Light dark matter ( \simeq 1 - 30 ~ { } MeV ) particles pair produced in electron-positron annihilation e ^ { - } e ^ { + } \stackrel { \gamma } { \longrightarrow } \chi \bar { \chi } inside the supernova core can take away the energy released in the supernova SN1987A explosion .
Working within the formalism of q -deformed statistics ( with the average value of the supernovae core temperature ( fluctuating ) being T _ { SN } = 30 ~ { } MeV ) and using the Raffelt ’ s criterion on the emissivity for any new channel \dot { \varepsilon } ( e ^ { + } e ^ { - } \rightarrow \chi \overline { \chi } ) \leq 10 ^ { 19 } ~ { } { erg% ~ { } g ^ { -1 } s ^ { -1 } } , we find that as the deformation parameter q changes from 1.0 ( undeformed scenario ) to 1.1 ( deformed scenario ) , the lower bound on the scale \Lambda of the dark matter effective theory varies from 3.3 \times 10 ^ { 6 } TeV to 3.2 \times 10 ^ { 7 } TeV for a dark matter fermion of mass m _ { \chi } = 30 ~ { } MeV .
Using the optical depth criteria on the free streaming of the dark matter fermion , we find the lower bound on \Lambda \sim 10 ^ { 8 } ~ { } TeV for m _ { \chi } = 30 ~ { } MeV .
In a scenerio , where the dark matter fermions are pair produced in the outermost sector of the supernova core ( with radius 0.9 R _ { c } \leq r \leq R _ { c } , R _ { c } ( = 10 ~ { } km ) being the supernova core radius or the radius of proto-neutron star ) , we find that the bound on \Lambda ( \sim 3 \times 10 ^ { 7 } TeV ) obtained from SN cooling criteria ( Raffelt ’ s criteria ) is comparable with the bound obtained from free streaming ( optical depth criterion ) for light fermion dark matter of mass m _ { \chi } = 10 - 30 MeV .

Keywords : Dark matter , Supernova cooling , q -deformed statistics , free-streaming ,