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A. D. Challinor coauthored, with A. N. Lasenby, analytic corrections to the thermal Sunyaev-Zeldovich effect in hot galaxy clusters. The calculation uses an expansion of the Kompaneets equation to include relativistic effects in scattering by electrons.

The resulting change in spectral shape matters when combining X-ray and cosmic microwave background observations to infer cluster properties or a Hubble constant. A small correction to the physical signal can produce a systematic shift in a derived parameter.^[ref]

A. N. Lasenby coauthored a theoretical calculation of relativistic corrections to the thermal Sunyaev-Zeldovich effect with A. D. Challinor. The effect arises when hot electrons change the cosmic microwave background spectrum through scattering.

Their calculation extends the Kompaneets equation and expands the spectral distortion in the dimensionless electron temperature $k_B{T}_e/(m_e{c}^2)$. It connects the accuracy of a cluster measurement to the validity of the nonrelativistic approximation used to interpret it.^[ref]

Arno Allan Penzias and Robert Woodrow Wilson discovered an unexplained microwave antenna signal at Bell Laboratories. Their measurement compared a horn-reflector antenna with a cooled reference and separately estimated atmospheric emission, instrumental losses and ground pickup.

The remaining excess was approximately isotropic, unpolarized and seasonally stable. Those checks made a purely local instrumental explanation less plausible. Interpreted as cosmic microwave background, the signal connected a calibrated radio-noise measurement to the thermal history of the cosmos; the original single-frequency result did not yet establish the full blackbody spectrum.^[ref]

For the measured mean cosmic microwave background temperature, the baryon-to-photon ratio and cosmological baryon density obey the displayed conversion. Big Bang nucleosynthesis, especially deuterium, measures this density during the first minutes, while acoustic peak ratios measure it near recombination. Agreement tests the thermal history and approximate conservation of baryon number and photon entropy between these epochs. It is not a claim that every observed light-element abundance has no residual tension.

Thermal photon number density is proportional to $T^3$. A sudden photon heating factor $(1+r{)}^{1/4}$ therefore raises photon number by $(1+r{)}^{3/4}$ and dilutes the baryon-to-photon ratio by its inverse when baryon number is conserved. Comparing Big Bang nucleosynthesis and cosmic microwave background baryon-density determinations can test entropy injection between the two epochs, subject to changes in the expansion rate and nuclear abundances.

Charles L. Bennett led the final nine-year WMAP analysis of cosmic microwave background maps. The work combined sky temperature and polarization measurements while modeling Galactic foreground emission and the response of the observing beams.

The angular patterns constrain cosmological parameters through a physical model of the early Universe. The paper compared the resulting Lambda-CDM model with independent observations, including baryon acoustic oscillations. Its quoted cosmic age and expansion rate are fitted consequences of those combined data and assumptions, rather than direct readings from a single sky image.^[ref]

FIRAS compared sky radiation with an adjustable internal blackbody using an interferometer and cold bolometers. An external blackbody could cover the sky entrance for measurement calibration. The team adjusted the internal temperature to minimize the difference signal.^[ref]

Comparisons across detectors and mirror strokes tested instrumental errors; sky mapping separated Milky Way emission. John C. Mather's lecture reports a final root-mean-square spectral departure of 50 parts per million of peak brightness. This is neither the preliminary 1990 precision nor a relative error in every spectral bin. Calibration drift and nonideal reference emission required correction.^[ref]

The advantage of a null measurement is that a small difference can be resolved without assigning the same absolute precision to a large detector output. Nevertheless, matching an internal reference is meaningful only after its relation to an independently calibrated blackbody has been tested. A bound on departures from a Planck spectrum constrains energy redistribution after thermal equilibrium; it does not require the cosmic microwave background to have exactly the same temperature in every direction.

Hot electrons can transfer energy to cosmic microwave background photons through Compton scattering. For a fixed electron fraction $x_e$ and electron temperature well above the radiation temperature, the optically thin gas cooling time scales approximately as $(1+z{)}^{-4}/x_e$. Its volume loss scales with electron density, not with $n_H^2$, because the radiation density is independent of local gas density. The process tends toward the radiation temperature; it heats rather than cools electrons colder than that radiation.

The Cosmic Background Explorer connected two questions: does the cosmic microwave background follow a Planck spectrum, and how does its temperature vary across the sky? John C. Mather and George F. Smoot shared the 2006 Nobel Prize in Physics for these discoveries.^[ref] The FIRAS and COBE DMR comparisons below explain why measuring spectral shape and measuring angular structure require different information.

The cosmic microwave background is nearly isotropic thermal electromagnetic radiation. Its spectrum connects radio receiver measurements with the universe's thermal history: redshift preservation of a thermal photon spectrum cools a blackbody spectrum as space expands. The 1965 Princeton analysis described this relation.^[ref]

A radiometer cannot label a received photon as cosmic, atmospheric or instrumental. Its measured power combines the incoming radiation with the receiver response. The useful question is which comparisons and controls can isolate a sky contribution. This makes the Penzias-Wilson microwave background experiment an example of both observational cosmology and measurement calibration.

The cosmic microwave background last-scattering surface is the shell from which the photons observed in the cosmic microwave background last scattered around photon decoupling.

The cosmic microwave background quadrupole is the degree-two spherical harmonic part of the observed cosmic microwave background sky. It has five real coefficients. The axisymmetric example $Q{P}_2(\cos \theta )$ selects only one orientation and shape from this space; $Q$ is its chosen amplitude, not a normalization for every possible cosmic microwave background quadrupole.

The blackbody temperature of freely propagating cosmic microwave background photons scales inversely with the scale factor: cosmological redshifting changes every frequency by the same factor and preserves the Planck spectrum shape. Its present value is approximately $2.73\mathrm{K}$. Measuring that absolute spectral temperature is distinct from measuring temperature fluctuations about it.

The cosmic microwave background temperature is about $2.725\mathrm{K}$, corresponding to the displayed thermal energy. The conversion uses $k_B\simeq 8.61733\times 10^{-5}eV/K$. Thus $2.35\mathrm{meV}$ would correspond to about $27.3\mathrm{K}$, rather than the present CMB. Primary references are NASA's CMB temperature measurements and NIST's Boltzmann constant in electronvolts per kelvin.

The cosmic neutrino background last-scattering surface is the locus from which relic neutrinos that reach an observer underwent their final appreciable scattering. Although neutrino decoupling occurs before photon decoupling, a massive neutrino travels slower than light after becoming nonrelativistic, so its last-scattering surface need not be farther away than the cosmic microwave background last-scattering surface.

At the cosmological baryon density favored by deuterium and the cosmic microwave background, standard Big Bang nucleosynthesis predicts more lithium-7 than is directly inferred from the Spite plateau, typically by a factor of a few. Stellar depletion, atmosphere systematics and possible new microscopic physics must be distinguished before interpreting this discrepancy as a failure of the cosmological framework.

For freely propagating radiation with the usual distance duality relation, a fixed emitted bolometric surface brightness is observed smaller by $(1+z{)}^4$. One factor each comes from photon energy and arrival rate, and two from the relation of projected angular area to luminosity distance. In a fixed observed photometric passband, spectral shifts add a band-dependent correction; the bolometric power is not automatically the monochromatic one. A distortion created by scattering the local cosmic microwave background has a different redshift comparison, as in the thermal Sunyaev-Zeldovich effect.

Cosmology investigates the large-scale structure, expansion and history of the universe by connecting observations with physical models. A redshift and an apparent brightness do not themselves provide a unique expansion history: source calibration and the relation between geometry and distance enter the inference.

The cosmic microwave background, Type Ia supernovae and baryon acoustic oscillations probe different physical processes and epochs. A viable model must explain their combination using consistent assumptions about gravity, matter and initial conditions. Comparing such independent observables helps expose degeneracies that remain when one data set alone is fitted by an FLRW metric.

Dale J. Fixsen studies precision measurements of the cosmic microwave background. He led an analysis of FIRAS data from COBE that compared the sky signal with a precise blackbody spectrum, constraining departures from that spectral shape.^[ref]

His 2009 study independently recalibrated FIRAS using WMAP and combined published measurements of the background temperature. It distinguishes the temperature from that recalibration from the value obtained by combining the literature: $2.72548\pm 0.00057\mathrm{K}$. The distinction makes the origin of an extremely precise quoted number traceable.^[ref]

The entropy density of a photon gas is its entropy per physical volume, $s=S/V$. For homogeneous equilibrium vacuum radiation with two states of electromagnetic polarization and zero photon chemical potential,The first equality is the entropy density at zero chemical potential, obtained from the thermodynamic Euler relation $U=TS-pV+{\mu }_{\gamma }{N}_{\gamma }$ with internal energy $U=uV$. The second uses the photon gas equation of state, $u=a_R{T}^4$ and $p=u/3$. If an independently constrained photon number gives a nonzero chemical potential, the general entropy density is $(u+p-{\mu }_{\gamma }{n}_{\gamma })/T$ and $u=a_R{T}^4$ need not hold. This distinction explains why the absence of a conserved number matters for entropy as well as for the spectral shape. ^[ref]

A short first law of thermodynamics check, for neighboring equilibrium states with ${\mu }_{\gamma }=0$, isThe thermodynamic Euler relation fixes the additive normalization. Equivalently, this radiation entropy tends to zero with temperature at fixed volume; no contribution from the containing walls is included. Interpreting $TdS$ as transferred heat and $pdV$ as boundary work requires a reversible thermodynamic process. For a reversible thermodynamic process that is also an adiabatic process, there is no entropy production or entropy exchange, soAn irreversible adiabatic process can produce entropy even without heat exchange, so the word adiabatic alone is insufficient to infer $T^{3}V=\text{constant}$.

For an analytic example, expand an initially equilibrated photon gas reversibly from $V_0$ to $8V_0$, keeping only the same two photon states of electromagnetic polarization. Its temperature halves, its energy density becomes $u_0/16$, its entropy density of a photon gas becomes $s_0/8$, and its total entropy stays $S_0$. The internal energy halves rather than remaining fixed: the missing half leaves as work against radiation pressure,This is a stipulated reversible radiation model, not a report of a piston experiment. Maintaining a real cavity along this path requires controlling wall heat exchange and including the wall heat capacity in any actual apparatus balance.

For a computed scale at stipulated $T=2.725\mathrm{K}$, $s\simeq 2.04122\times 10^{-14}\mathrm{J}{\mathrm{K}}^{-1}{\mathrm{m}}^{-3}$. Combining this with the photon number density givesThis is the total equilibrium radiation entropy divided by its mean photon number, not a separately assigned entropy carried by each labeled particle. During the ideal expansion, $n_{\gamma }\propto T^3$ and $T^{3}V$ is fixed, so the mean number $N_{\gamma }=n_{\gamma }V$ also stays constant. This follows from the path; it was not imposed as the independent number constraint used to establish chemical potential.

The beautiful connection is that expansion lowers each typical photon energy and dilutes the radiation in just the proportions needed to retain its equilibrium spectral shape and total entropy. For the cosmic microwave background, homogeneous expansion of the universe has physical volume proportional to the cube of the scale factor; fixed-species cosmological entropy conservation therefore gives $T\propto a^{-1}$. This describes preservation of an already established Planck photon distribution and connects the FIRAS spectral observation in photon gas to an entropy prediction; FIRAS did not perform a direct entropy measurement. When other equilibrated species disappear and transfer energy into radiation, use the effective number of relativistic entropy degrees of freedom, $g_{\ast s}$: the conserved quantity for the combined sector is $g_{\ast s}{T}^{3}V$, under the corresponding equilibrium and no-entropy production assumptions. A changing species population, separate decoupled sectors or entropy production requires an appropriate entropy balance rather than the pure photon gas formula. ^[ref]