Beginning of Matter, Start to Existence Itself

When the inequality

μ+3p/c2 >0 ⇔ w > −1/3

is satisfied, one obtains directly from the Raychaudhuri equation

3S ̈/S = -1/2 κ(μ +3p/c²) + Λ

the Friedmann-Lemaître (FL) Universe Singularity Theorem, which states that:

In a FL universe with Λ ≤ 0 and μ + 3p/c² > 0 at all times, at any instant t₀ when H₀ ≡ (S ̇/S)₀ > 0 there is a finite time t_∗: t₀ − (1/H₀) < t_∗ < t₀, such that S(t) → 0 as t → t_∗; the universe starts at a space-time singularity there, with μ → ∞ and T → ∞ if μ + p/c² > 0.

This is not merely a start to matter – it is a start to space, to time, to physics itself. It is the most dramatic event in the history of the universe: it is the start of existence of everything. The underlying physical feature is the non-linear nature of the Einstein’s Field Equations (EFE): going back into the past, the more the universe contracts, the higher the active gravitational density, causing it to contract even more. The pressure p that one might have hoped would help stave off the collapse makes it even worse because (consequent on the form of the EFE) p enters algebraically into the Raychaudhuri equation with the same sign as the energy density μ. Note that the Hubble constant gives an estimate of the age of the universe: the time τ₀ = t₀ − t_∗ since the start of the universe is less than 1/H₀.

This conclusion can in principle be avoided by a cosmological constant, but in practice this cannot work because we know the universe has expanded by at least a ratio of 11, as we have seen objects at a redshift 6 of 10, the cosmological constant would have to have an effective magnitude at least 11³ = 1331 times the present matter density to dominate and cause a turn-around then or at any earlier time, and so would be much bigger than its observed present upper limit (of the same order as the present matter density). Accordingly, no turnaround is possible while classical physics holds. However energy-violating matter components such as a scalar field can avoid this conclusion, if they dominate at early enough times; but this can only be when quantum fields are significant, when the universe was at least 10¹² smaller than at present.

Because T_rad ∝ S⁻¹, a major conclusion is that a Hot Big Bang must have occurred; densities and temperatures must have risen at least to high enough energies that quantum fields were significant, at something like the GUT energy. The universe must have reached those extreme temperatures and energies at which classical theory breaks down.

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