Hadron Binding in de Sitter Space

Cosmic Expansion Effects: In de Sitter space with exponential expansion, the Hubble parameter \(H\) creates an effective repulsive force that can overcome binding forces between particles.

Bound State Disruption: For sufficiently large \(H\), the expansion prevents hadrons from combining into nuclei by stretching the potential well faster than binding interactions can act.

Length Scale Competition: Binding occurs only if the characteristic size of the bound state is much smaller than the Hubble radius \(H^{-1}\). When \(H^{-1}\) approaches hadronic scales (\(\sim 1\) fm), binding becomes impossible.

Effective Potential Modification: The relative motion equation gains a repulsive term:

\(\ddot{r} = -\frac{\nabla V}{m} + H^2 r\)

where \(H^2 r\) acts as a cosmic repulsion.

Energy Scale Criterion: Binding fails when the Hubble parameter exceeds the binding energy scale:

\(H \gtrsim \frac{\Delta E}{\hbar}\)

For QCD scales (\(\Delta E \sim 200\) MeV), this requires extremely large \(H\).

Schrödinger Problem Analogy: This is equivalent to solving the Schrödinger equation with modified potential:

\(V_{\text{eff}}(r) = V_{\text{binding}}(r) - \frac{1}{2} m H^2 r^2\)

The repulsive \(-H^2 r^2\) term destroys bound states when dominant.

Quantum Mechanical Interpretation: The cosmological expansion creates an "inverted harmonic oscillator" potential that overwhelms the binding potential, making localized bound states impossible.

Conclusion: In de Sitter space with large Hubble parameter \(H\), hadrons cannot form nuclei because cosmic expansion introduces effective repulsion that destroys bound states. This represents a cosmological modification of the quantum mechanical binding problem where the confining potential is overwhelmed by expansion effects.