What is the explanation for rapid inflation just after the big bang? I often hear about cosmic inflation occurring rapidly after the Big Bang, but have never came across any explanation for it. What supposedly caused this inflationary period in the early universe? Also, my assumption is that the laws of physics, i.e. light speed and relativity, did not exist during inflation, but were after effects. Is that a correct assumption? Thanks!
With normal matter, the strength of gravity depends on the stress-energy tensor, which, in an isotropic homogeneous universe, has trace of $\rho + 3p$.  Note that the pressure from relativistic particles (photons and neutrinos) enhances the mass density in slowing down expansion.   In inflation theory, it is assumed that there was a scalar field (perhaps) in a small section of the universe that began in a false vacuum state. That is, the field's energy density remained roughly constant as the universe began to inflate, and therefore the total energy of this field grew as the volume of the universe grew.  The field's pressure was negative and greatly overwhelmed the inward force of the density of normal matter and pressure. It is this negative pressure that rapidly expanded this section of the universe by many factors.
Energy was conserved because work was done by the gravitational field as it pressed inward against the field's outward pressure; the gravitational potential became very negative, while the energy density in the field became very large.  Eventually, the field decayed away, heating the particles, which resulted in the traditional Big Bang conditions of a hot expanding universe.
Throughout this theory, standard physics is expected to hold with just the addition of this new scalar field.  It is introduced because it helps to explain why the entire visible universe today is so homogeneous even though there are subregions that were previously thought to have not been in causal contact until quite recently.  It also explains why the universe is very nearly flat.  It converted a small section of the universe to extremely large radii and thus small curvature.
You can read about this from one of its inventors here