From the formation of subatomic particles to the formation of atoms to the nuclear fusion in stars to the explosions of supernovae, the process of conversion of rest-mass energy from fields into momentum divergence carried outward in all directions by massive particles and photons has been ongoing in the history of the expanding mass distribution we call the Universe. This is just the same process that occurs in every kind of explosion we are familiar with, from firecrackers to H-bombs. One might suppose that this process could make at least some contribution to the expansion of the Universe, but it is never discussed in Big Bang models.
Perhaps the reason for this is that this momentum divergence has no effect on the expansion because the expanding mass distribution is infinite in extent, so the resulting incoming momentum at any point in the distribution is equal in all directions and thus cancels. However, how are we so sure that the distribution is infinite? When I asked a cosmologist about this about 15 years ago, I was told that the uniformity of the COBE data ruled out a finite mass distribution, and the "cosmological principle" compels us to believe that what we see around us as reflected by the COBE data must accurately describe the mass distribution of the Universe as a whole. However, I'm skeptical about this for two main reasons.
The first reason is that if we had invoked a "cosmological principle" to predict the mass distribution of the Universe in 1910, when it was thought that the distribution of stars around us in the Milky Way was representative of the entire Universe because we knew of no other galaxies, then we would have been very wrong. Of course he history of astronomy also has earlier examples of our underestimation of the scale and nature of the entire mass distribution. So how do we know that is now safe to apply a "cosmological principle"? Why can't we be part of a very large but finite expanding distribution, possibly with other neighbors like it?
The second reason I'm skeptical is that more recent measurements of the microwave background are apparently showing more significant departures from homogeneity/isotropy than COBE did. Does this make it more likely that our expanding mass distribution is finite?
If the distribution is finite, then momentum divergence from rest-mass energy conversion may be a factor in the expansion. The "Big Bang" might just be the biggest example of an ordinary, though extended, explosion with particles, atoms, and stars forming, exploding, then contributing to next-generation stars that incorporate the outward momentum. In this model, the expansion rate would naturally accelerate as more momentum diverge is released.
This model of the Big Bang as an ordinary extended explosion, if it could work, seems far simpler than a Universe that inflates as if by magic due to an unknown force, then accelerates its expansion due to a massive quantity of unknown repulsive energy. I've noticed for decades that cosmologists just seem to keep adding new variables as bolt-ons to preserve their models when new results require explanation, and it is disturbing that they argue that these additional parameters should be considered strengths of their models, ignoring the principle that if a theory is not falsifiable then it becomes useful only as a way of organizing results, not achieving an understanding that can lead to predictions.