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Friedmann equations

The Friedmann equations are a set of coupled ordinary differential equations in cosmology that describe the evolution of a homogeneous and isotropic universe according to general relativity.^[1] They relate the Hubble parameter, which measures the expansion rate, to the universe's energy density through matter and radiation, the vacuum energy (via the cosmological constant), and its spatial curvature.^[1] Formulated by Russian mathematician and physicist Alexander Friedmann in 1922, these equations provide the foundational dynamical framework for the Big Bang model and predictions about the universe's past, present, and future geometry.^[2] Friedmann derived the equations by applying Einstein's field equations of general relativity to the Friedmann–Lemaître–Robertson–Walker (FLRW) metric, which assumes spatial uniformity and isotropy on large scales.^[1] The first Friedmann equation expresses the square of the Hubble parameter H = \dot{a}/a (where a(t) is the scale factor describing relative distances) as H^2 = \frac{8\pi G}{3} \rho - \frac{k c^2}{a^2} + \frac{\Lambda c^2}{3}, with \rho as the energy density from matter, radiation, and other non-vacuum components, G as the gravitational constant, c as the speed of light, k as the curvature parameter (k = +1 for closed, k = 0 for flat, k = -1 for open geometries), and \Lambda as the cosmological constant representing vacuum (dark) energy.^[1] A second equation, the acceleration equation, follows from the first and the continuity equation for energy conservation: \frac{\ddot{a}}{a} = -\frac{4\pi G}{3} \left( \rho + \frac{3p}{c^2} \right) + \frac{\Lambda c^2}{3}, where p is pressure, revealing how the universe's expansion can accelerate or decelerate based on its contents.^[1] Published in Zeitschrift für Physik amid skepticism from Einstein (who initially questioned the non-static solutions), Friedmann's work in his 1922 paper "Über die Krümmung des Raumes" proposed three possible cosmic evolutions: an expanding universe from a singularity, a contracting one, or an oscillating model.^[2] Independently rediscovered by Georges Lemaître in 1927 and later connected to Edwin Hubble's 1929 observations of galactic redshifts, the equations gained acceptance and now underpin key cosmological parameters like the density parameter \Omega = \rho / \rho_c (where critical density \rho_c = 3H^2 / (8\pi G)) and observations from cosmic microwave background data indicating a nearly flat universe (\Omega \approx 1).^[1] Their inclusion of dark energy via \Lambda has been crucial for explaining the universe's observed acceleration since 1998.^[1]

Background and Assumptions

FLRW Metric

The Friedmann–Lemaître–Robertson–Walker (FLRW) metric provides the standard geometric framework for modeling a homogeneous and isotropic universe on large scales, as required by the cosmological principle. This principle asserts that the universe appears the same from any location (homogeneity) and in any direction (isotropy) when averaged over sufficiently large distances, simplifying the description of cosmic evolution.^[3] The metric was first introduced by Alexander Friedmann in 1922 as a solution to Einstein's field equations for a spatially curved, expanding universe, independently rediscovered by Georges Lemaître in 1927, and fully generalized by Howard Robertson and Arthur Walker in the 1930s to encompass all possible spatial geometries consistent with these symmetries.^[4]^[5] In comoving coordinates, where galaxies are at fixed spatial positions and only recede due to expansion, the FLRW line element takes the form

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

where t is the cosmic time coordinate (proper time for comoving observers), r, \theta, \phi are the comoving spatial coordinates with r as the dimensionless radial coordinate and d\Omega^2 = d\theta^2 + \sin^2 \theta \, d\phi^2 the metric on the unit sphere, a(t) is the dimensionless scale factor describing the relative expansion of space as a function of time (normalized such that a(t_0) = 1 today), and k is the spatial curvature parameter.^[4]^[5] The proper distance between comoving points is then d_p = a(t) \int_0^r \frac{dr'}{\sqrt{1 - k r'^2}}, which scales with a(t) and reflects the expansion.^[6] The curvature parameter k determines the global geometry of the three-dimensional spatial hypersurfaces: k = +1 for a closed universe with positive curvature (finite volume, like a 3-sphere), k = 0 for a flat (Euclidean) universe with zero curvature and infinite volume, and k = -1 for an open universe with negative curvature (hyperbolic, also infinite volume).^[4] These choices ensure the metric respects the symmetries of homogeneity and isotropy, reducing the general 10 independent components of the spacetime metric tensor in general relativity to a single dynamical function, a(t), which encodes the universe's expansion history.^[6] This simplification arises directly from the imposed symmetries, making the FLRW metric the foundational ansatz for deriving dynamical equations in cosmology.

Key Assumptions

The Friedmann equations rely on the cosmological principle, a foundational assumption in modern cosmology that the universe appears the same from any location and in any direction when observed on sufficiently large scales. This principle encompasses two key aspects: homogeneity, which implies that the density of matter and energy is uniform across space on cosmic scales, and isotropy, which means there is no preferred direction in the universe's large-scale structure. These assumptions simplify the application of general relativity to cosmology by allowing the use of a single, symmetric metric to describe the geometry of spacetime.^[7]^[8] The matter and energy content of the universe is modeled using the stress-energy tensor of a perfect fluid, which assumes that the fluid is isotropic and has no viscosity or heat conduction, enabling a straightforward description of pressure and density for components such as ordinary matter, radiation, and dark energy. This perfect fluid approximation captures the collective behavior of these constituents without accounting for microscopic details, aligning with the homogeneity and isotropy requirements.^[9]^[10] Gravity is treated as the dominant force governing the universe's evolution through Einstein's general theory of relativity, with small-scale inhomogeneities—such as galaxies and clusters—neglected in favor of an averaged, smooth distribution on cosmological scales. The expansion of the universe is solely parameterized by a time-dependent scale factor a(t), which describes the relative separation of comoving coordinates, under the condition of no anisotropic stresses that could introduce directional dependencies. These assumptions collectively underpin the Friedmann equations by realizing them geometrically through the Friedmann-Lemaître-Robertson-Walker (FLRW) metric.^[7]^[11]

Derivation of the Equations

Einstein Field Equations in Cosmology

The Einstein field equations, formulated by Albert Einstein in 1915, relate the geometry of spacetime to the distribution of matter and energy within it. In their standard form with a cosmological constant, they are expressed as

G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu},

where G_{\mu\nu} is the Einstein tensor, \Lambda is the cosmological constant, g_{\mu\nu} is the metric tensor, G is the gravitational constant, c is the speed of light, and T_{\mu\nu} is the stress-energy tensor.^[12] These equations provide the foundational framework for general relativity applied to cosmology, enabling the description of the universe's large-scale structure and evolution.^[13] In cosmological models, the Friedmann–Lemaître–Robertson–Walker (FLRW) metric is assumed to describe a homogeneous and isotropic universe, serving as the geometric input to the field equations. The stress-energy tensor for the cosmic content is modeled as that of a perfect fluid, given by

T_{\mu\nu} = (\rho + p) u_\mu u_\nu + p g_{\mu\nu},

where \rho is the energy density, p is the pressure, and u^\mu is the four-velocity of the fluid elements, normalized such that u^\mu u_\mu = -1 in units where c=1. For comoving observers in the FLRW framework, u^\mu = (1, 0, 0, 0), simplifying the tensor to a diagonal form with components T^0_0 = -\rho and T^i_j = p \delta^i_j.^[12]^[14] To apply the field equations, the Ricci tensor R_{\mu\nu} and scalar R = g^{\mu\nu} R_{\mu\nu} must be computed from the FLRW metric using the Christoffel symbols, which encode the connections in curved spacetime. The non-zero Christoffel symbols for the FLRW metric include terms like \Gamma^t_{rr} = \frac{\dot{a} r}{1 - k r^2}, \Gamma^r_{rt} = \frac{\dot{a}}{a}, and spatial components involving the scale factor a(t) and curvature parameter k. These lead to the time-time component of the Ricci tensor R_{00} = -3 \frac{\ddot{a}}{a} and spatial components such as R_{ii} = \left[ \frac{\ddot{a}}{a} + 2 \left( \frac{\dot{a}}{a} \right)^2 + 2 \frac{k}{a^2} \right] g_{ii}, where dots denote derivatives with respect to cosmic time t. The Ricci scalar then follows as R = 6 \left[ \frac{\ddot{a}}{a} + \left( \frac{\dot{a}}{a} \right)^2 + \frac{k}{a^2} \right].^[13]^[12] The Einstein tensor components are obtained by G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R. Inserting these into the field equations, the \mu = \nu = 0 (time-time) component simplifies due to the diagonal metric and fluid tensor, yielding an equation relating the Hubble-like expansion rate \left( \frac{\dot{a}}{a} \right)^2, the curvature term \frac{k}{a^2}, and the energy density \rho, modulated by \Lambda. Similarly, the trace-reversed form or the spatial components (e.g., ii) produce an equation involving the acceleration \frac{\ddot{a}}{a}, \rho, and p. These reductions, first systematically derived by Alexander Friedmann in 1922 using Einstein's equations on the spatially curved metric, bridge general relativity to the dynamical evolution of the universe without presupposing specific matter content beyond the perfect fluid assumption.^[14]

First Friedmann Equation

The first Friedmann equation relates the Hubble parameter, which measures the expansion rate of the universe, to the total energy density, spatial curvature, and the cosmological constant. Derived by Alexander Friedmann in 1922 from the Einstein field equations for a homogeneous and isotropic universe described by the Friedmann–Lemaître–Robertson–Walker (FLRW) metric, it provides a fundamental constraint on cosmic dynamics.^[15] The equation is expressed as

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

where H is the Hubble parameter, a(t) is the scale factor describing the relative size of the universe as a function of cosmic time t, \dot{a} = da/dt, \rho is the total energy density, k is the curvature parameter (k = +1 for closed, k = 0 for flat, and k = -1 for open universes), G is Newton's gravitational constant, c is the speed of light, and \Lambda is the cosmological constant.^[16]^[15] Physically, this equation represents Friedmann's "energy equation," balancing the "kinetic energy" of expansion on the left-hand side against contributions from gravitational attraction due to energy density, the geometric effect of spatial curvature, and the repulsive influence of the cosmological constant. In a Newtonian analogy, it mirrors the conservation of total energy for a shell of particles expanding under self-gravity, where the \rho term acts like a binding potential, the curvature term corresponds to the total mechanical energy (positive for hyperbolic expansion, zero for parabolic, and negative for oscillatory), and \Lambda provides an additional outward-driving term akin to negative pressure. The total energy density \rho encompasses all forms present in the universe, including matter (\rho_m), radiation (\rho_r), and the effective density from the cosmological constant treated as \rho_\Lambda = \Lambda c^2 / (8\pi G), allowing \Lambda to be absorbed into \rho for a unified description. The gravitational constant G sets the strength of density-driven collapse, while c ensures dimensional consistency in relativistic units, particularly in the curvature term where it relates geometric and dynamical scales. Friedmann's original derivation, incorporating \Lambda as introduced by Einstein, highlights the equation's roots in general relativity without relying on non-relativistic approximations.^[16]^[15]

Second Friedmann Equation

The second Friedmann equation, also known as the acceleration equation, governs the second derivative of the scale factor a(t), describing the rate of change of the cosmic expansion rate. It takes the form

\frac{\ddot{a}}{a} = -\frac{4\pi G}{3} \left( \rho + \frac{3p}{c^2} \right) + \frac{\Lambda c^2}{3},

where \ddot{a} is the acceleration of the scale factor, G is the gravitational constant, \rho is the total energy density, p is the isotropic pressure, c is the speed of light, and \Lambda is the cosmological constant.^[17] This equation arises as a consequence of the Einstein field equations applied to the Friedmann–Lemaître–Robertson–Walker (FLRW) metric and complements the first Friedmann equation by providing dynamical information on acceleration rather than the instantaneous Hubble rate.^[8] The physical interpretation of the equation reveals that ordinary matter and radiation contribute to deceleration of the expansion, as the term \rho + 3p/c^2 > 0 for these components yields a negative \ddot{a}/a, effectively acting as an attractive gravitational force.^[1] In contrast, the cosmological constant term \Lambda c^2/3 > 0 provides a repulsive effect that can drive acceleration if it dominates, as observed in the current universe.^[1] This balance determines whether the universe's expansion slows, halts, or speeds up over time. The pressure p is often parameterized via the equation of state p = w \rho c^2, where w is the dimensionless equation-of-state parameter specific to each cosmic component.^[18] For non-relativistic matter (dust), w = 0, leading to \rho + 3p/c^2 = \rho > 0 and deceleration; for radiation, w = 1/3, yielding \rho + 3p/c^2 = 2\rho > 0 and stronger deceleration; and for the cosmological constant, w = -1, resulting in \rho + 3p/c^2 = -2\rho < 0 and acceleration.^[18] This equation is a special case of the Raychaudhuri equation, which describes the evolution of the expansion scalar \theta = 3\dot{a}/a in general relativity for geodesic congruences, reduced here to the FLRW symmetry with vanishing shear and vorticity.^[19] It can also be obtained by differentiating the first Friedmann equation and applying the conservation of the energy-momentum tensor, \dot{\rho} + 3(\rho + p/c^2)\dot{a}/a = 0, ensuring consistency with thermodynamic principles.^[19]

Forms and Parameters

Dimensionless Scale Factor

In cosmology, the scale factor a(t) describes the relative expansion of the universe as a function of cosmic time t. To render this quantity dimensionless and facilitate comparisons across different epochs, it is conventionally normalized such that a(t_0) = 1 at the present time t_0. This choice implies that a(t) < 1 for past times t < t_0, corresponding to a smaller, denser universe, and a(t) > 1 for future times t > t_0 in an expanding model.^[20]^[21] This normalization establishes a direct link between the scale factor and observable phenomena, particularly the cosmological redshift z of light emitted from distant sources at time t_e. The relation is given by $1 + z = 1 / a(t_e), which quantifies how the universe's expansion stretches wavelengths, shifting spectral lines to longer (redder) values as light travels to us.^[22]^[23] The expansion rate at any epoch is captured by the Hubble parameter H(t) = \dot{a}(t) / a(t), whose current value H_0 = H(t_0) serves as a fundamental observational benchmark.^[1] By adopting this dimensionless convention, the Friedmann equations—which relate the evolution of a(t) to the universe's energy content—become easier to integrate numerically and analyze parametrically, enabling straightforward comparisons of the expansion history from early times to the present. This approach enhances the interpretability of cosmological data without altering the underlying physics.^[1]^[24]

Critical Density

The critical density, denoted \rho_c, represents the threshold energy density of the universe required for a flat spatial geometry, derived by setting the curvature parameter k = 0 and the cosmological constant \Lambda = 0 in the first Friedmann equation.^[1] This yields the expression

\rho_c = \frac{3 H^2}{8 \pi G},

where H is the Hubble parameter at a given epoch and G is the gravitational constant.^[1] For the present-day universe, with the Hubble constant H_0 \approx 70 km s^{-1} Mpc^{-1}, the critical density is \rho_{c0} \approx 9.2 \times 10^{-27} kg m^{-3}.^[25] Physically, \rho_c serves as a benchmark that classifies the universe's geometry and ultimate fate based on the total energy density \rho: if \rho < \rho_c, the universe is open with negative curvature and expands indefinitely; if \rho > \rho_c, it is closed with positive curvature and eventually recollapses; and if \rho = \rho_c, it is flat with zero curvature and expands forever at a decelerating rate.^[26] Since \rho_c is proportional to H^2, it evolves with cosmic time as the expansion rate H changes, increasing in the early universe when H was larger and decreasing as expansion proceeds.^[1]^[27]

Density Parameter

The density parameter \Omega quantifies the contribution of each energy component to the total energy density of the universe relative to the critical density \rho_c = 3H^2/(8\pi G), where H is the Hubble parameter and G is the gravitational constant. For a specific component i, it is defined as \Omega_i = \rho_i / \rho_c.^[28] The total density parameter is then \Omega_\mathrm{tot} = \Omega_m + \Omega_r + \Omega_\Lambda + \Omega_k, where the subscripts denote matter (m), radiation (r), dark energy (\Lambda), and curvature (k), respectively.^[29] The curvature term is given by \Omega_k = -k c^2 / (H^2 a^2), with k the spatial curvature parameter (k = 0, +1, -1 in units where the radius of curvature is absorbed into a), c the speed of light, and a the scale factor.^[29] A universe with \Omega_\mathrm{tot} = 1 is spatially flat, corresponding to k = 0, as the Friedmann equation requires the sum of all density contributions to balance exactly with the critical density for zero curvature.^[29] Observations indicate that the present-day values (denoted with subscript $0) are \Omega_{m0} \approx 0.315, \Omega_{\Lambda 0} \approx 0.685, \Omega_{r0} \ll 1(specifically\sim 9 \times 10^{-5}including photons and relativistic neutrinos), and\Omega_{k0} \approx 0within measurement uncertainties, consistent with a flat\Lambda$CDM model.^[30] The individual density parameters \Omega_i evolve with the scale factor a according to the conservation laws derived from the continuity equation \dot{\rho} + 3H(\rho + p) = 0, which governs the dilution of energy densities during cosmic expansion. For matter with equation-of-state parameter w_m = 0, \rho_m \propto a^{-3}, so \Omega_m(a) decreases as a increases in epochs dominated by other components. Radiation with w_r = 1/3 scales as \rho_r \propto a^{-4}, causing \Omega_r(a) to drop rapidly after early times. Dark energy, modeled as a cosmological constant with w_\Lambda = -1, has constant \rho_\Lambda, leading to \Omega_\Lambda(a) growing toward dominance at late times. The curvature term \Omega_k(a) \propto a^{-2}, which diminishes relative to other contributions as the universe expands. These scalings reflect how the relative importance of each component shifts over cosmic history, with \Omega_\mathrm{tot}(a) = 1 preserved in a flat universe.^[31]

Cosmological Models

Interpretation of the Equations

The curvature parameter k in the Friedmann equations dictates the intrinsic geometry and topology of the three-dimensional spatial hypersurfaces in the universe. When k = +1, the geometry is closed and spherical, implying a finite but unbounded volume with positive spatial curvature, akin to the surface of a sphere. For k = 0, the geometry is flat and Euclidean, with zero curvature and infinite extent. In the case of k = -1, the geometry is open and hyperbolic, featuring negative curvature and also infinite volume.^[32]^[33] Dynamically, the Friedmann equations govern the evolution of the universe through the Hubble parameter H(a), which depends on the scale factor a and describes the rate of expansion at different epochs. Starting from the Big Bang singularity where a = 0, H(a) encodes the transition from rapid early expansion to the slower, ongoing expansion today, and projects future behaviors such as continued deceleration or acceleration based on energy content. This framework allows reconstruction of the universe's expansion history by integrating over a, revealing how gravitational effects from matter and other components influence the overall trajectory.^[1]^[34] Observationally, the Friedmann equations underpin key tests of cosmology, including Hubble's law, which states that the recession velocity v of nearby galaxies is proportional to their proper distance d, given by v = H_0 d where H_0 is the present-day Hubble constant. This linear relation emerges directly from the uniform expansion predicted by the equations and has been verified through redshift-distance measurements of galaxies and supernovae. Additionally, the equations enable estimation of the universe's age t_0 via the integral t_0 = \int_0^1 \frac{da}{a H(a)}, which integrates the expansion rate from the Big Bang to the present, yielding values consistent with stellar and globular cluster ages when combined with measured densities.^[35] The density parameter \Omega, defined as the ratio of actual density to critical density, briefly quantifies how various energy components contribute to the dynamics encoded in H(a). However, the Friedmann equations apply primarily to large-scale structures under the assumptions of homogeneity and isotropy, as encoded in the Friedmann-Lemaître-Robertson-Walker metric; they lose validity on small scales where these assumptions fail, such as in galactic dynamics or local gravitational fields. Furthermore, the equations predict singularities where densities and curvatures diverge, marking the Big Bang and potential future crunches, at which point classical general relativity breaks down and quantum effects become essential, though a complete quantum gravity theory remains unresolved.^[1]

Role of the Cosmological Constant

The cosmological constant, denoted by \Lambda, is incorporated into the Friedmann equations as an effective constant energy density \rho_\Lambda = \frac{\Lambda c^2}{8\pi G}, where G is the gravitational constant, contributing a term that remains invariant under cosmic expansion unlike matter or radiation densities. This component has an equation of state parameter w = -1, implying negative pressure p_\Lambda = -\rho_\Lambda c^2, which uniquely drives repulsive gravitational effects in general relativity. In the Friedmann framework, \Lambda modifies both the first equation, relating the Hubble parameter to total energy density including \rho_\Lambda, and the second equation, where it provides a positive contribution to the acceleration \frac{\ddot{a}}{a} = -\frac{4\pi G}{3}\left( \rho + \frac{3p}{c^2} \right) + \frac{\Lambda c^2}{3}. Historically, Albert Einstein introduced the cosmological constant in 1917 to enable a static universe solution in general relativity, balancing gravitational attraction with a repulsive term to prevent collapse. After Edwin Hubble's 1929 observations revealed an expanding universe, Einstein reportedly regarded \Lambda as his "biggest blunder," leading to its temporary abandonment in favor of matter-dominated models. However, the constant was reintroduced in the late 1990s following observational evidence from Type Ia supernovae indicating accelerated expansion, with key results from the High-Z Supernova Search Team in 1998 showing that a positive \Lambda best fits the dimming of distant supernovae compared to a decelerating model.^[36] In modern cosmology, the cosmological constant dominates the late-time evolution of the universe, overcoming the deceleration induced by matter and radiation in earlier epochs to produce accelerated expansion today.^[30] As of Planck 2018, measurements yield a density parameter \Omega_\Lambda \approx 0.69; recent DESI 2024 results confirm \Omega_\Lambda \approx 0.70 within the \LambdaCDM model, though suggesting a mild preference for evolving dark energy.^[37] While the standard \LambdaCDM model assumes a fixed constant, alternatives such as quintessence propose a dynamic scalar field \phi with a slowly varying potential that can mimic w \approx -1 but allows for possible time evolution, potentially resolving fine-tuning issues in the constant's magnitude.^[38] This \Lambda-domination leads to exponential growth of the scale factor a(t) \propto \exp\left(\sqrt{\frac{\Lambda c^2}{3}}\, t\right) in the asymptotic future.

Dust and Radiation Models

In Friedmann models dominated by non-relativistic matter, often termed "dust" due to its pressureless nature (equation of state parameter w = 0), the energy density scales as \rho_m \propto a^{-3}, where a(t) is the scale factor. This scaling arises because the volume of the expanding universe dilutes the number density of matter particles while their individual masses remain constant. For a flat (k=0) universe filled solely with dust, the Friedmann equation yields an analytic solution where the scale factor evolves as a(t) \propto t^{2/3}. This parametric form implies a decelerating expansion, with the Hubble parameter H(t) \propto t^{-1} and the deceleration parameter q = 1/2 > 0. In contrast, radiation-dominated Friedmann models feature relativistic particles with w = 1/3, leading to an energy density that scales more rapidly as \rho_r \propto a^{-4}. The additional factor of a^{-1} beyond the volume dilution accounts for the redshift of photon wavelengths (or equivalently, the decrease in energy per particle) during expansion. For a flat radiation-only universe, the scale factor follows a(t) \propto t^{1/2}, resulting in H(t) \propto t^{-1} and q = 1 > 0, again indicating deceleration but at a faster initial rate than in the dust case. These models are particularly relevant to the early universe, where radiation energy density exceeded that of matter until the epoch of equality. The transition from radiation domination to dust domination occurs at the matter-radiation equality redshift z_{\rm eq} \approx 3390, when the energy densities of the two components are comparable (\rho_m \approx \rho_r). At this epoch, the scale factor is a_{\rm eq} \approx 1/(1 + z_{\rm eq}) \approx 3 \times 10^{-4}, marking the shift from radiation-driven to matter-driven expansion dynamics. Observationally, the cosmic microwave background (CMB) provides key evidence for the radiation-dominated era, as its blackbody spectrum and acoustic peak structure encode the tight coupling between photons and baryons prior to recombination, influenced by the lingering effects of early radiation domination. In the later dust-dominated phase, galaxy clustering patterns reflect the growth of density perturbations under matter control, with the power spectrum's turnover scale directly tied to the horizon size at equality.

Analogies and Solutions

Newtonian Analog

The Newtonian analog to the Friedmann equations provides an intuitive derivation of cosmic expansion using classical mechanics, treating the universe as a homogeneous sphere of uniform density expanding under self-gravity. Consider a spherical region of the universe with physical radius R(t) = a(t) r, where a(t) is the scale factor describing the expansion and r is a fixed comoving radius. A test particle of unit mass at the surface experiences gravitational attraction solely from the mass M enclosed within the sphere, as contributions from exterior shells cancel by Newton's shell theorem. The enclosed mass M = \frac{4\pi}{3} \rho(t) R^3 remains constant for non-relativistic matter, with \rho(t) the density scaling as \rho(t) \propto a^{-3}.^[39] Conservation of mechanical energy for the test particle governs its radial motion: the sum of kinetic energy per unit mass \frac{1}{2} \left( \frac{dR}{dt} \right)^2 and gravitational potential energy per unit mass -\frac{GM}{R} equals a constant total energy E. This yields the energy equation

\frac{1}{2} \left( \frac{dR}{dt} \right)^2 - \frac{GM}{R} = E.

Substituting R = a r and \frac{dR}{dt} = \dot{a} r gives \frac{1}{2} \dot{a}^2 r^2 - \frac{GM}{a r} = E. Multiplying through by $2/r^2 yields \dot{a}^2 - \frac{2 G M}{a r^3} = \frac{2 E}{r^2}. Since M = \frac{4\pi}{3} \rho a^3 r^3, the middle term simplifies to \frac{8\pi G}{3} \rho a^2, so \dot{a}^2 = \frac{8\pi G}{3} \rho a^2 + \frac{2 E}{r^2}. Dividing by a^2 then produces

\left( \frac{\dot{a}}{a} \right)^2 = \frac{8\pi G}{3} \rho + \frac{2 E}{r^2 a^2},

where the constant term \frac{2 E}{r^2} is identified with -\frac{k c^2}{a^2}, and k parameterizes the curvature (positive for closed, zero for flat, negative for open universes). This matches the form of the first Friedmann equation in general relativity for pressureless dust.^[1]^[39] This analog assumes non-relativistic particle speeds (v \ll c) and neglects pressure gradients, restricting its validity to matter-dominated (dust) universes where pressure p = 0. It fails for radiation-dominated eras, where p = \rho/3 and relativistic effects alter the dynamics, or when expansion approaches the speed of light. The Newtonian framework also overlooks intrinsic spacetime curvature beyond the effective k term, requiring general relativity for a complete description.^[7] Historically, this heuristic insight predates full general relativistic treatments and was formalized post-1922 by W. H. McCrea and E. A. Milne, who demonstrated its equivalence to the relativistic Friedmann equation for homogeneous cosmologies, highlighting the robustness of the expansion law across gravity theories.

Pure Component Solutions

The Friedmann equations admit exact analytic solutions when the universe is dominated by a single energy component, simplifying the dynamics to a tractable form that reveals key behaviors such as expansion history and singularities. These solutions assume a homogeneous and isotropic universe described by the Friedmann-Lemaître-Robertson-Walker metric, with the energy content characterized by a constant equation-of-state parameter w = p / (\rho c^2), where p is pressure and \rho is energy density. For dust (non-relativistic matter, w = 0) and radiation (relativistic particles, w = 1/3), the solutions describe decelerating expansion from a hot Big Bang, while for a cosmological constant (\Lambda-dominated, w = -1), the universe undergoes eternal exponential expansion. Curvature introduces additional complexity, particularly in closed models leading to recollapse. In a flat (k = 0) dust-dominated universe, the first Friedmann equation \left( \frac{\dot{a}}{a} \right)^2 = \frac{8\pi G \rho}{3} integrates to yield the scale factor

a(t) \propto t^{2/3},

with the Hubble parameter

H(t) = \frac{2}{3t}.

This solution originates from the parametric integration of the equations under the assumption of pressureless matter, where \rho \propto a^{-3}. The expansion decelerates over time, and the model features a Big Bang singularity at t = 0, where a \to 0 and densities diverge.^[1]^[40] For a flat radiation-dominated universe, the energy density scales as \rho \propto a^{-4} due to both dilution and redshift, leading to the solution

a(t) \propto t^{1/2},

and

H(t) = \frac{1}{2t}.

This reflects the dominance of relativistic components in the early universe, with even stronger deceleration than in the dust case. As with the dust solution, a Big Bang singularity occurs at t = 0.^[1]^[40] In a flat \Lambda-dominated universe, the constant vacuum energy density drives accelerated expansion, resulting in de Sitter spacetime where

a(t) \propto e^{H t},

with H constant and given by H = \sqrt{\Lambda c^2 / 3}. This eternal expansion lacks a Big Bang singularity if \Lambda is the sole component, though in realistic cosmologies it emerges after matter and radiation dilution.^[1]^[40] Curvature effects are prominent in non-flat models; for a closed (k = +1) dust-dominated universe, the Friedmann equation cannot be solved algebraically but admits a parametric cycloid form:

a(\theta) = A (1 - \cos \theta), \quad t(\theta) = B (\theta - \sin \theta),

where A and B are constants related to the total mass and critical density, and \theta is a development angle parameter. The universe expands to a maximum scale factor at \theta = \pi before recollapsing, culminating in a Big Crunch singularity at \theta = 2\pi, where a \to 0 and t reaches a finite maximum. This oscillatory behavior highlights the role of positive spatial curvature in reversing expansion.^[1]^[41]

Mixed Density Solutions

In multi-component universes, the Friedmann equation describing the expansion rate incorporates contributions from matter, radiation, a cosmological constant, and spatial curvature, expressed in dimensionless form as

H^2 = H_0^2 \left[ \Omega_{m0} a^{-3} + \Omega_{r0} a^{-4} + \Omega_{\Lambda 0} + \Omega_{k0} a^{-2} \right],

where H is the Hubble parameter at scale factor a, H_0 is the present-day Hubble constant, and the \Omega_{i0} are the present-day density parameters for each component normalized by the critical density.^[42]^[43] To compute key quantities such as the age of the universe or lookback time, the Friedmann equation must be integrated as

t = \int_0^a \frac{da'}{a' H(a')},

which generally lacks a closed-form analytic solution for arbitrary combinations of components and instead requires numerical evaluation; in specific cases, such as the flat \LambdaCDM model without radiation, the integral can be expressed in terms of elliptic integrals of the first, second, or third kind.^[44]^[45] The standard \LambdaCDM model, featuring cold dark matter (\Omega_{m0}), radiation (\Omega_{r0}), and a cosmological constant (\Omega_{\Lambda 0}), provides the current concordance paradigm for cosmology, with the total density parameter \Omega_{\rm tot} = \Omega_{m0} + \Omega_{r0} + \Omega_{\Lambda 0} + \Omega_{k0} \approx 1 consistent with observations of the cosmic microwave background and large-scale structure.^[42] For practical computations in mixed-density scenarios, exact solutions are often infeasible, leading to the use of numerical integration schemes; however, approximate methods such as series expansions or parametric fits facilitate analytic insights, particularly at early times when radiation or matter dominates or at late times when the cosmological constant prevails.^[46]^[45]

Historical Context

Origins and Friedmann's Work

In 1922, Russian mathematician and physicist Alexander Friedmann derived solutions to Einstein's general relativity field equations that described a dynamic, expanding universe, directly challenging the prevailing static cosmological models of the time.^[47] In his seminal paper, Friedmann applied the field equations to a homogeneous and isotropic universe, demonstrating that non-static solutions were mathematically permissible even for positive spatial curvature.^[47] A key innovation was his introduction of a time-dependent scale factor, denoted as R(t), which represented the evolving radius of curvature in the universe's geometry, allowing for scenarios of expansion from a singularity or oscillatory behavior.^[47] These insights, published in Zeitschrift für Physik, marked the first rigorous cosmological application of general relativity to predict a non-stationary cosmos.^[47] This work emerged in the post-World War I era, when general relativity had gained widespread acceptance following the 1919 solar eclipse observations confirming light bending, enabling broader exploration of its cosmological implications in Europe and Russia.^[48] Friedmann's analysis built on Einstein's 1917 static universe model but showed that such stability was not required, opening the door to evolving models without ad hoc assumptions. Independently, in 1927, Belgian physicist and priest Georges Lemaître arrived at similar expanding universe solutions using the same relativistic framework, further validating the approach though initially overlooked.^[48] Friedmann's paper initially faced skepticism; Albert Einstein, committed to a static universe, published a note claiming a mathematical error in Friedmann's calculations shortly after its appearance.^[49] However, after reviewing Friedmann's response, Einstein retracted his criticism in 1923, acknowledging the validity of the solutions, though he continued to view them as physically uninteresting until 1931, when he fully embraced expanding models and abandoned the cosmological constant.^[50]^[48]

Subsequent Developments

Following Edwin Hubble's 1929 observation of the recession of galaxies, which provided empirical support for the expanding universe predicted by the Friedmann equations, subsequent tests in the 1930s sought to verify the relativistic nature of this expansion.^[51] Richard Tolman's surface brightness test, proposed in 1930, predicted that in an expanding Friedmann–Lemaître–Robertson–Walker (FLRW) universe, the observed surface brightness of distant galaxies should dim as (1+z)^{-4}, where z is the redshift; although early observational attempts in the 1930s, such as those by Hubble and Tolman, were inconclusive due to technological limitations, later studies in the late 20th century confirmed the predicted dimming, supporting the expanding universe model and ruling out static alternatives.^[52]^[53] By the 1940s and 1950s, the steady-state theory emerged as an alternative to the evolving density models of Friedmann cosmology, proposed by Hermann Bondi, Thomas Gold, and Fred Hoyle in 1948, which incorporated continuous matter creation to maintain constant density despite expansion, challenging the Big Bang interpretation until later evidence favored the latter.^[54] The 1960s brought decisive validation for the hot Big Bang model derived from Friedmann equations through the discovery of the cosmic microwave background (CMB) radiation by Arno Penzias and Robert Wilson in 1965, interpreted as relic radiation from the early universe at a temperature of about 2.7 K, consistent with predictions from nucleosynthesis and the time-dependent expansion. This observation marginalized steady-state models and solidified the Friedmann framework, with subsequent COBE and WMAP missions in the 1990s and 2000s mapping CMB anisotropies to constrain density parameters. In the 1980s, Alan Guth's 1981 proposal of cosmic inflation introduced a brief exponential expansion phase driven by a scalar field, resolving fine-tuning issues in Friedmann models such as the flatness problem—where the total density parameter Ω_tot must be finely tuned near 1 for the observed universe—by dynamically attracting trajectories toward flatness during inflation. The 1990s marked the reintroduction of the cosmological constant Λ into Friedmann equations to explain accelerating expansion, evidenced by type Ia supernovae observations in 1998 from the Supernova Cosmology Project and High-Z Supernova Search Team, which showed distant supernovae fainter than expected in a matter-dominated decelerating universe, implying Λ > 0 and Ω_Λ ≈ 0.7. Precision measurements from the Planck satellite, culminating in 2018 results (incorporating data up to 2013), confirmed a spatially flat universe with Ω_tot = 1.000 ± 0.002 within the ΛCDM model, tightly constraining parameters like matter density Ω_m ≈ 0.315 and providing percent-level accuracy on the expansion history.^[30] Beyond the standard ΛCDM paradigm, recent extensions explore dynamical dark energy to address unresolved tensions, such as the Hubble constant (H_0) discrepancy, where local measurements (≈73 km/s/Mpc) exceed CMB-inferred values (≈67 km/s/Mpc) by over 5σ, as confirmed by recent JWST data through 2025.^[55] Models like quintessence, where dark energy evolves via a scalar field rather than a constant Λ, have been proposed to alleviate this, though no consensus resolution exists, with ongoing surveys like DESI and Euclid probing potential deviations in the Friedmann equation's dark energy term. For instance, the Dark Energy Spectroscopic Instrument (DESI) first-year results in 2024 tightened constraints on the equation of state for dark energy, while maintaining the H_0 tension, with ongoing data from Euclid expected to further probe deviations.^[37]

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None
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