  The idea of Dark Matter (DM) with self interaction was invoked to resolve a
number of discrepancies between the simulation based predictions by
collisionless cold DM and the astrophysical observations on galactic and
subgalactic scales. Evidences for self interaction would have striking
implications for particle nature of DM. In order to reconcile such
astrophysical observations for self interaction with particle properties for
DM, we consider the general scenario of self interacting Dirac fermionic DM,
$\chi$. Also since the exact particle physics model for DM is yet to be probed,
we simply adopt the effective model independent framework for DM self
interaction which occurs via the most general effective 4-fermion operators
invariant under both Lorentz and CPT transformations. From the thorough
investigation of the interrelations among the parameters in this framework,
namely, the effective DM self couplings ($G_{i}$), DM mass ($m_{\chi}$) and
relative velocity ($v_{\rm rel}$), it can be inferred that $G_{i}$ decrease
with increasing $m_{\chi}$ for a given DM self interaction strength. Moreover,
for few types of effective operators the values of $G_{i}$ fall off with
increasing $v_{\rm rel}$ while they remain roughly constant for a wide range of
$v_{\rm rel}$ for other cases. In addition, the parameter space in this
framework is constrained by the claimed observational results of ${\sigma \over
m_{\chi}}$ on cluster scales (Abell 3827, Bullet Cluster) after averaging the
DM self interaction cross sections over DM velocity distribution in the
cluster. This puts interesting constraints on the values of effective DM self
couplings for different fermionic DM masses for various effective operators
(scalar, vector, etc.) of DM self interactions in this scenario. Some other
implications of DM effective self interaction are also discussed in this model
independent framework.
