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Cosmological isotropy means that coarse-grained spatial observations or their statistics are unchanged by rotations about an observer. In an FRW metric it holds about every comoving location. cosmological isotropy about a single location is weaker than spatial homogeneity: a radially inhomogeneous model centred on that location can also have an isotropic sky.

The cosmological principle assumes that the universe is statistically homogeneous and isotropic on sufficiently large spatial scales. It applies to a coarse-grained description, not to each galaxy or local void. Combined with the Copernican principle, cosmological isotropy around us supports a homogeneous description rather than a specially centred model. It motivates an FRW metric, whose expansion is governed by a few time-dependent quantities. Observations sample only a finite past light cone, so they support this hypothesis without proving global spatial homogeneity beyond the observed region.

The official notes, PDF p. 27, handwritten p. 20 derive the early-time photon count, its vanishing limit, and the failure of thermal equilibrium when the count is of order one. Two details need supplementing.

First, their constant $N_{\gamma 0}$ is the normalization obtained by extrapolating the early-time particle horizon formula to $z=0$. It is not the count within today's actual particle horizon. Let $N_0=N_{\gamma }(t_0)$ and let $r_{h0}$ be that horizon's physical radius. Using the paper's early-time approximation givesFor a present matter-dominated approximation, $r_{h0}\simeq 2c/H_0$. Alternatively, retaining both components in the flat model exactly, ${\Omega }_R=1/(2+z_{\mathrm{eq}})$, and integrating the particle horizon givesThe second boxed formula uses the exact two-component normalization and the radiation-dominated early-time limit. Its large-$z_{\mathrm{eq}}$ leading term agrees with the first expression and $r_{h0}\simeq 2c/H_0$.

Second, a particle horizon containing only order-one photons cannot support a smooth, locally representative radiation density: discrete-particle fluctuations are of the same order as the mean, and a single photon's direction is strongly anisotropic. Thus the fluid description assuming spatial homogeneity and cosmological isotropy is not self-consistent on that causal scale. This does not by itself disprove an ensemble-averaged cosmological symmetry; it limits the extrapolation of the smooth thermal equilibrium description.

On large scales the appropriate background is the spatially flat FLRW metric,Here $t$ is cosmic time, the proper time of a comoving observer; $R(t)$ is the scale factor; and $(r,\theta ,\varphi )$ are dimensionless comoving coordinates. With this convention $R$ has units of length. A fixed comoving radial interval has instantaneous proper length $R(t)dr$. The cosmological principle assumes spatial homogeneity and cosmological isotropy on the scales where this background description applies.

The general FLRW metric instead has $d{r}^2/(1-k{r}^2)$ in its radial part, with normalized $k=+1,0,-1$ describing positive, zero or negative spatial curvature of an FLRW universe. The best simple description takes $k=0$; observations constrain spatial curvature to be small rather than proving it identically zero. The late accelerated expansion is encoded by the time dependence of $R$, determined by the stress-energy tensor through the Friedmann equations, rather than by an additional term in the FLRW metric. Small-scale galaxies and other inhomogeneities require perturbations of this background.

The actual large-scale expansion is well described by the Lambda-CDM model: radiation in cosmology, ordinary baryonic matter, cold dark matter and late-time dark energy coexist. A component becomes dominant through continuous changes in their density ratios, rather than every previous component disappearing at an instantaneous phase boundary. Matter pressure is small in its nonrelativistic regime, but radiation persists, and massive neutrinos interpolate between relativistic and nonrelativistic behavior.

There is also a late accelerated era, absent from a model ending permanently in S3. It is consistent with nearly constant vacuum energy, whose negative pressure eventually dominates because pressureless matter dilutes as $R^{-3}$. The scale factor then ceases to follow the pure $t^{2/3}$ Einstein-de Sitter universe law.

An early phase of cosmic inflation is a well-motivated model rather than a directly observed exact $w=-1$ phase. Successful cosmic inflation must end and transfer its energy to matter and radiation through reheating; realistic models generally have slowly varying $H$ and $w$ close to, rather than exactly equal to, $-1$. Extrapolating a strictly constant $H$ to arbitrarily small $R$ is not a measurement of the universe's past duration. Finally, spatial homogeneity and cosmological isotropy apply statistically on large scales: galaxies, gravitational waves and other perturbations are absent from the exact FLRW metric, and small spatial curvature remains an observational possibility.

The Friedmann-Lemaître-Robertson-Walker metric implements the cosmological principle: on the scales being modelled, the spatial slices have spatial homogeneity and cosmological isotropy. There is a family of comoving observers whose orthogonal slices share one cosmic time and one scale factor. Each spatial slice is a locally maximally symmetric space, with time-independent normalized $k$.

The matter is represented by a perfect fluid in general relativity, at rest relative to the comoving observers, with energy density $\varrho (t)$ and pressure $p(t)$ depending only on cosmic time. Its stress-energy tensor isThere is no preferred spatial direction, so there is no anisotropic stress or spatial heat flux in that rest frame. Gravity obeys the Einstein field equations; the equations in this paper contain no separate cosmological constant, although a component with $p=-\varrho$ can represent one. An equation of state relating $p$ and $\varrho$ closes the evolution equations. Spatial homogeneity and cosmological isotropy are idealizations of the cosmological background; they do not assert that individual local structures are uniform.

Spatial homogeneity makes $\rho$ independent of position and makes the two-particle number density depend only on the separation. Isotropy removes its directional dependence. ConsequentlyHere $g_2$ is a radial distribution function, with the finite-$N$ normalization inherited from a; statistical independence is unnecessary. In particular, the translated pair in c has separation $\mathbf{R}$ for every $s$, so its $s$ integral is simply ${\rho }^2{g}_2(R,t)$.

Isotropy of the random velocity $\mathbf{c}=\mathbf{v}-\mathbf{u}$ makes its second velocity moment proportional to the identity. Taking its trace fixes the proportionality constant:Thus $\theta$ is the one-component random velocity variance; for unit mass in thermal equilibrium, $\theta =k_{B}T$. A Maxwell-Boltzmann velocity distribution is not required for the pressure tensor to be isotropic.

For the interaction contribution, the angular integral is $\int d\Omega {\hat{R}}_i{\hat{R}}_j=(4\pi /3){\delta }_{ij}$, by isotropy and its trace. With $d^{3}R=R^{2}dRd\Omega$, c becomesCombining the two pressure tensors gives the virial pressure from pair correlations:For a repulsive pair potential, the outward radial force is $-{\varphi }^{\prime }(R)\ge 0$. Since $g_2\ge 0$, the second term in $p$ is nonnegative, and positive when occupied separations feel nonzero repulsion. It expresses additional momentum transfer through interactions. The displayed integral assumes a differentiable pair potential and convergence; a hard core requires a contact formula or a limiting argument.

For a statistically homogeneous matter distribution, the cosmological two-point correlation function of the mean-zero density contrast $\delta =(\rho -\bar{\rho })/\bar{\rho }$ isSpatial homogeneity removes dependence on $\mathbf{x}$, and isotropy reduces the separation argument to its magnitude. For discrete galaxies, the galaxy two-point correlation function equivalently specifies the excess pair probability at distinct positions:Consequently the mean density of other galaxies, conditioned on a galaxy at the origin, is $\bar{n}[1+\xi (r)]$. The self-pair at zero separation is excluded from this definition; its Dirac delta contribution is the galaxy power spectrum shot noise of the unsmoothed number-density field.

The strongest evidence for large-scale cosmological isotropy is the nearly direction-independent blackbody temperature of the cosmic microwave background. After removing the dipole associated mainly with our peculiar velocity, its residual temperature fluctuations are of order $10^{-5}$. Nearly isotropic counts of distant radio sources and galaxies, after correcting for sky coverage and selection effects, provide independent evidence. Local galaxies, clusters and voids clearly are not isotropic structures; the claim concerns a statistical description on sufficiently large scales.

Large redshift surveys show that the mean galaxy density stabilizes as increasingly large volumes are sampled, and that the excess pair correlations weaken at large separations. This supports statistical homogeneity beyond the characteristic clustering scales. The consistency of large-scale observations in different directions and the success of a common FRW metric description are additional checks.

Isotropy around one observer does not alone prove spatial homogeneity: a radially varying distribution centered on that observer could look isotropic. The Copernican principle supplies the assumption that we occupy no preferred position; comparable isotropy about typical locations then supports the cosmological principle. Surveys observe a finite past light cone, with source evolution and selection, so they cannot establish homogeneity everywhere on a simultaneous spatial slice. The evidence supports large-scale statistical isotropy and homogeneity, with substantial small-scale structure and an explicit typical-observer assumption.

The cosmological principle assumes statistical spatial homogeneity and cosmological isotropy on sufficiently large scales. Spatial homogeneity means that coarse-grained statistical properties do not depend on location; cosmological isotropy means that they do not depend on direction. The claim concerns a large-scale description, allowing individual galaxies, clusters and voids. It motivates the FRW metric and makes cosmic expansion describable by a single scale factor and a small number of matter variables.

Its adoption combines empirical large-scale regularity with the Copernican principle, the assumption that our location is typical. The resulting models provide a coherent account of cosmic expansion, the cosmic microwave background and primordial abundances. Neither the Einstein field equations alone nor cosmological isotropy around just one observer proves global spatial homogeneity: a spherically symmetric universe specially centred on that observer would also look isotropic. Observations of a finite past light cone support the principle over sampled scales without establishing it beyond the observed region.

Three supporting observations probe complementary aspects. First, after the observer-motion dipole and local foregrounds are removed, the cosmic microwave background is extremely isotropic, with intrinsic temperature fluctuations of order $10^{-5}$ and a nearly identical thermal spectrum across the sky. Second, suitably corrected all-sky counts of distant radio sources and extragalactic background intensities show no large preferred direction; Galactic obscuration and survey selection must be accounted for. Third, three-dimensional galaxy redshift surveys show that clustering and voids average toward statistically similar distributions in sufficiently large regions: large-volume number counts approach volume scaling and correlations weaken on large separation scales. The first two primarily test cosmological isotropy, while the third adds evidence for large-scale spatial homogeneity. The principle is a well-supported large-scale working hypothesis whose observational and typicality assumptions remain explicit.

Relative positions and velocities measured from galaxy $A$ are $r_B-r_A$ and $v_B-v_A$. Spatial homogeneity says that the same position-to-relative-velocity rule applies from every galaxy. Thus $v(r_B-r_A)=v(r_B-r_O)-v(r_A-r_O)$. Taking $r_O=0$ and putting $r_B=x+y$, $r_A=y$ gives additivity $v(x+y)=v(x)+v(y)$. For a continuous physical velocity field, additivity implies rational homogeneity and, by continuity, real homogeneity. Thereforefor a position-independent matrix $H$. Regularity is necessary to exclude pathological discontinuous additive functions.

Decompose the given matrix as $H=(5D/t)I+(D/t)W$, whereIts symmetric part is isotropic: no cosmological shear, expansion scalar $\nabla \cdot v=15D/t$, and linear scale factor proportional to $t^{5D}$ for $t>0$. Its antisymmetric part is rigid rotation with angular velocityThe vorticity is $2\mathbf{\omega }$. Comoving separations obeyThus all distances grow by the same power, volumes grow as $t^{15D}$, and the pattern rotates about the fixed axis $(1,-2,1)$ through angle $\sqrt{6}D\log (t/t_0)$. Homogeneity alone permits this rotation; full cosmological isotropy would not.

The cosmological principle gives spatial homogeneity and cosmological isotropy for the large-scale spatial slices. About any chosen comoving observer, spherical coordinates therefore separate a radial direction from the angular directions. In particular,is the metric tensor on the unit sphere. Its appearance expresses cosmological isotropy, rather than a privileged centre of the universe. The factor $(1-K{r}^2{)}^{-1}$ permits the radial geometry to have nonzero spatial curvature of an FLRW universe.

The scale factor $a(t)$ changes every spatial separation between fixed comoving coordinates by the same multiplicative factor. For example, the radial proper distance in cosmology between fixed radial labels at a fixed cosmic time isExpansion or contraction through this single function preserves spatial homogeneity and cosmological isotropy; direction-dependent scale factors would generally spoil the latter. The Friedmann-Lemaître-Robertson-Walker metric specifies this possible evolution geometrically; a dynamical theory is needed to determine $a(t)$.

The constant $K$ fixes the sign and normalization of the spatial curvature of an FLRW universe. The spatial scalar curvature is ${}^{(3)}\mathcal{R}=6K/a^2(t)$. With dimensionless $r$ and a suitably normalized scale factor, the familiar choices are $K=+1,0,-1$ for spherical, flat, and hyperbolic spatial geometry. A rescaling of $r$ changes the normalization of $K$ and $a$ together; its sign has invariant geometric significance. The local metric tensor alone does not fix global topology.

The coordinates share a cosmological, comoving construction: $r,\theta ,\varphi$ label the worldlines of the smooth cosmic fluid, and $t$ is the common cosmic time measured as proper time along those worldlines. Observers following that fluid keep their spatial labels while the scale factor changes their physical separations. The coordinate $t$ itself increases; it is not another constant spatial label. Thus the question's final phrase is understood as asking for their common adaptation to the cosmological fluid, rather than suggesting that all four coordinates are time independent or have the same units.

A fundamental observer follows the coarse-grained cosmological matter, with fixed comoving coordinates. The cosmological principle asserts spatial homogeneity and cosmological isotropy on large scales for this family of observers. A moving observer generally sees a preferred direction, for example a Doppler dipole in the cosmic microwave background, even when the underlying cosmology is isotropic.

In an FLRW metric, these worldlines are orthogonal to homogeneous spatial slices. Synchronize their clocks to define cosmic time $t$, which is their proper time. Along a fixed spatial label, $d{s}^2=c^{2}d{t}^2$, giving $g_{tt}=c^2$. Orthogonality of the spatial coordinate directions to the observers' four-velocity gives $g_{ti}=0$. Thus, in the $+---$ metric signature,where the spatial matrix $g_{ij}$ is negative definite. Spatial homogeneity and cosmological isotropy further restrict it to $-a(t{)}^2{\gamma }_{ij}(\mathbf{x})$, with $\gamma$ a constant-curvature spatial metric.

The scale factor $a(t)>0$ measures how spatial lengths in an FLRW metric change with cosmic time $t$. Choose $a$ to be a dimensionless quantity with $a(t_0)=1$ at a reference epoch, usually today. For two comoving observers, their fixed radial comoving distance $\chi$ and their proper distance in cosmology $d(t)$ satisfyThe proper distance in cosmology is measured along a spatial slice at one cosmic time; it is not the distance that a travelling photon covers between emission and reception. Only ratios of the scale factor have a normalization-independent meaning: replacing $a$ by $Ca$ and $\chi$ by $\chi /C$ leaves $d$ unchanged.

For example, take $\chi =100\mathrm{Mpc}$, where $\mathrm{Mpc}$ means a million parsecs. At $a=0.5,1,1.5$, the same pair of comoving observers has proper distance in cosmology $50,100,150\mathrm{Mpc}$ respectively. Every such separation changes by the same factor, although more distant pairs gain more distance. The illustration treats the marked objects as ideal comoving observers; internal motions in galaxies need not follow this background scaling.

The beauty of the scale factor is that one function describes the changing spatial ruler for every comoving observer. Spatial homogeneity gives no preferred centre of cosmic expansion: in a flat spatial patch, relative positions obey $\mathbf{r}-{\mathbf{r}}_O=a(\mathbf{x}-{\mathbf{x}}_O)$, so recentering on any comoving observer preserves Hubble's law. The evolution of the scale factor is determined by the Friedmann equations, while its ratio between emission and reception determines cosmological redshift.