In Fig. 20.4(b) we can see how the presence of gravity in an expanding universe affects the relationship between the Hubble time (1/H0) and the true age of the universe. The presence of gravity means that the expansion has been slowing, so the universe was expanding faster in the past. That means it took less time to reach its current size than we would estimate from the current expansion rate. From Fig. 20.4(b) we see that H0 is proportional to the current slope of R(t).

Having established that the universe is expanding, we would now like to ask whether that expansion will continue forever. In other words, is the universe open or closed? We would like to have some quantity that we can measure to tell us. If we look back to our analogy of the ball thrown up in the air, if we know the position and velocity of the ball at some time, we also need to know its acceleration to know if the ball has sufficient energy to escape. Since the ball is slowing down, we want to know the deceleration of the ball.

For the universe, we define a deceleration parameter, whose value will tell us whether the universe is open or closed. We would like to define this parameter so that it is dimension-less (just as the scale factor is dimensionless), and is independent of the time t0 that we choose for our reference epoch. The latter requirement says that the parameter should depend on quantities such as R/R and R/R. With these ideas in mind, we define the deceleration parameter as q =

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