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## IV. STAU NL SP PHENOMENOLOGY 

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 

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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 

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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] 

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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] 

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where Ω ˜ τ is the thermal abundance of stau. The total gravitino abundance is thus 

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 

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