Spatial geometry. An embedding of the three-sphere is

w = A cos χ, z = A sin χ cos θ, x = A sin χ sin θ cos ϕ, y = A sin χ sin θ sin ϕ .

Taking differentials and collecting terms gives the induced metric tensor

d ℓ^{2} = A^{2} d χ^{2} + A^{2} sin^{2} χ (d θ^{2} + sin^{2} θ d ϕ^{2}) .

(2)

Put

\overset{r}{ˉ} = A sin χ

. Then

d \overset{r}{ˉ} = A cos χ d χ

, so

d ℓ^{2} = \frac{d r ˉ ^{2}}{1 - r ˉ ^{2} / A ^{2}} + \overset{r}{ˉ}^{2} d Ω^{2}, d Ω^{2} = d θ^{2} + sin^{2} θ d ϕ^{2} .

(3)

The areal radius

\overset{r}{ˉ}

is not a global radial coordinate on the entire three-sphere: its maximum occurs at the equator and two hemispheres share each smaller value. Use separate charts there; the coordinate singularity at

\overset{r}{ˉ} = A

is not a physical singularity.

For Euclidean space and hyperbolic space of curvature

- 1/ A^{2}

, respectively,

d ℓ_{0}^{2} = d \overset{r}{ˉ}^{2} + \overset{r}{ˉ}^{2} d Ω^{2}, d ℓ_{-}^{2} = \frac{d r ˉ ^{2}}{1 + r ˉ ^{2} / A ^{2}} + \overset{r}{ˉ}^{2} d Ω^{2} .

(4)

For the latter,

\overset{r}{ˉ} = A sinh χ

gives

d ℓ_{-}^{2} = A^{2} [d χ^{2} + sinh^{2} χ d Ω^{2}]

. In all three cases the induced two-metric at fixed

\overset{r}{ˉ} = \overset{ˉ}{R}

\overset{ˉ}{R}^{2} d Ω^{2}

. Its surface area is therefore

A (\overset{ˉ}{R}) = \int_{0}^{2 π} d ϕ \int_{0}^{π} \overset{ˉ}{R}^{2} sin θ d θ = 4 π \overset{ˉ}{R}^{2} .

(5)

This equality compares the same areal radius, not the same proper radial distance. For a proper radial distance

ℓ

, the corresponding areas are

4 π A^{2} sin^{2} (ℓ / A)

4 π ℓ^{2}

, and

4 π A^{2} sinh^{2} (ℓ / A)

Homogeneity, isotropy and redshift. The FRW metric has constant-time spatial slices of constant curvature

k / a^{2} (t)

, with

k = + 1, 0, - 1

selecting the three geometries above. Their isometries act transitively on points and isotropically about every point, implementing the cosmological principle. Multiplication by one common time-dependent scale factor preserves these properties. With the source's

k = \pm 1

normalization,

r

is dimensionless and

a

has units of length; using a dimensionless scale factor instead rescales the spatial curvature parameter. The origin of the spherical coordinates is arbitrary; it does not identify a physical centre. Cosmic time is the proper time of the comoving observers, and the absence of time-space cross terms makes their worldlines orthogonal to the homogeneous slices.

For a radial null geodesic between comoving endpoints,

\int_{t_{E}}^{t_{0}} \frac{c d t}{a ( t )} = \int_{0}^{R} \frac{d r}{1 - k r ^{2}} .

(6)

The next wave crest travels between the same endpoints. Subtracting the two path integrals, for wave periods short compared with the expansion time, gives

δ t_{0} / a_{0} = δ t_{E} / a_{E}

. Thus the cosmological redshift is

1 + z = \frac{λ _{0}}{λ _{E}} = \frac{ν _{E}}{ν _{0}} = \frac{a _{0}}{a _{E}} .

(7)

Both observers' clocks are cosmic proper-time clocks. Peculiar velocities and local gravitational-potential shifts have been left out of this homogeneous calculation.

Angular size and brightness. For a galaxy small compared with its distance, the angular diameter distance is

D_{A} = a_{E} R

and its angular diameter is

Δ θ ≃ \frac{d}{a _{E} R} = \frac{( 1 + z ) d}{a _{0} R} .

(8)

Here the emitted proper diameter is measured transverse to the line of sight; radial proper distance is a different integral when

k \neq = 0

. Ignoring absorption, lensing caustics and intrinsic evolution, photon-number conservation spreads an isotropic-equivalent luminosity across the sphere

4 π a_{0}^{2} R^{2}

. Each photon loses energy by

1 + z

, and its arrival interval is stretched by the same factor. The received bolometric flux is

F = \frac{L}{4 π a _{0}^{2} R ^{2} ( 1 + z ) ^{2}}, D_{L} = (1 + z) a_{0} R = (1 + z)^{2} D_{A} .

(9)

The disc's proper projected area is

A_{d} = π d^{2} /4

, and its observed solid angle is

Ω_{d} ≃ A_{d} / D_{A}^{2}

. Therefore its surface brightness is

I_{obs} = \frac{F}{Ω _{d}} = \frac{L}{4 π A _{d}} (1 + z)^{- 4} = \frac{L}{π ^{2} d ^{2}} (1 + z)^{- 4} .

(10)

This is cosmological surface-brightness dimming. The normalization uses

L

as isotropic-equivalent luminosity along the viewing direction; another disc emission law changes that normalization but not the redshift power. This is a bolometric result. In a specified observed frequency band the emitted spectrum is sampled at a shifted frequency and requires a spectral correction.

The distinction in telescope aperture and resolved surface brightness matters here. A larger telescope collects more photons and improves angular resolution, but cannot restore the incoming radiance lost by cosmological dimming. For aperture diameter

D

and a diffraction resolution element of solid angle

Ω_{res} \sim (λ / D)^{2}

, the collected power from uniform resolved brightness scales as

I_{obs} A_{tel} Ω_{res} \propto I_{obs} λ^{2},

(11)

independent of

D

. This is consistent with conservation of etendue. It explains why increasing aperture while keeping every gain in resolution does not increase counts per native resolution element from a smooth extended source. It does not make larger telescopes useless: at a fixed angular bin the counts grow as

D^{2}

, so a photon-noise-limited or uniform-sky-noise-limited signal-to-noise ratio grows roughly as

D t_{exp}

. They also resolve smaller structures and improve point-source sensitivity. Aperture, exposure and angular binning help measure a dim galaxy, without changing its intrinsic