The paradigm (i.e., a grand vaguely specified theory) of inflation was introduced in 1979--1981 and significant evidence as accrued for it. But alternative theories exist.
By the by, the false vacuum universe may also be the multiverse. But that is a very speculative theory. Maybe theory of everything (TOE) (if it is ever discovered) will prove or disprove the multiverse paradigm.
Immediately after this phase, mass-energy of the observable universe consisted of the matter (i.e., the aforementioned primoridal light elements), cosmic background radiation (CBR) (which was a blackbody radiation field in in thermodynamic equilibrium with the matter), cosmic neutrino background, dark matter and and dark energy (which is a name for our ignorance of the cause of the accelerating universe).
The fluctuations led to clumping where density was high and declumping where it was low It was a case of the rich getting richer and the poor getting poorer.
The overall process is called structure formation (AKA large-scale structure formation).
Structure formation created galaxies, galaxy clusters, galaxy superclusters, galaxy filaments, and cosmic voids: i.e., all the large-scale structure of the universe which nowadays is often called cosmic web which is NOT quite a synonym for large-scale structure since it is the particular form of large-scale structure we actually have.
The galaxies, in fact, formed in gravitational wells of dark matter which we call dark matter halos.
Of course, in the galaxies, there was star formation.
The expansion of the universe continues to spread apart gravitationally unbound systems.
Within gravitationally bound systems (i.e., galaxies and galaxy clusters), the galaxies continue to evolve by galaxy mergers and galaxy quenching (i.e., the turning off of star formation). See galaxy formation and evolution.
UNDER CONSTRUCTION BELOW
The Big Bang and evolution since then form a well established theory: it would be extremely surprising if that theory were just plain wrong.
Cosmic evolution before the Big Bang (i.e., the multiverse and inflation) is much more speculative---one of those deep mysteries of modern physics.
It wouldn't be extremely surprising if the multiverse and inflation were just plain wrong.
Theories have different statuses as mentioned above.