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The angular sound horizon compares two lengths using the same distance convention. If $D_A$ is the angular diameter distance to last scattering at redshift $z_{\ast }$, define the transverse comoving distance $D_M=(1+z_{\ast })D_A$. A comoving sound horizon $r_s$ has proper length $r_s/(1+z_{\ast })$ at emission, so its angular size is ${\theta }_{\ast }=r_s/D_M$. Mixing a comoving ruler with $D_A$ would introduce an erroneous factor $1+z_{\ast }$. In a curved FLRW metric, $D_M$ is generally different from the radial comoving distance.

For a stipulated example, $r_s=144\mathrm{Mpc}$ and $D_M=14000\mathrm{Mpc}$ give ${\theta }_{\ast }=0.010286$ radians, or $0.589^{\circ }$, and an acoustic scale ${\ell }_A=305.4$. Multiplying both lengths by $1.1$ gives $158.4\mathrm{Mpc}$ and $15400\mathrm{Mpc}$ with exactly the same ${\theta }_{\ast }$ and ${\ell }_A$. This example establishes a ratio degeneracy, without claiming that any arbitrary rescaling corresponds to an allowed cosmological model. Equivalently,Other observables are needed to separate changes that preserve the ratio.

The approximate adiabatic acoustic-peak spacing is ${\ell }_A$. The first observed temperature peak is shifted by gravitational driving and angular projection of density and Doppler acoustic sources; it need not occur at ${\ell }_A$. In particular, neither the illustrative $305.4$ nor the BOOMERANG acoustic-peak measurement of $197\pm 6$ should be substituted for the other.

The cosmological baryon density enters through the baryon loading parameter and cosmological recombination, changing relative acoustic amplitudes as well as the sound horizon. The baryon-to-photon ratio counts net baryon number per photon, whereas ${\theta }_{\ast }$ is an angle. Constraining cosmological baryon density requires the pattern of relative peak heights within a model, with other parameters fitted jointly. Big Bang nucleosynthesis provides a separate abundance test. Baryon-density consistency between nucleosynthesis and the microwave background is consequently a test of one common cosmic history, rather than two direct measurements of the same angle. A first peak by itself cannot determine every component of the cosmological density parameter or uniquely fix spatial curvature of an FLRW universe.

For a uniform spherical cluster of depth $\ell \propto \theta d_A$, thermal bremsstrahlung gives received bolometric flux $S_X\propto n_e^2{T}_e^{1/2}{\ell }^3/d_L^2$, while the Rayleigh-Jeans law limit of the thermal Sunyaev-Zeldovich effect gives $|\Delta T|\propto n_e{T}_e\ell$. Eliminating $n_e$ and using $d_L=(1+z{)}^2{d}_A$ yields the displayed angular diameter distance. A model for the density profile fixes its numerical coefficient. Clumping, nonsphericity and temperature structure can bias this distance because the two observables weight the gas differently.

Gravitational lensing constrains a galaxy cluster's projected mass independently of its light output or dynamical equilibrium. The lensing convergence is projected surface mass density divided by ${\Sigma }_{\mathrm{crit}}=c^2{D}_s/(4\pi G{D}_d{D}_{ds})$, using the appropriate angular diameter distances. Strong gravitational lensing constrains the central mass through multiple-image geometry, and weak gravitational lensing constrains larger-scale projected mass through coherent image distortions. Converting these observations into a three-dimensional mass needs geometrical modelling and the relevant lensing degeneracies.

For a flat universe with a single constant equation-of-state parameter $w>-1$, $H(z)=H_0(1+z{)}^{\alpha }$. Its angular diameter distance is $H_0^{-1}[(1+z{)}^{-\alpha }-(1+z{)}^{-1}]/(1-\alpha )$, with logarithmic limit $H_0^{-1}\log (1+z)/(1+z)$ at $\alpha =1$. Differentiation gives the displayed unique maximum, interpreted as $z_m=e-1$ at $\alpha =1$. The distance vanishes at both endpoints $z=0$ and $z\to \infty$. As $w\downarrow -1$, the maximum moves to infinite cosmological redshift.

The Einstein angular radius is the angular form of the Einstein radius: $R_E=D_l{\theta }_E$ in the lens plane. Its displayed value applies to a point-mass gravitational lens with the angular diameter distances defined above. It depends on source distance as well as lens mass and position. At fixed distances, multiplying $M$ by four doubles ${\theta }_E$.

At exact alignment, the vector thin gravitational lens equation is $0=(1-{\theta }_E^2/r^2)\mathbf{\theta }$. For a nonzero image direction it requires $r={\theta }_E$ and leaves the direction unrestricted: all azimuths form an Einstein ring. The signed equation's roots $\theta =\pm {\theta }_E$ are only a diameter cut through that circle. A source of finite angular size makes an annulus or arcs with finite received light; the infinite point-source lensing magnification at alignment is an idealization. An Einstein ring expresses rotational symmetry and source alignment, not a circular arrangement of emitting stars.

A horizon-length patch on the matter-radiation equality surface has physical size $a_{\mathrm{eq}}{\tau }_{\mathrm{eq}}$ in a flat universe with conformal time measured from the Big Bang. Dividing by its angular diameter distance $a_{\mathrm{eq}}({\tau }_0-{\tau }_{\mathrm{eq}})$ gives its small angular size. For the radiation-matter scale factor in conformal time, this is $(\sqrt{2}-1)\sqrt{{\Omega }_{r,0}}/[1-\sqrt{2{\Omega }_{r,0}}]$. A patch diameter is twice the angle associated with one horizon radius. The particle horizon is an integral over past evolution and must not be silently replaced by the instantaneous Hubble radius.

At fixed conformal time and fixed radial coordinate, the metric gives transverse physical separation $D=a({\tau }_1)r_1\theta$ to first order in the angle. Thus the angular diameter distance isThe coordinate $r_1$ is the transverse comoving radius appearing in the angular metric coefficient. It is not the radial geodesic distance $r_{\mathrm{curv}}\mathrm{arsinh}(r_1/r_{\mathrm{curv}})$; replacing one by the other would omit the curvature dependence of angular projection.

Divide the physical sound radius by the angular diameter distance $a_{\ast }{r}_{\ast }$. The sound-horizon angle in an open matter universe isThe characteristic acoustic multipole is approximately half a wavelength across this angle:Thus negative curvature moves the acoustic feature to smaller angular scales and larger multipoles. The first temperature peak is of order $200/\sqrt{{\Omega }_m}$ in this approximation; precise peak positions include the sound-speed history, radiation effects and acoustic phase shifts, so the acoustic-scale multipole is not an exact peak-location formula.

The cosmological perfect-fluid continuity equation gives $\rho \propto a^{-3(1+w)}$. In a flat universe the Friedmann equation then gives $H(z)=H_0(1+z{)}^{\alpha }$, where $\alpha =3(1+w)/2>0$. Writing $u=1+z$, the angular diameter distance obtained from the luminosity distance isIt is positive for $u>1$, starts at zero and tends to zero as $u\to \infty$ for every $\alpha >0$, including $\alpha <1$ where the radial comoving distance itself is unbounded. Differentiation gives the constant-equation-of-state angular diameter distance maximum:For $\alpha \ne 1$ the stationary equation is $u^{\alpha -1}=\alpha$, which has exactly one solution greater than one. For $\alpha =1$ the derivative has the sign of $1-\log u$. Hence this stationary point is always the unique maximum.

To determine the sketch's shape, differentiate $\log (1+z_m)=\log \alpha /(\alpha -1)$:with limiting value $-1/2$ at $\alpha =1$. The inequality follows because $\log \alpha -1+1/\alpha$ has its unique minimum zero at one. Thus $z_m$ decreases from infinity as $w\downarrow -1$ to $\sqrt{3}-1$ as $w\uparrow 1$. Useful points are $z_m=e-1$ at $w=-1/3$, $5/4$ at $w=0$, and $1$ at $w=1/3$. The endpoint $w=-1$ is excluded: there $d_A=z/[H_0(1+z)]$ increases to an asymptote and has no finite turning point.

Use the flat matter-only Einstein-de Sitter approximation for the distance-redshift relation, neglecting radiation corrections at cosmological recombination. Let $z_{\ast }$ be the cosmological recombination redshift. Adiabatic photon cooling gives $1+z_{\ast }=T_{\ast }/T_0$. The Hubble rate is $H(z)=H_0(1+z{)}^{3/2}$, so the comoving radial distance to last scattering isThe angular diameter distance is $D_A={\chi }_{\ast }/(1+z_{\ast })$. A proper Hubble radius $H_{\ast }^{-1}=H_0^{-1}(1+z_{\ast }{)}^{-3/2}$ therefore subtends the small angleThis is the angular Hubble-radius scale in an Einstein-de Sitter universe; the present Hubble constant cancels.

Taking $T_{\ast }=0.3\mathrm{eV}$ and the literal printed $T_0=2.35\mathrm{meV}$ gives $1+z_{\ast }=127.7$ and ${\theta }_H\simeq 0.0486$ radians, or $2.8$ degrees. As noted above, that present CMB energy is too large by ten. With the corrected $T_0=0.235\mathrm{meV}$, $1+z_{\ast }=1277$ andwhich gives the familiar order-one-degree scale within the assumed cosmology. Using $0.28\mathrm{eV}$ to mark a few-per-cent ionization fraction changes this only modestly. These numbers refer to the Hubble-radius scale, not its diameter. In a matter-only universe the particle-horizon radius is $2H^{-1}$ and would give twice the angle; the acoustic sound horizon is a different length again.

Use the small-angle, thin gravitational lens equation geometry. An undeflected ray seen at angular position $\theta$ would have transverse source position $D_s\theta$. Bending that ray through physical angle $\hat{\alpha }$ at the lens changes its source intercept by $D_{ds}\hat{\alpha }$. Thusand the required physical deflection isAlso $\beta =\eta /D_s$. Define the reduced deflection $\alpha$ as the angular difference $\theta -\beta$. Dividing the first equation by $D_s$ givesThese are angular diameter distances; in a cosmological spacetime one must not replace $D_s$ by $D_d+D_{ds}$ as if the distances were ordinary collinear Euclidean lengths.

Separate the observables before combining their parameter constraints. The baryon fraction is included within total matter: for the benchmark, ${\Omega }_{\mathrm{CDM}}={\Omega }_m-{\Omega }_b=0.255$, rather than adding ${\Omega }_b$ to ${\Omega }_m$ a second time. With $h=H_0/(100\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1})=0.7$, the physical density combinations areFor baryons, Big Bang nucleosynthesis predicts primordial light-element abundances as a function of the baryon-to-photon ratio; deuterium is especially sensitive. Independently, baryon inertia changes the relative heights of compressional and rarefaction peaks in the cosmic microwave background through the baryon loading parameter. Agreement of these two physical-density estimates supports a baryon abundance near the quoted value. For example, the three-year WMAP cosmological analysis found ${\Omega }_b{h}^2=0.02229\pm 0.00073$ in its fitted model.

For total matter, matter-radiation equality affects the microwave-background peak pattern and fixes a characteristic turnover in the matter power spectrum. galaxy clustering therefore constrains matter density, with corrections for galaxy bias and peculiar-velocity distortions. Cluster baryon fractions combine observed gas and stars with total masses inferred dynamically or by gravitational lensing; comparison with the cosmic baryon fraction also constrains ${\Omega }_m$. Galactic rotation, cluster dynamics and gravitational lensing give independent evidence that visible baryons alone cannot supply the gravitating mass. The precise ${\Omega }_m$ estimate comes from quantitative joint fits, rather than from a rotation curve by itself. Different 2006 data combinations gave somewhat different best fits, so $0.3$ is a rounded benchmark rather than a universal exact fit.

For a cosmological constant, standardized high-redshift Type Ia supernovae measure the shape of the luminosity-distance relation. In a flat matter-vacuum model,The observed supernova dimming relative to simple decelerating matter models favours late accelerated expansion. The original evidence is given by Riess and collaborators' 1998 supernova analysis. Combining the distance curve with matter constraints selects a positive vacuum fraction close to the quoted benchmark; it gives $q_0={\Omega }_m/2-{\Omega }_{\Lambda }=-0.55$.

For curvature, the angular scale of the Cosmic microwave background acoustic peaks compares the sound horizon with the angular diameter distance to last scattering. This strongly constrains spatial geometry, but curvature and late-time expansion parameters can partly compensate one another. Independent distances, notably baryon acoustic oscillations and supernova distances, help break that degeneracy. The 2005 SDSS acoustic-peak detection supplies a low-redshift standard ruler complementary to the microwave-background ruler. In combination, these observations support near-flat geometry and the closure relation ${\Omega }_m+{\Omega }_{\Lambda }+{\Omega }_k\simeq 1$.

The benchmark is supported by mutually constraining abundance, clustering, angular-scale and distance measurements. Its uncertainties are correlated and depend on model assumptions. An exactly zero curvature parameter has no meaningful percentage error; observational evidence establishes consistency with zero and an absolute uncertainty, not a literal fractional accuracy better than ten per cent on zero.

A comoving coordinate labels a position relative to the homogeneous expansion. For two nearby comoving positions separated by $\Delta \chi$, their proper distance in cosmology at time $t$ is $a(t)\Delta \chi$. Thus a fixed comoving separation expands physically with the scale factor, while a fixed proper separation corresponds to a shrinking comoving separation. Normalize $a_0=1$, so $a(z)=1/(1+z)$.

At a common cosmological redshift, a transverse proper size $l$ subtends $\theta \simeq l/D_A$, where $D_A$ is the angular diameter distance. At $z=3$, a $5$ comoving Mpc separation has proper size $5/(1+3)=1.25$ Mpc. HenceThe proper-size object has the larger angular size, by a factor of four. This comparison does not require a numerical angular diameter distance, since it cancels.

Cosmic shear is the small, statistically correlated distortion of distant galaxy images by weak gravitational lensing from intervening large-scale matter inhomogeneities. An individual galaxy has an unknown intrinsic shape, so the signal is measured through ensemble shape correlations rather than by assuming every source is circular. The angular and redshift dependence of these correlations probes the projected matter fluctuations, their growth and the angular diameter distance relation, constraining dark matter, cosmological density parameters and dark energy. Gravitational lensing responds to all gravitating matter rather than just luminous tracers. Intrinsic alignments and shape-measurement biases must be modeled separately from the lensing signal; the physical and statistical basis is described in arxiv.org/abs/1411.0115 .

An object at matter-radiation equality lies at comoving radial distance $\chi ={\tau }_0-{\tau }_{\mathrm{eq}}$ in the flat FRW metric. Its angular diameter distance is $D_A=a_{\mathrm{eq}}\chi$, while one particle horizon radius has physical size $a_{\mathrm{eq}}{\tau }_{\mathrm{eq}}$. Its small angular radius is consequentlyThe last step uses ${\tau }_{\mathrm{eq}}\ll {\tau }_0$. If ${\Omega }_{\Lambda }/{\Omega }_m\gg 1$, the matter-vacuum conformal age integral tends to $I$: its integrand behaves as $y^{-1/2}$ at zero and $y^{-2}$ at infinity. Inserting the two conformal times givesThe negative matter exponent follows directly from ${\Omega }_m^{-1}/{\Omega }_m^{-1/3}$. This agrees with the original PDF; the TeX transcription drops the minus sign. If desired, $z=y^3$ evaluates the constant through the beta function and Gamma function:The infinite-upper-limit approximation requires vacuum dominance much stronger than in the illustrative parameters of part (a), so it is an asymptotic formula rather than a precision angular prediction for those values.

The distance modulus gives $m_V-M_V=5{\log }_{10}(d/10\mathrm{pc})$. At the magnitude limit, $m_V-M_V=17-(-21)=38$, soThus the stated sample reaches a typical galaxy to approximately $400$ megaparsecs. This uses the assumed absolute magnitude, neglects the specified spectral correction, and equates luminosity distance, angular diameter distance, and proper distance as permitted. It does not imply completeness for galaxies fainter than the adopted typical luminosity.

The physical radius comes from the angular diameter distance, not the luminosity distance. In the specified flat cosmology,Using the supplied distance-integral approximation at $z=1$ and $c\simeq 3.00\times 10^5\mathrm{km}{\mathrm{s}}^{-1}$ gives $D_A\simeq (c/70)(0.8/2)=1714\mathrm{Mpc}$. The angular radius in radians is $(7/12)\pi /(180\times 3600)=2.828\times 10^{-6}$. ThusThis is a proper radius at the observation epoch; its comoving counterpart would be twice as large. The supplied approximate integral is sufficient here; integrating the specified Hubble parameter numerically gives a slightly smaller radius, about $4.82\mathrm{kpc}$.

The cosmological perfect-fluid continuity equation gives ${\rho }_r={\rho }_{r,0}{a}^{-4}$ and ${\rho }_m={\rho }_{m,0}{a}^{-3}$. In conformal time, $\dot{a}=a^{\prime }/a$ and $H=a^{\prime }/a^2$. Thus the flat Friedmann equation isChoose the expanding branch and put the Big Bang at $\tau =0$. Integration givesInverting it supplies the radiation-matter scale factor in conformal time:The early linear term gives radiation domination, while the late quadratic term gives matter domination. The solution presumes the stated radiation-plus-matter model, without a cosmological constant.

At matter-radiation equality, $a_{\mathrm{eq}}={\Omega }_{r,0}/{\Omega }_{m,0}$. Since ${\Omega }_{r,0}+{\Omega }_{m,0}=1$,In units $c=1$, the comoving particle horizon at equality is ${\tau }_{\mathrm{eq}}$; its physical size is $a_{\mathrm{eq}}{\tau }_{\mathrm{eq}}$. The angular diameter distance to the equality surface is $a_{\mathrm{eq}}({\tau }_0-{\tau }_{\mathrm{eq}})$. Therefore the horizon angle at matter-radiation equality corresponding to one horizon length isFor ${\Omega }_{r,0}=10^{-4}$, this is $4.20\times 10^{-3}$ radians, or $0.241^{\circ }$, about $14.4$ arcminutes. If quoting the full diameter of a horizon-sized patch, the result is twice this, $0.481^{\circ }$. The small-angle approximation is excellent. Expanding in $\sqrt{{\Omega }_{r,0}}\ll 1$ would give $4.14\times 10^{-3}$ radians; retaining both cosmic components near equality avoids an inaccurate pure-matter extrapolation there. The scale factor cancels between physical size and angular diameter distance, and $H_0$ cancels from the ratio.

If cosmological recombination occurs only a modest expansion after equality, its causal patch still subtends a small part of the sky. Widely separated regions of the observed cosmic microwave background have nearly the same temperature even though their past particle horizons did not overlap in the ordinary decelerating history. Local thermalization after the Big Bang therefore cannot explain this large-angle uniformity. This is the horizon problem; inflation supplies an earlier connected region that can grow to encompass the observed sky.