Constraints on the Reheating Temperature in Gravitino Dark Matter Scenarios

Considering gravitino dark matter scenarios, we study constraints on the reheating temperature of inﬂation. We present the gauge-invariant result for the thermally produced gravitino yield to leading order in the Standard Model gauge couplings. Within the framework of the constrained minimal supersymmetric Standard Model (CMSSM), we ﬁnd a maximum reheating temperature of about 10 7 GeV taking into account bound-state eﬀects on the primordial 6 Li abundance. We show that late-time entropy production can relax this constraint signiﬁcantly. Only with a substantial entropy release after the decoupling of the lightest Standard Model superpartner, thermal leptogenesis remains a viable explanation of the cosmic baryon asymmetry within the CMSSM.


INTRODUCTION
The observed flatness, isotropy, and homogeneity of the Universe suggest that its earliest moments were governed by inflation [1,2]. The inflationary expansion is followed by a phase in which the Universe is reheated. The reheating process repopulates the Universe and provides the initial conditions for the subsequent radiationdominated epoch. We refer to the reheating temperature T R as the initial temperature of this early radiationdominated epoch of our Universe.
The value of T R is an important prediction of inflation models. While we do not have evidence for temperatures of the Universe higher than O(1 MeV) (i.e., the temperature required by primordial nucleosynthesis), inflation models can point to T R well above 10 10 GeV [2,3].
In this Letter we consider supersymmetric (SUSY) extensions of the Standard Model in which the gravitino G is the lightest supersymmetric particle (LSP) and stable because of R-parity conservation. The gravitino LSP is a well-motivated dark matter candidate. As the gauge field of local SUSY transformations and the spin-3/2 superpartner of the graviton, the gravitino is an unavoidable implication of SUSY theories including gravity [4]. Its interactions are suppressed by inverse powers of the (reduced) Planck scale M P = 2.4 × 10 18 GeV. Its mass m e G results from spontaneous SUSY breaking and can range from the eV scale up to scales beyond the TeV region [5].
While any initial population of gravitinos must be diluted away by the exponential expansion during inflation [6], gravitinos are regenerated in scattering processes of particles that are in thermal equilibrium with the hot primordial plasma. The efficiency of this thermal production of gravitinos during the radiation-dominated epoch is sensitive to T R [7,8,9,10,11]. Since the resulting gravitino density Ω TP e G is bounded from above by the dark matter density Ω dm , upper bounds on T R can be derived [8,12,13,14,15]. These bounds can be compared with predictions of the reheating temperature T R from inflation models. Moreover, T R is important for our understanding of the cosmic baryon asymmetry. For example, successful standard thermal leptogenesis [16] requires T R 10 9 GeV [17].
We update the T R limits using the full gauge-invariant result for the relic density of thermally produced gravitinos, Ω TP e G , to leading order in the Standard Model gauge couplings [11]. This allows us to illustrate the dependence of the bounds on the gaugino-mass relation at the scale of grand unification M GUT ≃ 2 × 10 16 GeV.
We consider gravitino dark matter scenarios also in the framework of the constrained minimal supersymmetric Standard Model (CMSSM) in which the gaugino masses, the scalar masses, and the trilinear scalar interactions are assumed to take on the respective universal values m 1/2 , m 0 , and A 0 at M GUT . Taking into account gravitinos from thermal production and from late decays of the lightest Standard Model superpartner, we provide new upper bounds on the reheating temperature in the (m 1/2 , m 0 ) plane for various values of m e G . Previous studies of T R constraints within the CMSSM used the result of [10] to explore the viability of T R 10 9 GeV [13,14]. Our study presents also scans for T R as low as 10 7 GeV based on the result of [11] which includes electroweak contributions to thermal gravitino production [18].
In the considered CMSSM scenarios with the gravitino LSP, the next-to-lightest SUSY particle (NLSP) is either the lightest neutralino χ 0 1 or the lighter stau τ 1 . 1 Because of the extremely weak interactions of the gravitino, the NLSP typically has a long lifetime before it decays into the gravitino. If these decays occur during or after big-bang nucleosynthesis (BBN), the Standard Model particles emitted in addition to the gravitino can affect the abundance of the primordial light elements. Indeed, these BBN constraints disfavor the χ 0 1 NLSP for m e G 100 MeV [13,14,21]. For the slepton NLSP case, the BBN constraints associated with hadronic/electromagnetic energy injection have also been estimated and found to be much weaker but still significant in much of the parameter space [13,14,15,21].
Only recently, it has been stressed that bound-state formation of long-lived negatively charged particles with the primordial nuclei can affect BBN [22,23,24,25]. With the charged long-lived stau NLSP, these boundstate effects also apply to the considered gravitino dark matter scenarios. In particular, a significant enhancement of 6 Li production has been found to imply severe upper limits on the τ 1 NLSP abundance prior to decay [22] which strongly restricts the mass spectrum in the τ 1 NLSP case [26]. For generic parameter regions of the CMSSM, we show that this constraint disfavors T R > 10 7 GeV and thereby successful thermal leptogenesis.
Entropy production after decoupling of the NLSP and before BBN can weaken the BBN constraints significantly [27]. At the same time, the gravitino density is diluted which relaxes the bounds on T R . We show explicitly the effect of entropy production on the T R bounds. Here we consider the cases of late-time entropy production before and after the decoupling of the NLSP. Indeed, a relaxation of the T R bounds can render models of inflation with T R > 10 7 GeV viable in CMSSM scenarios with gravitino dark matter. Since also a baryon asymmetry generated in the early Universe is diluted, the temperature required by thermal leptogenesis increases in a cosmological scenario with late-time entropy production. Still, we find that a sufficient amount of entropy production after NLSP decoupling and before BBN can revive successful thermal leptogenesis.

THERMAL GRAVITINO PRODUCTION
Gravitinos with m e G 1 GeV have decoupling temperatures of T e G f 10 14 GeV, as will be shown below. We consider thermal gravitino production in the radiationdominated epoch starting at T R < T e G f assuming that inflation has diluted away any initial gravitino population. 2 For T R < T e G f , gravitinos are not in thermal equilibrium with the post-inflationary plasma. Accordingly, the evolution of the gravitino number density n e G with cosmic time t is described by the following Boltzmann 2 Taking a conservative point of view, we do not include gravitino production before the radiation-dominated epoch. However, inflaton decays, for example, can lead to a sizable yield of nonthermally produced gravitinos depending on the inflation model; cf. [28,29] and references therein.
where H denotes the Hubble parameter. The collision term C e G involves the gaugino mass parameters M i , the gauge couplings g i , and the constants c i and k i associated with the gauge groups U(1) Y , SU(2) L , and SU(3) c as given in Table I. In expression (2) the temperature T provides the scale for the evaluation of M i and g i . The given collision term is valid for temperatures sufficiently below the gravitino decoupling temperature, where gravitino disappearance processes can be neglected. A primordial plasma with the particle content of the minimal SUSY Standard Model (MSSM) in the high-temperature limit is used in the derivation of (2).
The collision term (2) results from a consistent gaugeinvariant finite-temperature calculation [11,18] following the approach used in Ref. [10]. Thus, in contrast to the previous estimates in [7,8], the expression for C e G is independent of arbitrary cutoffs. Note that the fieldtheoretical methods of [30,31] applied in its derivation require weak couplings, g i ≪ 1, and thus high temperatures T ≫ 10 6 GeV.
Assuming conservation of entropy per comoving volume, the Boltzmann equation (1) can be solved to good approximation analytically [10,32]. At a temperature T low ≪ T R , the resulting gravitino yield from thermal production reads where the constants y i are given in Table I. These constants are obtained with the Hubble parameter describing the radiation-dominated epoch, H rad (T ) = g * (T )π 2 /90 T 2 /M P , the entropy density s(T ) = 2π 2 g * S (T ) T 3 /45, and an effective number of relativistic degrees of freedom of g * (T R ) = g * S (T R ) = 228.75. We evaluate g i (T R ) and M i (T R ) using the one-loop evolution described by the renormalization group equation in the MSSM: with the respective gauge coupling at the Z-boson mass, g i (m Z ), and the β (1) i coefficients listed in Table I. Without late-time entropy production, the gravitino yield from thermal production at the present temperature T 0 is given by The resulting density parameter of thermally produced gravitinos is  [10] for M 3 = m 1/2 , which was used to study T R constraints on gravitino dark matter scenarios in Refs. [13,14,15]. We find that (3) exceeds the yield derived from [10] by about 50%; cf. [11]. The dashed (blue in the web version) horizontal line indicates the equilibrium yield which is given by the equilibrium number density of a relativistic spin 1/2 Majorana fermion, n eq  In the analytical expression (3) we refer to T R as the initial temperature of the radiation-dominated epoch. So far we have not considered the phase in which the coherent oscillations of the inflaton field φ dominate the energy density of the Universe, where one usually defines T R in terms of the decay width Γ φ of the inflaton field φ. To account for the reheating phase, we numerically integrate (1) together with the Boltzmann equations for the energy densities of radiation and the inflaton field,

ÌÈ
respectively; for details see Appendix F of Ref. [34]. With our result for the collision term (2), we find that the gravitino yield obtained numerically is in good agreement with the analytical expression (3) for the associated numerically obtained gravitino yield is described by the analytical expression obtained after sub- (3). While we focus on scenarios in which the gravitino is stable, the yield (3) is also crucial to extract cosmological constraints in scenarios with unstable gravitinos. Based on the result of [10] and taking into account thermal gravitino production during reheating, the following fitting formula was used to study constraints from decaying gravitinos in Refs. [33,34,35]: where T R was defined via Γ φ = 3 H rad (T R ). Comparing (13) with our result after the matching of the T R definitions, we find that our result exceeds the m e Gindependent yield (13) (13) is used for m e G as small as 100 GeV in Refs. [33,34,35]. As can be seen in Fig. 1, the actual yield for m e G = 100 GeV is thereby underestimated by about an order of magnitude. Accordingly, the T R bounds given in [33,34,35] are underestimated in the region m e G < 1 TeV.

CONSTRAINTS ON TR
The reheating temperature T R is limited from above in the case of a stable gravitino LSP since Ω TP e G cannot exceed the dark matter density Ω dm [8,12,13,14,15]. In this paper, we use [36,37] Ω 3σ dm h 2 = 0.105 +0.021 −0.030 (14) as obtained from the measurements of the cosmic microwave background (CMB) anisotropies by the Wilkinson Microwave Anisotropy Probe (WMAP) satellite. 3 In Fig. 2 we show the resulting upper limits on T R as a function of m e G . On the gray band, the thermally produced gravitino density (7) is within the nominal 3σ range (14). The upper (lower) gray band is obtained for M 1,2,3 = m 1/2 at M GUT with m 1/2 = 500 GeV (2 TeV). The dashed lines show the corresponding constraints for the exemplary non-universal scenario [39]   The T R limits shown in Fig. 2 are conservative bounds that do only depend on m e G and the M i values at M GUT . Once details of the SUSY model realized in Nature are known, one will be able to refine the limits by including contributions to Ω e G from NLSP decays. In the next section, we will account for this non-thermal gravitino production in a systematic way within the framework of the CMSSM. 4 Note that the dotted curve shown for m 1/2 = 500 GeV in Fig. 2 is by about a factor of 4 more severe than the T R limits shown in Figs. 5 and 6 of Ref. [14] in the region m e G 10 GeV in which Ω TP e G governs the limits. It seems to us that the gluino mass in (1.2) of Ref. [14] was accidentally evaluated at the scale µ = T R rather than at the scale µ ≃ 100 GeV; see Sec. 5 of [10].

CONSTRAINTS ON TR IN THE CMSSM
In the CMSSM, one assumes universal soft SUSY breaking parameters at M GUT . The CMSSM yields phenomenologically acceptable spectra with only four parameters and a sign: the gaugino mass parameter m 1/2 , the scalar mass parameter m 0 , the trilinear coupling A 0 , the mixing angle tan β in the Higgs sector, and the sign of the higgsino mass parameter µ.
Assuming A 0 = 0 for simplicity, the lightest Standard Model superpartner is either the lightest neutralino χ 0 1 or the lighter stau τ 1 . Indeed, most CMSSM investigations assume that χ 0 1 is the LSP that provides dark matter; cf. [40] and references therein. The parameter region in which m e τ1 < m e χ 0 1 is usually not considered because of the severe upper limits on the abundance of stable charged particles [37]. In the gravitino LSP case, m e τ1 < m e χ 0 1 can be viable since the lightest Standard Model superpartner is unstable [13,14,25,41].
With the gravitino LSP, the lightest Standard Model superpartner is the NLSP that decays into Standard Model particles and one gravitino LSP. For m e G 1 GeV, the NLSP decays after its decoupling from the thermal plasma. 5 Thus, the relic density of the associated nonthermally produced gravitinos reads [42] Ω NTP where m NLSP is the mass of the NLSP and Y NLSP (T 0 ) and Ω NLSP h 2 are respectively the yield and the relic density that the NLSP would have today, if it had not decayed. In Fig 4 GeV from chargino and Higgs searches at LEP [37]. The leftmost dotted (blue in the web version) line indicates the LEP bound m e τ1 > 81.9 GeV [37]. For tan β = 30, tachyonic sfermions occur in the low-energy spectrum at points in the white corner labeled as "tachyonic." We employ the FORTRAN program SuSpect [43] to calculate the low-energy spectrum of the superparticles and the Higgs bosons, where we use m t = 172.5 GeV for the top quark mass. Assuming standard cosmology, the yield Y NLSP (T 0 ) is obtained 5 The NLSP freezeout temperature can be estimated from its mass: T NLSP f m NLSP /20 [12]. Thus, T R ≫ T NLSP f for T R > 10 6 GeV which is considered in this Letter. from the Ω NLSP h 2 values provided by the computer program micrOMEGAs [44].
The contours shown in Fig. 3 are independent of m e G and T R . Therefore, they can be used to interpret the results shown in the figures below. Note the sensitivity of both Y e τ1 (T 0 ) and m e τ1 on tan β. By going from tan β = 10 to tan β = 30, Y e τ1 (T 0 ) decreases by about a factor of two at points that are not in the vicinity of the dashed line, i.e., that are outside of the τ 1 -χ 0 1 coannihilation region. While m e τ1 becomes smaller by increasing tan β to 30, the tan β dependence of m e χ 0 1 is negligible. Let us now explore the parameter space in which Now, T R and m e G appear in addition to the traditional CMSSM parameters. We focus on m e G 1 GeV since the soft SUSY breaking parameters of the CMSSM are usually assumed to result from gravity-mediated SUSY breaking. However, we do not restrict our study to fixed relations between m e G and the soft SUSY breaking parameters such as the ones suggested, for example, by the Polonyi model.
In Fig. 4 the light, medium, and dark shaded (green in the web version) bands show the (m 1/2 , m 0 ) regions that satisfy the constraint (17) as obtained in the limit m τ → 0. For the χ 0 1 NLSP, we calculate τ e χ 0 1 from the expressions given in Sec. IIC of Ref. [21].
The τ NLSP contours in Fig. 4 illustrate that the NLSP decays during/after BBN. Successful BBN predictions therefore imply cosmological constraints on m e G , m NLSP , and Y NLSP [13,14,15,21]. Indeed, it has been found that the considered χ 0 1 NLSP region is completely disfavored for m e G 1 GeV by constraints from late electromagnetic and hadronic energy injection [13,14,21,25]. In the τ 1 NLSP region, the constraints from electromagnetic and hadronic energy release are important but far less severe than in the χ 0 1 NLSP case. Thus, much of the τ 1 NLSP region was believed to be cosmologically allowed [13,14,15,21].
Recently, this picture has changed. It has been found that bound-state formation of long-lived negatively Ø Ò ¬ ½¼ ¼ ¼ charged τ 1 's with primordial nuclei can catalyze the production of 6 Li significantly [22,25]. Indeed, in most of the τ 1 NLSP parameter space, the associated bounds are much more severe than the ones from late energy injection. Only for ττ 1 10 3 s and m e G 40 GeV, the constraints from hadronic energy release can become more severe than the ones from catalyzed 6 Li production [25,26]. We thus consider both the constraint from catalyzed 6 Li production derived in [22] and the one from late hadronic energy injection derived in [15]. 6 For the constraint from bound-state effects on 6 Li production, we adopt the bounds given in Fig. 4 of Ref. [22] as ττ 1 -dependent upper limits on the yield of the negatively charged staus, Y NLSP /2. These bounds are obtained assuming a limiting primordial abundance of [45] The resulting constraint disfavors the τ 1 NLSP region to the left of the long-dash-dotted (red in the web version) line shown in Fig. 4. For the constraint from late hadronic energy injection, we use the upper limits on Y NLSP that are given in Fig. 11 of Ref. [15]. These limits are derived from a computation 6 For details on the other BBN bounds and the additional CMB bounds, we refer the reader to the detailed investigations presented in Refs. [13,14,21,41,46]. of the 4-body decay of the stau NLSP into the gravitino, the tau, and a quark-antiquark pair. 7 They are based on the severe and conservative upper bounds on the released hadronic energy (95% CL) obtained in [34] for observed values of the primordial D abundance of (n D /n H ) mean = (2.78 +0.44 −0.38 ) × 10 −5 (severe), (n D /n H ) high = (3.98 +0. 59 −0.67 ) × 10 −5 (conservative).
In Fig. 4 the associated constraints are shown by the short-dash-dotted (blue in the web version) lines. The D constraint disfavors the region between the corresponding lines in panel (b) and the region above the corresponding lines in panels (c) and (d). In panel (a) the D constraint does not appear. 8 Remarkably, one finds in each panel of Fig. 4 that the highest T R value allowed by the considered BBN constraints is about 10 7 GeV. The bands obtained for T R 10 8 GeV are located completely within the region 7 The 3-body estimate of the hadronic energy release given in Ref. [21] leads to overly restrictive limits, as shown in Ref. [15]. 8   With the e τ1 NLSP, the region to the left of the long-dash-dotted (red in the web version) line is cosmologically disfavored by bound-state effects on the primordial 6 Li abundance [22]. The effects of late hadronic energy injection on the primordial D abundance [15] disfavor the e τ1 NLSP region between the short-dash-dotted (blue in the web version) lines in panel (b) and the one above the corresponding lines in panels (c) and (d). The e χ 0 1 NLSP region above the dashed line, in which m e χ 0 1 < m e τ 1 , is cosmologically disfavored by the effects of late electromagnetic/hadronic energy injection on the abundances of the light primordial elements [13,14,21,25,41]. disfavored by the 6 Li bound. In previous gravitino dark matter studies within the CMSSM that did not take into account bound-state effects on the primordial 6 Li abundance, much higher temperatures of up to about 10 9 GeV were believed to be allowed [11,13,14]. 9 The constraint T R 10 7 GeV remains if we consider larger values of tan β. This is demonstrated in Fig. 5 for tan β = 30, A 0 = 0, µ > 0, and m e G = m 0 . The shadings (colors in the web version) and line styles are identical to the ones in Fig. 4.
Let us comment on the dependence of the considered BBN constraints on the assumed primordial abundances of D and 6 Li. As can be seen in Figs. 4 and 5, the constraint from late hadronic energy release is quite sensitive on the assumed primordial D abundance. In contrast, even if we relax the restrictive 6 Li bound on Y NLSP /2 by two orders of magnitude, we still find T R 10 7 GeV. For example, the 6 Li constraint relaxed in this way would appear in Fig. 4 (b) as an almost vertical line slightly above m 1/2 = 3 TeV.
While the constraint T R 10 7 GeV is found for each of the considered m e G relations, one cannot use the 6 Li bound to set bounds on m e τ1 without insights into m e G . The 6 Li bound disappears for τ e τ1 10 3 s [22] which is possible even for m e τ1 = O(100 GeV) provided m e G is sufficiently small; see (18). However, the constraints on T R become more severe towards small m e G as is shown in Fig. 2. Thus, the constraint T R 10 7 GeV cannot be evaded by lowering m e G provided T R < T e G f . An upper limit on T R of 10 7 GeV can be problematic for inflation models and baryogenesis scenarios. This finding can thus be important for our understanding of the thermal history of the Universe.

CONSTRAINTS ON TR WITH LATE-TIME ENTROPY PRODUCTION
The constraints shown above are applicable for a standard thermal history during the radiation-dominated epoch. However, it is possible that a substantial amount of entropy is released, for example, in out-of-equilibrium decays of a long-lived massive particle species X [2,51]. 10 If X lives sufficiently long, it might decay while its rest mass dominates the energy density of the Universe. The associated evolution of the entropy per comoving volume, S ≡ s a 3 , is described by [2,51] together with the Boltzmann equation (10) for φ = X and the Friedmann equation governing the evolution of the scale factor of the Universe a. Here Γ X and ρ X denote respectively the decay width and the energy density of X. Thus, the temperature after the decay can be expressed in terms of Γ X , T after ≡ 10 g * (T after )π 2 1/4 which satisfies Γ X = 3H rad (T after ). Indeed, primordial nucleosynthesis imposes a lower limit on this temperature [55,56,57,58]: In Fig. 6 we show the evolution of S, a 3 ρ X , and a 3 ρ rad for two exemplary scenarios respecting (22). The scale factor a is normalized by a I ≡ a(10 GeV) = 1 GeV −1 and the temperature dependence of g * is taken into account as determined in [59]. For ρ X (10 GeV) = 0.1 ρ rad (10 GeV) and T after = 6 MeV, S increases by a factor of ∆ = 100 as shown by the corresponding solid line. For ρ X (10 GeV) = 8 ρ rad (10 GeV) and T after = 4.9 MeV, S increases by a factor of ∆ = 10 4 as shown by the corresponding dotted (blue in the web version) line.  We restrict our study to entropy production at late times, T before ≃ T low ≪ T R , so that the thermal production of gravitinos is not affected. To work in a model independent way, we assume that the production of gravitinos and NLSPs in the entropy producing event, such as the direct production in decays of X, is negligible. 11 Moreover, in this section, we focus on scenarios in which the decoupling of the NLSP is not or at most marginally affected by entropy production, i.e., either T R ≫ T after ≫ T NLSP f or ρ rad ≫ ρ X for T T NLSP f . Thus, the thermally produced gravitino yield and-in the case of entropy production after NLSP decoupling-also the non-thermally produced gravitino yield are diluted: In the case of late-time entropy production before the decoupling of the NLSP, we parameterize this by writing In this case, Y NLSP (T 0 ) and thereby Ω NTP e G and the BBN constraints remain unaffected. 11 The constraints discussed below shall therefore be considered as conservative bounds. For studies of gravitino production during an entropy producing event, we refer to [60] and references therein.
Conversely, in the case of late-time entropy production after the decoupling of the NLSP (and before BBN) both, Y TP e G (T 0 ) and Y NLSP (T 0 ), are reduced: Accordingly, Ω TP e G and Ω NTP e G become smaller and the BBN constraints can be relaxed.
In Fig. 7 we show how late-time entropy production before (left) and after (right) NLSP decoupling affects the 6 Li constraint and the region in which 0.075 ≤ Ω e G h 2 ≤ 0.126 for T R = 10 9 GeV. The (m 1/2 , m 0 ) planes are considered for tan β = 10, A 0 = 0, µ > 0, m e G = 100 GeV (upper panels) and m e G = m 0 (lower panels). The dark shaded (green in the web version) region is obtained without late time entropy production, δ = ∆ = 1. The medium and light shaded (green in the web version) bands are obtained with a dilution of Ω TP e G (Ω TP e G + Ω NTP e G ) by δ = 10 (∆ = 10) and δ = 100 (∆ = 100), respectively. The dot-dashed (red in the web version) line illustrates that the 6 Li bound is independent of δ, as shown in the panels on the left-hand side, and becomes weaker (i.e., moves to the left) with increasing ∆, as shown in the panels on the right-hand side. Other curves and regions are identical to the ones in the corresponding panels of Fig. 4. Note that we do not show the D constraint on late hadronic energy injection since it is not sensitive to δ and vanishes already for ∆ = 10; an exception is the severe D constraint which still appears for ∆ = 10 in panel (d). BBN constraints on χ 0 1 NLSP scenarios with entropy production after NLSP decoupling will be studied elsewhere.
Comparing panels (b) and (d) of Fig. 4 with panels (a) and (c) in Fig. 7, we find that a dilution factor of δ = 10 (100) relaxes the T R bound by a factor of 10 (100). Since the BBN constraints are unaffected by δ, the cosmologically disfavored range of NLSP masses cannot be relaxed. With the dilution after NLSP decoupling, the relaxation of the T R constraints is more pronounced. Here also the cosmologically disfavored range of NLSP masses can be relaxed [27]. However, as can be seen in panels (b) and (d) of Fig. 7, the 6 Li bound is persistent. With a dilution factor of ∆ = 100, large regions of the (m 1/2 , m 0 ) plane remain cosmologically disfavored. For ∆ 10 4 , however, the 6 Li bound can be evaded as will be shown explicitly below. Figure 7 shows that inflation models predicting, for example, T R = 10 9 GeV become allowed in the CMSSM with gravitino dark matter for δ = ∆ ≈ 100. Here it is not necessary to have late-time entropy production in the somewhat narrow window between NLSP decoupling and BBN. This is different for the viability of thermal leptogenesis in the considered scenarios (T e G f > T R ) and for collider prospects as discussed below.

THERMAL LEPTOGENESIS IN THE CMSSM WITH GRAVITINO DARK MATTER
The constraint T R 10 7 GeV obtained in the considered CMSSM scenarios for a standard cosmological his-tory strongly disfavors thermal leptogenesis. However, if entropy is released after NLSP decoupling, a dilution factor of ∆ ≃ 10 4 can render thermal leptogenesis viable for T R ≃ 10 13 GeV.
Standard thermal leptogenesis usually requires T R 10 9 GeV [17]. However, late-time entropy production dilutes the baryon asymmetry which is generated well before NLSP decoupling, Therefore, the baryon asymmetry before entropy production must be larger by a factor of ∆ in order to compensate for the dilution. For ∆ ≃ 10 4 , this can be achieved in the case of hierarchical neutrinos for M R1 ∼ T R ≃ 10 13 GeV, as can be seen in Fig. 7 (a) of Ref. [61] and in Fig. 2 of Ref. [62]. Here M R1 is the mass of the lightest among the heavy right-handed Majorana neutrinos. In Fig. 6 the dotted (blue in the web version) lines show a scenario in which a dilution factor of ∆ = 10 4 is generated in the out-of-equilibrium decay of a heavy particle X. Because of ρ X (10 GeV) = 8 ρ rad (10 GeV), the Hubble rate can be enhanced already during the decoupling phase of the NLSP, which leads to an increase of T NLSP f and Y NLSP (T NLSP f ). In the results shown below, we account for this by using a modified version of the micrOMEGAs code. 12 After entropy production, the net effect is still a significant reduction of Y NLSP (T 0 ). For the same initial conditions, ∆ = 2 × 10 4 -and thereby an additional reduction of Y NLSP (T 0 ) by a factor of twocan be achieved by lowering T after from 4.9 MeV down to 2.5 MeV.
We consider these two scenarios for tan β = 30, A 0 = 0, µ > 0, and m e G = m 0 , in Fig. 8. Here the shaded (green in the web version) bands indicate the region in which 0.075 ≤ Ω e G h 2 ≤ 0.126 for T R = 10 13 GeV and ∆ = 10 4 (dark) and 2 × 10 4 (medium). In addition, the corresponding evolution of the 6 Li bound is shown by the dotdashed (red in the web version) lines. For ∆ = 10 4 , the regions below the associated two rightmost curves and to the right of the associated leftmost curve are allowed. For ∆ = 2 × 10 4 , the cosmologically allowed region is the τ 1 NLSP region below the line labeled accordingly. The gray regions are identical to the ones in Fig. 5.
We find that the 6 Li bound cannot be evaded for the tan β = 10 scenarios even for ∆ = 2×10 4 since Y NLSP (T 0 ) becomes larger. However, the 6 Li bound given in Fig. 4 of Ref. [22] depends linearly 13 on the assumed limiting primordial abundance (19) that is subject to uncertainties; cf. Ref. [50]. Accordingly, for a limiting abundance that is a factor of two above the value given in (19), one obtains the 6 Li bound labeled with ∆ = 2 × 10 4 in Fig. 8 for the scenario with tan β = 30 and ∆ = 10 4 .
Scenarios with successful thermal leptogenesis in the τ 1 NLSP region are located preferably on the dark-shaded 12 The Y NLSP contours shown in Fig. 3 do not apply in this section. 13 We thank M. Pospelov for bringing this point to our attention. The effect of entropy production after NLSP decoupling for TR = 10 13 GeV and ∆ ≥ 10 4 in the (m 1/2 , m0) plane for tan β = 30, A0 = 0, µ > 0, and m e G = m0. The shaded (green in the web version) bands show the region in which 0.075 ≤ Ω e G h 2 ≤ 0.126 for ∆ = 10 4 (dark) and 2 × 10 4 (medium). The dot-dashed (red in the web version) lines illustrate the corresponding evolution of the 6 Li bound. For ∆ = 10 4 , the regions below the associated two rightmost curves and to the right of the associated leftmost curve are allowed. For ∆ = 2 × 10 4 , the region below the line labeled accordingly is cosmologically allowed.
(dark green in the web version) band and in the white corner to its left, in which even slightly higher values of T R are possible for ∆ = 10 4 . For T R = 10 13 GeV and ∆ ≫ 10 4 , the generated baryon asymmetry is diluted too strongly in order to explain the observed baryon asymmetry.

CONCLUSION
Using the full gauge-invariant result for Ω TP e G to leading order in the Standard Model gauge couplings [11], we have studied bounds on T R from the constraint Ω e G ≤ Ω dm . Our results take into account the dependence of Ω TP e G on the masses of the gauginos associated with the Standard Model gauge group SU(3) c ×SU(2) L ×U(1) Y . This has allowed us to explore the dependence of the T R bounds on the gaugino-mass relation at the scale of grand unification M GUT .
Within the CMSSM, we have explored gravitino dark matter scenarios and the associated T R bounds for m e G 1 GeV and for temperatures as low as 10 7 GeV. Taking into account the restrictive constraint from bound-state effects of long-lived negatively charged staus on the primordial 6 Li abundance [22], we find that T R 10 7 GeV is the highest cosmologically viable temperature of the radiation-dominated epoch in case of a standard thermal history of the Universe. This imposes a serious constraint on model building for inflation. Moreover, thermal leptogenesis seems to be strongly disfavored in the considered regions of the CMSSM parameter space.
With late-time entropy release, the obtained limit T R 10 7 GeV can be relaxed. For example, the dilution of the thermally produced gravitino yield by a factor of 10 relaxes the T R bound by about one order of magnitude in regions where Ω TP e G dominates Ω e G . In the case of entropy production after NLSP decoupling, the yield of the NLSP prior to its decay, Y NLSP , is reduced so that the BBN constraints can be weakened. Although the 6 Li bound is persistent, we find that it disappears provided Y NLSP is diluted by a factor of ∆ 10 4 .
We have discussed the viability of thermal leptogenesis in a cosmological scenario with entropy production after NLSP decoupling. We find that successful thermal leptogenesis can be revived in generic regions of the CMSSM parameters space for M R1 ∼ T R ≃ 10 13 GeV and ∆ 10 4 , where M R1 is the mass of the lightest among the heavy right-handed Majorana neutrinos.
Remarkably, for a dilution factor of ∆ 10 4 , the τ 1 NLSP region with m e τ1 200 GeV reopens as a cosmologically allowed region in the CMSSM with the gravitino LSP. A long-lived τ 1 in this mass range could provide striking signatures of gravitino dark matter at future colliders [63,64,65,66].