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Coordinate time

Coordinate time is the temporal component of a coordinate system used to label events in spacetime, particularly within the frameworks of special and general relativity, where it provides a conventional measure for ordering occurrences rather than directly representing physical duration experienced by observers.^[1] Unlike proper time, which quantifies the invariant interval along an object's worldline as measured by its own clock, coordinate time depends on the chosen reference frame and can vary due to relative motion or gravitational fields.^[1] In mathematical terms, the relationship between infinitesimal proper time d\tau and coordinate time dt is given by the spacetime metric, such as d\tau^2 = dt^2 - dx^2 - dy^2 - dz^2 (in units where c=1) for flat Minkowski space in special relativity.^[1] In special relativity, coordinate time corresponds to the synchronized time across an inertial frame, where simultaneity is relative, and it equals proper time only for observers at rest in that frame.^[2] General relativity extends this concept to curved spacetime, emphasizing covariance: all coordinate systems, including their time labels, are equally valid for describing physics, with no absolute frame privileged, as per the equivalence principle.^[3] For instance, in noninertial frames like those near Earth, coordinate time incorporates adjustments for acceleration and gravity, differing from inertial coordinate time through metric components that account for time dilation.^[4] Practically, coordinate time underpins astronomical time scales, such as Barycentric Dynamical Time (TDB), which serves as the independent variable for ephemerides in solar system dynamics within general relativity's framework.^[5] This realization, approximating barycentric coordinate time (TCB) with periodic corrections, ensures precise synchronization for space missions, light propagation, and celestial mechanics calculations.^[5]

Fundamental Concepts

Coordinate Time vs. Proper Time

In special relativity, coordinate time serves as a global parameter that labels events within a chosen spacetime coordinate system, typically measured by a network of synchronized clocks at rest relative to an inertial observer, and remains independent of the motion of individual clocks or observers.^[6] This time coordinate, often denoted as t, provides a frame-dependent measure of temporal progression across space, allowing events separated in position to be assigned simultaneous timestamps through synchronization conventions.^[2] In contrast, proper time represents the invariant spacetime interval along the worldline of a specific observer or particle, measured directly by a single clock traveling along that path, and is independent of the choice of coordinate system.^[2] Denoted as \tau, it quantifies the duration experienced locally by the clock, unaffected by relative motion between frames, and serves as the fundamental measure of aging or physical processes for that observer.^[7] The distinction arises because proper time is an intrinsic property tied to the geometry of spacetime, while coordinate time is an extrinsic label imposed by the observer's frame.^[6] Albert Einstein introduced the foundational concepts of this distinction in his 1905 paper on special relativity, where he demonstrated through time dilation that the time measured by a moving clock differs from the synchronized time in a stationary frame, a difference that becomes particularly pronounced when considering acceleration in non-inertial frames.^[8] Hermann Minkowski formalized proper time as an invariant in his 1908 reformulation of special relativity into a four-dimensional spacetime framework, emphasizing its role as the proper duration along timelike paths.^[7] A classic illustration of this difference is the twin paradox, where one twin remains on Earth while the other travels at relativistic speeds and returns; the traveling twin accumulates less proper time due to their accelerated worldline, aging slower than the Earth-bound twin, whose proper time coincides with the coordinate time in the inertial frame.^[2] In Minkowski space, this relationship is captured by the differential form for proper time:

d\tau = \sqrt{dt^2 - \frac{dx^2 + dy^2 + dz^2}{c^2}}

where dt is the infinitesimal coordinate time interval, dx, dy, dz are spatial displacements, and c is the speed of light, showing that proper time is always shorter than or equal to coordinate time, with equality only for stationary observers.^[7]

Clock Synchronization in Relativity

In special relativity, clock synchronization establishes a coordinate time across an inertial reference frame, but the procedure reveals fundamental limitations due to the theory's postulates. Henri Poincaré first discussed the synchronization of moving clocks in his 1905 memoir, where he introduced the concept of "local time" derived from light signal exchanges between observers, highlighting the conventional nature of simultaneity in relative motion.^[9] Albert Einstein elaborated on this in his seminal 1905 paper, defining simultaneity for spatially separated events through the assumption that light propagates at constant speed c in all directions within the frame.^[10] The Einstein synchronization convention provides the standard procedure for inertial frames: two clocks at positions A and B are synchronized if a light signal emitted from A at coordinate time t_A arrives at B at t_B = t_A + \frac{d}{c}, and the return signal from B arrives back at A at t_A' = t_B + \frac{d}{c}, ensuring the round-trip travel time is symmetric and light speed appears isotropic.^[10] This method assumes the one-way speed of light is c, a convention that aligns with the principle of relativity but is not empirically verifiable without prior synchronization assumptions.^[11] However, special relativity introduces the relativity of simultaneity, where clocks synchronized in one inertial frame appear desynchronized in another frame moving relative to the first, as the transformation of time coordinates depends on spatial separation and relative velocity.^[12] This frame-dependence means no absolute simultaneity exists, rendering synchronization a local convention rather than a universal truth.^[10] Extending synchronization to accelerated frames, where inertial methods fail due to non-constant velocity, requires alternative protocols. The radar method uses a single clock to emit and receive light or radio signals from a distant point, assigning coordinate time to events based on the midpoint of emission and reception times, scaled by the assumed round-trip light speed; this defines an approximate synchronization for the observer's instantaneous rest frame.^[13] Complementarily, slow clock transport involves moving a clock at low velocity from one location to another, where the accumulated proper time during transport approximates the coordinate time difference, yielding synchronization equivalent to the Einstein convention in the low-speed limit and avoiding direct reliance on light signals.^[14] These approaches provide practical consistency in non-inertial settings but inherit the relativity of simultaneity when comparing across frames. A key practical application arises in the Global Positioning System (GPS), where satellite clocks must synchronize with Earth-based coordinate time despite orbital velocities causing special relativistic effects. GPS employs atomic clocks pre-adjusted for relativistic time dilation effects, including both velocity-induced and gravitational contributions, and uses two-way signal ranging to maintain synchronization, ensuring positional accuracy within meters by aligning satellite time readings to the International Atomic Time scale on Earth.^[15]

Mathematical Foundations

Formulation in Special Relativity

In special relativity, the spacetime of flat Minkowski geometry provides the foundational framework for defining coordinate time. The line element in Minkowski spacetime is given by the metric

ds^2 = -c^2 dt^2 + dx^2 + dy^2 + dz^2,

where dt represents the infinitesimal coordinate time interval in a given inertial frame, c is the speed of light, and dx, dy, dz are spatial differentials.^[16] This metric, introduced by Hermann Minkowski, treats time as a coordinate on equal footing with space, forming a four-dimensional continuum where coordinate time t parameterizes events along worldlines.^[16] Coordinate time transforms between inertial frames via the Lorentz transformation, which accounts for the relativity of simultaneity. For two frames where the primed frame moves at velocity v along the x-axis relative to the unprimed frame, the time component is

t' = \gamma \left( t - \frac{v x}{c^2} \right),

with \gamma = \frac{1}{\sqrt{1 - v^2/c^2}}.^[10] This transformation, derived from the constancy of the speed of light and the principle of relativity, demonstrates that coordinate time is frame-dependent: events simultaneous in one frame (dt' = 0) are not in another unless x = 0.^[10] The relationship between coordinate time and proper time \tau—the time measured by a clock along its worldline—arises from the metric for timelike paths (ds^2 < 0). For a clock moving at velocity \mathbf{v}(t) in the coordinate frame, the proper time interval is d\tau = dt \sqrt{1 - v^2/c^2}, leading to the integrated form

\tau = \int \sqrt{1 - \frac{v^2(t)}{c^2}} \, dt

along the worldline.^[16] This integral shows that proper time accumulates more slowly than coordinate time for moving observers, a direct consequence of the invariance of the spacetime interval. For constant velocity v, the time dilation formula simplifies to \Delta \tau = \Delta t / \gamma = \Delta t \sqrt{1 - v^2/c^2}, where \Delta t is the coordinate time elapsed in the rest frame of the coordinates.^[10]^[16] Consider an example: a clock moving at constant speed v = 0.8c relative to an inertial frame, with coordinate time \Delta t = 10 years elapsing in that frame. Here, \gamma = 1 / \sqrt{1 - (0.8)^2} = 1.6667, so the proper time is \Delta \tau = 10 / 1.6667 \approx 6 years. This calculation illustrates how coordinate time overestimates the clock's elapsed proper time by the factor \gamma > 1.^[10]

Formulation in General Relativity

In general relativity, spacetime is described by a pseudo-Riemannian manifold equipped with a metric tensor g_{\mu\nu}, which defines the line element as

ds^2 = g_{\mu\nu} \, dx^\mu \, dx^\nu,

where the coordinates x^\mu include the coordinate time t (typically as x^0 = ct) alongside spatial coordinates, and the metric components encode the curvature induced by mass and energy.^[17] This formulation generalizes the flat Minkowski metric of special relativity to curved spacetimes, allowing coordinate time to serve as a global parameter for events in asymptotically flat regions.^[18] The proper time \tau experienced by an observer along a timelike worldline is obtained by integrating the infinitesimal interval

d\tau = \sqrt{ -\frac{g_{\mu\nu} \, dx^\mu \, dx^\nu}{c^2} },

which extends the special-relativistic definition to account for gravitational effects through the metric's dependence on position.^[17] For observers at rest in the coordinate system (i.e., dx^i = 0 for spatial indices i), this reduces to a relation between proper time and coordinate time influenced solely by the temporal component g_{00}. A prominent example of gravitational time dilation arises in the Schwarzschild metric, which describes the exterior spacetime around a spherically symmetric, non-rotating mass M:

ds^2 = -\left(1 - \frac{2GM}{c^2 r}\right) c^2 dt^2 + \left(1 - \frac{2GM}{c^2 r}\right)^{-1} dr^2 + r^2 d\Omega^2.

For stationary observers at fixed radial coordinate r > 2GM/c^2, the proper time elapses as

d\tau = dt \sqrt{1 - \frac{2GM}{c^2 r}},

demonstrating that clocks deeper in the gravitational potential run slower relative to coordinate time measured at infinity.^[18]^[19] In static metrics like Schwarzschild, the coordinate time t is associated with a timelike Killing vector field \xi = \partial_t, which generates time translations and remains orthogonal to spatial hypersurfaces, ensuring the metric's time-independence.^[20] This vector is timelike in the asymptotically flat region (r \to \infty), where g_{00} \to -1, allowing coordinate time to approximate proper time for distant observers and facilitating the matching of local physics to global coordinates.^[21] To illustrate the distinction, consider radial light signals (null geodesics, ds^2 = 0) in the Schwarzschild solution. For an outgoing photon from radius r_e to infinity, the coordinate time interval is

\Delta t = \int_{r_e}^\infty \frac{dr}{c \left(1 - \frac{2GM}{c^2 r}\right)},

which diverges logarithmically as r_e approaches the event horizon at r_s = 2GM/c^2, whereas the proper time for a stationary emitter at r_e is finite but redshifted. Conversely, an infalling observer measures a finite proper time to reach the horizon, while coordinate time for the signal remains undefined inside r_s. This highlights how coordinate time captures asymptotic behavior without directly corresponding to local clock readings.^[22]

Implementation and Measurement

Observational and Experimental Methods

Atomic clocks serve as the primary standards for realizing coordinate time in relativistic frameworks, with cesium-beam and hydrogen maser clocks providing the necessary precision for both terrestrial and space-based applications. Cesium atomic clocks, based on the hyperfine transition frequency of the cesium-133 atom at approximately 9.192 GHz, achieve long-term stability and accuracy essential for defining scales like International Atomic Time (TAI), where they contribute to the weighted ensemble average through primary frequency standards such as NIST-F2, with an accuracy of about 10^{-16}. Hydrogen maser clocks, utilizing the 21 cm hyperfine transition of neutral hydrogen, offer superior short-term stability on the order of 10^{-15} over integration times of 100 seconds, making them ideal for high-precision timing in very long baseline interferometry (VLBI) and satellite missions, where they maintain phase coherence for coordinate time synchronization. These clocks incorporate relativistic corrections for gravitational redshift and velocity effects to ensure alignment with coordinate time definitions in general relativity. Interferometric techniques and pulsar timing arrays provide astronomical-scale verification of coordinate time predictions, leveraging the stability of millisecond pulsars as celestial clocks to test relativistic effects over vast distances. Millisecond pulsars, rotating hundreds of times per second with pulse arrival times predictable to within nanoseconds over years, enable measurements of Shapiro delay and orbital decay that confirm general relativistic time dilation and periastron advance, as demonstrated in analyses of binary systems like PSR B1913+16. Pulsar timing residuals, after accounting for interstellar dispersion and solar system ephemeris, yield tests of the strong equivalence principle with precision approaching 10^{-5} of the predicted relativistic effects, using arrays such as the North American Nanohertz Observatory for Gravitational Waves (NANOGrav). These methods validate coordinate time scales by comparing observed pulse phases against models incorporating barycentric coordinate time (TCB). The Hafele-Keating experiment in 1971 provided early empirical confirmation of coordinate time variations due to both special and general relativistic effects, using four cesium-beam atomic clocks flown on commercial airliners around the world. Clocks on eastward flights lost 59 ± 10 ns relative to ground stations, while those on westward flights gained 273 ± 7 ns, combining kinematic and gravitational contributions, with results agreeing with relativistic predictions to within experimental uncertainties of about 10%. Complementing this, the Gravity Probe A rocket experiment in 1976 launched a hydrogen maser clock to 10,000 km altitude, measuring a gravitational redshift of 4.5 × 10^{-10} over two hours, consistent with general relativity to 0.01% precision, as the clock rate increased by the predicted factor (1 + gh/c²) where h is altitude. These terrestrial and suborbital tests established the feasibility of coordinate time realization under dynamic conditions. Satellite-based astrometric missions like Hipparcos and Gaia further verify coordinate time scales through precise measurements of stellar positions and proper motions, incorporating relativistic light propagation delays in their data reduction. The Hipparcos mission (1989–1993) achieved parallax accuracies of 1 milliarcsecond by modeling observations in the barycentric celestial reference system, applying corrections for annual aberration and gravitational deflection using TCB, which ensured positional consistency at the microarcsecond level. Similarly, the Gaia mission, operational since 2013, processes billions of stellar observations with a relativistic astrometric model that includes coordinate time-dependent effects like the gravitomagnetic field, yielding proper motion precisions of 0.1 milliarcsecond per year and validating time scales through consistency checks against pulsar-based ephemerides. Modern experimental setups achieve relativistic corrections in coordinate time realization at precisions of parts in 10^{15}, enabling tests of fundamental physics beyond classical limits. For instance, comparisons of optical clocks, such as single-ion and lattice types separated by millimeter to centimeter heights, demonstrate gravitational redshift with fractional frequency shifts on the order of 10^{-17} per meter, as in early measurements of (7.8 ± 1.3) × 10^{-17} and more recent lattice clock tests approaching 10^{-18}. Error analyses in cesium fountain clocks reveal systematic uncertainties from blackbody radiation and relativistic Doppler shifts below 10^{-16}, with hydrogen masers in VLBI networks maintaining timing residuals under 10 femtoseconds after corrections, thus supporting coordinate time scales with uncertainties far below astronomical signal levels.

Practical Applications

In global navigation satellite systems (GNSS) such as GPS, coordinate time forms the basis for precise positioning and timing, requiring relativistic corrections to account for gravitational and velocity effects on satellite clocks. These corrections include a net adjustment of approximately 38 microseconds per day to compensate for the faster rate of satellite clocks relative to ground clocks, arising from weaker gravitational fields and orbital velocities.^[23] Additionally, the Sagnac effect due to Earth's rotation and periodic variations from orbital eccentricity, with amplitudes around 46 nanoseconds, are incorporated into the coordinate time calculations to ensure synchronization accuracy within nanoseconds. GPS time itself serves as a continuous coordinate time scale in the Earth-centered inertial frame, enabling reliable broadcast of time signals for user receivers.^[24] For deep space missions, coordinate time is critical for signal timing and navigation, particularly through the use of Barycentric Coordinate Time (TCB) to model propagation delays over vast distances. In the Voyager program, which has operated since 1977, TCB-based ephemerides compute one-way light times exceeding 20 hours for Voyager 1, ensuring accurate command sequencing and data return via the Deep Space Network.^[25] Similarly, the New Horizons mission to Pluto and beyond relies on barycentric coordinate time for ranging and Doppler measurements, achieving position accuracies better than 0.3 AU from the solar system barycenter during interstellar phases. In satellite telecommunications networks, coordinate time synchronization prevents errors in data transmission, particularly in time-division multiple access (TDMA) systems where precise timing aligns bursts across geostationary or low-Earth orbit constellations. GNSS-derived coordinate time, such as GPS time, provides the reference for clock alignment, mitigating delays from signal propagation and relativistic effects to maintain bit error rates below 10^{-10}.^[26] This ensures seamless global coverage for services like mobile backhaul and broadcasting, where even microsecond offsets could disrupt packet synchronization.^[27] Astronomy employs coordinate time in ephemerides to determine planetary positions with high fidelity, as seen in the Jet Propulsion Laboratory's DE430 model, which integrates orbits using Barycentric Dynamical Time (TDB), a scaled version of TCB. DE430 spans from 1550 AD to 2650 AD and achieves positional accuracies of 1 meter for inner planets over decades, facilitating precise predictions for observations and mission planning.^[28] This coordinate time framework accounts for relativistic light-time corrections, enabling accurate modeling of celestial events like planetary conjunctions. Future applications of coordinate time leverage quantum technologies for enhanced precision in relativistic environments, including optical lattice clocks that achieve stabilities below 10^{-18}, surpassing current atomic standards. These clocks, using trapped neutral atoms in laser lattices, promise to refine coordinate time in distributed networks for deep space communication and gravitational mapping.^[29] In quantum-enhanced relativistic networks, entanglement-shared optical clocks could probe spacetime curvature on Earth-scale baselines, supporting tests of general relativity with sensitivities to height differences of centimeters.^[30]

Established Coordinate Time Scales

Geocentric and Terrestrial Scales

Geocentric and terrestrial time scales provide standardized coordinate times referenced to Earth's center of mass or surface, incorporating relativistic corrections to account for gravitational and rotational effects in the geocentric frame. These scales form the foundation for both scientific computations, such as ephemerides, and civil timekeeping, ensuring uniformity across applications like astronomy and global navigation. International Atomic Time (TAI) is a continuous, high-precision time scale realized by the International Bureau of Weights and Measures (BIPM) through the weighted average of readings from over 450 atomic clocks maintained by institutions worldwide. It serves as the primary realization of the SI second on the rotating geoid and acts as the foundational coordinate time scale for deriving other geocentric times, with its stability ensured by primary frequency standards and its accuracy by a subset of the most precise clocks. TAI began continuous computation in 1958, retroactively defined from January 13, 1958, at 0h UT1, and is disseminated via BIPM Circular T publications.^[31] Geocentric Coordinate Time (TCG) is the relativistic coordinate time defined in the Geocentric Celestial Reference System (GCRS), with its origin at the geocenter (Earth's center of mass)^[32]; it represents the proper time for a hypothetical clock at infinite distance from Earth, unaffected by local gravitational fields. TCG was established by the International Astronomical Union (IAU) in 1991 as part of resolutions on relativistic reference systems. Its rate is adjusted relative to TAI through a linear scaling that accounts for Earth's gravitational potential, given by the defining constant L_G = 6.969290134 \times 10^{-10}, such that the difference accumulates as \Delta t = L_G \times ( \text{JD}_{TT} - 2443144.5 ) seconds, where JD is the Julian Date in Terrestrial Time (TT) and 2443144.5 corresponds to the epoch 1977 January 1, 0h TT. This adjustment ensures TCG runs faster than terrestrial realizations by approximately 0.7 microseconds per day due to the absence of gravitational redshift in its definition.^[33] Terrestrial Time (TT) is a scaled version of TCG specifically designed for astronomical observations and ephemeris calculations, providing a uniform time coordinate in the geocentric system that aligns closely with atomic time for practical use. Defined by IAU Resolution A4 in 1991 and refined in 2000, TT differs from TCG by a prescribed constant rate to maintain continuity with earlier ephemeris times, expressed as

TT = TCG - L_G \times (JD_{TT} - 2443144.5)

in seconds, ensuring the scale unit matches the SI second as realized by TAI plus a fixed offset of 32.184 seconds (TT = TAI + 32.184 s) at the 1977 epoch. This scaling eliminates secular drift, making TT suitable for modeling Earth's motion without ongoing relativistic offsets. TT is realized as TT(BIPM), computed annually by BIPM from international atomic clock data for enhanced long-term stability.^[31] Coordinated Universal Time (UTC) is the civil time standard derived from TAI by the insertion or deletion of leap seconds, as determined by the International Earth Rotation and Reference Systems Service (IERS), to maintain alignment with solar time (UT1) within 0.9 seconds. This adjustment compensates for irregularities in Earth's rotation, with leap seconds announced in advance and typically added at the end of June or December; as of November 2025, 37 leap seconds have been introduced since 1972. UTC is disseminated globally via radio signals and network time protocols, serving as the basis for international time zones and everyday applications while preserving the uniform second of TAI.^[34] Relativistic adjustments to these geocentric scales account for Earth's oblateness and rotation, which influence local coordinate times through variations in gravitational potential and velocity-dependent effects. Earth's oblateness, characterized by the dynamical ellipticity J_2 \approx 1.0826 \times 10^{-3}, perturbs the geopotential, leading to a quadrupole correction in the metric that affects time dilation by up to several nanoseconds per day at the equator compared to poles; this is incorporated into the definition of L_G as the average potential at the geocenter and further refined for local proper times via the post-Newtonian parameter \Phi / c^2, where \Phi includes oblateness terms. Rotational effects introduce a velocity term -v^2 / (2c^2) in the proper time formula for clocks on the surface, causing a slowing of about 100 ns per day due to equatorial speeds of 465 m/s, while the Sagnac effect in rotating frames adds path-dependent delays for signal propagation; these are modeled in the IERS Conventions for transforming TT to local terrestrial reference frame times. Such corrections ensure consistency between coordinate times and observed clock rates in applications like satellite navigation.^[35]^[15]

Barycentric and Cosmological Scales

Barycentric Coordinate Time (TCB) serves as the standard time coordinate within the Barycentric Celestial Reference System (BCRS), which is centered at the solar system's barycenter and aligns with the International Celestial Reference System (ICRS) at infinite distance. TCB is the coordinate time in the Barycentric Celestial Reference System (BCRS), with its unit defined as the proper time interval of an ideal clock at rest at infinity in the BCRS, unaffected by solar system gravitational fields, and it functions as the independent argument for the relativistic equations of motion of solar system bodies.^[36] This timescale is independent of Earth's motion, providing a uniform temporal framework for describing the dynamics across the solar system. The relativistic foundation of TCB relies on the post-Newtonian formalism, which approximates general relativity for weak gravitational fields and slow velocities relevant to the solar system. In this framework, TCB incorporates the gravitational potentials of the Sun and planets, ensuring that the metric in the BCRS includes post-Newtonian corrections to the coordinate time, such as those from the isotropic parameterized post-Newtonian gauge. The IAU 2000 resolutions establish TCB such that its rate is adjusted relative to atomic time scales, with the linear scaling factor chosen so that TCB and Geocentric Coordinate Time (TCG) coincide at the Earth's geocenter on the 1977 January 1 epoch. Barycentric Dynamical Time (TDB) is derived from TCB through a periodic adjustment to align with Earth's orbital period, making it suitable for use in ephemerides and dynamical theories referenced to the solar system barycenter. Specifically, TDB differs from TCB by small, periodic variations arising from relativistic effects, approximated as TDB = TCB − ΔTDB, where ΔTDB is a periodic function with amplitude on the order of 1.5 milliseconds, ensuring TDB remains close to Terrestrial Time (TT) over long periods.^[37] This adjustment facilitates practical astronomical computations by avoiding secular drifts while preserving the relativistic accuracy of TCB. In applications such as exoplanet detection via transit methods, TDB provides the precise timing reference needed for Barycentric Julian Dates (BJD_TDB), enabling accurate determination of orbital parameters. For instance, transit timing variations (TTVs) in systems like those observed by Kepler require BJD_TDB corrections to achieve sub-minute precision, revealing additional planets through deviations in predicted transit epochs. This use of TDB ensures that light travel time across the solar system and relativistic effects are accounted for, yielding reliable ephemerides for exoplanetary orbits.^[38] On galactic and cosmological scales, coordinate time is often defined in the rest frame of the cosmic microwave background (CMB), which serves as a universal reference for isotropic expansion and large-scale structure studies.^[39] In this frame, cosmic time corresponds to the proper time measured by comoving observers, providing a synchronized temporal coordinate across the observable universe that underpins models of structure formation from CMB anisotropies.^[40] This CMB-based coordinate time facilitates analyses of galaxy distributions and the universe's age, estimated at 13.80 billion years, by aligning observations with the homogeneous, isotropic Friedmann-Lemaître-Robertson-Walker metric.^[39]

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We started by folding the PDC-MAP Kepler long-cadence data, with the BJDTDB timings, using the ephemeris of NASA Exoplanet Archive in order to obtain a good ...
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[PDF] 22. Big-Bang Cosmology - Particle Data Group
Dec 1, 2023 · using the microwave background and large-scale structure: t0 = (13.80 ± 0.02)Gyr [24], where the extra accuracy comes at the price of ...
[39]
Monopole fluctuation of the CMB and its gauge invariance
where the coordinate (time) lapse δ η o represents the difference of the ... CMB rest frame provides an absolute frame for all observers in the Universe.