Linearly Expanding Universe

• A linearly expanding universe is defined by a scale factor that grows proportionally with cosmic time, implying a uniform expansion rate.
• The model arises from various frameworks such as modified gravity, conformal cosmologies, and dynamical 3-space theories, offering alternatives to the ΛCDM paradigm.
• Observational data from Type Ia supernovae, BAO, and cosmic chronometers often show competitive fits with linear expansion models, underscoring its potential in explaining cosmic dynamics.

A linearly expanding universe is a cosmological model in which the scale factor grows linearly with cosmic time, typically expressed as $a(t) \propto t$ or, in some generalized settings, as $a^2(t) \propto t$. Such models fundamentally contrast with the standard ΛCDM paradigm, where the expansion rate evolves nonlinearly through various cosmological epochs (radiation-, matter-, and dark energy-dominated eras). Linear expansion arises in diverse theoretical contexts, including modified gravity, scale-invariant scalar-tensor dynamics, conformally invariant cosmologies, dynamical 3-space approaches, and as an approximation over limited time intervals in standard cosmology. The linear expansion scenario has significant implications for observational cosmology, the interpretation of the Hubble law, and the resolution of several outstanding theoretical problems.

1. Theoretical Foundations of Linear Expansion

Linear expansion emerges most directly from models where the cosmic scale factor is proportional to cosmic time: $a(t) = t/t_0$. This behavior arises under several distinct theoretical frameworks:

• Equation of State $w = -1/3$:

In the Friedmann equation for a flat universe, setting the active gravitational mass to zero,

$\rho + 3p = 0,$

leads to $a(t) \propto t$ (Monjo, 15 Aug 2024). This is referred to as the coasting, Milne-like, or $R_h=ct$ cosmology.

• Hyperconical and Embedding Models:

The hyperconical model, treating the universe as a 4D hypersurface embedded in 5D Minkowski space, produces a scale factor linear in cosmic time, with expansion governed not by the matter content, but by the geometry of the embedding. This background metric is independent of local energy-momentum (Monjo, 2017, Monjo, 15 Aug 2024).

• Conformal Basis and Energy Conservation:

By adopting a conformal metric, $ds^2 = \Omega^2(t)(-dt^2 + d\vec{x}^2)$, and enforcing global energy conservation, the conformal scale factor must grow linearly: $a^{(conf)}(t) = t/t_H$, with total (vacuum) energy content remaining constant (Greben, 31 Jul 2025). This requirement forces the spatial volume $V(t) \propto t^3$.

• Jordan-Brans-Dicke (JBD) Scalar-Tensor Theories:

In a scale-invariant JBD framework with a quartic Jordan potential, the late radiation-dominated era yields $a^2(t) \propto t$—that is, linear in $a^2(t)$—with a radiation equation of state $p/\rho = 1/3$. Introduction of matter (density $\propto 1/a^3$) in such models produces deceleration, breaking strict linearity (Arik et al., 2018).

• Dynamical 3-Space Theory:

Here, spatial expansion is described by a velocity field $v(\vec{r}, t) = H(t)\vec{r}$ arising as a solution to the dynamical equation for space. In the limit of negligible matter, $a(t) \propto t$ appears as the "natural" time dependence of the Hubble flow (Rothall et al., 2013).

• Milne-like Universes and Modified Gravity:

The Milne model, with $a(t) = t$, is naturally obtained in spacetimes of negative spatial curvature or in five-vectors gravitational theories. Assigning a vacuum equation of state $w = -1/3$ drives linear expansion without the need for inflation (Cherkas et al., 2018).

2. Historical Context and the Hubble Law

The concept of a linearly expanding universe is deeply rooted in the original interpretation of the Hubble law. Lemaître's 1927 integration of Einstein's equations with observational redshift–distance data led to the pivotal relation

$v = H d$

where $v$ is the recession velocity, $d$ is the proper distance, and $H$ is the Hubble constant (Nussbaumer et al., 2011, Nussbaumer, 2013). In local observations or for small redshifts, this relation is a direct consequence of a locally linear expansion, and forms the basis for early and modern cosmological distance determinations (Pössel, 2017).

While the standard ΛCDM cosmology modifies the global scaling, the linear Hubble law is retained as a first-order approximation for $z \ll 1$, with the scale factor expressed as

$a(t) \approx a_0 \left[1 + H_0 (t-t_0)\right].$

Thus, the "linear expansion" paradigm is intrinsic to both historical developments and practical cosmological inference.

3. Observational Consequences and Model Comparisons

Linear expansion models have been extensively tested against cosmological data, often with competitive or comparable statistical fits to the concordance ΛCDM cosmology:

• Type Ia Supernovae, Quasars, Clusters, BAO, and Cosmic Chronometers:

The hyperconical (linearly expanding) model, with $a(t) = t/t_0$, fits SNe Ia (Pantheon+), quasar, cluster, BAO, and chronometer Hubble data with $\chi^2$ values similar to ΛCDM. The luminosity distance–redshift relation, especially, is naturally explained (Monjo, 2017, Monjo, 15 Aug 2024).

• Angular Size–Redshift Relation:

In static Euclidean or phenomenological linearly redshifted models with $d_A(z) = (c/H_0)z$, the observed $\theta(z) \propto z^{-1}$ dependence of galaxy angular sizes is matched without invoking strong size evolution, in contrast to standard models that require galaxies at $z\approx3$ to be $\sim6$ times smaller at fixed luminosity (Lopez-Corredoira, 2010). This avoids unphysical surface brightness and velocity dispersion values inferred in ΛCDM for high-$z$ galaxies (Lerner, 2018).

• Cosmic Microwave Background, Sound Horizon, and BAO:

Linear expansion models produce different evolution in the horizon size and comoving scales. Milne-like cosmologies, due to the absence of a finite particle horizon and constant comoving Hubble radius, naturally sidestep the "horizon problem" and the requirement for inflation (Cherkas et al., 2018). However, the use of the BAO scale as a distance ruler is ambiguous in linear expansion, as the bump separation emerges from the Jeans length modified by model parameters and does not directly map to a single "sound horizon" scale (Zhang et al., 2021).

• Linear Redshift–Distance Relation and Surface Brightness:

In static linear Hubble law or SEU (static Euclidean universe) models, a simple $d = cz/H_0$ prescription suffices to match the surface brightness–redshift and angular size–redshift data across a range of $z$, challenging the standard interpretation that attributes such trends exclusively to cosmological expansion (Lerner, 2018).

4. Cosmological Dynamics and Matter Content

The dynamics of strict linear expansion require the active gravitational mass density—the source term in the Friedmann equations, $\rho + 3p$—to vanish. This leads to the effective equation of state $w = -1/3$, which is atypical for standard matter, radiation, or a cosmological constant. Several theoretical interpretations have been developed:

• Dominance of Vacuum Energy in a Conformal Metric:

The conformal basis approach demonstrates that, with vacuum energy density $\epsilon$ and global energy conservation, the effective energy density $\epsilon^{(eff)}(t) = (t_H^3/t^3)\epsilon$ governs the expansion rate. Matter and radiation are then introduced as perturbations, contributing secondary corrections to the expansion, with induced terms potentially linked to "dark matter" and cosmic acceleration phenomenology (Greben, 31 Jul 2025).

• Independence of Background Metric from Local Matter:

Embedding models dissociate the global metric from the local energy-momentum, allowing the universe to expand inertially (linearly) regardless of the matter content. The local application of GR recovers standard gravitational dynamics, but the background (hyperconical) metric sets the expansion (Monjo, 15 Aug 2024).

• Scalar-Tensor and Scale-Invariant Gravity:

In Jordan–Brans–Dicke cosmology with scale-invariant quartic potential, linear expansion arises naturally in the late radiation-dominated era if the scalar field evolves as $\phi(t) \propto 1/t$ and $a^2(t) \propto t$, satisfying $p/\rho = 1/3$. Matter admixture introduces deceleration (Arik et al., 2018).

• Dynamical 3-Space:

In the limit of negligible matter, the space self-interaction parameter $\alpha$ (found to be close to the fine structure constant) alters the expansion equation insignificantly, so $a(t) \propto t$ holds robustly (Rothall et al., 2013).

Linear expansion, therefore, has a universal dynamical appeal in theories where the expansion rate is decoupled from the local matter–energy content, or where specific symmetry or conservation requirements fix the expansion law.

5. Quantum, Thermodynamic, and Laboratory Analogues

Quantum corrections, thermodynamical considerations, and experimental analogs provide complementary insights:

• Renyi and Generalized Entropy Corrections:

Incorporation of quantum-modified entropy formulas, such as the Renyi entropy, into the first law of thermodynamics on the cosmic horizon produces modified Friedmann equations with additional logarithmic terms. These corrections can induce epochs of nearly linear expansion, particularly during transitions between radiation and matter domination, or in early-universe inflationary phases (Fazlollahi, 2022).

• Scalar Casimir and Quantum Fields:

In a linearly expanding FRW universe, the quantum vacuum state responds to boundaries (Casimir plates) with boundary-induced expectation values and Casimir forces that decay as power laws at large separations, in contrast to exponential decay in static backgrounds. The vacuum energy-momentum tensor exhibits both diagonal and off-diagonal (energy flux) components, reflecting the dynamical, expanding geometry (Saharian et al., 2018).

• Laboratory Simulations:

Certain features of expansion, such as the redshifting of phonon modes and "Hubble friction," have been experimentally emulated in rapidly expanding ring-shaped Bose–Einstein condensates, yielding laboratory analogs of cosmological linear expansion and allowing detailed paper of nonlinear energy cascades reminiscent of preheating in inflationary scenarios (Eckel et al., 2017).

6. Challenges, Controversies, and Open Questions

Despite the economy and observational competitiveness of linear expansion models, several fundamental questions persist:

• Zero Active Mass and the Need for Non-Standard Matter:

The "zero active gravitational mass" condition ($\rho + 3p = 0$) is not achieved by standard matter, radiation, or a cosmological constant alone, demanding theoretical constructs either in modified gravity, exotic fluids, or global geometric settings (Monjo, 15 Aug 2024).

• Redshift Mechanism in Phenomenological Linear Models:

Static Euclidean models with a linear Hubble law, while successful in matching some scaling relations, lack a physical explanation for the origin of cosmological redshift, often postulating phenomenological 'tired light' or quantum mechanisms that remain unproven (Lopez-Corredoira, 2010, Hartnett, 2011).

• Distinguishing Apparent Versus Real Acceleration:

In some embedding scenarios, the observed acceleration inferred from supernova luminosity distances is interpreted as an artifact of coordinate projection (stereographic inhomogeneity), not as evidence for dark energy or a cosmological constant (Monjo, 15 Aug 2024).

• Energy Conservation, Thom Catastrophe, and Dark Energy:

In models with spatially varying $G$ or inclusion of dark energy, adding a repulsive acceleration (e.g., from dark energy) induces a "Thom catastrophe," whereby the solution structure and conservation laws qualitatively change, and limiting behaviors with $\Lambda \to 0$ do not approach the inertia-driven solution (Christodoulou et al., 2019).

• Statistical Interpretation of BAO and the Sound Horizon:

The scenario of linear expansion challenges the use of the sound horizon as a cosmic standard ruler, as the Gaussian random nature of the primordial field and evolving Jeans length obscure a unique association, complicating BAO-based distance estimates (Zhang et al., 2021).

7. Summary Table: Linear Expansion Scenarios and Key Features

Theoretical Context	Scale Factor $a(t)$	Key Governing Mechanism/Assumption	Comments/Observational Fit
Standard ΛCDM (locally for $z \ll 1$)	$a(t) \approx a_0 + H_0 t$	Local (Taylor expanded) FLRW dynamics	Only valid for short time intervals
Hyperconical/Embedding Models	$a(t) = t/t_0$	Geometry set by embedding, not matter	Competitive with SNe Ia, BAO, CC
Conformal/Energy Conservation Basis	$a^{(conf)}(t) = t/t_H$	Vacuum energy + global energy conservation	$H_0^{-1} = t_0$; fits SNe, CMB
JBD, Quartic Potential	$a^2(t) \propto t$	Scalar field dynamics, scale-invariant phase	Decelerated with added matter
Dynamical 3-Space	$a(t) \propto t$	Intrinsic spatial dynamics; α-term small	Predicts cosmic filament networks
Milne (-like), $w_v = -1/3$	$a(t) \propto t$	Vacuum energy with EOS $w = -1/3$	Inflation-free, solves horizon prob.

• (Lopez-Corredoira, 2010) "Angular size test on the expansion of the Universe"
• (Monjo, 2017) "Study of the observational compatibility of an inhomogeneous cosmology with linear expansion according to SNe Ia"
• (Pössel, 2017) "The expanding universe: an introduction"
• (Arik et al., 2018) "Inflation and Linear Expansion in the Radiation Dominated era in Jordan-Brans-Dicke Cosmology"
• (Rothall et al., 2013) "Dynamical 3-Space: Black Holes in an Expanding Universe"
• (Lerner, 2018) "Observations contradict galaxy size and surface brightness predictions that are based on the expanding universe hypothesis"
• (Zhang et al., 2021) "The nonlinear equation of correlation function of galaxies in the expanding universe and the solution in linear approximation"
• (Fazlollahi, 2022) "Renyi Entropy Correction to Expanding Universe"
• (Greben, 31 Jul 2025) "A conformal basis for cosmology with energy conservation"
• (Monjo, 15 Aug 2024) "What if the universe expands linearly? A local general relativity to solve the 'zero active mass' problem"
• (Christodoulou et al., 2019) "Universal expansion with spatially varying $G$"
• (Cherkas et al., 2018) "Plasma Perturbations and Cosmic Microwave Background Anisotropy in the Linearly Expanding Milne-like Universe"
• (Henriksen et al., 1 Jul 2024) "An Event Horizon 'Firewall' Undergoing Cosmological Expansion"
• (Nussbaumer et al., 2011, Nussbaumer, 2013) (historical/theoretical context for discovery of linear expansion and the Hubble law).

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The linearly expanding universe paradigm, whether as a fundamental cosmological model, a limiting case, or an observationally competitive alternative, continues to inform debates on cosmic expansion, the role of matter-energy content, horizon and flatness problems, and the interpretation of cosmological data. It often exposes theoretical challenges in energy conservation, projection effects, and the necessity—or lack thereof—of dark energy and inflation in the early universe.