Negative Cosmological Constant and the Impossibility of Structure

This analysis explores the specific condition where a negative cosmological constant (Λ) renders the universe so hostile that no structures, not even subatomic particles like hadrons, can form.

The Fundamental Conflict

The formation of any structure is a battle between attractive local forces (like gravity and the strong nuclear force) and the global expansion or contraction of the universe. A negative cosmological constant acts as a powerful, global attractive force, dramatically accelerating the collapse of the universe. To prevent all structure formation, this "collapse pressure" must be so intense that it overwhelms the local forces that would otherwise bind particles together.

The Critical Comparison: Timescales

The answer is found by comparing two fundamental timescales.

1. The Gravitational Collapse Timescale

This is the time for the universe to recollapse to a "Big Crunch," driven by the negative Λ. A stronger, more negative Λ leads to a faster collapse. This timescale is approximated by the formula: t_collapse ≈ 1 / (c * √|Λ|).

2. The Hadron Formation Timescale

This is the time required for quarks and gluons to condense into stable hadrons like protons and neutrons. In the early universe, this process occurred when the universe was approximately 10⁻⁶ to 10⁻⁵ seconds old. This is the minimum timescale needed for the strong nuclear force to create the most basic building blocks of matter.

The Condition for No Structure

For no hadrons to form, the universe must begin its recollapse before quarks can coalesce. The critical condition is therefore:

t_collapse ≤ t_hadron

If the collapse timescale is shorter than the hadron formation timescale, the universe collapses into a singularity before protons and neutrons can ever exist.

The Numerical Threshold

Using the condition above and a representative hadron formation time of t_hadron ≈ 10⁻⁵ seconds, we can calculate the critical value for the cosmological constant.

Setting the timescales equal: 1 / (c * √|Λ|) ≈ 10⁻⁵ s

Solving for |Λ| gives an approximate threshold value of:

|Λ| ≈ 10⁻⁷ m⁻²

Conclusion

The degree to which a negative cosmological constant prevents all structure formation is determined by its magnitude. When |Λ| > 10⁻⁷ m⁻², the collapse timescale of the universe drops below approximately 10⁻⁵ seconds.

In such a universe, the relentless global attraction is the dominant physical phenomenon from the first instant. The entire cosmic lifecycle—from Big Bang to Big Crunch—is over in microseconds. The universe never cools down or exists long enough for the quark-gluon plasma to transition into hadrons. The formation of any structure, from hadrons to galaxies, becomes impossible.