0 50 100 150 200 250 300 0.01 0.1 1 10 100 1000 10 4 m$_{Τ}$ GLYPH<OverTilde> GLYPH<LBracket1> GeV GLYPH<RBracket1> Stau Lifetime GLYPH<LBracket1> s GLYPH<RBracket1> 8 9 10 11 12 13 14 0.6 0.8 1.0 1.2 1.4 log$_{10}$ GLYPH<LBracket1> M GLYPH<Slash1> GeV GLYPH<RBracket1> Mass Ratio 5 FIG. 5: Lifetime of stau plotted against the stau mass, for m 3 / 2 = 1 MeV (red solid curve), m 3 / 2 = 10 MeV (brown dashed), m 3 / 2 = 0 . 1 GeV (purple dot-dashed) and m 3 / 2 = 1 GeV (orange dotted). FIG. 4: The stau/neut ralino mass ratio m ˜ $_{τ}$/m ˜ χ 0 plotted against the messenger sc ale M , for λ = 2 (solid red), λ = 3 (dashed brown), λ = 4 (dot-dashed purple), λ = 5 (dotted orange) from above, wit h tan β = 10. The vertical dashed line indicates the mini mal messenger scale in our model, M$_{∗}$ = 9 . 0 × 10 9 GeV. For large enough λ stau becomes lighter than neutralino. MeV ≲ m$_{3}$$_{/}$$_{2}$ ≲ 0 . 6 GeV. The model prediction (16) is safely within this range. The origin of relic gravitino is either thermal or non- thermal productions. The thermal production is due to scattering in the thermal plasma in the radiation domi- nated era after inflation. The relic abundance from this contribution is computed as [26–28] In the previous sect ion we have seen that the hidden sector renormalization can change the low energy mass spectrum. Here we describe phenomenological implica- tions when the NLSP is the lightest stau. We note that cosmological constraints on the mass parameters in the stau NLSP scenario are less strin- gent than in the neutralino NLSP case [5]. In BBN, stau produces a bound state with $^{4}$He that tends to overproduce primordial $^{6}$Li through catalytic reaction $^{4}$He ˜ τ − + D → $^{6}$Li + ˜ τ − [21, 22] (see also [5, 23, 24]), giving upper bound for the stau lifetime 3 where T$_{R}$ is the reheating temperature and h is the Hub- ble constant in the unit of 100 km Mpc − $^{1}$s − $^{1}$. The non- thermal production of gravitino is due to decay of the NLSP 4 [38, 39]. The abundance of non-thermally pro- duced gravitino is estimated as [40–43] where Ω th ˜ τ is the thermal abundance of stau. The total gravitino abundance is thus 4 The productions of gravitino from decay of moduli, inflation, and Polonyi fields have also been discussed [29–37]. 3 This constraint may be relaxed by considering substantial left- right mixing of the stau mass eigenstates [25]. IV. STAU NL SP PHENOMENOLOGY Ω TP ˜ G h 2 ≃ 0 . 3 ( T$_{R}$ 10 $^{10}$GeV )( 100GeV m$_{3}$$_{/}$$_{2}$ )( M$_{3}$ 1TeV ) 2 , (18) Ω NTP ˜ G h 2 = m$_{3}$$_{/}$$_{2}$ m$_{˜}$ τ Ω th ˜ τ h 2 ≃ 0 . 02 ( m$_{3}$$_{/}$$_{2}$ 100GeV )( m$_{˜}$ τ 1TeV ) , (19) τ$_{˜}$ τ = Γ − $^{1}$(˜ τ → τ ˜ G ) ≃ 48 πm 2 3 / $_{2}$M 2 P m 5 ˜ τ ( $^{1}$− m 2 3 / 2 m 2 ˜ τ ) − 4 ≃ 5 . $^{7}$× 10 4 ( m$_{3}$$_{/}$$_{2}$ 1 GeV ) 2 ( 100 GeV m$_{˜}$ τ ) 5 ≲ $^{5}$× 10 3 seconds . (17) Ω$_{˜}$ $_{G}$h 2 = Ω TP ˜ G h 2 + Ω NTP ˜ G h $^{2}$, (20) Ω$_{˜}$ $_{G}$h 2 ≤ Ω$_{CDM}$ h 2 ≃ 0 . $^{1131}$± 0 . 0034 . (21) which is constrained by the observed dark matter density [1] For stau mass m$_{˜}$ τ ≈ 130 GeV, the BBN bound for the gravitino mass m$_{3}$$_{/}$$_{2}$ ≲ 0 . 6 GeV restricts the non- thermally produced gravitino abundance (19) to be ex- tremely small, Ω NTP ˜ G h 2 ≲ 10 − $^{5}$, indicating that most of Fig. 5 shows the stau lifetime τ$_{˜}$ τ plotted against the stau mass m$_{˜}$ τ for different values of gravitino mass m$_{3}$$_{/}$$_{2}$ . The upper bound of the stau lifetime τ$_{˜}$ τ ≲ 5 × 10 3 con- strains the gravitino mass from above. For our typical range of stau mass m$_{˜}$ τ ≈ 130 GeV, the gravitino mass is m$_{3}$$_{/}$$_{2}$ ≲ 0 . 6 GeV. Together with the condition that gravitino behaves as cold dark matter, a successful cos- mological scenario with stau NLSP and gravitino cold dark matter requires the gravitino mass to be within 0 . 1