In high school, we learn to prove congruent triangles from first principles; similarly in cosmology, indirect yet powerful arguments can be made on the basis of little or no dynamics. A singularity in general relativity is characterised by 'geodesic incompleteness' where time-like and null geodesics can't be extended to the infinite past or future but terminate after a finite proper time boundary. The Friedman equations hold that in an increasingly homogeneous and isotropic universe (such as ours), there was once a time when distances between particles was zero. And since Einstein's field equations hold that a given distribution of energy and momentum is proportional to the geometric properties of spacetime (particularly its curvature), the initial zero volume causes the curvature to become infinite. Crucial to the understanding of the singularity is the notion of a geodesic, which is the worldline of the shortest path traced by a particle only subject to gravitation (the curvature of space time) and not any other force; because the worldline is shorter when not accelerated, a geodesic may also be defined as the worldline where the four acceleration is zero (in the language of special relativity). Like the moon orbiting around the earth, it is just tracing the shortest path via curved spacetime ie. a geodesic. A geodesic is to a sphere, what a straight line is to a flat surface; a straight line and a geodesic are both the shortest distance between two points, but unlike straight lines, geodesics are not infinite in length (they are closed and always circle back on themselves) and they are never parallel. On a globe, only the equator and lines of longitude are geodesics. If we follow the path of a particle back in time to the point of infinite density, the geodesic will terminate, and the particle will cease to exist and will not anymore be part of spacetime. Identifying unextendable geodesics is used to identify singularities in theorems. So if the universe is expanding, in fact, accelerating in its expansion, does it follow that we can extrapolate back to a point when all galaxies and clusters were packed into a single point? In the 1960s, it was thought that any form of matter would adhere to the strong energy condition (suggests a tendency for geodesics to merge to guarantee that gravity is attractive:
pc^2+3p>0 ) and because the universe is not entirely homogenous and isotropic as the Friedman-Robertson-Walker metric holds (it is only so on large scales), it is theoretically possible to avoid singularities if particles of mass miss each other when you trace their worldlines back in time (big bounce). Hawking and Penrose, assuming a strong energy condition (and causality conditions on global structure like no closed time-like curves to prevent time travel) proved that you can have geodesic incompleteness in a space-time that is not completely homogenous and isotropic, thus the presence of singularities is extremely generic even in black holes. But inflationary space-times don't abide by the strong energy condition and in fact violate it, so the Hawking-Penrose singularity theorems didn't apply to them.
The kinematic incompleteness theorem of Borde, Guth and Vilenkin proved that inflationary space-times are not past geodesically complete and this indicates under general relativity an initial singularity. They showed an integral going from
a(t) distant past to now is bounded (only condition assumed is Hubble constant over zero) and is purely geometric, not assuming any dynamics such as energy conditions. The momentum of an object or test particle travelling on a geodesic is red-shifted in an expanding universe and extrapolating back into the past causes the geodesic to blue-shift, their theorem showed that the blue-shift reaches the velocity of light in a finite proper time (or affine parameter for photons) showing such a trajectory to be geodesically incomplete. Inflation is an exponential phase of de Sitter expansion, a full de Sitter space can't be past-eternal as it would experience a contracting phase preceding it that would cause tiny perturbations preventing the future expansion of the universe. Eternal inflation and cyclic models (which lead to thermal death) are both past geodesically incomplete and the emergent model (which assumes a closed and static universe in the asymptotic past) can collapse quantum mechanically, so they can’t be infinite into the past, even multidimensional brane models can't be extended indefinitely into the past. So its safe to say the universe had a beginning that is if you don't consider the
subtleties...
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