Cosmic time

Cosmic time, or cosmological time, is the time coordinate employed in Big Bang models of physical cosmology to measure the progress of the universe's evolution since its origin, specifically as the proper time experienced by observers at rest relative to the expanding cosmic background.^[1]^[2] In the standard Friedmann–Lemaître–Robertson–Walker (FLRW) metric describing a homogeneous and isotropic universe, cosmic time t is defined as the time measured by fundamental observers moving along the Hubble flow—meaning they have no peculiar velocity and are comoving with the cosmic microwave background (CMB)—with clocks synchronized across spacelike hypersurfaces orthogonal to their worldlines.^[3]^[2] This synchronization relies on the cosmological principle, which posits uniformity on large scales, and the Weyl postulate that the cosmic fluid follows non-intersecting geodesics without rotation.^[4]^[2] The scale factor a(t), which quantifies the universe's expansion, evolves with cosmic time according to the Friedmann equations, linking t to observable quantities like redshift z via $1 + z = 1/a(t).^[3] Cosmic time provides a universal timeline for key cosmological epochs, such as the Big Bang singularity [at

t](/page/AT&T) = [0](/page/0)

, matter-radiation equality at approximately 51.7 kyr, and recombination at about 372.6 kyr, enabling precise calculations of the universe's current age—estimated at 13.8 billion years based on CMB data.^[3]^[1] It distinguishes itself from local proper time by accounting for the global expansion, avoiding relativity issues in defining simultaneity across vast distances, and serves as the reference for lookback time, which measures the light-travel duration to distant objects.^[2]^[4] In this framework, cosmic time underpins models of structure formation, dark energy evolution, and the overall history of the universe from inflation to the present acceleration.^[3]

Conceptual Foundations

Challenges to Absolute Time

In Newtonian mechanics, time is defined as absolute, true, and mathematical, flowing equably and uniformly in itself without relation to anything external, and independent of the motion or position of observers. This concept posits time as a universal backdrop, unaffected by physical processes or observers, serving as a fixed measure for all events. Isaac Newton articulated this view in the Scholium to the Definitions in his Philosophiæ Naturalis Principia Mathematica (1687), where he distinguished it from relative, apparent, and common time, which varies with human perception or motion.^[5] Critiques of absolute time emerged from philosophical perspectives emphasizing relationalism. Gottfried Wilhelm Leibniz, in his correspondence with Samuel Clarke (a defender of Newtonian ideas) from 1715 to 1716, argued that time is not an independent substance but a relational order of successive events among coexisting things, lacking meaning without reference to changes in the world. Similarly, Ernst Mach, in The Science of Mechanics (1883), rejected Newton's absolute time as metaphysical and unobservable, proposing instead that time be understood relationally through the measurable changes and dependencies among physical phenomena in the universe. These relational views challenged the substantival nature of time, suggesting it derives its reality from interactions rather than existing in isolation.^[6]^[7] A key issue with absolute time arose from problems of simultaneity in moving reference frames, illustrated by thought experiments considering events like lightning strikes observed from both a stationary platform and a passing train. Such scenarios demonstrate that what appears simultaneous to one observer may not to another in relative motion, undermining the universality of Newtonian time. This relativity of simultaneity, along with time dilation effects where moving clocks tick slower relative to stationary ones, was rigorously established by Albert Einstein in his seminal 1905 paper "On the Electrodynamics of Moving Bodies," which replaced absolute time with a framework where time is intertwined with space and observer motion in special relativity. These developments were later extended in general relativity to account for gravitational influences on time.^[8]

Proper Time in General Relativity

In general relativity, proper time \tau represents the duration measured by an idealized clock following a timelike worldline, which is the path of a massive particle or observer through spacetime. This invariant quantity is the integral \tau = \int d\tau along the worldline, where each infinitesimal element d\tau quantifies the "experienced" time independent of the coordinate system. The spacetime geometry is described by the metric tensor g_{\mu\nu}, with the line element ds^2 = g_{\mu\nu} \, dx^\mu \, dx^\nu. For timelike paths where ds^2 < 0, the proper time interval is derived as

d\tau = \frac{1}{c} \sqrt{-ds^2},

or equivalently, ds^2 = -c^2 d\tau^2 + dl^2, where dl^2 encompasses the spatial separation orthogonal to the time direction. This formulation arises from the requirement that proper time maximizes the interval along geodesics, the shortest paths in spacetime for massive objects, contrasting with the absolute time in Newtonian mechanics where simultaneity is universal across all observers.^[9]^[10] Coordinate time t, such as the time parameter in a chosen chart like Schwarzschild coordinates, labels events but does not correspond to the physical ticking of clocks; it varies with the observer's frame and gravitational field. In curved spacetime, proper time \tau differs from t due to both velocity and gravitational effects, leading to time dilation where clocks in stronger fields or relative motion elapse more slowly relative to distant standards. This distinction is fundamental, as proper time is a scalar invariant under general coordinate transformations, while coordinate time is frame-dependent.^[11] A key example occurs in the Schwarzschild metric, describing spacetime around a non-rotating black hole, where for a stationary observer at radial coordinate r > r_s (with r_s = 2GM/c^2 the Schwarzschild radius), the proper time relates to coordinate time by d\tau = \sqrt{1 - r_s/r} \, dt. This gravitational redshift effect causes clocks near the black hole to tick slower compared to those at infinity, as verified in observations like atomic clocks on Earth versus GPS satellites. Another illustration is uniform acceleration in flat spacetime using Rindler coordinates (\eta, \xi), where the metric is ds^2 = -(1 + g\xi/c^2)^2 c^2 d\eta^2 + d\xi^2 + dy^2 + dz^2; here, the proper time for an observer at fixed \xi with constant proper acceleration a = c^2 / (\xi + c^2/g) satisfies d\tau = (1 + g\xi/c^2) d\eta, demonstrating how acceleration mimics gravitational time dilation in an inertial frame.^[12]

Definition in Cosmological Models

Formal Definition

In the standard cosmological model, cosmic time t is defined as the proper time measured by fundamental observers who are comoving with the Hubble flow, meaning they remain at fixed spatial coordinates in the expanding universe. These observers experience no peculiar velocity relative to the overall expansion, and their clocks tick according to the time coordinate in the Friedmann-Lemaître-Robertson-Walker (FLRW) metric, which describes a homogeneous and isotropic universe.^[13]^[14] The FLRW line element formalizes this as

ds^2 = -c^2 \, dt^2 + a(t)^2 \left[ \frac{dr^2}{1 - k r^2} + r^2 \, d\Omega^2 \right],

where t denotes cosmic time, a(t) is the dimensionless scale factor that encodes the expansion history, k is the spatial curvature parameter (k = -1, 0, +1), r is the comoving radial coordinate, and d\Omega^2 = d\theta^2 + \sin^2 \theta \, d\phi^2 spans the angular part. For comoving observers at rest (dr = d\theta = d\phi = 0), the line element simplifies to ds^2 = -c^2 \, dt^2, confirming that dt directly measures their proper time. This setup ensures cosmic time serves as a synchronous parameter labeling uniform hypersurfaces across the universe, facilitating the description of its global evolution.^[13]^[15] The validity of cosmic time relies on the hypersurface orthogonality of the comoving observers' worldlines, as postulated by Weyl, which guarantees that these geodesics are everywhere orthogonal to spacelike hypersurfaces of simultaneity without vorticity or acceleration relative to the expansion. In homogeneous and isotropic universes governed by the cosmological principle, this choice of time coordinate is unique up to the overall normalization of the metric, providing a canonical framework for modeling the universe from the Big Bang singularity at t = [0](/page/0), where a(t) \to [0](/page/0).^[13]^[16]

Comoving Frame and Coordinates

In the Friedmann–Lemaître–Robertson–Walker (FLRW) models of cosmology, comoving coordinates provide a reference frame where the spatial positions of galaxies remain fixed despite the universe's expansion. These coordinates, typically denoted as (r, θ, φ) in spherical form, label points that are carried along with the overall expansion, such that galaxies occupy constant values of these coordinates over time. The physical distances between such points increase proportionally to the scale factor a(t), a dimensionless function of cosmic time t that quantifies the relative expansion of the universe at different epochs. Fundamental observers are defined as those at rest within this comoving frame, meaning they have no peculiar velocity relative to the average matter distribution and are thus aligned with the cosmic microwave background (CMB) rest frame. These observers follow geodesic worldlines with zero acceleration in the FLRW spacetime, serving as the standard reference for measuring the homogeneity and isotropy of the universe. Their four-velocity u^\mu is aligned with the coordinate time direction \partial / \partial t, normalized such that u^\mu u_\mu = -1. In practice, most cosmological analyses identify fundamental observers with the CMB frame, where the CMB appears isotropic to within small perturbations of order $10^{-5}. The spatial geometry in the comoving frame is characterized by the curvature parameter k, a dimensionless constant that takes discrete values of -1, 0, or +1. For k = +1, the universe has positive spatial curvature, resulting in a closed geometry analogous to the surface of a three-sphere, which is finite but unbounded. When k = 0, the spatial sections are flat with Euclidean geometry, extending infinitely without curvature. A value of k = -1 corresponds to negative curvature, yielding an open hyperbolic geometry that is also infinite in extent. This parameter influences the overall topology and the relation between comoving distances and physical observables, though current measurements from the CMB indicate a nearly flat universe with |k| approaching zero. Physically, the expansion of the universe in this framework is understood as the growing separation between fixed comoving points due to the time evolution of a(t), rather than the bulk motion of galaxies through pre-existing space. Galaxies are effectively stationary in comoving coordinates, with any observed recession velocities arising from the metric's scale factor rather than local dynamical effects. This interpretation avoids superluminal motion issues for distant objects, as the expansion rate is a global property of spacetime itself. Cosmic time corresponds to the proper time elapsed along the worldlines of these fundamental observers.

Cosmological Time Scales

Age of the Universe

In the standard ΛCDM cosmological model, the age of the universe, denoted as t_0, represents the proper time elapsed since the Big Bang for an observer at rest in the comoving frame. This scalar quantity is computed by integrating the inverse of the expansion rate over the scale factor a, from the initial singularity (a = 0) to the present (a = 1):

t_0 = \frac{1}{H_0} \int_0^1 \frac{da}{a^2 H(a)/H_0},

where H_0 is the present-day Hubble constant, and the normalized Hubble parameter is given by

\frac{H(a)}{H_0} = \sqrt{\frac{\Omega_m}{a^3} + \Omega_\Lambda + (1 - \Omega_\mathrm{tot}) a^{-2}}.

Here, \Omega_m is the present matter density parameter (including baryons and cold dark matter), \Omega_\Lambda is the dark energy density parameter, and \Omega_\mathrm{tot} = \Omega_m + \Omega_\Lambda + \Omega_k accounts for curvature (\Omega_k = 0 in a flat universe). This integral encapsulates the cumulative effect of gravitational slowing in the early matter-dominated era and acceleration from dark energy in later epochs.^[17] Measurements from the Planck satellite's 2018 analysis of cosmic microwave background anisotropies provide the benchmark estimate for t_0, yielding $13.797 \pm 0.023 billion years under the base ΛCDM model assuming flatness. This has been confirmed by the Atacama Cosmological Telescope's 2025 analysis, yielding 13.8 billion years with 0.1% uncertainty. This result relies on parameters such as H_0 = 67.36 \pm 0.54 km s⁻¹ Mpc⁻¹, \Omega_m = 0.3153 \pm 0.0073, and \Omega_\Lambda = 0.6847 \pm 0.0073, derived from temperature and polarization power spectra combined with baryon acoustic oscillation data. These values reflect a universe transitioning from radiation- and matter-dominated phases to dark energy dominance around redshift z \approx 0.3.^[18] The computed age depends sensitively on the density parameters: higher \Omega_m shortens t_0 by enhancing early deceleration, while greater \Omega_\Lambda lengthens it through late-time acceleration; non-zero curvature (\Omega_k \neq 0) further modulates the integral, with positive curvature implying a younger universe. In the historical Einstein–de Sitter model—a flat, matter-only universe with \Omega_m = 1 and \Omega_\Lambda = 0—the age simplifies analytically to t_0 = \frac{2}{3 H_0}, or approximately 9.2 billion years using modern H_0 values, underscoring how dark energy extends the timeline beyond matter-dominated predictions. This model, proposed in 1932, served as a reference for early cosmological interpretations before observations favored ΛCDM. Uncertainties in t_0 arise primarily from the Hubble tension, a discrepancy between CMB-inferred H_0 (e.g., Planck's 67.4 km s⁻¹ Mpc⁻¹) and local measurements like those from the SH0ES team using Cepheid-calibrated supernovae, which yield H_0 \approx 72.6–73 km s⁻¹ Mpc⁻¹ as of 2024 JWST observations. Adopting the higher SH0ES value would reduce the age to about 12.9 billion years, highlighting a potential inconsistency in expansion history that challenges ΛCDM consistency at the 4–5σ level. As of 2025, the tension remains unresolved, increasingly referred to as a crisis, with ongoing JWST efforts refining distance ladders.^[17]^[20]

Lookback Time

Lookback time, denoted as t_L(z), represents the duration of cosmic time that has elapsed since light was emitted from a source at redshift z and received by an observer at the present epoch (z = 0). It quantifies the light-travel time across an expanding universe, providing a measure of when distant events occurred in the cosmic timeline. Unlike proper distance, which describes the physical separation between emitter and observer at a given epoch, lookback time emphasizes the temporal interval affected by the universe's expansion history. In the standard \LambdaCDM model, lookback time is computed via the integral

t_L(z) = \int_0^z \frac{dz'}{(1 + z') H(z')},

where H(z) = H_0 \sqrt{\Omega_m (1 + z)^3 + \Omega_\Lambda + (1 - \Omega_\mathrm{tot})(1 + z)^2} is the Hubble parameter at redshift z, H_0 is the present-day Hubble constant, \Omega_m is the present matter density parameter, \Omega_\Lambda is the present dark energy density parameter, and \Omega_\mathrm{tot} = \Omega_m + \Omega_\Lambda + \Omega_k accounts for total density including curvature \Omega_k. For a flat universe (\Omega_k = 0, \Omega_\mathrm{tot} = 1), the expression simplifies by omitting the curvature term. This formulation arises from the Friedmann-Lemaître-Robertson-Walker metric, integrating the scale factor's evolution along null geodesics for photons. A representative example illustrates its scale: in a flat \LambdaCDM cosmology with H_0 \approx 70 km s^{-1} Mpc^{-1}, \Omega_m = 0.3, and \Omega_\Lambda = 0.7, the lookback time to z = 1 (approximately halfway to the particle horizon) is about 7.7 billion years. This places the emission event roughly halfway through the universe's current age of approximately 13.8 billion years. Lookback time enables precise timing of key cosmic events, such as the formation of early galaxies by combining it with stellar age estimates from spectroscopy. It also supports analyses of supernova explosions, particularly Type Ia events, by mapping their delay-time distributions relative to star formation epochs to probe nucleosynthesis and dark energy evolution.^[21]

Observational Connections

Relation to Redshift

In cosmology, the redshift z observed for light emitted at cosmic time t_\mathrm{em} and received at the present cosmic time t_0 is directly tied to the evolution of the scale factor a(t), which parametrizes the expansion of the universe over cosmic time. Specifically, the redshift is defined as z = \frac{\lambda_\mathrm{obs} - \lambda_\mathrm{em}}{\lambda_\mathrm{em}} = \frac{a(t_0)}{a(t_\mathrm{em})} - 1, where \lambda_\mathrm{obs} and \lambda_\mathrm{em} are the observed and emitted wavelengths, respectively.^[22] This relation establishes a one-to-one correspondence between the redshift and the ratio of scale factors at emission and observation, reflecting how the universe's expansion history, governed by cosmic time, stretches the light's wavelength.^[22] The origin of this cosmological redshift lies in the expansion of space itself, rather than relative motion through space. As photons propagate from distant sources, the intervening space expands, causing the photon's wavelength to increase proportionally to the scale factor: \lambda(t) \propto a(t). This geometric effect accumulates over the light's travel from emission at t_\mathrm{em} to observation at t_0, resulting in the observed stretching without energy loss to Doppler-like motion of the source.^[22] In the framework of the Friedmann-Lemaître-Robertson-Walker metric, this follows from the null geodesic equation for light in an expanding universe, where the proper distance between comoving points grows with a(t).^[22] A kinematic interpretation provides an intuitive understanding, particularly at low redshifts, by viewing the redshift as the cumulative effect of infinitesimal recession velocities along the photon's path. In this picture, each small segment contributes a Doppler shift dz \approx H(t) \, dl / c, where H(t) = \dot{a}(t)/a(t) is the Hubble parameter at time t and dl is the proper distance element. Integrating along the path yields \ln(1 + z) = \int_{t_\mathrm{em}}^{t_0} H(t) \, dt, which approximates to z \approx \int_{t_\mathrm{em}}^{t_0} H(t) \, dt for small z.^[22] This equivalence holds because the scale factor evolution satisfies da/a = H(t) \, dt, linking the kinematic sum directly to the geometric stretching.^[22] Observational evidence strongly confirms this time-redshift connection through the cosmic microwave background (CMB), the relic radiation from the early universe. The CMB temperature scales inversely with the scale factor, T \propto 1/a(t), implying that photons emitted when the universe was smaller appear hotter in the past. Measurements show the CMB at a redshift z_\mathrm{CMB} \approx 1100, corresponding to the epoch of recombination around 380,000 years after the Big Bang, when the universe cooled sufficiently for neutral hydrogen to form and decouple photons.^[23] This high-redshift signature validates the expansion's impact on cosmic time scales.^[23]

Integration with Expansion History

Cosmic time serves as the fundamental parameter in the Friedmann equations, which govern the dynamical evolution of the universe's scale factor a(t) in a homogeneous and isotropic cosmology. The first Friedmann equation relates the Hubble parameter H(t) = \dot{a}/a to the energy density \rho, spatial curvature k, and cosmological constant \Lambda as follows:

\left( \frac{\dot{a}}{a} \right)^2 = H^2 = \frac{8\pi G}{3} \rho - \frac{k c^2}{a^2} + \frac{\Lambda}{3},

where G is the gravitational constant and c is the speed of light. This equation, derived from Einstein's field equations applied to the Friedmann-Lemaître-Robertson-Walker metric, describes how the expansion rate changes over cosmic time t, with \rho encompassing contributions from radiation, matter, and dark energy.^[17] The universe's expansion history divides into distinct epochs defined by the dominant energy component, each spanning specific intervals of cosmic time. In the radiation-dominated era, from the Big Bang until matter-radiation equality at approximately 52,000 years (redshift z ≈ 3400), relativistic particles and photons drive the expansion with \rho \propto a^{-4}, leading to a(t) \propto t^{1/2}. This transitions to the matter-dominated epoch around that time, persisting until matter-dark energy equality at z ≈ 0.3 (about 10.3 billion years after the Big Bang, corresponding to a lookback time of approximately 3.5 billion years), where non-relativistic matter dominates with \rho \propto a^{-3} and a(t) \propto t^{2/3}. Currently, the universe is in the dark energy-dominated phase (since z ≈ 0.3), where \Lambda or a similar component causes accelerated expansion with \rho nearly constant, yielding a(t) \propto \exp(H t) asymptotically.^[17] Cosmic time integrates into distance measures that quantify the expansion's spatial extent. The comoving distance \chi to a source emitting at time t_e (observed at t_0) is given by \chi = \int_{t_e}^{t_0} c \, dt / a(t), which can be recast in terms of redshift z as \chi = \int_0^z c \, dz' / H(z'), remaining fixed in the comoving frame as the universe expands.^[24] For a flat universe (k=0), the luminosity distance d_L relates to \chi by d_L = (1+z) \chi, enabling inferences about expansion history from supernova observations. The particle horizon, representing the maximum comoving distance light has traveled since t=0, thus defines the observable universe's boundary at \chi_h \approx 14.4 Gpc, corresponding to a proper radius of approximately 46.5 billion light-years today.^[17]