## Inflationary scenarios

In the case of inflation, the idea is that the exponential expansion obliterates the dependence on the initial conditions [1], so we would not need to know exactly how the Universe began - just that it was inflating. To lose all memory of the initial state would require an infinite amount of exponential expansion, which leads to the notion of ever-lasting or eternal inflation [2, 3]. The original argument for eternal inflation went as follows. Consider a massive scalar field in a spatially infinite expanding universe. Suppose the field is nearly constant over several horizon regions on a spacelike surface. In an infinite universe, there will always be such regions. The scalar field will have quantum fluctuations. In half the regions, the fluctuations will increase the field; in the other half, they will decrease it. In the half where the field jumps up, the extra energy density will cause the universe to expand faster than in the half where the field jumps down. After a certain proper time, more than half the regions will have the higher value of the field, because the high-field regions will expand faster than the low-field regions.

Thus the volume-averaged value of the field will rise. There will always be regions of the universe in which the scalar field is high, so inflation is eternal. The regions in which the scalar field fluctuates downwards will branch off from the eternally inflating region and exit inflation. Because there will be an infinite number of exiting regions, advocates of eternal inflation get themselves tied in knots on what a typical observer would see. So even if eternal inflation worked, it would not explain why the Universe is the way it is.

In fact, the argument for eternal inflation that I have outlined has serious flaws. First, it is not gauge-invariant. If one takes the time surfaces to be surfaces of constant volume increase rather than surfaces of constant proper time, the volume-averaged scalar field does not increase. Second, it is not consistent. The equation relating the expansion rate to the energy density is an integral of motion. Neither side of the equation can fluctuate because energy is conserved. Third, it is not covariant. It is based on a 3 + 1 split. From a 4-dimensional view, eternal inflation can only be de Sitter space with bubbles. The energy-momentum tensor of the fluctuations in a single scalar field is not large enough to support a de Sitter space, except possibly at the Planck scale, where everything breaks down. For these reasons -lack of gauge-invariance and covariance and inconsistency - I do not believe the usual argument for eternal inflation. However, as I shall explain later, I think the Universe may have had an initial de Sitter stage considerably longer than the Planck timescale.