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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.

Gravitational lensing maps source directions into observed image directions because spacetime curvature deflects null geodesics. In the weak-field limit the deflection by a point mass at impact parameter $b$ isFor a thin lens, angular diameter distances $D_l,D_s,D_{ls}$ relate the angles throughA point lens has Einstein angleand in a one-dimensional source alignment its two images satisfyExact alignment gives an Einstein ring for an ideal circular lens; finite source size and departures from symmetry alter the ideal image.

The mapping's Jacobian determines area magnification and parity. Its determinant can vanish on critical curves, whose source-plane images are caustics; a source near a caustic can produce highly stretched arcs or changing image multiplicity. In geometric optics, gravity is achromatic and preserves surface brightness after the usual redshift factors, while changing the apparent solid angle and total flux. Frequency-dependent source size, extinction and plasma propagation can modify an observed broadband image without changing this basic gravitational law.

Strong lensing produces multiple images, arcs and relative time delays. In angular variables the delay surface is proportional toup to an irrelevant additive constant. The geometrical path and gravitational potential delay both contribute. Monitoring a variable multiply imaged source can constrain the time-delay distance, provided the lens mass distribution and external convergence are modeled.

Weak lensing produces small coherent distortions in background galaxy shapes. Convergence describes isotropic focusing and shear describes anisotropic stretching. In a thin-lens convention,Statistical shear correlations probe projected matter structure and cosmological growth; intrinsic alignments, shape noise and selection require control. Compact masses can instead produce microlensing magnification changes when individual image separations are unresolved.

Lensing weighs both luminous and dark gravitating matter and can reveal mass concentrations without requiring hydrostatic equilibrium. It is nevertheless a projected and model-dependent inference: the mass-sheet degeneracy, line-of-sight structure and uncertain source redshifts can mimic changes in inferred mass or distance. Strong, weak and microlensing therefore constrain different scales and complement dynamical, photometric and cosmological measurements rather than automatically recovering a unique three-dimensional mass map.

redshift. An incoming radial null ray satisfies $d\chi =-cdt/R(t)$. Consecutive wave crests follow the same comoving path, so the endpoint changes in the fixed path integral obey $d{t}_0/R_0=d{t}_e/R_e$. Comoving clocks read cosmic time; hence their measured periods and frequencies satisfyThe physical wavelength stretches with the scale factor during propagation. It is not necessary to assign a unique global inertial-frame recession velocity to a distant source.

Radial coordinate and cosmological look-back time. Differentiating $1+z=R_0/R(t)$ gives $dz/dt=-(1+z)H(z)$. Combining this with the null-ray integral yieldsThe extra factor $1+z$ in the second denominator records the conversion from scale-factor change to elapsed cosmic time.

proper-motion distance. A small transverse displacement at emission has proper length $d\ell =R_{e}S(\chi )d\theta$. Define its source-frame transverse speed as $v=d\ell /d{t}_e$; cosmological time dilation makes $d{t}_0=(1+z)d{t}_e$. ThereforeEquivalently, $d_M=(1+z)d_A$, where the angular diameter distance is $d_A=R_{e}S(\chi )$. The observer-time versus source-time distinction is essential. This calculation uses the stated transverse speed convention and the homogeneous background, without additional peculiar radial relativistic Doppler effect or gravitational lensing.

redshift shell volume at the present epoch. The induced spatial metric tensor on the present slice gives $d{V}_0=R_0^3{S}^2(\chi )d\chi d\Omega$. Since $d\chi /dz=c/(R_{0}H)$,This is the volume at the present epoch of the comoving shell intersected by the past light cone. Its proper volume at emission is smaller by $(1+z{)}^3$; a conserved population's proper number density is larger by the same factor, preserving its count.

Einstein-de Sitter cumulative number counts. In flat geometry $S(\chi )=\chi$. Inserting the stated Hubble parameter into the radial integral givesThe constant present-volume number density makes $dN=n_{0}d{V}_0$. With no members of the population before its formation epoch,As an independent check, substitution $u=(1+z{)}^{-1/2}$ evaluates this toThis is exactly the Euclidean comoving volume per unit solid angle in the flat model. It assumes a complete count with the specified fixed population; luminosity selection, mergers or subsequent number-density evolution would require additional factors.

A point-like astronomical source has an angular extent negligible for the calculation or measurement being considered. The ideal mathematical limit concentrates its luminosity at one source-plane position. In gravitational lensing, this approximation permits divergent formal magnifications on a caustic; a finite source averages the magnification over its emitting area. An unresolved source need not have negligible size compared with the lens's caustic scale.

The two resolved quasar images of 0957+561 provide an observational question: are they separate sources, or different light paths from one source? In May 1979, spectrophotometry used the McDonald Observatory 2.7 m reflecting telescope, the UVITS spectrograph, and an intensified dissector scanner at the f/9 focus of the Cassegrain reflector. A 600-groove-per-millimetre diffraction grating produced 4100-7100 ångström scans sampled every 2 ångströms, with about 7 ångström resolution over most of that range. Two spectrograph slits, about 52 arcseconds apart, recorded object and sky background simultaneously for subtraction. Interleaving observations of the two images reduced instrumental and atmospheric gain changes.^[ref]

The readout was received light versus wavelength for each image. Closely matching absorption line redshifts supported a common source and absorber; differences in continua and line strengths motivated differential interstellar extinction. Spectral agreement supplied evidence for gravitational lensing, rather than proving the point-mass gravitational lens model. The authors discussed a physical double and discrepant optical and radio light ratios. Their note added in proof attributed the redder spectrum to foreground galaxy contamination and emphasized an extended mass distribution.^[ref]

The gravitational lensing approximation in which deflection is concentrated in a plane whose thickness is small relative to the relevant observer-lens-source distances. The reduced angular deflection includes the distance ratio. A projected surface mass density then determines the lensing potential and the image-to-source mapping. This is distinct from a refracting thin optical lens.

When gravitational lensing gives small angular distortions without generic multiple imaging, its deflection field changes source positions, magnification and shear. Angular derivatives of the lensing potential determine these distortions. Ensemble galaxy-shape distortions and cosmic microwave background remapping probe the foreground gravitational field.

The average of the two conformal Newtonian gauge potentials controls scalar gravitational lensing. A conformal rescaling makes the spatial optical metric flat and its time component $1+4\Psi$ to first order. Vanishing scalar anisotropic stress makes $\varphi =\psi$ in general relativity, but lensing depends on their sum even when they differ.