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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 the velocity convention $v_{\gamma }=-{\Theta }_1$ and $\dot{\tau }<0$, momentum-conserving Thomson scattering gives ${\dot{v}}_b+\mathcal{H}{v}_b+k\psi =(\dot{\tau }/R)(v_b-v_{\gamma })$. Here $R$ is the baryon loading parameter. Its collision force is opposite to the photon force when both are weighted by their enthalpies.

A deliberately restricted forced harmonic oscillator shows two effects of the baryon loading parameter without duplicating the full photon Boltzmann hierarchy. Use conformal time $d\eta =dt/a$ in seconds, a comoving wavenumber $k$ in inverse metres, constant $R=3{\rho }_b/(4{\rho }_{\gamma })$, and equal constant dimensionless Newtonian gauge gravitational potentials $\Phi =\Psi \ne 0$. The energy densities ${\rho }_b$ and ${\rho }_{\gamma }$ use the same units. Neglect expansion drag, Silk damping and evolution of the potentials. The photon-baryon sound speed is $c_s=c/\sqrt{3(1+R)}$. Writing ${\Theta }_0$ for the photon monopole temperature perturbation, the photon-baryon acoustic oscillator reduces toFor $y=({\Theta }_0+\Psi )/\Psi$ this gives the displayed forced equation. Constant loading and the neglected expansion drag make this a local teaching model, not a self-consistent expanding cosmology with $R$ fixed forever.

Choose $y(0)=1/3$ and $y^{\prime }(0)=0$, the static matter-era Sachs-Wolfe combination normalized by $\Psi$. With $q=k{c}_s\eta$, direct substitution givesAt $R=0$, successive extrema have equal magnitudes $1/3$. At the stipulated $R=0.6$, $y(\pi )=-23/15$ whereas $y(2\pi )=1/3$. The shifted equilibrium $y=-R$ makes alternating extrema unequal. This is an elementary illustration of how the baryon loading parameter changes acoustic amplitudes; these extrema are neither measured peak heights nor a complete Cosmic microwave background power spectrum. Comparing curves at fixed $q$ isolates the response; at fixed $k$ and $\eta$, increasing $R$ also reduces the photon-baryon sound speed and hence the phase.

For example, holding $c{\eta }_{\ast }=300\mathrm{Mpc}$ gives sound horizons $r_s=173.2\mathrm{Mpc}$ at $R=0$ and $136.9\mathrm{Mpc}$ at $R=0.6$. Their ratio is $1/\sqrt{1.6}$. Each term of the forced equation has units of inverse seconds squared because $y$ and $R$ are dimensionless and $k{c}_s$ is an inverse time. In the actual photon-baryon fluid, $R$ grows with the scale factor and the sound horizon is the integral of the evolving photon-baryon sound speed.

The first mechanism is Silk damping: finite photon mean free paths allow radiation to diffuse out of compressions, while velocity slip and radiation shear dissipate acoustic motion. A Fourier diffusion equation has $X^{\prime }=-D(\tau )k^{2}X$, hence $X\propto \exp [-k^2\int Dd\tau ]$. For the polarization-inclusive tightly coupled plasma, a useful expression isThe positive scattering rate is $\dot{\kappa }=a{n}_e{\sigma }_T$, and $R$ is the baryon loading parameter. The two terms represent heat conduction from velocity slip and photon shear viscosity. The mode amplitude has an approximately Gaussian cutoff in $k$; power is suppressed by $e^{-2k^2/k_D^2}$. The formula assumes tight coupling and slowly varying coefficients before decoupling.

The second mechanism is finite-width last-scattering damping. Photons emerge from a range of times and depths, so rapidly varying sources are averaged rather than observed at one sharply defined phase. Approximate the normalized cosmological visibility function by a Gaussian of width ${\sigma }_{\tau }$. For a radial plane-wave phase,The associated power factor is $e^{-(k\mu {\sigma }_{\tau }{)}^2}$. An acoustic factor $e^{\pm ik{c}_s\tau }$ is likewise averaged, replacing $k\mu$ by $k(\mu \pm c_s)$ in the constant-sound-speed approximation. Thus fine radial structure and rapid acoustic time oscillations are suppressed once their phase changes substantially across the visibility width. This is an averaging effect after the perturbations exist; Silk damping physically erases them by diffusion. Angular projection and the evolving source determine the final isotropic damping envelope, so the radial $\mu$ dependence must not be silently discarded.

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.

An initially adiabatic cosmological perturbation leaves the local photon-to-baryon number ratio unchanged. Since radiation density scales as the fourth power of temperature and nonrelativistic particle number as the third, its density contrasts obey ${\delta }_b=3{\delta }_{\gamma }/4$. With the corrected common-velocity continuity equations,so the adiabatic relation persists in tight coupling.

Use the paper's ratio $R=4{\bar{\rho }}_{\gamma }/(3{\bar{\rho }}_b)$, which is the reciprocal of the commonly used baryon loading parameter. Multiply the photon Euler equation by $R$ and add the baryon Euler equation. The collision forces cancel exactly:This cancellation works even with the common erroneous $k^{-2}$ in the printed forces, but a physical collision rate uses the repaired forces in part (i). At leading tight-coupling order put ${\theta }_{\gamma }={\theta }_b=\theta$ and substitute ${\delta }_b=3{\delta }_{\gamma }/4$. ThenDefine the pressure-to-inertia sound speed, neglecting the very small baryon thermal pressure, byThe combined velocity equation consequently becomesThis is the requested photon-baryon acoustic oscillator system. Photon pressure supplies the restoring force, while both photon enthalpy and baryon mass supply inertia. The retained $c_{sb}^2$ term adds the small thermal-baryon correction rather than being part of the quoted leading $c_s^2$.

In the decoupled perfect fluid in general relativity limit, photons have $w=1/3$ and no anisotropic stress, while cold baryons have $w\simeq 0$ with a small pressure correction $c_b^2{\delta }_b$. The density equations express particle and energy conservation; Thomson scattering transfers momentum rather than leading-order rest-frame energy. The photon Euler equation consequently gains a drag toward the baryon velocity potential, and the baryon Euler equation gains the equal-and-opposite drag weighted by their inertias.

Let ${\Gamma }_T=a{n}_e{\sigma }_T>0$. With the velocity potential used in part (i), these forces are $-{\Gamma }_T({\theta }_{\gamma }-{\theta }_b)$ and $-R{\Gamma }_T({\theta }_b-{\theta }_{\gamma })$, with $R=4{\bar{\rho }}_{\gamma }/(3{\bar{\rho }}_b)$. Multiplication by the photon and baryon enthalpies, $4{\bar{\rho }}_{\gamma }/3$ and ${\bar{\rho }}_b$, makes their sum zero. Note that this paper's $R$ is the reciprocal of the usual baryon loading parameter. Thomson scattering also relaxes the photon quadrupole, giving the $-{\Gamma }_T{\sigma }_{\gamma }$ term in the stipulated simplified hierarchy. This quadrupole is an anisotropic stress, omitted by a perfect-fluid model; finite relaxation creates Silk damping.

The printed baryon density sign and collision normalization are inconsistent with part (i). The density equation must be ${\delta }_b^{\prime }-k^2{\theta }_b=-h^{\prime }/2$. As a direct check, with $h^{\prime }=0$, nonzero common velocity, and initially ${\delta }_{\gamma }=4{\delta }_b/3$, the printed plus sign gives $({\delta }_{\gamma }-4{\delta }_b/3{)}^{\prime }=8k^2\theta /3$, violating the tight-coupling adiabatic condition immediately. Likewise division of the collision forces by $k^2$ belongs to neither this potential convention nor the usual common velocity-divergence convention; the physical relaxation rate is ${\Gamma }_T$, independent of $k$. Both printed drag terms still cancel in the weighted sum used below, so that particular algebraic combination is unaffected by this normalization error.

For a nonrelativistic distribution, $T^{ij}=\int d^{3}p{p}^i{p}^{j}f/E$ is down by $v^2$ relative to the rest-mass energy density, and the corresponding anisotropic stress is also suppressed. The streaming terms coupling successive angular moments are proportional to $kv$, and stresses from the velocity distribution disappear in the cold limit. Thus, on scales where velocity dispersion is negligible, density and bulk velocity suffice; the photon Boltzmann hierarchy has no analogous small-speed suppression. This argument needs a cold distribution or a controlled small-dispersion approximation. Collisionless nonrelativistic matter with appreciable dispersion or multistreaming need not be an exact perfect fluid, and near photon decoupling the higher photon moments cannot all remain negligible.

Throughout this question dots denote conformal time derivatives, and $\mathcal{H}=\dot{a}/a$ is the conformal Hubble parameter. The monopole of the photon Boltzmann hierarchy gives ${\dot{\Theta }}_0+k{\Theta }_1/3=\dot{\varphi }$. With the specified velocity convention, the photon continuity equation is thereforeThe dipole equation is ${\dot{\Theta }}_1+k(2{\Theta }_2/5-{\Theta }_0)=\dot{\tau }({\Theta }_1+v_b)+k\psi$. Thus the photon Euler equation isIn particular, $\dot{\tau }<0$ makes the scattering term damp the photon-baryon velocity slip; it does not amplify it.

The linear momentum density of the photon-baryon fluid is weighted by enthalpy, not merely energy density:Elastic Thomson scattering exchanges momentum internally. Its photon force density is $(4{\bar{\rho }}_{\gamma }/3)\dot{\tau }(v_{\gamma }-v_b)$, so the baryon force density is its negative. Dividing by ${\bar{\rho }}_b$ gives the baryon Euler equation with Thomson dragThis $R$ is the baryon loading parameter, the baryon-to-photon inertia ratio. The two collision terms cancel exactly when the equations are weighted by their enthalpies. Background conservation gives ${\bar{\rho }}_b\propto a^{-3}$ and ${\bar{\rho }}_{\gamma }\propto a^{-4}$, so $\dot{R}=\mathcal{H}R$.

With $v_b=v_{\gamma }$ at leading order and $c_s^2=1/[3(1+R)]$, the photon continuity equation and total Euler equation give ${\ddot{\delta }}_{\gamma }+\dot{R}{\dot{\delta }}_{\gamma }/(1+R)+k^2{\delta }_{\gamma }/[3(1+R)]=4\ddot{\varphi }+4\dot{R}\dot{\varphi }/(1+R)-4k^2\psi /3$. The baryon loading parameter changes inertia and shifts the gravitational equilibrium. The leading equations omit diffusion damping.