Regular Cosmological Expansion

• Regular cosmological expansion is defined as the nearly uniform increase in the distances between comoving points, driven by energy conservation and conformal geometric principles.
• It employs analytic methods using conformal and kinematic approaches to derive redshift–distance relations without introducing new dynamical entities like dark energy.
• Alternative approaches involving metric regularity, cyclic models, and quantum corrections provide robust predictions that align with observational tests such as the Hubble law and CMB measurements.

Regular cosmological expansion refers to the large-scale, systematic increase in proper distances between comoving points in spacetime, typically modeled by a uniform or statistically homogeneous and isotropic metric such as the Friedmann–Robertson–Walker (FRW) or its generalizations. “Regularity” may refer to linearity or smoothness in expansion laws, absence of singularities, or the existence of geometric or physical mechanisms guaranteeing the well-defined, non-pathological evolution of the universe’s scale. It is foundational both to classic cosmological modeling and to current theoretical debates on expansion-driven phenomena, cosmic acceleration, and the interplay with local systems.

1. Conformal and Kinematic Approaches: Geometry as the Driver of Expansion

Recent analyses highlight that many key properties of cosmological expansion—such as the Hubble law, redshift–distance relations, and observed frequency shifts (e.g., the Pioneer anomaly)—can be obtained from conformal geometry, rather than strictly from the dynamical content of Einstein’s equations. In the pure-kinematic approach based on the conformal structure of spacetime (Tomilchik et al., 2010), generalizing the clock synchronization procedure of special relativity to an expanding geometry yields a quadratic time inhomogeneity: $t' \approx t + \tfrac{1}{2} H_0 t^2,$ inducing a universal acceleration-like kinematic effect,

$W = c H_0,$

manifest at all scales. This framework allows the derivation of explicit redshift dependencies for cosmological distances: $$R(z) = R_\text{limit} \frac{(z+1)^2 - 1}{(z+1)^2}, \quad R_\text{limit} = 2c/H_0,$$ together with a velocity–redshift relation

$V(z) = c \frac{(z+1)^2 - 1}{(z+1)^2 + 1},$

recovering the standard linear Hubble law for $z \ll 1$. Notably, this approach links the background acceleration to observed phenomena without invoking new dynamical causes (such as dark energy), implying that the acceleration and even local anomalies are natural consequences of the kinematic consequences of conformal geometry.

2. Energy Conservation and the Conformal Basis

A complementary line of development is the proposal that regular expansion, particularly the linear expansion law, is not a direct consequence of local Einsteinian dynamics applied via the FRW formalism, but emerges from global energy conservation when the cosmological background is formulated using the conformal vacuum metric (Greben, 31 Jul 2025). Here, the dominant metric is

$$ds^2 = \Omega^2(t) (-dt^2 + d\sigma^2),\qquad \Omega(t) = \frac{t_H}{t},$$

where $t_H = \sqrt{3/(8\pi G\epsilon)}$ is set by vacuum energy density $\epsilon$. In this picture, the effective energy density, including the Jacobian, diverges as $t \rightarrow 0$: $$\epsilon^{\text{eff}}(t) = \frac{t_H^3}{t^3} \epsilon.$$ Global energy conservation governing the total energy $E_V = - \int d^3x\, \sqrt{|g|}\, T_0^{\,0}$, combined with the dilution law for the effective density, enforces a strictly linear expansion of the universe: $$a_\text{conf}(t) = \frac{t}{t_H}, \qquad H(t) = \frac{1}{t},$$ leading to $t_0 = H_0^{-1}$ at the present epoch, agreeing with observed cosmic chronometry and redshift-luminosity data. This approach provides alternative explanations for the supernova redshift relation

$d_L = c\,t_0\,(1 + z)\ln(1 + z)$

and the CMB temperature evolution $T_\text{CMB}(z) \sim (1+z) T_\text{CMB}(0)$, matching empirical measurements. Matter and radiation, treated as perturbations, induce secondary corrections (with matter terms enhanced by a factor $9/4$ and radiation entering with $-1/2$), plausibly mimicking the phenomenology of dark matter and expansion acceleration.

3. Regularity through Metric Structure, Topology, and Dynamics

Regular cosmological expansion is also achieved by enforcing metric regularity via global topological or dynamical constraints. For example, cyclic “regular” universes constructed on the principle of Haar measure regularization on $S^3$ (Mashkevich, 2011) employ a periodic radius function

$R(t) = R_{\max} \sin^4(v t),$

ensuring that spacetime is free of metric singularities (no big bang/crunch) and cycles between finite maximum and minimum volumes. The system’s regularity is reinforced by defining dark matter as a non-particle compensational tensor field, which “smooths away” potential quantum jumps in matter’s energy–momentum distribution, and by introducing a metrodynamical equation that governs the coupled evolution of the spatial metric and its compensational field. Such constructions are characterized by only two constants (the gravitational constant and cosmic period), with no structural singularities.

Alternatively, models where the speed of light and Newton’s constant are time-dependent and spatial topology is that of a 3-sphere (k = 1) also achieve regular expansion without initial or final singularities (Shu, 2010). In these settings, cosmic acceleration and deceleration regimes emerge naturally as geometric properties tied to the evolution of the conversion factors between mass, length, and time, not to ad hoc additions such as dark energy.

4. Quantum and Inflationary Modifications to Regularity

The structure and stability of regular expansion is sensitive to quantum and inflation-inspired model corrections. When quantum corrections are derived using perturbative gravity and the Schwinger–Keldysh formalism, the leading effect is a minute “quantum friction,” producing a correction to the deceleration parameter of order $q^Q_0 \sim 10^{-46}$—entirely negligible for all observational purposes (Broda, 2013). The expansion is therefore extraordinarily robust against such quantum modifications. In contrast, models with a delayed response in the Friedmann equation (Choudhury et al., 2011) can yield an early rapid accelerated phase (even in ordinary matter- and radiation-dominated epochs), followed naturally by a transition to standard expansion. Such mechanisms obviate the need for scalar field–driven inflation and demonstrate the sensitivity of regular expansion to the dynamical form of the cosmic response function.

Regularity at very high energy densities is also enforced by extending gravity into Riemann–Cartan or gauge-theoretic spacetime, allowing the presence of torsion (Minkevich, 2013). In these models, singularities are regularized through “bounces” at maximal energy density, with all physically meaningful quantities (energy density, Hubble parameter, curvature and torsion invariants) remaining finite through the turnaround.

5. Local Manifestations and Constraints: From Bound Systems to Physical Observers

A key question is the extent to which regular cosmological expansion impacts local (sub-galactic or even laboratory) systems. For exact Lemaître–Tolman–Bondi solutions, it is demonstrated that even strongly bound two-body systems embedded within a dust-dominated FLRW cosmological background ultimately participate in cosmic expansion and become comoving with the universal substratum (Bochicchio et al., 2011), challenging the notion that only sufficiently large (weakly bound) systems follow the Hubble flow.

A more general geometric analysis reveals that the influence of expansion on local dynamics is encoded as a shift in inertial structure, which formally manifests as a $-\ddot{a}/a$ term in local equations of motion for particles, but not as a friction or drag force (Giulini, 2013). The effect is only significant at relatively large spatial scales (galaxy clusters). In gravitationally bound or laboratory systems, corrections are minuscule or emerge only at order $H_0^2$, and can observationally manifest, for instance, in the Doppler tracking of spacecraft (e.g., the Pioneer anomaly) or as minute modifications to resonator frequencies (Widom et al., 2015, Spengler et al., 2021). Notably, theory indicates that for physically relevant observer classes (geodesic or Kodama observers), all expansion effects in local laboratory experiments are quadratic in $H_0$, placing them several orders of magnitude below detectability.

Although superficially the emergence of an acceleration scale $a_0 \sim c H_0$ in galaxy rotation curves suggests a possible expansion-induced origin, detailed perturbative analysis in scalar-tensor and spinor-tensor theories shows that only even powers of the Hubble constant survive in the local gravitational potential (Schiffer, 2014). Absent a theoretical mechanism for generating $O(H_0)$ corrections (as would be needed for MOND phenomenology), regular expansion is irrelevant for local dynamics in these frameworks.

6. Inhomogeneities, Averaging, and Backreaction

The regularity of cosmic expansion is robust against inhomogeneities under certain statistical conditions, as shown by analysis of “Swiss-cheese” models embedding Szekeres or other inhomogeneous solutions into an FRW background (Lavinto et al., 2013). Provided the inhomogeneities are smoothly matched and the monotonicity of radius functions is preserved, the average expansion rate remains essentially identical to the background FRW rate. When this monotonicity is lost, one can engineer “Tardis regions” (locally larger volumes inside the same boundary), yielding significant “backreaction” that increases the averaged expansion rate. Such constructions demonstrate that, while regular expansion is a robust mean-field prediction, local geometrical structures can modulate, but not fundamentally disrupt, the large-scale predictability of cosmological expansion.

7. Observational and Physical Consequences

Multiple observational tests support, challenge, or nuance the notion of regular cosmological expansion. Classical probes—CMB temperature vs. redshift, time dilation, Hubble diagram, Tolman surface brightness, angular size and UV surface brightness tests—offer mixed evidence, often requiring strong ad hoc galaxy evolution for consistency with standard expansion, except for the Alcock–Paczynski test, which is essentially independent of such evolutionary effects (Lopez-Corredoira, 2015). This test finds agreement with both the standard $\Lambda$CDM expansion and certain static (tired-light) alternatives, underscoring the empirical subtleties inherent in resolving the true nature of cosmic expansion.

On the theoretical front, swampland conjectures restrict scalar-field–driven expansion to an order-one number of e-folds, at odds with cyclic cosmological scenarios requiring $\sim 100$ e-folds for viability—posing deep consistency challenges to the regularity and long-term duration of expansion phases under string-inspired constraints (Corianò et al., 2020).

Quantum effects in expanding backgrounds can also be harnessed as practical probes: quantum communication protocols between harmonic oscillators via quantum fields experience an increase in classical channel capacity proportional to particle production induced by regular expansion, with sensitivity to the cosmic equation of state and curvature coupling (Lapponi et al., 5 Aug 2024).

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In sum, regular cosmological expansion emerges as both a geometric and a physically universal phenomenon: encoded in the conformal symmetry of spacetime, driven by global energy conservation, enforced by topological or tensorial constraints, robust against quantum and dynamical corrections, and deeply interwoven with the observational signatures that define modern cosmology. Theoretical modeling consistently points to regular, nearly linear expansion as the dominant large-scale behavior, while local and small-scale systems are largely immune to expansion-induced effects except via indirect, highly suppressed channels. These findings reinforce the empirical and conceptual coherence of regular expansion as both a classical and quantum cosmological principle.