Non-Singular Bouncing Cosmology from Hyperbolic Field Space Geometry

A ghost-free, NEC-satisfying cosmological model that replaces the Big Bang singularity with a geometric bounce.

This repository contains the code and LaTeX source for the paper:

"Non-Singular Bouncing Cosmology from Hyperbolic Field Space Geometry" by Oleksandr Kravchenko, OkMath Organization

Abstract

We investigate a two-field cosmological model in a closed ($k=+1$) universe where the field space is endowed with a hyperbolic geometry. We demonstrate that the curvature of the field space introduces a kinetic coupling that exponentially suppresses the scalar field kinetic energy, allowing the spatial curvature to dominate and trigger a non-singular bounce. Crucially, the model satisfies the Null Energy Condition (NEC) throughout, with the bounce driven entirely by the positive spatial curvature—not by exotic physics.

Key Results

• Ghost-free: The model has positive-definite kinetic matrix for all finite field values
• NEC-satisfying: $\rho + p \geq 0$ throughout the evolution
• Non-singular: Scale factor remains positive: $a(t) \geq a_{\min} > 0$
• Predictions: $r \approx 0.003-0.005$, $n_s \approx 0.96-0.97$, $f_{\rm NL}^{\rm local} \sim 1$

The Mechanism

The key equations in a closed universe ($k = +1$):

Friedmann constraint: $$H^2 = \frac{\rho}{3M_{\rm Pl}^2} - \frac{1}{a^2}$$

Acceleration equation: $$\dot{H} = -\frac{\rho + p}{2M_{\rm Pl}^2} + \frac{1}{a^2}$$

The $+1/a^2$ term from spatial curvature enables $\dot{H} > 0$ even when the NEC is satisfied!

Files

• main.tex - LaTeX source of the paper
• Bouncing_Cosmology_by_Oleksandr_Kravchenko_OkMathOrg.pdf - Compiled paper with embedded figures
• numerical_solution.py - Python code for numerical solutions
• verify_christoffel.py - Symbolic verification of field-space geometry
• bounce_closed.pdf - Figure 1: Bounce solution
• kinetic_suppression.pdf - Figure 2: Kinetic suppression mechanism

Dependencies and Installation

• Python 3.x
• numpy, scipy, matplotlib (for numerical solution)
• sympy (only for symbolic verification script)

Install via pip:

pip install numpy scipy matplotlib sympy

Running the Code:

python numerical_solution.py

This will generate all figures and print a summary of the bounce solution.

Note: Variable names in numerical_solution.py will confuse you 😸

Reference

If you use this code in your research, please cite the following paper:

Non-Singular Bouncing Cosmology from Hyperbolic Field Space Geometry Oleksandr Kravchenko arXiv:2511.18522 [gr-qc]
https://arxiv.org/abs/2511.18522

Citation (BibTeX)

@article{Kravchenko2025,
    title = {Non-Singular Bouncing Cosmology from Hyperbolic Field Space Geometry},
    author = {Kravchenko, Oleksandr},
    year = {2025},
    eprint = {2511.18522},
    archivePrefix = {arXiv},
    primaryClass = {gr-qc},
    url = {https://arxiv.org/abs/2511.18522}
}

Contact

For questions or comments, please open an issue or contact the author.