We present limits for the compactification scale in the theory of Large Extra Dimensions (LED) proposed by Arkani-Hamed, Dimopoulos, and Dvali. We use 11 months of data from the Fermi Large Area Telescope (Fermi-LAT) to set gamma ray flux limits for 6 gamma-ray faint neutron stars (NS). To set limits on LED we use the model of Hannestad and Raffelt (HR) that calculates the Kaluza-Klein (KK) graviton production in supernova cores and the large fraction subsequently gravitationally bound around the resulting NS. The predicted decay of the bound KK gravitons to γγ should contribute to the flux from NSs. Considering 2 to 7 extra dimensions of the same size in the context of the HR model, we use Monte Carlo techniques to calculate the expected differential flux of gamma-rays arising from these KK gravitons, including the effects of the age of the NS, graviton orbit, and absorption of gamma-rays in the magnetosphere of the NS. We compare our Monte Carlo-based differential flux to the experimental differential flux using maximum likelihood techniques to obtain our limits on LED. Our limits are more restrictive than past EGRET-based optimistic limits that do not include these important corrections. Additionally, our limits are more stringent than LHC based limits for 3 or fewer LED, and comparable for 4 LED. We conclude that if the effective Planck scale is around a TeV, then for 2 or 3 LED the compactification topology must be more complicated than a torus.
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