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Variable speed of light

Variable speed of light (VSL) theories propose that the speed of light in vacuum, traditionally considered a universal constant in special relativity, may vary across space or over cosmic time, potentially resolving longstanding issues in cosmology and quantum gravity. These hypotheses emerged as alternatives to cosmic inflation, suggesting that an initially much higher speed of light in the early universe allowed for greater causal connectivity, thereby explaining the observed uniformity of the cosmic microwave background without invoking an exponential expansion phase. Unlike standard models where c is fixed at approximately 299,792 km/s, VSL frameworks modify fundamental equations, such as Maxwell's, to accommodate dynamic variations while preserving key experimental tests of relativity in the present epoch. The conceptual roots of VSL trace back to mid-20th-century physics, but systematic modern development began in the late 1990s through works by researchers including John Moffat, João Magueijo, and Andreas Albrecht. Motivated by the —why distant regions of the universe appear thermally equilibrated despite never having been in causal contact under constant c—VSL models posit a where c decreases dramatically shortly after the , expanding light cones retroactively to enable information exchange across the . This approach also addresses the by naturally driving the density parameter Ω toward unity through the varying c, reducing the need for extreme initial fine-tuning required in Friedmann-Robertson-Walker cosmologies. Additionally, VSL has been linked to phenomena, such as "doubly special relativity," where at Planck scales exhibits effective Lorentz invariance breaking, potentially explaining anomalies in high-energy particle propagation. Key VSL models include bimetric theories, which assign different propagation speeds to and ; color-dependent variants altering c for photons of varying energy; and brane-world scenarios from where extra dimensions induce variability. Observational support remains tentative but includes early hints (now controversial and largely disputed) of a redshift-dependent α, with 2001–2003 measurements suggesting Δα/α ≈ -0.72 × 10⁻⁵ at z ≈ 0.5–3.5, interpretable as tied to c variations since α ∝ 1/c; later analyses find no significant change. Ultra-high-energy cosmic rays exceeding the Greisen-Zatsepin-Kuzmin cutoff may also signal energy-dependent c, though constraints from and atomic clock comparisons limit past changes to |Δc/c| ≲ 10⁻⁴ over the last billion years. Overall, VSL offers a parsimonious framework for , predicting a scale-invariant, Gaussian spectrum consistent with data, while avoiding inflation's reliance on unverified scalar fields.

Fundamentals of VSL

Definition and physical meaning

Variable speed of light (VSL) theories propose that the in , denoted by c, is not a fixed universal constant but varies as a function of time, position , scale, or cosmic , either as a fundamental dynamical parameter or an effective quantity emerging from underlying physics. This variation contrasts with the foundational postulate of , where c is for all observers, serving as the universal . In VSL frameworks, such changes to c can arise from modifications to the properties, scalar fields influencing electromagnetic propagation, or altered metrics, potentially simplifying theoretical descriptions when units are chosen such that c is non-constant. Physically, c carries multiple interpretations in VSL contexts: it represents the propagation speed of electromagnetic waves, the maximum causal limit for information transfer between events, or a coefficient in the spacetime metric that governs geodesic paths. As the propagation speed, a varying c implies that photons follow dispersion relations modified from the standard E = pc (for massless particles), such as E = p \, c(t) in temporally varying cases, where the energy E and momentum p of photons scale with the local or epoch-dependent c(t). More generally, VSL can introduce energy-dependent modifications like E^2 f_1^2(E; \lambda) - p^2 f_2^2(E; \lambda) = m^2, where f_1 and f_2 are functions incorporating the variation, altering how light travels compared to massive particles. In terms of causality, c defines the boundaries of light cones in spacetime; a varying c distorts these null cones, expanding or contracting the region of causally connected events and potentially allowing superluminal effective speeds for light relative to a fixed gravitational metric, while preserving local Lorentz invariance in some formulations. For instance, in bimetric approaches, photons may propagate along null geodesics of an effective electromagnetic metric distinct from the gravitational one, leading to tilted or widened light cones that reflect differing speeds for light and gravity. This variability distinguishes VSL from special and general relativity, where c is invariant and sets the scale for intervals via the , ensuring the same laws of physics in all inertial frames without preferred directions or times. In VSL, such invariance may hold only locally or be broken globally, requiring modified that account for the changing scale, such as incorporating a position-dependent factor in the between coordinates. Consequently, units of time, , and rescale with c, interpreting variations as shifts in measurement standards rather than absolute changes, though some models treat c as a dynamical field reflecting evolving fundamental interactions. These interpretations maintain the core geometric role of null geodesics for light paths but adapt them to a non-uniform , emphasizing c's role as both a physical limit and a descriptor.

Implications for relativity and causality

In variable speed of light (VSL) theories, the variation of c fundamentally challenges the foundational postulate of that the speed of light is invariant in all inertial , leading to a breakdown of global Lorentz invariance. Instead, these models often introduce a preferred cosmological in which c = c(t) evolves with , necessitating modified Lorentz transformations that depend on the local value of c. This breaking of Lorentz invariance can manifest as the emergence of a preferred or , altering the between space and time and potentially allowing for anisotropic effects in and . However, some formulations aim to preserve local Lorentz invariance at each epoch by treating c(t) as constant on hypersurfaces of constant time, ensuring that physics remains relativistic within small scales while permitting global variations. The variability of c raises significant concerns for , as the light cones defining causal boundaries would evolve over time, potentially permitting superluminal signaling relative to later epochs. In the early , a dramatically higher c (e.g., exceeding the current value by factors of $10^{30} or more) expands past light cones, enabling causal connections across regions that would otherwise be disconnected in standard relativity, thus addressing issues like the without . This temporal variation risks tachyonic instabilities or closed timelike curves if not carefully constrained, as signals propagating at the local c(t) might appear to violate when viewed from frames with different c, though most VSL models mitigate this by enforcing subluminal propagation within each local frame and avoiding acausal loops through to matter. Such dynamics highlight the tension between VSL and the strict of Minkowski , where fixed light cones prevent information from traveling backward in time. VSL can be interpreted through unit rescaling as equivalent to variations in other fundamental constants or effective , rather than a literal change in light's propagation speed. For instance, an effective c(t) = c_0 f(t), where f(t) is a scaling function and c_0 is the current value, is mathematically indistinguishable from rescaling Planck's constant or the unit of time, preserving the form of physical laws while altering their numerical values across cosmic . This equivalence arises because c enters dimensionally in the definitions of units; varying c while keeping \hbar fixed is akin to a global clock adjustment, which simplifies comparisons but complicates direct measurements of constancy. Broader implications extend to the foundational equations of physics, where c(t) modifies the structure of and Einstein's field equations. In , the covariant form becomes \partial_\mu (c F^{\mu\nu}) = 4\pi j^\nu, leading to a time-dependent \partial^2 A / \partial t^2 = c^2(t) \nabla^2 A for the , which alters dispersion and propagation without violating local . For , minimal substitution in Einstein's equations yields G_{\mu\nu} = (8\pi G / c^4(t)) T_{\mu\nu}, introducing non-conservation of energy-momentum due to terms involving \dot{c}/c, which can mimic effects or resolve flatness problems through dynamical adjustments. These modifications underscore how VSL reframes not as an absolute framework but as an emergent property tied to cosmic scales.

Historical proposals

Early background and motivations

In the , discussions surrounding the provided early precursors to ideas about a variable , though these were primarily local and non-cosmological in nature. George Gabriel Stokes proposed in 1845 that the could be fully dragged along by moving matter, such as the , which would imply that the speed of light relative to an observer varies depending on the motion of the medium through which it propagates. This ether drag hypothesis, building on earlier partial drag ideas by , suggested that light's velocity could differ in moving media compared to a stationary , challenging the notion of an absolute constant speed but remaining tied to classical ether models rather than universal variation. Similarly, William Thomson () and Peter Guthrie Tait speculated in 1874 on the possibility of light's speed varying as a , at a time when c held no privileged role in physics, reflecting a broader willingness to treat propagation speeds as mutable. By the early , introduced a more formal consideration of variable light speed in the context of gravity, predating the full development of . In his 1911 paper, Einstein proposed that the decreases in a to account for the predicted of spectral lines, deriving a formula where c varies as c(1 + φ/c²), with φ as the . This idea, though later superseded by the and spacetime curvature in 1915, marked an influential shift toward viewing c as potentially non-constant under gravitational influence, motivating further exploration of its variability. During , amid debates over the nature of cosmological , some physicists invoked varying c as an alternative explanation to galactic recession or "tired light" hypotheses, suggesting that a decreasing over cosmic distances could mimic the observed stretching without requiring an expanding . Philosophically, these early ideas stemmed from the view that the speed of light was not inherently fundamental or immutable, especially before special relativity elevated c to a cornerstone of spacetime structure. Proponents argued that physical constants might evolve with the universe's development, avoiding ad hoc assumptions about their permanence and allowing for a more dynamic cosmology. A key catalyst in the 1950s was Paul Dirac's 1937 large numbers hypothesis, which posited that the gravitational constant G decreases over cosmic time to explain coincidences between atomic and astronomical scales, thereby inspiring considerations of variation in other fundamental constants, including the speed of light. This pre-Dicke era lacked rigorous formal theories but laid groundwork through such practical motivations as reconciling redshift observations and philosophical openness to evolving laws.

Dicke's 1957 proposal

In 1957, proposed a novel framework for gravitation and cosmology that eschewed the principle of equivalence, instead describing gravitational effects through a flat with a variable (VSL). In this model, the speed of light c varies spatially near masses, acting like a \epsilon = c_0 / c = 1 + 2GM / (r c^2), where c_0 is the speed far from masses, G is the , M is the mass, and r is the distance; this formulation successfully accounted for the observed deflection of light by at 1.75 arcseconds. Cosmologically, Dicke extended this to a time-varying c(t) that decreases over cosmic history, proportional to $1/t in an expanding universe, mimicking the slowing of atomic clocks and linking local measurements to global cosmic evolution. The model's formulation tied the variation in c to atomic timescales, such that changes in speed directly affect : the relative shift is given by \delta \nu / \nu = -\delta c / c, reflecting how a decreasing c leads to in emitted as photons propagate through evolving cosmic conditions. In this setup, the cosmic factor's of change is governed by \dot{R}(t) = c(t), allowing for a dynamic where propagation adjusts to resolve causal disconnects without invoking expansion-driven mechanisms alone. Dicke aimed to address key cosmological puzzles, including the large s of quasars, which he interpreted through "rod shortening" effects from varying c rather than pure Doppler or expansion shifts, yielding a redshift formula z + 1 = (t_2 / t_1)^{1/4} for radiation-dominated epochs. This approach also tackled the by permitting faster light travel in the early universe, enabling causal contact across vast distances without reliance on steady-state cosmology, while connecting to the flatness issue via adjusted dynamics. The proposal integrated with scalar-tensor gravity ideas, foreshadowing the Brans-Dicke theory, where a modulates c and embodies by tying gravitational strength to the universe's total mass distribution. While Dicke's ideas garnered initial interest for their elegance in linking , , and cosmology—particularly in relation to Dirac's large number hypothesis—the model saw limited adoption due to insufficient and challenges in reconciling with observations like particle number conservation.

Late 20th-century developments

In the , physicist John W. Moffat developed a bimetric of that introduced two distinct metrics: one governing the propagation of with a constant speed c_m and another for with a variable speed c_l. This framework allowed for a varying while maintaining consistency with in local frames, aiming to address cosmological issues like the through non-constant propagation. During the 1980s, , Geoffrey Burbidge, and Jayant V. Narlikar proposed the quasi-steady-state cosmology (QSSC), an alternative to the standard model that incorporated a varying alongside other evolving physical constants to better fit observational data on distributions and cosmic microwave background features. In this model, the universe undergoes periodic expansions and contractions with matter creation, where adjustments to G helped reconcile the theory with observations and large-scale structure without relying on . Jean-Pierre Petit contributed to VSL ideas in 1988 with a gauge cosmological model featuring a variable light velocity tied to evolving fundamental constants like the h and G. In Petit's approach, characteristic lengths (such as Compton and Schwarzschild radii) scale with the cosmic scale factor R(t), leading to c \propto 1/R and enabling interpretations of redshifts as arising from secular variations in these constants rather than solely from expansion. A significant breakthrough came in 1998 when Andreas Albrecht and João Magueijo introduced a VSL model where the speed of light varies inversely with the cosmic scale factor, c \propto 1/a(t), to resolve the horizon problem without invoking inflation. This variation allows distant regions of the early universe to achieve thermal equilibrium via faster light travel, while the modified Friedmann equation becomes H^2 = \frac{8\pi G}{3} \rho \left( \frac{c}{c_0} \right)^4, where H is the Hubble parameter, \rho is the energy density, G is the gravitational constant, c_0 is the current speed of light, and the (c/c_0)^4 term accounts for the enhanced effective density in a radiation-dominated era with varying c. Their work demonstrated that such a model could also alleviate the flatness problem and suppress magnetic monopoles, providing a Lorentz-violating alternative to inflationary cosmology.

Key VSL models and frameworks

Bimetric and varying-e theories

Bimetric gravity theories introduce a framework for variable speed of light (VSL) by employing two distinct metrics to describe different sectors of physics, thereby allowing the effective to vary without fundamentally breaking Lorentz invariance in local frames. In this approach, one metric, g_{\mu\nu}, governs the dynamics of and massive , while a second metric, \gamma_{\mu\nu} (often denoted as \hat{g}_{\mu\nu}), is associated with and massless particles. The speed of light emerges as the ratio of proper distances measured in these metrics, c = \frac{ds_{\gamma}}{ds_g}, enabling c to vary spatiotemporally as a function of the relative scaling between the metrics. This formulation was pioneered by John W. Moffat in 2002, who incorporated an interaction term in the action to couple the two metrics, ensuring consistency with in the appropriate limits while permitting VSL effects on cosmological scales. The interaction between the metrics is typically mediated by a scalar field \phi, which dynamically adjusts the conformal relation \gamma_{\mu\nu} = \Omega^2(\phi) g_{\mu\nu}, where \Omega(\phi) determines the variation in c. This scalar-tensor extension leads to modified field equations, including a wave equation for \phi: \gamma^{\mu\nu} \nabla_\mu \nabla_\nu \phi + K V'[\phi] = 0, where V[\phi] is the potential and K is a coupling constant. Such models preserve local Lorentz invariance for each metric separately, meaning observers in the gravitational frame experience standard relativity, while the electromagnetic frame allows for varying light propagation speeds. This duality provides a mechanism for global VSL without local violations of causality or equivalence principles. Complementing bimetric approaches, varying-\epsilon models treat the electric permittivity of the , \epsilon(t), as a time-dependent quantity driven by a , leading to an effective VSL through the relation c(t) = 1 / \sqrt{\epsilon(t) \mu_0}, assuming constant magnetic permeability \mu_0. Jean-Pierre Petit proposed such a framework in 1988, where a modulates the fundamental constants, including \epsilon, to achieve varying c, h, and G in a cosmological context. This variation implies modified Maxwell equations, such as \nabla \cdot \mathbf{D} = \rho with \mathbf{D} = \epsilon(t) \mathbf{E}, altering propagation and energy-momentum conservation in expanding universes. These modifications affect electromagnetic wave equations, yielding dispersion relations where the phase velocity aligns with the local c(t), influencing and luminosity distance in . By scaling \epsilon(t) inversely with the cosmic scale factor in early epochs, these models facilitate superluminal signal propagation across cosmic horizons without invoking . The primary advantage of both bimetric and varying-\epsilon theories lies in maintaining local Lorentz invariance and gauge symmetry for , while permitting global variations in c to address cosmological discrepancies, such as the , through adjusted causal structures.

Non-Lorentz invariant approaches

Non-Lorentz invariant approaches to variable speed of light (VSL) theories explicitly violate Lorentz invariance by introducing an energy-dependent speed of light, c(E), often motivated by effects where high-energy particles propagate differently from low-energy ones. This violation typically arises in a preferred , leading to anisotropic propagation and modified at Planck scales, without requiring a full theory of . In the flat-space limit of such models, the speed of light varies with , providing a phenomenological description of Lorentz-breaking effects in quantum gravity contexts. A common formulation modifies the standard relativistic dispersion relation to E^2 = p^2 c^2 \left(1 + \xi \left(\frac{E}{M_{\mathrm{Pl}}}\right)^n \right), where E is , p is , c is the low-energy , \xi is a dimensionless of order unity, M_{\mathrm{Pl}} is the Planck mass, and n is typically 1 or 2 depending on the model. This energy dependence implies that high-energy photons travel faster or slower than low-energy ones, altering their arrival times over cosmic distances and potentially leading to observable delays in high-energy signals. One seminal example is the Albrecht-Magueijo model, which in its flat-space limit derives an energy-dependent c(E) from a time-varying c(t) framework, breaking Lorentz invariance to accommodate quantum gravity-inspired modifications. Another key development is 2000 covariant formalism, which incorporates a varying c within a preferred frame while maintaining , allowing for Lorentz-violating effects in gravitational theories. These approaches have been applied to gamma-ray bursts, where energy-dependent propagation could manifest as spectral lags between high- and low-energy photons. Theoretically, such non-invariant VSL models emerge as effective descriptions from underlying quantum gravity frameworks, including where Lorentz violation arises from low-energy approximations of string modes, and where discrete spacetime leads to modified relations at high energies. These connections provide a bridge to unification efforts without fully resolving quantum gravity inconsistencies. Building briefly on historical proposals, these frameworks extend early VSL ideas by emphasizing explicit Lorentz breaking for high-energy phenomenology.

Recent extensions (2000s–2025)

In the 2000s, João Magueijo and John D. Barrow developed variable speed of light (VSL) models where the speed of light c is proportional to the T in the early , allowing c to be significantly higher during the hot, dense phases to address cosmological issues without invoking . These models posit that as the cools, c decreases proportionally with T, preserving local Lorentz invariance while altering global cosmology. A key 2003 publication highlighted how such VSL variations serve as alternatives to inflationary scenarios, enabling horizon crossing and flatness without , and producing scale-invariant perturbations. During the 2010s, the minimally extended VSL (MEVSL) framework emerged, incorporating a \phi(t) such that c = c_0 \exp(\kappa \phi), where \kappa is a , to minimally alter while allowing time-dependent c. This approach maintains compatibility with and by adjusting the metric minimally, ensuring local physics remains consistent with observations. A 2024 review synthesized these developments, emphasizing MEVSL's potential to reconcile VSL with standard cosmological probes like and the without violating Lorentz invariance at low energies. From 2020 to 2025, advancements focused on non-Lorentz invariant VSL theories, with a 2024 review in Classical and Quantum Gravity re-deriving frameworks that break Lorentz symmetry explicitly to resolve cosmological puzzles, such as the , through direction-dependent c variations. This work evaluated implications for and , highlighting testable predictions in high-energy regimes. In 2025, an preprint integrated VSL into the Friedmann-Lemaître-Robertson-Walker (FLRW) metric, interpreting variations as changes across hypersurfaces, yielding the ds^2 = -c(t)^2 dt^2 + a(t)^2 dr^2 (for flat space), which reframes c(t) as a dynamical lapse function influencing expansion dynamics. Emerging research in 2025 linked VSL to gravitational waves and vacuum speed variations, proposing that vacuum density fluctuations induce local c changes, potentially altering gravitational wave propagation speeds by up to 15% relative to electromagnetic waves, as constrained by pulsar timing data. These extensions suggest VSL could manifest in detectable deviations during strong-field events, bridging cosmology with multimessenger astronomy.

Connections to varying physical constants

Dimensionless quantities like the fine-structure constant

The fine-structure constant, denoted as \alpha, is a dimensionless quantity that characterizes the strength of the electromagnetic interaction between elementary charged particles. It is defined by the formula \alpha = \frac{e^2}{4\pi \epsilon_0 \hbar c}, where e is the elementary charge, \epsilon_0 is the vacuum permittivity, \hbar is the reduced Planck's constant, and c is the speed of light in vacuum. In the context of variable speed of light (VSL) theories, a variation in c directly impacts \alpha unless compensated by corresponding changes in the other parameters, establishing a fundamental link between VSL and the constancy of dimensionless constants. In models where c varies independently while e, \epsilon_0, and \hbar remain fixed, \alpha changes inversely with c, yielding the relative variation \delta \alpha / \alpha = -\delta c / c. This relationship arises directly from the definition of \alpha and has been explored in VSL frameworks to interpret observational data on \alpha evolution. Early analyses of absorption spectra suggested a possible time variation in \alpha of \delta \alpha / \alpha \approx (0.5 to $1) \times 10^{-5} per unit redshift, which some VSL models attributed to historical changes in c rather than variations in e or \hbar. However, more recent measurements as of 2025, including JWST observations of emission-line galaxies at redshifts up to z ≈ 9.5, find no significant variation with constraints |Δα/α| ≲ (0.2 ± 0.7) × 10^{-4} over cosmic history, tightening limits and questioning earlier hints. Such interpretations align with bimetric VSL theories, where the effective adjusts cosmologically, inducing coupled variations in \alpha. Variations in \alpha induced by VSL have significant implications for atomic physics and astrophysics. In atomic spectra, \alpha governs the fine-structure splitting of energy levels, which scales as \alpha^2 times the gross structure; thus, even small changes in \alpha would shift spectral line positions and intensities, altering the observable signatures of distant or early-universe atoms. For stellar evolution, these effects propagate to influence electromagnetic processes within stars, such as opacity in radiative transfer and rates of electron-capture reactions in stellar cores, potentially modifying nucleosynthesis outcomes and evolutionary tracks compared to standard models assuming constant \alpha. Dimensionless constants like \alpha serve as critical tests of the universality of fundamental laws, as their values are independent of unit choices and scale, distinguishing them from dimensionful constants that inherently involve measurement scales.

Dimensionful quantities like the gravitational constant

The G possesses dimensions of length cubed per mass per time squared (\mathrm{m}^3 \mathrm{kg}^{-1} \mathrm{s}^{-2}), whereas the c has dimensions of length per time (\mathrm{m} \mathrm{s}^{-1}). In variable speed of light (VSL) theories, a time- or space-dependent c demands coordinated adjustments to other dimensionful constants like G to ensure dimensional consistency and preserve key physical scales. For instance, the , a fundamental unit combining quantum and gravitational effects, is expressed as l_p = \sqrt{\frac{\hbar G}{c^3}}, where \hbar is the reduced ; if \hbar remains fixed and l_p is to be invariant, variations in c require G to scale proportionally to c^3. Such linkages arise naturally in frameworks where fundamental constants evolve together to avoid inconsistencies in field equations or unit systems. Paul Dirac's 1937 proposal for a time-varying G, motivated by observed large-number coincidences between atomic scales and cosmological parameters, laid early groundwork for connecting varying dimensionful constants to cosmic evolution and has since inspired hybrid VSL models incorporating G- c links. In the Brans-Dicke scalar-tensor theory of , introduced in , a dynamical \phi modulates the effective gravitational strength via G_\mathrm{eff} \propto 1/\phi, while extensions to VSL allow \phi to influence both G and c through coupled field equations, enabling consistent propagation of gravitational and light signals in curved . These scalar-mediated variations ensure and local Lorentz invariance in modified regimes, with \phi's evolution tied to the cosmic expansion. Certain VSL frameworks introduce an effective G_\mathrm{eff} adjusted relative to a reference value to maintain dimensionless ratios in gravitational interactions amid c's variation. This scaling helps preserve the form of Newtonian potentials or relativistic corrections, such as the r_s = 2 G M / c^2, thereby stabilizing collapse criteria and perturbation growth rates during cosmic . Without such adjustments, unchecked c variations could disrupt the balance between gravitational clustering and causal horizons, altering galaxy formation timelines. John Moffat's scalar-tensor gravity models within bimetric VSL frameworks provide a representative example, linking G's evolution to the scalar field's coupling with the varying light speed to resolve cosmological puzzles, implying enhanced gravitational strength in epochs of higher c and influencing density contrasts without invoking inflation.

Cosmological applications

Addressing the horizon problem

The horizon problem in standard Big Bang cosmology stems from the observed uniformity of the cosmic microwave background (CMB) radiation across angular scales larger than 1 degree, which requires that causally disconnected regions in the early universe achieved thermal equilibrium. In the conventional model with constant speed of light c, the particle horizon—the comoving distance light could travel from the Big Bang to the time of CMB decoupling—is far too small to allow causal contact between points separated by the full extent of the observable universe today, as these regions expand beyond the light-travel limit during the universe's radiation-dominated phase. Variable speed of light (VSL) models address this issue by positing that c was much larger during early epochs, enabling photons to propagate across vastly greater distances and establish causal connections before the universe's expansion isolated those regions. In the seminal Albrecht-Magueijo framework, a non-Lorentz VSL theory, the speed of light scales inversely with the factor as c \propto 1/a^n with n \approx 1, where a is the cosmic factor; this results in c being exponentially higher when a is small, corresponding to the hot, dense early universe. The core mechanism involves modified null geodesics for light propagation, where the comoving particle horizon size is enlarged to d_h = \int_0^t \frac{c(t')}{a(t')} \, dt' , which exceeds the standard horizon scale due to the boosted c(t') in the integral's early-time contribution, allowing the entire observable universe to fit within a single causal domain at early times. This enhanced horizon ensures the requisite thermalization for CMB isotropy without invoking exotic physics, while the model's perturbation dynamics predict the observed slight CMB anisotropies by suppressing large-scale fluctuations through the varying c, consistent with a scale-invariant spectrum emerging naturally. As an alternative to cosmic , VSL circumvents the need for a finely tuned field and its associated , which must be precisely adjusted to produce the correct number of e-folds of expansion; instead, the variation in c alone resolves the horizon issue through a simpler modification to general relativity's propagation rules, though it requires specifying the functional form of c(t) to match detailed cosmological data.

Impacts on inflation and

In variable speed of light (VSL) theories, a varying c can replicate the key effects of slow-roll by modifying the and perturbation generation in the early , potentially obviating the need for an field. Models proposed by Magueijo and Barrow in 2003 illustrate how a rapid decrease in c immediately following produces a scale-invariant spectrum of Gaussian primordial fluctuations, aligning with observations from the (WMAP). This variation in c addresses the horizon and flatness problems while ensuring the perturbations seed large-scale structure formation without fine-tuning. Such VSL inflationary scenarios also impact post-inflationary processes, including reheating and , by altering particle interaction rates and thermalization dynamics due to the changing . For instance, a decreasing c can enhance out-of-equilibrium conditions necessary for generation, potentially facilitating electroweak through modified velocities. These effects provide a unified framework for early-universe evolution, though they require careful calibration to avoid conflicts with constraints. Regarding , VSL models offer an effective variation in the \Lambda by incorporating time-dependent c(t) into the Friedmann , where the term scales with factors like (c/c_0)^2, modifying the behavior of existing components such as . In the 2014 analysis by Qi et al., parameterizations such as c(t) = c_0 a^n(t) (with a the scale factor) were used to test VSL effects on late-time Hubble expansion using , BAO, and data, yielding fits comparable to standard \LambdaCDM but with tight constraints indicating negligible variation (n \approx 0). Recent extensions in 2025, such as the minimally extended VSL (meVSL) within the Friedmann-Lemaître-Robertson-Walker (FLRW) framework, further tie VSL to accelerated expansion by reinterpreting via lapse function variations, yielding deviations in density perturbations that could explain Hubble tension without . The Higgs-dilaton coupling model by Nguyen also generates VSL that reproduces supernova Ia observations (Pantheon catalog) in an Einstein-de Sitter universe, effectively bypassing while aligning with data and alleviating the coincidence between matter and acceleration epochs. These developments highlight VSL's potential as a parsimonious alternative for late-time , though they remain theoretical proposals requiring further observational tests as of November 2025.

Experimental constraints and tests

Observational evidence and limits

Observational data from Type Ia supernovae constrain variations in the at intermediate redshifts, with analyses of light curves from samples spanning z ≈ 0 to 2 indicating that |δc/c| ≲ 10^{-2} at z ≈ 1, consistent with a constant speed within observational precision. measurements from Planck further limit early-universe VSL effects, requiring near-constancy of c during recombination (z ≈ 1100) to reproduce the observed temperature and polarization power spectra, with deviations parameterized by b ≈ 0 at the 1σ level in minimally extended VSL models. Recent 2025 analyses of the minimally extended VSL (meVSL) model using updated Planck data confirm b ≈ 0, reinforcing no evidence for VSL. Constraints from absorption spectra and the Oklo natural provide tight bounds on historical variations. Quasar absorption lines at z = 0.5–3.5 yield Δα/α consistent with 0 within ~10^{-6} over cosmological lookback times, interpretable as δc/c < 10^{-6} assuming fixed electron charge and Planck's constant. Similarly, isotopic ratios from the Oklo reactor, active ≈2 billion years ago (z ≈ 0.15), constrain δc/c < 10^{-7} over that interval, again via α sensitivity, with no evidence for change. Recent reviews as of 2024, along with 2025 updates, summarize these astrophysical limits, reporting no positive evidence for and an upper bound on the present-day rate of |dc/c| / dt < 10^{-17} yr^{-1} from combined cosmological datasets. Some gamma-ray burst observations, such as time delays in high-energy photons from , hint at energy-dependent propagation effects that could mimic VSL, but analyses confirm no significant deviation from constant c across energies up to TeV scales, rendering such hints inconclusive.

Laboratory and astrophysical probes

Laboratory experiments have provided stringent tests of the constancy of the speed of light, particularly through modern implementations of the Michelson-Morley experiment using cryogenic optical resonators and Fabry-Pérot cavities. These setups compare the resonance frequencies of orthogonal cavities rotated relative to the Earth's velocity through the cosmic microwave background, searching for directional anisotropies that would indicate a variable speed of light. In one such experiment, the resonance frequencies of two sapphire resonators were monitored, yielding no detectable anisotropy and constraining the parameter describing light speed isotropy to better than 2 × 10^{-15} at the 1σ level. A more recent fiber-based Fabry-Pérot interferometer achieved even higher precision, limiting any anisotropy in light propagation to less than 10^{-17}, consistent with no variation in the speed of light. These laboratory probes demonstrate the local invariance of c to extraordinary accuracy, placing tight bounds on VSL models that predict directional dependence. Astrophysical observations offer complementary tests by probing light propagation over cosmic distances. Pulsar timing arrays, such as NANOGrav, monitor millisecond pulsars to detect nanohertz gravitational waves and constrain their propagation speed relative to light. Analysis of pulsar timing residuals from the NANOGrav 15-year dataset, combined with international PTA data, shows no deviation in the overlap reduction function expected from differing speeds, limiting |v_GW / c - 1| to within 10^{-3} for the stochastic gravitational wave background. Similarly, Fermi Large Area Telescope observations of gamma-ray bursts search for energy-dependent dispersion in photon arrival times, which could arise from VSL or Lorentz-violating effects. Data from four bright GRBs constrain the quantum gravity energy scale parameter to E_QG > 6.9 × 10^{19} GeV for linear LIV models, implying no detectable speed variation for photons across a wide range. High-energy multimessenger events provide further probes of speed equality across particles. Gravitational wave detections by and , particularly from a , arrived nearly simultaneously with gamma-ray emission observed by Fermi, constraining the speed of relative to light to |v_GW - c| / c < 10^{-15}. data from high-energy astrophysical sources, including searches around events, complement this by testing speeds. While no coincident s were found with / events, analyses of diffuse flux and time-of-flight measurements from atmospheric and cosmic sources limit |v_ν - c| / c to below 3 × 10^{-6} for energies up to PeV, supporting equality between and speeds within current sensitivities. These 2020s results from combined datasets tighten bounds on VSL scenarios involving particle speed differences. Future laboratory and space-based experiments aim to probe temporal variations in c with unprecedented precision. Proposed missions like the Ensemble in Space (ACES) on the will deploy cold-atom and clocks to test the constancy of constants, including c, through comparisons with ground clocks via links, potentially detecting variations at the 10^{-16} level per year. Similarly, concepts for optical clocks on solar-orbiting satellites, such as the proposed mission, would monitor frequency drifts sensitive to scalar or varying constants, offering direct tests of c's temporal stability over mission durations. These initiatives build on current constraints from the previous section's observational limits, focusing on controlled, long-baseline measurements.

Criticisms and theoretical challenges

Issues with Lorentz invariance

Variable speed of light (VSL) theories frequently necessitate the introduction of a preferred frame to define the variation of the , which directly contravenes the that assert the of all inertial frames and the invariance of the laws of physics across them. This reliance on a preferred frame, often aligned with the cosmic in cosmological models, results in physics that is explicitly frame-dependent, where physical quantities such as particle relations or electromagnetic exhibit directional dependence relative to the preferred direction. Such violations of Lorentz invariance undermine the foundational symmetry principles of modern physics, potentially allowing for absolute notions of rest and motion that have been empirically disfavored since the Michelson-Morley experiment. Theoretically, VSL models encounter significant incompatibilities with the symmetries of the , as the breaking of Lorentz invariance can induce violations of , a cornerstone conserved in quantum field theories. Moreover, modifications to the Lorentz structure in VSL frameworks may introduce ghosts—fields with negative —or tachyonic instabilities that destabilize the or lead to runaway solutions in the . These issues arise particularly in non-covariant implementations, where altered relations for particles can propagate anomalies through the field equations, compromising the unitarity and stability required for consistent theoretical descriptions. Non-covariant VSL models, such as those employing a hard breaking of Lorentz , predict anisotropic effects in cosmic propagation, including direction-dependent speeds for light or , which would manifest as deviations from the observed of the . For instance, in preferred-frame VSL cosmologies, the variation of c introduces subtle anisotropies proportional to the rate of change of c and relative velocities, conflicting with the high degree of seen in the . Attempts to mitigate these Lorentz invariance issues include covariant embeddings of VSL, such as those proposed by Magueijo, which redefine covariance and local Lorentz invariance to accommodate a varying c through a scalar field while preserving general coordinate invariance. However, even these formulations face challenges from no-go theorems, which argue that genuine variations in fundamental constants like c are indistinguishable from mere changes in units or lead to inconsistencies in the low-energy effective theory without additional structure. Such theorems, exemplified by analyses showing that VSL effects can be absorbed into redefinitions of spacetime metrics, highlight persistent obstacles to fully reconciling VSL with relativistic principles.

Compatibility with quantum field theory

Variable speed of light (VSL) theories face significant challenges in reconciling with quantum field theory (QFT), which fundamentally relies on the constancy of the speed of light c and full Lorentz invariance as cornerstones of special relativity. In standard QFT, c serves as the universal speed limit for causal propagation of fields and particles, ensuring microcausality and the absence of faster-than-light signaling. VSL proposals, by allowing c to vary with time, space, or energy, typically introduce Lorentz invariance violations (LIV), which can disrupt these principles and lead to inconsistencies such as acausal effects or instabilities in particle interactions. For instance, energy-dependent variations in c may permit superluminal propagation for high-energy particles, potentially violating unitarity or causality in QFT calculations. One major issue arises from the need to maintain gauge invariance in the , where c is embedded in the structure of electromagnetic and other interactions. Traditional VSL models often require introducing new dynamical fields or modifying the , which can break gauge symmetries unless carefully tuned. John Moffat has argued that VSL can mitigate certain QFT pathologies, such as those arising from future horizons in spacetimes or , by altering propagation and avoiding singularities in quantum corrections. However, this comes at the cost of , which conflicts with the local Lorentz invariance assumed in QFT on flat , potentially leading to non-renormalizable divergences or altered amplitudes. To address these incompatibilities, some formulations propose VSL as an effective low-energy phenomenon emerging from . In approaches like doubly special relativity or deformed , c becomes energy-dependent through modified relations, such as E^2 = p^2 c^2 + \lambda E^3 / M_{Pl}, where \lambda is a dimensionless and M_{Pl} is the Planck ; this preserves an invariant maximum speed while allowing variations at high energies, potentially integrable with QFT via effective field theory expansions. João Magueijo's bimetric VSL theories introduce two metrics—one for and one for —allowing variable c without fully breaking Lorentz invariance for fields, though quantum consistency requires further analysis. Recent minimally extended VSL (meVSL) models achieve better compatibility by treating variations in c as coordinate effects via the lapse function in , without new fields, and by covariantly evolving the Planck constant \hbar alongside c to preserve quantum relations like the during cosmological expansion. This ensures thermodynamic consistency and gauge invariance in the , as the effective c remains a non-dynamical scalar under diffeomorphisms. Nonetheless, full quantum treatment in curved spacetimes remains an open challenge, with ongoing work exploring VSL in or to resolve ultraviolet divergences without invoking . Experimental bounds from gamma-ray bursts and cosmic rays provide indirect tests of these signatures, constraining VSL parameters to within fractions of the Planck scale.

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