Cyclic and Emergent Scenarios

A repeated scale factor is not yet a cyclic universe, and an infinite coordinate past is not yet a past-complete spacetime. Cyclic and emergent scenarios must control entropy, anisotropy and perturbations over arbitrarily many epochs while supplying a regular transition law. Their strongest tests are cumulative: a tiny amplification or entropy increase per cycle can dominate after many repetitions.

Helpful background. Einstein Equations fix the classical background constraints, Relativistic Matter and Compact Objects supplies entropy and energy-condition language, Causality and Global Structure supplies completeness tests, Relativistic Cosmology supplies perturbation transfer, and Mathematical Relativity clarifies global extension and stability.

Repetition, emergence and conformal succession

A strictly periodic cosmology would satisfy $a(t+T)=a(t)$ together with periodic matter and perturbation data. An ekpyrotic cyclic model instead joins hot expansion, dark-energy dilution, slow contraction and a bounce. Some constructions repeat $H$, density and temperature while the scale factor grows by a fixed factor from cycle to cycle; they are locally recurrent but not periodic in $a$.

An emergent universe has no prior contraction. It approaches a nonzero static or loitering radius as $t\to-\infty$ and later exits into expansion. The original closed emergent construction approaches an Einstein-static state and can be geodesically complete, but it requires a special balance of positive curvature, matter and scalar potential Ellis–Maartens 2004, §§ II–IV.

Conformal cyclic cosmology proposes a different succession: the asymptotic conformal future of one aeon is identified with the conformal big-bang boundary of the next. This is neither an ordinary finite-$a$ bounce nor a periodic FLRW solution. It requires a dynamical account of which dimensionful matter data survive the conformal matching and how perturbations cross it.

Entropy makes exact cycles exceptional

Let $S_n$ be the thermodynamic entropy in a comoving domain after the $n$th reheating. Irreversible processes imply

$$S_{n+1}=S_n+\Delta S_n,\qquad \Delta S_n\ge0.$$

If every cycle returned to the same physical volume and temperature, increasing $S_n$ would contradict exact recurrence. A model can instead enlarge the physical volume assigned to the next observable patch. At fixed post-bounce entropy density $s_\star$,

$$\frac{a_{n+1}}{a_n} =\left(\frac{S_{n+1}}{S_n}\right)^{1/3}.$$

The “new kind” of cyclic universe uses net growth of $a$ so that total entropy increases globally while entropy inside a later Hubble patch is reset by dilution Ijjas–Steinhardt 2019, § 3.2. This is a coherent entropy-dilution mechanism, but its net expansion is also relevant to past completeness.

Black holes, relics and vacuum transitions sharpen the same issue. Diluting them beyond one observer’s horizon is not erasing global entropy. A complete multicycle claim must state the global spatial domain, which degrees of freedom are coarse-grained, and whether horizons merely hide records that still gravitate.

Anisotropy and perturbations accumulate

During contraction, shear scales as $a^{-6}$. Ekpyrotic matter with $w\gt1$ grows faster and can suppress homogeneous shear relative to its own density, but a repeated scenario must recover that smoothing basin every cycle and control inhomogeneous mixmaster behavior. The bounce must then remain stable for the resulting data class, not only for an exactly FLRW trajectory.

Perturbations provide a compact multicycle check. For each Fourier mode, write the linear state immediately after successive bounces as

$$\mathbf X_{n+1}(k)=M_{k,n}\mathbf X_n(k)+\mathbf N_n(k),$$

where $M_{k,n}$ is the transfer matrix through cycle $n$ and $\mathbf N_n$ represents newly generated fluctuations. A periodic background does not make perturbations periodic. If the same physical linear problem repeats, one may set $M_{k,n}=M_k$. A spectral radius of $M_k$ greater than one then gives a growing mode; if it is below one, new fluctuations can still approach a stochastic steady state rather than repeat exactly. For net-growing cycles, the physical wave number $k/a$ changes from cycle to cycle even at fixed comoving $k$. One must generally study the product of the actual $M_{k,n}$, or state the rescaling that makes a repeated operator possible. Matching must be derived from the field equations through the transition.

The spectral radius is the largest absolute value of an eigenvalue. Even a value exactly equal to one needs care: a nontrivial Jordan block can accumulate perturbations polynomially. For example,

$$M=\begin{pmatrix}1&\alpha\\0&1\end{pmatrix}, \qquad M^n=\begin{pmatrix}1&n\alpha\\0&1\end{pmatrix}.$$

Both eigenvalues are one, but the first component grows linearly whenever $\alpha X_{0,2}\ne0$. A stationary stochastic limit for eigenvalues strictly inside the unit circle additionally requires appropriate noise statistics, such as independent identically distributed noise with finite variance; it does not follow from the background alone.

Brandenberger and Peter point out that a four-dimensional cyclic repetition can reprocess an already scale-invariant spectrum into an unacceptable one unless successive cycles probe new scales Brandenberger and Peter 2017, § VI, pp. 22–23 (PDF). That is a negative result for naive repetition, not for every ekpyrotic construction.

Past completeness is an integral condition

Borde, Guth and Vilenkin relate positive averaged expansion along a past-directed null or timelike geodesic to finite affine or proper length, without assuming an energy condition Borde–Guth–Vilenkin 2003, Eqs. (1)–(6), pp. 1–2. For a generalized FLRW null ray the direct test remains

$$\Delta\lambda\propto\int_{-\infty}^{t_0}a(t)\,dt.$$

Net expansion per cycle means $a$ decreases geometrically toward the infinite past. If each cycle has bounded duration and shape after rescaling, the affine contributions form a convergent geometric series, so a nonsingular repeating background can still be past null incomplete.

The precise averaging hypothesis matters. Garcia-Saenz, Hua and Zhao exhibit a complete FLRW counterexample to an overly literal formulation of the original average and replace it with a uniform positive lower bound on every finite past segment Garcia-Saenz et al. 2024, §§II–III (PDF). A separate result of theirs states that a smooth, nonconstant generalized FLRW scale factor that is past geodesically complete and has $H(t_f)\gt0$ at some finite time must have an earlier interval with $\dot H\gt0$ Garcia-Saenz et al. 2024, §V, Theorem 4, p. 4 (PDF). This is a conditional statement about expansion history; it does not require the specific nonzero Einstein-static past described above. A theorem diagnosis must specify the geodesic, averaging interval, lower bound, and any assumption of expansion; “BGV applies” is not a substitute for the affine integral.

Results at the stated cutoff

The literature, theoretical and observational evidence below is assessed through 2026-08-19. The competing model classes evade different premises and do not inherit one another’s successes.

Scenario or result	Evidence or mechanism	Scope and limitation
Ekpyrotic cyclic contraction	$w\gt1$ suppresses homogeneous shear; entropic modes can seed curvature perturbations	Requires a stable bounce, a repeated smoothing basin and bounded multicycle transfer
Net-growing cyclic model	Expansion between cycles dilutes entropy, matter and black holes from a later Hubble patch	Not strictly periodic in $a$; geometric past scaling can make null affine length finite
Emergent Einstein-static past	Nonzero limiting radius can give past completeness and no particle-horizon problem	Linear scalar stability needs restricted sound speed; homogeneous Bianchi IX modes destabilize the static state
Conformal succession of aeons	Conformal matching avoids an ordinary contracting bounce	Matter-scale matching and quantitative primordial predictions remain model-dependent
CMB circles and Hawking-point searches	High-resolution Planck/WMAP search found no statistically significant claimed features	Null result constrains tested templates and search pipeline; it does not disprove every conformal-cyclic realization
Averaged-expansion incompleteness	Uniformly positive finite-segment average bounds past affine or proper length	Does not cover histories whose average tends to zero or changes sign; extension beyond an incomplete patch remains separate

The Einstein-static stability boundary is concrete: inhomogeneous adiabatic scalar modes are neutrally stable for $c_s^2\gt1/5$, while homogeneous Bianchi IX perturbations are unstable Barrow et al. 2003, §§ III–VI. The CMB null row is likewise bounded: Bodnia and collaborators removed bright/dark-pixel artifacts and found neither significant low-variance circles nor significant Hawking points in their Planck and WMAP tests Bodnia et al. 2024, §§ 2–5. These are useful discriminants, not a universal verdict.

The evidence does not establish a controlled past-eternal cosmology. Some model classes supply exact backgrounds, some control selected linear modes, and some evade a particular incompleteness hypothesis; no one of those achievements proves all the others. The unresolved axes are global entropy, nonlinear genericity, the transition’s EFT cutoff, perturbation accumulation and a distinctive likelihood-level prediction.

What would change the assessment

A complete demonstration should evolve entropy, curvature, shear and gauge-invariant perturbations through many cycles or through the entire emergent phase, using one declared action and derived transition rule. It should report geodesic integrals, stability spectra, strong-coupling scales, initial-data basin and observational transfer. A single-cycle background plot cannot establish a past-eternal attractor.

The interpretation could change after a controlled multicycle solution demonstrates bounded perturbations and entropy accounting for a broad initial-data class; after a sharpened completeness theorem or counterexample settles a disputed averaging condition; or after a distinctive cyclic or emergent prediction is detected with foreground and look-elsewhere controls.

Exercises

Suppose each cycle has duration $T$ and the scale factor profile is multiplied by $q\gt1$ toward the future. Estimate the past null affine length and explain the tension between entropy dilution and completeness.

Solution

Label cycles into the past by $n=0,1,2,\ldots$. Their scale factors are proportional to $q^{-n}$, so each contributes a null affine interval proportional to $Tq^{-n}$. The sum is $T\sum_n q^{-n}=T/(1-q^{-1})$, which is finite: the past-directed null geodesic is incomplete under the bounded-profile assumptions. The same net growth can dilute entropy density while total entropy grows. Thus entropy dilution does not by itself provide past completeness.

References

• Barrow, John D., George F. R. Ellis, Roy Maartens, and Christos G. Tsagas. “On the Stability of the Einstein Static Universe.” Classical and Quantum Gravity 20 (2003): L155–L164. DOI. arXiv:gr-qc/0302094.
• Bodnia, Eve, Vlad Isenbaev, Kellan Colburn, Joe Swearngin, and Dirk Bouwmeester. “The Quest for CMB Signatures of Conformal Cyclic Cosmology.” Journal of Cosmology and Astroparticle Physics 2024, no. 05 (2024): 009. DOI. arXiv:2208.06021.
• Borde, Arvind, Alan H. Guth, and Alexander Vilenkin. “Inflationary Spacetimes Are Incomplete in Past Directions.” Physical Review Letters 90 (2003): 151301. DOI. arXiv:gr-qc/0110012.
• Brandenberger, Robert, and Patrick Peter. “Bouncing Cosmologies: Progress and Problems.” Foundations of Physics 47 (2017): 797–850. DOI. Open PDF.
• Ellis, George F. R., and Roy Maartens. “The Emergent Universe: Inflationary Cosmology with No Singularity.” Classical and Quantum Gravity 21 (2004): 223–232. DOI. arXiv:gr-qc/0211082.
• Garcia-Saenz, Sebastian, Junjie Hua, and Yunke Zhao. “Geodesic Completeness, Cosmological Bounces, and Inflation.” Physical Review D 110 (2024): L061304. DOI. arXiv:2405.04062.
• Ijjas, Anna, and Paul J. Steinhardt. “A New Kind of Cyclic Universe.” Physics Letters B 795 (2019): 666–672. DOI. arXiv:1904.08022.