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Use ${\rho }_r,{\rho }_m$ as energy per volume, as specified in the prose, and let $R$ have dimensions of length so that the positive curvature magnitude $k$ is dimensionless. The Friedmann equation for spatial curvature $-k$ isFor an energy density the cosmological perfect-fluid continuity equation is $\dot{\rho }+3H(\rho +P)=0$. Nonrelativistic matter has negligible pressure and radiation has $P_r={\rho }_r/3$. Thus the four contributions to $H^2$ scale respectively asFor nonzero positive densities and $\Lambda >0$, the usual possible sequence is radiation domination, matter domination, curvature domination, then cosmological-constant domination. A distinct era for each is not guaranteed: coefficients may cause curvature to be overtaken by the cosmological constant before it ever dominates. If $\Lambda =0$, curvature dominates asymptotically; negative $\Lambda$ does not give a late positive vacuum-dominated expansion.

The printed auxiliary relation $P_r={\rho }_r{c}^2/3$ uses mass-density notation, in conflict with the stated energy-density definition. It is consistent if ${\rho }_r$ there is understood as energy density divided by $c^2$; in that convention the density term in the Friedmann equation is $8\pi G\rho /3$. The two conventions must not be mixed.

In the prescribed matter-dominated approximation, neglect curvature, radiation, and the cosmological constant throughout the calculation. The Friedmann equation and ${\rho }_m\propto R^{-3}$ give $\dot{R}\propto R^{-1/2}$, hence $R\propto t^{2/3}$ and $H=2/(3t)$. For $H_0=70\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}=2.269\times 10^{-18}{\mathrm{s}}^{-1}$,The particle horizon is a proper distance on a constant-time slice, not simply $ct$:ThereforeThese are estimates within a spatially flat, matter-only model, rather than the measured age and particle horizon of the actual late-time universe.

Normalize the scale factor to $a(t_0)=1$ and set $a_d=10^{-3}$. In the same matter-only model $a=(t/t_0{)}^{2/3}$, soIts present proper size after expansion with the scale factor isThe present comoving distance to decoupling isThe angular diameter distance is $D_A=a_d{\chi }_d$, so the angular size of a transverse length equal to one particle horizon radius isIf “the horizon” denotes the diameter of the whole causally connected ball, its angular diameter is twice this, approximately $3.74^{\circ }$. The degree-scale conclusion is the same. Neglecting early radiation is an explicit approximation inherited from the question.

Source: www.whatdotheyknow.com/request/oxford_physics_past_papers_2/response/2340696/attach/9/Physics%20Papers%202020.pdf?cookie_passthrough=1

Source bundle pages 70–73; the cover identifies Trinity Term 2021 despite the attachment filename. Use coordinates $x^{\mu }=(ct,x,y,z)$, metric signature $(-,+,+,+)$, and $A^{\mu }=(\varphi /c,\mathbf{A})$. For the electromagnetic field tensor use $F^{0i}=E_i/c$ and $F^{ij}={ϵ}_{ijk}{B}_k$. Prescribed external fields are treated classically, neglecting radiation reaction in the trajectories; radiation losses are computed separately where requested.

Let $z^{\mu }(\tau )$ be the point electric charge's world line, with proper time $\tau$ and four-velocity $U^{\mu }=d{z}^{\mu }/d\tau =\gamma (c,\mathbf{u})$. For the field event $x^{\mu }$, choose the retarded source event: $R^{\mu }=x^{\mu }-z^{\mu }({\tau }_{\mathrm{ret}})$ is future-directed and null. Write $R^{\mu }=(R,R\mathbf{n})$, where $R=|\mathbf{x}-\mathbf{z}(t_{\mathrm{ret}})|=c(t-t_{\mathrm{ret}})$. In the instantaneous rest inertial frame of that source event,Consequently the covariant expression matching this Coulomb potential isThe denominator is a Lorentz scalar, so the right-hand side is a four-vector with the required instantaneous-rest components. This describes the retarded potential in Lorenz gauge, with no added free electromagnetic wave. Lorentz covariance alone does not establish the boundary conditions or exclude additional terms depending on four-acceleration. The physical input is the retarded point-source solution of Maxwell equations: the potential at the retarded event depends on the instantaneous four-current, while its derivatives generate the acceleration field. Equivalently, evaluating the retarded source integral gives a Jacobian $1/(1-\mathbf{n}\cdot \mathbf{\beta })$ from the light-cone constraint; this fixes that input for arbitrary smooth subluminal motion.^[ref]

In any inertial frame, put $\mathbf{\beta }=\mathbf{u}/c$ and $\kappa =1-\mathbf{n}\cdot \mathbf{\beta }$. Then $-R\cdot U=\gamma cR\kappa$, and the Liénard–Wiechert potential isDifferentiate using $\mathbf{E}=-\nabla \varphi -{\partial }_t\mathbf{A}$ and $\mathbf{B}=\nabla \times \mathbf{A}$, including the position and time dependence of the retarded time. For example, $\partial t_{\mathrm{ret}}/\partial t=1/\kappa$ and $\nabla t_{\mathrm{ret}}=-\mathbf{n}/(c\kappa )$. The resulting Liénard-Wiechert field isHere the dot denotes a derivative with respect to source coordinate time. The first term is the velocity or non-radiative field: it falls as $R^{-2}$ and exists even without acceleration. The second is the acceleration or radiation field, falling as $R^{-1}$ and transverse to $\mathbf{n}$. Its Poynting flux falls as $R^{-2}$, so its outward power integrated over a large sphere stays finite as its radius tends to infinity. The first term's squared-field flux instead vanishes in that limit. This large-distance distinction defines radiation; a moving near field can carry local energy without constituting radiation escaping to infinity.

A steady planetary surface energy balance equates absorbed incoming radiation plus upward internal heat flux with outgoing thermal radiation. For local incident stellar radiative flux $F_r$, surface albedo $\alpha$, upward conductive heat flux $F_c$ and thermal emissivity $ϵ$,The reflected part $\alpha F_r$ has already been subtracted. Equivalently, incident plus internal inputs equal reflected plus emitted outputs: $F_r+F_c=\alpha F_r+ϵ\sigma T_s^4$. Including reflected radiation again alongside an already absorbed input would count it twice. This simple surface equation omits downward atmospheric long-wave radiation and turbulent or latent heat fluxes; those require additional terms when present.

For an airless spherical planet with uniform temperature, intercepted stellar power is $\pi R^2{F}_{\star }$, while the emitting area is $4\pi R^2$. A whole-planet balance therefore uses mean incident flux $F_{\star }/4$ and Bond albedo ${\alpha }_B$:when internal heating is negligible. The factor four comes from geometry, not an extra loss mechanism. A nonuniform temperature field requires averaging $T_s^4$, rather than raising the mean temperature to the fourth power. In a planetary ice shell, a nonzero basal conductive contribution can keep the surface warmer than stellar heating alone predicts.

Loss of a rotating neutron star's angular momentum and rotational energy through electromagnetic torque and a magnetized outflow. For constant moment of inertia $I$, differentiating $E_{\mathrm{rot}}=I(2\pi /P{)}^2/2$ gives the displayed positive loss rate when $\dot{P}>0$. It is not the same as the measured radiation luminosity in any one observing band.

Radiation transports energy away from a source through propagating fields or emitted particles. Electromagnetic radiation includes radio waves, visible light, X-rays and gamma rays; alpha particles, beta particles and neutrons provide particle-radiation examples. The word alone does not specify an energy, a hazard, or a detection method.

An experiment must distinguish the incident field from its interaction with the detector. A photodetector measures a light-induced response, while an ionization chamber collects charge created in a gas. Neither instrument responds equally to every kind of radiation. Ionizing radiation is classified by its ability to remove bound electrons under the relevant interaction conditions, rather than simply by whether the source is radioactive.

The angular multipole of order two in a local radiation intensity distribution. A monopole is isotropic and a dipole has one directional axis; a quadrupole has a second-order angular pattern. Thomson scattering of an incident radiation quadrupole can generate linear polarization, which is central to Cosmic microwave background polarization.

A radiative-convective boundary separates a region transporting energy mainly by radiation from a region where convection carries an appreciable fraction. In a homogeneous, optically thick fluid, compare the radiative temperature gradient required to carry the flux with the adiabatic temperature gradient. When the required gradient is steeper, a displaced parcel can remain warmer and more buoyant than its surroundings, making the radiative state unstable under the Schwarzschild criterion.

A cooling giant planet commonly has a convective deep interior beneath a radiative layer. Stellar heating can push their boundary deeper and slow the interior's heat loss. Multiple convective regions are possible, and composition gradients modify the stability test; the boundary is therefore not a universal fixed pressure or an assurance that everything below it convects.

The study of biological responses to radiation. For ionizing radiation, the deposited energy and its microscopic distribution affect molecular damage and subsequent cellular responses, so equal absorbed doses need not produce identical biological outcomes.

Ravi Kumar Kopparapu coauthored estimates of stellar habitable zones using radiative-convective climate models. The boundaries depend on stellar radiation and atmospheric assumptions; being inside a modeled zone is a climatic possibility criterion rather than evidence that a planet is inhabited.^[ref]

The reheating temperature characterizes the thermal radiation bath after a nonthermal component transfers its energy to other particles and those particles reach thermal equilibrium. Use natural units and the reduced Planck mass $M=(8\pi G{)}^{-1/2}$. In the sudden decay of a dominant nonrelativistic relic, decay rate $\Gamma$ and the Friedmann equation give ${\rho }_R\simeq 3M^2{\Gamma }^2$ at $H\simeq \Gamma$. Equating this with equilibrium radiation energy density givesHere $g_{\ast ,\rho }$ counts the effective relativistic energy degrees of freedom, so the formula can be implicit when the particle content changes with temperature. For example, increasing $\Gamma$ by a factor four doubles this estimate at fixed $g_{\ast ,\rho }$.

The residual bath just before decay can be much colder, while gradual reheating can temporarily produce a maximum bath temperature above the eventual radiation-domination value. Prompt energy transfer is also distinct from prompt thermalization. A scalar scalar potential alone therefore does not fix $T_R$ without interactions and a transfer history.^[ref]

Angular concentration and Doppler enhancement of radiation from a relativistically moving emitter. Frequency, photon arrival rate and apparent solid angle transform together; inferred brightness depends on viewing angle and the emission geometry.

Robert B. Leighton coauthored The Feynman Lectures on Physics, Volume I by Richard Feynman and others with Richard Feynman and Matthew Sands. Its radiation treatment connects the motion of a electric charge with a transverse electromagnetic field observed far away. The component of acceleration perpendicular to the viewing direction determines the angular pattern in the nonrelativistic example. This links an observable light pattern to source motion, rather than introducing radiation only as a formula for total power.^[ref]

Sebastian Bathiany coauthored an analysis explaining why increasing carbon dioxide can cool the middle atmosphere while contributing to warming elsewhere. The work compares radiative regimes rather than treating a gas as having one temperature effect independent of its surroundings.^[ref]

The physical interpretation depends on the balance of absorbed and emitted radiation, the atmospheric structure and the level at which energy escapes. This is a concrete application of radiative transfer to climate.

Consider unit energy of incident radiation travelling downward into a conservatively scattering slab. Let $t$ be its unscattered transmission fraction. Of the fraction $1-t$ that scatters, suppose a fraction $a$ continues exactly forward and the remaining fraction $1-a$ has an isotropic outgoing distribution. Approximate every scattered contribution as escaping without a second interaction.

The isotropic part splits equally between upward and downward directions. The escape fractions are thereforeThey sum to one, as required by conservation of energy without electromagnetic absorption. For example, $t=0.6$ and $a=0.5$ give $f_{\uparrow }=0.1$ and $f_{\downarrow }=0.9$. Exactly forward scattering does not remove downward energy in this model; treating all scattering as backward loss would give a different result.

For a specified direct path, exponential attenuation gives $t=e^{-{\tau }_s}$ with scattering optical depth ${\tau }_s$. The single-escape assumption becomes unreliable when repeated scattering is important. A thick slab generally requires the angular scattering phase function and the full radiative transfer equation, rather than this two-branch bookkeeping.

The sudden-decay approximation replaces a dominant nonrelativistic species $X$ by radiation near $H=\Gamma$, neglecting expansion during the conversion. Use natural units, reduced Planck mass $M=(8\pi G{)}^{-1/2}$ and relic mass $m$. Define the pre-decay yield $Y=n_X/s_{-}$, where $n_X$ is its number density and $s_{-}=(2{\pi }^2/45)g_{\ast ,s}{T}_{-}^3$ is the residual bath's entropy density. Relic domination givesAfter conversion and prompt thermalization, conservation of energy giveswhich defines the reheating temperature. The entropy count $g_{\ast ,s}$ and energy count $g_{\ast ,\rho }$ have different definitions and need not be identical. The yield in the first equation is defined before the newly produced entropy dilutes it.

If the relic energy greatly exceeds the residual radiation energy, the post-decay temperature can greatly exceed $T_{-}$. The related entropy production by decay of a dominant relic follows by comparing the two equilibrium baths at the same scale factor. The assumptions are relic domination, nonrelativistic relics and rapid conversion/thermalization; a gradual decay or incomplete equilibrium requires coupled evolution instead.

Temperature measurement infers temperature from a reproducible physical response of a thermometer. Examples include a liquid column's thermal expansion, a sensor's electrical resistance and the thermal radiation emitted by a surface. Calibration connects the response to specified reference temperatures or a physical temperature scale.

A contact probe must exchange heat with the object, so its reading depends on time and thermal coupling. A small probe can respond rapidly, while a large probe can noticeably disturb a small sample. Heat conducted along the probe stem and electrical self-heating can shift the reading away from the object's undisturbed temperature.

A useful immersion experiment is to insert a probe progressively farther into a well-stirred bath and wait after each movement. A reading that changes with immersion depth reveals a heat-leak or equilibration problem. Repeating a calibration does not by itself remove such measurement-condition errors.

The transition frequency between two stationary energy levels is the positive frequency satisfying $h\nu =|E_f-E_i|$. In absorption, the system gains this energy from the radiation; in emission, it loses it. The corresponding angular frequency is $\omega =2\pi \nu$.

An energy difference specifies where a spectral line can occur, but does not by itself guarantee that the transition is strong or allowed. The interaction and its selection rules determine the transition amplitude. External fields, motion and finite lifetimes can shift or broaden the measured line.