Probing the BSM Physics with CMB Precision Cosmology: an Application

JHEP02(2018)118 Springer February 8, 2018 January 31, 2018 February 20, 2018 : : : Received Accepted Published inﬂation model to exemplify 2 R CMB observables assuming a generic Published for SISSA by r https://doi.org/10.1007/JHEP02(2018)118 and s n b [email protected] , . 3 and Yuki Watanabe 1801.05736 a The Authors. c Cosmology of Theories beyond the SM, Supergravity Models, Supersymmetric

The cosmic history before the BBN is highly determined by the physics that , [email protected] Gunma College, Gunma 371-8530, Japan E-mail: Physics Division, National Technical University15780 of Zografou Campus, Athens, Athens, Greece Department of Physics, National Institute of Technology, b a Open Access Article funded by SCOAP Keywords: Standard Model ArXiv ePrint: the improved CMB predictions thatlution a yields. unified Our description analysis of underlinesthat the the early can importance universe be of cosmic the evo- viewed,searches CMB for to precision supersymmetry measurements some or other extend, BSM as theories. complementary to the laboratory experimental duction in order topossible be alternative cosmologically cosmic viable. histories onnon-thermal In the stage this after paper cosmic inflation. wesymmetry quantify We breaking analyze the schemes TeV assuming effects and the especially ofWe neutralino complement multi-TeV the super- and our gravitino analysis dark considering matter the scenarios. Starobinsky observationally. Ongoing and futureprovide precision us measurements with of significant the informationcosmological CMB about predictions observables the of can pre-BBN different era BSM andtivated BSM hence scenarios. theory possibly Supersymmetry and test is it the require a is different particularly often cosmic the mo- histories case that with different specific superymmetry reheating breaking temperatures schemes or low entropy pro- Abstract: operates beyond the Standard Model (BSM) of particle physics and it is poorly constrained Ioannis Dalianis Probing the BSM physicscosmology: with CMB an precision application to supersymmetry JHEP02(2018)118 21 24 35 . On the other hand, 22 4 18 11 29 dark pre-BBN period supergravity inflationary models 26 – 1 – (super)gravity inflationary model 2 2 R inflation R 2 24 R 19 + 38 20 R 15 and the 12 predictions for particular supersymmetry breaking ex- inflation 2 r 6 2 R 18 R X . This uncertainty could be minimized if the physics that operates and ∗ 30 s N supergravity inflation n 2 R 1 amples for the Starobinsky + R 5.2.1 The shift in the scalar5.2.2 spectral index and The the tensor-to-scalar ratio entropy production 5.2 The 5.3 Distinguishing the 4.3 The diluter4.4 field The maximum possible dilution due to a scalar condensate 5.1 The Starobinsky 3.3 Axino dark matter 4.1 Low reheating4.2 temperature Late entropy production 2.1 The shift in the scalar spectral index and tensor-to-scalar ratio due to3.1 late Gravitino dark3.2 matter Neutralino dark matter inflation that takes place at energytions scales thanks much to higher the than presence thecosmic of BBN phase the gives that quasi-de concrete introduces Sitter an predic- horizon. uncertaintythe at It number the inflationary of is predictions e-folds actually parametrized the by beyond dark pre-BBN the Standard Model of particle physics (BSM) was known. Indeed, different BSM 1 Introduction The cosmic evolution beforemuch unknown. the Big To date Bangvery there Nucleosynthesis early are (BBN) universe no and period, direct after that observational probes inflation can that be is can called constrain this 6 Discussion and conclusions 5 A concrete example: the 4 Alternative cosmic histories and supersymmetry 3 Supersymmetric dark matter cosmology Contents 1 Introduction 2 CMB observables and the post-inflationary evolution JHEP02(2018)118 ) ], 11 . , r 16 s 0 n r < ] motivate 3 and ], Core+ [ ]. The absence σ 15 21 , 20 for the assumed BSM 006 at 1 . 0 ]. ) values can indicate us the and such a measurement will 2 ) value accounts for an indirect , ) observables is a measurement , r ∗ s 1 3 032 ] and hence one could in principle . , r n N − 0 s ( 13 r n – 12 [ − . 4 = 0 1 = 2 ]. From the inflation phenomenology point h − 14 s n DM – 2 – cosmic selection criterion ], will improve significantly on this direction. The 19 is of the order of 10 r and ) and tensor-to-scalar ratio s ∗ ], PIXIE [ is modified by the details of the dark pre-BBN stage [ n ]. The current resolution of the temperature and polarizartion N ∗ 18 ( s N 2 , n 1 ], PRISM [ 17 ) plane, whereas from the particle physicist point of view, for a given predictable , r respectively [ s n σ Definitely, the idea that the CMB studies may probe energy scales well above the TeV From the experimental side, there is no signal that supports the supersymmetry hy- We aim at this work to show how one can systematically extract non-trivial information The fact that the the ultra high energy scales of cosmic inflation andis indirectly not to the a dark new pre-BBN one.impact period. There of are BSM numerous physics, ofmainly and seminal either in works from in particular the the supersymmetry, inflationary literature on that model the examine building the CMB or power from spectrum the dark matter perspective. of signals arouses increasing concernproblem that supersymmetry suggesting does that not supersymmetry, fully ifthan solve realized, the the may hierarchy lay TeV at scale.(LHC) energy at scales CERN Multi may much find TeV higher no supersymmetry BSMelusive signal implies for and the an that fiducial unspecified BSM theincreasing physics long scenarios Large sensitivity will of time. Hadron remain the CMB Collider However probes from has opened the up telescopic a rich observational phenomenological side, window to the duration of non-thermal phase afterexamine inflation whether and different in supersymmetry thisearly breaking paper cosmic schemes we evolution. can use fit this in information to this picture ofpothesis until the today, see e.g. a recent analysis of searches at the LHC [ account for a substantial leap forward at the observationalabout side. the BSM physicssupersymmetry via since we the consider CMB itthe as precision a terrestrial measurements. compelling colliders. BSM We theory that mostly A remains focus elusive precise from on knowledge the of the ( anisotropies of the CMB probes,support although or unprecedented, has exclude not the been differentthe BSM powerful enough physics proposed to schemes. next There generation areCMB-S4 CMB promising [ prospects experiments, that such assensitivity the forecasts LiteBIRD for [ of the BSM effectsprecision on measurements provide the us cosmic withphysics. evolution. a Planck collaboration In has other constrainedspectrum words, the spectral and we tilt the can value of tensor-to-scalar sayat the that ratio curvature 2 power the at ( measure of the reheating temperature ofexamine the the universe [ cosmology of theories beyondnon-trivial the extensions Standard of Model the of Einstein particleof gravity physics view, [ as for well a as given concretethe BSM ( scenario a predictive inflationary modelinflationary can scenario be spotted the on precise measurement of the ( and the observed dark matter abundance Ω us to investigate this small buton non-zero the residual tentative dependence BSM of the physics.of inflationary the In predictions spectral most index of the inflationary models, a precise measurement scenarios often imply a different cosmic evolution in order to satisfy the BBN predictions JHEP02(2018)118 12, the . ] and ] for a 29 = 0 27 2 h decrease DM ]. Actually, the null 22 ] for a recent analysis on 28 ) observables, see e.g. [ , r ) observables can break due to the s ], we find that most of parameter n N 26 ( – r 24 constraint reconciles only with particular ) and 12 . 0 N ( – 3 – s . n ]. Therefore, assuming that the LSP is part of the 2 and the supersymmetry breaking scale as unknown h 23 rh LSP T Ω which may greatly differ to the simple scenario of a single the ) precision measurements we utilize existing results on supersymmetric cosmology Departing from the minimal field content analysis, i.e. the MSSM, the overabundance In order to extract information about the BSM supersymmetric scenarios from the The degeneracy between supersymmetric inflation models and with their non- , r s n symmetry breaking sector suchDM as abundance the if messengers. theye.g. Extra decay due fields late to can coherently and however oscillating dominateare scalars the or rather scalars energy common that density cause andcommon of thermal well examples inflation. the are motivated early Such in the fields universe moduli, many supersymmetry BSM breaking schemes fields, such the as saxion, supersymmetry; etc. Here relevant analysis on non-thermal neutralinoFIMP dark dark matter. matter and [ problem in general deteriorates. Indeed,from the the dark perturbative matter and abundancefrom receives non-perturbative thermal contributions decay scatterings, processes thermal and of non-thermal the decays of inflaton fields field coming [ from the super- tive in this work is thatshould the parameter not space be that faced yields an asspace excessive a dark that cosmologically matter favours forbidden abundance a one different but,LSP cosmic on abundance history the for implies contrary, the as either veryproduction. a a early parameter Both low universe. cases reheating Namely, excessive have temperature a after non-trivial inflation impact or on low ( entropy quantities. We estimatesparticle the mass neutralino parameter and space.natural, gravitino As split LSP a and abundances rule highsupersymmetry by of scale breaking. scanning thumb supersymmetry we the As when adoptspace expected, the we of supersymmetric see classification scan theories of e.g. the yields quasi- [ an possible excessive energy dark matter scales abundance. of Our perspec- ( aiming at an analysis basedMSSM plus on the assumptions gravitino as is minimalresults. the as necessary We possible. minimal a set-up We priori consider that consider that gives the the the most conservative dark matter in theradiation universe domination histories and smooth radiation phase afterthis the work, inflaton is decay. that Anthe the interesting supersymmetry features point, breaking that of patterns. stimulates the radiation dominated phase depend on the details of supersymmetric versions in terms ofdifferent the post-inflationary evolution. The thermalin evolution general of much a different when supersymmetricLHC supersymmetry plasma results is is push realized the in sparticles nature massof [ bounds the to thermal larger dark values that matter spoil scenario the [ nice predictions framework often without anytrajectory change may remain in intact the byefficiently inflationary the stabilized. presence dynamics of since Moreover, additionalmatter it the supersymmetric is cosmology inflationary fields often that focus are theneglecting on case other the features that of dark studies the of matter scalar supersymmetric power density spectrum. dark parameter fitting, Ω However, successful inflation models can be consistently embedded into a supergravity JHEP02(2018)118 (2.2) (2.1) , and , since Pl M supergravity diluter ... 2 + precision mea- 2 R due to the dark r )) ∗ ∗ N k/k and s )(ln( n 2 k ln . This prospect, though very inflation and /d s 2 n 2 R d 6)( / , ∗ α )+(1 N ∗ . In general one can assume that the scale − s hypothesis for a particular inflation model. k/k n inflation is also performed. In the last section ) ln( – 4 – k 2 ) = 1 ∗ ln R k ( /d s s n dn BSM-desert 2)( / Hence precision cosmology can offer us complementary con- exclude or verify supersymmetry by 1+(1 − values coming from the unknown value of s n r is the only field beyond the MSSM and gravitino that we consider. any of this sort of scalars and explicitly refer to it as ∗ and supergravity k k cannot X 2 inflation model and we compare the and R 2 s s diluter n R A inflation is used as a specific example to demonstrate a full computation of 2 R ) = k is the scalar amplitude and the powers of the expansion are the scalar spectral ( , the running and the running of the s R s A P n The organization of the paper is the following. In section 2 we parametrize the un- Apparently one where index dependence of the spectral index to be given at leading order by the expression It is convenient to expandas the power spectra of the dimensionless curvature perturbation predictions of the we outline the mainfuture idea theoretical and and the observational method prospects. proposed in this work and we2 comment on the CMB observables and the post-inflationary evolution that are necessary forthe the implications estimation of various of supersymmetry theogy breaking and dilution patterns examine magnitude. to the the In features earlyStarobinsky universe of section cosmol- the 4 possible we alternativethe analyze cosmic spectral histories. index In and section tensor-to-scalar 5 ratio the shift. A comparison between the theoretical certainty in the pre-BBN era. We computerespect the to shift the in dilution magnitude theresults in spectral of a index neutralino, general and gravitino BSM tensor-to-scalar and context. ratio briefly In with section the 3 axino we cosmology overview regarding key the LSP yield, to note that the terrestrialare experiments, sensitive such as to colliders low and scaleto direct supersymmetry high detection whereas scale experiments, the supersymmetry. CMBstraints observables to are the more parameter sensitive space ofchallenging, is the supersymmetric actually theories a feasible possibility. surement, nevertheless one candifferently, indeed rule support out the the presenceThis so-called of is BSM a physics minimal or,colliders but probe to only put undoubtedly a it an smallsupersymmetry exciting part or of possibility the any given vast other energy the BSM scales fact scale up that to may the lay terrestrial Planck anywhere Mass, in between. It is also exciting what we actually measure onour analysis, the the CMB is theFinally, in diluter order impact to on performthe the a Starobinsky expansion complete history. calculation In ofinflation the predictions spectral by index taking value into we account consider the effects of the post-inflationary phase. we collectively label JHEP02(2018)118 , 1 − is a ) (2.7) (2.5) (2.6) (2.3) (2.4) ∗ ∗ 3. The H N ∗ / a ) is recast due to the = 1 = ( 2.4 . 1 ). The − dark ∗ dark ∗ k w dark /a ˜ N ) reads end a , into ] implies an uncertainty 2.2 dark w 30 4 [ 3 ¯ , end BBN dark ρ = ln( − 0 ρ ˜ eq ), the relation ( N 0 1 H ), the time inflation ends (end), BBN H ln ∗ H T N. − ) ∗ 0 eq Hdt a ∆ rad ∼ H ( H the e-folds number of the radiation 2 ˜ / end N dark ) ∗ t 0 t s eq w end eq a R 1 + a n 2 rad ρ ∗ α H ˜ stands for the e-folds number that take V N 4 X Pl − field in the period after the decay of the ≡ , that is within the accuracy of the future 1 MeV eq ˜ eq N X a ∗ BBN M a (1 ˜ 3(1 + ¯ X & h a N + N = T ≡ – 5 – rh 10) ln end ˜ 2 BBN running that for the eq. ( N N 1 4 value. It carries the information of how much the a − a 15. We can split the s N s during inflation, to the size of the present Hubble ∆ = + n (1 n ∗ end α ∼ 1 BBN end a a O a − a ∗ value one relates the size of the scale 0 = N dark H ∗ ∗ s ˜ H = 1 has been assumed. We call this period N k 0 n = 0 and the minimum to be around 41 for ¯ ln N − 0 a ∆ ∗ ≡ H 6= k 0 dark is of size a ln w s ) stands for the e-folds number of the postinflationary reheating dark − dark n ˜ N 7 w end . the number of e-folds from the end of inflation until the beginning of /a 66 exits the Hubble radius, rh ∗ dark ≈ a k ˜ CMB scale has been stretched since the inflationary era. The uncertainty ∗ N 10 the ∆ ], 1 N 3 − − ∗ stands for the average value of the equation of state parameter during the dark [ k = ln( 1 comes mainly from the post-accelaration stage and induces an uncertainty on is the number of e-folds remaining till the end of inflation after the moment 1 ∗ − 0 ∗ ∼ rh dark ˜ N H N N w ] to be around 56 for ¯ N 2 After plugging in the value for the ratio To explicitly estimate the dark ˜ inflaton and before BBN. into [ where period until the completedominated decay era of that the preceded inflaton, place the during BBN the and domination of an arbitrary as thermal inflation orN stiff fluid domination, weobservational can uncertainty for estimate temperatures the maximumat value the of e-folds of the inflation about ∆ where ¯ pre-BBN period, and ¯ lack of observational evidences ofsuper-cooled the transition conditions to during the inflation. radiation dominated Unless phase exotic from forms the of matter are assumed, such where the subscripts refer tothe the time time of BBN horizon takes crossing place(0). ( (BBN), We the define radiation-matter equalitythe (eq) BBN and the present time which exited the Hubbleradius radius For ∆ observations. the pivot scale critical quantity that determines the observable on the the spectral index value given by the where JHEP02(2018)118 , ] = X X H 31 (2.8) (2.9) = rad has a (2.11) (2.10) w ∗ inf then the N 1 − inf , = 0 since ¯ and the reheating dec X and the measured dom X . ρ ρ rh 1 rad w rad − N N ln . The crucial quantity is factor to mark explicitly . ) rh X X + ∆ N w dark dark w 002Mpc . X 3 ¯ . N N = 0. Note that the N Pl ∆ − rh M 1 = 0 − and after substituting numbers for N 12(1 + ¯ + ∆ inf ∗ s ∗ k Γ value, end rh A = V ρ ∗ p N X N 4 ln N / ) = it is ∆ , that is the e-folds number and the spectral 1 1 4 4 ∆ Pl = ∆ − 4 / + 1 M ) – 6 – ∗ being the effective number of relativistic species ∗ domination era on the spectral index value. We rh for a scalar condensate that coherently oscillates dark ∗ rh 2 ˜ g ∗ rh 3 N X π ln ∗ g 2 − , 2 g 90 π 1 4 a (24 dark /π / constant for a scalar field with sufficiently flat potential ∝ w ∗ 4 = 8 + rh 3 ¯ V end . X ≈ (30 ρ , with ρ ρ 4 rh − ]. We also introduced the ∆ / 60 X rh T 1 end 1 ρ ∗ 2 ) = . Here, we prefer to remain agnostic about the identity of ρ , ≈ g into the contributions from the inflationary reheating, the X- ∗ ln 1 ≡ k X ∗ collaboration pivot scale, ) ( = N rh R thermal reference values rh dark P w dark 089 [ w . N max 3 ¯ N the T of the inflaton which determines the reheating temperature. Assuming ) we get ∆ − 0 Planck = inf 1 H ) = 3 (th) s 0 12(1 + ¯ s n a rh A ( T = / is the energy density of the thermal plasma right after the decay of the scalar 10 ∗ and rh k N dec X ρ (th) entropy production ∆ It is however well possible that after the inflaton decay the evolution of the universe N 3 and / 2.1 The shift in the scalar spectral indexLet and us tensor-to-scalar now ratio estimate due thecall to impact late of the that we collectively label but we do utilize its propertycan to dominate cause the efficient energy dilution density andits of low the entropy energy production. universe density over The stored. radiation duein It to a is the quadratic slower potential redshift and of that causes thermal inflation. upon thermalization. could have been episodic withthermal additional reheating era events could after have inflation. startedthesis Hence after caused the by the cosmic other last than reheating the stage inflaton before scalar primordial field, for nucleosyn- instance a modulus or a flaton [ The maximum temperature possibleently is achieved when in the instantlogarithmic reheating dependence scenario. on Appar- X. In principle, for atemperature concrete and after predictable inflation inflationary can model bethe the decay estimated ¯ rate and Γ hence thethat ∆ the decay andreheating temperature the is thermalization found by occur equating (and instantaneously omitting at order one the coefficients) time Γ Γ 1 where the uncertainty of the dark pre-BBN era on the We have split thedomination ∆ and the pre-BBN radiation domination period. It is ∆ the ratio We adopted the value ln(10 Utilizing the relation JHEP02(2018)118 ]. = 6 N h 38 and [ /N (th) s (2.16) (2.12) (2.13) (2.14) (2.15) α 4 X n N N ∆ . + O correction can (th) (th) 2 X . In the above β 4 is at most of N ). Since precision ,N N F 2 = N X (th) ∆ N 2.2

N N /N ) , . 3 ∆ 2 X 5 = N 1, the N rh N ( N β 00 (th) N , the spectral index β ∆ F β N . ∆ 1 4 X + 000 3 α/N − N # + N 3 3 , β , 1, terms of order − ∆ 1 00 000 N N 0 (th) end β − < (th) s , β X β ρ 0 V n 1 = 1 (th) 18 N β − O (th) ln − ] there is no common form for the next- N ∆ (th) − β s + (th) s 1 4 N 32 2 n ) /N + s 2 N + inflation model where the parameters ≡ N X 00 n 2 ( N are estimated at and s + 2 2 is a constant and absorb possible complicated β N β (th) – 7 – n 1 3 can also be a slowly varying function of , n R 000 α X X + 3 (th) β X α + ln (th) N N N N N , α 1 4 N = N ∆ ∆ N 00 0 ∆ . Otherwise, if N β | − -dependent terms. β β 1 and ∆ 2 − , α 8 + N 0 . + − 0 ), for some inflation models, may involve parameters of > β /N ) function. In section 5 we will explicitly estimate the shift = 1 β the contribution to the spectral index shift is found to be X ) (th) s β, β ) are determined only after a particular inflation model is − N (th) X N 2.14 = 60 N n ( N ( β N N N ∆ β ( +3 > β β = " β ), and (th) ∗ = − X N N 2.8 N (th) s and ∆ ]. A phenomenological way to parametrize it is based on the large n (th) α/N α d/dN α 33 ]. Here we assume that − = ,N 1 33 X 6= 0, after Taylor expanding the (th) s N ) are roughly of order − n ]. One can see that the next-to-leading correction X ∆ denotes = ) value is shifted by an amount ∆ N − 35 2.16 s β 000 n F (th) ) have a particular form. If ∆ At leading order the scalar tilt is generally given by the equation ( ∆ N N ( ( s in eq. ( be important, howeverclasses such [ a behavioraccuracy is and not for foundsubdominant in with respect any to of the the known universality The “ expressions, given thanand ∆ smaller have been neglected. We have also assumed that the terms in the parentheses where 1 β n considered. In principle theIn parameter addition the expansionthe potential ( [ behaviors in the arbitrary in the spectral index for the Starobinsky to-leading term [ expansion The parameters is expected to increase in theDue future to it the is large worthwhile number to consider of next-to-leading inflationary corrections. models [ dilution effects. It is at leading order where, following eq. ( index values respectively if there is no late entropy production after the inflaton decay, i.e. JHEP02(2018)118 . = , dec s due X and   n T (2.19) (2.20) (2.18) γ X 0 α ≡ (th) β N N . We also rh X the energy + = T 0. This case N 1 (th) | − X ) ) reads ∆ N Γ → 1 (2.17) (th) s N ( n 0 ≥ rad 2.20 β matter with exotic − ˜ N 1 = X dec dom X X dom . X 0 t T T 2 β X 2 ˜ , α D 0 and βγ ' ln (th) − → field oscillations dominate the 1 3 N p dec X dom = rh X ! X T ≡ ˜ T N N | X ) we obtain the shift in the spectral ) ) dec X ) # dom X scalar cosmic evolution have to be ˜ ρ D ρ X about the minimum of a effectively N ( dom dom X ln X X D 2.15 ln β that a given inflation model yields, the T T 4 scalar reheats the universe at the time ( ( / 1 s ∗ 1 12 = (th) s g g X α n (th) s ) ) β ) = 3 ) n , − dec X dec X X 1 dec – 8 – X dom ˜ X , the dilution magnitude is estimated to be T T N (th) γ T ( ( T 1 4 ( N s ∗ X (

∗ g g ∗ = , due to a post-inflationary dilution of the thermal is the running of the spectral index at g g = N 2 . Notice that at leading order the ( =0 | 15 is achieved when h p X 2 X 1 " in the expansion ( accuracy. , elements of the ∼ γ/α (th)   N 2 limit values can be extended. /N = 1 + X coherently oscillates ln N . ) X X X ∆ ) h = ˜ N 1 3 = 0. After plugging in the dilution magnitude we get ˜ s D X D N N N X (th) s | ( = 0. In such a case, at the cosmic time N X = ln /n 1 ln n after | β before w − 1 3 s X S ] − X α S γ n − 2 w 3 field decays and reheats the universe with temperature denote the entropy density right before and after the decay of the X N = ∆ N = 1 when no dilution takes place. Overall, the size of the ∆ 2 | ∆ ) X count the total number of the effectively massless degrees of freedom X 1 + = (1 = 0 case corresponds to an uninterrupted radiation phase following s X s N s n α/N after ≡ X g is the temperature that the D α S (th) s − = ) returns the shift in the spectral index due to a pre-BBN dilution of N n X dec D X − and and )[(1 is larger than that of the plasma and the universe enters a scalar dominated era T (th) s 2.20 1 ∗ . It is n N g 1 X , where ( scalar domination reads decay. The − X − − β X before Γ X S N X = ' s ∆ Plugging in the thermal reference value Substituting ∆ In order to specify the ∆ 1 n s − = 1 + α ∆ − mention that the threewithout last significant terms cost in in the the brackets of the aboveexpression equation ( can be neglected where 1 γ the post-inflationary reheating. Ifbarotropic someone parameter assumes the the ∆ presence of index, with accuracy plasma The maximum value of thecorresponds ∆ to the maximum dilutionenergy scenario density where the of theof universe BBN. right The after ∆ the end of high scale inflation until the onset where we considered that ¯ for the energyequal. density and The entropy respectivelyH and can beto the taken to be approximately Considering instant decay of the scalar where field. The specified. When thequadratic potential scalar it is ¯ density of that dilutes any pre-existing abundancesof of the the relativistic degrees of freedom at the time JHEP02(2018)118 . X is ˜ the D and α/N (2.22) (2.23) (2.21) dec , where dom X = X T when the T SC

2 , T X /dN domination N FD X ˜ ln D scenario and are X d + ∆ ln + 1 3 TI | ≡ X flaton # due to period of thermal N s n dec X dom . X ) is sufficiently good, i.e one . = ∆ T T 4 , 20 ) ) ) ) simply by replacing the ln dec α X which means that the spectrum 2.20 10 FD | T dom dec thermal inflation X X dom 3 X 2 ∼ X 2.20 T AN T T T ( ( N ( X 4 4 + / / D 1 ∗ ' 1 ∗ implies that the first slow roll parameter g N g negative 1 1 4 " ) − 3. The dilution magnitude maximizes when is dec / X due to thermal inflation has an upper bound ln 1) 15 or T α/N s dom X 1 3 3 – 9 – FD X 2 and the shift in the tensor-to-scalar ratio due to N − n ∼ T ) and it can efficiently dilute any overabundant − T ( D + α X # ln 2( N 2.17 = 1 1 + = 16 ' 2 s dom T X r ) = n ' T FD | N ) ( ) X FD X 2 N dom D T X ( ]. Due to Yukawa interactions the flaton can be trapped at the T , is given again by the eq. ( 4 ( / 4 31 1 ∗ / value for flaton domination is FD g 1 | ∗ ] s g X n 38 " N ]. of the flaton dominates over the background radiation energy density and 0 36 , see Fig 1. It is remarkable that the shift in V = ln FD X writes [ ˜ D FD 2 | X an integration constant coming from the differential equation N 10 in order that the cosmological density perturbation remain intact. Nevertheless, ˙ A H/H ∆ ∼ Finally, let us comment on the shift in the tensor-to-scalar ratio. The phenomenological The ratio of the relativistic degrees of freedom accounts for a small correction and one Apart from scalar condensates, several BSM construction, mostly supersymmetric − TI = ˜ where At first order in slow roll we have thermal inflation, ∆ with the ln inflation can resurrect ruledin out supergravity inflationary [ models such as the minimal hybridparametrization inflation of the scalar tilt the flaton field decays veryN slowly. The ∆ the dilution magnitude canby be a many scalar orders of condensaterelic magnitude domination such larger ( than as the dark dilution matter caused particles. Essentially, the shift in the spectral index due to and each term can beTI and written SC in stand a for compact thermal form inflation and ∆ scalarcan condensate see respectively. that it isthe thermal actually inflation ∆ phase is followed by a scalar condensate domination phase, e.g. when Respectively the ∆ perature minimum. The flatonwe finally consider decays instant at reheating. the temperature The that dilution we magnitude denote due to the a relatively flat potential.generally called Such flatons fields [ canorigin of realize the the field spacevacuum energy by thermal effects. Ata some period temperature of that thermal inflation wethermal starts. denote trap Thermal has inflation become ends too at weak the and temperature the flaton field starts oscillating about its zero tem- tilt becomes more redillustrated when in dilution figure of 1. theper radiation mile, The plasma even precision takes for of place; the the this extreme case expression behavior ∆ is ( ones, predict the presence of singlets under the Standard Model symmetries that have the thermal plasma. We see that the ∆ JHEP02(2018)118 1 10 the ), a ∼ and . α (2.24) X 57 that is GeV TI | 2.24 N 0.975 55 X 57 for the 12 (th) N , 50 inflation (left 10 due to scalar N ) depends on 56 increases SC = 2

, i.e. 0.970 X R rh 56 X 45 ' X Linear T D 55 2.23

N N TI s ∆ (th) n 40 1 )∆ 0.965 50 − N , 4] (th) SC /

X 45 ) N ( N 0.960 X (1) corrections to the dilution 0 Plateau 10. In order to estimate the r ∆ w

O 3 ¯ = 20 translates into ∆ 40 TI ∼ 1 − X ' − − X 0.955 α ) D inﬂation and N Plateau & Linear , 2 (th) = [(1 R 5 0 αAN N 0.950 25 20 15 10 X (

10 10 ˜

r + into ∆ 10 10 10 10 N iuinmagnitude Dilution 1 − ) is obtained. The scaling ( 13 − ) 54 for X 1) X ˜ requires to go beyond the approximate relation 3, either due to a scalar condensate domination D inﬂation model with the next-to-leading order ' / – 10 – ( N 4 − = 10 r 2 X − 3, initiates at the moment of the complete inﬂaton ∆ α for scalar condensate domination and the ∆ 54 ˜ X R / D (th) 10 − 2( D = ∆ N 3. A dilution of size 0.965 ∼ = 1 50 BBN 2 r / r (th) = ln 3 and duration X /T w / N X D rh 16 1 ( (th) GeV T r , can be explicitly determined by the dynamics and the full N r ln < Scalar-Condensate 9

45 = (th) 0.960 ∼ X = r s 10 w N n X r = N rh ∆ ). For ∆ T , Thermal Inflation Thermal 40 2 (th) for the Starobinsky 0.955 R N r ( r = . The shift in the spectral index value and the dilution magnitude and consider corrections at second order in slow roll. In section 5 we will explicitly 0.950 (th) r 5 5 1 20 15 10 - Summarizing, the duration of a non-thermal phase is encoded in the number of e-folds 10

between the moment a relevant mode exits the horizon and the end of inﬂation. If the 10 10 10 10

= 16 . Moreover, accuracy of order ∆ iuinmagnitude Dilution thermal e-folds number interactions of the inflaton field.folds If by not, the a amount non-thermal ∆ phaseand changes a the aforementioned prolonged e- dilutionshift e.g. in of the size spectral index and the tensor-to-scalar ratio one has to know the non-thermal phase with ¯ tensor-to-scalar ratio value. N radiation domination era, where decay and continues without break until the BBN epoch then the e-folds number, called here the potential that implements inflation.A Different potentials yield different values for r estimate the ∆ corrections taken into account. The general conclusion is that, according to eq. ( where or thermal inflation, the relation ∆ magnitude are expected due toat the ultra uncertainty high at energies. the number of the relativistic degrees of freedom a non-thermal phase is ∆ condensate domination (SC) and duepanel), to thermal general inflation plateau (TI) and forthe the linear dilution Starobinsky inflationary is given potentials by (rightconstraint the panel). ratio for The thermal maximumproduction inflation. number after of infaton Thegeneral red decay. plateau and dots linear It show potential is respectively the (red e-folds dots). number Order if there is no entropy Figure 1 JHEP02(2018)118 model, 2 ) and the R (th) , given by the s N n value, that cor- ( s GeV. The knowl- ) the result is far ], and in the right n (th) s 12 37 2.3 = n = 10 (th) s n GeV [ rh 9 T 10 ) as illustrated in the figure 1. ) the ∼ . Although a rough estimation of 2.20 r rh 2.14 T 12 relates the amount of the dilution . and ∆ = 0 s n 2 h assuming that the LSP is part of the dark matter DM – 11 – model that predicts 2 according to the pre-BBN cosmology implied by the BSM R r and ∆ ). This is possible only after an inflationary model and the param- s ) can be estimated and hence the spectral index shift ∆ n 5 TeV and suggest that we should depart from scenarios with natural X . 2.13 N ), is obtained. In figure 1 we illustrate the shift in the spectral index due to ∆ . We will estimate the minimum dilution size dictated by the requirement − 2.20 12 and determine the expected shift in the spectral index and tensor-to-scalar . (th) 0 N ) or ( ≤ ( both characterized by a fiducial reheating temperature s 2 n h φ 2.15 There are several fundamental theoretical reasons to believe that supersymmetry is From a more bottom-up approach, the postulation of a non-thermal phase during the = ∝ LSP s a symmetry ofscale nature. that supersymmetry For issparticles the realized. exceed devotee The 1 of directsupersymmetry superpartner supersymmetry paying LHC-limits the the price for of central all pushing the question colored amount is of tuning the at the MSSM to less than contains dark matter withwith relic the density dark Ω matterTeV supersymmetric production. scenarios, we In will this overviewthat the section, the expected focusing LSP dilution yield. is onnon-trivially We generally TeV will influenced required, and stress out hence by especially the the multi- universe. CMB post inflationary inflationary observables expansion should history be of the supersymmetric 3 Supersymmetric dark matterIn cosmology the previous sectionratio we due computed to the shift post-inflationary in entropy the production. spectral index The and tensor-to-scalar fact that the present universe different supersymmetry breaking schemes in the universe Ω ratio when a particularcomplements inflation the model, description of which the in early section universe 5 evolution. is the Starobinsky and estimate the ∆ theory at hand. Examplesof of BSM the cosmic universe processes arefollowing connected we with the will the dark consider expansion the matter history pre-BBN supersymmetric cosmology, production BSM such scenario as and and possible the determine non-thermal features stages, baryogenesis of that processes. allow the the accommodation In of the pre-BBN era is not enoughthe to spectral determine index the ∆ shiftfrom can accurate be and cannot done consistently constrain bymethod the the is early to approximate universe choose cosmic expression an history. ( inflationtion The model best of that the is in early accordance universe with a (e.g. particular a BSM supersymmetric, descrip- stringy or modified gravity framework) panel a Starobinsky-like potential with non-gravitationalV interactions and a linear potential edge of these inflatonresponds features to enables the the red explicit dotsshifts calculation in the of the spectral the plots. index value A according scalar to condensate the domination formula or ( thermal inflation eters describing reheating aren chosen. Then from eq.eq. ( ( a non-thermal phase that iswe implemented considered after the reheating Starobinsky and before BBN. In the left panel given by the eq. ( JHEP02(2018)118 ], the 39 12 is . only = 0 , in which ] or Mini- 2 h 43 ] in which all DM 46 , 45 , unrelated to the weak m , see e.g [ . The stability of the LSP dark ] for a review, and possibly it is the 49 stable Quasi-natural supersymmetry ], there are strong arguments based on 47 , in which only the scalar supersymmetric ]. These results motivate us to assume that 48 is small, while in the High Scale supersymmetry – 12 – ]. There are also the Mega-Split [ β ). The gravitino is naturally the LSP in gauge me- , while gauginos and higgsinos are lighter, possibly 42 ˜ m – m − High-Scale supersymmetry 40 : From (i) thermal scatterings in MSSM and messengers : from scatterings (i) with the MSSM plasma, (ii) with the (eV -parity conservation even when the scale of supersymmetry -parity violating models have been actually constructed and GeV. O R R 12 Split supersymmetry GeV when tan 8 is 10 m is constrained by the Higgs mass value according to the details, of each m 30 TeV range. ii) ] scenarios. iii) Finally the − , which can be thermal or non-thermal, receives contributions from many sources. value can be up to 10 44 2 1 percent level. However, the absence of BSM signals in the LHC rules out (iv) perturbative and non-perturbative decay offields. the inflaton field, (v) decay of moduli thermalized messenger fields. plasma, (ii) decays of sfermions and the NLSP, (iii) decays of the messenger fields, -parity. If the R-parity is violated then the cosmological constraint Ω h Given the supersymmetry breaking scheme the stability of the LSP puts strong con- For our analysis it is critical that the LSP is BSM physics scenarios with unnatural supersymmetry are still very appealing. Gauge m R − 2 / 5 3 Thermal gravitinos (freeze out) Non-thermal gravitinos (freeze in) . The gravitino is theacquire a supersymmetric mass partner in the ofdiated range supersymmetry the of breaking graviton models (GMSB), in seeLSP supergravity [ in Split and and it High scaleΩ can supersymmetry frameworks. The relic density of the gravitinos straints on the thermalthe history basic of the relevant cosmological universe.scenarios aspects necessary In for and the the following results goals subsections of of we our the overview 3.1 analysis. gravitino and neutralino Gravitino LSP dark matter have interesting phenomenological implications [ GUT models that support the breaking is well above thethe electroweak LSP scale lifetime [ is much largerof than the the age dark of matter. the universe and thus the LSP is constituent the ˜ matter is assured by thethe presence of araised discrete for symmetry the of LSP. Although the supergravity Lagrangian, Split [ supersymmetric particles have massesscale. around a The commonsupersymmetry scale ˜ breaking ˜ scenario.value allowed for Roughly ˜ in the Split supersymmetry the maximum different supersymmetry breaking scenariosspectrum have features into been three categorized representative cases: accordingsupersymmetric i) to particles the are mass heavierin than the the 1 weak scale,particles but have masses not of too thewith far order from masses of it, near ˜ about the weak scale [ electroweak scale supersymmetry and not supersymmetry in general. coupling unification, the presence of aprocesses stable and dark the matter stringy particle, the UVthe possible completion baryogenesis of electroweak the scale. low energy Supersymmetry theories do may not appear link SUSY at with higher energy scales. In ref. [ 0 JHEP02(2018)118 . 2 / 2 > 1 m m / 4 g 32 f.o. (3.2) (3.1) 3 6 L /T 5 LSP rh H T 2 and 4 dilution for / D 3 3 10 − 8 8 GeV, G GeV 6 12 10 9 8 , while in the right 0 2 ˜ g 6 3 10 required ∼ m = m @ rh 4 = ,T Log ˜ L f . is the gluino mass evalu- X 5 2 m sc 3 D 2 - H ˜ g ˆ γ / MSSM(sc) 3 , m 2 0 6 Pl Log Y 2 T Pl M 2" in the thermalized early universe 1 i 5 - is achieved when only the MSSM M ˜ . " 2 2 0 10 2 m / / / 2 3 3 -

3 9 8 7 6 5 4

, where ≡

f n

g m

GeV m 10 m Log D @

) 3 - ! π 3 ˜ ) g 1 2 ]. A key quantity is the gravitino produc- / µ 48 2 3 ( ). If the reheating temperature is below the 50 /m ˜ 2 g (b) . Thermal production of helicity m ' ˜ g 3 m 3.4 ) m ˜ – 13 – G 3

> (TeV m32 mg i 1 + 2 L

→ = 10 2 i ˜ 5 ˜ 2 f 6 Pl LSP ( H GeV T m M 0 / 10 D 1 2 . / 0 MSSM 3 8 GeV, G Γ GeV m ∼ 9 0 2 6 3 10 sc γ = m @ rh 4 GeV ,T Log L 14 2 gravitinos have been taken into account. The contributions to the gravitino X / 5 2 10 1 D - H × 0 Log 2 , in scatterings with thermalized Standard Model particles and sparticles 2" sc ∼ - γ " . Density and contour plot of the decadic logarithm of the 2 GeV reheating temperature after inﬂation and gravitno the stable LSP. In the left panel 10 /

f.o. 9 3

3 7 6 5 4 9 8

GeV m m Log D @

= the gravitino yield from MSSM thermal scatterings is

The less model independent estimation of the Ω > T = 10 2 / f.o. rh 3 rh decay width into gravitinos is nearly the same for both gauginos and sfermions The gravitinos obtainT aated thermal at the distribution reheating temperature, viaT see eq. interactions ( withFurthermore, the the heavier MSSM sparticles MSSM are unstable for and will decay to gravitinos. The evolves according to the Boltzmanntion equation rate, [ of the required dilutioncorrespond is to given underabundance, by hence the to contour no numbers dilution onto contour the area. density plot. Negative numbers sector is considered. The gravitino number density degenerate spectrum for thepanel sfermions there and is gauginos a wasgravitinos split from considered, spectrum scatterings with in theNLSP plasma, to non-thermal production helicity from decaysabundance of have sfermions been and conditionally the added,is i.e. achieved in the the total parts abundance of the is contour replaced that by thermal the equilibrium thermal one. The magnitude of the logarithm (a) Figure 2 T JHEP02(2018)118 . ]. ], 55 68 mess (3.4) (3.5) (3.3) – M 66 ∼ ], and the , e.g due to 2 2 # / / 56 3 f.o. 3 [ 3 ], or due to the T 5 26 − i ˜ 10 f GeV − 5 m ] 6 . We only assume that 10 ]. In general the extra − , 58 , 49 10 i 57 2 & / . In addition, the inflaton ) cannot be final because we f.o. 3 diluter 2 . 2 T / 3 3.3 GeV 2 Pl mess 2 ∗ m λ M 5 Φ TeV) 10 2 /g / / m ˜ g 2 3 m 270 m the gravitinos for broad range of values ]. ( 2 | N 46 h 2 π 52 / 2 , 3 / (eff) Φ 3 51 288 [ G GeV m /m | + 3 7 rh – 14 – ' 6 T equilibrate 2 ) 3 GeV / 10 ˜ − 3 G GeV ˜ × rh m G T . 5 12 , and the relic density parameter reads , unless there is a late entropy production. For example, ∝ 2 → 2 GeV 10 − 2 / ∼ 4 / 3 h . 3 ], or due to feeble couplings in the supersymmetry breaking sector [ 2 2 10 (th) NLSP / Γ(Φ − 12 3 54 / 2 . Ω 2 h / 0 / 2 (th) 3 3 / MSSM(dec) 3 12 Ω mess 2 . GeV ], or a high temperature decoupling of the messenger fields [ m Y / ]. Therefore, the estimation of the gravitino relic abundance based 0 3 M 53 NLSP + 71 the final Ω m . The total MSSM contribution to the gravitino yield is sc – 2 4 m ˜ ], from the decay of the supersymmetry breaking field, see e.g. [ 1 . ˆ γ f 0 69 " + 65 1 the thermal scatterings of messengers contribute to gravitino relic density g, that we do not identify and collectively call it – ∼ ∼ result could decrease in the case that extra fields interrupt the thermal phase, 2 X 59 2 / 2 2 MSSM(sc) 3 = ˜ / − h 3 Y h 2 2 mess / / i = ˜ MSSM 3 λ mess 3 12 . Ω 0 The Ω Apart from particular cases, the above gravitino yield ( It is though possible that the supersymmetry breaking sector leads to a suppressed Ω 2 1 / MSSM -symmetry restoration [ 3 scenarios may become viable possibilities.by the The scalar tentative low entropy production is caused R dynamics of the sgoldstino field [ or other moduli [ solely on the MSSM sectorconservative gives value a for model the independent Ω albeit an underestimated and hence e.g. due to the dominationperatures. of Thanks a to non-thermal the scalar dilution the field gravitino that cosmologically produces problematic supersymmetric entropy at low tem- Also gravitinos are produced duringthe the inflaton preheating [ stage via its non-perturbative decay of where the freeze out temperatureEven is if here equal towith the messenger Ω mass scale, perturbative decay produces non-thermal gravitinos with rate [ of the Yukawa coupling at therelic messenger gravitino superpotential, density parameter reads neglected sources beyond theingredient MSSM. for The all supersymmetry thefields consistent breaking only supersymmetric sector increase BSM is scenarios athermalized [ messengers necessary fields generically The gravitino relic abundancein sourced figure by 2 the andthe MSSM reheating 3. and temperature then messenger In theexpression fields the gravitino with is relic case dependence abundance illustrated Ω that is the given by gravitino a is much the different only sparticle with mass below Y where JHEP02(2018)118 ) 2 ]. / 8 > tot 3

m m 3.6 6 32 g 72 (3.6) , L 56 5 LSP ) value is H 12 is the con- 2 10 3.6 D 2 / mess 3 GeV, G GeV 4 9 0 the contribution of 2 6 3 10 dilution for gravitino = m o @ rh 2 4 / ,T Log (th) 3 L X Ω 2 5 D required - H , 2 right panel 0 / Log the gravitinos are thermal (heavy SB 3 temperature, increase the Ω 2" 2 - / " sum + Ω f 3 10 2 is the contribution of the infationary T -

/ 9 8 7 6 5 4 3

inf 3

g /

GeV m m Log D @ is the contribution of the supersymme-

= inf 3 left panel 2 / + Ω SB 3 2 (b) / mess 3 conditional – 15 – L m > m 32 g + Ω 14 that produces low entropy modiﬁes the result ( GeV. In the LSP H 9 X 12 2 5 / 3 MSSM = 10 Ω 10 D n rh ) is not strictly exact: it is well possible that contributions GeV. T 8 8 GeV is the most representative example of a WIMP dark matter 3.6 min 0 2 0 6 3 GeV, Thermal G χ = 10 ' m 9 @ 2 4 / 10 3 tot Log mess = Ω rh 2 5 M value. ,T - L 2 X 0 / in order to emphasize that it is sourced by processes taking place before the is the contributions of the MSSM (scatterings and decays), Ω D (th) 3 H 2 2" / < 3 - Log 2 " / . Density and contour plot of the decadic logarithm of the 10 MSSM 3

3 7 6 5 4 9 8

GeV m m Log D @

Finally, the presence of a scalar decay. X 3.2 Neutralino darkThe matter lightest neutralinoand ˜ an appealing candidate thanks to its main merit: the thermal production mechanism. from non-thermal decays, that takebeyond place the below Ω the as will be discussedrenamed in Ω the section 4. In such a case the density parameter ( perturbative and non-perturbative decay andtry Ω breaking field. It ismay called result conditional in sum because anthermalized the overestimate messengers simple of modifies add of the the eachWe gravitino gravitino contribution mention production abundance. that from the the For sum MSSM example ( sector the [ presence of the gravitino relic density parameter is the where Ω tribution of messengers (scatterings and decays), Ω gravitinos can thermalizemessengers due plus MSSM to is considered, the assumingdo a messenger not small enough thermalize sector). messenger by coupling the soscale that interactions In is gravitinos with the taken messengers, to but be only due to the MSSM. Theit messengers interacts too weakly with the other fields, e.g. via gravitational interactions. Therefore (a) Figure 3 LSP and reheating temperature JHEP02(2018)118 . 3 , 0 ' = − ˜ 2 χ # He 0 4 0 ˜ 10 χ (3.9) (3.7) (3.8) m f.o. ˜ GeV, χ 3 Y . 7 T × − ∝ c the non- 10 10 0 = 3

production ]. The ˜ (th) χ v c 0 0 and the tem- 2 ˜ χ χ GeV the grav- / 0 76 ]. Focusing on , 3 ˜ σ χ 8 H

71 m 75 m TeV 10 – = = > 2 69 / 0 + ˜ f 3 ˜ 30 TeV and for smaller 2 χ . 3 m − ] for details. The gravitino c/m i MeV ˜ mass squared, Ω f = 10 76 GeV 4 , 0 ]. If the reheating temperature 7 / m (Radiation-domination) 2 1 χ / 3 42 10 # 75 ]. Specific neutralino types, such [ , ) m 0 2 23 i 57 / ), where χ dec 75 g 3 2 . T − 2 ( i GeV, or large enough sfermion masses, for 10 h ∗ X 2 10 g / 2 0 1 ˜ " c/ (th) χ , 2 12 / for a radiation dominated universe, or 3 − 2 + Ω / ] which has been excluded by collider searches. 2 GeV) 3 – 16 – GeV / 10 5 5 3 2 2 m 73 Y . / h 10 , 10 3 GeV / for mostly wino ˜ 2 GeV to avoid BBN complications [ and the neutralinos reach a thermal equilibrium. In 5 2 ∼ 24 / m 2 TeV) + ln( / ]. The gravitinos decay when Γ 4 2 3 3 / (dec) 0 / − 10 0 Y ˜ + 0 ˜ f 3 f.o. χ ˜ 42 10 χ ˜ m [ χ Y ( 8 6 > m . 14 GeV or the sfermions very massive, / = 10 + Ω > T 2 5 2 2 c / / ln( 14 10 2 ], or the annihilation mechanism can be invoked to match the 3 GeV h = 6 / rh 9 dec 3 10 × − m 74 T 2 T / and 5 10 GeV) dec 3 > 5 decouples from the thermal bath at a freeze-out temperature 28 0 T & χ 0 rh 2 " 10 / ' the neutralinos reach thermal and chemical equilibrium and the relic χ T / rh MSSM(sc) 3 2 0 T / f Ω 0 ˜ 3 χ x f.o. ˜ χ m T m TeV 2 annihilation cross section. In scenarios with split spectrum it is 0 ( ) for a gravitino dominated early universe. Unless the reheating temperature / ˜ χ 8 0 3 2 / χ m 3 ] and the supersymmetry breaking field or other moduli [ 10 m ∼ m 65 × = 2 , where – to data, but in general a rather heavy neutralino cannot be a viable thermal relic. (4 f 2 − 2 0 / values the bound becomes much severer, see e.g. [ h 2 h 57 ˜ χ The gravitinos, that are unstable, are produced via thermal scatterings, non-thermal The neutralino ˜ 2 /x & 0 2 h / 0 ˜ / Ω dec χ 0 3 12 ˜ ˜ 3 χ f ˜ . χ T Ω 0 Thus one finds 3 is particularly high itinos do not dominate over the radiation, and the neutralino relic density parameter reads abundance implies that it must be m decay populates the universe withdecay neutralinos. promptly Heavy so enough gravitinos, that the opposite case, the neutralinoslibrium produced and by the either graviton have decay are a out yield of chemical equi- Apparently, it has to be ton [ the MSSM sectoring the temperature, gravitinos dominatem the universe eitherperature for after large decay is enough reheat- density parameter is UVWhen insensitive the sparticles and masses depends laydisfavored on well and above ususally the the non-thermal TeV production ˜ scale scenariosvia the are the thermal considered, neutralino decay e.g scenario of ˜ is heavy gravitinos. decays of sfermions and possible decays of scalars beyond MSSM such as the infla- m relativistic ˜ for a mostly higgsino ˜ is larger than masses lie in theIn 50-100 GeV addition range the [ directthe and neutralino with indirect mass detectionas about experiments the the shrink higgsino, electroweak the scale seeΩ [ parameter e.g. [ space of The thermal neutralino scenario however works best provided that the squark and slepton JHEP02(2018)118 0 χ and GeV . The 0 (3.10) 5 ˜ 2 χ 0 f.o. 10 ˜ χ scenario, , because /m > 1 < T 2 / . 2 ∝ 3 / dec 3

m T v 0 ) compared to the ˜ χ 2 dilution for neutralino / . The gravitino mass is σ 3

2 / dec 3 /T branching ratio T 0 domination) f.o. ˜ χ required T − annihilation scenario ˜ G ). It is ( 2 / dec 3 T 25. If gravitinos dominate the early ( . 2 times heavier than the gravitinos and hence / > H = 86 1 3 ∗

(b) g v 0 2 ˜ χ / – 17 – 3 GeV σ

5 m and the relic density parameter is for this case, 0 ˜ 10 χ ]. The neutralinos annihilate until their number

n v 77 0 ˜ χ 0 ˜ χ σ

m TeV H/ , that is enhanced by the ratio ( were taken to be 5 3 10 2 0 1, then the relic density parameter of the non-thermally produced GeV the sfermions are 10 f.o. / ˜ behaves as an attractor and determines the relic abundance of χ dec ∼ 3 ∼ T = 46 for sfermions heavier than the gravitino and 12 GeV to avoid BBN constraints. 0 2 /T 5 0 ˜ crit χ 2 ˜ N − crit χ 0 / n 10 f.o. ˜ h 3 n χ = 10 0 T , hence for a wino-like neutralino at the TeV scale and ˜ > (n-th) χ D 2 rh 12 ]. Nevertheless it hardly works when one departs from the TeV scale . 2 / Ω T / is produced non-thermally during the low entropy production caused by the 3 3 0 3 0 77 0 ˜ m (th) χ m field will be discussed, and in section 5 we will consider the production of ˜ χ ∝ . Density and contour plot of the decadic logarithm of the runs up to X ) = Ω i 2 / dec 3 In the section 4 the non-thermal scenario, that is often-called T 0 ( ˜ (ann) χ neutralino. It is also much constrained from the indirectwhere detection the experiments. ˜ diluter from the supersymmetry breaking field aiming at a complete analysis. Ω thermal abundance. Thisthe is an critical appealing value scenario,neutralino called (mostly wino)abundance LSP, making [ it independent of the primordial gravitino relic It is possibleachieve that a chemical the equilibriumH non-thermally for produced neutralinospair-annihilation from can the takedensity place gravitino becomes decay [ degrees of freedom at universe that is neutralinos is the neutralino yield is dominatedright by bottom the corner decay of of thetaken gravitinos plot to produced the be from neutralinos sfermion thermalize decays; due in to the large where (a) Figure 4 stable LSP. In the left paneltemperature the is neutralino abundance is the thermal one. In the right the reheating JHEP02(2018)118 2 ]. ) for a 82 2 (4.1) (4.2) /f ) (3.11) a ]. It is /f 80 GeV 12 bound, . 11 GeV , , = 0 ˜ 11 f ˜ f 2 h (10 , m , m 2 2 ˜ g / GeV)(10 DM 3 4 ˜ MSSM(sc) a 10 < m TeV) / < m 2 + Ω rh / ] for some recent results on µ/ 0 3 ( T ˜ χ . If the axino is the LSP it ( 5 84 7 a − − NLSP 10 Ω 10 ∼ ∼ ˜ a , m NLSP m ]. It freezes out at high temperatures ], see also [ δ m rh gets generally severely violated when the T 79 83 + , 2 (DFSZ) ) are either positive or zero, depending on γ , m ˜ a ˜ δ h (KSVZ) 78 Y ˜ a ˜ δ rh ˜ 2 γ, Y ˜ / DM f T 2 – 18 – 3 the axion decay constant. At lower temperatures (dec) constraint. ˜ / β, TeV the axino dark matter is also cosmologically for its condensate decay can produce late entropy ˜ ˜ m γ f 3 a m & bound if the reheating temperature is rather low or f α, X ], and 2 LSP ˜ a 2 ˜ / f + Ω where 81 β 3 h m m ˜ a m Y 2 DM the superpotential Higgs/Higgsino parameter, see e.g [ ˜ / g ) and (˜ 3 , where µ 2 m 2 2 / m / ˜ β 3 MSSM(sc) 3 GeV) / m Ω ˜ a 2 0 / GeV) ˜ ˜ α 3 α χ m 2 α, β, γ, δ 12 ( ˜ / a m 8 m 3 10 m m 10 / ∝ ∝ a 2 0 f × ' / ˜ χ 3 8 ˜ a . Ω Ω 2 Ω GeV( ∼ 11 below the NLSP freeze out temperature. We note that the two body decay of 10 2 / dec 3 ∼ The predicted supersymmetric dark matter overdensity for “unnatural” supersymme- For axino mass not much smaller than the NLSP, the axino dark matter case is quite T ˜ f.o. MSSM(sc) a ˜ a where the exponents ( the dark matter production mechanism considered. try can be reconciled with the Ω and The overview of thesuggests predicted that the relic observational densitysparticle value of masses of increase. supersymmetric Ω For dark gravitino anda matter neutralino general LSP in scaling one with section can collectively respect 3 write to down the mass parameters and temperature saxion can play the rôleof thethat diluter successfully decreases the LSPthe abundance reheating [ temperature and the Ω 4 Alternative cosmic histories and supersymmetry the DFSZ axion model where similar to the neutralino LSP.problematic For since its relicand density the parameter generally essential violates conclusionis the is required Ω that, for in the general, axino a dark special matter thermal scenario history as of well. the universe Remarkably in these models, the for a squark to anΩ axino is subdominantfor for the gluino KSVZ masses axion less model, see than e.g squark [ mass [ it can bedominated produced universe, from the axino thermal relicthermal scatterings density scatterings, parameter and is the the decays. gravitino sum decay of and In the the contributions that NLSP from decays case, for a radiation In the sake ofaxino completeness dark matter. of the In supersymmetry,with basic the an LSP axion extra solution scenarios, to scalar, weis the the briefly strong a CP comment saxion well problem here and motivated comes T on a dark the fermion, matter the candidate axino [ ˜ 3.3 Axino dark matter JHEP02(2018)118 ) 3 ]. ∼ rh / T the 4.2 2 1 85 / 12 is 3 rh ∼ . < T m alterna- m X ) and ( = 0 w ]. Finally, for domination 4.1 2 to processes 2 86 [ / h 0 3 X ˜ χ 7 rh DM T are the freeze-out GeV. For gravitino ∝ 5 f.o. LSP 10 T LSP sufficiently large that sources . . In order to quantify this and the LSPs never reach 2 r (or more than one scalar) with . rh and X ]. We have also called 0 T 2 X f.o. X ˜ χ LSP and m 85 T m s ∼ n . Apparently, in the MSSM the ] for a brief overview on the topic. X f.o., new particle. The maximum reheating LSP rh ρ T 87 rh with barotropic parameter decay and reach chemical equilibrium X T X X m < T where ), see [ respectively [ (TeV) or for the particular case that ˜ 4 rh ) O LSP /T – 19 – . m f.o. LSP & T f.o., new LSP LSP ( T rh (TeV) and ˜ ( m T × and this has implications for the relic LSP density [ / In this case, the dark matter production due to processes O 2 f.o. that dominated the energy density of the universe before the inflaton dec LSP (th) LSP X and > T T the decay rate of the inf . On the other hand, if 3 rh Γ LSP T X LSP dec X m × m T X inflation. In this case, the reheating temperature at the expressions ( we mean that the radiation domination phase after inflation was inter- (th) LSP rh X T reheating temperature scenarios those with is frozen during inflaton oscillations with the , is small enough or if it is the result of the decay of a weakly coupled 1, where Γ inf X low 12 can be satisfied for . . dec X = 0 T /H < . On the other hand, for neutralino LSP the UV-sensitivity of the Ω 2 X = h Both for gravitino and neutralino LSP, the observational bound Ω We call Low reheating temperatures may be caused by the a scalar field rh decay temperature, 2 NLSP , is taken to be from the TeV scale up to the energy scale of the reheating temperature. T DM ˜ Note that these scenarios that can reconcile heavy supersymmetryrelatively with long the lifetime, observational Γ decay, e.g if the some extra e-folds of is temperatures for X a chemical equilibrium thenif the the relic LSPs density are has producedthen a non-thermally the dependence from relic the Ω density reads Ω cosmic phase is a decayingfor particle Γ dominated phase, hencetemperature entropy is is gradually greater produced that If the LSPs reachroughly chemical given equilibrium by before Ω reheating, the relic LSP energy density, is and it is mostly determined at the freeze outgenerally temperature violated for Ω where the Boltzmann suppression may play a critical rôle. Note that the LSP the yield from thermal scatterings decreasesand when the the NLSP-decays reheating to temperature decreases, gravitinosm account for the leading contributionthat to Ω take place at high temperatures is small. The neutralino abundance is IR-sensitive 4.1 Low reheating temperature The reheating temperature ofdecay the universe rate, after Γ inflation canscalar unrelated be to rather the low inflaton. ifsensitive to the the inflaton maximum temperature gets suppressed. effect we consider in our analysistry below breaking different schemes. cosmic histories We and follow different thebreaking supersymme- base scenarios line framework with of either thespectrum. benchmark gravitino The supersymmetry or scale of neutralino supersymmetrym LSP breaking, and represented by degenerate the or general sfermion split mass mass cosmic history takes place iftive supersymmetry is cosmic a history symmetry ofrupted nature. or By delayed the by term adominated the cosmic energy era, density of where thesection a early 2 fluid universe. such a As cosmic discussed era in impacts the the introduction observable values and in late entropy production takes place. Remarkably, both solutions imply that an alternative JHEP02(2018)118 > the (B) (A) (4.3) rh LSP m GeV, for bound, is 9 < T 10 2 and DM / f.o. 3 & values indicates T ˜ m rh ) , T > , r s (TeV) requires the re- n 6= 1 rh ( TeV . For O and the relic neutralino T 2 X / 0 ∼ 3 > ˜ 2 χ D m m m LSP m and or ˜ m . Interestingly enough, the opposite ) implies that, if there no late entropy then depends on the maximum temperature 2 4.3 GeV due to BBN constraints on the late ˜ / m 8 3 ˜ (TeV) and Y m . . 10 O and the neutralinos from the gravitino decay . rh rh & T 0 – 20 – rh f.o. ˜ χ rh T m > thermal abundance freezes out at the temperature T < T 0 ) beyond the TeV scale, see figure 4. Moreover TeV > T ]. . χ ] falls into this category, although there the reheating (TeV) is in the probing range of terrestrial colliders ˜ m 0 2 , ˜ O / χ 76 dec f.o. , 3 88 LSP , m T . 75 LSP 12 increases with the ˜ . 52 < T , 0 m ( / and LSP rh 2 51 GeV. Such reheating temperatures mean that the dark matter T h < m 2 m 13 / 3 10 = 1 then 12 work mainly for the TeV neutralino dark matter scenario and share ∼ . X (TeV) receives extra contributions from the gravitino late decays. If the grav- D O Φ 0 ˜ χ = 0 m Y (TeV) and 2 the measurement of the reheating temperature via the h the sparticle mass scale. ∼ O DM If If m m and the Hence, scenarios with high reheating temperature generally require an extra scalar One concludes that scenarios with reheating temperatures In the neutralino LSP case, the ˜ The abundance that is more sensitive to UV processes is that of the gravitino. In the To this end, one draws the general conclusion that the gravitino or neutralino LSP relic • • 0 (TeV) are compatible only if late entropy production takes place. The above remarks are f.o. ˜ χ field that causes dilution. where ˜ disfavoured for reheating temperatures decaying gravitino abundance [ O synopsised in the following conditions, T itino is heavy enoughequilibrate. it can Generally be the neutralinodensity abundance is increases as too largescale supersymmetry for with ˜ neutralino LSP, although compatible with the Ω MSSM framework, the leading contributionafter to reheating and thefigure decays 2 of the and heavygravitino 3, abundance thermalized is the sfermions. the Ω thermal Aslate one, sfermion illustrated and decays. in in particular the cases it may be enhanced due to If the inflaton decay reheatedinflaton the mass universe it isgeneration generally processes expected that that take place atof high the temperatures dark are matter critical abundance. for the determination limit ˜ and detection experiments, a factinvestigation. that manifests the complementarity of the4.2 cosmological Late entropy production Otherwise the darkscenario matter of is EeV gravitino overabundant. [ temperature is Let not low. us Remarkably mentionproduction, the relation that ( thea very lower interesting bound for the supersymmetry breaking scale the common feature that density for “unnatural” supersymmetry ˜ heating temperature after inflation to be below or about the supersymmetry breaking scale, bound Ω JHEP02(2018)118 ], ); LSP dec 89 X (4.5) (4.6) (4.4) , ], not /T 85 [ 75 = 0 and 0 f.o. ˜ χ 0 f.o. T ˜ χ decay reads domination, ( X LSP × X X < T . In the case (ii) 0 (th) ˜ χ 0 dec ) hold. If not, then X (th) Y ˜ χ decay should fit the T Y dec X = 0 scenario becomes . The X T X ( decreases the initial LSP X LSP 12 implies X . < H 0 , D i . times and supplements it with ≤ σv X X , LSP h 2 dec X decay. In order to specify the Ω 0 h D 2 Ω m T ˜ χ − , LSP and radiation has to be solved X n that may be important in particular h ≡ LSP X < LSP dilution throughout the text, necessary X LSP and was appropriately diluted by the . The LSP yield from the 12 Ω the . Br LSP 0 dec X LSP X ] for detailed analytic results. For gravitino dec 3 2 X T of the diluter into two LSPs (directly or via ≡ is lighter than LSP then it is Br T and (ii) 89 + Ω = , before 0 X – 21 – required f.o. min ˜ X χ X LSP X 86 , s < LSP D LSP D Ω n 85 < T ≥ ≡ 1 can produce neutralinos even for → X dec X D T X LSP < LSP Y Ω /H < decay temperature X X is referred as the X the LSP abundance . On the other hand, if the dilution min X X < LSP suppressed interactions. These kind of scalars, that are common in su- D Pl M is subdominant the observed dark matter has to be produced by processes decay is not free from constraints. It must decay before the BBN [ ). The X X LSP 2.20 decay generates Standard Model radiation only. The Br Y depends on the branching ratio Br The X particles for the times Γ X LSP to give at most a critical density of LSPoverproduce particles LSPs today. and notsimple overproduce but late quite decaying unnatural particlesthe case such that as the gravitinos. In the observed dark matter abundance. The constraint Ω which determines theindex ( dilution magnitude and consequently the shift in the spectral If the taking place atdecay higher of the temperatures scalar than abundance to negligible levels, then the LSP production from the scenarios without, however, modifying theΩ conclusions of the currentcascade analysis. decays) Finally, and the the pair annihilations take place until thethis neutralino corresponds yield reaches to the the value so-calledwith TeV annihilation mass scenario scale. and Let works usX mostly mention here for that wino-like the LSP radiation producedwhich from accounts the decay for of the an extra contribution to Ω the system of the threeand interacting we cosmic refer fluids the reader of or to axino references LSP [ the above expression generallycheck whether applies. the For the conditions neutralino (i) LSP onein should the also case (i) the neutralinos might reach a thermal equilibrium value the contribution from the diluter decay where we labeled Ω In supersymmetric theories genericallyvery weak exist or scalar fieldspergravity with and rather superstring flat theories,either potentials are due here and to collectively its labeledlations nearly about constant the potential vacuum, energy dilutes or the due LSP to abundance the energy stored in its oscil- 4.3 The diluter field JHEP02(2018)118 X (4.7) decays X domiation X 1 the . The decay rate ∼ >> >> to light degrees of >> >> >> >> , then the dilution c is forbidden due to dec X , , , 8 8 8 X 0 ). For a gravitationally y 1, Gravitino LSP 10 1, Thermal Neutralino LSP 10 1, Thermal Gravitino LSP 10 LSP ˜ /T χ ======0 Dilution Bound c c c c c c 4.7 m . For χ condensate, with respect to lifetime. In the area above 2 dom 1 non-gravitational decay X / & T 1 X or ˜ X ) X ' ˜ Pl G m c ˜ regardless the diluter branching X G M X D → parameter area. The solid and dashed (Γ 4 . For X / , 2 1 / 3 2 − Pl 3 X 9 necessary m M 10 90) excluded π / c GeV decay rate and channels. Actually, the details GeV) ∗ 4 9 ) is 5 g 2 – 22 – 10 8 X GeV. = 10 π 4.6 = 9 10 ( / X rh X T Γ ' field has Yukawa-like coupling that maximizes the diluter M = 10 7 according to the parametrization ( L X Excluded dec X since the channel rh 10 8 T T LSP is the reheating temperature caused by the inflaton decay. GeV m H LSP 12, hence it is an = 10 mass bound 6 . 4 MeV ( produces LSPs or other late decaying particles the relevant rh X ' c 0 m LSP BBN lower BBN m for = c 1 T 10 2 m ∼ X X > with the supersymmetry breaking scale. A late time entropy pro- > < m X X dec 5 X X m T /D 10 = 1 and 2 , where c h rh < m 4 < LSP correlated 10 < T LSP decay temperature is m scalar can be parametrized as decay do not change any of the conclusions synopsised in the conditions (A) and . The maximum possible dilution size, caused by a scalar dom X X T X X 9 6 1000 1 12 15 If the decay of the 10 10 inflation and we take into account the 10 10

1000

2 a osbeDilution Possible Max and the gravitationally and channels exist; for example if the duction takes place when theera radiation at dominated era gets interruptedFor by an an oscillating scalar fieldof the the dilution magnitude is ratios, and this is a key point of4.4 this work. The maximum possibleIf dilution the due diluter to massmagnitude is a is about scalar or condensate larger than the LSP mass, branching ratios have toexamined be in considered. the context ThisR of is each a model. model Inof dependent the the issue next and section(B). should we The consider be minimum the amount supergravity of dilution ( supersymmetry breaking scale and natural if kinematic constraints. the LSP mass, forconservative assumption gravitino LSP (black,the brown) lines and it neutralino islines LSP Ω correspond (blue). to Wedecaying have diluter (c=1), made the the thermalspoils gravitino the scenario BBN (brown solid predictions. line) The is plot excluded demonstrates because the the decrease of the dilution efficiency for large Figure 5 JHEP02(2018)118 . 2 = = = / 3 X (4.8) (4.9) rh (4.10) (4.11) m T min X , yields > m X , or for a D rh and , gives the only if 2 rh 0 T ˜ χ ≥ rh T m < T ) has to be used field is below the = ˜ g 2 ∼ , . m X 3.10 is explicitly written. 1 X dec(min) 1 X , dom , reaches a maximum X c 1 m ≤ T X /T ≤ 2 D rh ≤ / . T 3 2 12 12 2 . and . / 0 m 0 12 = 3 . / / rh X X exists which obviously must be 2 0 2 m T value is achieved for Γ / h D D h GeV cannot be reconciled with X 2 2 X max X 0 9 / h ˜ D < χ = < 3 D . 2 10 the expression ( dec / Ω /D X X Ω . For thermalized gravitinos instead, < 3 2 0 T 2 GeV for thermalized LSP gravitinos. m h ˜ Ω . χ / < dom < X 6 3 GeV T 8 m rh < 10 < LSP 1, that is, if T > m 2 10 ), and the parameter / 2 . 5 / × rh relic abundance is determined at the freeze out 3 1 or for GeV rh ) that typical reheating temperatures X GeV 2 3.1 7 T 12. / 9 – 23 – T 0 sc 9 . 3 m γ χ < 10 10 m 4.10 > m c 2 GeV = 0 2 ˜ GeV for / f / 8 7 3 2 2 3 2 . For the borderline case that / m / h 7 m 10 mass is obtained. 10 5 π m 20. If the decay temperature of the 8 1 or for 0 × / DM 1, see eq. ( / > 0 χ 7 X ˜ χ 0 2 & comes from the decays of gravitinos which are respectively 0 ˜ / χ is dropped out in the above relation it must be GeV ˜ m 3 GeV c χ m 0 1. Assuming that gravitinos are mainly produced by thermal 7 sc m 2 5 , 2 X ˜ < χ m m rh / γ ∼ y 1 10 T X ∼ 10 , ˆ c 0 = c f.o. ˜ sc χ 2 2 ˆ γ . We see from ( T / X / ) says that the abundance of gravitino LSPs produced from thermal 1 2 1 / c c and 2 eq 3 / 4.8 MSSM(sc) 3 ) applies and the maximum possible dilution magnitude gives the following 2 Pl 3.4 = Ω ) becomes more severe for = Ω /M GeV imply a mass bound 2 2 / / 12 < 3 3 LSP < 3 4.10 10 m the dilution magnitude due to an oscillating scalar field, ) the neutralinos number density will get diluted. For thermally produced neutralinos When the neutralino is the LSP the ˜ In particular, for gravitino LSP the lowest π − 4 0 9 LSP f.o. ˜ χ c/ Thus, neutralino masses cosmological scenarios where anleading oscillating scalar contribution field to dilutes Ω produced the from thermal sfermion plasma. decays for and If another the constraint for the ˜ T the maximum possibleconstraint dilution magnitude, for Although such heavy gravitinosplasma, hardly thermalized get messengers thermalized canbound via bring ( interactions them with to the thermal equilibrium. MSSM Again here,temperature the that is constraint where Ω 10 The constraint becomes morenon-gravitational scalar severe if ˆ the formula ( where Ω Note that although the The constraint ( scatterings in the plasma isan possible oscillating to scalar get field diluted that to obtains observationally mass acceptable from values the by supersymmetry breaking scatterings then the maximum possiblethe dilution lower value, bound m value. Consequently, a minimum valuebelow for the the Ω observational value Ω ( freedom then it is Γ JHEP02(2018)118 ' (or k (5.1) (5.2) (5.3) , the ln ϕ LSP thermal /d m t ). In such gravity and dn 2 2.21 ), R 2 ∗ . ) observables will N 2 , r (2 s / Pl . n 3 inflation model is par- 2 ∗ − 2 12 ϕ/M N R 2 3 ) measurement can give us = = q t and important constraint on , r ∗ − s n e n . . The inflationary predictions of 2 − Pl R 1 extra M 2 , r 5 2 Pl 2 m ∗ − 2 Pl 2 M N 12 10 M 2 × + 3 m . ' − (super)gravity inflationary model R 3 4 1 k 2 , 2 s Pl − ∗ ' 2 – 24 – R ln gravity and supergravity inflation models will be M dn d m 2 − + ) gravity model described by the Lagrangian R = 16 = ∂ϕ∂ϕ R model new predictions for the ( R with the thermalized degrees of freedom, that regulate ( ∗ 1 2 L r 2 f , observables. The Starobinsky 1 X − R ∗ − ] we get r 2 e 1 N is not a gravitationally decaying scalar due to the necessary R inflation 2 Pl 2 − X 2 and ] is an R M s 90 n = 1 − s = n L 1 ] at leading order are given by the following expressions of the primordial − 1 since the e 91 c takes place and then the dilution size is given by the expression ( theory [ . 3 ∗ 2 In this section we will apply the previously obtained results on the This correlation among the dilution size due to scalar oscillations and the R /N 3 spectra and tensor-to-scalar-ratio Also, the tensor spectral− tilt and running are respectively From the CMB normalization [ the This theory is conformallyscalaron, minimally equivalent coupled to to the gravity Einstein gravity with a scalar field reviewed. For the supergravity be derived, depending onthe the two models supersymetry will breaking be scheme, compared. and the5.1 phenomenology of The Starobinsky The Starobinsky model [ supergravity inflation modelsexpected in values order for to the ticularly perform motivated because a it full isit placed estimation is in of the self center the evident ofIn theoretically that the the likelihood a following, contour, preliminaries similar nonetheless of analysis the can be performed for any other inflation model. turbations that grew intotypical the model CMB of anisotropies. inflationand that If the naturally the expected explains early tensor thephysical perturbations Universe evidences statistical then is for the properties described the precision of by radiation ( the a dominated density era before the epoch of nucleosynthesis. the decay rate. Thea gravitational more diluter generic scenario one. is certainly a less model5 dependent and A concrete example:Inflation the is the leading paradigm for explaining the origin of the primordial density per- alternatively the supersymmetry breakingthese scale) scenarios, is see an figureinflation 5. Thea constraints case on it the is LSPpresence mass of Yukawa can couplings of be raised if JHEP02(2018)118 . ' = ] 75, ) is . 963 thus rh (5.8) (5.6) (5.7) . (5.4) (5.5) 33 ρ 2 ∗ end . Also, 2.12 ρ 2 = 0 . /N = 106 value that /N X 4) ]. Thus the is the decay rh (th) s / ∗ N ) reads [ ϕ n 37 g (th) s ∆ Γ . n = (3 − 2.14 ≡ 4 . ∗ / = 0, [ 1 ) inf rh N ), that is rh w and the number of the rh GeV ∗ . thermal 100 T g 9 2 ln( 2, hence 5.3 rh 2 . X / T 10 0 ) has been neglected because N N , where Γ ∗ ≈ ∆ H ) ln . /N GeV inflation the expression ( . The − 81 + 3 = 1 3 9 ∗ . 1 changes that could shed light on 2 end 0 N 10 + inf R ∼ /ρ rh plateau potential changes very slowly end + ∗ = 54 ρ ρ 2 ∼ 2 ln(54 X V decays after it has oscillated excessively / rh 2 R N X ∗ N Pl 100 g ϕ N ln − 4 ln( M ∆ / = 54 into the eq. ( 1 12 ln inf − = 1 Γ – 25 – 1 2 = (th) 12 ) p R . Since the corrections at second order in slow roll 2

N N 4 2 2 + R ( / | 1 2 N 60 it is 1 (th) rh ∗ − /N R ) reads end . The energy density at the end of reheating, V 1 β . In total, for the N N − ρ 4 Pl rh ∆ rh ' we get for the first slow roll parameter + = ∗ 2.10 ∗ ). In addition the ln M g 2 inflation model predicts at leading order is found when we ∗ g 2 2 V 1 4 2 11 ξ = 45 R R R 2 | N ≡ 90 π − /N α ∗ ∗ + R ) N ∗ N 10 − and rh e-folds number = T × ln ( 2 ∗ 3 2 ln(54 = 1 /N 1 4 . R g | / 1 3 s n rh ' 9 + T . ' − GeV. For the thermal 1 + 1 , is determined by the reheating temperature ) . 9 4 V 2 rh η value and for T end 10 − = 55 ϕ rh ϕ 2 ( ∼ ∗ = 2 R g | ∗ R given by the expression ( rh ∗ V T 30) N After the end of the inflationary expansion the inflaton is a homogeneous condensate rh / 2) 4 ln 2 / N / π at the scalar tilt willgoing not at be next-to-leading negligible order inthe we the could pre-BBN future cosmic probe it history. ∆ is For crucial the to Starobinsky go model to the order 1 expression ( its value isthe less standard than Starobinsky 0.1substitute for the relevant valuesIn of terms the of themodel read e-folds number, the other two slow roll parameters for the Starobinsky 1 with the In the above equation the logarithmic correction 1 Assuming only Standard Model degrees ofthus freedom, at that energy scales it is The reheating temperature is estimatedrate by equating of Γ the scalar graviton, The energy density(3 of the( inflaton atdegrees the of freedom endrecast of into inflation is found to be equation of state during∆ reheating is to good approximation zero, ¯ of scalar gravitons. The scalaronwith universally gravitational interacts strength with and other the elementaryThe particles inflaton lifetime only perturbative of decay the processmany scalaron can times is be about rather computed. the longphase minimum and of evolves its as potential. a The pressureless universe during matter scalaron dominated oscillation phase and the effective value of the JHEP02(2018)118 . , ) − γ m V b and V , η (5.9) η (5.13) (5.12) (5.10) (5.11) S V ( r and 1 = 2 . = 2 + ln 2 + ) T − r becomes large N s ≡ − n ) but it is more ), the gravitino S is not integrated . Next we review a m C e T ) and 5.13 (th) s . V n . 68 + 3 ln , ξ and the real vector ). 2 . = 0 V ¯ ). This is achieved by the R ) where s 5.2 2 − 037 M , η . terms and kinematic terms ( 4 n V 5.1 0 . R V 3 ]. We mention that attention η 2 4 1 ( − ( N R s ζ m O 110 = 0 = n 3 + – + 3 2 ) 2 2 = + supergravity inflation model. R 97 s 3 V

, η − s n N 2 ¯ R ) n R − − (th) s C R 965 α . 2 = 24 (1 4 m s 4 − scalar tilt value is obtained 23 = 0 α − 2 ]. The real component of + 1 R ) relation reads – 26 –

2 s + (8 ) 116 n s 2 V – ( (th) s n ]. In this stage a non-linear realization of supersym- η r θ E n . One may work directly with ( thermal values for the 4 − 0034 and has to be zero and hence ) and 113 = m d . r b 112 r N = 12 , X Z 3(1 r = 0 N ) the 2 P and 2 − the 111 and s R M 5.8 r 3

1 + ln n 3 the tensor-to-scalar ratio and running read . M ]. During inflation the universe undergoes a quasi de Sitter phase ) and − 3 /N (2 (th) 3 V r 3 92 = η /N ( 18 N order. In particular for the Starobinsky model it is L supergravity inflation O ]. Eventually one finds the effective model ( 3 − 2 + ) when expanded to components yields 2 R 96 2 V /N , value is 17% smaller than the value obtained at leading order. Furthermore, = 54 in eq. ( 12 N + r ) was obtained from the expressions ] 5.13 Cη 94 = , (th) 2 larger than the leading order prediction. We also take at next-to-leading order R 96 r – 93 N 5.12 − ), and a pair of auxiliary fields: the complex scalar h α m 2 92 V the Euler-Mascheroni constant. η ψ γ 6) The old-minimal supergravity multiplet contains the graviton ( If nature is successfully described by the Standard Model of particle physics and the inflation model then the ∆ = / 2 ˜ G which implies that supersymmetryand is it broken, can the be mass integratedmetry of out during the [ inflation sgoldstino isout possible due to [ theinflation non-linear [ realization and it is the only dynamic degree of freedom during ( Lagrangian ( for the “auxiliary” fields convenient to turn to thestandard dual supergravity description [ in terms of two chiral superfields: Modifications and further properties canshould be be paid found to in the [ temeprature full in couplings of each the of inflaton these field models that since may yield not a all different of reheating them are pure supergravitational. 5.2 The The embedding ofsuperspace the approach Starobinsky consists of model reproducingaction of the [ Lagrangian inflation ( in old-minimal supergravity in a (19 with R and estimate the expected going to accuracy level 1 The eq. ( written up to 1 that is 2 Note that the Plugging Also, going to order 1 JHEP02(2018)118 ] ∼ has 2 117 / 3 (5.15) (5.14) X m may be totally Φ . and the reheating inf > m GeV 2 m 9 / 3 ≡ 10 ], where a new class of R- m Φ 96 ∼ m Pl is usually satisfied. The super- M model are found to be identical to ). In addition, the reheating phase Φ sector with the MSSM and a basic 2 ) means the supergravity and non- 2 , where R 5.3 2 / 2 R models with Pl < m 3 inf 5.6 sugra-inf 2 2 Γ Br R / /M 3 Φ 3 Φ p rh m 4 T m m 0 / , the gravitino yield due to the direct decay of supergravity is that at the end of inflation the 3 2 and the sub-inflation supersymmetry breaking c 1 2 model ( 2 – 27 – Λ , i.e. the sgoldstino, although it can play the rôle Φ = = 2 predictions ( ) R / Z 2 4 R 2 rh | / inf ] 3 T R Z > m ( Y 90 ∗ 2 117 g / − | inflation models are completely degenerate in terms of sugra-inf 2 3 Assuming a simple supersymmetry breaking sector, with 2 π | 2 m and supergravity 3 Z R 2 | R models was considered, it was found that it is possible inflation = = 2 2 R R SB K + sugra R | . We can directly apply the analysis and the results of the previous case. The fist case is realized when the minimum of the inflationary 2 rh and 2 T R 0 / 3 W supergravity scenario can be distinguished in two basic cases: the ultra high + 2 > m R Φ FZ m without invoking any matter superfields. The new properties of these models which = In the case that the inflaton field vacuum is supersymmetric an extra field is required The The fact that the reheating temperature is found to be about the same with that The inflationary predictions for the supergravity and gravity inflation to the abundances of superparticles. Any late decaying scalar field, e.g stringy moduli, can be the diluter field. Φ 3 field contribution becomes important. In such ultra high scale supersymmetry breaking SB 2 2 m the inflaton Φ is calculated to be [ to break supersymmetry, andsymmetry the breaking spurion condition field called of the diluter itto overproduces play LSPs the and rôleof theW an diluter. extra scalar that we generically label scenarios the superparticles possiblyreheating play temperature no may rôleduring notModel the be particles. thermal sufficient Hence, evolution the to sinceindistinguishable excite the unless the gauginos superpartners or some of moduli the fields are Standard much lighter than the gravitino. potential breaks supersymmetry. Particularly insymmetry the violating model of [ and supersymmetry breaking2 to originate fromdistinguish the them supercurvature from the and R-symmetric S obtain supersymmetry breaking sector. LetR us first examine the implications of the supergravity scale supersymmetry breaking scale supergravity versions of the the inflationary predictions. However,after the details the of decay theR of expansion history the of inflaton thesections universe should by break minimally completing the the degeneracy supergravity between the supergravity- predicted in the non-supersymmetric the non-supersymmetric Starobinsky is much similar and thethe inflaton inflaton decay decay rate channels roughly weredecay the identified rate same. and was Indeed, parametrized the as in branchingtemperature Γ the ratios was work calculated. estimated of to The [ be total JHEP02(2018)118 , 2 ) 2 yield , and / (5.18) (5.20) (5.21) (5.16) (5.17) (5.19) Φ 3 Φ Z ˜ G m m m /m 2 Z / 2 1 ] m / ) Z Z the branching 1 Φ 2 ) / m 2 m 2 3 m 117 / / Z 3 3 m particles there are Φ , m 2 2 / Z 3 Φ Φ m / 1 ( 1 rh ) m m ) 0 T is produced as particles Φ ( Φ 2 Z m πc Z m 2 . Otherwise, the gravitino Z m 2 / 1 16 / 3 . Z m 3 − ) m 0 = 2 m Pl m is found to be [ , 2 . 3 Z 2 πc dominated universe it is / / Z Z 3 2 Φ 2 Z πM Z 3 m Z (48 for m m Br 96 Br for (3 0 for for (3 2 Φ ' is rather model dependent. The rh Z πc 2 dec 1 Z T m 2 ) to be Z m 2 / 3 T 4 48 / 2 2 Z inf 3 3 / Φ 3 4 4 decays dominantly into a pair of gravitinos 3 × m m 2 5.16 m m / Φ 2 × 4 ) = Z 3 Φ Z m m – 28 – m m × Φ Z ZZ 16 = 2 field are rather model dependent and it is possible m m ) =    ˜ 16 = 2 → G Z ˜ ×          G Z (cond) 2 s × Pl n / Z → Φ 2 3 0 Br(Φ M Y m Z x πc = 1 c Γ( 48 r inflation the above contribution to the gravitino abundance is with the partial decay rate enhanced by the factor ( π ' condensate can be computed if the initial amplitude of oscillations 2 1 ) field. The supersymmetry breaking field Φ 32 R ˜ Z (particle) 2 G 4 / Z Z ˜ / 3 G 1 < m Y → Z modes, produced by the inflationary de-Sitter phase, which may store a rh ∗ into a pair of gravitinos. In addition to the incoherent m Z g 90 the branching ratio maximizes, Br 2 Z π Br(Φ Φ the reheating temperature after the decay of the inflaton and Br mass and couplings are known. Assuming a 2 / ]. In the case that the spurion field is heavier than the inflaton, ≡ 3 is determined by the dominant decay channel, here the anomaly induced pro- rh Z 2 ' T 0 / m < m 117 c inf 3 2 Apart from direct gravitino production from inflaton decays, gravitinos are produced / inf 2 3 / Br , the 3 Y 0 z and gravitinos and the LSPszero are generally temperature found VEV to of be the overabundant. scalar The initial value and where ratio of the the coherent significant amount of energy. The precisefrom VEV the of decay of the Thus, the gravitino yield as a decay product of particle Considering the generic decaywhen channel the via the decay of the by the decay of inflaton with branching ratio For the supergravity small. The cess [ m yield is calculated from the branching ratio ( with branching ratio JHEP02(2018)118 . < in Pl X M LSP (5.22) (5.23) (5.24) = 54. 2 ) m Z where (th) /m N dec X 2 field depends / 3 , X > T m . ) field gets over the 3( 12 and sub-critical g .   √ dom X + ˆ X T depends on the dilution = 0 053 X . = 2 s 2 0 D i n h z − h e-folds number (ln p LSP dynamics are left unspecified ) 2 , is X 300 X N SB 3. The ∆ W / 1 + thermal X ) ˜ 019 ∆ D . g for scalar condensate for thermal inflation (0 + ˆ the vacuum energy of the flaton field 2 = ln =0 X . Keeping only the relevant terms and after p X 0 field domination induces a shift in the spectral D X V   dec X – 29 – N X X T (ln , N 4 , rh − ∆ 4 T and / 4 10 1 2 − GeV and the decay temperature of the is the initial amplitude of the oscillations in a potential 4. × 0 9 / 0 10 dom Pl V X inflation 2 ) the size of the shift due to a non-thermal phase that lasts )] x 10 0 ∗ T the shift in the scalar tilt reads M × g − x due to the change of the number of the effective degrees of 2 e-folds after the inflaton decay for the Starobinsky model is 3 2 , and the superpotential, 30 dec 3 ∼ g X √ π . R g 2.20 T X 6 SB ( + ˆ rh ) = N ∗ − ˆ g K T      X , 965 due to a change in the ∆ . /g = 1 X ) D ' s − is trapped near the origin during inflation. The zero temperature VEV, D n ( 4] = 0 dom s Z X / dom ∆ about the minimum and X = ln ) n T T ( 2 field dominates the energy density of the universe if X ∆ ∗ X (th) s g . w ˜ X n D X 3 ¯ scalar does not dominate the energy density of the universe. In the analysis 2 X ln[ that dilutes the LSP abundance at the critical Ω LSP − m Z plus a correction ˆ ≡ the Starobinsky X m 2 g ] the scalar X ) = D < In the following we assume benchmark sparticle mass patterns and we estimate the = [(1 117 X X ( X ˜ analyzing ln where ˆ For a scalar condensate domination it issize ∆ freedom at the temperatures index value According to the formula ( N radiation energy density. The V the case of thermal inflation.Starobinsky The inflation reheating is temperature foron Starobinsky its and full supergravity interactions. The non-thermal the temperature that the energy density stored in the oscillating 5.2.1 The shift in the scalar spectralThe index diluter and the tensor-to-scalar ratio for neutralino dark matter scenarios.labeled We generallyvalues. assume the Particular presence hiddenexcept of sector for an details the requirement concerning extra theproduction. the scalar diluter not This is to achieved overproduce is LSPsm if at the the branching time ratio of to LSPs late is entropy very suppressed or of [ dictated by the Kähler, corresponding shift in the spectral indexdark and the tensor-to-scalar matter ratio in density order the to predicted be in accordance with observations. We assume gravitino and that the JHEP02(2018)118 ) 4 − ) for 5.12 (5.25) (5.26) ) since 5.9 .

5.26

clrratio scalar to

tensor . 4 - - ) . Nevertheless X 3 ) g − N (th) ) and ( ∆ + ˆ N ( ) yields, agrees with X 5.24 0.0050 0.0045 0.0040 0.0035 0.0030 D O 3 and keeping only the ) precision measurement ) given by the eq. ( 5.26 / s (ln ) + , r n 3 g 14 s ( 4 − r n inﬂation model reads + ˆ 2 X 10 L 2 = ` N g X (th) R (1), hence the change in the num- × r 12 , ∆ D N 2 O X ) measurement of the order of 10 . r D ( . H inﬂation model. The solid line corresponds s r g 10 ) observables of the two base case dark n 2 = 1 + 8 + 36 ), that the eq. ( R ˆ , r g 3 D ) 3 = (ln s , , r X / g L n X X 8 ` ), and the dashed line corresponds to no change, g D X + ˆ N @ (th) D , ˜ ( D dec X X ∆ – 30 – X r N T D Log ( D H 6 correction in the expressions ( ∗ s g = ln + ∆ n g (ln = s 5 X ) = 24 (th) n − 4 N r th ) = 10 10 , 2 N × ( R dom X r 9 2 ). T . ( ˆ g − ∗ , g ) predictions for particular supersymmetry breaking exam- X X r ) = 3 D ). 0 N ˆ g ( = 54 and ∆ s , , i.e. dec ∆ X n X T dec and X ( −

D 0.966 0.964 0.962 0.960 0.958 0.956

(th) ∗ T s ( clrtilt scalar g + ∆ r N n (th) ∆ and ) = N (th) s ( . The scalar tilt (in black) and tensor-to-scalar ratio (in orange) values when post- n r ples dom X dom X = = T T ) and one can safely neglect the ˆ precision with the value one gets from the relation ( , s 5 r ∗ Regarding the effective degrees of freedom, it is ˆ The shift in the tensor-to-scalar ratio is found by expanding the expression ( r 4 n g − ∆ − 5.2.2 The In this subsection wematter explore the scenarios impact of on supersymmetry, ( the gravitino and the neutralino, when the initial ber of degrees of freedom(10 requires accuracy at the the expected accuracy of theobserving future that, CMB probes in will principle beeffective degrees of at of the least, order freedom of at oneis 10 the certainly can thermal important plasma additionally and from determine exciting, the see the ( figure number 6. of the We have verified that the10 value for Substituting relevant terms, the above expression for the Starobinsky i.e. the Figure 6 inflationary dilution is considered for theto Starobinsky a change oftimes factor 10 in the number of effective degrees of freedom in the energy density at the JHEP02(2018)118 Z /m (5.28) (5.27) dec Z 12. We ). If the . T field that 0 2 / Z Z 3 5.19 ≤ . 2 4 / , h 2)Br 1 / 4 ! / (3 LSP 1 ˜ 2 g 2 Z − ]. ∼ m m ! 2 4 ˜ 2 g / 2 Z Z 3 118 − , m m Y decay, with branching ratio 1 4 56

Z ﬁeld necessary to decrease the − 2 1 X ]. In such a scenario the super-

˜ g is required to dilute the thermally 2 49 m TeV X ˜ g m TeV 2 ) and in general is found to be subleading / 3 ], it is the imaginary part of the 2 GeV / m 5.17 7 68 – The gravitino is the LSP if the supersymmetry – 31 – scalar to gravitinos is given by eq. ( 2 / Z 66 1 Z m TeV scalar field and the inflaton. The inflaton contribution Z 4 Z / m TeV 1 ] 4 ∗ / g 15 1 67 and the MSSM mass pattern is necessary. 2 ∗ / g 15 ∼ 3 field decays dominantly into MSSM fields with non-gravitational 2 m − , 2 Z h / Z domination. In order the precise dilution size to be determined the Z 3 m Ω 12 Z . 0 760MeV ' the mass of the bino. We mention that it is also possible that the spurion field ˜ g dec Z T m inflation. The decay rate of the will be part of the dark matter in the universe with yield 2 2 Let us now consider four benchmark mass patterns for the supersymmetry breaking Before proceeding with the survey of particular examples, let us mention that the / R Z 3 decay produces late entropy then the gravitinos from the knowledge of the sector plus theconsider MSSM, the with presence of different messengerLSP sizes relic fields density of and and the which supersymmetry dominantly diluter decays breaking to visible scale. sector fields and We not to also gravitinos. gravitino relic density parameter violatesin the the observational bound TeV and unless sub-TeV theproduced gravitinos scale. sparticles and lay the Another energy scalar storedfield, in field the oscillations in of case the supersymmetry of breaking where does not dominate the energy density due to thermal effects [ to the gravitino abundance isin given by eq. ( Z Br and relic density parameter [ The LSP gravitinos arefrom produced the from non-thermal thermal decay of scatterings the and decays in the plasma and symmetry breaking interactions. Following realistic models [ decays last and the dominantthe channel decay is temperature onto given a by pair of gauginos, in particular binos, with any other inflationary modelphase, the after reheating the temperature appropriate and the adjustments inflaton regarding fieldExample the branching ratios. I: reheating gravitino darkbreaking matter. is mediated morestandard efficiently paradigm to the is MSSM the than gauge to mediation the scenario supergravity sector. [ The consider both thermal andscalar non-thermal decays. dark We matter examine production differentwe from and quantify the illustrative how supersymmetry hot the breaking plasma expectedthermal schemes and values and for post-reheating the phase inflationarymention dictated observables change that by due this the to analysis, a universal that non- constraint probes Ω cosmologically a BSM scheme, can be applied to conditions for the hot Big Bang are set by the supergravity Starobinsky inflation. We JHEP02(2018)118 . 4 − not fol- . field GeV , see ) say 10 rh 6 Z hidden 10 X × 10 . The 4 5.28 10 > T gravitinos . condensate & ∼ r Z GeV 2 mess mess ], or when the 8 h ) and ( 2 M / M 56 10 < 3 . This dilution can ] it is possible that and ∆ . Messengers get 4 5.27 3 ' rh and 26 10 − features for the . & 10 non-thermally < T 4 mess 12 bound implies that the . . The shift in the spectral × − X GeV 0 2 M D 6 10 dec X mess ≤ would have a relic density pa- & × 10 2 | /T and M s h . There are scenarios in the 2 n 10 ' / dom and the scalar spurion field 3 X ∆ | Z and & T non-minimal 2 r rh GeV = 0. We mention that these scenarios, m / ' 4 3 r ∼ abundance due to interactions with the X < T 10 ˜ f D ). The Ω and ∆ . The diluter can be either a flaton field = few TeV ' > m 1 GeV and produces 3 10 , m 3.4 . mess Z Z Z − – 32 – ]. We also assume that the messenger coupling is 4 ' 10 with M − m m m 10 = 0 and ∆ 10 X 118 & dec × ∼ s Z ∼ ∼ , GeV dominate the energy density of the universe. The n T 5 ˜ × 5 ˜ ˜ f g g 56 X 7 & ] and their relic density would be Ω m m m 10 D 12 bound implies that the thermally produced gravitinos | not . & s 56 0 [ times the observational bound. In order the ∼ ∼ ∼ ' n r ˜ ˜ 5 ˜ 1, so that the gravitinos do not get thermalized. The gravitinos g ˜ g f f ≤ ∆ . field has to be a flaton and cause thermal inflation. In this | 6 , see below eq. ( ] and generally assume thermal equilibrium 2 scalar oscillations generally have a large enough amplitude and does 4 decays at h X , m , m , m , m 2 Z 10 and ∆ 56 Z / Z , 3 3 ∼ scatterings of thermalized messengers mess − 2 λ 54 GeV GeV GeV , h 10 2 4 3 2 / × < 3 53 , 10 10 10 3 field does not receive thermal corrections because the messengers are not and figure 8. dominate the energy density of the universe. Equations ( ). The Ω 26 = few GeV ' ' ' & 1 . For example when the messengers masses lay in the range Z | 3.4 2 2 2 2 s / / / / n does 3 3 3 3 ∆ require dilution and predict ∆ in their original versions, workscale. better Features when of supersymmetry these is scenarios broken are about currently the tested TeV by the LHC experiments. tures [ sector and the goldstinomessenger does coupling not is controlled residegravitinos by have in the the VEV right a of abundance. single another These field supersymmetry chiral breaking [ superfield schemes do [ not to get diluted the case, the shift| in the spectralm index andliterature tensor-to-scalar ratio that are reconcile respectively gravitino cosmology with high reheating tempera- The thermalized. The Z that the spurion that exceed about 10 that causes thermal inflationdominates the or energy a density of scalara the condensate. universe dilution shortly after In size the the toratio reheating later in be are order respectively case realized. such the The shiftm in the spectral index and tensor-to-scalar thermalized and gravitinos obtain a thermalized messengers eq. ( are sufficiently diluted if rameter Ω thermally produced gravitinos are sufficientlybe diluted caused if by scalarindex condensate and tensor-to-scalar ratio are respectively m messengers get thermalized since lows the finite temperature minimumdominate without the sizable energy oscillations density [ and hencesmall enough, does produced from m The above benchmark examples for the gravitino dark matter scenario are synopsized 4. 3. 2. 1. in the table JHEP02(2018)118 2 Z R scalar scalar (5.29) Th Th Th Z Origin X Non-th min min min | | | 12 GeV. In turn, r ) the neutralinos 0034 . ∼ 0038 0044 0041 . . . . Here we assume the 4.11 0 0 0 dec Z T . field decays is estimated Z GeV max max max ), which generally accounts 2 | | | 7 s / 3 965 0 n . 10 GeV m 5.17 0 963 960 962 . . . produces late entropy if displaced 0 0 0 ' 2 Z / Z 5 supergravity inflation, and by the de- ∗ 2 max max max , m | | | 54 N R Z 51 46 49 . Contrary to the GMSB case the GeV Z 8 m GeV 10 6 For gravity or anomaly mediation of supersym- decay dominate the energy density and decay min min min | X | | – 33 – 1 10 4 6 10 Z D 10 10 10 ' GeV 9 ˜ f ) that decreases the LSP relic density via late entropy 10 m × X , and the total decay rate yields the decay temperature ∼ LSP 2 3 4 2 4 ( / 2 10 3 10 10 10 2 / ' / 3 where the neutralino has an annihilation cross section few orders m 3 m dec Z m prediction for gravitino LSP and a gauge mediation scheme for the T Z r GeV are impossible to get diluted by the oscillations of the ˜ 4 5 6 4 f 7 m 10 10 10 10 m GeV, 2 GeV producing neutralinos that annihilate rapidly and acquire a relic 10 and . 3 0 s > scalar that decays to gravitinos at the temperature ˜ 4 4 5 3 g n 0 10 ∼ ˜ χ Z 10 10 10 10 m 2 . field. In cases # 1, 2 and 3 dilution is required to decrease the LSP abundance below / m dec GeV not to spoil BBN predictions at the time of decay. The gravitinos are pro- 3 0 4 4 6 3 5 Z T Z field oscillations dilute the thermal plasma then the gravitinos coming from the ˜ χ . The 10 10 10 10 m Z annihilation scenario of magnitude higher that the conventionalby value. the The universe is generallythe dominated gravitinos produced fromat the m . The neutralinos are generally found to be overabundant when supersymmetry Let us now consider benchmark mass patterns for the supersymmetry breaking sector 2 1. 1 2 3 4 / # dec 3 with mass and thermal inflation is required. plus the MSSM, characterized mainly by split and quasi-natural sparticle mass spectrum. decay are the leading sourceT of dark matter neutralinos atbreaks the gravitino at decay temperature energiesthe beyond presence the of TeVproduction. a scale diluter We and field mention dilution that according is to required. the general Hence constraint ( we assume of gravitinos, when If the duced non-thermally by the decayfor of a the subleading contribution inflaton, in seecay the eq. of framework the ( of supersymmetry breaking scalaroscillations are field not thermally damped andfrom generally the the zero temperature minimum.by The considering temperature the that the various partial decay rates. The dominant decay channel is into a pair Example II: neutralino dark matter. metry breaking the gravitino massdecay is populates naturally heavier the than universe theabove with neutralinos. 10 neutralinos. The gravitino Here we assume the gravitino mass to be Table 1 supergravity model. In the cases #messengers 1, and 2 MSSM and fields 4 while the inbreaking gravitinos the are case produced # from 3 thermal from scatterings thethe of non-thermal observational decay of bound. the supersymmetry InThe the masses case are # in 4 GeV non-minimal units. hidden sector features have been assumed. JHEP02(2018)118 . . 4 8 − 10 10 Th Th ∼ × Origin Non-th Non-th field and 2 2 h Z & 0 ˜ < χ r min min min | | | of chemical and is required. It is r 0034 . and ∆ X 0036 0036 0042 ), is Ω out . . . . In this example we 3 . In this example the scalar field is not dis- 0 0 0 − 3.10 Z 10 after the decay of grav- , but it can many orders GeV 2 GeV × max max max 1 | | | 9 8 s 10 965 0 n . & 10 10 & 0 964 964 961 | . As a last example we consider . . . s ' 0 0 0 ' D n and they are Z ∆ Z | . This dilution magnitude induces GeV 8 ∗ 5 max max max is required. The case # 4 is the standard , m , m | | | 54 10 ). Here again the rôleof the diluter is N scalar condensate are sufficiently diluted. 10 52 52 48 X 3.9 & Z > GeV GeV oscillations can be many orders of magnitude ) TeV then a diluter scalar X 8 7 Z . This is a phenomenologically viable example min min min X . | | | Z D > 4 1 ( – 34 – 4 neutralinos thermalize 2 2 8 10 m 10 , see section 3.2. The resulting LSP abundance − 0 − D ˜ χ 10 10 10 scenario assuming that the ' 10 ∼ ' 2 10 / m ˜ dec ˜ and ˜ f 3 f f × ) × 8 m /T m 0 m 2 f.o. ˜ χ non-thermal phase is required (although a non-thermal 0 & 3 3 5 3 ]. This minimum value of the dilution magnitude yields LSP & f.o. ˜ ∼ ∼ ∼ χ ( r T r 10 10 10 10 2 2 2 27 no 0 > T / / / ˜ χ 3 3 3 2 0 m / ˜ m m m (th) χ dec 3 and ∆ and ∆ T field and the gravitinos and it is thermal WIMP ˜ prediction for neutralino LSP and anomaly/gravity mediation scheme for 6 8 7 5 f 3 3 r = Ω − − Z dominated early universe becomes in turn gravitino dominated at 10 10 10 10 m GeV, GeV, GeV, 10 10 12 can be satisfied, see eq. ( GeV. For TeV and sub-TeV scale neutralinos the observational bound . Z and 3 3 5 0 2 0 4 × × ˜ 6 8 7 5 / (ann) χ s but the dilution size due to 3 4 1 10 10 10 10 n ∼ 10 10 10 10 2 m & & 2 ' ' ' ∼ 10 | | h s s 0 0 0 7 9 8 5 Z ˜ ˜ ˜ & n n χ χ χ . The dec DM Z supergravity model. In the case # 1 the neutralino annihilate after the decay of gravitinos, 10 10 10 10 ∆ ∆ m not constrained by terrestrial experiments. m the conventional placed from the zero temperatureof minimum the and never universe, dominates hence the energy density The LSP abundance hasis to possible be only if decreased theThermal eight gravitinos orders inflation and the of is magnitudea required down shift with and in this the| spectral index and tensor-to-scalar ratio respectively at least of size tensor-to-scalar ratio respectively at least of size m neutralinos are produced from thekinetic gravitino equilibrium. decay The LSP relic density, given by eq. ( itinos. A T Ω played by the of magnitude larger.tion experiments. This This scenario dilution is magnitude currently induces a tested shift by in LHC the spectral and index direct and detec- detection experiments [ | m assume that the density Ω can fit the observed valuethe and here gravitinos. the We rôleof the note diluterD that is if played by the larger. This scenario is currently tested and constrained by the LHC and indirect 2 R 2. 4. 3. 1 2 3 4 # Table 2 the while in case #the 2 gravitino neutralinos decay acquire are athermal overabundant thermal WIMP and abundance. scenario. a The In diluter masses the are case in # GeV 3 units. the neutralinos from JHEP02(2018)118 ] 2 ). R N 117 ( infla- decay r 2 = = 0 and X R r s case, where n 2 R ) and N ( s n = supergravity inflation is s 2 n R ]. This is a very interesting 121 ) make manifest. , In the case of supergravity 120 5.21 inflation can be viewed as complementary to 4 2 inflation models predict the same reheat- 1 MeV. A non-thermal phase that dilutes R 2 ∼ R ) and ( supergravity inflationary models 2 supergravity inflation model with observations. – 35 – 5.20 BBN R 2 T R ), ( 12 constraints imply that the thermal cosmic history . 5.17 and supergravity ]. and the 2 = 0 R 122 2 2 GeV until h 9 R GeV and the same expressions for the 10 9 DM supergravity automatically alters the details of the thermal history is not ruled out in general). In this scenario it is ∆ ∼ 10 2 0 R rh f.o. ˜ ∼ χ models depends on the energy scale and the pattern of supersymmetry T T 2 rh R T ] that focused on the initial conditions of the two models. , sparticles will be constituents of the thermalized plasma of the reheated rh 119 and figure 8. forest observations [ = 0. This supersymmetry breaking scheme is currently tested at LHC, direct and 2 α r m < T ∆ indirect detection experiments. phase before The fact that the The BBN and the Ω Let us finally note that the LSP particles produced from the gravitino and The present comparison of the 4 sparticles and supersymmetry breaking fieldsthe are absent, two allows inflationary the discrimination models. between model dependent Considering and only conservative the analysis, the MSSM conditions degrees (A) of and freedom (B)the of as analysis section of the 4, [ less when we conclude that theand degeneracy supergravity breaking of thebreaking, inflationary see predictions figure between 7 the and 8. and possibly the expansion history of the universe compared to the simple for both gravitino andcompatible neutralino with the LSP cosmological scenarios. observations onlynot if Therefore, perpetual the from thermal historythe of the supersymmetric universe is thermalmatter relics particles and can fully potentially reconcileThe supplements the required dilution the generally universe increases with with increasing dark the sparticle masses. Henceforth, after inflation, as the expressions ( is influenced by theto change be of overabundant the infurther supersymmetry contributions the breaking when greatest pattern. the part supersymmetry of The breaking the LSP field MSSM is is parameter found taken into space account and [ it receives However, the degeneracy between the twothe models reheating that stage appears during breaks thetion, after accelerating if the and inflaton ˜ decay. universe. In addition toand thermal the processes, supersymmetry the breaking presence field of the produce supergravitational a inflaton significant number of gravitino particles Lyman- 5.3 Distinguishing the The supersymmetric and non-supersymmetric ing temperature, are warmer than thestreaming LSPs length produced of fromout the thermal small scatterings LSP scale and dark cosmologicalpossibility this matter, perturbations, that changes could see which the provide e.g. further has free constraintsand [ to the lifetimes these considered effect scenarios, here though of yield the free mass potentially scales streaming washing lengths that are not in conflict with the The above benchmark examples forthe the table neutralino dark matter scenario are synopsized in JHEP02(2018)118 (5.30) GeV. The 9 . 0034 are the s . n = 10 rh ) relation for the = 0 s T r n ( r = ) contour in figure 8, it GeV. The information that r , 9 , r s 965 and . n = 10 0034 . = 0 0 rh s T > n ∗ r ) values. From a different point of view, dilution magnitude as a function of the gravitino , r for a reheating temperature s ˜ g value, magnified 1000 times on the contour labels, n s m – 36 – not to exceed the observational bounds. The density- n = 2 ) observables could indicate cosmologically viable h ˜ f required 2 inflationary model predicts / , r m 3 965 and s 2 , which is the characteristic . 3 n 0 R ) s < n ) ∗ − k ( s 4(1 n / 23 − 2 ) inflationary model, see figure 8. The s n 2 R − . This is a compound plot consisting of 3D graph and a density-contour plot. The 3D = 3(1 r the precise measurementsupersymmetry of breaking the patterns. ( Althoughof it may the not dark be possible matter,is to possible see specify to the the constrain identity proximitybreaking significantly of parameter and space. the even rule spots out on a the great ( part of the supersymmetry and Starobinsky reference thermal values. A knowledgewould of allow the details us of to the supersymmetry accurately breaking predict sector the ( one extracts from thiscan graph be is compatible that with the supersymmetric CMB models data (e.g. only quasi-natural, for split, particulartrue, high values for imply scale) the that scalar the tilt supergravity Figure 7 graph shows the decadic logarithmLSP of and the gaugino-sfermion massesdilution with is calculated by requiringcontour the plot Ω demonstrates the changefor in inflationary the models that predict a reheating temperature JHEP02(2018)118 2 4) R , and 3 , 3 2 − , 10 0.99 ∼ s δn or the SUGRA- 2 R ). Last but not least, 0.98 . 5.30 ) cannot “prove” or disprove general requirement to fit the model is targeted with a fiducial respectively. If the future CMB , r s 2 2 n R and 0.97 plus “desert” 1 4 ) can be obtained if one simply assumes s 4 2 ** n 1 1 5.30 * * * 3 * 3 * – 37 – * 2 0.96 inflation model 2 R 12 that in turn implies the result ( inflation model, a gravitational modulated reheating was ) contour plane from Planck-2015 in the pink, and the schematic . 0 2 , r a roughly continuous thermal phase with reheating temperature, s R 0.95 ≤ n 2 h plus LSP and the green asterisks to the four benchmark models (#1 GravitinoLSP NeutralinoLSP LSP 2 Ω . The red asterisks correspond to the predictions of the four benchmark models R

3 forecast constraints from a future CMB probe with sensitivity − gravitino 0.94 model rather than the LSP, as explained in the text and tables σ 10 ] and non-Gaussianity was additionally predicted. In the present work the 2 × 0.1 R 4 0.01

depicted with the dotted and dashed ellipsis. The

123

0.001

∼ GeV. The selection of the dashed ellipsis area will exclude a large class of supersymmetry

. Constraints on the ( 4) with 3 Tensor clrratio scalar to 9 - - , r − 3 10 , 10 neutralino 2 Let us also mention that a similar result to ( , ∼ ∼ rh postulation of a non-thermal phase hasuniversal been constraint motivated by the we emphasize again that thethe precise measurement existence of of the supersymmetry. ( of It freedom, will that only supersymmetry indicate or the any presence other BSM of scenario extra will scalar be degrees challenged to explain. the presence of extraexample, scalars for that a dominate the supergravity assumed energy [ density of the early Universe. For the SUGRA- experimental probes select the areainflation inside model the is dashed ellipsis selected T then either the models that predict aif too the large dotted LSP abundance ellipsismuch for limited area that and is reheating extra selected temperature. scalar then particles On the should the duration be contrary, present of above the the thermal TeV phase scale, before hence supporting the BBN is Figure 8 illustration of 2 δr value of (#1 with JHEP02(2018)118 ∼ s . The /n r s X n w dark and 6= 0 N field s r n ∆ X ∆ , s n values for the e-folds ∆ respectively. ) ) if th Diluter th ( s ( (th) r n r thermal and X ˜ rh D (th) s rh T n , , w , due to a scalar condensate or a flaton # dof (th) ) which can distinguish different inflation (th) s values. However, our ignorance about the s X INF N n n r D ( Γ r – 38 – = and r s n ,.. LSP 0 Ω ˜ χ ˜ G, ) are not strictly scale invariant, hence important information , r s n value is possible to have been shifted by the amount ∆ SUSY s M n is the contour line Inflation Model Selection ∗ ) line that corresponds to what we called N s n early universe cosmic era. Any understanding of the cosmic processes that ( r from the expected thermal value, ). The = s h . This graph demonstrates the analysis followed in this work to cosmologically probe BSM dark Future CMB probes QN, Split, HS r n r 2.20 6) , is rather strong and will become significant as the accuracy on the observations SUSY ∗ − Motivated by the advertised sensitivity of the future CMB probes in this paper we The inflationary paradigm can be used as a concrete and compelling framework for N (1 number, scalar tilt and tensor-to-scalar ratio, quantified the effectvalue of ( a genericO primordial non-thermal phase on the spectral index independent of models. Furthermore, if inflation isinflation followed by model a predicts continuous thermal a phasethe specific then relevant number a modes concrete for exit the theon horizon number the and of the e-folds end between of the inflation, moment hence predicts a specific spot be obtained. the theoretical determination ofreheating the process and the subsequenton evolution of the universe, encodedare in expected the dependence to be further improved the next decades. An inflationary prediction that is take place before the BBNoperates will at provide us that withcosmic energy critical era scales. insights is into the through Oneessential microphysics the fact that significant precision is prospect that measurement the of to ( about the contemplate the CMB the background observables expansion early rate dark and the reheating temperature of the universe can 6 Discussion and conclusions The cosmic energy windowtered from to about the current 1 observational MeV probes up andcalled the to a corresponding the timescale can inflationary be energy reasonably scale is shut- Figure 9 scenarios. JHEP02(2018)118 ) s 12. n . ( r = 0 = 2 r h ) values due DM , r s n supergravity inflation and we along a contour line 2 r R ) and we broadly related it with the and ∆ , r s s n n ) observables. Our findings point out that , r s observables concerns the expansion rate of the n – 39 – r and 12 then the expected change in the ( s . 0 n 2 h LSP Undoubtedly any non-trivial cosmological information about the BSM physics is of ma- A complete understanding of the pre-BBN thermal phase and the CMB observables The most direct cosmological implication of supersymmetry is that the LSP expected to Moving a step further we applied our general results to study the observational con- directly tested at the terrestrial colliders. jor importance. Certainly theof results of figure this 9, cosmological analysis, cannotinformation illustrated discover that in or we the get disprove graph from supersymmetry. the The only concrete cosmological be excited either thermallythe or sake non-thermally of after completeness we theperformed considered end a in of theoretical this the estimation paperthe inflationary of the ultra-TeV phase. the scale ( supersymmetry For leavesobservables. a This more clear fact cosmological isios imprint particularly can on exciting be the because cosmologically CMB falsified high while scale the supersymmetric low scenar- mass range supersymmetric scenarios are requires the knowledge ofcessfully the provided initial by the condition inflationaryinflation for theory. and the In the thermal this subsequent Big work reheatingity we Bang, stage. inflationary suggested models which Actually a are are it unified degenerate,the is suc- study in often non-supersymmetric of terms the versions. of case the that However, inflationary supergrav- the observables, with supersymmetric their degrees of freedom can growing conflict with colliderhave data not and assumed that direct theverse. detection LSP experiments. If accounts it for In is theto our actually bulk a analysis dark Ω non-thermal we stage matter becomes component greater. in the uni- We find that a non-thermalsupersymmetry phase UV or completes low the reheatingeffect Standard temperatures of Model are the of generally different particle expansion requireddifferent physics. histories supersymmetry if on We breaking quantified the schemes. the ( scale In supersymmetry this paper since we low mostly scale focused supersymmetry on models ultra-TeV with thermal WIMPs are in produced non-thermally during the dark early cosmic era. be stable and hence contributesquantity to that the we dark estimate matter density. inamine The different how LSP classes it abundance of can is be supersymmetry the cosmologically breaking key reconciled schemes with and the ex- observational value Ω Although it lacks any experimentalsistently support, accommodates high it energy provides processes an suchActually, as appealing inflation the framework and fact that dark matter con- that production. the the Planck LHC mass, probes while onlythe supersymmetry a systematic may cosmological small lay examination part of anywhere supersymmetric ofbe in scenarios. the cosmologically between, Supersymmetry manifest vast strongly can energy if motivates supersymmetric scales degrees up of to freedom get thermally excited or is an indirect observation ofsince a it non-thermal has phase to and be connects attributed cosmology to to a microphysics BSMsequences scalar on field the domination. CMBmotivated of theories a that supersymmetric is universe. extensively Supersymmetry used is to one describe of the the very most early universe evolution. field domination. The observation of non zero ∆ JHEP02(2018)118 ) , r 6= 0 s n r is necessary rh T and the reheating rh w ]. ] suggests. Moreover the under- 37 124 inflation model is well understood, a 2 R ). A thorough understanding of the re- 2.10 – 40 – ] for a review. We should mention here that the 125 ), which permits any use, distribution and reproduction in CC-BY 4.0 to be estimated, see eq. ( This article is distributed under the terms of the Creative Commons rh ]. Presumably, the synergy of different cosmological surveys will enable ˜ N 126 From the observational side, future CMB primary anisotropy measurements should From the theoretical side a more complete analysis should also take into account any medium, provided the original author(s) and source are credited. the Greek government.Scientists The (B) work No. of 16K17712. YW is supported by JSPSOpen Grant-in-Aid Access. for Young Attribution License ( Acknowledgments We thank Fotis Farakos, Alexments Kehagias on and the Jun’ichi draft. YokoyamaStrengthening for Post The discussions Doctoral work and Research, co-financed of com- by ID the is European Social supported Fund by ESF the and IKY Scholarship Programs for as DECIGO [ a leap forward inthat precision operates cosmology beyond the giving Standardthan us, Model can at of be particle the physics, obtained same at at energy time, CERN. scales access much to higher the physics play a decisive rôleinprograms, probing such the as pre-BBN thenificantly cosmic direct to observation era. this of endeavorimprinted tensor as Complementary in well. perturbations, observational the should gravitational Informationheating wave contribute energy on spectrum scales, sig- the which in can thermal the be probed history frequencies by corresponding after future space-based to inflation laser the is interferometers such reheating process can also bringtemperature out new of observables the that can universe, furtheroscillatory see constrain epoch e.g. the and [ reheating the reheatingfact process that of makes the the results obtained in section 5 reliable [ temperature, as e.g. the thermalstanding leptogenesis of scenario several [ distinct stagesthe universe in in the a reheating radiation dominated processin phase that order at leads some a reheating to more temperature thermalizatione-folds accurate of number value for the equation of state parameter ticular inflation model isthe possible. BSM desert In scenario suchfigure and a 8 indicate case illustrates. possible our features analysis of has candidate BSM the theories, power asbaryognesis to the rule scenarios out and thematter-antimatter asymmetry details in of the universe, thermalization seems process. to have a The critical generation dependence of on the the rate cannot be revealed anddeviate it from is their only subject thermalwe to values focused interpretations. then on Nonetheless, new supersymmetry, if physics thoughthe the exists any ( event in BSM of high scenario detection energies. cantogether of be with In primordial analyzed possible this gravitational accordingly. features paper waves, In of that the is tensor an power spectrum, observation of then the selection of a par- very early universe. The identity of the matter content that controls the cosmic expansion JHEP02(2018)118 , D ] 03 ]. Phys. , 05 , ] , JCAP SPIRE , IN Phys. Rev. ]. JCAP , ][ , ]. ]. ]. J. Low. Temp. ]. Phys. Rev. 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