Codex @codex  0

Vacuum density is constant for a true cosmological constant. Hence $0.3(1+z{)}^3=0.7$ givesmatter-vacuum density equality is not the same event as the start of accelerating expansion: neglecting radiation, acceleration begins when ${\rho }_m=2{\rho }_{\Lambda }$.

Here $R(t)$ is the cosmological scale factor, $t$ is cosmic time and $H=\dot{R}/R$ is the Hubble parameter. With $R$ a length, $k=+1,0,-1$ labels positively curved, flat and negatively curved spatial sections. This $k$ is unrelated to Question 1's specific orbital energy. The density $\rho$ is mass-equivalent energy density: $\rho c^2$ has energy-per-volume units. In the displayed Friedmann equation $\Lambda$ has units ${\mathrm{s}}^{-2}$; the geometric cosmological constant is ${\Lambda }_{\mathrm{geom}}=\Lambda /c^2$. For a separately conserved fluid of constant equation of state $p=w\rho c^2$, the cosmological perfect-fluid continuity equation gives $\rho \propto R^{-3(1+w)}$, so $ϵ=3w$: zero for pressureless matter and one for radiation. With several fluids, sum their densities in the expansion equation and evolve each separately.

Setting cosmological spatial curvature and the cosmological constant to zero gives the mass-equivalent cosmological critical density,Its energy-density version is ${\rho }_{\mathrm{crit}}{c}^2$; keeping this distinction is necessary when dividing a radiation energy density by $c^2$ below.

In an Einstein-de Sitter universe, $H^2\propto R^{-3}$ integrates to $R\propto t^{2/3}$ and $H=2/(3t)$. With $1+z=R_0/R$,Here $z_1>z_2$ are the two epoch redshifts relative to today. This is elapsed cosmic time, not comoving distance divided by $c$.

For collisionless preservation of a Planck spectrum, use frequency redshift as well as photon-number conservation. Let $f=R/R_i$, so a frequency bin $d\nu$ now corresponds to $d{\nu }_i=fd\nu$ and ${\nu }_i=f\nu$ initially. Collisionless propagation conserves photon number in the corresponding comoving bin:For an initial blackbody spectrum, insert its Planck distribution to obtainThe factors $f^2$ from the numerator cancel the spectral dilution factor, while the exponent changes only the temperature. Integrating gives photon number density proportional to $T^3\propto R^{-3}$ and radiation energy density proportional to $T^4\propto R^{-4}$. Conservation of total photon number alone would not guarantee a blackbody spectrum shape; the frequency-bin mapping and absence of injection or spectral redistribution are essential.

For the numerical density comparisons, the supplied Hubble constant is $H_0=2.33312\times 10^{-18}{\mathrm{s}}^{-1}$. Taking $G=6.67430\times 10^{-11}{\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$ and $c=2.99792458\times 10^8\mathrm{m}{\mathrm{s}}^{-1}$ givesThe radiation specified here is cosmic microwave background photons only. Adding a relic neutrino density would change the photon-only matter-radiation equality result. The rounded present matter and vacuum fractions sum to one without this small photon contribution; the requested pairwise density comparisons do not require silently renormalizing them.

For the final unlabelled flight-time request, insert the case-(c) value $1+z_c=(0.7/{\Omega }_{\gamma 0}{)}^{1/4}=10.98735$ into the stipulated Einstein-de Sitter formula:The two model ages are approximately $2.486\times 10^8$ and $2.478\times 10^5$ years. This uses exactly the instructed matter-only model with the same $H_0$. It is not an exact age difference in the matter-plus-vacuum cosmology; a high-redshift matter-era normalization using $H_0\sqrt{{\Omega }_{m0}}$ would be a different approximation.

A positive cosmological constant contributes a constant vacuum energy density and negative pressure. In the time-unit convention of Question 2,It therefore causes accelerating expansion once its negative-pressure effect outweighs the decelerating components. Its density stays constant while matter dilutes, so a small early contribution can dominate late. For an isolated positive-vacuum component the scale factor grows exponentially. The cosmological constant problem is the mismatch between the small inferred vacuum density and generic quantum-field vacuum-energy expectations, rather than a failure of the background solution itself.

The key direct expansion evidence came from Type Ia supernova distance-redshift measurements. After light-curve and colour standardization, distant events were fainter than expected in simple decelerating matter-only models, indicating larger luminosity distances. Riess and collaborators' 1998 analysis and Perlmutter and collaborators' independent 42-supernova study favoured an accelerating low-matter-density cosmology with a positive vacuum contribution. Dust, source evolution, selection effects, photometric calibration and peculiar velocities must be controlled: faintness alone is not a model-independent measurement of $\Lambda$.^[ref][ref]

cosmic microwave background acoustic geometry and independent matter-density constraints help break the supernova parameter degeneracy: nearly flat geometry with substantially less than critical matter requires another component within the FLRW model. large-scale structure of the universe and distance measurements provide further consistency tests. The combined evidence supports a dark-energy contribution compatible with a constant; accelerating expansion by itself does not prove exactly constant vacuum energy or rule out every alternative gravity model.

The Friedmann-Lemaître-Robertson-Walker metric describes homogeneous, isotropic spatial slices. Here $t$ is cosmic time, equal to the proper time of a comoving observer; $R(t)$ is the length-valued scale factor; and $\chi ,\theta ,\varphi$ are dimensionless comoving radial, polar and azimuthal coordinates. The speed of light is $c$, andfor positive, zero and negative cosmological spatial curvature. The convention used makes $d{s}^2$ a squared time interval, with signature $(+---)$; multiplying the metric by $c^2$ gives the more usual squared-length convention. At fixed $t$, radial proper length is $R(t)d\chi$, while a sphere has area $4\pi R^2(t)S^2(\chi )$.

luminosity distance is defined by $F=L/(4\pi d_L^2)$ for bolometric radiative flux $F$ and intrinsic luminosity $L$. Radial null propagation gives $d\chi =\pm cdt/R(t)$. Applying the same path integral to two successive wave crests gives $\Delta t_0/R_0=\Delta t_e/R_e$, hence $1+z=R_0/R_e$. Each photon loses an energy factor $1+z$, and their arrival rate loses another. For transparent propagation and isotropic emission,The two redshift factors have distinct physical origins; geometrical dilution alone is insufficient.

With $H=\dot{R}/R$, differentiating $1+z=R_0/R$ yields $dt/dz=-[(1+z)H(z){]}^{-1}$. Therefore the radial comoving coordinate isThis is independent of any particular matter model; the transverse comoving distance is $R_{0}S(\chi )$, which equals $R_0\chi$ only for flat space.

For pressureless matter, the continuity equation ${\dot{\rho }}_m+3H{\rho }_m=0$ integrates to ${\rho }_m={\rho }_{m0}(R_0/R{)}^3={\rho }_{m0}(1+z{)}^3$. The cosmological constant is kept separately from this matter density. In the flat Friedmann equation, defineThe flat dust-vacuum Hubble parameter is thereforeIn the equation's convention ${\rho }_m$ is mass-equivalent energy density and $\Lambda$ has units ${\mathrm{s}}^{-2}$; the geometrical cosmological constant convention instead writes $c^2{\Lambda }_{\mathrm{geom}}$. The model contains dust and vacuum energy with negligible radiation over the relevant late epochs: “matter-dominated” here does not require matter to exceed vacuum energy today.

The cosmic microwave background supplies a standard ruler through its acoustic scale: the angular positions of the Cosmic microwave background acoustic peaks constrain cosmological spatial curvature, while their relative heights constrain matter and baryon densities. Together with independent distance information, these favour a nearly flat Lambda-CDM model containing substantial cold dark matter and vacuum energy. The cosmic microwave background alone has a geometrical degeneracy between parameters affecting distance to last scattering; its analysis also includes early radiation, which the late-time formula above omits. The nine-year WMAP results were already available as a December 2012 preprint.^[ref]

A standardized Type Ia supernova at appreciable redshift is fainter, hence farther in luminosity distance, than in a matter-only Einstein-de Sitter universe with the same local expansion scale. Fitting the supernova Hubble diagram supports late acceleration; in flat dust plus vacuum, $q_0={\Omega }_{m0}/2-{\Omega }_{\Lambda 0}$ is negative for ${\Omega }_{\Lambda 0}>1/3$. Light-curve and colour corrections, extinction, calibration, selection bias and possible population evolution must be controlled. The supernova evidence and cosmic microwave background constraints are complementary; together they favour this model without proving that every alternative dark-energy model is impossible.^[ref][ref]

For the final comparison, assume $z>0$, a common $H_0$ and $0<{\Omega }_{\Lambda 0}<1$ in model (a). ThenThus $H_c<H_a<H_b$ and the integrands obey $1/H_b<1/H_a<1/H_c$. Because all three models are flat, compare the normalization-independent distance $R_0\chi =c\int d{z}^{\prime }/H(z^{\prime })$, rather than separately normalized coordinates. The luminosity-distance ordering of flat dust-vacuum universes is $d_{L,b}<d_{L,a}<d_{L,c}$, henceModel (b) gives the largest received flux, model (c) the smallest. Equalities replace the strict ordering at the endpoint values of the vacuum fraction; all have the same leading local Hubble law as $z\to 0$.

Definitions and pressure convention. $R(t)$ is the scale factor, $t$ cosmic time and dots time derivatives. $G$ is the gravitational constant, $c$ the speed of light, $\rho$ the mass-equivalent density of the material fluid and $k=+1,0,-1$ the spatial-curvature sign in this normalization. $\Lambda$ here has units of inverse time squared, rather than inverse length squared. Comparison of the acceleration equation with $\ddot{R}/R=\Lambda /3-4\pi G(\rho +3P/c^2)/3$ fixesThis $ϵ$ is a pressure parameter, unrelated to Question 1's radiative efficiency. pressureless matter has $P=0$ and $ϵ=0$. When solving the generalized model below, $ϵ$ is assumed constant.

Evidence for a positive vacuum component. Standardized Type Ia supernovae measure the luminosity-distance–redshift relation. Their corrected apparent magnitudes at cosmological distances imply greater distances than a decelerating matter-dominated model of comparable parameters, favoring late accelerated expansion. Light-curve standardization, interstellar extinction, selection bias and possible population evolution must be controlled rather than treating every supernova as an identical candle.^[ref][ref]

The cosmic microwave background and baryon acoustic oscillation measurements constrain geometry and matter density in combination. Near spatial flatness together with a subcritical matter density leaves room for an additional smooth component; cosmic ages and structure growth supply consistency tests. A positive cosmological constant fits these observations within the standard framework. They do not establish that every allowed dark-energy theory must be a strictly constant vacuum energy; the inference depends on the cosmological and gravitational model.

pressureless matter evolution. Set $k=0$ and $ϵ=0$. Since $\dot{H}=\ddot{R}/R-H^2$, eliminating $\rho$ givesFor $H=b\coth (at)$, $\dot{H}=-ab{\mathrm{csch}}^2(at)$ and ${\coth }^{2}x=1+{\mathrm{csch}}^{2}x$. ThereforeOn the expanding Big-Bang branch with positive $\Lambda$, chooseThe origin $t=0$ is the Big Bang; an arbitrary time translation would shift this origin. Integrating gives $R\propto {\sinh }^{2/3}(at)$. At early times $H\sim 2/(3t)$, while at late times $H\to \sqrt{\Lambda /3}$.

Age and the density fractions. Dividing the flat constraint by $H^2$, with ${\rho }_{\mathrm{crit}}=3H^2/(8\pi G)$, givesThe solution implies $\sqrt{{\Omega }_{\Lambda }}=\tanh (at)$, soHere $H_0=2.3331\times 10^{-18}{\mathrm{s}}^{-1}$, using the printed megaparsec conversion. Taking one year as $365.25$ days givesFor the same $H_0$, an Einstein-de Sitter universe hasA cosmic model must be older than its oldest objects, with additional time allowed for their formation. Old stellar populations are dated through stellar-evolution fitting, particularly the globular cluster main-sequence turnoff, with uncertainties from distances, composition and input stellar physics. A pre-examination globular-cluster age constraint on cosmology obtained a $95\%$ lower cosmic-age limit of $11\mathrm{Gyr}$ after allowing for formation time. This conflicts with the $9.05\mathrm{Gyr}$ model but is compatible with the $13.09\mathrm{Gyr}$ calculation.^[ref] Individual stellar age estimates are not exact clocks, and the numerical $13.09\mathrm{Gyr}$ is the result for the specified $H_0$ and matter fraction, not a claim to be a contemporary best-fit age.

Constant nonzero pressure parameter. Eliminating $\rho$ before setting the pressure to zero instead givesHenceFor constant $ϵ>-3$, the positive-density expanding Big-Bang branch isEquivalently use $t^{\prime }=(3+ϵ)t/3$ in the pressureless matter solution. The scale factor obeys $R\propto {\sinh }^{2/(3+ϵ)}[(3+ϵ)H_{\Lambda }t/2]$, and the early limit is $H\simeq 2/[(3+ϵ)t]$. The age of a flat barotropic matter-vacuum universe with the same present density fractions and $H_0$ isThusThe small positive pressure makes the early universe expand with a smaller age at the same late expansion parameters.

The acceleration transition. Normalize the present scale factor to one. Constant $ϵ$ and the continuity equation give $\rho ={\rho }_0(1+z{)}^{3+ϵ}$. Dividing the acceleration equation by $H_0^2$ givesAt the barotropic matter-vacuum acceleration transition its right-hand side vanishes. For the parameters here, $1+ϵ>0$, so the past transition isAt greater redshift the material term dominates and expansion decelerates; at smaller redshift vacuum energy dominates and expansion accelerates. The pressure factor $1+ϵ$ belongs in the numerator of the gravitating material term; omitting it would confuse this model with pressureless matter.

cosmic inflation is an early interval of accelerated expansion, $\ddot{R}>0$. For expansion this means a shrinking comoving Hubble radius, $d[(RH{)}^{-1}]/dt<0$. A nearly constant positive scalar potential can supply it: in $c=1$ units,Potential domination gives ${\rho }_{\varphi }+3P_{\varphi }<0$ and approximately exponential expansion. Sustaining the interval requires sufficiently slow evolution, not just setting the initial field velocity to zero on a steep potential.

The horizon problem is the near equality of conditions in widely separated microwave-background regions that could not equilibrate in the standard decelerating hot Big Bang history. cosmic inflation permits the currently observed patch to descend from an earlier region small enough to have been causally connected, before that patch is stretched to enormous size. The claim concerns a causal history extended before the conventional hot phase; merely accelerating a collection of already disconnected regions would not explain their common temperature.

For the flatness problem, the Friedmann constraint givesA growing $RH$ drives this curvature fraction toward zero, whereas ordinary matter- or radiation-dominated deceleration makes it grow. cosmic inflation therefore suppresses observable curvature without requiring an equally extreme initial tuning within the later hot evolution.

Pre-existing monopoles and other unwanted relics are also diluted by the volume increase: a conserved particle number density falls as $e^{-3N}$ over $N$ e-folds. The transition to the hot universe must reheat efficiently, and its temperature must not regenerate an excluded relic population. The required number of e-folds is model and reheating dependent; “about sixty” is a useful scale for many scenarios rather than a universal exact requirement.

cosmic inflation also supplies a mechanism for perturbations. quantum fluctuations are stretched beyond the Hubble radius, and nearly scale-invariant comoving curvature perturbations can later seed cosmic structure. Simple single-field slow-roll models commonly predict approximately adiabatic, nearly Gaussian fluctuations, with a small spectral tilt; tensor perturbations would provide an additional test. Acoustic coherence and the observed scalar spectrum are compatible with this account, but they do not single out one inflaton potential. cosmic inflation addresses causal homogeneity, curvature and relic dilution while motivating the initial perturbations; it does not automatically solve the cosmological constant problem or determine reheating.

The cosmic microwave background has an almost perfect blackbody spectrum at about $2.725\mathrm{K}$. This is strong evidence for an earlier hot, dense phase in which radiation thermalized, followed by expansion that redshifted the photon spectrum while preserving its Planck form. COBE FIRAS compared the sky with a precision reference blackbody; instrument temperature, gain, foregrounds and calibrator emissivity were crucial controls. Agreement with a blackbody constrains substantial late energy injection or photon-production distortions.^[ref]

After subtracting the dominant dipole associated mainly with observer motion, anisotropies at the level of roughly $10^{-5}$ reveal primordial structure. COBE DMR established large-angular-scale anisotropy; multifrequency measurements and independent maps were needed to distinguish it from Galactic emission and instrumental artifacts.^[ref] Approximate isotropy supports a homogeneous background model under the usual cosmological assumptions, but isotropy at one observing position is not alone a proof of spatial homogeneity everywhere.

The temperature CMB angular power spectrum has multiple acoustic peaks. Their coherent pattern follows oscillations in the pre-recombination photon-baryon fluid. Peak positions compare the sound horizon with angular diameter distance and hence constrain geometry together with expansion parameters. Relative peak heights test baryon inertia, matter–radiation equality and gravitational-potential evolution, motivating both baryonic and nonbaryonic matter. Cosmic microwave background diffusion damping suppresses small-scale power. Cosmic microwave background polarization and temperature–polarization correlations provide further tests of scattering, acoustic phase and reionization; gravitational lensing remaps the observed pattern and probes intervening structure.

By the time of this examination, the Planck 2013 temperature and lensing analysis supported a spatially flat six-parameter model with cold dark matter and a cosmological constant. This is a joint fit within an explicit model, not an inference that each peak uniquely measures one parameter.^[ref] Foreground separation requires frequency coverage, sky masks, beam and noise characterization, and consistency across instruments. The primary microwave background has geometric and other parameter degeneracies, so acoustic-ruler, supernova and structure observations are valuable complements. The spectrum establishes a thermal hot-universe history; anisotropy and polarization constrain its geometry, contents and primordial perturbations. None of these observations by itself demonstrates a unique cosmic inflation mechanism.

Use the geodesic radial comoving coordinate, in which the Friedmann-Lemaître-Robertson-Walker metric has the formOn a slice of constant cosmic time, the radial proper-length element is $R(t)\mathrm{d}\chi$. ThereforeIf one starts from a curvature coordinate with radial element $R\mathrm{d}r/\sqrt{1-k{r}^2}$, then $\chi ={\int }_0^r\mathrm{d}{r}^{\prime }/\sqrt{1-k{r}^{\prime 2}}$; the curvature coordinate $r$ itself is not generally the $\chi$ used above.

For a comoving galaxy, $\dot{\chi }=0$, so $\dot{D}=\dot{R}\chi =HD$. The definitions areThis recession law concerns proper distances on common cosmic-time slices. It is not a local special-relativistic relative velocity measurement; a large recession rate is not a signal exceeding the local speed of light.

For a comoving volume $V=V_0{R}^3$, total fluid energy is $E=\rho c^{2}V$. Adiabatic energy conservation gives $\mathrm{d}E+P\mathrm{d}V=0$. Expanding,henceHere $\rho$ is mass-equivalent energy density. Radiation loses energy both by dilution and by pressure work; pressureless matter only dilutes in this description.

In the time-unit convention for the cosmological constant, it represents a uniform vacuum contribution with pressure equal to minus its energy density. There is a units convention in the printed field equations: their term is $\Lambda /3$, so the symbol $\Lambda$ there has units of inverse time squared. If the geometrical constant is instead given in inverse length squared, this symbol equals $c^2{\Lambda }_{\mathrm{geom}}$. The curvature parameter $k$ labels constant-curvature spatial slices; with a conventional scale-factor normalization it is $+1,0,-1$ for positively curved, flat and negatively curved slices. It fixes local curvature sign, not every possible global spatial topology.

For a flat fluid-only model without a cosmological constant and with $P=w\rho c^2$, the two field equations giveThe two printed cases and their scale-factor laws are worked under the literal inline labels below.

Inflation is accelerated expansion, $\ddot{R}>0$, equivalently a decreasing comoving Hubble radius $(RH{)}^{-1}$ on an expanding branch. An approximately constant positive vacuum-like energy can give nearly exponential expansion. Enough inflation can put regions now widely separated inside a common earlier causal patch, suppress $|k|/(RH{)}^2$ and dilute unwanted relics such as monopoles. These are the horizon, flatness and relic-abundance motivations. Reheating is needed to populate the subsequent hot universe. Quantum fluctuations can seed primordial perturbations, but inflation alone is not a universal solution to the initial singularity or a proof that all initial conditions inflate.

Now adopt the printed natural units $c=\hslash =1$. For the Homogeneous massive-scalar equation in an expanding universe,The continuity equation givesWhere $\dot{\varphi }\ne 0$, this implies the expanding-universe Klein-Gordon equationA regular canonical-field solution extends it through isolated turning points by continuity. Purely dividing the conservation equation does not determine a trajectory on an interval where $\dot{\varphi }$ is identically zero; canonical field dynamics, or variation of its action, supplies the equation there. In particular a constant nonzero massive field is not an exact freely evolving solution.

With $\Lambda =k=0$,ThereforeThis inflation condition for a homogeneous massive scalar means a potential-dominated field has sufficiently negative pressure to accelerate expansion. In quadratic chaotic inflation, a large initial field with $V=m^2{\varphi }^2/2$ can enter slow roll, with $3H\dot{\varphi }\simeq -m^2\varphi$ and $H^2\simeq 4\pi G{m}^2{\varphi }^2/3$. The potential slow-roll parameter is ${ϵ}_V=1/(4\pi G{\varphi }^2)$, so a large field relative to the gravitational scale can support many e-folds. The field eventually rolls far enough that this regime ends and oscillation and reheating become relevant. Large potential energy with arbitrary initial kinetic energy does not by itself guarantee an inflationary interval.

Differentiating the flat Friedmann equation and then using the scalar equation,For the expanding branch,This Nonincreasing Hubble rate for a canonical inflaton decreases strictly where the field rolls and can be momentarily stationary at a turning point; the statement “always decreasing” means nonincreasing. This does not conflict with acceleration, because $\ddot{R}/R=H^2+\dot{H}$ remains positive when ${ϵ}_H=-\dot{H}/H^2<1$.

At horizon exit, larger comoving wavelengths leave earlier, when $H$ is larger. For a canonical field in slow roll, with reduced Planck mass ${\overline{M}}_{\mathrm{Pl}}=(8\pi G{)}^{-1/2}$,The decreasing expansion rate gives a red tensor tilt at this order. The scalar curvature spectrum also contains the conversion from field fluctuations to curvature:Here $N=\ln R$ counts elapsed e-folds. Thus the Decreasing Hubble rate does not alone fix scalar tilt. Evolution of ${ϵ}_H$ matters. For the quadratic slow-roll model, ${\eta }_V={ϵ}_V$ and $n_s-1\simeq -6{ϵ}_V+2{\eta }_V=-4{ϵ}_V<0$. If $N_{\mathrm{rem}}$ e-folds remain until the end, leading estimates give $n_s\simeq 1-2/N_{\mathrm{rem}}$ and tensor-to-scalar ratio $r_{\mathrm{TS}}\simeq 8/N_{\mathrm{rem}}$, with small end-of-inflation corrections.

The relevant contemporary comparison is a nearly scale-invariant but red scalar spectrum. The February2015 Planck analysis reported $n_s=0.968\pm 0.006$ for its temperature, low-polarization and lensing combination, and an upper limit $r_{0.002}<0.11$ at95percent confidence. The scalar tilt is qualitatively consistent with a slowly evolving inflaton, but that result does not establish the quadratic potential; its relatively large tensor prediction was already disfavored against lower-tensor alternatives in that analysis.^[ref]

The Einstein tensor and its contractions areTo match the supplied Ricci tensor components, use the Riemann curvature tensor conventionThis convention also fixes the sign used for geodesic deviation above. In vacuum, tracing the Einstein field equations gives $-R+4\Lambda =0$. Substitution therefore yields $R_{\mu \nu }=\Lambda g_{\mu \nu }$.

Differentiate the angular field equation to obtainConsequently $R_{00}=\Lambda A=\Lambda g_{00}$ and $R_{11}=-\Lambda /A=\Lambda g_{11}$. The other angular component follows from $R_{33}={\sin }^2\theta R_{22}$. Thus the single angular equation implies all the displayed vacuum equations wherever this coordinate chart is regular.

Integrating $(rA{)}^{\prime }=1-\Lambda r^2$ gives $A=1+C/r-\Lambda r^2/3$. Match $g_{tt}\simeq 1+2\Phi$ with the Newtonian gravitational potential $\Phi =-GM/r$ in the weak-field overlap region to identify $C=-2GM$. Hence the Schwarzschild-de Sitter spacetime hasNegative $\Lambda$ instead gives the Schwarzschild--anti-de Sitter black hole case. This matching fixes the normalization locally; a spacetime with nonzero $\Lambda$ is not asymptotically flat.

For timelike geodesics in the orbital plane, take proper time as parameter and use $L=(A{\dot{t}}^2-{\dot{r}}^2/A-r^2{\dot{\varphi }}^2)/2$. The cyclic coordinates give $A\dot{t}=k$ and $r^2\dot{\varphi }=h$; the angular canonical momentum is $-h$ in this signature. The four-velocity normalization $2L=1$ then givesHere $k$ is conserved Killing energy per unit rest mass in the chosen time normalization, and $h$ is conserved angular momentum per unit rest mass. There is no general interpretation of $k$ as energy measured at an asymptotically flat infinity when $\Lambda \ne 0$.

For radial motion $h=0$, differentiation gives $2\dot{r}\ddot{r}=-A^{\prime }\dot{r}$. The radial Euler-Lagrange equation extends the resulting expression through a turning point where division by $\dot{r}$ would fail. ThusThis is the proper-time second derivative of the areal radius, not a nonzero four-acceleration of the freely falling particle. Positive $\Lambda$ contributes an outward term; negative $\Lambda$ reinforces inward attraction. Restoring units in the slow-motion, weak-field Newtonian limit givesThe extra term describes the gravitational effect of a cosmological constant, not drag or a contact force. The approximation requires $GM/(r{c}^2)\ll 1$, $|\Lambda |r^2\ll 1$ and speed much less than $c$.

The cosmological principle idealizes spatial slices as homogeneous and isotropic. The corresponding Friedmann-Lemaître-Robertson-Walker metric can be writtenHere $t$ is cosmic time, $a(t)$ has dimensions of length in this convention, and $\chi$ is a dimensionless comoving coordinate. The sign of $K$ gives positive, zero or negative spatial curvature. It does not alone determine global topology. A comoving pair at fixed radial coordinates has proper separation scaling with $a(t)$, so the Hubble parameter is $H=\dot{a}/a$, and cosmological redshift satisfies $1+z=a(t_{\mathrm{obs}})/a(t_{\mathrm{em}})$.

For a perfect fluid of mass-equivalent energy density $\rho$ and pressure $p$, Einstein's equations giveThe last equation is local energy conservation for constant $\Lambda$, with vacuum energy kept separate from $\rho ,p$. A component with $p=w\rho c^2$ has $\rho \propto a^{-3(1+w)}$: pressureless matter dilutes as $a^{-3}$ and radiation as $a^{-4}$, the additional radiation factor being photon redshift. With $K=\Lambda =0$, matter domination gives $a\propto t^{2/3}$ and radiation domination gives $a\propto t^{1/2}$. Positive vacuum energy alone gives exponential expansion, $a\propto e^{Ht}$ with $H=c\sqrt{\Lambda /3}$.

Geometry, contents and initial conditions together determine the expansion history. A closed matter-only model recollapses; an open matter-only model expands indefinitely. With a cosmological constant, positive spatial curvature no longer by itself implies recollapse. Present critical density is ${\rho }_c=3H_0^2/(8\pi G)$ and the density parameters obey ${\Omega }_m+{\Omega }_r+{\Omega }_{\Lambda }+{\Omega }_K=1$, where ${\Omega }_K=-K{c}^2/(a_0^2{H}_0^2)$. These models describe the large-scale background, while local structure and inhomogeneities require perturbations or more general metrics.

Use ${\rho }_r,{\rho }_m$ as energy per volume, as specified in the prose, and let $R$ have dimensions of length so that the positive curvature magnitude $k$ is dimensionless. The Friedmann equation for spatial curvature $-k$ isFor an energy density the cosmological perfect-fluid continuity equation is $\dot{\rho }+3H(\rho +P)=0$. Nonrelativistic matter has negligible pressure and radiation has $P_r={\rho }_r/3$. Thus the four contributions to $H^2$ scale respectively asFor nonzero positive densities and $\Lambda >0$, the usual possible sequence is radiation domination, matter domination, curvature domination, then cosmological-constant domination. A distinct era for each is not guaranteed: coefficients may cause curvature to be overtaken by the cosmological constant before it ever dominates. If $\Lambda =0$, curvature dominates asymptotically; negative $\Lambda$ does not give a late positive vacuum-dominated expansion.

The printed auxiliary relation $P_r={\rho }_r{c}^2/3$ uses mass-density notation, in conflict with the stated energy-density definition. It is consistent if ${\rho }_r$ there is understood as energy density divided by $c^2$; in that convention the density term in the Friedmann equation is $8\pi G\rho /3$. The two conventions must not be mixed.

In the prescribed matter-dominated approximation, neglect curvature, radiation, and the cosmological constant throughout the calculation. The Friedmann equation and ${\rho }_m\propto R^{-3}$ give $\dot{R}\propto R^{-1/2}$, hence $R\propto t^{2/3}$ and $H=2/(3t)$. For $H_0=70\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}=2.269\times 10^{-18}{\mathrm{s}}^{-1}$,The particle horizon is a proper distance on a constant-time slice, not simply $ct$:ThereforeThese are estimates within a spatially flat, matter-only model, rather than the measured age and particle horizon of the actual late-time universe.

The angle estimates the largest transverse region at decoupling that could have established causal contact after the big bang in the assumed decelerating model. The cosmic microwave background has nearly the same temperature in directions separated by much larger angles. This is the horizon problem: those regions have no shared causal history in that simple model, yet their thermodynamic conditions are strikingly alike. The particle horizon should be distinguished from the smaller sound horizon, which controls acoustic propagation in the primordial fluid.

An early epoch dominated by a sufficiently large positive effective cosmological constant gives $\ddot{R}>0$ and approximately $R\propto e^{Ht}$. During this cosmic inflation, the comoving Hubble length $c/(RH)$ shrinks. A patch that was initially in causal contact can expand to contain the whole presently observable cosmic microwave background surface, while expansion suppresses spatial curvature. A finite period of cosmic inflation followed by reheating and the usual hot expansion therefore explains the uniformity without superluminal local signals. An eternally large unchanged cosmological constant would prevent the later matter-dominated history; an exit from cosmic inflation and reheating are required. Sufficient duration and a suitable pre-inflationary causal patch are part of this explanation.

Set the spatial-curvature constant to zero in the stipulated matter-only Friedmann equation to get ${\rho }_c=3H^2/(8\pi G)$. Conservation of pressureless matter with $R(t_0)=1$ gives $\rho ={\rho }_0/R^3$, and thereforeChoosing the Big Bang as $t=0$ yields $R=(t/t_0{)}^{2/3}$ and $H=2/(3t)$. Consequently $\boxed{t_0=2/(3H_0)\simeq 9.3\mathrm{Gyr}}$ for the supplied $H_0$. This is the age of the flat pressureless model without a cosmological constant, not a determination in the current multi-component cosmological model.

The curvature constant specifies the intrinsic spatial curvature of homogeneous constant-time slices. Positive curvature has spherical geometry, zero curvature Euclidean geometry, and negative curvature hyperbolic geometry. A conventional coordinate normalization uses $k=+1,0,-1$, with the scale factor carrying the curvature radius; rescaling that factor changes the numerical magnitude assigned to a dimensional curvature coefficient.

The sign alone does not determine the future expansion once a cosmological constant is admitted. Nor does negative or zero local curvature by itself determine the global topology: quotient spaces can share the same local curvature.

For conserved nonrelativistic matter, ${\rho }_M={\rho }_{M0}{a}^{-3}$, where $a=R/R_0$ and $a_0=1$. Substitution into the Friedmann equation giveswithThe source writes the dimensionless normalized factor as $R$ in its final expression; this is equivalent to setting $R_0=1$ consistently in the curvature coefficient. Evaluating today gives ${\Omega }_M+{\Omega }_k+{\Omega }_{\Lambda }=1$. Here the printed $\Lambda$ already has dimensions of inverse time squared. If the cosmological constant is instead defined geometrically with inverse-length-squared units, replace this $\Lambda$ by ${\Lambda }_{\mathrm{geom}}{c}^2$.

With zero curvature and no cosmological constant, the present normalization requires ${\Omega }_M=1$. Thus $\dot{a}=H_0{a}^{-1/2}$, and choosing the big bang as $t=0$ givesFor an empty expanding model, ${\Omega }_k=1$ and the curvature must be negative. The equation becomes $\dot{a}=H_0$, givingThe latter is the Milne slicing of flat spacetime; its zero-scale-factor origin is not a matter-density curvature singularity. In the matter model, gravity decelerates expansion, so the earlier slope was greater than the present slope and the same present size was reached in less time. With the stipulated $H_0$, the two model ages are approximately $8.69$ and $13.0$ Gyr.

The Penrose singularity theorem states that a globally hyperbolic spacetime with a noncompact Cauchy hypersurface, a closed trapped surface, and the null convergence condition is future null-geodesically incomplete. Time reversal replaces the trapped surface by an anti-trapped surface and gives past incompleteness.

In his 1965 paper Gravitational Collapse and Space-Time Singularities, Roger Penrose used trapped surfaces and global causal structure to address whether nonspherical perturbations could avert a breakdown during gravitational collapse. Quasars supplied the opening astrophysical motivation. The paper used metric signature $(+,-,-,-)$ and a stronger timelike local-energy assumption; the statement above uses the modern null convergence condition.^[ref]

For this version, take a smooth connected four-dimensional spacetime with a time orientation. A Cauchy hypersurface meets every inextendible causal curve exactly once; its existence expresses global hyperbolicity. The hypersurface must be noncompact, meaning not compact; this is a global spatial condition, not an assumption about the size of the collapsing body. The null convergence condition is $R_{ab}{k}^a{k}^b\ge 0$ for every null vector $k$. Through the Einstein field equations it follows from the null energy condition; a cosmological constant drops out of the null contraction.^[ref]

The conclusion is that at least one future-inextendible null geodesic has a finite upper endpoint of its affine parameter. It neither asserts infinite mass density or divergent curvature at every endpoint, nor says that such an endpoint is visible. Locating an event horizon requires additional global information; the theorem is not a horizon-finding rule for arbitrary spacetimes. Weak cosmic censorship conjecture is an additional issue.^[ref]

The supermassive black hole account describes the motivating observational evidence from stellar orbits at the Galactic centre. Such measurements infer a compact gravitating mass. They do not observe a trapped surface, measure a geodesic's complete future, or experimentally prove the Penrose singularity theorem.

For $T_{ab}=(\rho +p)u_a{u}_b+p{g}_{ab}$ and nonzero null $k$, contraction gives $T_{ab}{k}^a{k}^b=(\rho +p)(u\cdot k{)}^2$. A timelike fluid velocity has nonzero scalar product with every nonzero null vector, so the null energy condition is exactly the displayed inequality. Individual nonnegativity of pressure or density is unnecessary for null focusing. A cosmological constant term has zero null contraction.