Yes, time dilation plays a significant role in allowing the universe to reach its current cosmic size without violating the theory of special relativity. The expansion of the universe is governed by the theory of general relativity, which takes into account both space and time.
According to general relativity, the fabric of spacetime itself can expand, causing the distances between galaxies to increase over time. As the universe expands, the light emitted by distant objects, such as galaxies, also undergoes a stretching of its wavelength, resulting in a phenomenon known as cosmological redshift.
As objects move away from each other due to the expansion of space, the relative velocities between them increase. This increase in velocity leads to time dilation, as predicted by special relativity. In other words, as galaxies move away from us, the light they emit experiences time dilation, causing it to appear redshifted when observed from our reference frame.
Time dilation due to the expansion of the universe ensures that the cosmic size can increase without violating the principles of special relativity. It allows for a consistent framework where objects can recede from each other at speeds greater than the speed of light (as measured by their relative velocities), without violating the universal speed limit set by special relativity.
It is important to note that this cosmic expansion and the associated time dilation apply to the large-scale structure of the universe and the vast distances between galaxies. On smaller scales, such as within galaxies or within our solar system, the effects of time dilation due to motion are negligible and can be ignored for practical purposes.