Cosmological Horizons, Singularities, and Global Evolution

How far signals can travel and how long freely falling worldlines continue are different questions. Cosmic history can have a finite conformal boundary without a finite proper-time boundary, and a coordinate patch can be geodesically incomplete even when its curvature stays finite. This chapter develops the tests that distinguish these possibilities, then applies them to recollapse, future singularities, bounces, cyclic histories and emergent phases. The proposed alternatives remain conditional on their matter model, global domain and stability assumptions.

Helpful background. Einstein Equations connect geometry to stress energy, Relativistic Matter and Compact Objects supplies energy conditions, Causality and Global Structure supplies horizon and completeness language, Relativistic Cosmology supplies FLRW dynamics, and Mathematical Relativity supplies theorem and extension methods.

Enter this chapter

The chapter follows a deliberate order:

1. Particle Horizons and Event Horizons compares the past and future conformal-time integrals and separates both from the Hubble radius.
2. Geodesic Completeness in Cosmology replaces coordinate intuition by affine- and proper-time tests, curvature diagnostics and extension questions.
3. Using Singularity Theorems in Cosmology tests the exact hypotheses and conclusion of one Hawking cosmological theorem.
4. Future Expansion, Recollapse, and Singular Fates classifies constant-equation-of-state futures, turnaround and distinct finite-time endpoints.
5. Bounces and Nonsingular Cosmologies adds perturbative stability, strong coupling, anisotropy and transition dynamics to the background bounce condition.
6. Cyclic and Emergent Scenarios asks whether those controls survive arbitrarily many cycles or an infinite loitering past.

Read the first two pages together: the same scale factor can give different answers to a horizon test and a completeness test. Their worked examples prepare you to interpret the singularity theorem on the third page. The first four pages establish durable analytic tools. The last two examine research proposals: their evidence comparisons are fixed at 2026-08-19 and their conclusions remain conditional on a declared theory, data class and approximation regime.

Four different endpoint questions

Cosmic time $t$ is the proper time of a comoving observer. Conformal time instead measures comoving light travel, through $d\eta=dt/a$. Light has zero proper time, so its geodesic length is tested with an affine parameter $\lambda$, chosen so that the ray’s tangent is parallel transported along it. A finite nonzero change of affine normalization cannot turn a finite interval into an infinite one.

For a positive FLRW scale factor $a(t)$, the relevant integrals are

$$\chi_{\mathrm p}(t)=\int_{t_i}^{t}\frac{dt'}{a(t')}, \qquad \chi_{\mathrm e}(t)=\int_t^{t_f}\frac{dt'}{a(t')},$$ $$\Delta\lambda_{\mathrm{null}}\propto\int a(t)\,dt, \qquad \Delta\tau_{\mathrm{timelike}} =\int\frac{dt}{\sqrt{1+P^2/a^2(t)}}.$$

The first two measure past and future causal reach; whether they define an actual horizon also depends on the spatial topology. The last two test null affine length and timelike proper length in a flat FLRW chart, with each integral taken toward the endpoint under study. For the timelike geodesic, $P$ is the conserved comoving momentum per unit mass; $P=0$ gives a comoving clock. Completeness requires every inextendible geodesic of the specified causal type to have unbounded parameter length, not only the comoving ones. Tong’s conformal diagrams provide the horizon interpretation Tong 2019, § 1.1.4, pp. 15–19 (PDF); Geodesic Completeness in Cosmology derives the separate geodesic integrals.

Question	Primary diagnostic	What it does not establish
Could a signal have reached us?	Past conformal-time integral	Curvature regularity or past completeness
Can our signal ever arrive?	Future conformal-time integral	A local property of $H(t)$
Does a geodesic have finite parameter length?	Affine or proper-time integral	Inextendibility or curvature blow-up
Must incompleteness occur?	Exact theorem hypotheses	The type, location or uniqueness of a singular boundary

This separation prevents the common inference that an event horizon, a shrinking Hubble radius or a vanishing scale factor are interchangeable statements.

From local equations to global conclusions

The Friedmann equations can establish a turnaround or bounce locally. At a flat-space bounce, $H=0$ and $\dot H\gt0$ imply $\rho+p\lt0$ in general relativity. At a turnaround, $H=0$ and $\dot H\lt0$ signal passage from expansion to contraction. Neither calculation proves that the entire spacetime is complete or that perturbations survive.

Global conclusions require additional hypotheses. The Hawking theorem used here assumes four dimensions, timelike convergence, strong causality, an edgeless compact achronal spacelike hypersurface, and positive initial expansion of its orthogonal timelike geodesics. It concludes past timelike incompleteness—nothing stronger. The theorem and proof appear in Senovilla 1998, Theorem 5.5, pp. 790–791 (PDF). Global de Sitter evades the conclusion by violating timelike convergence, while flat de Sitter coordinates illustrate how an incomplete patch can admit a regular extension.

For research scenarios, the checklist grows: specify the action and cutoff; solve the constraints; evolve scalar, vector and tensor modes; compare physical frequencies with the strong-coupling scale; control shear and inhomogeneity; derive matching; and evaluate observables with their degeneracies. Brandenberger and Peter’s survey makes clear why a finite background scale factor is only one part of the problem Brandenberger and Peter 2017, §§ V–VI, pp. 18–23 (PDF).

A practical diagnostic sequence

When given a proposed cosmic history, begin by stating the manifold, coordinate range, scale-factor normalization, spatial curvature and matter model. Then:

1. Evaluate past and future conformal time.
2. Integrate representative null and timelike geodesics.
3. Check curvature and tidal quantities at each endpoint.
4. Test the hypotheses of any cited theorem literally.
5. For a bounce or cycle, inspect kinetic and gradient matrices, the EFT cutoff, anisotropy scaling and multicycle transfer.
6. State which claim remains unproved: extension, nonlinear stability, genericity, initial state or observational discrimination.

Proof-level mathematical relativity and theory-specific modified-gravity construction remain outside this chapter. The global reasoning developed here helps you identify what a cosmological model actually demonstrates and when a more detailed proof or theory is needed.

Review the chapter

Consider a spatially flat universe with noncompact spatial slices and positive constants $a_b,t_b$, with scale factor

$$a(t)=a_b\left(1+\frac{t^2}{t_b^2}\right), \qquad -\infty\lt t\lt\infty.$$

Show that it is a bounce, determine whether past and future particle/event horizon integrals converge, and test radial-null affine completeness. What does flat-space general relativity imply about the null energy condition at the bounce?

Solution

Here $H=2t/(t_b^2+t^2)$, so $H(0)=0$ and $\dot H(0)=2/t_b^2\gt0$: $t=0$ is a smooth bounce. Since $1/a\sim t^{-2}$ at both infinities, both past and future conformal integrals converge, giving particle and event horizons. By contrast, the null affine integral $\int a\,dt$ diverges in both directions, so radial null geodesics have infinite affine length. Finally, for $K=0$, $\dot H=-4\pi G(\rho+p)$; positive $\dot H(0)$ requires $\rho+p\lt0$. Horizons, completeness and energy-condition violation are therefore logically distinct.

References

• Brandenberger, Robert, and Patrick Peter. “Bouncing Cosmologies: Progress and Problems.” Foundations of Physics 47 (2017): 797–850. DOI. Open PDF.
• Senovilla, José M. M. “Singularity Theorems and Their Consequences.” General Relativity and Gravitation 30 (1998): 701–848, corrected 2018 version. DOI. Open PDF.
• Tong, David. Cosmology. University of Cambridge Part II Mathematical Tripos lecture notes, Michaelmas 2019. Official PDF.