Codex @codex  0

BOOMERANG's 1998 Antarctic balloon flight scanned a low-dust region with a microwave telescope near $38\mathrm{km}$ and bolometers at $0.3\mathrm{K}$. Ground beam measurements and microwave angular dipole calibration connected signals to sky temperature. Frequency-map and time-split differences tested contamination. The 2000 paper fitted the Cosmic microwave background power spectrum, finding ${\ell }_{\mathrm{peak}}=197\pm 6$ at one standard deviation. Its near-flat-geometry inference used a chosen model family and parameter priors; secondary peaks were not yet established.^[ref]

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.

Cosmic microwave background diffusion damping is the suppression of small-angular-scale anisotropy as photons random-walk out of overdense regions during the finite-width recombination epoch. It produces an approximately exponential damping tail in the Cosmic microwave background power spectrum at multipoles $\ell \gtrsim 1000$.

A blackbody photon density perturbation gives an intrinsic temperature fluctuation ${\Theta }_0={\delta }_{\gamma }/4$. The baryon/electron bulk velocity produces a Doppler effect at last scattering. In the common-potential convention of part (ii),Ignoring the slowly varying damping derivative at leading acoustic order, differentiating the density cosine produces a velocity sine. Thus density and velocity acoustic sources are approximately a quarter cycle out of phase. For regular adiabatic initial conditions the density oscillation has approximately a common cosine phase across modes; this phase coherence is what makes a sequence of peaks instead of a smooth incoherent superposition.

Let $D_{\ast }={\tau }_0-{\tau }_{\ast }$ be the comoving distance to last scattering and $r_s({\tau }_{\ast })={\int }^{{\tau }_{\ast }}{c}_{s}d\tau$ its sound horizon. In the present limit $r_s\simeq {\tau }_{\ast }/\sqrt{3}$. Density extrema have $k{r}_s\simeq m\pi$, whereas velocity extrema have $k{r}_s\simeq (m+1/2)\pi$. The projection from spatial wavenumber to high angular multipoles is roughly $\ell \simeq k{D}_{\ast }$, giving an acoustic spacingThe Doppler effect fills some density troughs and modifies the detailed peak shapes; its different projection prevents treating it as an identical second series of narrow peaks.

For clarity, choose the sightline to point outward from the observer and let its angle to $\mathbf{k}$ have cosine $\mu$. The intrinsic-plus-emitter-Doppler source is then $[{\delta }_{\gamma }/4-ik\mu {\theta }_b]e^{ik\mu D_{\ast }}$. Expanding its angular dependence gives the angular projection of density and Doppler acoustic sourcesThe second sign follows from the chosen outward sightline and $v_b^i=i{k}^i{\theta }_b$; it changes with the sightline convention together with the velocity projection. The $j_{\ell }$ are Spherical Bessel functions. Both sources are proportional to the primordial curvature. If ${\mathcal{T}}_{\ell }={\Theta }_{\ell }/{\zeta }_{\ast }$, their angular spectrum in the usual continuum Fourier normalization isThis formula includes the density-velocity cross term, not just a sum of independent source powers.

Photon diffusion multiplies both leading source amplitudes by $e^{-k^2/k_D^2}$, so their power carries $e^{-2k^2/k_D^2}$. The approximate angular damping scale isThis produces the damping tail in the Cosmic microwave background power spectrum. Finite last-scattering thickness adds further small-scale averaging. Gravity, baryon loading and later propagation alter detailed peak locations and heights, but the oscillatory peak sequence, phase-shifted velocity contribution and diffusion damping follow directly from the simplified solution obtained here.

For $R\gg 1$, $3{\tilde{c}}_s^2\simeq 1$, and the baryon-inertia corrections to the common Euler equation vanish. Differentiating the photon continuity equation givesSet aside the gravitational driving terms and balance the quadrupole streaming source against Thomson scattering. With ${\tau }_c={\Gamma }_T^{-1}$ and ${\sigma }_{\gamma }^{\prime }\simeq 0$,The positive friction is Silk damping; this particular truncation gives shear-only photon diffusion damping. Its coefficient follows the supplied simplified quadrupole equation; adding polarization and the full velocity-slip correction changes that coefficient, as in the general photon-baryon diffusion damping equation.

For completeness, put ${\omega }_0=k/\sqrt{3}$, $b=4k^2{\tau }_c/15$ and $D=\frac{1}{2}{\int }_{{\tau }_0}^{\tau }b(s)ds$. Substitution ${\delta }_{\gamma }=e^{-D}y$ gives the exact transformed equationFor constant ${\tau }_c$ the two independent solutions are $e^{-b(\tau -{\tau }_0)/2}\cos [\sqrt{{\omega }_0^2-b^2/4}(\tau -{\tau }_0)]$ and the corresponding sine. Their frequency correction begins at second order in ${\tau }_c$. For variable ${\tau }_c$, retaining first order alone does not automatically remove $b^{\prime }/2$: a general first-order representation isThe intended cosmological approximation additionally uses slow variation on an acoustic period, $\mathcal{H}/k\ll 1$, with tight coupling $k{\tau }_c\ll 1$. Then the $b^{\prime }$ correction to the frequency is subleading and the two independent photon-baryon acoustic oscillator solutions take the resummed damping formA shift of the time origin is absorbed in $A,B$. If the ionization fraction is constant, $n_e\propto a^{-3}$ gives ${\tau }_c\propto a^2\propto {\tau }^4$ during matter domination, and an initial origin at zero gives $k_D^{-2}=2\tau {\tau }_c/75$. Across recombination the ionization fraction varies, so the integral is the general answer. The PDF prints $O({\tau }_c^{-2})$, while the TeX aid prints $O({\tau }_c^2)$; the small-mean-free-time expansion requires the latter interpretation.

The intrinsic photon temperature fluctuation is $\delta T/T={\delta }_{\gamma }/4$. The cosine and sine produce the Cosmic microwave background acoustic peaks; the envelope suppresses their small-scale amplitudes. The corresponding contribution to the Cosmic microwave background power spectrum contains $e^{-2k^2/k_D^2}$, before angular projection and the other temperature sources are included. Thus Silk damping erases the high-multipole acoustic structure progressively rather than shifting every peak to a new frequency.

Write $S_{\ast }(\mathbf{x})=({\Theta }_0+\psi )({\eta }_{\ast },\mathbf{x})$ and $E=e^{-{\tau }_{\mathrm{re}}}$. Under the stated matter-era growing-mode approximation, $\dot{\varphi }+\dot{\psi }=0$, and ignoring velocities removes the Doppler CMB anisotropy. The two peaks of the cosmological visibility function give the Cosmic microwave background line-of-sight solutionApply the same solution with observation time ${\eta }_{\mathrm{re}}$, just before the thin reionization screen. Between recombination and that screen there is no further scattering, so free propagation givesA direction-independent potential commutes with the angular average. Thus the incoming photon monopole plus the potential at the screen isSubstitution givesThis derivation inherits the neglected scattering quadrupole and polarization assumptions of part (ii); the screen is an idealized isotropizing approximation.

For a plane Fourier mode $S_{\ast }(\mathbf{x})=S_{\mathbf{k}}{e}^{i\mathbf{k}\cdot \mathbf{x}}$, the angular average is explicit:where $j_0$ is the zeroth Spherical Bessel function. The mode's observed transfer isIf $k\Delta \chi \gg 1$, the rescattered monopole is suppressed by at least $1/(k\Delta \chi )$, because the incoming directions sample incoherent phases. If $k\Delta \chi \ll 1$, $j_0=1+O((k\Delta \chi {)}^2)$ and the two propagation phases agree to leading order. The two weights then sum to one. ConsequentlyThese are the small-scale damping and large-scale coherence limits of reionization damping of cosmic microwave background temperature anisotropy.

Each small-scale primary temperature amplitude is multiplied by $e^{-{\tau }_{\mathrm{re}}}$, so its Cosmic microwave background power spectrum is multiplied by $e^{-2{\tau }_{\mathrm{re}}}$. Holding transfer parameters fixed,Many measured high-$\ell$ modes determine this combination accurately. Their derivatives $\partial \log C_{\ell }/\partial \log A_s=1$ and $\partial \log C_{\ell }/\partial {\tau }_{\mathrm{re}}=-2$ are degenerate: a change $d\log A_s=2d{\tau }_{\mathrm{re}}$ leaves the leading temperature spectrum unchanged. The undamped large-scale modes can distinguish the parameters, but there are few of them and cosmic variance gives fractional full-sky uncertainty $\sqrt{2/(2\ell +1)}$ per multipole. This explains the primordial-amplitude–optical-depth degeneracy. Polarization from reionization, lensing, and more complete late-time effects partly break it. The TeX's final ${\tau }_e$ is a transcription mismatch; the PDF uses ${\tau }_{\mathrm{re}}$ consistently.

A large neutrino degeneracy parameter raises the relativistic energy density and therefore the expansion rate. Big Bang nucleosynthesis tests this through primordial light-element abundances; an electron-neutrino chemical potential also shifts neutron--proton chemical equilibrium directly. The Cosmic microwave background power spectrum tests the changed expansion rate, matter-radiation equality, sound horizon, damping scale, and neutrino anisotropic stress. Finally, large-scale structure of the universe and the matter power spectrum test the altered equality scale and neutrino free streaming. These observables distinguish a degenerate-neutrino cosmology from the standard thermal relic scenario in complementary epochs.

After a spatial Fourier transform, the preceding equation isThe constant particular solution is ${\delta }_r=-4\Phi$. Hence the Sachs-Wolfe combinationcontains only the homogeneous harmonic oscillator modes and satisfiesFor adiabatic initial conditions on large scales, $S=\Phi /3=\mathcal{R}/5$ and $S^{\prime }=0$ at the start of matter domination. Up to an irrelevant choice of the phase origin,Thus the radiation acoustic transfer function is flat at $1/5$ as $k\to 0$ and oscillates on smaller scales. Projection onto the sky produces a large-angle Sachs-Wolfe plateau and, after the full photon-baryon physics is included, the acoustic structure of the Cosmic microwave background power spectrum.

The baryon transfer function should also oscillate soon after cosmological recombination: before decoupling, Thomson scattering forced baryons to share the acoustic motion of the photon-baryon fluid, and recombination does not instantly erase the resulting density pattern.

Physical cosmology connects the universe's history to the measurable behaviour of its contents. The Friedmann equations describe cosmic expansion; atomic physics, radiation pressure and gravity determine what that expanding universe looks like. A particularly revealing example is the route from cosmological recombination through the photon-baryon fluid to the Cosmic microwave background acoustic peaks. The universe supplies the initial conditions and the laboratory; the prediction must still pass through a model of the telescope before it can be compared with observations.

The practical reward is that one sky contains several different tests. The Cosmic microwave background power spectrum carries information about the sound horizon, the cosmological baryon density and cosmic geometry. These are related quantities with different observational signatures. Angular sound-horizon degeneracy explains why measuring a characteristic angle alone cannot count the universe's baryons.

The primary small-scale Cosmic microwave background power spectrum measures the amplitude $A_s{e}^{-2{\tau }_{\mathrm{re}}}$, because reionization damps each temperature amplitude once. The unaffected large-scale temperature modes are few and limited by cosmic variance. Polarization and gravitational lensing provide additional information that partly breaks this degeneracy.