Borde–Guth–Vilenkin theorem

The Borde–Guth–Vilenkin theorem, or the BGV theorem, is a theorem in physical cosmology which deduces that any universe that has, on average, been expanding throughout its history cannot be infinite in the past but must have a past spacetime boundary. The theorem does not assume any specific mass content of the universe and it does not require gravity to be described by Einstein field equations. It is named after the authors Arvind Borde, Alan Guth and Alexander Vilenkin, who developed its mathematical formulation in 2003. The BGV theorem is also popular outside physics, especially in religious and philosophical debates. Here is an example of derivation of the BGV theorem for an expanding homogeneous isotropic flat universe (in units of speed of light c=1). Which is consistent with ΛCDM model, the current model of cosmology. However, this derivation can be generalized to an arbitrary space-time with no appeal to homogeneity or isotropy. The Friedmann–Lemaître–Robertson–Walker metric is given by where t is time, xi (i=1,2,3) are the spatial coordinates and a(t) is the scale factor. Along a timeline geodesic xi = constant, we can consider the universe to be filled with comoving particles. For an observer with proper time τ following the world line xμ(τ), has a 4-momentum , where is the energy, m is the mass and p=|p| the magnitude of the 3-momentum. From the geodesic equation of motion, it follows that where pf is the final momentum at time tf. Thus where ti is an initial time, is the Hubble parameter, and γ being the Lorentz factor. For any non-comoving observer γ>1 and F(γ)>0. The expansion rate averaged over the observer world line can be defined as Assuming it is follows that Thereby any non-comoving past-directed timelike geodesic satisfying the condition , must have a finite proper length, and so must be past-incomplete. Alternative models, where the average expansion of the universe throughout its history does not hold, have been proposed under the notions of emergent spacetime, eternal inflation, and cyclic models.