According to current scientific understanding, the speed of gravity is believed to be the same as the speed of light. This means that gravitational interactions propagate at the speed of light. However, it's important to clarify that this refers to the speed at which changes in the gravitational field propagate, not the speed at which gravitational waves themselves travel.
Now, regarding the concept of inflation in the early universe, inflationary theory suggests that the universe underwent a period of rapid expansion in its earliest moments. During this inflationary epoch, it is postulated that the universe expanded faster than the speed of light, leading to a significant increase in its size.
The important thing to note is that during inflation, it is not matter or energy that is moving through space faster than light, but rather the expansion of space itself. In this scenario, the fabric of space itself is stretching, carrying distant regions away from each other at speeds greater than the speed of light. This expansion is not bound by the restrictions of special relativity, which prohibits objects with mass from traveling at or exceeding the speed of light.
During inflation, the laws of physics, including the behavior of gravity, can be modified. It is believed that during this period, gravitational effects were also subject to the rapid expansion of space. While gravity is presumed to have mediated the expansion, the details of how gravity exactly operated during inflation are still an active area of research.
Therefore, the assumption that the speed of gravity is equal to the speed of light does not necessarily conflict with the concept of inflation in the early universe, where space itself expanded faster than the speed of light.