Probing the BSM physics with CMB precision cosmology: an application to supersymmetry

Journal of High Energy Physics, Feb 2018

Abstract The cosmic history before the BBN is highly determined by the physics that operates beyond the Standard Model (BSM) of particle physics and it is poorly constrained observationally. Ongoing and future precision measurements of the CMB observables can provide us with significant information about the pre-BBN era and hence possibly test the cosmological predictions of different BSM scenarios. Supersymmetry is a particularly motivated BSM theory and it is often the case that different superymmetry breaking schemes require different cosmic histories with specific reheating temperatures or low entropy production in order to be cosmologically viable. In this paper we quantify the effects of the possible alternative cosmic histories on the n s and r CMB observables assuming a generic non-thermal stage after cosmic inflation. We analyze TeV and especially multi-TeV super-symmetry breaking schemes assuming the neutralino and gravitino dark matter scenarios. We complement our analysis considering the Starobinsky R2 inflation model to exemplify the improved CMB predictions that a unified description of the early universe cosmic evolution yields. Our analysis underlines the importance of the CMB precision measurements that can be viewed, to some extend, as complementary to the laboratory experimental searches for supersymmetry or other BSM theories.

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Probing the BSM physics with CMB precision cosmology: an application to supersymmetry

JHE Probing the BSM physics with CMB precision cosmology: an application to supersymmetry Ioannis Dalianis 0 1 3 Yuki Watanabe 0 1 2 Standard Model 0 Gunma College , Gunma 371-8530 , Japan 1 15780 Zografou Campus , Athens , Greece 2 Department of Physics, National Institute of Technology 3 Physics Division, National Technical University of Athens The cosmic history before the BBN is highly determined by the physics that operates beyond the Standard Model (BSM) of particle physics and it is poorly constrained observationally. Ongoing and future precision measurements of the CMB observables can provide us with signi cant information about the pre-BBN era and hence possibly test the cosmological predictions of di erent BSM scenarios. Supersymmetry is a particularly motivated BSM theory and it is often the case that di erent superymmetry breaking schemes require di erent cosmic histories with speci c reheating temperatures or low entropy production in order to be cosmologically viable. In this paper we quantify the e ects of the possible alternative cosmic histories on the ns and r CMB observables assuming a generic non-thermal stage after cosmic in ation. We analyze TeV and especially multi-TeV supersymmetry breaking schemes assuming the neutralino and gravitino dark matter scenarios. We complement our analysis considering the Starobinsky R2 in ation model to exemplify the improved CMB predictions that a uni ed description of the early universe cosmic evolution yields. Our analysis underlines the importance of the CMB precision measurements that can be viewed, to some extend, as complementary to the laboratory experimental searches for supersymmetry or other BSM theories. Cosmology of Theories beyond the SM; Supergravity Models; Supersymmetric - BSM physics with CMB precision entropy production Gravitino dark matter Neutralino dark matter Axino dark matter CMB observables and the post-in ationary evolution 2.1 The shift in the scalar spectral index and tensor-to-scalar ratio due to late Supersymmetric dark matter cosmology The ns and r predictions for particular supersymmetry breaking ex 5.3 Distinguishing the R2 and the R2 supergravity in ationary models 6 Discussion and conclusions 1 Introduction 2 3 3.1 3.2 3.3 4.1 4.2 4.3 4.4 5.1 5.2 4 Alternative cosmic histories and supersymmetry Low reheating temperature Late entropy production The diluter eld X The maximum possible dilution due to a scalar condensate very early universe period, that can be called dark pre-BBN period. On the other hand, in ation that takes place at energy scales much higher than the BBN gives concrete predictions thanks to the presence of the quasi-de Sitter horizon. It is actually the dark pre-BBN cosmic phase that introduces an uncertainty at the in ationary predictions parametrized by the number of e-folds N . This uncertainty could be minimized if the physics that operates beyond the Standard Model of particle physics (BSM) was known. Indeed, di erent BSM { 1 { scenarios often imply a di erent cosmic evolution in order to satisfy the BBN predictions and the observed dark matter abundance DMh2 = 0:12 [1, 2]. The fact that the N is modi ed by the details of the dark pre-BBN stage [3] motivate us to investigate this small but non-zero residual dependence of the in ationary predictions on the tentative BSM physics. In most of the in ationary models, a precise measurement of the spectral index ns(N ) and tensor-to-scalar ratio r(N ) value accounts for an indirect measure of the reheating temperature of the universe [4{13] and hence one could in principle examine the cosmology of theories beyond the Standard Model of particle physics as well as non-trivial extensions of the Einstein gravity [14]. From the in ation phenomenology point of view, for a given concrete BSM scenario a predictive in ationary model can be spotted on the (ns; r) plane, whereas from the particle physicist point of view, for a given predictable in ationary scenario the precise measurement of the (ns; r) observables is a measurement of the BSM e ects on the cosmic evolution. In other words, we can say that the (ns; r) precision measurements provide us with a cosmic selection criterion for the assumed BSM physics. Planck collaboration has constrained the spectral tilt value of the curvature power spectrum and the tensor-to-scalar ratio at ns 1 = 0:032 0:006 at 1 and r < 0:11 at 2 respectively [1, 2]. The current resolution of the temperature and polarizartion anisotropies of the CMB probes, although unprecedented, has not been powerful enough to support or exclude the di erent BSM physics schemes. There are promising prospects that the proposed next generation CMB experiments, such as the LiteBIRD [15], Core+ [16], CMB-S4 [17], PRISM [18], PIXIE [19], will improve signi cantly on this direction. The sensitivity forecasts for ns and r is of the order of 10 3 and such a measurement will account for a substantial leap forward at the observational side. We aim at this work to show how one can systematically extract non-trivial information about the BSM physics via the CMB precision measurements. We mostly focus on the supersymmetry since we consider it as a compelling BSM theory that remains elusive from the terrestrial colliders. A precise knowledge of the (ns; r) values can indicate us the duration of non-thermal phase after in ation and in this paper we use this information to examine whether di erent supersymmetry breaking schemes can t in this picture of the early cosmic evolution. From the experimental side, there is no signal that supports the supersymmetry hypothesis until today, see e.g. a recent analysis of searches at the LHC [20, 21]. The absence of signals arouses increasing concern that supersymmetry does not fully solve the hierarchy problem suggesting that supersymmetry, if realized, may lay at energy scales much higher than the TeV scale. Multi TeV supersymmetry implies that the Large Hadron Collider (LHC) at CERN may nd no BSM signal and the ducial BSM physics scenarios will remain elusive for an unspeci ed long time. However from the telescopic observational side, the increasing sensitivity of the CMB probes has opened up a rich phenomenological window to the ultra high energy scales of cosmic in ation and indirectly to the dark pre-BBN period. De nitely, the idea that the CMB studies may probe energy scales well above the TeV is not a new one. There are numerous of seminal works in the literature that examine the impact of BSM physics, and in particular supersymmetry, on the CMB power spectrum mainly either from the in ationary model building or from the dark matter perspective. { 2 { However, successful in ation models can be consistently embedded into a supergravity framework often without any change in the in ationary dynamics since the in ationary trajectory may remain intact by the presence of additional supersymmetric elds that are e ciently stabilized. Moreover, it is often the case that studies of supersymmetric dark matter cosmology focus on the dark matter density parameter tting, neglecting other features of the scalar power spectrum. DMh2 = 0:12, The degeneracy between supersymmetric in ation models and with their nonsupersymmetric versions in terms of the ns(N ) and r(N ) observables can break due to the di erent post-in ationary evolution. The thermal evolution of a supersymmetric plasma is in general much di erent when supersymmetry is realized in nature [22]. Actually, the null LHC results push the sparticles mass bounds to larger values that spoil the nice predictions of the thermal dark matter scenario [23]. Therefore, assuming that the LSP is part of the dark matter in the universe the LSPh2 . 0:12 constraint reconciles only with particular radiation domination histories which may greatly di er to the simple scenario of a single and smooth radiation phase after the in aton decay. An interesting point, that stimulates this work, is that the features of the radiation dominated phase depend on the details of the supersymmetry breaking patterns. In order to extract information about the BSM supersymmetric scenarios from the (ns; r) precision measurements we utilize existing results on supersymmetric cosmology aiming at an analysis based on assumptions as minimal as possible. We consider that the MSSM plus the gravitino is the necessary minimal set-up that gives the most conservative results. We a priori consider the Trh and the supersymmetry breaking scale as unknown quantities. We estimate the neutralino and gravitino LSP abundances by scanning the sparticle mass parameter space. As a rule of thumb we adopt the classi cation of quasinatural, split and high scale supersymmetry when we scan the possible energy scales of supersymmetry breaking. As expected, see e.g. [24{26], we nd that most of parameter space of supersymmetric theories yields an excessive dark matter abundance. Our perspective in this work is that the parameter space that yields an excessive dark matter abundance should not be faced as a cosmologically forbidden one but, on the contrary, as a parameter space that favours a di erent cosmic history for the very early universe. Namely, excessive LSP abundance implies either a low reheating temperature after in ation or low entropy production. Both cases have a non-trivial impact on (ns; r) observables, see e.g. [27] for a relevant analysis on non-thermal neutralino dark matter and [28] for a recent analysis on FIMP dark matter. Departing from the minimal eld content analysis, i.e. the MSSM, the overabundance problem in general deteriorates. Indeed, the dark matter abundance receives contributions from the perturbative and non-perturbative decay processes of the in aton eld [29] and from thermal scatterings, thermal and non-thermal decays of elds coming from the supersymmetry breaking sector such as the messengers. Extra elds can however decrease the DM abundance if they decay late and dominate the energy density of the early universe e.g. due to coherently oscillating scalars or scalars that cause thermal in ation. Such elds are rather common and well motivated in many BSM schemes such as supersymmetry; common examples are the moduli, supersymmetry breaking elds, the saxion, etc. Here { 3 { we collectively label X any of this sort of scalars and explicitly refer to it as diluter, since what we actually measure on the CMB is the diluter impact on the expansion history. In our analysis, the diluter is the only eld beyond the MSSM and gravitino that we consider. Finally, in order to perform a complete calculation of the spectral index value we consider the Starobinsky R2 in ation model and we compare the R2 in ation and R2 supergravity in ation predictions by taking into account the e ects of the post-in ationary phase. Apparently one cannot exclude or verify supersymmetry by ns and r precision measurement, nevertheless one can indeed support the presence of BSM physics or, to put it di erently, rule out the so-called BSM-desert hypothesis for a particular in ation model. This is a minimal but undoubtedly an exciting possibility given the fact that terrestrial colliders probe only a small part of the vast energy scales up to the Planck Mass, MPl, and supersymmetry or any other BSM scale may lay anywhere in between. It is also exciting to note that the terrestrial experiments, such as colliders and direct detection experiments, are sensitive to low scale supersymmetry whereas the CMB observables are more sensitive to high scale supersymmetry. Hence precision cosmology can o er us complementary constraints to the parameter space of the supersymmetric theories. This prospect, though very challenging, is actually a feasible possibility. The organization of the paper is the following. In section 2 we parametrize the uncertainty in the ns and r values coming from the unknown value of N due to the dark pre-BBN era. We compute the shift in the spectral index and tensor-to-scalar ratio with respect to the dilution magnitude in a general BSM context. In section 3 we overview key results of neutralino, gravitino and brie y the axino cosmology regarding the LSP yield, that are necessary for the estimation of the dilution magnitude. In section 4 we analyze the implications of various supersymmetry breaking patterns to the early universe cosmology and examine the features of the possible alternative cosmic histories. In section 5 the Starobinsky R2 in ation is used as a speci c example to demonstrate a full computation of the spectral index and tensor-to-scalar ratio shift. A comparison between the theoretical predictions of the R2 and supergravity R2 in ation is also performed. In the last section we outline the main idea and the method proposed in this work and we comment on the future theoretical and observational prospects. 2 CMB observables and the post-in ationary evolution It is convenient to expand the power spectra of the dimensionless curvature perturbation as PR(k) = As k k ns 1+(1=2)(dns=d ln k) ln(k=k )+(1=6)(d2ns=d ln k2)(ln(k=k ))2+::: where As is the scalar amplitude and the powers of the expansion are the scalar spectral index ns, the running and the running of the ns. In general one can assume that the scale dependence of the spectral index to be given at leading order by the expression ns(k ) = 1 N ; { 4 { (2.1) (2.2) where N is the number of e-folds remaining till the end of in ation after the moment the pivot scale k exits the Hubble radius, N Rttend Hdt = ln(aend=a ). The N is a critical quantity that determines the ns value. It carries the information of how much the observable k 1 CMB scale has been stretched since the in ationary era. The uncertainty on the N comes mainly from the post-accelaration stage and induces an uncertainty on the spectral index value given by the ns running that for the eq. (2.2) reads ns = N N 2 = (1 ns)2 N : k a0H0 = a aend aBBN aeq H aend aBBN aeq a0 Heq H0 Heq ; For which exited the Hubble radius H 1 during in ation, to the size of the present Hubble ns is of size O(1 10)h , that is within the accuracy of the future HJEP02(18) where the subscripts refer to the time of horizon crossing ( ), the time in ation ends (end), the time BBN takes place (BBN), the radiation-matter equality (eq) and the present time (0). We de ne N~dark the number of e-folds from the end of in ation until the beginning of the BBN N~dark ln aBBN aend 1 3(1 + wdark) ln end ; BBN where wdark stands for the average value of the equation of state parameter during the dark pre-BBN period, and wdark 6= 1 has been assumed. We call this period dark due to the lack of observational evidences of the transition to the radiation dominated phase from the super-cooled conditions during in ation. Unless exotic forms of matter are assumed, such as thermal in ation or sti uid domination, we can estimate the maximum value of the N~dark to be around 56 for wdark = 0 and the minimum to be around 41 for wdark = 1=3. The observational uncertainty for temperatures T & 1 MeV TBBN [30] implies an uncertainty at the e-folds of in ation about N 15. We can split the N~dark into (2.3) (2.4) (2.5) (2.6) (2.7) N~dark = N~rh + N~X + N~rad where N~rh = ln(arh=aend) stands for the e-folds number of the postin ationary reheating period until the complete decay of the in aton, N~rad the e-folds number of the radiation dominated era that preceded the BBN and N~X stands for the e-folds number that take place during the domination of an arbitrary X eld in the period after the decay of the After plugging in the value for the ratio aeqHeq=(a0H0), the relation (2.4) is recast in aton and before BBN. into [2] N 66:7 ln k a0H0 + ln 1 4 V 2 MP4l end 1 4 3wdark N~dark : { 5 { Utilizing the relation PR(k ) = V =(24 2 the ratio k =(a0H0) we get MP4l) = As and after substituting numbers for N value ln(1010As) = 3:089 [1, 2]. We also introduced the the uncertainty of the dark pre-BBN era on the N value, = 0:002Mpc 1 and the measured Ndark factor to mark explicitly Ndark 1 4 3wdark N~dark = Nrh + NX + Nrad : (2.9) end rh ; NX = 1 3wX 12(1 + wX) ln dom X dec X ; (2.10) where dec is the energy density of the thermal plasma right after the decay of the scalar X. In principle, for a concrete and predictable in ationary model the wrh and the reheating temperature after in ation can be estimated and hence the Nrh. The crucial quantity is the decay rate inf of the in aton which determines the reheating temperature. Assuming that the decay and the thermalization occur instantaneously at the time in1f then the reheating temperature is found by equating (and omitting order one coe cients) inf = H, Trh = 2 90 g rh 1=4 p infMPl : (2.11) in a quadratic potential and X that causes thermal in ation. The maximum temperature possible is achieved in the instant reheating scenario. Apparently when Trh = Tmax = e1n=4d(30= 2 g rh)1=4 it is Nrh = 0. Note that the N has a logarithmic dependence on g rh, with g rh being the e ective number of relativistic species upon thermalization. It is however well possible that after the in aton decay the evolution of the universe could have been episodic with additional reheating events after in ation. Hence the cosmic thermal era could have started after the last reheating stage before primordial nucleosynthesis caused by other than the in aton scalar eld, for instance a modulus or a aton [31] that we collectively label X. Here, we prefer to remain agnostic about the identity of X but we do utilize its property to cause e cient dilution and low entropy production. The X can dominate the energy density of the universe over radiation due to the slower redshift of its energy density stored. It is X / a 3 for a scalar condensate that coherently oscillates constant for a scalar eld with su ciently at potential 2.1 The shift in the scalar spectral index and tensor-to-scalar ratio due to late entropy production Let us now estimate the impact of the X domination era on the spectral index value. We call N (th) and n(sth) the thermal reference values, that is the e-folds number and the spectral { 6 { (2.12) (2.13) (2.14) index values respectively if there is no late entropy production after the in aton decay, i.e. dilution e ects. It is at leading order where, following eq. (2.8), N = N (th) NX ; n(th) = 1 s =N (th) ; N (th) = 60:8 + ln (th) + ln 1 4 1 4 V (th) end Nrh : At leading order the scalar tilt is generally given by the equation (2.2). Since precision is expected to increase in the future it is worthwhile to consider next-to-leading corrections. Due to the large number of in ationary models [32] there is no common form for the nextto-leading term [33]. A phenomenological way to parametrize it is based on the large N expansion (N ) are determined only after a particular in ation model is considered. In principle the parameter can also be a slowly varying function of N [38]. In addition the expansion (2.14), for some in ation models, may involve parameters of the potential [33]. Here we assume that is a constant and absorb possible complicated behaviors in the arbitrary (N ) function. In section 5 we will explicitly estimate the shift in the spectral index for the Starobinsky R2 in ation model where the parameters and (N ) have a particular form. If NX 6= 0, after Taylor expanding the ns N (th) ns(N (th)) value is shifted by an amount ns ns n(sth), ns = 1 N N(thX) + NX N (th) 2 + NX N (th) 3# + F (N )=N 2jN=N(th) and NX , the spectral index n(sth) = NX ; N (th) (2.15) F NX ; N (th) = 0N N 3 NX + 2 +3 0N + 3 1 00N 2 1 18 0N + 000N 3 4 1 00N 2 N X3 N 5 N X2 + N 4 N=N(th) : (2.16) The \000 denotes d=dN and expressions, given than ; 0, 00, 000 are estimated at N = N (th). In the above NX > 1 and NX =N (th) < 1, terms of order O N X4 =N 6 and smaller have been neglected. We have also assumed that the terms in the parentheses in eq. (2.16) are roughly of order . Otherwise, if 0; 00; 000 1, the F correction can be important, however such a behavior is not found in any of the known universality classes [35]. One can see that the next-to-leading correction accuracy and for NX > the contribution to the spectral index shift is found to be (N )=N 2 is at most of h subdominant with respect to the -dependent terms. { 7 { In order to specify the NX , elements of the X scalar cosmic evolution have to be speci ed. When the scalar X coherently oscillates about the minimum of a e ectively quadratic potential it is wX = 0. In such a case, at the cosmic time tdXom X1 the energy density of X is larger than that of the plasma and the universe enters a scalar dominated era that dilutes any pre-existing abundances of the relativistic degrees of freedom at the time of the X decay. The X eld decays and reheats the universe with temperature T rh X TXdec. Considering instant decay of the scalar X, the dilution magnitude is estimated to be DX 1 + Sbefore Safter = 1 + gs(TXdec) g (TXdom) TXdom g (TXdec) gs(TXdom) T dec ' X T dom X T dec X 1 (2.17) where Sbefore and Safter denote the entropy density right before and after the decay of the X eld. The g and gs count the total number of the e ectively massless degrees of freedom X for the energy density and entropy respectively and can be taken to be approximately equal. The T dec is the temperature that the X scalar reheats the universe at the time where we considered that wX = 0. After plugging in the dilution magnitude we get NX = ln 1 3 NX NX = ns = 1 The maximum value of the 15 is achieved when N~rh ! 0 and N~rad ! 0. This case corresponds to the maximum dilution scenario where the X eld oscillations dominate the energy density of the universe right after the end of high scale in ation until the onset of BBN. The NX = 0 case corresponds to an uninterrupted radiation phase following the post-in ationary reheating. If someone assumes the presence of X matter with exotic barotropic parameter the NX limit values can be extended. Substituting NX = 13 ln D~ X in the expansion (2.15) we obtain the shift in the spectral index, with accuracy j nsj=ns . 1h, due to a post-in ationary dilution of the thermal n(th) 2 s 3 ln D~ X 4 (N )jN=N(th) , 0 = 0(N )jN=N(th) and ns)N 2] 1jN=N(th) . Notice that at leading order the (2.20) reads n(sth))2 = is the running of the spectral index at N (th). We also ns = mention that the three last terms in the brackets of the above equation can be neglected without signi cant cost in the h accuracy. Plugging in the thermal reference value n(sth) that a given in ation model yields, the expression (2.20) returns the shift in the spectral index due to a pre-BBN dilution of { 8 { (2.18) (2.19) 0 3 5 ; (2.20) the thermal plasma. We see that the ns is negative which means that the spectrum tilt becomes more red when dilution of the radiation plasma takes place; this behavior is illustrated in gure 1. The precision of the expression (2.20) is su ciently good, i.e one per mile, even for the extreme case NX 15 or DX Apart from scalar condensates, several BSM construction, mostly supersymmetric ones, predict the presence of singlets under the Standard Model symmetries that have a relatively at potential. Such elds can realize the thermal in ation scenario and are generally called atons [31]. Due to Yukawa interactions the aton can be trapped at the origin of the eld space by thermal e ects. At some temperature that we denote T dom the vacuum energy V0 of the aton dominates over the background radiation energy density and a period of thermal in ation starts. Thermal in ation ends at the temperature T2 when the thermal trap has become too weak and the aton eld starts oscillating about its zero temperature minimum. The aton nally decays at the temperature that we denote T dec and we consider instant reheating. The dilution magnitude due to the aton X domination is DXFD ' 1 + (TXdom)4 T23TXdec ' (TXdom)4 T23TXdec : Respectively the NX value for aton domination is NX jFD = ln " g1=4(TXdom) TXdom # g1=4(T2) T2 + ln 1 3 " g1=4(TXdom) TXdom # g1=4(TXdec) T dec X 1 3 ln D~ XFD ; N~TI and each term can be written in a compact form NX jFD = NX jTI + NX SC, where TI and SC stand for thermal in ation and scalar condensate respectively. The ratio of the relativistic degrees of freedom accounts for a small correction and one can see that it is actually NX jFD ' ln DXFD=3. The dilution magnitude maximizes when the thermal in ation phase is followed by a scalar condensate domination phase, e.g. when the aton eld decays very slowly. The N due to thermal in ation has an upper bound 10 in order that the cosmological density perturbation remain intact. Nevertheless, the dilution magnitude can be many orders of magnitude larger than the dilution caused by a scalar condensate domination (2.17) and it can e ciently dilute any overabundant relic such as dark matter particles. Essentially, the shift in the spectral index due to thermal in ation, with the ln D~ XFD, see Fig 1. It is remarkable that the shift in ns due to period of thermal in ation can resurrect ruled out in ationary models such as the minimal hybrid in ation nsjFD, is given again by the eq. (2.20) simply by replacing the ln D~ X in supergravity [36]. Finally, let us comment on the shift in the tensor-to-scalar ratio. The phenomenological parametrization of the scalar tilt ns = 1 =N implies that the rst slow roll parameter = H_ =H2 writes [38] where A an integration constant coming from the di erential equation +d ln =dN = At rst order in slow roll we have r = 16 and the shift in the tensor-to-scalar ratio due to 40 TI Plateau 45 SC 40 condensate domination (SC) and due to thermal in ation (TI) for the Starobinsky R2 in ation (left panel), general plateau and linear in ationary potentials (right panel). The maximum number of the dilution is given by the ratio Trh=TBBN for scalar condensate domination and the constraint for thermal in ation. The red dots show the e-folds number if there is no entropy production after infaton decay. It is N (th) ' 54 for R2 in ation and N (th) ' 56; 57 for the general plateau and linear potential respectively (red dots). Order O(1) corrections to the dilution magnitude are expected due to the uncertainty at the number of the relativistic degrees of freedom NX jTI . 10 at ultra high energies. a non-thermal phase is NX = ln D~ X =3, either due to a scalar condensate domination or thermal in ation, the relation r = r(D~ X ) is obtained. The scaling (2.23) depends on the potential that implements in ation. Di erent potentials yield di erent values for and A. Moreover, accuracy of order r 10 4 requires to go beyond the approximate relation r = 16 and consider corrections at second order in slow roll. In section 5 we will explicitly estimate the r for the Starobinsky R2 in ation model with the next-to-leading order corrections taken into account. The general conclusion is that, according to eq. (2.24), a non-thermal phase with wX < 1=3 and duration N~X = [(1 Summarizing, the duration of a non-thermal phase is encoded in the number of e-folds N between the moment a relevant mode exits the horizon and the end of in ation. If the radiation domination era, where w = 1=3, initiates at the moment of the complete in aton decay and continues without break until the BBN epoch then the e-folds number, called here thermal e-folds number N (th), can be explicitly determined by the dynamics and the full interactions of the in aton eld. If not, a non-thermal phase changes the aforementioned efolds by the amount NX ln DX =3. A dilution of size DX = 20 translates into NX 1 and a prolonged dilution e.g. of size DX = 1013 into NX 10. In order to estimate the shift in the spectral index and the tensor-to-scalar ratio one has to know the N (th) that is given by the eq. (2.13). This is possible only after an in ationary model and the parameters describing reheating are chosen. Then from eq. (2.14) the n(sth) = ns(N (th)) and the ns = ns(N (th) NX ) can be estimated and hence the spectral index shift ns, given by the eq. (2.15) or (2.20), is obtained. In gure 1 we illustrate the shift in the spectral index due to a non-thermal phase that is implemented after reheating and before BBN. In the left panel we considered the Starobinsky R2 model that predicts Trh 109 GeV [37], and in the right panel a Starobinsky-like potential with non-gravitational interactions and a linear potential V / edge of these in aton features enables the explicit calculation of the n(sth) value, that corboth characterized by a ducial reheating temperature Trh = 1012 GeV. The knowlresponds to the red dots in the plots. A scalar condensate domination or thermal in ation shifts the spectral index value according to the formula (2.20) as illustrated in the gure 1. From a more bottom-up approach, the postulation of a non-thermal phase during the pre-BBN era is not enough to determine the r. Although a rough estimation of the spectral index shift can be done by the approximate expression (2.3) the result is far from accurate and cannot consistently constrain the early universe cosmic history. The best method is to choose an in ation model that is in accordance with a particular BSM description of the early universe (e.g. a supersymmetric, stringy or modi ed gravity framework) and estimate the r according to the pre-BBN cosmology implied by the BSM theory at hand. Examples of BSM cosmic processes connected with the expansion history of the universe are the dark matter production and the baryogenesis processes. In the following we will consider the supersymmetric BSM scenario and determine features of the pre-BBN cosmology, such as possible non-thermal stages, that allow the accommodation of di erent supersymmetry breaking schemes assuming that the LSP is part of the dark matter in the universe. We will estimate the minimum dilution size dictated by the requirement LSPh2 0:12 and determine the expected shift in the spectral index and tensor-to-scalar ratio when a particular in ation model, which in section 5 is the Starobinsky R2 model, complements the description of the early universe evolution. 3 Supersymmetric dark matter cosmology In the previous section we computed the shift in the spectral index and tensor-to-scalar ratio due to post-in ationary entropy production. The fact that the present universe contains dark matter with relic density DMh 2 = 0:12 relates the amount of the dilution with the dark matter production. In this section, focusing on TeV and especially multiTeV supersymmetric scenarios, we will overview the expected LSP yield. We will stress out that the dilution is generally required, hence the CMB in ationary observables should be non-trivially in uenced by the post in ationary expansion history of the supersymmetric universe. There are several fundamental theoretical reasons to believe that supersymmetry is a symmetry of nature. For the devotee of supersymmetry the central question is the scale that supersymmetry is realized. The direct superpartner LHC-limits for all colored sparticles exceed 1:5 TeV and suggest that we should depart from scenarios with natural supersymmetry paying the price of pushing the amount of tuning at the MSSM to less than 0:5 1 percent level. However, the absence of BSM signals in the LHC rules out only the electroweak scale supersymmetry and not supersymmetry in general. BSM physics scenarios with unnatural supersymmetry are still very appealing. Gauge coupling uni cation, the presence of a stable dark matter particle, the possible baryogenesis processes and the stringy UV completion of the low energy theories do not link SUSY with the electroweak scale. Supersymmetry may appear at higher energy scales. In ref. [39], di erent supersymmetry breaking scenarios have been categorized according to the mass spectrum features into three representative cases: i) Quasi-natural supersymmetry, in which supersymmetric particles are heavier than the weak scale, but not too far from it, about in the 1 30 TeV range. ii) Split supersymmetry, in which only the scalar supersymmetric particles have masses of the order of m~, while gauginos and higgsinos are lighter, possibly with masses near the weak scale [40{42]. There are also the Mega-Split [43] or MiniSplit [44] scenarios. iii) Finally the High-Scale supersymmetry, see e.g [45, 46] in which all supersymmetric particles have masses around a common scale m~, unrelated to the weak scale. The m~ is constrained by the Higgs mass value according to the details, of each supersymmetry breaking scenario. Roughly in the Split supersymmetry the maximum value allowed for m~ is 108 GeV when tan is small, while in the High Scale supersymmetry the m~ value can be up to 1012 GeV. For our analysis it is critical that the LSP is stable. The stability of the LSP dark matter is assured by the presence of a discrete symmetry of the supergravity Lagrangian, the R-parity. If the R-parity is violated then the cosmological constraint raised for the LSP. Although R-parity violating models have been actually constructed and have interesting phenomenological implications [47], there are strong arguments based on GUT models that support the R-parity conservation even when the scale of supersymmetry breaking is well above the electroweak scale [48]. These results motivate us to assume that the LSP lifetime is much larger than the age of the universe and thus the LSP is constituent of the dark matter. Given the supersymmetry breaking scheme the stability of the LSP puts strong constraints on the thermal history of the universe. In the following subsections we overview the basic relevant cosmological aspects and results of the gravitino and neutralino LSP scenarios necessary for the goals of our analysis. 3.1 Gravitino dark matter The gravitino is the supersymmetric partner of the graviton in supergravity and it can acquire a mass in the range of O(eV m~ ). The gravitino is naturally the LSP in gauge mediated supersymmetry breaking models (GMSB), see [49] for a review, and possibly it is the LSP in Split and High scale supersymmetry frameworks. The relic density of the gravitinos 3=2h2, which can be thermal or non-thermal, receives contributions from many sources. Thermal gravitinos (freeze out): from scatterings (i) with the MSSM plasma, (ii) with the thermalized messenger elds. Non-thermal gravitinos (freeze in): From (i) thermal scatterings in MSSM and messengers plasma, (ii) decays of sfermions and the NLSP, (iii) decays of the messenger elds, (iv) perturbative and non-perturbative decay of the in aton eld, (v) decay of moduli elds. L (a) -10 0 -5 2 5 m 3 2 > m g Ž 2 6 8 Ž 8 0 Density and contour plot of the decadic logarithm of the required dilution for Trh = 109 GeV reheating temperature after in ation and gravitno the stable LSP. In the left panel panel there is a split spectrum with mf~ = 103mg~. Thermal production of helicity degenerate spectrum for the sfermions and gauginos was considered, mf~ = mg~, while in the right 3=2 and 1=2 gravitinos from scatterings in the plasma, non-thermal production from decays of sfermions and the NLSP to helicity 1=2 gravitinos have been taken into account. The contributions to the gravitino abundance have been conditionally added, i.e. in the parts of the contour that thermal equilibrium is achieved the total abundance is replaced by the thermal one. The magnitude of the logarithm of the required dilution is given by the contour numbers onto the density plot. Negative numbers correspond to underabundance, hence to no dilution contour area. The less model independent estimation of the 3=2 is achieved when only the MSSM sector is considered. The gravitino number density n3=2 in the thermalized early universe evolves according to the Boltzmann equation [50]. A key quantity is the gravitino production rate, sc, in scatterings with thermalized Standard Model particles and sparticles sc 0:1 1 + T 6 MP2l mg2~( ) ! 3m23=2 0:1 T 6 MP2l ^sc : The gravitinos obtain a thermal distribution via interactions with the MSSM for Trh > T3f.=o2. 2 1014GeV m3=2=GeV 2 (TeV=mg~3 )2, where mg~3 is the gluino mass evaluated at the reheating temperature, see eq. (3.4). If the reheating temperature is below the T3f.=o2. the gravitino yield from MSSM thermal scatterings is Y3M=2SSM(sc) 10 3 Trh=T3f.=o2. . Furthermore, the heavier MSSM sparticles are unstable and will decay to gravitinos. The decay width into gravitinos is nearly the same for both gauginos and sfermions ~ MSSM(~i ! i G~) ' 48 m23=2MP2l ; i 1 m5 (3.1) (3.2) where ~i = g~; f~. The total MSSM contribution to the gravitino yield is Y3M=2SSM = Y3M=2SSM(sc) + Y3M=2SSM(dec), and the relic density parameter reads MSSM 3=2 The gravitino relic abundance sourced by the MSSM and messenger elds is illustrated in gure 2 and 3. In the case that the gravitino is the only sparticle with mass below the reheating temperature then the gravitino relic abundance is given by a much di erent expression with dependence Apart from particular cases, the above gravitino yield (3.3) cannot be nal because we neglected sources beyond the MSSM. The supersymmetry breaking sector is a necessary ingredient for all the consistent supersymmetric BSM scenarios [49]. In general the extra elds only increase1 the nal 3=2, unless there is a late entropy production. For example, thermalized messengers elds generically equilibrate the gravitinos for broad range of values of the Yukawa coupling at the messenger superpotential, mess & 10 6 10 5 [56], and the relic gravitino density parameter reads (th) 3=2 where the freeze out temperature is here equal to the messenger mass scale, T3f.=o2. Even if mess with 3m=e2ssh2 1 the thermal scatterings of messengers contribute to gravitino relic density 0:4 Mmess=104GeV GeV=m3=2 (mg~=TeV)2. In addition, the in aton perturbative decay produces non-thermal gravitinos with rate [57, 58] ( ! G~G~) ' jG(e )j2 288 m5 m23=2MP2l : Also gravitinos are produced during the preheating stage via its non-perturbative decay of the in aton [59{65], from the decay of the supersymmetry breaking eld, see e.g. [66{68], or other moduli [69{71]. Therefore, the estimation of the gravitino relic abundance based solely on the MSSM sector gives a model independent albeit an underestimated and hence conservative value for the 3=2. The 3=2 result could decrease in the case that extra elds interrupt the thermal phase, e.g. due to the domination of a non-thermal scalar eld that produces entropy at low temperatures. Thanks to the dilution the gravitino cosmologically problematic supersymmetric scenarios may become viable possibilities. The tentative low entropy production is caused by the scalar X that we do not identify and collectively call it diluter. We only assume that 1It is though possible that the supersymmetry breaking sector leads to a suppressed 3=2, e.g due to R-symmetry restoration [53], or a high temperature decoupling of the messenger elds [26], or due to the dynamics of the sgoldstino eld [54], or due to feeble couplings in the supersymmetry breaking sector [55]. (3.3) (3.4) Mmess. (3.5) 9 8 4 3 0 12 8 4 6 14 m 3 2 > m g Ž DV 7 Ž mf 6 Ž g 9 8 4 3 -10 Ž 10 2 2 -5 0 4 LSP and reheating temperature Trh = 109 GeV. In the left panel the gravitinos are thermal (heavy gravitinos can thermalize due to the messenger sector). In the right panel the contribution of messengers plus MSSM is considered, assuming a small enough messenger coupling so that gravitinos do not thermalize by the interactions with messengers, but only due to the MSSM. The messengers scale is taken to be Mmess = 108 GeV. it interacts too weakly with the other elds, e.g. via gravitational interactions. Therefore the gravitino relic density parameter is the conditional sum tot 3=2 ' min n MSSM + 3=2 3m=e2ss + i3n=f2 + 3S=B2 ; (th)o 3=2 (3.6) 3=2 where MSSM is the contributions of the MSSM (scatterings and decays), 3m=e2ss is the contribution of messengers (scatterings and decays), i3n=f2 is the contribution of the infationary perturbative and non-perturbative decay and 3S=B2 is the contribution of the supersymmetry breaking eld. It is called conditional sum because the simple add of each contribution may result in an overestimate of the gravitino abundance. For example the presence of thermalized messengers modi es the gravitino production from the MSSM sector [56, 72]. We mention that the sum (3.6) is not strictly exact: it is well possible that contributions from non-thermal decays, that take place below the T3f=2 temperature, increase the tot 3=2 beyond the (3t=h2) value. renamed X decay. Finally, the presence of a scalar X that produces low entropy modi es the result (3.6) as will be discussed in the section 4. In such a case the density parameter (3.6) value is < 3=2 in order to emphasize that it is sourced by processes taking place before the 3.2 Neutralino dark matter The lightest neutralino ~0 is the most representative example of a WIMP dark matter and an appealing candidate thanks to its main merit: the thermal production mechanism. ing temperature, Trh & 5 mf~ & 2 perature after decay is The thermal neutralino scenario however works best provided that the squark and slepton masses lie in the 50-100 GeV range [24, 73] which has been excluded by collider searches. In addition the direct and indirect detection experiments shrink the parameter space of the neutralino with mass about the electroweak scale [23]. Speci c neutralino types, such as the higgsino, see e.g. [74], or the annihilation mechanism can be invoked to match the ~0 h2 to data, but in general a rather heavy neutralino cannot be a viable thermal relic. The neutralino ~0 decouples from the thermal bath at a freeze-out temperature T ~f.0o. = m ~0 =xf , where xf ' 28 ln(m ~0 =TeV) + ln(c=10 2), where c=m2~0 = relativistic ~0 annihilation cross section. In scenarios with split spectrum it is c = 3 for a mostly higgsino ~0 and c = 10 2 for mostly wino ~0 [42]. If the reheating temperature is larger than T ~f.0o. the neutralinos reach thermal and chemical equilibrium and the relic the nondensity parameter is UV insensitive and depends on the ~ 0 mass squared, (th) ~0 / m2~0 . When the sparticles masses lay well above the TeV scale the thermal neutralino scenario is disfavored and ususally non-thermal production scenarios are considered, e.g ~0 production via the decay of heavy gravitinos. The gravitinos, that are unstable, are produced via thermal scatterings, non-thermal decays of sfermions and possible decays of scalars beyond MSSM such as the in aton [57{65] and the supersymmetry breaking eld or other moduli [69{71]. Focusing on the MSSM sector the gravitinos dominate the universe either for large enough reheat1014(m3=2=105GeV)1=2 GeV, or large enough sfermion masses, 108(m3=2=105GeV)5=6 [42]. The gravitinos decay when 3=2 = H and the temT3d=e2c = 6:8 m3=2 105GeV 3=2 " 10:75 g (T3d=e2c) #1=4 MeV : (3.7) m ~0 m3=2 Thus one nds ~0 Apparently, it has to be m3=2 > 104 GeV to avoid BBN complications [75, 76]. The 4He abundance implies that it must be Y3=2 . 10 12; for m3=2 = 10 30 TeV and for smaller m3=2 values the bound becomes much severer, see e.g. [57, 75, 76] for details. The gravitino decay populates the universe with neutralinos. Heavy enough gravitinos, m3=2 decay promptly so that T3d=e2c > T ~f.0o. and the neutralinos reach a thermal equilibrium. In the opposite case, the neutralinos produced by the graviton decay are out of chemical equilibrium and either have a yield Y ~0 Y3=2 for a radiation dominated universe, or Y ~0 ' 3T3d=e2c=(4m3=2) for a gravitino dominated early universe. Unless the reheating temperature is particularly high Trh > 1014 GeV or the sfermions very massive, mf~ > 108 GeV the gravitinos do not dominate over the radiation, and the neutralino relic density parameter reads 3=2 MSSM(sc)h2 + 3=2 f~(dec)h2 + (~t0h)h2 (Radiation-domination) : (3.8) m ~0 TeV " Trh=2 109GeV + 105GeV m3=2 2 X g i i m ~ fi 107GeV 3 + m ~0 TeV taken to be m3=2 > 105 GeV to avoid BBN constraints. stable LSP. In the left panel the neutralino abundance is the thermal one. In the right the reheating temperature is Trh = 1012 GeV the sfermions are 103 times heavier than the gravitinos and hence the neutralino yield is dominated by the decay of gravitinos produced from sfermion decays; in the right bottom corner of the plot the neutralinos thermalize due to large T3d=e2c. The gravitino mass is universe that is D3=2 neutralinos is where i runs up to N = 46 for sfermions heavier than the gravitino and T3d=e2c < T ~f.0o.. The degrees of freedom at T ~f.0o. were taken to be g = 86:25. If gravitinos dominate the early 1, then the relic density parameter of the non-thermally produced (n-th) ~0 It is possible that the non-thermally produced neutralinos from the gravitino decay achieve a chemical equilibrium for n ~0 ~0 v > H(T3d=e2c). It is ~0 v / 1=m2~0 and H(T3d=e2c) / m33=2, hence for a wino-like neutralino at the TeV scale and m3=2 > 105 GeV pair-annihilation can take place [77]. The neutralinos annihilate until their number density becomes nc~r0it 3H= ~0 v and the relic density parameter is for this case, ~0 (th) ~0 T ~f.0o.=T3d=e2c , that is enhanced by the ratio (T ~f.0o.=T3=2) compared to the thermal abundance. This is an appealing scenario, called annihilation scenario, because the critical value nc~r0it behaves as an attractor and determines the relic abundance of neutralino (mostly wino) LSP, making it independent of the primordial gravitino relic abundance [77]. Nevertheless it hardly works when one departs from the TeV scale neutralino. It is also much constrained from the indirect detection experiments. In the section 4 the non-thermal scenario, that is often-called branching ratio scenario, where the ~0 is produced non-thermally during the low entropy production caused by the diluter X eld will be discussed, and in section 5 we will consider the production of ~0 from the supersymmetry breaking eld aiming at a complete analysis. 3.3 In the sake of completeness of the basic LSP scenarios, we brie y comment here on the axino dark matter. In supersymmetry, the axion solution to the strong CP problem comes with an extra scalar, the saxion and a fermion, the axino ~a. If the axino is the LSP it is a well motivated dark matter candidate [78, 79]. It freezes out at high temperatures T f.o. a~ 1011GeV(fa=1012GeV)2, where fa the axion decay constant. At lower temperatures it can be produced from thermal scatterings and decays. In that case, for a radiation dominated universe, the axino relic density parameter is the sum of the contributions from thermal scatterings, the gravitino decay and the NLSP decays ma~ a~ ' m3=2 3=2 MSSM(sc) + f~(dec) 3=2 + ma~ mNLSP NLSP + a~ MSSM(sc) ; (3.11) for T3d=e2c below the NLSP freeze out temperature. We note that the two body decay of a squark to an axino is subdominant for gluino masses less than squark mass [80]. It is MSSM(sc) a~ 2:8 108(ma~=GeV)Ya~ where Ya~(KSVZ) for the KSVZ axion model, see e.g [81], and Ya~(DFSZ) 10 7(Trh=104GeV)(1011GeV=fa)2 the superpotential Higgs/Higgsino parameter, see e.g [82]. For axino mass not much smaller than the NLSP, the axino dark matter case is quite similar to the neutralino LSP. For ma~ & TeV the axino dark matter is also cosmologically problematic since its relic density parameter generally violates the DMh2 = 0:12 bound, and the essential conclusion is that, in general, a special thermal history of the universe is required for the axino dark matter scenario as well. Remarkably in these models, the saxion can play the r^ole of the diluter X for its condensate decay can produce late entropy that successfully decreases the LSP abundance [83], see also [84] for some recent results on the reheating temperature and the LSP constraint. 4 Alternative cosmic histories and supersymmetry The overview of the predicted relic density of supersymmetric dark matter in section 3 suggests that the observational value of DMh2 gets generally severely violated when the sparticle masses increase. For gravitino and neutralino LSP one can collectively write down a general scaling with respect to the mass parameters and temperature and 3=2 / m3=2 mg~ m3=2 ~0 / m~~0 m3=2 ~ m ~ f m3=2 m ~ f m3=2 ~ ~ Trh ; Trh ; m3=2 < mg~; m ~ ; f m ~0 < m3=2; m ~ f (4.1) (4.2) where the exponents ( ; ; ; ) and ( ~; ~; ~; ~) are either positive or zero, depending on the dark matter production mechanism considered. The predicted supersymmetric dark matter overdensity for \unnatural" supersymmetry can be reconciled with the DMh2 bound if the reheating temperature is rather low or HJEP02(18) late entropy production takes place. Remarkably, both solutions imply that an alternative cosmic history takes place if supersymmetry is a symmetry of nature. By the term alternative cosmic history we mean that the radiation domination phase after in ation was interrupted or delayed by a cosmic era, where a uid X with barotropic parameter wX < 1=3 dominated the energy density of the early universe. As discussed in the introduction and in section 2 such a cosmic era impacts the observable values ns and r. In order to quantify this e ect we consider in our analysis below di erent cosmic histories and di erent supersymmetry breaking schemes. We follow the base line framework of the benchmark supersymmetry breaking scenarios with either gravitino or neutralino LSP and degenerate or split mass spectrum. The scale of supersymmetry breaking, represented by the general sfermion mass m~ , is taken to be from the TeV scale up to the energy scale of the reheating temperature. Low reheating temperature The reheating temperature of the universe after in ation can be rather low if the in aton decay rate, inf, is small enough or if it is the result of the decay of a weakly coupled scalar unrelated to the in aton.2 In this case, the dark matter production due to processes sensitive to the maximum temperature gets suppressed. We call low reheating temperature scenarios those with Trh . 105 GeV. For gravitino LSP the yield from thermal scatterings decreases when the reheating temperature decreases, and the NLSP-decays to gravitinos account for the leading contribution to mNLSP. On the other hand, for neutralino LSP the UV-sensitivity of the 3=2 for m3=2 ~0 to processes that take place at high temperatures is small. The neutralino abundance is IR-sensitive and it is mostly determined at the freeze out temperature T ~f.0o.. Both for gravitino and neutralino LSP, the observational bound generally violated for mLSP > O(TeV) and m~ < Trh. DMh2 = 0:12 can be satis ed for mLSP . O(TeV) or for the particular case that m~ Trh where the Boltzmann suppression may play a critical ro^le. Note that the X domination cosmic phase is a decaying particle dominated phase, hence entropy is gradually produced for X =H < 1, where X the decay rate of the X particle. The maximum reheating temperature is greater that T dec and this has implications for the relic LSP density [85]. If the LSPs reach chemical equilibrium before reheating, the relic LSP energy density, is Apparently, in the MSSM the roughly given by (th) LSP Tr3hTLf.SoP.=(TLf.SoP., new)4 where TLf.SoP., new and TLf.SoP. are the freeze-out temperatures for Trh mLSP and Trh & mLSP respectively [85]. We have also called Trh the X decay temperature, T dec. On the other hand, if Trh X a chemical equilibrium then the relic density has a dependence mLSP and the LSPs never reach LSP / Tr7h [86]. Finally, if the LSPs are produced non-thermally from the X decay and reach chemical equilibrium then the relic density reads (TLf.SoP.=Trh), see [87] for a brief overview on the topic. Note that these scenarios that can reconcile heavy supersymmetry with the observational 2Low reheating temperatures may be caused by the a scalar eld X (or more than one scalar) with relatively long lifetime, X inf that dominated the energy density of the universe before the in aton decay, e.g if the X is frozen during in aton oscillations with the X m2X X2 su ciently large that sources some extra e-folds of X in ation. In this case, the reheating temperature at the expressions (4.1) and (4.2) is Trh = TXdec. Also, going to order 1=N 3 the tensor-to-scalar ratio and running read r = 12 N 2 18 N 3 (2:1 + ln N ) 2 N 2 + 1 N 3 and s = ( 0:68 + 3 ln N ) : (5.9) Plugging N (th) = 54 in eq. (5.8) the thermal scalar tilt value is obtained that is 2h larger than the leading order prediction. We also take at next-to-leading order r(th) R2 = 0:0034 and (th) s R2 = Note that the r value is 17% smaller than the value obtained at leading order. Furthermore, going to accuracy level 1=N 3 the r = r(ns) relation reads r 3(1 ns)2 + ns)3 = 0 : 23 4 (1 the Euler-Mascheroni constant. (19=6) V2 with The eq. (5.12) was obtained from the expressions ns = ns( V ; V ; V ) and r = r( V ; V ) written up to 1=N 3 order. In particular for the Starobinsky model it is ns 2C V2 + O( V3 ) and r = 12 V2 + (8 24C) V3 + O( V4 ) where C If nature is successfully described by the Standard Model of particle physics and the R2 in ation model then the NX has to be zero and hence ns = n(sth). Next we review and estimate the expected ns and r values for the R2 supergravity in ation model. The R + R2 supergravity in ation The embedding of the Starobinsky model of in ation in old-minimal supergravity in a superspace approach consists of reproducing the Lagrangian (5.1). This is achieved by the action [92{96] L = 3MP2 Z d m2 RR + 3m4 R R Modi cations and further properties can be found in [97{110]. We mention that attention should be paid to the full couplings of the in aton eld that may yield a di erent reheating temeprature in each of these models since not all of them are pure supergravitational. The old-minimal supergravity multiplet contains the graviton (eam), the gravitino (G~ = m), and a pair of auxiliary elds: the complex scalar M and the real vector bm. Lagrangian (5.13) when expanded to components yields R2 terms and kinematic terms for the \auxiliary" elds M and bm. One may work directly with (5.13) but it is more convenient to turn to the dual description in terms of two chiral super elds: T and S and standard supergravity [92]. During in ation the universe undergoes a quasi de Sitter phase which implies that supersymmetry is broken, the mass of the sgoldstino S becomes large and it can be integrated out [111, 112]. In this stage a non-linear realization of supersymmetry during in ation is possible [113{116]. The real component of T is not integrated out due to the non-linear realization and it is the only dynamic degree of freedom during in ation [93, 94, 96]. Eventually one nds the e ective model (5.2). (5.10) (5.11) The in ationary predictions for the supergravity R2 model are found to be identical to the non-supersymmetric Starobinsky R2 predictions (5.3). In addition, the reheating phase is much similar and the in aton decay rate roughly the same. Indeed, in the work of [117] the in aton decay channels were identi ed and the branching ratios calculated. The total decay rate was parametrized as sugra-inf = c0m3 =MP2l, where m temperature was estimated to be minf and the reheating TrhjsugraR2 = 90 2g (Trh) 1=4 p sugra-infMPl 109 GeV : (5.14) HJEP02(18) Y3i=n2f = 2m 3Trh Bri3n=f2 The fact that the reheating temperature is found to be about the same with that predicted in the non-supersymmetric R2 model (5.6) means the supergravity and nonsupergravity versions of the R2 in ation models are completely degenerate in terms of the in ationary predictions. However, the details of the expansion history of the universe after the decay of the in aton should break the degeneracy between the supergravityR2 and gravity R2. We can directly apply the analysis and the results of the previous sections by minimally completing the supergravity R2 sector with the MSSM and a basic supersymmetry breaking sector. Let us rst examine the implications of the supergravity R2 in ation to the abundances of superparticles. The R2 supergravity scenario can be distinguished in two basic cases: the ultra high scale supersymmetry breaking m3=2 > m and the sub-in ation supersymmetry breaking scale m > m3=2 case. The st case is realized when the minimum of the in ationary potential breaks supersymmetry. Particularly in the model of [96], where a new class of Rsymmetry violating R + R2 models was considered, it was found that it is possible in ation and supersymmetry breaking to originate from the supercurvature and obtain m3=2 2m without invoking any matter super elds. The new properties of these models which distinguish them from the R-symmetric R2 supergravity is that at the end of in ation the S eld contribution becomes important. In such ultra high scale supersymmetry breaking scenarios the superparticles possibly play no ro^le during the thermal evolution since the reheating temperature may not be su cient to excite the superpartners of the Standard Model particles. Hence, the R2 and supergravity R2 models with m3=2 > m may be totally indistinguishable unless gauginos or some moduli elds are much lighter than the gravitino. In the case that the in aton eld vacuum is supersymmetric an extra eld is required to break supersymmetry, and the condition m3=2 < m is usually satis ed. The supersymmetry breaking spurion eld called Z, i.e. the sgoldstino, although it can play the ro^le of the diluter it overproduces LSPs and an extra scalar that we generically label X has to play the r^ole of the diluter.3 Assuming a simple supersymmetry breaking sector, with WSB = F Z + W0 and KSB = jZj2 the in aton is calculated to be [117] jZj4= 2, the gravitino yield due to the direct decay of 3Any late decaying scalar eld, e.g stringy moduli, can be the diluter eld. (5.15) with branching ratio Bri3n=f2 Br( ! G~G~) ' 48 c0 1 8 >>16 > < > > for mZ (m m3=2)1=2 for (3m3=2m )1=2 mZ m The c0 is determined by the dominant decay channel, here the anomaly induced process [117]. In the case that the spurion eld is heavier than the in aton, m m3=2 < m the branching ratio maximizes, Bri3n=f2 ' (48 c0) 1. Otherwise, the gravitino yield is calculated from the branching ratio (5.16) to be (5.16) mZ , and Y3i=n2f ' 90 2g rh For the supergravity R2 in ation the above contribution to the gravitino abundance is small. Apart from direct gravitino production from in aton decays, gravitinos are produced via the decay of the Z eld. The supersymmetry breaking eld Z is produced as particles by the decay of in aton with branching ratio (5.17) (5.18) (5.19) (5.20) (5.21) Br( ! ZZ) = Considering the generic decay channel the Z decays dominantly into a pair of gravitinos when m3=2 mZ < m with the partial decay rate enhanced by the factor (mZ =m3=2)2, Thus, the gravitino yield as a decay product of particle Z is found to be [117] Y3Z=2(particle) = s 2nZ = 2 2 4m 3Trh Br3Z=2 = m2Z Trh ; 16 c0m3 where Trh the reheating temperature after the decay of the in aton and Br3Z=2 the branching ratio of the Z into a pair of gravitinos. In addition to the incoherent Z particles there are the coherent Z modes, produced by the in ationary de-Sitter phase, which may store a signi cant amount of energy. The precise VEV of Z is rather model dependent. The G~ yield from the decay of the Z condensate can be computed if the initial amplitude of oscillations z0, the Z mass and couplings are known. Assuming a Z dominated universe it is Y3Z=2(cond) = 2 3 TZdec 4 mZ Br3Z=2 ; and gravitinos and the LSPs are generally found to be overabundant. The initial value and zero temperature VEV of the scalar Z eld are rather model dependent and it is possible that the Z scalar does not dominate the energy density of the universe. In the analysis of [117] the scalar Z is trapped near the origin during in ation. The zero temperature VEV, dictated by the Kahler, KSB, and the superpotential, WSB, is hzi = 2p3(m3=2=mZ )2MPl. In the following we assume benchmark sparticle mass patterns and we estimate the corresponding shift in the spectral index and the tensor-to-scalar ratio in order the predicted dark matter density to be in accordance with observations. We assume gravitino and neutralino dark matter scenarios. We generally assume the presence of an extra scalar labeled X that dilutes the LSP abundance at the critical LSPh2 = 0:12 and sub-critical values. Particular hidden sector details concerning the X dynamics are left unspeci ed except for the requirement the diluter not to overproduce LSPs at the time of late entropy production. This is achieved is if the branching ratio to LSPs is very suppressed or mLSP < mX < 2mLSP. The shift in the scalar spectral index and the tensor-to-scalar ratio for the Starobinsky R2 in ation The diluter X eld dominates the energy density of the universe if T dom > T dec where X X TXdom 8 > : >< p3xM0 Pl 1=4 Trh ; 2 320g V0 for scalar condensate for thermal in ation the temperature that the energy density stored in the oscillating X eld gets over the radiation energy density. The x0 is the initial amplitude of the oscillations in a potential V (X) = m2X X2 about the minimum and V0 the vacuum energy of the aton eld X in the case of thermal in ation. The reheating temperature for Starobinsky and supergravity Starobinsky in ation is Trh 109 GeV and the decay temperature of the X eld depends on its full interactions. The non-thermal X eld domination induces a shift in the spectral index value n(sth) = 0:965 due to a change in the thermal e-folds number N (th) = 54. According to the formula (2.20) the size of the shift due to a non-thermal phase that lasts NX = [(1 NX e-folds after the in aton decay for the Starobinsky model is ns = 6:3 2 2 NX 4 X (0:019 p=0 NX ) p 3 0:0535 : (5.22) (5.23) For a scalar condensate domination it is NX = ln D~ X =3. The ns depends on the dilution analyzing ln D~ X = ln DX + g^ the shift in the scalar tilt reads size DX plus a correction g^ due to the change of the number of the e ective degrees of freedom at the temperatures T dom and TXdec. Keeping only the relevant terms and after X ns(DX ; g^) = 2 10 4 (ln DX + g^) 1 + (ln DX + g^) ; (5.24) 2 300 where g^ ln[g (TXdom)=g (TXdec)]=4. 0.966 0.964 ilt0.962 t r c s la0.960 a i.e. g (TXdom) = g (TXdec). in ationary dilution is considered for the Starobinsky R2 in ation model. The solid line corresponds to a change of factor 10 in the number of e ective degrees of freedom in the energy density at the times TXdom and TXdec, i.e. g (TXdom) = 10 g (TXdec), and the dashed line corresponds to no change, The shift in the tensor-to-scalar ratio is found by expanding the expression (5.9) for the r, Substituting N (th) = 54 and NX = ln D~ X =3 = (ln DX + g^)=3 and keeping only the relevant terms, the above expression for the Starobinsky R2 in ation model reads r(DX ; g^) = 3:9 10 5 (ln DX + g^) 1 + 8:2 10 3(ln DX + g^) : (5.26) for ns = n(sth) + ns(DX ; g^). We have veri ed that the value r(th) + r(DX ; g^), that the eq. (5.26) yields, agrees with 10 4 precision with the value one gets from the relation r = r(ns) given by the eq. (5.12) Regarding the e ective degrees of freedom, it is g^ . O(1), hence the change in the number of degrees of freedom requires accuracy at the ns (r) measurement of the order of 10 4 (10 5) and one can safely neglect the g^ correction in the expressions (5.24) and (5.26) since the expected accuracy of the future CMB probes will be of the order of 10 3. Nevertheless observing that, in principle at least, one can additionally determine the number of the e ective degrees of freedom at the thermal plasma from the (ns; r) precision measurement is certainly important and exciting, see gure 6. 5.2.2 The ns and r predictions for particular supersymmetry breaking examples In this subsection we explore the impact on (ns; r) observables of the two base case dark matter scenarios of supersymmetry, the gravitino and the neutralino, when the initial conditions for the hot Big Bang are set by the supergravity Starobinsky in ation. We consider both thermal and non-thermal dark matter production from the hot plasma and scalar decays. We examine di erent and illustrative supersymmetry breaking schemes and we quantify how the expected values for the in ationary observables change due to a nonthermal post-reheating phase dictated by the universal constraint mention that this analysis, that probes cosmologically a BSM scheme, can be applied to any other in ationary model after the appropriate adjustments regarding the reheating phase, the reheating temperature and the in aton eld branching ratios. LSPh2 0:12. We Example I: gravitino dark matter. The gravitino is the LSP if the supersymmetry breaking is mediated more e ciently to the MSSM than to the supergravity sector. The standard paradigm is the gauge mediation scenario [49]. In such a scenario the supersymmetry breaking Z eld decays dominantly into MSSM elds with non-gravitational interactions. Following realistic models [66{68], it is the imaginary part of the Z eld that decays last and the dominant channel is onto a pair of gauginos, in particular binos, with the decay temperature given by TZdec ' 760MeV The LSP gravitinos are produced from thermal scatterings and decays in the plasma and from the non-thermal decay of the Z scalar eld and the in aton. The in aton contribution to the gravitino abundance is given by eq. (5.17) and in general is found to be subleading in R2 in ation. The decay rate of the Z scalar to gravitinos is given by eq. (5.19). If the Z decay produces late entropy then the gravitinos from the Z decay, with branching ratio Br3Z=2 will be part of the dark matter in the universe with yield Y3Z=2 (3=2)Br3Z=2TZdec=mZ and relic density parameter [67] Z 3=2 mg~ 2 1 4 mg2~ ! 1=4 m2Z where mg~ the mass of the bino. We mention that it is also possible that the spurion eld does not dominate the energy density due to thermal e ects [56, 118]. Before proceeding with the survey of particular examples, let us mention that the gravitino relic density parameter violates the observational bound unless the sparticles lay in the TeV and sub-TeV scale. Another scalar eld X is required to dilute the thermally produced gravitinos and the energy stored in the oscillations of the supersymmetry breaking eld, in case of Z domination. In order the precise dilution size to be determined the knowledge of the mZ , m3=2 and the MSSM mass pattern is necessary. Let us now consider four benchmark mass patterns for the supersymmetry breaking sector plus the MSSM, with di erent sizes of supersymmetry breaking scale. We also consider the presence of messenger elds and the diluter X eld necessary to decrease the LSP relic density and which dominantly decays to visible sector elds and not to gravitinos. 1. m3=2 ' 102 GeV; mf~ mZ ' 104 GeV and Mmess ' 108 GeV. The messengers get thermalized since Mmess < Trh and the scalar spurion eld Z follows the nite temperature minimum without sizable oscillations and hence does not dominate the energy density [56, 118]. We also assume that the messenger coupling is small enough, mess 1, so that the gravitinos do not get thermalized. The gravitinos produced from scatterings of thermalized messengers would have a relic density parameter 3<=2h 2 104, see below eq. (3.4). The 3=2h2 index and tensor-to-scalar ratio are respectively j nsj & 2 10 3 and thermally produced gravitinos are su ciently diluted if DX & 104. This dilution can be caused by scalar condensate X with DX ' T dom=TXdec. The shift in the spectral X r & 4 10 4. 0:12 bound implies that the mg~ mZ > m3=2 and Mmess < Trh. Messengers get thermalized and Z does not dominate the energy density of the universe. The gravitinos obtain a thermal equilibrium abundance due to interactions with the thermalized messengers [56] and their relic density would be 3<=2h 2 1010, see 0:12 bound implies that the thermally produced gravitinos eq. (3.4). The 3=2h2 are su ciently diluted if DX & 1010. The diluter can be either a aton that causes thermal in ation or a scalar condensate. In the later case the X dominates the energy density of the universe shortly after the reheating in order such a dilution size to be realized. The shift in the spectral index and tensor-to-scalar ratio are respectively j nsj & 5 mZ ' 106 GeV and Mmess > Trh. The Z eld does not receive thermal corrections because the messengers are not thermalized. The Z scalar oscillations generally have a large enough amplitude and Z does dominate the energy density of the universe. Equations (5.27) and (5.28) say that the spurion Z decays at T dec Z ' 1 GeV and produces non-thermally gravitinos that exceed about 106:5 times the observational bound. In order the Z condensate to get diluted the X eld has to be a aton and cause thermal in ation. In this case, the shift in the spectral index and tensor-to-scalar ratio are respectively eld eld mf~ mZ = few TeV. There are scenarios in the literature that reconcile gravitino cosmology with high reheating temperatures [26, 53, 54, 56] and generally assume non-minimal features for the hidden sector. For example when the messengers masses lay in the range Mmess . 106 GeV and the goldstino does not reside in a single chiral super eld [56], or when the messenger coupling is controlled by the VEV of another eld [26] it is possible that gravitinos have the right abundance. These supersymmetry breaking schemes do not require dilution and predict r = 0. We mention that these scenarios, in their original versions, work better when supersymmetry is broken about the TeV scale. Features of these scenarios are currently tested by the LHC experiments. The above benchmark examples for the gravitino dark matter scenario are synopsized in the table 1 and gure 8. 104jmin 1010jmin 106jmin 1 51jmax 46jmax 49jmax 54 0:963jmax 0:960jmax 0:962jmax 0:965 r 0:0038jmin 0:0044jmin 0:0041jmin 0:0034 supergravity model. In the cases # 1, 2 and 4 the gravitinos are produced from thermal scatterings of messengers and MSSM elds while in the case # 3 from the non-thermal decay of the supersymmetry breaking Z eld. In cases # 1, 2 and 3 dilution is required to decrease the LSP abundance below the observational bound. In the case # 4 non-minimal hidden sector features have been assumed. The masses are in GeV units. Example II: neutralino dark matter. For gravity or anomaly mediation of supersymmetry breaking the gravitino mass is naturally heavier than the neutralinos. The gravitino decay populates the universe with neutralinos. Here we assume the gravitino mass to be above 105 GeV not to spoil BBN predictions at the time of decay. The gravitinos are produced non-thermally by the decay of the in aton, see eq. (5.17), which generally accounts for a subleading contribution in the framework of R2 supergravity in ation, and by the decay of the supersymmetry breaking scalar eld Z. Contrary to the GMSB case the Z scalar oscillations are not thermally damped and generally the Z produces late entropy if displaced from the zero temperature minimum. The temperature that the Z eld decays is estimated by considering the various partial decay rates. The dominant decay channel is into a pair of gravitinos, when mZ m3=2, and the total decay rate yields the decay temperature TZdec ' 4 109GeV mZ 108GeV 5=2 GeV m3=2 : (5.29) If the Z eld oscillations dilute the thermal plasma then the gravitinos coming from the Z decay are the leading source of dark matter neutralinos at the gravitino decay temperature T3d=e2c. The neutralinos are generally found to be overabundant when supersymmetry breaks at energies beyond the TeV scale and dilution is required. Hence we assume the presence of a diluter eld X that decreases the LSP relic density via late entropy production. We mention that according to the general constraint (4.11) the neutralinos with mass m ~0 > 107 GeV are impossible to get diluted by the oscillations of the X scalar and thermal in ation is required. Let us now consider benchmark mass patterns for the supersymmetry breaking sector plus the MSSM, characterized mainly by split and quasi-natural sparticle mass spectrum. 1. m ~0 . 103 GeV, m3=2 mf~ ' 106 GeV; mZ ' 107 GeV. Here we assume the annihilation scenario where the neutralino has an annihilation cross section few orders of magnitude higher that the conventional value. The universe is generally dominated by the Z scalar that decays to gravitinos at the temperature T dec 12 GeV. In turn, the gravitinos produced from the Z decay dominate the energy density and decay 102jmin 102jmin 108jmin 1 52jmax 52jmax 48jmax 54 0:964jmax 0:964jmax 0:961jmax 0:965 r 0:0036jmin 0:0036jmin 0:0042jmin 0:0034 the R2 supergravity model. In the case # 1 the neutralino annihilate after the decay of gravitinos, while in case # 2 neutralinos acquire a thermal abundance. In the case # 3 the neutralinos from the gravitino decay are overabundant and a diluter X is required. The case # 4 is the standard thermal WIMP scenario. The masses are in GeV units. HJEP02(18) density T ~f.0o.=T3d=e2c , see section 3.2. The resulting LSP abundance can t the observed value and here the ro^le of the diluter is played by the Z eld and the gravitinos. We note that if m ~0 > TeV then a diluter scalar X is required. It is D & 102 but the dilution size due to Z oscillations can be many orders of magnitude larger. This scenario is currently tested and constrained by the LHC and indirect detection experiments [27]. This minimum value of the dilution magnitude yields j nsj & 1 mf~ ' 108 GeV; mZ ' 109 GeV. In this example we assume that the T3d=e2c > T ~f.0o. and neutralinos thermalize after the decay of gravitinos. T dec Z DMh2 A Z dominated early universe becomes in turn gravitino dominated at 104 GeV. For TeV and sub-TeV scale neutralinos the observational bound 0:12 can be satis ed, see eq. (3.9). Here again the ro^le of the diluter is played by the Z eld and the gravitinos and it is D & 102, but it can many orders of magnitude larger. This scenario is currently tested by LHC and direct detection experiments. This dilution magnitude induces a shift in the spectral index and tensor-to-scalar ratio respectively at least of size j nsj & 1 10 3 and r & 2 10 4. 3. m ~0 ' 105 GeV, m3=2 mf~ ' 107 GeV; mZ ' 108 GeV. In this example the neutralinos are produced from the gravitino decay and they are out of chemical and kinetic equilibrium. The LSP relic density, given by eq. (3.10), is The LSP abundance has to be decreased eight orders of magnitude down and this is possible only if the gravitinos and the Z scalar condensate are su ciently diluted. Thermal in ation is required with DX & 108. This dilution magnitude induces a shift in the spectral index and tensor-to-scalar ratio respectively at least of size <~0 h2 not constrained by terrestrial experiments. . This is a phenomenologically viable example 4. m ~0 ' 103 GeV, m3=2 mf~ mZ > 105 GeV. As a last example we consider the conventional thermal WIMP scenario assuming that the Z scalar eld is not displaced from the zero temperature minimum and never dominates the energy density of the universe, hence no non-thermal phase is required (although a non-thermal phase before T ~f.0o. is not ruled out in general). In this scenario it is r = 0. This supersymmetry breaking scheme is currently tested at LHC, direct and indirect detection experiments. The above benchmark examples for the neutralino dark matter scenario are synopsized in the table 2 and gure 8. Let us nally note that the LSP particles produced from the gravitino and X decay are warmer than the LSPs produced from thermal scatterings and this changes the free streaming length of the LSP dark matter, which has the e ect of potentially washing out small scale cosmological perturbations, see e.g. [120, 121]. This is a very interesting possibility that could provide further constraints to these scenarios, though the mass scales and lifetimes considered here yield free streaming lengths that are not in con ict with the Lyman- forest observations [122]. Distinguishing the R2 and the R2 supergravity in ationary models The supersymmetric and non-supersymmetric R2 in ation models predict the same reheating temperature, Trh 109 GeV and the same expressions for the ns = ns(N ) and r = r(N ). However, the degeneracy between the two models that appears during the accelerating and the reheating stage breaks after the in aton decay.4 In the case of supergravity R2 in ation, if m~ < Trh, sparticles will be constituents of the thermalized plasma of the reheated universe. In addition to thermal processes, the presence of the supergravitational in aton and the supersymmetry breaking eld produce a signi cant number of gravitino particles after in ation, as the expressions (5.17), (5.20) and (5.21) make manifest. The BBN and the DMh2 = 0:12 constraints imply that the thermal cosmic history is in uenced by the change of the supersymmetry breaking pattern. The LSP is found to be overabundant in the greatest part of the MSSM parameter space and it receives further contributions when the supersymmetry breaking eld is taken into account [117] for both gravitino and neutralino LSP scenarios. Therefore, R2 supergravity in ation is compatible with the cosmological observations only if the thermal history of the universe is not perpetual from Trh 109 GeV until TBBN 1 MeV. A non-thermal phase that dilutes the supersymmetric thermal relics and potentially supplements the universe with dark matter particles can fully reconcile the R2 supergravity in ation model with observations. The required dilution generally increases with increasing the sparticle masses. Henceforth, we conclude that the degeneracy breaking of the in ationary predictions between the R2 and supergravity R2 models depends on the energy scale and the pattern of supersymmetry breaking, see gure 7 and 8. The fact that the R2 supergravity automatically alters the details of the thermal history and possibly the expansion history of the universe compared to the simple R2 case, where sparticles and supersymmetry breaking elds are absent, allows the discrimination between the two in ationary models. Considering only the MSSM degrees of freedom as the less model dependent and conservative analysis, the conditions (A) and (B) of section 4, when 4The present comparison of the R2 and supergravity R2 in ation can be viewed as complementary to the analysis of [119] that focused on the initial conditions of the two models. graph shows the decadic logarithm of the required dilution magnitude as a function of the gravitino LSP and gaugino-sfermion masses with mf~ = mg~ for a reheating temperature Trh = 109 GeV. The dilution is calculated by requiring the 3=2h2 not to exceed the observational bounds. The densitycontour plot demonstrates the change in the ns value, magni ed 1000 times on the contour labels, for in ationary models that predict a reheating temperature Trh = 109 GeV. The information that one extracts from this graph is that supersymmetric models (e.g. quasi-natural, split, high scale) can be compatible with the CMB data only for particular values for the scalar tilt ns. true, imply that the supergravity R2 in ationary model predicts ns(k ) < 0:965 and r > 0:0034 ; (5.30) and r = 3(1 23=4(1 ns)3, which is the characteristic r = r(ns) relation for the Starobinsky R2 in ationary model, see gure 8. The ns = 0:965 and r = 0:0034 are the reference thermal values. A knowledge of the details of the supersymmetry breaking sector would allow us to accurately predict the (ns; r) values. From a di erent point of view, the precise measurement of the (ns; r) observables could indicate cosmologically viable supersymmetry breaking patterns. Although it may not be possible to specify the identity of the dark matter, see the proximity of the spots on the (ns; r) contour in gure 8, it is possible to constrain signi cantly and even rule out a great part of the supersymmetry breaking parameter space. o i t a o s n e0.001 T 2 3 1 4 ****** 3 12 4 forecast constraints from a future CMB probe with sensitivity 10 3 depicted with the dotted and dashed ellipsis. The R2 model is targeted with a ducial value of r 4 10 3. The red asterisks correspond to the predictions of the four benchmark models (#1; 2; 3; 4) with gravitino LSP and the green asterisks to the four benchmark models (#1; 2; 3; 4) with neutralino LSP, as explained in the text and tables 1 and 2 respectively. If the future CMB experimental probes select the area inside the dashed ellipsis then either the R2 or the SUGRA-R2 in ation model is selected plus a roughly continuous thermal phase with reheating temperature, 109 GeV. The selection of the dashed ellipsis area will exclude a large class of supersymmetry models that predict a too large LSP abundance for that reheating temperature. On the contrary, if the dotted ellipsis area is selected then the duration of the thermal phase before the BBN is much limited and extra scalar particles should be present above the TeV scale, hence supporting the SUGRA-R2 model rather than the R2 in ation model plus \desert". Let us also mention that a similar result to (5.30) can be obtained if one simply assumes the presence of extra scalars that dominate the energy density of the early Universe. For example, for a supergravity R2 in ation model, a gravitational modulated reheating was assumed [123] and non-Gaussianity was additionally predicted. In the present work the postulation of a non-thermal phase has been motivated by the general requirement to t the universal constraint LSPh2 0:12 that in turn implies the result (5.30). Last but not least, we emphasize again that the precise measurement of the (ns; r) cannot \prove" or disprove the existence of supersymmetry. It will only indicate the presence of extra scalar degrees of freedom, that supersymmetry or any other BSM scenario will be challenged to explain. Future CMB probes INF ; wrh if In ation Model Selection Trh n(sth) r(th) Ndark wX SUSY MSUSY LSP DX eld QN, Split, HS G~; ~0,.. # dof Discussion and conclusions The cosmic energy window from about 1 MeV up to the in ationary energy scale is shuttered to the current observational probes and the corresponding timescale can be reasonably called a dark early universe cosmic era. Any understanding of the cosmic processes that take place before the BBN will provide us with critical insights into the microphysics that operates at that energy scales. One signi cant prospect to contemplate the early dark cosmic era is through the precision measurement of the CMB observables ns and r. The essential fact is that the (ns; r) are not strictly scale invariant, hence important information about the background expansion rate and the reheating temperature of the universe can be obtained. The in ationary paradigm can be used as a concrete and compelling framework for the theoretical determination of the ns and r values. However, our ignorance about the reheating process and the subsequent evolution of the universe, encoded in the dependence on N , is rather strong and will become signi cant as the accuracy on the observations are expected to be further improved the next decades. An in ationary prediction that is independent of N is the contour line r = r(ns) which can distinguish di erent in ation models. Furthermore, if in ation is followed by a continuous thermal phase then a concrete in ation model predicts a speci c number for the number of e-folds between the moment the relevant modes exit the horizon and the end of in ation, hence predicts a speci c spot number, scalar tilt and tensor-to-scalar ratio, N (th), ns (th) and r(th) respectively. on the r = r(ns) line that corresponds to what we called thermal values for the e-folds Motivated by the advertised sensitivity of the future CMB probes in this paper we quanti ed the e ect of a generic primordial non-thermal phase on the spectral index value (2.20). The ns value is possible to have been shifted by the amount ns=ns O(1 6)h from the expected thermal value, ns (th), due to a scalar condensate or a aton eld domination. The observation of non zero r along a contour line r = r(ns) is an indirect observation of a non-thermal phase and connects cosmology to microphysics since it has to be attributed to a BSM scalar eld domination. Moving a step further we applied our general results to study the observational consequences on the CMB of a supersymmetric universe. Supersymmetry is one of the most motivated theories that is extensively used to describe the very early universe evolution. Although it lacks any experimental support, it provides an appealing framework that consistently accommodates high energy processes such as in ation and dark matter production. Actually, the fact that the LHC probes only a small part of the vast energy scales up to the Planck mass, while supersymmetry may lay anywhere in between, strongly motivates the systematic cosmological examination of supersymmetric scenarios. Supersymmetry can be cosmologically manifest if supersymmetric degrees of freedom get thermally excited or produced non-thermally during the dark early cosmic era. The most direct cosmological implication of supersymmetry is that the LSP expected to be stable and hence contributes to the dark matter density. The LSP abundance is the key quantity that we estimate in di erent classes of supersymmetry breaking schemes and examine how it can be cosmologically reconciled with the observational value DMh2 = 0:12. We nd that a non-thermal phase or low reheating temperatures are generally required if supersymmetry UV completes the Standard Model of particle physics. We quanti ed the e ect of the di erent expansion histories on the (ns; r) and we broadly related it with the di erent supersymmetry breaking schemes. In this paper we mostly focused on ultra-TeV scale supersymmetry since low scale supersymmetry models with thermal WIMPs are in growing con ict with collider data and direct detection experiments. In our analysis we have not assumed that the LSP accounts for the bulk dark matter component in the universe. If it is actually LSPh2 to a non-thermal stage becomes greater. 0:12 then the expected change in the (ns; r) values due A complete understanding of the pre-BBN thermal phase and the CMB observables requires the knowledge of the initial condition for the thermal Big Bang, which are successfully provided by the in ationary theory. In this work we suggested a uni ed study of in ation and the subsequent reheating stage. Actually it is often the case that supergravity in ationary models are degenerate, in terms of the in ationary observables, with their the non-supersymmetric versions. However, the supersymmetric degrees of freedom can be excited either thermally or non-thermally after the end of the in ationary phase. For the sake of completeness we considered in this paper the R2 supergravity in ation and we performed a theoretical estimation of the (ns; r) observables. Our ndings point out that the ultra-TeV scale supersymmetry leaves a more clear cosmological imprint on the CMB observables. This fact is particularly exciting because high scale supersymmetric scenarios can be cosmologically falsi ed while the low mass range supersymmetric scenarios are directly tested at the terrestrial colliders. Undoubtedly any non-trivial cosmological information about the BSM physics is of major importance. Certainly the results of this cosmological analysis, illustrated in the graph of gure 9, cannot discover or disprove supersymmetry. The only concrete cosmological information that we get from the ns and r observables concerns the expansion rate of the very early universe. The identity of the matter content that controls the cosmic expansion rate cannot be revealed and it is only subject to interpretations. Nonetheless, if the (ns; r) deviate from their thermal values then new physics exists in high energies. In this paper we focused on supersymmetry, though any BSM scenario can be analyzed accordingly. In the event of detection of primordial gravitational waves, that is an observation of r 6= 0 together with possible features of the tensor power spectrum, then the selection of a particular in ation model is possible. In such a case our analysis has the power to rule out the BSM desert scenario and indicate possible features of candidate BSM theories, as the gure 8 illustrates. From the theoretical side a more complete analysis should also take into account baryognesis scenarios and the details of thermalization process. The generation of the matter-antimatter asymmetry in the universe, seems to have a critical dependence on the temperature, as e.g. the thermal leptogenesis scenario [124] suggests. Moreover the understanding of several distinct stages in the reheating process that leads to thermalization of the universe in a radiation dominated phase at some reheating temperature Trh is necessary in order a more accurate value for the equation of state parameter wrh and the reheating e-folds number N~rh to be estimated, see eq. (2.10). A thorough understanding of the reheating process can also bring out new observables that can further constrain the reheating temperature of the universe, see e.g. [125] for a review. We should mention here that the oscillatory epoch and the reheating process of the R2 in ation model is well understood, a fact that makes the results obtained in section 5 reliable [37]. From the observational side, future CMB primary anisotropy measurements should play a decisive r^ole in probing the pre-BBN cosmic era. Complementary observational programs, such as the direct observation of tensor perturbations, should contribute signi cantly to this endeavor as well. Information on the thermal history after in ation is imprinted in the gravitational wave spectrum in the frequencies corresponding to the reheating energy scales, which can be probed by future space-based laser interferometers such as DECIGO [126]. Presumably, the synergy of di erent cosmological surveys will enable a leap forward in precision cosmology giving us, at the same time, access to the physics that operates beyond the Standard Model of particle physics, at energy scales much higher than can be obtained at CERN. Acknowledgments We thank Fotis Farakos, Alex Kehagias and Jun'ichi Yokoyama for discussions and comments on the draft. The work of ID is supported by the IKY Scholarship Programs for Strengthening Post Doctoral Research, co- nanced by the European Social Fund ESF and the Greek government. 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Ioannis Dalianis, Yuki Watanabe. Probing the BSM physics with CMB precision cosmology: an application to supersymmetry, Journal of High Energy Physics, 2018, 118, DOI: 10.1007/JHEP02(2018)118