There is a theorem due to Borde, Guth, and Vilenkin which might be taken as evidence for a beginning of time.<\/p>\n
Roughly speaking, this theorem<\/a> says that in any expanding cosmology, spacetime has to be incomplete to the past.\u00a0 In other words, the BGV theorem tells us that while there might be an “eternal inflation” scenario where inflation lasts forever to the future, inflation still has to have had some type of beginning in the past.\u00a0 BGV show that “nearly all” geodesics hit some type of beginning of the spacetime, although there may be some which can be extended infinitely far back to the past.<\/p>\n If we assume that the universe was always expanding, so that the BGV theorem applies, then presumably there must have been some type of initial singularity.<\/p>\n The fine-print (some readers may wish to skip this section):<\/strong> More precisely, let \\(H\\) be the Hubble constant<\/a> which says how rapidly the universe is expanding.\u00a0 In general this is not a fully coordinate-invariant<\/a> notion, but BGV get around that by imagining a bunch of “comoving observers”, one at each spatial position, and defining the Hubble constant by the rate at which these observers are expanding away from each other.\u00a0 The comoving observers are assumed to follow the path of geodesics, i.e. paths through spacetime which are as straight as possible, that is without any acceleration.<\/p>\n Now let us consider a different type of geodesic\u2014the path taken by a lightray through spacetime.\u00a0 Now if the average <\/em>value \\(H_\\mathrm{avg}\\) along some lightlike geodesic is positive, then BGV prove that it must reach a boundary of the expanding region in a finite amount of time.\u00a0 In other words, these lightlike geodesics reach all the way back to some type of “beginning of time” (or at least the beginning of the expanding region of spacetime which we are considering).<\/p>\n We can also consider timelike <\/em>geodesics, describing the motion of particles travelling at less than the speed of light.\u00a0 For nearly all <\/em>timelike geodesics, if \\(H_\\mathrm{avg} > 0 \\) then that geodesic also begins at a beginning of time.\u00a0 However, the theorem only applies to geodesics which are moving at a finite velocity with respect to the original geodesics which we used to define \\(H_\\mathrm{avg}\\).\u00a0 The original set of observers is <\/em>allowed to extend back infinitely far back in time.<\/p>\n
\n[<\/strong>BGV do not need to assume that the universe is homogeneous (the same everywhere on average) or isotropic (the same in each direction on average).\u00a0 Although the universe does seem to be homogeneous and isotropic so far as we can tell, they don’t use this assumption.<\/p>\n