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An astronomer whose joint spectrophotometry of the Twin Quasar compared the spectra along its two apparent light paths. Closely matching absorption-line properties supported gravitational lensing: two visible images can correspond to one distant quasar viewed along paths bent by intervening matter.^[ref]

Gravitational lensing constrains a galaxy cluster's projected mass independently of its light output or dynamical equilibrium. The lensing convergence is projected surface mass density divided by ${\Sigma }_{\mathrm{crit}}=c^2{D}_s/(4\pi G{D}_d{D}_{ds})$, using the appropriate angular diameter distances. Strong gravitational lensing constrains the central mass through multiple-image geometry, and weak gravitational lensing constrains larger-scale projected mass through coherent image distortions. Converting these observations into a three-dimensional mass needs geometrical modelling and the relevant lensing degeneracies.

Dark matter is an inferred gravitating component whose mass is not accounted for by the observed luminous material. Galaxy rotation curves, motions in galaxy clusters and gravitational lensing probe its gravitational contribution through different measurements.^[ref]

For illustration, circular motion in a spherical mass distribution obeys $v_c^2=GM(<r)/r$. A roughly constant orbital speed over increasing radius therefore requires enclosed mass to keep increasing in this model. A realistic inference also models visible matter and the system's geometry; a rotation curve alone is not a direct identification of a new particle.

Dark-matter direct detection searches for interactions in terrestrial targets, while indirect detection of dark matter looks for secondary products. These tests address microscopic models rather than replacing the astronomical gravitational evidence.

Douglas Clowe coauthored a study comparing gravitationally inferred mass with the gas distribution in a colliding galaxy cluster. Gravitational lensing supplies information about the projected mass, while X-rays trace hot gas that can be slowed during the collision.^[ref]

The comparison is informative because collisionless and collisional components need not remain spatially coincident. The interpretation uses a model of gravity, geometry and measurement uncertainty rather than identifying all matter with the brightest visible component.

Edward Fomalont led a precise radio test of gravitational lensing near the Sun. The team used the Very Long Baseline Array to compare four radio sources over eighteen days in 2005, measuring changes in their relative apparent directions as the Sun moved across the sky.

Observations at several frequencies helped assess refraction in the solar corona. The 43 gigahertz data, sufficiently far from the Sun, largely avoided that disturbance. Fitting the gravitational signal gave the post-Newtonian parameter $\gamma =0.9998\pm 0.0003$, consistent with general relativity. The experiment illustrates why controlling propagation through plasma matters even in a test of gravity.^[ref]

A gravitational lens is a foreground distribution of matter whose gravitational field deflects light from a background source. Its projected surface mass density determines the mapping in the thin-lens approximation. The lens is the deflecting object; gravitational lensing is the resulting optical process.

Jacob Bekenstein developed tensor-vector-scalar gravity, a relativistic theory intended to realize modified Newtonian dynamics. The construction supplements the metric tensor with dynamical scalar fields and vector fields.

A nonrelativistic rule for stellar motion alone does not determine how light bends. His 2004 theory addressed both slow-particle dynamics and gravitational lensing, while examining the high-acceleration limit and Solar-System tests. This made the proposal a broader physical framework with multiple observational consequences, rather than merely a fit to a galaxy rotation curve.^[ref]

The signed magnification is the reciprocal determinant of the Jacobian matrix of the thin gravitational lens equation. Its absolute value gives the flux and solid-angle amplification; its sign gives image parity. Gravitational lensing preserves surface brightness in the geometric-optics approximation.

Manoj Kaplinghat studies connections between particle, dark matter and astronomical structure. The research programme described by his institution uses gravitational lensing and stellar motions to investigate matter distributions in galaxies and clusters.^[ref]

These measurements connect microscopic matter models with macroscopic dynamics. Inferring a particle's properties requires more than estimating total mass: the spatial distribution, velocities and possible interactions must also be consistent with the observations.

An astrophysicist whose work develops gravitational lensing as a probe of matter in the Universe. His review derives lensing relations from geodesic deviation and applies them to objects from stars to large-scale structure. Because gravity deflects light regardless of whether the lens emits light, the method can constrain otherwise unseen mass. The point-mass gravitational lens provides a concrete starting example, producing multiple images and a critical alignment that becomes an Einstein ring.^[ref]

Modified gravity changes the gravitational dynamics described by general relativity, for example through additional gravitational fields or curvature terms beyond the Einstein-Hilbert action. A proposed theory must specify both its equations and how matter couples to the geometry before trajectories, light propagation or cosmological evolution can be predicted.

The regime of a test matters. Linearized f(R) gravity about Minkowski spacetime can coincide with the Linearized Einstein equations for suitable derivatives of its defining function, while nonlinear terms still differ. Agreement in one weak-field calculation therefore need not identify the full theory. Comparisons involving orbital motion, gravitational lensing and gravitational waves constrain different combinations of dynamics, matter assumptions and observational geometry.

The Navarro--Frenk--White profile is an empirical model for the spherically averaged mass density of collisionless dark matter halos. Julio F. Navarro, Carlos S. Frenk and Simon D. M. White found that the same two-parameter shape fitted halos in their hierarchical-clustering N-body simulations over a broad resolved radial interval.^[ref]

Writing $x=r/r_s$, the profile iswith positive scale radius $r_s$ and characteristic density ${\rho }_s$. Its inner density cusp scales as $r^{-1}$ and its outer tail as $r^{-3}$. Integrating $4\pi r^2\rho$ gives the enclosed massAlthough the central density diverges, the enclosed mass tends to zero as $r^2$. At large radius the total mass diverges logarithmically, so a quoted halo mass requires a finite boundary convention, such as a specified mean enclosed overdensity. The halo concentration compares that boundary radius with $r_s$. The simulation fit is useful for consistent halo descriptions and gravitational lensing calculations; its empirical success does not require every halo or baryon-modified inner region to follow it exactly.

Type Ia supernovae are standardizable candles. Repeated multiband photometry measures peak apparent magnitude, light-curve width and colour; spectra identify the supernova class and determine redshift, often from the host galaxy. Nearby objects calibrate the relation between light-curve shape, colour and absolute luminosity. Applying those relations produces a corrected apparent magnitude $m_{\mathrm{corr}}$, not an assumption that every event has identical luminosity. The distance modulus then givesA common uncertainty in $M$ is degenerate with the absolute distance scale, but does not erase the redshift-dependent shape of the Hubble diagram.

luminosity distance probes the expansion history. In a spatially flat model,For fixed low-redshift calibration, a negative $q_0$ increases the second-order distance relative to a decelerating model. Objects at a given redshift are therefore fainter. A Hubble diagram spanning enough redshift can distinguish this curvature from the common magnitude intercept. In matter-plus-cosmological-constant models, $q_0={\Omega }_m/2-{\Omega }_{\Lambda }$; fitting the distances can test whether $q_0<0$. The two original supernova teams found distances supporting an accelerating expansion in that framework.^[ref][ref]

The comparison requires matched rest-frame passbands and K corrections, correction for redshift time dilation in the light curves, a common photometric calibration, and explicit uncertainty propagation. Dust extinction can imitate dimming; multicolour and infrared observations test reddening and the adopted extinction law. Evolution in the progenitor population or host-galaxy dependence can alter standardization, so one compares matched populations and fits host-dependent terms where warranted. Flux-limited discovery introduces Malmquist bias and other selection effects, which must be modelled through the survey detection efficiency. peculiar velocities matter at low redshift, while gravitational lensing and intrinsic scatter broaden the distant sample. Classification and outlier tests reduce contamination by other transient types.

Independent data reductions, nearby anchors, spectroscopic checks and a fit over the full redshift range distinguish an expansion-history signal from one instrumental offset. Complementary cosmological observations help separate matter density, curvature and dark-energy parameters. The acceleration inference comes from the relative distance-redshift relation, conditional on standardization and an expansion model. Distant supernovae integrate the past expansion history; they do not directly measure the instantaneous second time derivative of today's scale factor without such modelling.

Outside the core, $f^{\prime \prime }=0$ and the metric is locally flat: set $\tilde{\varphi }=(1-\lambda )\varphi$. Static test particles have no Newtonian-like attraction because $g_{00}$ is constant and ${\Gamma }^r{}_{00}=0$. Local trajectories are straight lines in locally unwrapped coordinates, so a very thin dark core is hard to identify through ordinary orbital accelerations or local tidal forces.

The global angular range is nevertheless $2\pi (1-\lambda )$ rather than $2\pi$. The missing angle $\delta =2\pi \lambda$ can give gravitational lensing with two nearly identical, essentially undistorted images of a background galaxy, and multiple such pairs aligned along the projected object. For a small deficit, a straight string and favourable geometry, their angular separation is of order $\delta D_{ls}/D_s$, with an additional inclination factor for a tilted string. The matching spectra and common line of image pairs distinguish this from an ordinary compact lens.

A moving cosmic string can also induce a step in the cosmic microwave background temperature across its path. That signature needs motion in addition to the static geometry supplied here. These are consequences of the exterior conical geometry; interpreting the inferred deficit as a positive mass per length requires the regular-axis/core assumptions distinguished in part b.

In a local inertial frame in vacuum, gravitational waves travel at $c$ in general relativity; there is no ordinary material index of refraction analogous to a transparent dielectric. Ordinary matter couples so weakly that laboratory prisms for gravitational waves are implausible. Propagation through a varying background gravitational field nevertheless changes ray directions and travel times, giving gravitational lensing and Shapiro time delay. In weak static coordinates $d{s}^2\simeq -(1+2{\Phi }_N/c^2)c^{2}d{t}^2+(1-2{\Phi }_N/c^2)d{\mathbf{x}}^2$, the null condition gives an effective coordinate index of refraction $n_{\mathrm{eff}}\simeq 1-2{\Phi }_N/c^2$. This useful description depends on coordinates and does not change the local propagation speed. Thus astronomical gravitational refraction is plausible, while a substantial everyday material effect is not.

Gravitational waves can exhibit interference. In linearized gravity, overlapping strains add, so coherent contributions can enhance or cancel the measured response. Independent astrophysical sources generally have unrelated phases and average to a stochastic background rather than stable fringes. More plausible coherent paths arise when gravitational lensing sends different parts of the same signal to the detector with different delays; fringes require sufficient overlap and coherence. The relevant quadratic radiative energy flux contains the cross terms even though the leading wave equation is linear.

Gravitational waves also exhibit diffraction: the geometrical optics ray description fails when a propagation structure or lens delay is not large compared with the wave's wavelength or period. For a compact lens of mass $M_L$, a useful dimensionless parameter is $w=4G{M}_L{\omega }_{\mathrm{gw}}/c^3$; $w$ of order one signals potentially appreciable wave effects, with cosmological lens redshift supplying an additional factor $1+z_L$ when needed. Astrophysical gravitational lensing can therefore produce diffraction and frequency-dependent amplification. Ordinary objects are poor opaque screens for gravitational waves, so everyday slit-and-screen demonstrations are implausible despite the large wavelengths, which are about $3000$ km at $100$ Hz.

In the thin-lens approximation for gravitational lensing, consider a ray crossing the lens plane at transverse distance $b$ above the aligned source-observer axis. With the observer at axial position $0$, the black hole at $L$ and the source at $D$, the segment joining the lens plane to the source has slope $-b/(D-L)$, while the segment joining the observer to the lens plane has slope $b/L$. The observer sees the ray at apparent angle $\theta \simeq b/L$ to the axis. When following the photon toward the observer, the two small angles to the axis have magnitudes $b/(D-L)$ and $b/L$. The physical bend is their sum:Substitute the Schwarzschild light deflection, $\delta \simeq 4GM/(b{c}^2)$, and use $b\simeq L\theta$:Every azimuth around the axis is equivalent by rotational symmetry. Isotropic emission therefore supplies rays at all these azimuths, and the ideal image is an Einstein ring centered on the black hole, with Einstein angular radius $\theta$ and angular diameter $2\theta$. A finite source size or imperfect alignment broadens or breaks the ideal ring.

The stated expression uses small angles, distant endpoints and weak-field approximation $b\gg 2GM/c^2$, with ordinary Euclidean distances. For cosmological gravitational lensing, use angular diameter distances $D_l,D_s,D_{ls}$ in ${\theta }^2=4GM{D}_{ls}/(c^2{D}_l{D}_s)$; generally $D_{ls}\ne D_s-D_l$.

The three probes constrain the same deep gravitational potential by different observations. A large member-galaxy velocity dispersion requires a large binding mass. Applying the virial theorem, with a gravitational scale radius $R$ and approximately isotropic orbits, gives a cluster velocity-dispersion mass estimate of order $M\sim R{\sigma }_v^2/G$, with a profile-dependent numerical coefficient. The inferred mass-to-light ratio is much larger than the stellar value. Membership, orbital anisotropy and equilibrium assumptions affect the precision.

The hot intracluster medium radiates through thermal bremsstrahlung and line emission. Its X-ray spectrum measures temperature, while its surface brightness constrains gas density. Confining this gas requires a potential with characteristic $GM/R\sim k_{B}T/(\mu m_p)$. Combining resolved profiles through the cluster hydrostatic mass estimator gives the total mass, not just the gas mass. The detected gas contributes substantial baryonic mass but does not normally account for the full gravitating mass. Nonthermal pressure or a merging cluster can invalidate a purely thermal hydrostatic equilibrium estimate.

Finally, gravitational lensing measures mass through light deflection. Strong gravitational lensing supplies multiple images and arcs; weak gravitational lensing measures coherent background-image distortion over larger radii. The lensing convergence is projected surface mass density divided by the critical surface density, irrespective of whether the material emits light. Cluster gravitational-lensing mass estimates agree broadly with the high masses implied by galaxy motions and hot gas, while having different equilibrium assumptions. Together these observations support a dominant nonluminous component in galaxy clusters, after accounting for their stars and observed gas.

A genuine nondegenerate bounce has ${\dot{a}}_{\ast }=0$ and ${\ddot{a}}_{\ast }>0$; using ${\ddot{a}}_{\ast }\ge 0$ also includes the limiting stationary case. Put $s=a_{\ast }^{-1}=1+z_{\ast }$, and define the present matter density parameter ${\Omega }_m=8\pi G{\rho }_0/(3H_0^2)$, ${\Omega }_{\Lambda }=\Lambda /(3H_0^2)$ and ${\Omega }_k=-k/H_0^2$. The dust-plus-cosmological-constant Friedmann equation and its acceleration equation becomeAt the bounce these giveFor a past bounce with $s>1$, eliminate ${\Omega }_k$ from the first equation:Subtracting ${\Omega }_m{s}^3/2$ and multiplying by the positive denominator givesThus the matter bound from a Friedmann bounce isStrict positive bounce acceleration makes this inequality strict. The zero of the Friedmann equation also requires positive curvature for positive matter and positive $\Lambda$.

An observed source at $z=6$ has $a_e=1/7$. Since the minimum scale factor cannot exceed the scale factor at emission, $z_{\ast }\ge 6$. The denominator of the bound increases with positive $z_{\ast }$, soThe observational lower bound is much larger. Cluster mass-to-light ratios, inferred independently from motions, gas temperatures and gravitational lensing, combined with the observed luminosity density imply substantial cosmic mass density. Measurements of the cosmic baryon fraction from representative clusters, combined with the baryon cosmological density parameter inferred from Big Bang nucleosynthesis, provide another estimate: ${\Omega }_m\simeq {\Omega }_b/f_b$. Large-scale clustering and peculiar velocities provide independent dynamical checks, although their interpretation must account for galaxy bias. These methods support matter densities of order a few tenths, rather than below one percent. The dust-plus-cosmological-constant bouncing model is incompatible with both a quasar at redshift six and the stated observational evidence for ${\Omega }_m\gtrsim 0.2$.