saxion condensate also dilutes axinos and gravitinos , the main decay channel is into a pair of standard model fermions via ∼ W PQ s is low enough that the gravitinos thermalized at early times are diluted. For V m s 2 R T < is the multiplicity of the fermion s f m N bosons, with the rate In the next sub-section we discuss aspects of the saxion oscillation matter-dominated In KSVZ models, the saxion mainly decays into a pair of gluons with a rate In DFSZ models with This large saxion condensate plays a crucial role in determining the dark matter abun- = 2 in this paper, as in the minimal supersymmetric DFSZ model. For a lighter DFSZ Z µ or scattering of theaxions generated thermalized near particles. the QCD For phase higher transition values are of diluted. era. In the following sub-sectionogy we for provide the a early very evolution simple of illustration of the a saxion possible condensate cosmol- — the “Decoupled Saxion”. and is given byeither the case, PQ breakingmost scale values of from freeze-in (FI) andprocess superpartners where dark from matter freeze-out. (DM) is The not in FI the mechanism thermal refers bath to but produced the from the decay Here the axion decay constant where Here we have assumed the decoupling and large tan saxion, mixing with the Higgs, with the rate where we have summedlimits. over the The PQ final charge statesof and the assumed PQ the breaking decouplingq and fields large are tan integers with absolute values as small as possible. We fix Higgs doublets carry PQ charge. or dance. It eventually dominatesentropy, the when energy Γ density of the universe and releases most of its The decay rate of the saxion, Γ JHEP07(2017)125 . , ) Pl NA 2.1 M (2.7) (2.8) (2.9) T ) give between 2.9 and the saxion T s to describe there are so R GeV to T NA 9 NA T NA T T is varied over its ) and ( up to some temper- is constrained to be 2.2 . At PQ s in the potential ( V R /a NA ˜ over a wide range of cases T 1 , such as the previously m T 10 TeV. Throughout this s < ∼ ∝ NA < T T T s < m 1 . From s NA R < m ] for a review) of 10 T T . Furthermore, constraints on the ), with ˜ 24 . , the formulae ( m A . 5 5 < ∼ th PQ T Y V s T 2 s R D T / ): the radiation density is dominated by the R 3 T NA , however, the decay/scattering of the sax- T s T m NA m – 4 – ' and we find it convenient to use ' ' ) s ], which determine the temperatures T eq D ( and small m 28 s T D – 2 R µ follows from requiring that the dilution of previously 25 /T NA 100 TeV. 4 NA T T < ]. On the other hand at temperatures above ' 23 NA NA s < T ], while for relic particles produced at some temperature 23 in ref. [ 8 this result gets corrected; however, we find / . For temperatures above 3 s s the dilution factor is less: /a s 1 R this MD era is non-adiabatic (MD > m > m T . Hence the results of this paper only depend on the low temperature aspects ∝ s ˜ could be arbitrarily complicated, for example from interactions during inflation or m NA NA T R T T ˜ < ∼ and m In DFSZ theories with large The condition that fixes In particular well before it decays, the saxion condensate must dominate the energy ,T If T 1 NA NA T ion is affected by thermal effects [ discussed below. entire range from its lowerand astrophysical the bound saxion (see mass [ isparameter varied space, over the the range observedgravitinos of and/or dark 10 gravitinos matter MeV produced results byin freeze-in from the processes, range diluting and 10 previously MeV thermalized thermalized gravitinos yields the observed temperature of matter-radiation equality We study gravitino dilution over a very wide range of parameters: as derived in ref.T [ the strength of the saxion condensate,For as decoupled it relic appears particles directly produced inthermalized the at gravitinos, gravitino temperatures saxion dilution decays above factor. yield a dilution factor density producing a matter dominatedature universe at products of recently decayedgiving saxions rather thanfew from saxion pre-existing decays red-shifted that radiation, thecondensate MD has era a is size adiabatic (MD of the cosmology, notT the > high temperature aspects.with the Thus thermal the bath. evolutionwe Nevertheless, of need for the the to saxion late parameterize evolution at the at size of the condensate. In this paper the processes relevant forpreviously computing thermalized the gravitinos, dark matter freeze-in abundance gravitino areThese production dilution of and processes misalignment all axions. occurtheory from at overproduction low of temperatures, freeze-inat axinos and superpartner freezeout similarly occur 2.1 The matter dominated era of the saxion condensate JHEP07(2017)125 I D s (2.11) (2.12) (2.13) (2.14) (2.10) (2.15) ) for all . In the 3 ) 2.1 s , the decay , the saxion R f )GeV and is . In the lower 18 osc /T 4 T , when the radi- – 10 NA T 2 µ − NA ( TeV T 14 ∼ 1 2 1 2 . D . The resultant reheating s 1 2 s s m that it acquired from some with a fermion m TeV TeV m we consider the “Decoupled ¯ the saxion energy density is I f s ˜ s      . For simplicity we do not show m 1 2 osc m ysf T × T . NA s s 2 , to be larger than 2 1 T 100GeV this thermal effect is irrelevant. / I Γ . m R 3 T > for osc s 4 ). Furthermore, in this cosmology, T 3 T 3 osc T 2 I T in the range of (10 T 2 s th Pl 3 y 2 s osc 2.13 GeV I 2 I Y GeV and/or T PQ 2 s s eq M m / ∼ 2 s s 10 V 12 T 3 Γ Rs m s 10 10 m ' – 5 – T . In figure 1 2 m s 2 ' M ' / 3 T ' ' 2 Pl , and the dilution factor is s GeV 2 m > m / Pl ρ M . The gravitino dark matter abundance constraint of 5 s NA Pl I s M T s NA R m 2 M T y with some large amplitude p /T if ∼ M ' ) GeV osc s T T 17 NA R osc T T = 10 T , D 16 (10 ' as a derived quantity given by ( , thermal effects determine I 2 s NA ) becomes T 2.7 ) then leads to 2.9 Taking the reheat temperature after inflation, which we will findcorrelated yields with very other large parameters values according of to of eq. ( eq. ( radiation from inflaton decay, with Hence, in this scenarioand it have is best to describe the strength of the saxion condensate by The universe becomes matter dominated by the saxion condensate at and the subsequent matter-dominatedation era bath becomes non-adiabatic becomes at dominated by saxion decay products rather than by the red-shifted field starts to oscillateprevious at era, so that during the adiabatic era following This could happen ifsionless the couplings saxion or couples through to suppressedjust higher the a dimension thermal operators. simple bath illustrative We via stress example, either that and this very is is small not dimen- necessary for the results of the next section. 2.2 Cosmology ofAs the a “Decoupled Saxion” particular exampleSaxion”, defined of by a the saxiontemperatures assumption back cosmology that to the at saxion potential is given by eq. ( (dissipation) rate of thetemperature saxion is is given given by by next Γ section we showpart the of constraint figure on thethe parameter contours of space required in figures For example, when the saxion has a Yukawa interaction JHEP07(2017)125 γ ) ZZ ), at /T , a (2.16) ] show − /f 38 W ]. decreases – . We use (0)(Λ era, which + 2 2 a 32 35 GeV h W DW 6 m a NA /N ) = PQ is used to compute T 8 (2.7) computed in V ( . 2 2 2 a γ/ √ ) m ) = 6 a ( = osc γ in DFSZ (KSVZ) theories. 11 for various misalignment (0) = 6 eV (10 . a a s /T f R m T was derived in ref. [ = 0 , one finds that Ω 2 ξ , = (Λ and h ) , when the Hubble rate is equal to PQ a ξ s ) s s V a R R ( R osc T ) takes into account the temperature T T T ) ( on a ξ ( osc 2 i ξ 2 Pl θ Λ. Recent lattice calculations [ 2 is the misalignment angle. Coherent oscilla- a M f < ] i increases. This explains the different overall θ ) 9 8 – 6 – a 32 T < T ( osc = PQ T s V = 1; otherwise, . We evaluate the axion energy density per entropy R obtained from lattice calculations. The region above ξ T

1 for γ , where ]. The axion field value is initially displaced from the NA i a s T Λ, θ ρ ], the saxion potential may be quadratic even for a large ≤ ) can be calculated for both DFSZ and KSVZ theories. 18 a – f ≤ )( 31 2.2 – ) T 16 4) when a ( ( osc a 29 in a range where the saxion decays dominantly to T s /m γ < ) m ) in eq. ( a ( 4 ( osc s T R ( T a m γ > ≡ ]). In the regions where the two index lines merge, the axion starts oscillating ) and the axion mass index 34 T shows the numerical result of the contours of Ω ( i . The solid (dashed) lines are for the axion mass index ]([ ξ θ 1 = 1, while Λ. In other words, if 8, in high temperature regimes. The red region is excluded by BBN because of the 33 hh Interestingly, including the dependence of We predict the axion abundance in terms of The required large field value may seem to be incompatible with the assumption of the ' DW γ late decays of the saxion. (increases) for slopes of the two index lines, while the detailed features arise due to the rapid change of axis refers to theFor set DFSZ, of we parameters assume necessaryand to compute ref. [ after its mass is alreadythat a the constant, i.e. axion mass index is well described by the dilute instanton gas approximation, N Figure angles and to the right of the contour is excluded by axion overproduction. The top (bottom) where dependence of the axionabove the mass. QCD scale WeT Λ. assume < The a mass simple takesthe power a axion law constant energy value, density today. The analytic formula of density at the end of entropy production [ the effect of saxionminimum dilution today [ by an amount tions of the axionthe field axion commence mass. at We temperature is assume that the the case axion forthe oscillation the calculation starts is during parameter independent the spaces MD of constrained by the axion abundance. In this case, a tree-level superpotential [ field value. 2.3 Misalignment axion contributionThe to axion is dark produced matter by the usual misalignment mechanism but now we need to consider row respectively. quadratic potential. Actually incorrections, models the where saxion potential the in saxionhence general the mass becomes assumption is breaks logarithmic generated down. at via a However, quantum in large models field where value, the and saxion mass is given by for saxions decaying to pairs of (gluons, electroweak and Higgs bosons) in the (top, bottom) JHEP07(2017)125 PQ V ) and , µ ˜ a , m s m . W GeV. If the mis- m 2 13 10 > s > ∼ ) across the QCD phase m T 8, the dependence on . ( PQ KSVZ (3.1) ∗ V g = 6 ), the bound is relaxed roughly 3 γ / 2.4 , we can derive an upper bound on 2 1 . We illustrate how these constrain the GeV ˜ G a 8 . However, gravitinos can be overproduced f + 10 PQ a V × – 7 – 8 . Note first that the saxion mass cannot be larger and W × 2 / m 3 2 m in comparison to the case with 3. Then the higgsino mass should be smaller than about > 300 MeV. 4 √ 6 GeV / s . 3 0 ) π/ m , the upper bound is relaxed. Since a fermion with a mass close s > W ' < T < s /m ) space, but to illustrate the wide range that allows thermalized GeV. This includes the cases where the PQ symmetry is unbroken m 2 m mis 12 , µ θ ˜ a < 10 ) compete with each other, resulting in a nearly vertical contour. We s T , m > ∼ ( s m ∗ g m . Assuming the fine-tuning in the misalignment angle is no more than 10%, PQ ) from the Boltzmann distribution with the full SM spectrum and the MSSM ), but choose a few illustrative values for the key parameters ( µ V T 8 ( . PQ . For ∗ 0 g ,V , since otherwise saxions decay into pairs of higgsinos and result in too large grav- ' 2 3.4 dominantly contributes to the decay rate in eq. ( µ / 3 µ s ) during the QCD phase transition. In the case of 3 We comment on the lower bound on the saxion mass. In KSVZ theories, in order for There are several relevant parameters. We show results for essentially complete ranges The axion abundance gives an upper bound on the higgsino mass in the DFSZ model. m / T m 1 ( − ∗ the saxion to decay before the BBN, is required. In DFSZ theorieswith the standard saxion model mass may fermions be through smaller its due to mixing its with effective couplings the Higgs. However, with such a study of the ( gravitino dark matter andparameter its space corresponding from rich other signals.for processes dominant creating We saxion examine gravitinos decays and constraints to axions on gluons in the (Higgs/electroweak sub-section bosons). B (C) we indicate where axionsresonance during are saxion overproduced, oscillations. by either early misalignmentof or ( parametric for other supersymmetry breaking parameters. Our aim is not to provide an exhaustive the observed dark matterepoch abundance over to a arise very from wideby the range reactions thermalized of occurring gravitinos at ofdecay the an to TeV early scale gravitinos, or andregion below: saxion where gravitino decays dark freeze-in, to matter axino ˜ arises freeze-in from and the primordially thermalized gravitinos. Similarly, to by a factor of (100 GeV 3 Thermalized gravitino darkIn matter this section we show that dilution by the late decay of a saxion condensate allows the averaged angle, 200 GeV for during inflation, restored afterto inflation, the or parametric resonance thesection effect axion from field saxion obtains oscillations. large fluctuation The due last case is discussed in The bound is stringentthan for 2 itino abundance. Then from the2 upper x-axis of figure the higgsino mass shouldalignment angle be takes smaller a than randomized value, about the 1 axion TeV abundance for should be evaluated with and effects of compute spectrum degenerate at 1transition, TeV, i.e. and 100 MeV we linearly interpolate g JHEP07(2017)125 (3.2) . The s /m 2 / 3 m ) that allow for PQ ,V 2 / 5 3 scales as 10 m PQ V ]. In models where the saxion , exceeds that allowed by total . 31 3 = 0.3 (upper axis) and KSVZ (lower axis) 2 – PQ h 10 V mis 29 2 s (GeV) m θ 2 1 3 m ) (GeV) s 11, from vacuum misalignment and dilution > . s s is excluded because the contribution to the m = 0 =1 4 2 h . This bound on mis a – 8 – θ 2 Pl and KSVZ:m 3 (10) MeV DFSZ M , 2 /m DFSZ:μ(μ O / 2

4 2 3

< 3 MeV > γ m 10

3 Rs s

π show the maximum ranges of ( W T ∝T m γ= 6.8 γ= 2.7 2 =

= 3 2 a 2m BBN by excluded m mis 2 tot θ 2 F = 1. The solid (dashed) lines refer to diﬀerent temperature dependence ]. To prevent this process requires 10 13 12 11 16 15 14 8 (2.7), as estimated by the lattice QCD calculations. 42 .

–

10 10 10 10 10 10

PQ ) ( GeV V N = 6 39 γ keV are excluded by warm dark matter constraints, while values above ≈ . Contours of the axion abundance, Ω below = 30 GeV/ 3 TeV and each panel applies to both KSVZ and DFSZ theories. Values 2 / s 3 The gray shaded region of ﬁgures m m supersymmetry breaking bound is saturated in modelsbroken, where and supersymmetry the and saxion PQmass obtains symmetry its is are mass simultaneously given at byassociated tree quantum loop level corrections, factors. [ the bound is stronger by coupling constants and thermalized gravitino dark matter fromfor saxion condensate dilution. Theof left/right panel is 100 GeV are possible as long as the gravitinovacuum remains energy the from LSP. the saxion potential, of order 3.1 The maximal parameterThe space unshaded regions of ﬁgure of the axion mass, mixing, the saxion takesneutrino away emission energy [ from supernovae and changes the duration of the Figure 1 from decay of the saxiontheories. condensate Here in we DFSZ set with JHEP07(2017)125 . 2 2 / 10

3 s GeV

(3.3) 2 m

KSVZ DFSZ 10

= < 3 MeV 10 GeV

= 10 is larger

= 4 TeV).

NA NA

and again

T T KSVZ µ Rs 1 PQ T 2 ]. In KSVZ excluded by BBN / V 1 s 43 scales as -1

m GeV 2

= 3 TeV 2 10 s µ giving a bound on

= 1 GeV

= 10 GeV

= 10 V -2

(GeV)

T 10

T T 3/2 m -3 2 Pl . M 10 2 3/2 2 Pl -4 m KSVZ/DFSZ: 5 s M >3m 10 2 W W, ZZ, hh dominates while in DSFZ theories m / that scales as 2 PQ ) for KSVZ (DFSZ with 2 3 V b ¯ -5 2 s → b gg m m PQ 10 s , this bound on 2 V → → gg -6 s s ( 9 Pl 10 16 15 14 13 12 11 10 PQ 10

→

10 10 10 10 10 10 10

M V PQ ) ( GeV V s is excluded because the reheat temperature = 2 TeV so that this bound on (1), we found that this decay mode does not → 4 µ O 2 s π 2 – 9 – 0 10 = = 1 for KSVZ. κ 0 and 288 κ 3

10 DW

= 10 GeV 2

) = NA

˜ T G giving a bound on

< 3 MeV

= 1 GeV ˜ dominates. For both panels, the red region excluded by BBN G

bb NA ¯

KSVZ

Rs T -1

→ T

= GeV 0.1 excluded by BBN

= 1 GeV → 10 s = 30 GeV

. We have taken NA

s NA s

, the red shading applies only to KSVZ theories. For the right

T T Γ( / -2 , is below 3 MeV destroying the success of BBN [ 1 s (GeV) 2 s 10 R 3/2 /m T = 3 TeV and in KSVZ theories m 2 -3 W W, ZZ, hh s µ GeV and does not appear.

= GeV 0.1 2 Pl m →

16 M NA T s -4 KSVZ DFSZ 2 3/2 m KSVZ/DFSZ: (1) parameter which depends on the couplings between the PQ breaking field 10 O >3m -5 2 PQ GeV and does not appear in the figure. In the left panel for DFSZ theories, the V 10 . The maximal parameter space for thermalized gravitino dark matter. In the left panel, 2 s is an 16 m = 2 TeV 0 -6 κ µ 9 that scales as 10 These bounds, leading to the gray and red excluded regions, are inherent to the sax- The saxion decays into a pair of gravitinos through its mixing with the sgoldstino field The red shaded region of figures 13 12 11 10 15 14 16 = 30 GeV and saxion decay is dominated by symmetry in the couplings. Even if

10 10 10 10 10 10 10

PQ ) ( s GeV V 2 PQ Z give additional constraints beyond the gray and red shadedion regions. condensate dilutiongravitinos mechanism or and axions cannot are frequently be important, evaded. and may Other lead to processes further producing constraints in the Here and the supersymmetry breaking field, and may be suppressed if there is an (approximate) dominant saxion decay is is larger than 10 or the mixing ofgravitinos the is axino with the gravitino. The decay rate of the saxion into a pair of from saxion decays, theories, where the dominantIn saxion both decay panels is ofpanel figure in DFSZ theories, theV dominant saxion decay is than 10 In the right panel with applies only to KSVZ theories. We set Figure 2 m JHEP07(2017)125 > → . ˜ a s (3.6) (3.4) (3.5) b f m m m , 1 2 have only 1 5 ˜ G f ν . 3 2 N / 3 1 5 m MeV 1 5 2 / 3 2 / m MeV 3 m MeV GeV PQ are avoided by taking V . In KSVZ theories saxions ) gives 12 ˜ a GeV 10 ˜ a NA 2.3 PQ T ), giving V GeV 12 ˜ a, PQ 1 2 ˜ V G . Here we provide a brief qualitative 10 12 2.5 4 s 10 → µ 3 2 m s 1 2 ) gives 3 2 s 2.4 . m s and figure µ – 10 – to dominate. Finally, there is the possibility that m TeV 3 100 GeV 3.3 gg 100 GeV and → , the decay rate of eq. ( 2 ). Requiring dilution to yield the observed dark matter s 75. . W 3.2 2.7 m µ 100 GeV . This means a large amount of entropy is injected only after 2 TeV 100 MeV also removes constraints from axino freeze-in. Effects on BBN ' > ) = 10 NA ' ˜ a s ) s T ) R of eq. ( ]. A stau LOSP also makes the BBN constraint mild; a gravitino m ), gives an analytic estimate for m T gg ( 44 the decay rate in eq. ( D ∗ 2.9 g → W 20 GeV , which shows the maximal allowed region, and consider them at length s m ( 2 ' 2 ) ¯ NA f < W W, ZZ, hh T f s ) plane, but they depend on other parameters, such as the axino mass, or on → m → s PQ s ( ( ,V 2 NA We find that previously thermalized gravitinos decouple from the thermal bath at a Contributions to gravitino dark matter from NA / T must be mildly suppressed for . A sufficiently large 3 T s m In all cases we take while for decay dominantly to gluons, with a rate given in eq. ( In DFSZ theories with temperature higher than these gravitinos stop interactingdiluted with by the the factor bath.abundance As via a eq. result, ( the gravitino abundance is during the oscillation of the saxionenhanced field inhomogeneities by in parametric the axion resonance.the field are very exponentially early However, cosmological theall evolution importance these of of constraints the can this saxionquantitatively be field, effect below avoided, and in depends they sections is on are model frequently dependent. important While and we discuss them LOSP mass, and 300 GeVto is already dark large matter. enoughgravitinos to provide are A a highly sub-dominant crucial contribution suppressedsubsequently feature diluted as by of saxion they our decays. occuraa We scheme also during note is a that that in matter KSVZ freeze-in dominated theories of the era decay both and are axinos and m arising from the lightestcan observable supersymmetric be particle made (LOSP)300 sufficiently freezeout GeV the and mild freezeout decay abundance by ismild having quite effects small; a on and sneutrino BBN neutrinosmass from [ LOSP. decay below With to 10 a GeV sneutrino is mass allowed. of The gravitino freeze-in abundance is controlled by the details of cosmological evolution atthem temperatures from figure far above thein the TeV next scale. two Hence sub-sections and weillustration in omit of figure how these other constraints can be avoided. ( JHEP07(2017)125 , 2 ), =

10 2 PQ µ PQ V . The <3MeV →h

= V

10 10

Rs I

˜ T s 2 ,

(10

3.1

= 1 GeV

1 GeV ) plane as

FOH 2

1 NA

M T excluded by BBN

= 10 GeV

V 4

-1

,V NA = V

T 2 s

10 = 10 = /

3 NA I

= 10 TeV T -2 m s (GeV) s 10 3/2 contours as in ﬁgure m -3 2 Pl M 10 KSVZ:m 2 3/2 NA gg T ). Throughout the entire al- in the ( -4 >3m 10 2 → 2 PQ V > 0.11 s -5 2 s 2 m h 10 FI ∼ ∼ G G 10) TeV in the (left, right) panel. The Ω -6 give contours of , in ﬁgure 9 10 16 15 14 13 12 11 10 4 = 1. W W, ZZ, hh 10

10 10 10 10 10 10 10

(

PQ ) ( GeV V = (1 NA DW T s gg/ N 2 m 10 – 11 – for →h and ﬁgure 10 3 ˜ 3 FOH 1 < 3 MeV

excluded by BBN GeV

Rs 2 T -1

excluded by BBN

= 1 GeV

10 GeV 10 = 10

= 1 TeV

T T -2 s

(GeV)

= V 10 I 10

(TeV) so that dilution of previously thermalized gravitinos by decay s 3/2 m O -3 is the initial saxion ﬁeld value in the “Decoupled Saxion” cosmology of 10

KSVZ:m <

2 s M

-4

/ 2 3

PQ 2 ). NA 10

T = V

> m 3 I

PQ -5

2 and the importance of these contours for axion parametric resonance is discussed 2.13 V

3.4 s

2 . Constraints on gravitino dark matter in KSVZ theories, with saxions decaying dom- 10 m 2.2 -6 9 10 In addition to gravitinos thermalized during an early epoch, the decays of supersym- Using the full Boltzmann equations for the cosmological evolution and dark matter 13 12 11 10 15 14 16

10 10 10 10 10 10 10

PQ ) ( GeV V metric particles to gravitinos also contributesThe to freeze-in the final process abundance is viametric the IR FI partners dominated mechanism. and becomes terminatesmasses. exponentially when Since suppressed the gravitinos as abundance interact of the with supersym- all temperature multiplets falls via below the goldstino their interaction, this FI by BBN and the consistencydot-dashed of (dashed) supersymmetry lines breaking are ofusing discussed eq. figure in ( section section in section 3.2 Further constraints inWe KSVZ explore theories further with constraintsout on of KSVZ the theories from LOSP.with First freeze-in numerical results of we given gravitinos perform inallowed and a figure parameter freeze- space similar is calculation the white of (unshaded) region. The red and gray regions excluded lowed region of the saxion condensate involvesdiscuss cosmology this of scenario the in TeV aUV era very theory, or general by later. framework, making Hence without the we the two are need key able assumptions to to specify listed at a the particular beginning of section II. 1 TeV (6 TeV) for the left (right) panel. We set dilution, we show thesolid/dashed numerical contours for results saxion for decays to Figure 3 inantly to gluons. In deriving the BBN constraint on freeze-out higgsinos, we take JHEP07(2017)125 . s PQ take V > m ˜ a . As a ) plane. ˜ a m = 6 TeV . NA m PQ µ T 4 excluded by ,V and . This implies 2 ). The former / 2 2 µ PQ 3 / / 3 3 V m in this region, the m m but not NA s R T and T and solely determined by , where the saxions decay are the same as in KSVZ PQ V ) and from BBN limits from PQ tot NA and 2 hh/ZZ/W W V / , the FI contribution is absent T F 3 3 µ 3 → m √ s ), is loop-suppressed relative to the = 300 GeV, while < are given in the ( s 2.5 m PQ NA . V T ˜ G 2 / a 3 are excluded by BBN due to late decays of m 3 , (magenta region at large 3 MeV excludes only the largest values of . we fix ˜ G – 12 – 4 < ). In DFSZ theories, this faster saxion decay rate h NA s T R ) and is insensitive to 2.3 → T ˜ 2.8 H ) and depends only on . s 2.8 R T . In figure , is increased from the left to the right panel, increases both 4 µ and 3 era or after saxion reheating for low enough NA . The saxion decay rate to gluons, eq. ( so that the region with produced via freeze-out. The subsequent decays of higgsinos place constraints W s R m ˜ ). Numerical solutions for contours of H T s 2 . The freeze-in of axinos (and gravitinos) from decays of the higgsino LOSP occurs > = 1 TeV because the higgsino LOSP decay is inefficient. However, with PQ s < m . The latter disappears in the lower panels because the higgsinos from FO decay before Axino FI and decay to gravitinos yields too much dark matter in the green regions at A key feature of DFSZ theories is additional production processes involving axinos, if As in the KSVZ case, DFSZ theories can have small regions at low While the gray shaded regions requiring The regions shaded in magenta in figure The FI abundance is proportional to the decay rate, which is enhanced for low gravitino V is not too large, giving the large green, purple and orange regions of figure µ m µ ˜ a ˜ a m low during the MD that the relevant dilution factorthe is freeze-in ( temperature and disappears in the left panelson because the higgsino decay rateBBN to to gravitinos depends axinos strongly which then decay harmlessly to m for decay rate to electroweak bosons,increases eq. ( over-closure from gravitino freeze-in (brown regionthe at decays low of freeze-out higgsinos, ( The white (unshaded)limited regions by give several constraints the as allowed discussed below. parameter space,theories, which the red is shaded region significantly from BBN limits on saxion decay are much milder, especially 3.3 Further constraints inThe DFSZ analysis theories for with KSVZby and comparing DFSZ figures theorieson has different both values similarities in and the differences, four as panels, seen such that in the upper (lower) panels, on high gravitino masses, whereearly. the decay The is late, higgsino and mass, the on higgsino low decay rate andright the and higgsino upward. FO abundance. The BBN constraint shifts to the region. Furthermore, since thedilution freeze-in of factor gravitinos is occurs givenresult, below by the brown eq. region is ( excludeddilution because for the saxion FI decays gravitinos, too regardless early to of provide sufficient higgsinos purposes, we assume a higgsino LOSP. masses and high parent particlefor masses. As shown in figure in the right panel, freeze-in gravitinos become the dominant source in the brown shaded contribution is determined mainly by the LOSP and hence the LOSP mass. For illustration JHEP07(2017)125 = 2 2 s 10 10

PQ PQ m = 1.

= = 1 GeV

1 GeV →hG

2 10 10

= = V 10 V 10 /

> 0.11

I I

NA NA DW

s 2

T T

PQ PQ ˜ h N followed by m FOH 1 1

. In the top

mis a = = V V

I I →sG excluded by BBN

GeV

GeV ˜ s s

2 2 a ˜ s = 150 GeV = 400 GeV FI > 0.11 → -1 -1

∼ ∼

a a

= = 10 2 10 ˜ H h 10 10

→

s→ hh,WW,ZZ

NA NA

excluded by BBN mis a

T T a Ω > 0.11 -2 -2 (GeV) (GeV) 2 h 10 10 FI 3/2 3/2 . In the top right panel > 0.11 ∼ ∼ a G m m 2 ˜ Ω -3 -3 G h = 2. For the left (right) two FI h 10 10 ∼ ∼ a G

> 0.11 Ω

q 2 2 Pl 2 Pl

→ h -4 -4

M M

∼ G ∼ 10 10 ˜ 2 3/2 2 3/2

∼ G

→ a s DFSZ:μ= 600 GeV,m Ω DFSZ:μ= 600m GeV, > 0.11 > 0.11 -5 -5 >3m >3m 2 2 h h 2 2 PQ PQ 10 10 FI FI V V ∼ ∼ ∼ ∼ G G G G 2 s 2 s Ω Ω m m -6 -6 . In the upper (lower) two panels we take 9 9 10 10 16 15 14 13 12 11 10 16 15 14 13 12 11 10 µ

10 10

10 10 10 10 10 10 10 10 10 10 10 10 10 10

PQ PQ ) ( ) ( GeV V GeV V followed by 2 , 2 2 1 ˜ 10 10 – 13 – Hh M ). The axion abundance is computed with ˜ → s 10 10 a

→hG

GeV GeV

2 2 < 3 MeV < 3 MeV < m 1 1

Rs Rs

= = 10 10 FOH T T

excluded by BBN excluded by BBN excluded by BBN

NA NA

= 150 GeV = 400 GeV

T T

PQ PQ -1 -1

∼ ∼ a a

= = 1 GeV 1 GeV 10 10

> 0.11 ˜

NA NA

= =

V 10 V 10 . This leads to excessive dark matter in the large orange regions,

T h I I

-2 -2 s s ˜ FI →hG (GeV) (GeV) G ∼ ∼ a G = 150 GeV > 0.11 ˜ 10 10 ˜ a Ω 2 ˜ a 3/2 3/2 →H h m m ∼ G a ˜ m -3 -3 → ∼ (

FI PQ PQ → ∼ s→a G 10 10 s s

Ω ˜

excluded by BBN H . In the lower two panels the axino is the NLSP and the only decay mode

= = V V

I I 2 Pl 2 Pl

-4 -4 s s M M 10 10 > m 2 3/2 2 3/2 > 0.11 DFSZ:μ= 200 GeV,m DFSZ:μ= 200m GeV, 2 -5 -5 >3m >3m h 2 2 , which is harmless. Note that, in both purple and magenta regions, the exclusion PQ PQ FI 10 10 ∼ ∼ . Constraints on gravitino dark matter in DFSZ theories with saxions of mass a G V V ˜ = 200 GeV (600 GeV) and we take 2 s 2 s G Ω m m a -6 -6 µ 9 9 10 10 10 13 12 11 16 15 14 10 13 12 11 16 15 14 hh/ZZ/W W In the lower two panels, the axino mass is suﬃciently small that a new saxion decay Axino FI and decay is excluded by BBN in the purple regions of ﬁgure

→ 10 10

= 400 GeV

10 10 10 10 10 10 10 10 10 10 10 10 10 10

PQ PQ ) ( ) ( GeV V GeV V a ˜ a → s is ˜ from BBN is coming from the long lifetime of thechannel opens decay up: to gravitinos at large m left panel the relevant decaythe chain higgsino is ˜ is heavier than the axino so the relevant decay chain is ˜ Figure 4 300 GeV decaying dominantly topanels Higgs and electroweak bosons, for JHEP07(2017)125 I s ) is I (3.9) (3.7) (3.8) πs 2 / inf ]. Once the , this applies H µ 2 46 , < 45 s . Saxion oscillations m , where saxions decay 2.2 > µ ˜ a , , the axion field takes spatially 2 > m . PQ , PQ ) 4 3 s V s V 2 ( m θ I osc s s ∆ . Additional fluctuations created once ) leading to excessive dark matter and 2 s . Since we require µN s e m 4 PQ = , as the Hubble scale is smaller. . V s ∼ ) I < m s s < ∼ PQ inf ( ˜ a 2 V s πs m PQ H 2 H – 14 – V m 2 θ ∼ ) ∼ ∆ (for (1). This can be seen easily by energy conservation. s 2 ) ( k s O ˜ G ( osc ˜ a ∼ δθ N θ < δθ → ρ s = 1, these domain walls are unstable and decay into axions. If ) oscillations, the fluctuation of the angular direction is given . To discuss the constraint from axion parametric resonance we s ( 2 DW osc . ) leading to BBN problems. s N ˜ a N m are small, ∆ (1)) is the growth rate per oscillation. Here the factor of ( via parametric resonance. When the saxion field has dropped from ]. If > m O s s PQ 54 – V m = m 47 µ ∼ ( (for 1, these domain walls are stable, dominate the energy density of the universe, and k µ e ˜ G > The saxion condensate oscillatesfrom about an the initial present amplitude vacuum larger in than a quadratic potential and undergone s However, this scaling breaks down for For illustration, we consider the “Decoupled Saxion” of section We have not displayed a panel for the spectrum • s falls below → DW ˜ s The energy density of the fluctuation is the primordial fluctuation ofthe the universe is angular radiation-dominated direction during produced these during early inflation. oscillations, Assuming The number of the oscillations grows at small roughly by where hence are excluded. significantly enhance the fluctuations ofaround the axion field modesto with physical wave numbers the fluctuation of thefluctuation of axion the field axion via fieldvarying becomes the random much parametric larger values, than resonance leadingtransition effect [ to [ the formationN of domain walls after the QCD phase add an assumption about evolution well above the TeV scale: In this case the saxion oscillation, through its coupling with the axion field, enhances regions from the upper andto lower a panels of small figure range of 3.4 A limitThe on constraints the derived saxion in the condensatedeclared previous from early sub-section axion in require parametric section only resonance the two mild assumptions a complementarity between a into axinos (giving an orangea over-closure purple region), BBN and region). axinos decay In into this higgsinos (giving case, the excluded regions are roughly the unions of these again showing the critical importance of the axino mass in DFSZ theories. Note there is JHEP07(2017)125 ). ). τ θ < (1), ( (4.1) term ; the O drops 3.10 (3.10) µ s PQ < V ) h, γ, Z ∼ PQ . Thus ∆ in eq. ( V 2 s I s ∼ s 2 s s . m ( ) at 2 ”, this condition is 1, the regions below δθ 3.8 > PQ 2 ). In this more general V ]. It is illuminating that / 3 , which is shown by pink 57 DW – m 1 2.13 N 100 keV = 10 55 ) and ( . I 5 s 3.7 (1-10) via eq. ( (1). For the only allowed parameter region has O NLSP parametrizes the size of the primordial TeV 4 ) LOSP. NA ∼ O m I R T s 3 4 τ δθ > ), I inf with labels “ 2 m πs H – 15 – 2 3.10 . The growth in the fluctuation cuts off as 4 = 1, the axion misalignment angle is randomized ' ln is a parametrization of the strength of the saxion PQ ˜ G 1 µ I 1, from figure DW . However, to phrase the constraint in a way that is s → and N > c PQ ), 3 < s < V V NLSP I DW PQ Γ s 3.10 V N , I ≡ s ˜ G ) or right-handed stau (˜ → 0 χ NLSP cτ , and the condition for domain walls not to be produced is . In four dimensional grand unified theories, symmetries which control the PQ PQ V V The conventional signals of displaced vertices involving gravitinos result from NLSP Note that this constraint is derived by evaluating ( In the DFSZ theory with (1) for fluctuations produced at a decay length given by Examples include theThis neutralino conventional signal (stau) applies NLSP whenever the decaying LOSP to decays to a the axino gravitino are and kinematically We discuss the following collidersand signals resulting kinks from involving LOSP decays gravitinosassume — or a displaced neutralino vertices axinos, (˜ and the saxion resonance.decay via For interactions suppressed illustration by we the mediation scale of supersymmetry breaking, with scale during inflation. It affects the bound only4 logarithmically. Signals 4.1 Displaced vertices, kinks and saxion resonances at colliders On the left handcondensation, side and of can eq. be ( formulation rewritten of the in bound, terms thechanged. of positions of In the the dashed log linesfluctuation and in of pink the the shaded middle angular regions of direction, are and eq. not should ( be replaced by the PQ symmetry breaking details of the evolutionthe prior saxion to condensate this oscillatesan about is initial the irrelevant. amplitude present larger Thus vacuumindependent than in the of a constraint the quadratic earlier applies potential saxion provided from evolution requires a reinterpretation of itemized at the beginning of this sub-section. large of the Higgs doublets must bethe broken constraint at from the unification parametric scale resonance [ also points towards a large PQ breaking scale. Below the dashed linesviolated and in we expect figures axion fluctuationsthe with dashed lines are excluded.and these For regions areshadings. subject to These the constraints constraint may given be in avoided figure in cosmologies that violate the assumption O below giving an upper bound on and cannot be larger than the energy density of the saxion oscillations, JHEP07(2017)125 the (4.3) (4.2) τ 4 γ/Z, W , ν ± H L τ , τ ± — with ˜ ± H ˜ 2 h/Z, W H 0 , ν ˜ ˜ a 1 , W ˜ − R χ 0 L ˜ τ τ ˜ ˜ B, H C . 2 2 ± ˜ H ˜ R a , ˜ τ 0 β ˜ ]. This is applicable to H 10 GeV tan PQ . Below the gray regions, 23 . Thus the decay of the ˜ a γ/Z V 12 R Pl τ 2 10 M 2 / LOSP 3 3 µ > m R ˜ m ˜ R a B GeV ˜ τ ˜ τ PQ V 10 ) can occur. In the lower left panel of GeV µ 10 LOSPs decaying to NLSP axinos. 4.1 3 PQ L H and the upper left panel of figure τ R V the axino may be heavier than the LOSP. 10 τ s 2 3 m boson may be observed as a displaced vertex. (b) (c) † ˜ R ˜ τ and ˜ ˜ a H H H τ – 16 – R 0 GeV Z 0 ˜ τ ˜ m 3 χ µ χ m 10 ˜ (e) (f) (g) ˜ a B R 2 ˜ τ 2 µ µ — with 2 2 q q through higgsino-axino mixing — (b) of table 1 ˜ a τ γ/Z h/Z 5 m 1 m → ' ' τ ˜ a ˜ a , except the upper left since we need ˜ ˜ a a → → 4 ˜ τ c 0 LOSP (a) (d) ˜ . Displaced signals from ˜ ˜ χ H cτ 0 Γ ˜ χ W 0 0 ˜ ˜ χ χ is the higgsino component of the neutralino LOSP [ ≡ ˜ B, ˜ a ˜ H Table 1 0 → boson — (a) of table ˜ 0 χ , this signal competes with LOSP decays to axinos, described below. ˜ χ Z C 4 cτ In DFSZ models with a neutralino LOSP, the neutralino decays to the axino and the We may also observe displaced vertices and/or kink signals if the axino is lighter DFSZ KSVZ be seen from theneutralino into smaller the suppression axino scale With and the the stau Higgs/ LOSP, ˜ where all panels of figure this decay mode is typically more efficient than that to the gravitino final state, as can the LOSP. Through thetheories, axino the LOSP interactions will with decay higgsinos into the (gauginos) axino for andHiggs/ DFSZ SM (KSVZ) particles. axino is heavier than theHence, LOSP, in and all in these figure casesfigure a displaced signal from ( than the LOSP. Any MSSM particles produced at the colliders will first cascade down to forbidden or sufficiently suppressed. In figure JHEP07(2017)125 , a a 3 3 , ) ˜ +˜ 2 LOSP γ/Z Z/h era and ( (10 keV) + R appearing τ τ — and is < m NA ˜ a 1 ∼ O or W 2 a / 3 +˜ < m m s W + m is emitted. τ ν ). For Z/h — (d) of table A.4 a — the latter has ) ˜ 1 γ/Z . On the other hand, a stau LOSP 5-6 for KSVZ, leaving a displaced vertex for these gravitinos rapidly become cold. 2 / gg 3 m — (g) of table , we predict a mixture of cold and warm dark – 17 – ˜ a . Since the lower limit on the saxion mass is of 4 W 4 τ . If FI gravitinos dominate DM, the warm DM bound ν and — through axino-bino mixing arising from the non- A 3 . The stau also has 3-body decays linearly suppressed by for DFSZ and 1 ]. At larger and 1 58 a ) ˜ ¯ ff/gg , for example near the boundary of the brown regions of figures . Remarkably, the saxion can be observed as a resonance despite boson resonances, taus, missing energy, and a saxion resonance. 2 Z/γ ˜ G / ( ), typically by a factor of 10 (keV) [ 3 Z → L O τ m 4.2 term, leaving a pure kink. For large stau masses this mixing becomes is in an appropriate range. The axino then travels some other distance and decays to the axino and SM particles, leaving a displaced vertex or > ∼ — (e) of table 2 ã NLSP ã D / and . The latter decay is observed as a kink where 3 → → cτ ˜ a 1 m R PQ is larger than a keV. These FI gravitinos give a significant component of DM τ hh/W W/ZZ/ V 2 LOSP LOSP before it finally decays to the gravitino and the saxion/axion, with the saxion / 3 cτ cτ becomes somewhat more stringent, as shown in eq. ( ˜ G ), as discussed in appendix m 2 → / and is dominant far enough from these regions. . For this range of parameter space, FI gravitinos are produced in the MD 3 A.5 On the other hand, gravitinos produced from the FI decays of higgsinos can be warm If the saxion is heavier than the axino then the axino decays invisibly to an axion and In KSVZ models, a neutralino LOSP decays to ( 4 4 m NLSP hence are diluted less than thermaleq. gravitinos, ( leading to the largeron free-streaming length of near the brown regionsmatter of from figures thermal and FI gravitinos, respectively. constraints if even if only at low and 4.2 A warm componentWe of have considered dark three matter sources forin gravitinos: by higgsino decoupling from decays, the anddilution thermal decays is bath, of always freeze- freeze-in large axinos. enough Given that the the observed thermally DM decoupled abundance, gravitinos satisfy warm DM cτ decaying to an appropriate its feeble coupling withmulti-jets the from SM. Higgs/ This particular decay mode has a distinctive feature of e.g. in the upperorder right (10 MeV, panel 1 of GeV)parameter figure for space. (DFSZ, KSVZ) Indistance theories, this this case, cana the kink occur LOSP if in is a produced wide at region the of collision point, travels a three-body phase space factors ordecay the of ratio the between LOSP the into electroweakand axinos and is SUSY sub-dominant scales. near The the gray-shaded regions of figures a gravitino. However, an interesting and unique signal arises when decays into zero electroweak quadratically suppressed by the electroweakbecomes vev, favored so — that (f) the ofthe 3-body electroweak table vev: final state at a displaced vertex. All of the above modes are suppressed by loop factors as well as signal. For large stau masses, the— stau (c) instead dominantly of decays table to observed as a displacedthat vertex. the Since one this in mode eq. is ( loop-induced the decay rate is smaller Similar to conventional gauge mediation with a stau NLSP, this decay may leave a kink JHEP07(2017)125 . 2 ) PQ ) h W (4.5) (4.6) (4.4) /V 2 i m m v 2 2 3 i q > < (1). In all i s s P m m . The analyses ∼ O = . The FI axinos 4 κ , κ 4 i observations. It has KSVZ DFSZ ( DFSZ ( α . Gravitinos that were 2 symmetry or fine tuning. NLSP 4 2 2 m / f Z 650 GeV s m m and 3 f , 3 N 2 2 15 log 4 . , 3 s , as required to forbid the decay of the 2 DW 10 PQ µ 4 N 650 GeV there is a sizable component of m 2 µ 1 + 0 2 πV h s ] for a review.) ≈ 2 κ < µ 3 2 64 m 100 GeV s 63 1 3 . = α 0 2 2 m – 18 – , the free streaming length of gravitinos becomes NLSP is much relaxed because the saxion decay to the ˜ a aa µ µ 2 2 2 q q m m κ → NLSP 1 s . κ 0 Γ 2 2 m 1) to be compatible with the current Planck constraint 650 GeV . κ κ 2 / 5 1 3 (0 . . . 3 0 0 0 O can arise from an approximate m            ] place an upper bound on the dark matter free streaming κ < ∼ κ = 62 1 Mpc – aa ] and can be approximated by s 60 ≈ → Γ 59 s [ Γ 2 ) / ), part of parameter space is accessible to the CMB-S4 experiment. ]. A small decay FS 3 dec λ forest [ 65 m are the PQ charge and vev of each PQ breaking field, leading to an effective ν 4.6 i 1 Mpc. As a result, for T α 6 [ v ( . ∗ < ∼ g ]. This decay rate depends on the model-dependent parameter 4 7 FS = 0 and λ i 64 = , q eff 32 N eff These warm gravitinos lead to possible signals in the Lyman- In DFSZ theories, the abundance of gravitinos from FI axinos can be comparable to N ∆ three cases of ( 5 Conclusions The main resultsthermalized of early this in paper the universe are and shown later in diluted from figures the decay of a saxion condensate In KSVZ theories, we take of ∆ For DFSZ theories, thevisible constraint sector on is moresaxion efficient; in into fact, a for pair of higgsinos, the Planck constraint is satisfied even if where number of relativistic neutrinos Axions may contribute to dark radiationion in can both KSVZ decay and to DFSZ a theoriestial pair because [ of the relativistic sax- axions via the trilinear coupling in the Kählerpoten- warm dark matter in the parameter space close tobeen the shown boundaries that of warm the darkbaryon green matter feedback regions. can may also solve the play small a scale role. structure4.3 (See problems [ although Dark radiation where the NLSP is thefrom higgsino Lyman- (axino) in thelength, left (right) panels of figure that of thermally decoupledbecome gravitinos non-relativistic near before the decaying greenthe thermal regions to gravitinos. of gravitinos, When figure givingindependent them of momenta larger than JHEP07(2017)125 > PQ V ) from a PQ ,V 2 / GeV, the saxion 3 , parametric reso- ) and saxion mass 9 keV. m ) plane; as large as PQ PQ > V , which may allow an 2 ,V PQ / 2 3 / PQ ,V 3 V 2 m / m 3 ), leaving displaced vertices (10) MeV, where both BBN for this well-known signal, for O m 1 R T ) parameter space is allowed, although . However, these additional constraints to be larger than 10 4 PQ PQ ,V V 2 / 3 – 19 – m for the saxion condensate to decay before BBN, as PQ , but puts strong constraints on ( V . In DFSZ theories the higgsino mass is bounded from PQ V 1 (1) GeV. , which typically has a larger region in DFSZ theories, although O PQ V = 1 is more easily obtained. allows for a saxion mass as small as DW order unity, some dilution is required, placing an upper bound on the PQ N . For this upper bound on V . Results are shown for a very wide range of ( 3 mis 2 θ 1, these large fluctuations lead to the formation of disastrous stable domain walls. > The saxion condensate alsoradiation decays abundance. to relativistic axions, leading to a non-zero dark decay and can be observed as a resonance. In some parameter regions gravitinos areSuch dominantly produced gravitinos via may freeze-in behave processes. as warm dark matter even if vertices. We have provided aexample, cosmology of with low-scale high gauge mediation. If the LOSP is heavierchain than of the MSSM axino, particles theand/or produced axion kinks. multiplet at participates colliders If in (table the the saxion decay is lighter than the axino it is produced through axino If the LOSP is lighter than the axino, it will decay to light gravitinos at displaced GeV and A combination of measurements, especially from displaced vertices or kinks at collid- Here we summarize possible signals of our scenario: We have also estimated the axion abundance from the misalignment angle. For a If the saxion begins its oscillation with a field value larger than In DFSZ theories the efficient interaction between the axion and Higgs multiplets In KSVZ theories a large part of the ( • • • • , and are independent of almost all UV model-dependence. 12 s DW ers, could constrain theindependent theory probe from and axion narrow physics. the prediction for decay rate, as shownabove accordingly. in figure This is avoided for large in KSVZ theories sufficiently small saxion10 decay rate, the axion abundance is also diluted. For upper bound on and astrophysical bounds are included. nance may induce large fluctuations ofN the axion field. In theories with domain wall number shown in figure mass must be larger than weakens the uppervariety bound of processes, on as shown bycan shading be in figure removed, for example by making the axino mass sufficiently large. The reduced constraints, but theseshown leave in large figure allowed regionsm in the ( there is a strong upper bound on provide an excellent candidate for dark matter; they are subject to several important JHEP07(2017)125 = < FI (A.5) (A.3) (A.4) (A.1) (A.2) DM T m ) becomes for freeze-in 2 for freeze- / 2 A.3 / . A sufficiently 3 FI x m dec T at any temperature gives a lower bound in eq. ( f 3 α p / 1 i . Although the freeze-in 4 /s / i 1 p − for those that decouple from . ]. Nonetheless, this bound is 4 ) . 4 3 / , , 3 dec 58 1 3 [ T 2) 2 th A.3 th / / Y Y 3 ' DM FI D th keV i ] that assumes thermal decoupling DM m x DM Y p DM m ρ ( 3 i 58 m m DM p 3 f 4 DM ∝ / ρ of relativistic Weyl fermions. For freeze-in s 1 i = m i s i th s DM n Y ' – 20 – m D (keV) freeze-in. i i DM p s 3 i D O 2, whereas i 4 Mpc p . Specifically, the ratio / m ). This gives the momentum . s 1 4 = ˜ 0 H dec = = A.1 m the entropy density, and the yield is parametrized by T 3 FI f ∼ 8 DM (keV) thermal decoupling f f x 3 f f p s s ρ s p = 1). We are concerned with the free-streaming length of ' O m FI FS i λ      p 1 ( > 5, 2 / − 3 < 2 m ∼ FI x and thus a lower bound on is the dilution factor, is substituted using eq. ( DM , with D D FI /m f For gravitinos that originate from higgsino decays at the freeze-in temperature In general, the DM abundance today is related to the initial one at production by , its energy at decoupling is of order the decoupling temperature /x p ˜ H dec The free-streaming length can be computed using eq. ( we expect a bound ofcan similar be order obtained applies. from Therefore, rescaling the the constraint result on of ref. [ a constant for eachin production than mechanism thermal and decoupling.freeze-in is is larger This enhanced by in implieson a comparison. that factor the of The free-streaming constraintgravitino length phase from space Lyman- in distribution the is different case from of that of thermally decoupled gravitinos, m the thermal bath at temperature where where in units of the(thermal thermal decoupling), equilibrium value dark matter so we study how dilution affects the momentum red-shift; thermal decoupling during ascenario affects RD the era. constraint. In this section, we investigates how the freeze-in If dark matter is initially thermalized andT decouples from the bath while relativistic, light dark matter particleradiation will equality, affect leading structure formation todifferent via when a dark its lower matter large bound is velocity produced at on from matter- freeze-in decays during a MD era instead of A Warm dark matter from freeze-in gravitinos JHEP07(2017)125 ]. ∼ ∼ 2 T / (A.6) (A.7) (A.8) 3 SPIRE m IN ]. [ Phys. Rev. , SPIRE IN . era so ) ][ h (1981) 513 A Constraining warm m 2 , one finds that > 4 B 188 s m , and . 3 3 FI Pl T astro-ph/0501562 ˜ M KSVZ DFSZ ( H,FI 5 [ s NA Nucl. Phys. Y 2 R T , FI T 2 a th 3 s 2 q f PQ Y 4 H 2 m in figures ) V µ ˜ π 2 3 s G α FI 32 m → T NA ˜ 10 ( H T    ∗ , FI gravitinos are overproduced. We investigate – 21 – Γ √ g 4 3 × (2005) 063534 1, gravitinos stay in thermal equilibrium until = p π th FI and th > ˜ Y Y ≈ Y H,FI 3 D 71 Y FI ' ), which permits any use, distribution and reproduction in 2 Supersymmetry, Cosmology and New TeV Physics ]. / H 5 NA 2 T / 7 2 / µ 2 3 SPIRE IN 15 Phys. Rev. m 4 [ , CC-BY 4.0 π = This article is distributed under the terms of the Creative Commons Dynamical Breaking of Supersymmetry (1982) 223 is necessarily less than unity because the FI yield cannot exceed the thermal 48 Lett. dark matter candidates includingLyman-alpha sterile forest neutrinos and light gravitinos with WMAP and the In the brown regions of figures , where they decouple as the higgsino LOSPs becomes non-relativistic and exponentially H. Pagels and J.R. Primack, M. Viel, J. Lesgourgues, M.G. Haehnelt, S. Matarrese and A. Riotto, E. Witten, ˜ H (10 keV) can lead to warm freeze-in gravitinos in the parameter space near the edges of [2] [3] [1] Attribution License ( any medium, provided the original author(s) and source areReferences credited. supported in part by the NationalGrant Science No. Foundation Graduate DGE Research Fellowship under 1106400.number DE-SC0010107. L.H. F.D. is is supported supported by theOpen by Simons Access. the Foundation. U.S. Department of Energy grant Acknowledgments We thank Masahiro Kawasakisupported and in Satoshi part Shirai byPhysics, the for of Director, useful the Office US discussion.the of Department National Science, Science of This Office Foundation Energy work under of under grants was High PHY-1316783 Contract and Energy DE-AC02-05CH11231 PHY-1521446. and and R.C. Nuclear by was m depleted. Using theO numerical results of the brown regions. Note that value. If the above expression gives which we use to derive the parameter space immediately aboveabundance these is brown regions sizable. so In that this the case, warm FI gravitinos gravitino freeze-in during the MD One can estimate the yield for freeze-in production by JHEP07(2017)125 D , Phys. B 120 B 120 , ] (1979) ]. 43 (1993) 4523 Phys. Rev. , Phys. Rev. SPIRE (in Russian), ]. , ]. (1983) 137 Phys. Lett. IN D 48 Phys. Lett. , ][ , ]. SPIRE ]. SPIRE IN B 120 IN ]. arXiv:0802.2487 [ ][ [ SPIRE Phys. Rev. Lett. SPIRE IN Phys. Rev. , IN , SPIRE Axion cosmology with its scalar ][ ]. IN [ ]. hep-ph/9510461 Phys. Lett. (1980) 493 [ , Freeze-In Dark Matter with Displaced SPIRE ]. IN Can Confinement Ensure Natural CP (2008) 123524 ]. Relaxing the Cosmological Bound on SPIRE ]. ][ (1980) 497] [ IN B 166 ]. [ arXiv:1506.07532 SPIRE (1991) 3465 ]. 31 Cosmological constraints on the light stable [ IN D 77 SPIRE (1996) 313 hep-ph/9803263 SPIRE ][ 67 IN [ Can decaying particles raise the upper bound on [ IN SPIRE – 22 – A Simple Solution to the Strong CP Problem with a New Detectors to Explore the Lifetime Frontier Supersymmetry breaking in the early universe [ Cosmology of the Invisible Axion SPIRE IN [ IN Supersymmetry, axions and cosmology B 383 (1981) 199 Nucl. Phys. ][ Yad. Fiz. (2015) 024 , Phys. Rev. Solving Cosmological Problems of Supersymmetric Axion (1998) 44 , 12 ]. Constraints Imposed by CP Conservation in the Presence of CP Conservation in the Presence of Instantons arXiv:1606.06298 (1977) 1791 (1993) 289 B 104 [ A Cosmological Bound on the Invisible Axion ]. (1987) 323 The Not So Harmless Axion hep-ph/9503303 Phys. Rev. Lett. B 437 Phys. Lett. [ , JCAP SPIRE , (1980) 260 [ D 16 , IN B 303 SPIRE [ ]. ]. 31 B 192 IN (2017) 29 hep-ph/0210256 ][ [ Phys. Lett. On Possible Suppression of the Axion Hadron Interactions , (1995) 398 Phys. Lett. SPIRE SPIRE ]. , IN IN Phys. Rev. Dilution of cosmological densities by saxino decay B 767 [ [ 75 Effects of decay of scalar partner of axion on cosmological bounds of axion Weak Interaction Singlet and Strong CP Invariance , Phys. Lett. ]. ]. (1977) 1440 , Phys. Lett. , SPIRE 38 IN [ (2003) 075011 SPIRE SPIRE IN hep-ph/9306293 IN 103 Invariance of Strong Interactions? Signatures at Colliders Harmless Axion Sov. J. Nucl. Phys. Axions the Peccei-Quinn scale? superpartner [ Phys. Lett. [ (1983) 127 (1983) 133 Rev. Lett. supermultiplet properties 68 Models in Inflationary Universe [ gravitino Lett. Instantons M. Dine, L. Randall and S.D. Thomas, T. Banks, M. Dine and M. Graesser, M. Kawasaki and K. Nakayama, R.D. Peccei and H.R. Quinn, R.D. Peccei and H.R. Quinn, T. Moroi, H. Murayama and M. Yamaguchi, M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, R.T. Co, F. D’Eramo, L.J. Hall and D. Pappadopulo, M. Dine, W. Fischler and M. Srednicki, A.R. Zhitnitsky, J.E. Kim, M. Kawasaki, T. Moroi and T. Yanagida, M. Hashimoto, K.I. Izawa, M. Yamaguchi and T. Yanagida, M. Dine and W. Fischler, J.P. Chou, D. Curtin and H.J. Lubatti, G. Lazarides, C. Panagiotakopoulos and Q. Shafi, J. Preskill, M.B. Wise and F. Wilczek, L.F. Abbott and P. Sikivie, J.E. Kim, D.H. Lyth, [9] [7] [8] [5] [6] [4] [22] [23] [19] [20] [21] [17] [18] [14] [15] [16] [12] [13] [10] [11] JHEP07(2017)125 ] , ] B , , ]. Phys. Phys. , , Phys. Rev. Phys. Rev. Phys. Lett. , Constraints , , ]. SPIRE Phys. Lett. ]. , IN hep-ph/0611350 lattice QCD ][ ]. [ ]. SPIRE SPIRE arXiv:1606.03145 IN IN ]. [ (2017) 054502 [ ][ SPIRE SPIRE = 2 + 1 ]. IN IN f (2008) 51 ][ SPIRE D 95 N ][ IN Topological susceptibility at high ]. 741 SPIRE ][ (2016) 498 IN (1988) 1797 [ ]. astro-ph/0402490 Axions in Gauge Mediation ]. SPIRE 60 [ IN Topological susceptibility in finite Phys. Rev. B 762 ]. ][ , SPIRE ]. arXiv:1606.07175 IN SPIRE [ IN ][ (1988) 188 hep-ph/0406072 (2005) 7 ]. SPIRE arXiv:1603.04439 -dependence from [ The topological susceptibility in finite ][ Dynamics of the Peccei Quinn Scale θ [ IN SPIRE Lect. Notes Phys. IN ][ ]. arXiv:1606.07494 – 23 – , Phys. Lett. SPIRE ][ [ B 203 , Hadronic decay of late - decaying particles and B 625 IN Phys. Rev. Lett. (2016) 021 Supersymmetric axion grand unified theories and their , ][ SPIRE 09 ]. IN [ arXiv:1702.00401 (2004) 103511 [ (2016) 69 ]. ]. hep-ph/0510091 (2016) 075001 Unified Models of the QCD Axion and Supersymmetry Breaking [ ]. Phys. Lett. Phys. Lett. JHEP arXiv:0906.5015 Constraints on Light Particles From Supernova Sn1987a SPIRE , Bounds on Exotic Particle Interactions from SN 1987a , , 539 [ D 70 IN SPIRE SPIRE D 94 [ arXiv:1512.06746 IN IN arXiv:0906.1273 SPIRE [ [ [ ][ (2017) 507 IN (2006) 66 Calculation of the axion mass based on high-temperature lattice quantum Axion cosmology, lattice QCD and the dilute instanton gas [ Nature Axion phenomenology and arXiv:1508.06917 , [ Fate of oscillating scalar fields in the thermal bath andCan their oscillating cosmological scalar fields decay into particles with a large thermal mass? Astrophysical axion bounds Axions from SN 1987a arXiv:1012.5380 Phys. Rev. (1987) 525 The New Mechanism of Baryogenesis and the Inflationary Universe B 921 Phys. Rev. , On the Role of Quasiparticles and thermal Masses in Nonequilibrium Processes , B 635 (2009) 125023 , (2016) 155 ]. ]. (1988) 1793 (1985) 243 03 (2009) 125017 60 B 193 D 80 (2016) 175 SPIRE SPIRE arXiv:1611.02411 IN IN on Axions from SN 1987a Big-Bang Nucleosynthesis [ Lett. Lett. temperature on the lattice chromodynamics temperature (2+1)-flavor QCD using gradient flow JHEP temperature QCD and axion cosmology [ predictions 752 D 80 Rev. Nucl. Phys. Phys. Lett. in a Plasma [ B 160 implications M.S. Turner, R. Mayle, J.R. Wilson, J.R. Ellis, K.A. Olive, D.N. SchrammM. and Kawasaki, G. K. Steigman, Kohri and T. Moroi, J.R. Ellis and K.A. Olive, G. Raffelt and D. Seckel, S. et Borsányi al., Y. Taniguchi, K. Kanaya, H. Suzuki and T. Umeda, P. Petreczky, H.-P. Schadler and S. Sharma, J. Frison, R. Kitano, H. Matsufuru, S. Mori and N. Yamada, R.T. Co, F. D’Eramo and L.J. Hall, S. et Borsányi al., C. Bonati et al., L.M. Carpenter, M. Dine, G. Festuccia and L. Ubaldi, K. Harigaya and J. Leedom, J. Yokoyama, M. Drewes, L.M. Carpenter, M. Dine and G. Festuccia, A.D. Linde, J. Yokoyama, G.G. Raffelt, [41] [42] [43] [39] [40] [37] [38] [35] [36] [32] [33] [34] [30] [31] [27] [28] [29] [25] [26] [24] JHEP07(2017)125 ] D 73 ] , ]. ] Phys. ]. , (2016) 031 SPIRE ]. Phys. Rev. ]. ]. IN , 01 ]. SPIRE ][ hep-th/9510146 IN [ SPIRE ][ Phys. Rev. Lett. SPIRE SPIRE ]. ]. hep-ph/9709202 IN , IN SPIRE JCAP IN hep-ph/0201018 [ ][ , IN ][ , ][ ]. forest data ][ arXiv:0812.0010 SPIRE SPIRE α [ (1996) 35 IN IN ][ ][ SPIRE (1997) 117 IN Lyman-alpha constraints on warm ]. ][ B 376 arXiv:1611.05028 Cosmic strings from preheating Big-Bang Nucleosynthesis and [ Warm dark matter as a solution to (2009) 012 arXiv:1604.01489 hep-ph/9903324 [ B 415 hep-ph/9608405 ]. SPIRE ]. [ 05 [ IN arXiv:1505.05024 arXiv:1305.5338 ][ [ [ ]. Reheating after inflation Towards the theory of reheating after Domain wall and isocurvature perturbation Phys. Lett. SPIRE SPIRE (2017) 005 hep-ph/9804429 , JCAP arXiv:0804.3745 IN [ Cosmological axion problem in chaotic Domain wall problem of axion and isocurvature IN Constraints on mixed dark matter from anomalous , [ [ 03 ][ (1997) 94 – 24 – SPIRE hep-ph/9704452 Phys. Lett. ]. [ , IN (2016) 023522 (2015) 155 (2000) 083510 (2013) 030 Gravitino or Axino Dark Matter with Reheat ][ Mass-splitting between haves and have-nots — symmetry JHEP 09 B 409 SPIRE 11 , Global Symmetries in Four-dimensions and Higher (1986) 21 D 94 IN holonomy and doublet triplet splitting hep-ph/9805209 D 61 ][ [ Topological defects formation after inflation on lattice Comments on cosmic string formation during preheating on Can topological defects be formed during preheating? 2 (1998) 083516 ]. GeV (2008) 065011 Structure formation in a mixed dark matter model with decaying G (1997) 3258 JHEP JCAP 16 , B 271 , hep-ph/9703354 10 D 58 [ SPIRE D 78 Phys. Lett. D 56 , IN Phys. Rev. Phys. Rev. , (1998) 262 arXiv:1306.2314 ][ , [ Phase transitions at preheating hep-th/9405187 Nucl. Phys. Phys. Rev. [ , B 440 (1997) 7597 Deconstruction, Phys. Rev. , Phys. Rev. , ]. ]. ]. ]. , D 56 (2013) 043502 SPIRE SPIRE SPIRE SPIRE arXiv:1412.1592 IN IN IN IN sterile neutrino: the 3.5[ keV X-ray line and the Galactic substructure strong lens systems Temperature as high as and on warm-plus-cold dark[ matter models [ vs. Grand Unified Theory the small scale crisis:88 New constraints from high redshift Lyman- problems in axion models Dimensions simulation lattice simulations Phys. Lett. fluctuations in chaotic inflation[ models Rev. [ inflationary universe Gravitino (1994) 3195 inflation A. Harada and A. Kamada, A. Kamada, K.T. Inoue and T. Takahashi, R.T. Co, F. D’Eramo and L.J. Hall, A. Boyarsky, J. Lesgourgues, O. Ruchayskiy and M. Viel, K. Harigaya, M. Ibe and M. Suzuki, M. Viel, G.D. Becker, J.S. Bolton and M.G. Haehnelt, M. Kawasaki, T.T. Yanagida and K. Yoshino, M.W. Goodman and E. Witten, E. Witten, S. Kasuya and M. Kawasaki, I. Tkachev, S. Khlebnikov, L. Kofman and A.D. Linde, S. Kasuya and M. Kawasaki, S. Kasuya and M. Kawasaki, I.I. Tkachev, S. Kasuya, M. Kawasaki and T. Yanagida, S. Kasuya, M. Kawasaki and T. Yanagida, L. Kofman, A.D. Linde and A.A. Starobinsky, L. Kofman, A.D. Linde and A.A. Starobinsky, M. Kawasaki, K. Kohri, T. Moroi and A. Yotsuyanagi, [61] [62] [59] [60] [57] [58] [54] [55] [56] [52] [53] [50] [51] [47] [48] [49] [45] [46] [44] JHEP07(2017)125 Cold dark ]. SPIRE IN ][ (2014) 12249 ]. 112 SPIRE IN ][ arXiv:1502.01589 [ Planck 2015 results. XIII. Cosmological – 25 – Proc. Nat. Acad. Sci. hep-ph/9608222 , (2016) A13 [ Cosmological implications of radiatively generated axion 594 ]. (1997) 209 SPIRE IN ][ B 403 Astron. Astrophys. , collaboration, P.A.R. Ade et al., Phys. Lett. , arXiv:1306.0913 matter: controversies on small[ scales scale Planck parameters K. Choi, E.J. Chun and J.E. Kim, D.H. Weinberg, J.S. Bullock, F. Governato, R. Kuzio de Naray and A.H.G. Peter, [64] [65] [63]