We have derived the first, fully cosmological, similarity solutions for cold dark matter ( CDM) halo formation in the presence of non-gravitational collisionality (i.e. elastic scattering), which provides an analytical theory of the effect of the self-interacting dark matter (SIDM) hypothesis on halo density profiles. Collisions transport heat inward, which flattens the central cusp of the CDM density profile to produce a constant-density core, while continuous infall pumps energy into the halo to stabilize the core against gravothermal catastrophe. This is contrary to previous analyses based upon isolated haloes, which predict core collapse within a Hubble time. These solutions improve upon earlier attempts to model the formation and evolution of SIDM haloes, offer deeper insight than existing N-body experiments, and yield a more precise determination of the dependence of halo density profile on the value of the CDM self-interaction cross-section. Different solutions arise for different values of the dimensionless collisionality parameter Q equivalent to sigma rho br(vir) alpha r(vir)/gamma(mfp), where sigma is the scattering cross- section per unit mass, rho b is the cosmic mean matter density, r(vir) is halo virial radius and gamma mfp is the collision mean free path. The maximum flattening of central density occurs for an intermediate value of Q, Q(th), at which the halo is maximally relaxed to isothermality. The density profiles with constant-density cores preferred by dwarf and low surface brightness (LSB) galaxy rotation curves are best fit by the maximally flattened (Q = Q(th)) solution. If we assume that dwarfs and LSB galaxies formed at their typical collapse epoch in Lambda CDM, then the value of sigma which makes Q = Q(th) is sigma similar or equal to 200 cm(2) g(-1), much higher than previous estimates, sigma similar or equal to [0.5 - 5] cm(2) g(-1), based on N-body experiments. If sigma is independent of collision velocity, then the same value sigma similar or equal to 200 cm(2) g(-1) would make Q > Q(th) for clusters, which typically formed only recently, resulting in relatively less flattening of their central density profile and a smaller core.
