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A hot, comparatively optically thin plasma region associated with an accretion flow. Thermal or nonthermal electrons can transfer energy to softer seed photons through inverse Compton scattering, producing a harder radiation component. Its geometry, heating and coupling to the cooler disc are model dependent; a hard radiation spectrum alone does not uniquely determine these properties.

The Earth's changing velocity relative to a halo changes the recoil-rate integral above a detector threshold. Its phase and energy dependence depend on the velocity distribution and interaction, so a sinusoidal fit is a model approximation. Seasonal instrumental or environmental backgrounds can also vary annually; energy radiation spectrum, multiple-hit samples, time stability and independent targets help test a proposed dark-matter interpretation.

Let the scale factor grow by $f$ from an initially thermal photon distribution. geodesic redshift gives ${\nu }_i=f\nu$, and conserving photons in corresponding comoving frequency bins gives $n_{\nu }(t)=f^{-2}{n}_{f\nu }(t_i)$. Inserting Planck's law leaves the same spectral form at temperature $T_i/f$. Total photon-number conservation alone would not prove that the entire radiation spectrum stays Planckian; frequency redshift and collisionless propagation are also required. Absorption, injection or spectral mixing can invalidate this one-temperature argument.

A contained neutrinoless event deposits approximately the nuclear Q-value in the two-electron sum, apart from recoil and losses. Ordinary two-neutrino double beta decay has a continuous electron-sum radiation spectrum. Resolution, escape losses and backgrounds broaden or mimic features, so a peak assignment requires calibrated response, sidebands and appropriate event topology. A null peak search is a limit on a decay rate under an efficiency/background model, not proof of Dirac neutrinos.

Multiplying a finite interferogram by a smooth window reduces sharp-edge artifacts. In the recovered radiation spectrum this convolves the true spectral density with the window's Fourier transform. Smoother windows generally lower sidelobes while broadening the main lobe, so a resolution figure must specify the window and criterion. Apodization does not recover unmeasured long-delay information.^[ref]

The metric source is a gravitational frequency shift. photons following null geodesics through a time-dependent spatial metric change momentum relative to the background comoving frame. The directional projection of that metric change gives the ${\mu }^2$ factor. It is not a collision term or continuing photon scattering after recombination. The normalization and sign belong to the specified definition of $h$ and $\Delta$; a distribution perturbation should not silently be identified with temperature without fixing that normalization.

Define $\mathcal{H}=\dot{R}/R$, where dots here mean conformal-time derivatives. In an Einstein-de Sitter background,A power law $h\propto {\tau }^p$ therefore satisfies $p(p-1)+2p=6$, or $(p-2)(p+3)=0$. Thuswith the required growing mode $h=A{\tau }^2$. This is a solution in the perturbation convention used in the question, rather than a statement that every gauge-invariant gravitational potential grows as ${\tau }^2$.

Use an integrating factor for the photon equation:For the growing mode $\stackrel{...}{h}=0$. For the polynomial-source photon transport solution, a particular solution is $B_h=-2i\mu \dot{h}/k+2\ddot{h}/k^2$: direct substitution gives ${\dot{B}}_h+ik\mu B_h=2{\mu }^2\dot{h}$. The initial-value solution is consequentlyThis also follows by integrating the source twice by parts. The apparent inverse powers of $k$ cancel in the complete long-wavelength limit; the integral form remains regular. At $\mu =0$, the source vanishes and the complete expression reduces to $\Delta ={\Delta }_R$.

Physically, the exponential transports the recombination pattern along photon trajectories; its direction-dependent phase projects a Fourier perturbation onto different angular multipoles. The extra terms are the accumulated gravitational contribution, here expressible as endpoint terms because the source has this special time dependence. For endpoint monopole and dipole in photon transport, at the observer the last two terms are respectively a directional dipole and an angular monopole for each mode; they do not supply a new sky pattern at multipoles $\ell \ge 2$. The propagating recombination combination contains the intrinsic perturbation and gravitational/velocity endpoint contributions. Individual terms depend on the perturbation gauge; observable temperature anisotropy requires the consistent emitter and observer contributions.

In pure matter domination the gauge-invariant scalar potentials are constant for a growing linear mode: the matter contrast grows as $R$ while the potential scales as its contrast divided by $R$. Thus growth of this $h$ is compatible with no late Integrated Sachs-Wolfe effect in Einstein-de Sitter. When dark energy suppresses matter growth, the potentials generally evolve; photons gain and lose unequal energies entering and leaving a potential well. In a convention where $\Phi ,\Psi$ are the dimensionless Newtonian-gauge potentials, the integrated temperature source is $\int (\dot{\Phi }+\dot{\Psi })d\tau$ along the light path.

Large-angle cosmic microwave background data can constrain that late contribution, but low-multipole power alone has large cosmic variance and depends on the primordial radiation spectrum, geometry and foreground treatment. Cross-correlating the temperature map with low-redshift galaxies or other matter tracers isolates a late-time connection; decaying wells typically correlate overdensities with hotter cosmic microwave background directions. Boughn and Crittenden compared WMAP temperature data with the HEAO1 hard X-ray background and examined possible foreground origins of the correlation. This tests potential evolution under a cosmological model, not dark energy independently of all gravity and curvature assumptions.^[ref]

Gravitational evidence requires much more nonbaryonic matter than can be supplied by ordinary luminous material or by the baryon abundance permitted by nucleosynthesis. A stable, electrically neutral massive particle is a plausible cold component. Thermal production followed by cosmological particle freeze-out gives an additional motivation: an annihilation rate near a weak-interaction scale can leave an appreciable relic abundance. This is a mechanism and search hypothesis, not proof that the dark matter must be a WIMP or have any particular elastic-scattering cross section.

The plotted regions are conditional WIMP interpretations of different measurements. The experiments use different targets and distinguish signal from background differently:

The plot also includes CRESST experiment, whose cryogenic calcium-tungstate modules combine phonon and scintillation-light signals. Different nuclei contribute to its recoil interpretation. Its excess region is another conditional candidate region, not an extra confirmed detection.^[ref]

Around ten GeV, the shaded regions favour cross sections roughly $10^{-41}$–$10^{-40}\mathrm{c}{\mathrm{m}}^2$, while the XENON100 solid curve and the improved low-mass CDMS curve cut into or exclude much of those regions under the common elastic, spin-independent WIMP-nucleus scattering interpretation expressed per nucleon. There is tension within that shared model; the plot does not show mutually confirmed discoveries. electron backgrounds, threshold response, nuclear-recoil quenching factors, channeling assumptions and the halo velocity distribution can shift interpretations. Changing the interaction model can also alter target comparisons, but cannot establish a WIMP by assertion. The 100-live-day XENON100 result itself reported candidates compatible with estimated background.^[ref]

For a rough dimensional event estimate, $1\mathrm{c}{\mathrm{m}}^2=10^{-4}{\mathrm{m}}^2$ givesIf a collider production cross section had this value, that small yield, before acceptance and backgrounds, would not support a likely discovery. The plotted nuclear-scattering cross section is not the LHC production cross section, however. Converting between them needs a mediator, couplings, parton distributions and production kinematics; a light stable WIMP would normally be inferred from missing transverse momentum with a visible recoil object, not a visible WIMP track. The mass makes pair production kinematically accessible, but the plot and integrated collider luminosity alone do not decide discovery probability.

For the relic estimate, define $M=m{c}^2$ and $\theta =k_{B}T$. The sharp-freeze-out relic dilution model givesAssume stable particles, no subsequent annihilation, and $RT=\mathrm{constant}$. Conserved comoving particle number then givesThe present rest-energy density and its critical fraction areThus the supplied toy model yields a cosmologically substantial density, of the right broad scale to motivate a dark-matter candidate. It is not an independent precision prediction: $P$ is imposed and the abundance depends exponentially on it. If entropy degrees of freedom change, entropy conservation instead gives an extra factor $g_{\ast s,0}/g_{\ast s,f}$ in $n_0$ when using the photon temperature. The species content needed to evaluate that correction is not supplied. A realistic calculation also follows residual annihilations with the abundance Boltzmann equation rather than stopping exactly at equilibrium freeze-out.

annihilation can produce gamma rays, charged antiparticles and neutrinos, directly or through decay and hadronization of the primary products. For self-conjugate dark matter of mass $m_{\chi }$, the annihilation rate per unit volume is $⟨\sigma v⟩{\rho }_{\chi }^2/(2m_{\chi }^2)$. For a photon yield $d{N}_{\gamma }/dE$, the optically thin flux from a chosen solid angle isUse consistent mass-energy units. The factor $1/2$ assumes identical self-conjugate particles; distinct particle-antiparticle populations require their separate number densities. A gamma-ray continuum with an endpoint near the particle mass is generic for many channels, whereas a narrow line from $\chi \chi \to \gamma \gamma$ or $\gamma Z$ would be especially distinctive but often loop suppressed. dwarf spheroidal galaxies offer high dark-matter content with limited conventional emission; the Galactic centre has a potentially larger signal and much more complicated foregrounds.

Charged Positrons and antiprotons are redistributed by magnetic fields and lose energy, so their radiation spectrum depend on propagation and cannot usually point directly to the source. annihilation electrons can also produce synchrotron and inverse-Compton radiation. dark matter captured in the Sun could produce escaping high-energy neutrinos, with the signal depending on capture, annihilation equilibrium and particle interactions. Early annihilation energy injection can alter recombination and cosmic microwave background anisotropies.

A persuasive interpretation joins particle-like spectral structure with a consistent density-dependent morphology and other channels. Pulsars, cosmic-ray interactions and unresolved sources can imitate continuum excesses. Density-profile uncertainties, substructure and velocity-dependent annihilation modify the predicted signal; a lack of a detectable line does not exclude all annihilating dark matter.

A Majorana neutrino is its own antiparticle, with a mass term that can violate lepton number by two units. It is not thereby a state of both readily observable helicities with equal weak rates: charged-current vertices still select chirality. In light-neutrino exchange, the needed helicity connection between two beta vertices supplies a neutrino-mass factor.

Ordinary two-neutrino double beta decay is $(A,Z)\to (A,Z+2)+2e^{-}+2{\bar{\nu }}_e$ and conserves lepton number. It can occur when the direct single-beta route is energetically forbidden or highly suppressed but the double change is allowed. neutrinoless double beta decay, $(A,Z)\to (A,Z+2)+2e^{-}$, violates lepton number by two. For the conventional light-Majorana exchange in neutrinoless double beta decay mechanism,Here $U$ is the lepton mixing matrix, $G_{0\nu }$ the phase-space factor in this convention and ${\mathcal{M}}_{0\nu }$ a nuclear matrix element. Complex phases can cancel contributions to the effective Majorana mass; oscillation mass differences alone do not fix it. Other lepton-number-violating mechanisms can also cause the process, so a rate is not a model-independent measurement of $m_{\beta \beta }$.

The experimental signature is a peak in the summed electron energy at the nuclear Q-value, apart from small recoil and energy losses. Two-neutrino decay instead gives a continuous sum-energy radiation spectrum because neutrinos carry away variable energy. Isotope enrichment, large exposure, good energy resolution and radiopure shielding improve sensitivity; gamma backgrounds, radon, surface events and the two-neutrino tail require calibration and sideband controls. Coincidence or track information can test whether the topology is compatible with two electrons. A later concrete example is GERDA's 2013 germanium-76 source-detector search, which used pulse-shape discrimination and background controls rather than identifying any isolated energy deposit as a decay.^[ref]

The restoring, damping and electric forces on the electron give $m_e\ddot{x}=-kx-b\dot{x}-eE(t)$. Dividing by $m_e$ gives a Lorentz oscillator model withFor the $e^{i\omega t}$ convention, substitute $x=\mathrm{Re}(x_0{e}^{i\omega t})$. Then $x_0=-e{E}_0/[m_e({\omega }_0^2-{\omega }^2+i\gamma \omega )]$ andThe time-averaged dissipated power isFor positive frequency the squared term is minimized to zero only at $\omega ={\omega }_0$. Thus the power maximum is exactly at the undamped natural frequency, even though the displacement maximum is slightly shifted. Here $P_{\max }=e^2|E_0{|}^2/(2b)$. The half-power equation is $|{\omega }_0^2-{\omega }^2|=\gamma \omega$, whose two positive roots areThis is the absorbed-power bandwidth of a viscously damped oscillator, exact within the model. Near resonance the profile is approximately a Lorentzian line shape. With $c=2.99792458\times 10^8\mathrm{m}{\mathrm{s}}^{-1}$,For this wavelength linewidth of a damped-oscillator power resonance we used ${\omega }_{+}{\omega }_{-}={\omega }_0^2$. The half-power wavelength endpoints are not exactly symmetric around ${\lambda }_{\mathrm{peak}}$, although that distinction is tiny here.

In a constant field the settled displacement is $x_s=-e{E}_{\mathrm{dc}}/k$. On sudden removal, with zero initial velocity, the free motion isThe electron oscillates about its original equilibrium, losing the previously stored spring energy. The period is $3.52\mathrm{fs}$, the amplitude decay time is $2/\gamma =0.319\mathrm{ps}$, and the cycle-averaged energy decay time is $1/\gamma =0.160\mathrm{ps}$. This is the effective classical model supplied, not a prediction of the full hydrogen line radiation spectrum.

In Fourier transform spectroscopy, a scanning Michelson interferometer measures intensity against optical path difference $\ell$ rather than selecting one narrow wavelength at a time. After subtraction of the constant background, an ideal stationary source gives an interferogramA cosine Fourier transform recovers the spectral power density $S(\nu )$ after instrumental calibration. A mirror displacement $x$ changes the round-trip path by $2x$; the zero-path-difference burst aligns the scan, and a reference laser can calibrate the displacement.

Finite maximum path difference $L$ multiplies the interferogram by a window and convolves the recovered radiation spectrum with that window's Fourier transform. Thus $\delta \nu$ is of order $c/L$; precise factors depend on scan range and resolution convention. Uniform path samples of separation $\Delta \ell$ need ${\nu }_{\max }<c/(2\Delta \ell )$ to avoid aliasing without bandpass information. interferogram apodization reduces sidelobes at the price of broader lines. The broad accepted optical throughput can be advantageous; the multiplex signal-to-noise advantage depends on detector noise and is not universal in photon-noise-limited measurements. Source drift during a scan, finite beam divergence, imperfect sampling and wave phase errors require control. The finite-window reasoning is the delay-range resolution in Fourier transform spectroscopy.^[ref]

A donor star can feed a compact companion through Roche-lobe overflow near the inner Lagrange point, or through capture of a stellar wind. The transferred gas usually carries angular momentum; when its circularization radius exceeds the accretor's effective inner radius it forms an accretion disc. Outward angular momentum transport permits inward mass flow, while dissipation converts gravitational binding energy into heat and radiation. For a compact object with a surface $L\sim GM\dot{M}/R$; for a black hole write $L=\eta \dot{M}{c}^2$, with efficiency determined by the flow and inner boundary. Optically thick disc regions emit a sum of thermal radiation spectrum components; a hot optically thin accretion corona can inverse-Compton scatter softer photons into a harder X-ray component. A black hole has no material surface on which to radiate a stellar boundary layer.

An explicit example is the wind-fed black-hole binary Cygnus X-1. Its massive stellar companion supplies gas, and the accretion flow produces strong X-ray emission.^[ref] Compact-object evidence comes from orbital dynamics rather than X-ray brightness alone. The single-lined binary mass function constrains the unseen mass; inclination, donor properties and distance refine it. Jerome A. Orosz and collaborators combined optical photometry, radial velocities, rotation and a distance constraint to obtain a black-hole mass of $14.8\pm 1.0M_{\odot }$ in their 2011 analysis. The inference depends on the stellar and orbital model, and that historical value should not be confused with a latest measurement.^[ref]

For a blob in a relativistic jet, take two emission events separated by time $\Delta t_e$ in the galaxy frame. The blob moves transversely by $u\Delta t_e\sin \theta$, while its second photon has $u\Delta t_e\cos \theta$ less distance to travel to the distant observer. Therefore the arrival interval is $\Delta t_{\mathrm{obs}}=\Delta t_e(1-\beta \cos \theta )$ andDifferentiating with respect to viewing angle gives $d{\beta }_{\mathrm{app}}/d\theta =\beta (\cos \theta -\beta )/(1-\beta \cos \theta {)}^2$. The maximum occurs at $\cos \theta =\beta$ and equals $\beta \gamma$. Hence apparent superluminal motion is possible precisely when $\beta \gamma >1$, or $\beta >1/\sqrt{2}$. This is an arrival-time effect; the material and every local signal still travel slower than light.

For $\gamma \gg 1$, $1-\beta \simeq 1/(2{\gamma }^2)$ and $1-\beta \cos \theta \simeq ({\gamma }^{-2}+{\theta }^2)/2$. Thus the Doppler factor isThe Lorentz-invariant specific intensity gives $I_{\nu }/{\nu }^3=I_{{\nu }^{\prime }}^{\prime }/{\nu }^{\prime 3}$. Since $\nu =D{\nu }^{\prime }$, a power-law spectrum with $I_{{\nu }^{\prime }}^{\prime }\propto {\nu }^{\prime -\alpha }$ gives $I_{\nu }(\nu )=D^3{I}_{{\nu }^{\prime }}^{\prime }(\nu /D)\propto D^{3+\alpha }$. This is Doppler boosting of a moving blob: the spectral flux density of an unresolved, intrinsically isotropic blob has the same boost. Comparing identical approaching and receding blobs, with no differential absorption,A spatially steady continuous jet can have a different boost exponent because the emitting material contributing at one arrival time differs; it must not silently replace the specified blob model.

Using the stipulated peak of the photon-photon pair production cross section, not the exact head-on threshold, givesThe relevant photons are near infrared. Interpret the quoted $F_2$ as target-band energy flux, or energy flux per logarithmic frequency interval, consistent with its units $\mathrm{W}{\mathrm{m}}^{-2}$; it is not a spectral flux density per hertz. For isotropic emission $L_2=4\pi d^2{F}_2$. In a stationary source of radius $R$, the order-of-magnitude photon energy density is $L_2/(4\pi R^{2}c)$, soThe luminosity variability size bound gives $R\lesssim c\Delta t$. Taking the largest allowed radius gives the smallest stationary-source optical depth in this estimate:This assumes the gamma rays and target photons occupy the same region, with an approximately isotropic target field and an order-unity geometric escape factor. A smaller region makes the gamma-ray compactness problem worse.

Here $d=3\times 10^{25}\mathrm{m}$, $R\sim 3\times 10^{11}\mathrm{m}$, and ${ϵ}_{20}=1.67\times 10^{-19}\mathrm{J}$. ConsequentlySuch a stationary co-spatial source would absorb almost all of its high-energy photons.

Let primes denote the blob rest frame and continue to ignore cosmological redshift. For fixed observed gamma-ray energy, ${ϵ}_1^{\prime }={ϵ}_1/D$. The co-moving target energy at the assumed cross-section peak is therefore ${ϵ}_2^{\prime }=D{ϵ}_{20}$. The internal pair-production target energy in a relativistic blob acquires a second Doppler shift when this same target population is observed along our line of sight:It would be incorrect to transform only the gamma ray while treating the internal target radiation as a stationary external field. The target photons responsible for collisions span directions in the blob frame; the observed energy here labels the corresponding part of the internally emitted radiation spectrum.

relativistic beaming both increases the allowed rest-frame source radius, $R^{\prime }\lesssim Dc\Delta t$, and lowers the inferred rest-frame luminosity, $L_{\log }^{\prime }\sim L_{\mathrm{iso},\log }/D^4$. The beamed gamma-ray optical depth then follows from ${ϵ}_2^{\prime }=D{ϵ}_{20}$:For an unbroken power-law spectrum $F_{\nu }\propto {\nu }^{-\alpha }$, $F_{\log }=\nu F_{\nu }\propto {\nu }^{1-\alpha }$, so ${\tau }^{\prime }\propto D^{-(4+2\alpha )}$. In particular, $\alpha =1$ gives the stated $D^{-6}$. Spectral breaks, an external target field or non-coincident emission regions change this inference. Under the requested scaling, transparency requiresThis minimum uses the most favourable nearly axial view, $D\simeq 2\gamma$. The exact axial relation gives ${\gamma }_{\min }=(D_{\min }+D_{\min }^{-1})/2\simeq 8.94$; a less favourable viewing angle requires a larger Lorentz factor. The numerical bound is conditional on the stipulated opacity scaling and internal-source geometry.

A radiation spectrum whose spectral flux density follows a power law over a stated frequency range. For $F_{\nu }\propto {\nu }^{-\alpha }$, the spectral energy distribution satisfies $\nu F_{\nu }\propto {\nu }^{1-\alpha }$. Thus $\alpha =1$ gives equal energy flux in equal logarithmic frequency intervals. A spectral break invalidates extrapolation with a single index.