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The cosmic microwave background is relic thermal radiation from the hot early universe. Its temperature anisotropy is expanded in spherical harmonics, with angular power spectrum $C_{\ell }^{TT}$. Thomson scattering of a local radiation quadrupole generates linear polarization around last scattering and again during reionization. Polarization maps are decomposed into E-mode polarization and B-mode polarization, giving spectra such as $EE$, $TE$ and $BB$ in addition to temperature information.

Cosmic microwave background acoustic peak positions compare the sound horizon with the angular-diameter distance to last scattering, constraining geometry and expansion history. Relative peak heights constrain baryon loading and matter-radiation equality; the damping tail constrains diffusion and other early-universe physics. The scalar spectral index and primordial scalar amplitude constrain inflationary perturbations. Coherent acoustic structure and temperature-polarization correlations support predominantly adiabatic initial perturbations. Large-angle E-mode polarization measures the reionization optical depth, helping break its degeneracy with the scalar amplitude inferred from small-scale temperature anisotropy.

Under a specified cosmological model, these data constrain ordinary and cold dark matter densities, the Hubble constant, spatial curvature and cosmic age. Complementary distance information helps break the geometrical degeneracy: temperature peaks alone do not uniquely establish a cosmological constant. The Planck 2013 parameter analysis used Planck temperature data and, in important combinations, WMAP low-multipole polarization; it should not be described as a full Planck polarization release.^[ref]

Linear scalar cosmological perturbations generate E-mode polarization but no primordial B-mode polarization in the simplest treatment. tensor cosmological perturbations can generate B-mode polarization, making them a possible probe of inflationary gravitational waves. gravitational lensing converts some E-mode power into B-mode polarization, however, and Galactic dust and synchrotron radiation also contribute polarized foregrounds. Multifrequency observations, foreground modelling, beam calibration and control of temperature-to-polarization leakage are therefore necessary. Polarization does not by itself establish a primordial tensor signal. temperature and polarization jointly test the expansion history and primordial perturbations, while complementary data and foreground control determine how uniquely parameters can be inferred.

For successive radial null rays from a comoving source, ${\int }_{t_1}^{t_0}cdt/R(t)$ equals the same spatial integral. Subtracting two nearby wavecrests gives $\delta t_0/R_0=\delta t_1/R_1$, hence the cosmological redshiftFor transparent propagation, photon energies are reduced by $1+z$ and arrival intervals increased by $1+z$. The observer-area sphere is $4\pi (R_{0}r{)}^2$, so the observed bolometric flux isThus the requested $D_L=(1+z)D$ uses $D=R_{0}r$, the present transverse proper/area distance. In a curved metric this is not the radial proper distance $R_0{\int }_0^{r}d{r}^{\prime }/\sqrt{1-K{r}^{\prime 2}}$; the two coincide for $K=0$. The source's unqualified “proper distance” needs this distinction.

Type Ia supernovae are standardizable candles: light-curve shape and colour calibrate their luminosity. Distant events are fainter, hence have larger inferred luminosity distances, than appropriate matter-only decelerating models predict. Their redshift–distance relation favours recent accelerated expansion. In general relativity, sufficiently negative pressure, $\rho +3P/c^2<0$, drives acceleration. Dust, evolution, calibration and selection are checked as possible systematics; a single faint supernova is not the argument.^[ref]

An independent complementary test combines the cosmic microwave background acoustic distance/near-flat geometry with matter-density and baryon acoustic oscillation constraints. Matter alone does not supply the required late-time expansion history; a dark-energy component fits the joint data. CMB geometry alone has curvature/expansion degeneracies and is not an assumption-free detection of a cosmological constant.^[ref]

In natural units $c=\hslash =1$, a regulated zero-point energy estimate isThe integral is ultraviolet divergent and requires renormalization; it is not a finite prediction from counting “virtual pairs.” An electroweak scale estimate gives $|{\rho }_{\mathrm{vac}}|\sim m_t^4=(174{)}^4\mathrm{Ge}{\mathrm{V}}^4\simeq 9.2\times 10^8\mathrm{Ge}{\mathrm{V}}^4$, or somewhat smaller with loop factors. Compared with $10^{-47}\mathrm{Ge}{\mathrm{V}}^4$ this is roughly $10^{56}$ too large.

The source's units are inconsistent: $10^{-47}\mathrm{Ge}{\mathrm{V}}^4$ is a vacuum energy density, whereas $8\pi G{\rho }_{\mathrm{vac}}$ has units $\mathrm{Ge}{\mathrm{V}}^2$. The comparison above consistently uses densities. Bare and quantum contributions must cancel to extraordinary accuracy if this naive scale estimate is representative: the cosmological constant problem. Its severity is not a measured value of an individual cutoff integral.

The metric uses signature $(+---)$ and a time-valued interval $ds$: a comoving clock has $ds=dt$. Here $t$ is cosmic proper time, $R(t)$ a scale factor with dimensions of length, $c$ the speed of light, $\chi$ a dimensionless comoving radial coordinate, and $\theta ,\varphi$ the usual polar and azimuthal angles. With curvature normalized to $k=+1,0,-1$, $S(\chi )=\sin \chi ,\chi ,\sinh \chi$, respectively. Homogeneity and isotropy apply to each constant-$t$ spatial slice. A comoving object's fixed coordinates do not imply a constant proper separation: spatial lengths scale with $R$.

Radial null propagation obeys $d\chi =cdt/R(t)$ in magnitude. Two neighbouring wave crests between the same comoving emitter and observer have the same radial integral, giving $\delta t_0/R_0=\delta t_e/R_e$. The wavelength redshift is consequentlyPeculiar Doppler shifts are excluded by the comoving assumption. Define angular diameter distance by $d_{\theta }={\ell }_{\perp }/\delta \theta$, where ${\ell }_{\perp }$ is proper transverse size at emission. The transverse metric gives ${\ell }_{\perp }=R_{e}S(\chi )\delta \theta$, henceThis distance is neither the photon travel time times $c$ nor the object's present proper distance.

The comoving particle horizon at $t_1$, with the initial boundary at the Big Bang, is ${\chi }_p(t_1)={\int }_0^{t_1}cdt/R(t)$. Its proper radial distance is $D_p=R(t_1){\chi }_p$; for a power-law scale factor $R=A{t}^p$, $0<p<1$,Thus a matter-only flat Einstein-de Sitter universe, $p=2/3$, has $D_p=3c{t}_1$, and the flat radiation-only case, $p=1/2$, has $D_p=2c{t}_1$. Strictly, Einstein-de Sitter denotes the pressureless matter model; the radiation calculation uses the flat zero-$\Lambda$ radiation analogue requested here.

For the source seen at time $t_1=t_{\mathrm{rec}}$, the radial coordinate is the integral from recombination to today, not the earlier horizon integral. In these flat models,Taking the linear horizon scale in the question to be $D_p$, its small angular extent isThereforeUsing the supplied times gives ${\theta }_{p,m}=0.02684\mathrm{rad}=1.54^{\circ }$ and ${\theta }_{p,r}=0.004244\mathrm{rad}=0.243^{\circ }$. There is a genuine radius/diameter convention to state: the supplied $3c{t}_1$ is a radial horizon distance. If angular diameter means the full diameter of a spherical horizon patch of radius $D_p$, the two answers are instead $2{\theta }_{p,m}=3.08^{\circ }$ and $2{\theta }_{p,r}=0.486^{\circ }$. Both conventions follow from the same distance calculation; a factor of two should not be hidden in the word horizon.

Matter-only evolution is the better of these two approximations because the universe is already matter dominated around recombination and evolves mostly through a subsequent matter era, rather than remaining radiation dominated to the present. Neither is exact: pre-recombination radiation changes the particle horizon, and late-time dark energy changes the distance to the last-scattering surface. No supplied ages have been replaced by later best-fit values.

The cosmic microwave background originates near last scattering and samples many horizon-sized regions. In an always-decelerating hot-Big-Bang model, widely separated last-scattering patches lack overlapping early causal histories, motivating the horizon problem. The measured acoustic angular scale uses the smaller sound horizon, not automatically this light-travel particle horizon; baryon-photon pressure changes the propagation speed. Inflation supplies an earlier accelerated expansion during which the comoving Hubble radius shrinks, allowing a patch with a common earlier causal history to expand to the size of the observable universe. Quantum perturbations can then seed later acoustic and density fluctuations. This does not require local signals to propagate faster than light, and the Hubble radius must not be equated indiscriminately with the particle horizon. CMB temperature and lensing spectra test the resulting perturbation and expansion model.^[ref]

Write the dimensionless scale factor as $a=R/R_0$ and the density contrast as $\delta =(\rho -\bar{\rho })/\bar{\rho }$. Gravity amplifies overdensities while expansion and pressure oppose growth. In a sub-Hubble Newtonian single-fluid approximation, a comoving Fourier mode obeysFor pressureless matter in an Einstein-de Sitter background, the growing solution is $\delta \propto t^{2/3}\propto a$ and the decaying solution is $t^{-1}$. Jeans support suppresses growth when the pressure term dominates. The equation is not a substitute for relativistic perturbation theory on horizon-sized scales.

cold dark matter clusters without baryonic pressure support. Sub-horizon growth is much weaker during radiation domination, and baryons are initially tied to the photon fluid, producing acoustic oscillations rather than unrestricted collapse. Horizon entry, radiation-matter equality, acoustic evolution and diffusion therefore give a scale-dependent transfer function. The primordial curvature spectrum and the later matter-density spectrum are not the same function: evolution and the gravitational constraints relate them. Late dark-energy domination reduces the growth rate.

When $|\delta |$ approaches unity, linear theory fails. Collapse, virialization, mergers and tidal interactions form a hierarchy of haloes. Gas must cool and lose angular momentum to make dense stellar systems; feedback then changes its supply and distribution. Simulations combine collisionless gravitational dynamics with increasingly complex gas physics rather than extending the linear growing mode indefinitely. Galaxy clustering measures biased tracers of matter; gravitational lensing and CMB constraints provide complementary tests. Approximately adiabatic, nearly scale-invariant initial perturbations explain the coherent acoustic pattern and supply seeds for this later growth.^[ref]

The cosmic microwave background has an almost perfect blackbody spectrum at about $2.725\mathrm{K}$. This is strong evidence for an earlier hot, dense phase in which radiation thermalized, followed by expansion that redshifted the photon spectrum while preserving its Planck form. COBE FIRAS compared the sky with a precision reference blackbody; instrument temperature, gain, foregrounds and calibrator emissivity were crucial controls. Agreement with a blackbody constrains substantial late energy injection or photon-production distortions.^[ref]

After subtracting the dominant dipole associated mainly with observer motion, anisotropies at the level of roughly $10^{-5}$ reveal primordial structure. COBE DMR established large-angular-scale anisotropy; multifrequency measurements and independent maps were needed to distinguish it from Galactic emission and instrumental artifacts.^[ref] Approximate isotropy supports a homogeneous background model under the usual cosmological assumptions, but isotropy at one observing position is not alone a proof of spatial homogeneity everywhere.

The temperature CMB angular power spectrum has multiple acoustic peaks. Their coherent pattern follows oscillations in the pre-recombination photon-baryon fluid. Peak positions compare the sound horizon with angular diameter distance and hence constrain geometry together with expansion parameters. Relative peak heights test baryon inertia, matter–radiation equality and gravitational-potential evolution, motivating both baryonic and nonbaryonic matter. Cosmic microwave background diffusion damping suppresses small-scale power. Cosmic microwave background polarization and temperature–polarization correlations provide further tests of scattering, acoustic phase and reionization; gravitational lensing remaps the observed pattern and probes intervening structure.

By the time of this examination, the Planck 2013 temperature and lensing analysis supported a spatially flat six-parameter model with cold dark matter and a cosmological constant. This is a joint fit within an explicit model, not an inference that each peak uniquely measures one parameter.^[ref] Foreground separation requires frequency coverage, sky masks, beam and noise characterization, and consistency across instruments. The primary microwave background has geometric and other parameter degeneracies, so acoustic-ruler, supernova and structure observations are valuable complements. The spectrum establishes a thermal hot-universe history; anisotropy and polarization constrain its geometry, contents and primordial perturbations. None of these observations by itself demonstrates a unique cosmic inflation mechanism.