Cosmic backgrounds: transport, spectra and acoustic phase

01A background is also produced, propagated and received

The cosmic microwave background is first a measurable electromagnetic signal. Public COBE/FIRAS spectral data provide an observational comparison with a nearly blackbody spectrum; spectral shape, temperature anisotropy and polarization impose separate requirements on an origin model. ASD interprets electromagnetic radiation as transverse open modes and can therefore investigate how spatial ordering, topological reconnection and exchange between matter and radiation leave a transverse background. A complete origin account still requires preparation, thermalization, propagation and reception, rather than a new physical name for the signal.

Free longitudinal modes may have a different background distribution, but their existence, production, thermalization and decoupling require their own coupling dynamics. An observed transverse microwave background neither establishes an observed longitudinal background nor equal temperatures or energy densities. We first explain the transverse effective transport already developed in the locked Chapter 5, then its connection to acoustic structure and detection.

02How mode occupation changes with the background

The manuscript uses the dimensionless mean scale $a_U(t)$ for macroscopic spatial stretching and sets $H=\dot a_U/a_U$, with dimensions of inverse time. Here $k$ initially denotes physical wave number. $n_A(k,t)$ is the mode occupation under homogeneous, isotropic conditions; $A=T,L$ labels transverse and longitudinal channels. Equation 5.5.8 is

The left-hand side relates occupation changes to wave number drift under stretching; along its characteristics, $\dot k=-Hk$. The collision term $C_A$ includes production, absorption, scattering and longitudinal–transverse conversion. $\Theta$ denotes an ordered spatial state and $\rho_b$ physical baryonic mass density. The transport structure still requires collision kernels and initial conditions from the actual interaction. It does not by itself determine thermalization efficiency or decoupling time.

Adopt constant $c,\hbar$, energy $\hbar\omega$ per mode and Bose occupation statistics as effective inputs. If production and absorption drive the electromagnetic chemical potential to zero, equation 5.5.9 gives

$T_\gamma$ is the background temperature parameter, $k_B$ Boltzmann's constant, and $u_\omega$ energy density per unit angular frequency, whose integral gives total radiation energy density. The second expression includes two electromagnetic transverse polarizations. It specifies spectral shape for a given temperature and equilibrium conditions; it does not independently predict that temperature. Elastic scattering or detailed balance at fixed particle number is not enough to ensure zero chemical potential. Energy quanta and occupation statistics are effective inputs here, not results of classical nearest-neighbour counting.

03Why a decoupled spectrum scales with size

After transparency, set $C_T=0$ in the interval where transverse collisions can be neglected. Integrating $\dot k=-Hk$ and using $\omega=ck$ gives $k\propto a_U^{-1}$ and $\omega\propto a_U^{-1}$. For an initial equilibrium spectrum, preserving the form of the exponent $\hbar\omega/(k_BT_\gamma)$ yields the temperature scaling. In addition to decreasing energy per mode, the mode number density is diluted as $a_U^{-3}$. Equation 5.5.10 is therefore

The constants carry the initial values at the reference epoch $a_U=1$. The equalities describe the selected collisionless model, not the prior thermalization process.

$u_\gamma$ is total transverse radiation energy density. This is a transport relation for prescribed mean stretching and constant $c,\hbar$. It does not determine the background dynamics of $a_U(t)$. A different constitutive rule for background-dependent action scales would require rebuilding energy transport and measured frequency matching together; a varying $\hbar$ cannot simply be inserted into a constant-coefficient result.

For an isotropic, gapless collection of free modes with $\omega=ck$, mean momentum flux gives $p=u/3$. Ordinary longitudinal thermal radiation thus has pressure. Being neutral or hard to detect does not make it a fully pressureless cold dark component. Dark matter research discusses a different gapped coherent model and its conditions.

04Acoustic oscillation belongs to coupled matter and radiation

The microwave background's acoustic structure involves the photon–baryon fluid before recombination, not vacuum longitudinal waves themselves. To compare propagation at different epochs, the manuscript defines the conformal propagation parameter $d\eta=dt/a_U$. In the acoustic equations below, $k$ now denotes comoving wave number, distinct from the earlier physical wave number. Let $\delta_{\gamma,k}$ denote the radiation density perturbation of that Fourier mode. Equation 5.5.11 is

Primes denote derivatives with respect to $\eta$. $c_s$ is the effective sound speed of the coupled fluid, not vacuum longitudinal speed $c$; $\mathcal A$ represents damping or background terms and $S_k$ the driving term. Baryon loading, stretching and scattering affect these coefficients, which require a joint fluid and radiation transport calculation. This is the minimal effective structure adopted in the manuscript, not a completed ASD angular power spectrum.

To remove the first derivative, substitute $\delta=e^{-\frac12\int\mathcal A d\eta}y$. Successive product differentiation cancels the first derivative terms and yields equation 5.5.12:

The substitution retains derivative corrections from variable coefficients. For a source-free oscillatory branch, or after separating the driven solution, assume a real frequency and slow variation $|\omega_{\rm eff}'|\ll\omega_{\rm eff}^2$. The homogeneous approximation in equation 5.5.13 is

$C_k,D_k$ are fixed by the initial state. A changing frequency affects amplitude as well as phase. Replacing a constant frequency with an integral while retaining constant amplitude misses terms. Constant-amplitude sine and cosine functions are exact solutions of the corresponding homogeneous equation only for constant $c_s$ and $\mathcal A=0$.

The superscript WKB identifies a first-order approximate basis of the homogeneous equation. Writing an equality for that basis does not supply an exact solution for arbitrary varying coefficients or for the driven background.

05From an acoustic length to an angular scale

Let $\eta_*$ denote last scattering. Define the comoving acoustic scale $r_s(\eta_*)=\int^{\eta_*}c_s d\eta$ and the comoving transverse distance $D_M$ to that surface. Comparing lengths as an angle requires the same distance convention for both. Under the phase approximation, equation 5.5.14 reads

$\ell_A$ is a dimensionless angular scale index, $\ell_n$ an approximate peak location and $\phi_n$ a phase shift due to gravitational driving, propagating components and projection. It is not the gravitational stretching variable $\varphi$. If physical angular diameter distance $D_A$ is used, the denominator must be the physical acoustic scale $a_U(\eta_*)r_s$. Peak heights, odd–even differences, diffusion damping, polarization and lensing require the full perturbation transport system, not one angular-scale relation.

The original Planck cosmological analysis provides several spectra for comparison. ASD must derive their inputs and transfer relations from its own longitudinal potential, matter formation and mean stretching dynamics. This page publishes no quantitative ASD fit of these spectra; a spectral form at a prescribed temperature does not establish a cosmological origin.

06Sources, reception and further work

The theoretical source is Chapter 5 of the final collected manuscript (ASD-BOOK-ddbade997b45), Part V, subsections 3–4, equations 5.5.8–5.5.14. COBE/FIRAS and Planck material identify observational targets without upgrading ASD evidence status. The page remains an exploratory, untested guide and does not substitute an unapproved action-scale revision for these constant-coefficient conventions.

Transverse reception must connect to electromagnetic propagation and material response. A candidate longitudinal background needs a separate mapping from field to receiver readings; see free longitudinal waves and the detection open problem. Once background occupation, production and apparatus response are specified, signals can be compared under the same observing protocol.