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An inference of a compact object's gravitating mass from specified tracers and a dynamical or radiation model. Resolved stellar or gas orbits, spectroscopic binary dynamics, and reverberation mapping supply complementary approaches with different distance, geometry and equilibrium assumptions. A short variability timescale constrains size but is not a unique mass measurement.

An inference from radiation reflected by an irradiated accretion disc, including iron-line and continuum features. Doppler effects, gravitational redshift and ray propagation determine its profile. Associating the innermost reflecting radius with the innermost stable circular orbit can constrain spin, conditional on illumination, ionization, abundance, inclination and disk structure.

An inference of black-hole angular momentum, usually quoted through $a_{\ast }$. Accretion-based techniques use the dependence of the stable inner orbit and relativistic radiation transfer on spin in the Kerr black hole model. Disk truncation, inclination, illumination and atmosphere assumptions can imitate a change of spin and must enter the uncertainty model.

The process by which a stalled outgoing shock in a collapsing stellar core regains sufficient energy to expel outer material. Neutrino heating and multidimensional flow can contribute, while dissociation and radiation drain energy. Collapse does not guarantee a successful ordinary supernova.

At constant radiative efficiency of black-hole accretion $\eta$ and the Eddington luminosity, a black hole grows exponentially with the Salpeter time $t_S=\eta t_E/(1-\eta )$, where $t_E={\sigma }_{T}c/(4\pi G{m}_p)$. Its doubling time is $t_S\log 2$. The factor $1-\eta$ accounts for the supplied rest energy that escapes as radiation instead of increasing the hole's mass.

Selecting an inner fiducial volume reduces externally produced radiation because outer xenon attenuates it before it reaches the selected target. This trades target mass for reduced background. It does not remove uniformly distributed internal radioactivity or every neutron recoil, so purification, veto information and calibration remain important.

For pressureless matter, a positive cosmological constant, flat spatial geometry and negligible radiation, the continuity equation gives ${\rho }_m\propto a^{-3}$. Substitution in the Friedmann equation yields the displayed Hubble parameter, with ${\Omega }_{m0}+{\Omega }_{\Lambda 0}=1$. radiation must be restored at early epochs; the formula does not require matter to be the largest present-day density component.

Rapidly varying high-energy sources can appear too dense in target photons to transmit their observed gamma rays. For an approximately isotropic stationary region with target-band luminosity $L_2$, radius $R$ and target photon energy ${ϵ}_2$, the target number density is $n\sim L_2/(4\pi R^{2}c{ϵ}_2)$. photon-photon pair production then gives optical depth $\tau \sim n\sigma R$. A luminosity variability size bound can make this very large. relativistic beaming both increases the inferred rest-frame size and reduces the inferred rest-frame radiation density, allowing escape. This radiation-opacity argument is distinct from gravitational stellar compactness and topological compactness.

A network whose characteristic long-string separation grows with the horizon scale can lose energy into loops and radiation. Its density can scale as $\mu /t^2$ in natural units. This is distinct from the $a^{-2}$ dilution of an ideal frozen noninteracting string network.

radiation emitted by an incoming particle before the hard collision. At a hadron collider an energetic recoil jet can help trigger and measure events whose nearly degenerate parent-daughter decays otherwise yield soft visible particles.

Start with an inflationary patch of physical radius $H_i^{-1}$ in units $c=1$. After $N$ e-folds and subsequent expansion, its present radius is $H_i^{-1}{e}^N{a}_0/a_{\mathrm{reh}}$. Requiring it to cover the present Hubble radius gives the displayed inequality. With instantaneous conversion to radiation, fixed relativistic degrees of freedom and a fixed ratio $H_i/H_{\mathrm{end}}$, energy conservation gives $H_i\propto T_{\mathrm{reh}}^2$, whereas adiabatic subsequent expansion gives $a_0/a_{\mathrm{reh}}\propto T_{\mathrm{reh}}$. Thus the required $N$ increases approximately as $\log T_{\mathrm{reh}}$. Holding $H_i$ fixed while varying the end energy independently is a different comparison and reverses that simple temperature-only trend. The inflationary energy history and post-inflationary entropy history must be specified.^[ref]

Let the relevant rest-frame photon-photon pair production energy product be $K={ϵ}_1^{\prime }{ϵ}_2^{\prime }$ for a specified collision-angle prescription. For fixed observed high-energy photon energy, ${ϵ}_1^{\prime }={ϵ}_{1,\mathrm{obs}}/D$, so the target has ${ϵ}_2^{\prime }=DK/{ϵ}_{1,\mathrm{obs}}$. Observing the same internally produced target population supplies a second Doppler factor, giving the displayed result. An external stationary radiation field requires its own angular transformation instead.

The photon phase-space occupation is a Lorentz scalar. For radiation propagating through vacuum, this makes specific intensity divided by cubed frequency invariant under a change of inertial observer. With Doppler factor $D$, $I_{\nu }(\nu )=D^3{I}_{{\nu }^{\prime }}^{\prime }(\nu /D)$. This describes corresponding ray directions; it does not assert that an angularly integrated radiation field is isotropic in both frames.

For a Newtonian circular orbit, $v_{\varphi }^2=GM/r$, so the specific orbital energy is $e=v_{\varphi }^2/2-GM/r=-GM/(2r)$. Half of the released potential energy remains in circular orbital motion. The binding-energy estimate between the specified edges is thereforeUsing $M_{\odot }=1.98847\times 10^{30}\mathrm{kg}$ and one Julian year $=3.15576\times 10^7\mathrm{s}$ gives $\dot{m}=6.30\times 10^{22}\mathrm{kg}{\mathrm{s}}^{-1}$. Thus the radiative luminosity scale is $L\sim 1.9\times 10^{38}\mathrm{W}$, approximately $1.9\times 10^{45}\mathrm{erg}{\mathrm{s}}^{-1}$ or $4.9\times 10^{11}{L}_{\odot }$. The black-hole mass cancels because both edge radii scale with $M$. The specified inner edge is $6R_s$; it must not be silently replaced with a different relativistic orbit radius.

For a finite viscous disc, the precise radiated power also depends on boundary work: torque transport of energy in a steady accretion disk means that orbital binding-energy release need not all be radiated inside the two stated edges. If the intended model is a radiatively efficient steady Keplerian accretion disk with a zero-torque inner boundary condition, integrate the one-face flux $F=3GM\dot{m}[1-\sqrt{r_{\mathrm{in}}/r}]/(8\pi r^3)$ over both faces. It givesThe difference from $L_{\mathrm{bind}}$ is the outward torque power at the outer edge. The question supplies neither boundary torque nor the external material-supply energy budget, so the two calculations state their assumptions explicitly: about $2\times 10^{38}$ W is the binding-energy estimate, while $1.37\times 10^{38}$ W is the finite-disc zero-inner-torque dissipation result. Both neglect radiation advection, winds and relativistic corrections as required for this model.^[ref]

A cold self-gravitating core initially contracts on a dynamical scale $t_{\mathrm{ff}}\sim (G\rho {)}^{-1/2}$ if cooling prevents pressure from halting collapse. Increasing optical depth eventually traps radiation and forms a pressure-supported first hydrostatic core. Dissociation of molecular hydrogen absorbs compressional energy, softens the equation of state and triggers a second protostellar collapse to a compact protostar. Rotation produces a disk; angular-momentum transport permits continuing accretion, while outflows remove some mass and angular momentum.

An accreting object's luminosity includes a contribution of order $L_{\mathrm{acc}}\sim GM\dot{M}/R$, with a coefficient set by where accretion energy is radiated or retained. After accretion subsides, gravitational contraction supplies internal luminosity over the Kelvin-Helmholtz cooling time $t_{\mathrm{KH}}\sim G{M}^2/(R{L}_{\mathrm{int}})$. Hydrostatic contraction heats the interior even while energy is radiated, as follows from the virial theorem and the negative heat capacity of a virialized gravitational system. accretion luminosity must not be substituted for $L_{\mathrm{int}}$ in the contraction time without an energy-budget justification.

Low-mass, nearly fully convective pre-main-sequence stars contract down a Hayashi track at roughly constant effective temperature and decreasing radius and luminosity. A developing radiative core can lead onto a Henyey track toward higher effective temperature; the route depends on mass and opacity. The deuterium-burning energy-generation rate can temporarily moderate contraction, but sustained core hydrogen burning marks arrival on the main sequence. Massive objects may begin hydrogen burning while still accreting, and substellar objects never establish long-lived hydrogen burning. An infrared spectral class reflects envelope and disk emission as well as inclination and interstellar extinction, so it is not a unique stellar age or interior evolutionary track.

Stellar radiation, stellar winds and supernovae return energy, momentum, mass and metals to surrounding gas. Heating, ionization, turbulent support and gas removal can reduce subsequent star formation; swept-up shells can also compress gas locally. Whether thermal energy drives an outflow depends on its radiative gas cooling time relative to expansion: radiative losses can remove energy while leaving substantial momentum transfer. Thus deposited energy alone does not specify the dynamical effect.

The galactic gas regulator has mass budget ${\dot{M}}_g={\dot{M}}_{\mathrm{in}}-(1-R+\eta )\Psi$, where $R$ is the returned fraction and $\eta ={\dot{M}}_{\mathrm{out}}/\Psi$ is the mass-loading factor. In a quasi-steady state, $\Psi \simeq {\dot{M}}_{\mathrm{in}}/(1-R+\eta )$. This connects feedback to the available gas supply. A wind's kinetic power ${\dot{M}}_{\mathrm{out}}{v}_w^2/2$ and momentum flux ${\dot{M}}_{\mathrm{out}}{v}_w$ must fit the source budget and gravitational binding cost. Gas that fails to escape may form a galactic fountain and later return; preferential metal loss changes chemical evolution.

Absorption against a galaxy's stellar continuum provides a concrete outflow test: compare interstellar sodium absorption with the systemic velocity and stellar absorption spectrum. A primary starburst survey found blueshifted interstellar absorption and an inclination dependence consistent with minor-axis outflows. Its saturated lines also illustrate why velocity detection is more direct than a unique column density or mass flux.^[ref] Deprojection, covering fraction, ionization, radius and the unseen hot phase limit total outflow-rate estimates.

An accreting supermassive black hole adds radiation, winds and astrophysical jets. Coupling these to the interstellar or circumgalactic medium can suppress cooling or expel gas, especially in deep potentials where stellar driving alone may be inadequate. Jet cavities, line-emitting gas and multiphase outflows test different channels; coexisting activity and low star formation alone do not prove a causal feedback history.

Small primordial fluctuations seed later structure. For adiabatic cosmological perturbations on scales outside the horizon, the uniform-density curvature perturbation is conserved when the non-adiabatic pressure perturbation and the decaying mode can be neglected. A density contrast outside the horizon is gauge-dependent, so its apparent power-law growth should not be confused with directly measured local collapse.

For subhorizon, pressureless matter, the linearized continuity equation, Euler equation and Poisson equation combine intoHubble expansion damps peculiar motion; self-gravity amplifies overdensities. During radiation domination, radiation pressure drives acoustic oscillations and cold dark matter growth after horizon entry is only logarithmic. During matter domination the growing and decaying solutions are $\delta \propto a$ and $\delta \propto a^{-3/2}$. Dark-energy domination later suppresses growth relative to the matter-era law.

For a pressure-supported fluid the Fourier-mode equation also contains $+c_s^2{k}^2\delta /a^2$, giving a Jeans length separating acoustic support from gravitational growth. Before recombination, baryons are tightly coupled to photons and oscillate rather than freely following dark matter. Silk damping from photon diffusion damps small-scale photon-baryon fluctuations; after decoupling, baryons can fall into dark-matter potential wells. free streaming smooths species-dependent scales and distinguishes cold from warm or hot dark matter.

Schematically, the linear matter power spectrum can be written $P_{\delta }(k,a)=A{k}^{n_s}{T}^2(k)D^2(a)$ under the usual primordial-curvature normalization. The cosmological transfer function records horizon entry, matter-radiation equality, acoustic physics and damping; the linear growth factor records later amplification. density contrasts approaching unity require nonlinear collapse and halo formation beyond this approximation. cosmic microwave background anisotropies and galaxy clustering test the same initial perturbations through different transfer histories; galaxy bias, redshift-space distortions and observational selection enter the galaxy comparison.

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 a homogeneous canonical scalar field, ${\rho }_{\varphi }={\dot{\varphi }}^2/2+V$ and $P_{\varphi }={\dot{\varphi }}^2/2-V$. Potential-energy domination gives $P_{\varphi }\simeq -{\rho }_{\varphi }$, accelerated expansion and a decreasing comoving Hubble radius. Hubble friction can permit slow roll down a sufficiently flat potential. Enough inflation stretches an initially small causally connected region to encompass our observable universe, suppresses the relative spatial-curvature term and dilutes unwanted massive relics. quantum fluctuations supply possible seeds for cosmological structure; a successful model must also end inflation and create the hot post-inflationary universe. These statements do not imply that arbitrary initial conditions inevitably inflate.

Use the stated reduced-Planck units throughout the field calculation. Differentiating the Friedmann constraint and substituting the field equation givesDivision assumes the nontrivial expanding branch $H>0$; the vacuum endpoint follows continuously.

During rapid oscillations, require $m\gg H$ and $|\dot{A}|\ll mA$. To leading order, $\dot{\varphi }\simeq -mA\sin mt$, while the Friedmann constraint gives $H_m^2\simeq m^2{A}^2/6$. ThusAveraging over many oscillations on a timescale still short compared with $H^{-1}$ givesThe quadratic-inflaton Hubble oscillations obey, to leading order in $H_m/m$,Differentiating this expression reproduces the cosine term, with smaller corrections from the slowly varying amplitude. Instantaneously $H$ remains nonincreasing because its exact derivative is $-{\dot{\varphi }}^2/2$; an oscillatory correction does not mean oscillatory growth of the total Hubble parameter.

The mean equation integrates toThe shift $t_{\ast }$ is set by matching to the earlier evolution, not necessarily the Big Bang origin of the full solution. Averaged kinetic and potential energies are equal, so $⟨P_{\varphi }⟩\simeq 0$, $⟨{\rho }_{\varphi }⟩\propto a^{-3}$ and $A\propto a^{-3/2}$. The coherent quadratic oscillations behave as pressureless matter, not radiation, even though they are a scalar field rather than a gas of cold particles. radiation would instead give $a\propto t^{1/2}$. The averaged approximation is controlled only after oscillations are fast compared with expansion.

reheating requires interactions beyond the isolated field equations above. The oscillating inflaton can decay into other particles, whose scattering and further decays establish a thermal bath. The perturbative inflaton reheating equations areExpansion redshifts the components differently; transfer becomes efficient around $H\sim \Gamma$, and radiation domination follows when the transferred relativistic energy dominates. For suitable couplings, nonperturbative preheating by parametric resonance can precede thermalization, with backreaction and rescattering affecting its completion. Neither the decay rate nor thermalization is specified by $m$ and $V$ alone. Instantaneous reheating idealizes all this as immediate conversion of the end-of-inflation energy to radiation.^[ref]

For instantaneous-reheating horizon matching, specify the comparison: take the initial patch to be of order $H_i^{-1}$, compare the same-shaped inflationary history with approximately fixed $H_i/H_{\mathrm{end}}$, and assume instantaneous reheating followed by adiabatic expansion. Ignoring changes in relativistic degrees of freedom, $a_0/a_{\mathrm{reh}}\propto T_{\mathrm{reh}}$, while energy conservation and the Friedmann equation give $H_{\mathrm{end}}\propto T_{\mathrm{reh}}^2$. The present size of this inflated patch therefore scales asAt fixed present Hubble radius, a higher instantaneous reheating scale requires more e-folds in this standard comparison, approximately $\Delta N\simeq \ln (T_2/T_1)$, up to slow scale and entropy-degree corrections. Hotter reheating gives more subsequent expansion, but the associated initial Hubble patch is smaller by a stronger power of temperature. If instead one artificially holds the initial physical patch or $H_i$ fixed while varying the end energy independently, the post-inflationary geometric effect alone would give the opposite trend. temperature without an inflation-scale comparison convention does not fix that alternative problem; this is why the matching assumptions must be stated.^[ref]

Cosmological variables and conventions. The scale factor $R(t)$ describes the relative separation of comoving observers; dots denote derivatives with respect to cosmic time $t$. The Hubble parameter is $H(t)=\dot{R}/R$, usually time dependent; “Hubble constant” conventionally means its present value. Here $\rho$ is mass-equivalent density, so $ϵ=\rho c^2$ is energy density, while $P$ is isotropic pressure. $G$ is the gravitational constant and $c$ the speed of light. The constant $K$ specifies cosmological spatial curvature: with $R$ a length, $K=k{c}^2$ and $k=+1,0,-1$ describe closed, flat and open spatial geometries. Alternative normalizations of $R$ move a constant factor into $K$. In the printed equations $\Lambda$ has units ${\mathrm{s}}^{-2}$: it is $c^2$ times a cosmological constant defined in inverse square metres. This time-unit convention for the cosmological constant prevents a missing factor of $c^2$.

Continuity and the first law. Differentiating the first Friedmann equation with constant $K$ and $\Lambda$, and then eliminating $\ddot{R}/R$ with the second, givesThus, on an expanding or contracting branch,The relation extends continuously to a regular turning point. For a physical volume $V(t)=V_{\ast }{R}^3(t)$ bounded by fixed comoving coordinates, $\dot{V}=3HV$. Multiplication by $c^{2}V$ givesThis adiabatic work in a comoving volume is the first law of thermodynamics for adiabatic evolution with fixed entropy and no additional chemical work. It expresses local energy conservation, including pressure work. It does not assert that $U$ is constant: radiation loses internal energy as it does work; negative pressure makes $U$ grow during expansion. A generic expanding spacetime has no global time-translation symmetry from which a conserved total energy follows.

Constant scalar-field energy density. Setting $\dot{\rho }=0$ in the continuity equation, with $H\ne 0$, givesFor a homogeneous canonical scalar field, in units $c=1$, $ϵ={\dot{\varphi }}^2/2+V(\varphi )$ and $P={\dot{\varphi }}^2/2-V(\varphi )$. A stationary potential-dominated field has precisely this equation of state. A general evolving scalar field need not have $w=-1$; the constant-density assumption is essential.

Expansion at the stipulated Higgs energy density. With zero explicit $\Lambda$ and zero $K$, constant ${ϵ}_H$ givesThe plus sign selects the expanding branch; there is also a contracting solution. The scalar-field energy is already included in $\rho$, so adding it again as an explicit cosmological constant would double count it. Using $1\mathrm{GeV}=1.602176634\times 10^{-10}\mathrm{J}$ gives ${ϵ}_H=2.4033\times 10^{45}\mathrm{J}{\mathrm{m}}^{-3}$, ${\rho }_H=2.6740\times 10^{28}\mathrm{kg}{\mathrm{m}}^{-3}$ and $H_H\simeq 3.87\times 10^9{\mathrm{s}}^{-1}$. ThereforeThis vacuum-dominated doubling time describes accelerated expansion in de Sitter spacetime, with $\ddot{R}/R=H_H^2>0$. The stipulated energy density exceeds the critical density in energy units used in Question 2 by $3\times 10^{54}$, so it cannot persist as the dominant component of the present universe. The calculation exhibits the cosmological constant problem: the absolute gravitational vacuum energy must be enormously smaller than such a particle-physics scale today. It does not establish that the Standard Model Higgs field actually remains constant or supplies a viable inflationary epoch.

The acoustic ruler and accelerated expansion. Before baryon drag becomes ineffective, pressure in the photon-baryon fluid drives acoustic waves around initial linear cosmological density perturbations. Their comoving sound horizon at the baryon drag epoch isAfter decoupling, this preferred separation survives in galaxy clustering as baryon acoustic oscillations: a peak in the two-point correlation function and oscillations in the matter power spectrum. Its value is calibrated by early-universe physics, commonly constrained by the cosmic microwave background. The SDSS luminous-red-galaxy measurement established a peak near $100h^{-1}\mathrm{Mpc}$; its use as a distance indicator depends on the ruler calibration rather than the length of an individual galaxy.^[ref]

For a ruler perpendicular to the line of sight and one parallel to it,where $D_M=(1+z)D_A$ is the transverse comoving distance and $D_A$ the angular diameter distance. These measure $D_M/r_d$ and $H{r}_d$, respectively. Comparing multiple redshifts tests the distance–redshift relation and expansion history. In particular,so sufficiently precise measurements of the shape of $H(z)$ can test $q<0$. In general relativity, accelerated expansion requires a sufficiently negative total $\rho +3P/c^2$; a positive cosmological constant supplies it.

A single acoustic angular scale does not independently determine acceleration, cosmological spatial curvature, the absolute sound horizon and the dark-energy equation of state. One combines redshift coverage with early-universe calibration and independent matter-density or distance information, comparing accelerating and nonaccelerating models. The galaxy selection function and redshift completeness must be modeled with random catalogues, survey covariance estimated with mock catalogues, and redshift-space distortions, galaxy bias and nonlinear broadening controlled. These tests distinguish a cosmological feature from a survey artifact. The evidence is the measured expansion history within an explicit cosmological framework, rather than the existence of an acoustic peak alone.