Is the expansion of the universe caused by time?
No, not really. Space and time came into existence during the first micro-seconds after the Big Bang. Dark energy is the most likely suspect now.
Something called Dark Energy is now the leading theory of what is driving expansion of the universe, though cosmologists themselves are not sure what Dark Energy might actually be. However, it must be some kind of force that has properties opposite to gravitational attraction which over billions of years would be trying to collapse the entire universe back into a singularity - but this doesn't seem to be happening.