Past exam of the particle Astrophysics course of the University of Cambridge

2016 paper: Public original PDF, physical page 829. Student-answer availability for this paper remains unverified.

Original source: Released original, physical PDF pages 829–831. All three questions and every essay option are covered.

The known active neutrinos are too light and too fast to supply the dominant cold dark matter. Their relativistic decoupling leaves substantial free streaming, which erases rather than seeds sufficiently small-scale structure; their allowed mass density is also inadequate for the dominant cold component. Changing a light active neutrino from Dirac to Majorana does not remove those problems. What can change the picture is an enlarged neutral-fermion sector containing sterile states with different masses, production and decay rates. A sufficiently long-lived sterile neutrino in a Majorana state can be a dark-matter candidate; Majorana character alone supplies neither stability nor the correct abundance.

For the spinor calculation, put $a=p_z/(E+m)$ and define $A=N(1-a)/2$, $B=N(1+a)/2$. Applying the given chiral projectors givesConsequentlywhere the last equality uses $E^2-p_z^2=m^2$. Same-chirality Lorentz scalar products instead vanish: for example ${\overline{U}}_{1L}{U}_{1L}=0$. The nonzero overlaps demonstrate the Lorentz scalar structures coupling left and right chirality. For $U$ to $U$ this is a particle-number-preserving Dirac coupling; for $U$ to the charge-conjugate $V$ sector it can describe a lepton-number-violating Majorana coupling.

There is a normalization subtlety: both displayed overlaps equal the on-shell mass parameter used to define the spinors. A chiral scalar spinor overlap verifies a coupling structure without fixing an independent Lagrangian coefficient $M_L$. Repeating the on-shell construction with mass parameter $M_L$ would instead give the corresponding overlap $M_L$; this is a different input, rather than a newly derived parameter. The independent coefficients of a Dirac mass term and a Majorana mass term are defined byThe Majorana terms connect a field with its charge conjugate and violate the corresponding continuous lepton phase symmetry by two units. With only unbroken Standard Model gauge symmetry, a left-handed Majorana mass also requires electroweak symmetry breaking or an appropriate higher-dimensional operator; it is not an arbitrary gauge-invariant bare mass for a lepton doublet.

Use an all-left-handed Weyl basis ${\mathbf{n}}_L=({\nu }_L,{\nu }_R^c{)}^T$. For real parameters the one-family Dirac-Majorana mass matrix isWriting $({\nu }_{1L},{\nu }_{2L}{)}^T=\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right)({\nu }_L,{\nu }_R^c{)}^T$ gives an orthogonal diagonalization, with $\tan 2\theta =2m/(M_R-M_L)$ up to the rotation-sign convention. Massive-particle chirality and helicity are not identical; the familiar left/right helicity language becomes an approximation in the relativistic regime.

The characteristic polynomial isIts roots areFor Majorana fermions the nonnegative physical masses, as explained by physical Majorana mass and signed eigenvalue, are $|{\lambda }_{\pm }|$ after a possible field rephasing; a negative real eigenvalue is not a negative physical rest mass. When $M_R\gg |m|,|M_L|$,The light state is predominantly ${\nu }_L-(m/M_R){\nu }_R^c$, while the heavy state is predominantly ${\nu }_R^c+(m/M_R){\nu }_L$. Each mass eigenstate combines its Weyl component and charge conjugate to form a Majorana field.

For $M_R=10^{12}\mathrm{GeV}$, the heavy mass is approximately $10^{12}\mathrm{GeV}$. In the type-I limit $M_L\simeq 0$,The seesaw light-mass estimate needs a Dirac scale: no unique numerical light mass follows because the Dirac mass and $M_L$ were not supplied. As an explicitly conditional illustration, choosing $m=100\mathrm{GeV}$ and $M_L=0$ gives $10\mathrm{eV}$; this is an assumed Dirac scale, not additional exam data or a prediction from $M_R$ alone. A heavy seesaw state is also not automatically stable enough to be dark matter.

The conventional minimal Pati-Salam model gauge group isThe group printed in this paper is different: it contains two $SU(4)$ factors and only one $SU(2)$. This is a discrepancy in the original PDF, not just in the transcription. The standard-model discussion below uses the conventional intended Pati-Salam model model. A literal theory with the printed product would require additional fermion representations and symmetry-breaking data that are not specified.

The Pati-Salam model unifies the three quark color charges with lepton number as a fourth color charge. A family forms $(4,2,1)$ left-handed and $(4,1,2)$ right-handed multiplets, including a right-handed neutrino. The extended color charge group contains quark-lepton gauge bosons, while the right-handed weak factor supplies right-handed charged currents after symmetry breaking. The Pati-Salam charge embedding isThe original paper explicitly develops the smaller local subgroup $SU(2{)}_L\times SU(2{)}_R\times SU(4{)}_C$, as well as a larger underlying global structure; neither distinction licenses silently replacing the printed group.^[ref] The product group is a quark-lepton unification scheme rather than a simple gauge group with one automatically common coupling. Breaking the right-handed and $B-L$ symmetries can generate a large $M_R$ and light/heavy Majorana neutrinos through the seesaw. The gauge group alone does not uniquely fix their masses or guarantee a dark-matter lifetime.

The most important laboratory test of Majorana character is neutrinoless double beta decay,with no outgoing neutrinos. Under light-neutrino exchange, the amplitude involvesThe coherent squared mixing elements allow phase cancellations. A null result constrains this combination within the assumed mechanism, rather than proving that all neutrinos are Dirac.

The observable is a peak in summed electron energy at the decay Q value, unlike the continuous two-neutrino spectrum. Enriched source material, large exposure, good energy resolution, underground operation and radiopure construction suppress backgrounds. germanium arrays use the source as a semiconductor detector; xenon time-projection chambers combine ionization, light and event-position information; bolometers measure deposited energy through a temperature rise. The GERDA collaboration's Phase-I study is a concrete example of an enriched-germanium source-detector search with background suppression.^[ref] energy calibration, resolution tails, radon and material backgrounds, cosmic-ray vetoes, signal-like event topology and sideband fits all need independent checks before a Q-value excess is convincing. Collider same-sign dilepton processes can test heavier Majorana states, but require a specified production and decay mechanism. Lepton-number violation, absolute-mass measurements and oscillation data provide complementary information; oscillations alone do not settle Majorana character.

supersymmetry relates bosons and fermions through fermionic symmetry generators. A main motivation is the Lorentz scalar hierarchy problem: related boson and fermion loop contributions cancel particular ultraviolet sensitivities in an exactly supersymmetric theory. soft supersymmetry breaking leaves residual sensitivity to the superpartner scale, so a very heavy spectrum does not automatically solve naturalness. Supersymmetric matter content can also improve gauge-coupling unification, and a stable neutral lightest superpartner can be a dark-matter candidate. None of these motivations is itself experimental evidence; supersymmetry must be broken if ordinary particles and their partners have different observed masses.

If R-parity is conserved, superpartners are produced in pairs and cascades end in a stable lightest supersymmetric particle. A neutralino LSP can escape the detector, giving missing transverse momentum. Colored squarks and gluinos can generate energetic jets, sometimes b-tagged jets and leptons, plus that imbalance. An inclusive supersymmetry search classifies broad event samples by jet and lepton multiplicities, transverse momentum sums and missing momentum, without requiring a fully reconstructed decay chain.

Background controls are part of the physics argument. Lost-lepton W and top events can be constrained using lepton collider control regions; invisible-Z production can be related to visible-Z or photon samples with appropriate transfer factors. Instrumental imbalance in multijet events needs detector-quality checks and data sidebands. An excess in a collider signal region should survive uncertainty in those extrapolations, efficiencies and luminosity, with independently defined validation regions and attention to the number of searches performed.

An exclusive supersymmetry search targets a more specific production mode and decay pattern. Electroweak chargino-neutralino production can yield three leptons plus missing momentum; a cascade through a slepton can yield a dilepton invariant-mass edge that constrains mass combinations. Such searches often have cleaner backgrounds but depend more strongly on branching fractions and intermediate masses. An endpoint does not necessarily determine every absolute mass without additional observables.

Different spectra require different strategies. For a compressed supersymmetric spectrum, small parent-daughter mass differences make the visible decay products soft. initial-state radiation can recoil against the pair and provide a trigger, supplemented by soft-lepton reconstruction. The ATLAS compressed-spectrum study explicitly uses small neutralino mass differences and initial-state radiation to define a dedicated search rather than applying a hard-lepton selection indiscriminately.^[ref] If R-parity is violated, the LSP can decay visibly, so high-multiplicity jet or lepton channels may matter more than large missing momentum. CMS separately studied hadronic and leptonic final states for this reason.^[ref] Long-lived particles instead motivate displaced vertices, disappearing tracks, abnormal ionization or time-of-flight measurements. Thus there is no single model-independent missing-energy cut that covers every supersymmetric spectrum.

For the hypothetical relic-density calculation, use the supplied distribution and retain its prefactor exactly. Put $M=m{c}^2$, $\Theta =k_{B}T$ and ${\Theta }_0=k_B{T}_0$. Thennumber density, denoted $N$ in the paper, is written $n$ here to distinguish it from a total particle count. The cosmological particle freeze-out temperature in energy units isAssume a stable nonrelativistic relic, negligible further annihilation and $aT=\mathrm{constant}$ after cosmological particle freeze-out, using sharp-freeze-out relic dilution. Its comoving particle number is conserved, soThe particle mass cancels from this present number density under the stated approximation. Its present rest-energy density is $M{n}_0$, not merely $n_0$:This is only 0.371% of the critical density, not a universe-closing component. The correct further comparison is with the measured dark-matter fraction, not an assumption that all critical density must be dark matter. In standard cosmology this abundance would be a small subcomponent, despite a possible collider discovery.

The exponential freeze-out sensitivity makes the result highly sensitive to $P$. Also, thermal entropy release means $aT$ is not exactly constant through the whole cosmic history. If total comoving entropy is conserved, the above number density acquires a factor $g_{\ast s,0}/g_{\ast s,f}$, involving entropy degrees of freedom. Those degrees of freedom and a full annihilation history are not given, so no precise correction is justified from the supplied data.

Establishing the nature of dark matter would require more than measuring a missing particle's collider mass. Measure its spin, charges, interactions, decay lifetime or stability limits, and annihilation or coannihilation processes; compare the resulting relic calculation with cosmological observations. Seek compatible direct nuclear-recoil signals and indirect annihilation or decay products, while controlling astrophysical and detector backgrounds. A collider-invisible lifetime need not be cosmologically long, and nonthermal production could change the abundance. A consistent particle identity, cosmological abundance and independent non-collider evidence are needed to connect the hypothetical discovery to the dark matter.

gravitational lensing measures the gravitational influence of matter without requiring that matter to shine. In general relativity a point mass produces small-angle deflectionFor a thin lens, with angular-diameter distances $D_d,D_s,D_{ds}$,Image multiplicity, distortion and magnification therefore constrain projected gravitating mass, with source and lens geometry measured separately.

Strong lensing gives multiple images, arcs and Einstein rings. Their positions and, where available, time delays constrain a central mass distribution. Weak lensing measures correlated small image distortions over a larger field. Using the critical surface density for gravitational lensing and reduced gravitational shear, the convergence and shear satisfyBecause intrinsic galaxy shapes are unknown, weak shear is inferred statistically using many sources. Point-spread-function anisotropy, shear calibration, photometric redshifts, intrinsic alignments and foreground contamination must be controlled.

An inferred lensing mass can be compared with the independently estimated stellar and hot-gas mass. The bullet Cluster dark-matter separation provides a particularly informative separation of components: the collisional X-ray-emitting gas is displaced from the galaxies, while lensing mass peaks approximately follow the less-collisional galaxy component. The primary analysis combined wide-field and Hubble imaging rather than inferring lensing mass from X-ray hydrostatic equilibrium.^[ref] This favors a dominant relatively collisionless unseen component. Translating a merger offset into a particle self-interaction cross section also requires a merger model and projection uncertainties; lensing does not directly reveal a WIMP's mass or quantum numbers.

Microlensing tests a different possibility: dark matter in compact objects. A compact foreground lens can create an unresolved, time-dependent magnification of a background star. Event frequency and duration constrain the abundance and mass range of compact lenses only after survey cadence, detection efficiency, blending, source sizes and halo geometry are included. Achromaticity and the expected light-curve shape help distinguish simple lensing from intrinsic variability. A null survey result does not exclude every compact-object mass at every location.

Weak-lensing correlations and small-scale strong-lens structure also test the matter power spectrum and halo substructure, potentially distinguishing cold from warm free-streaming behavior. Baryonic feedback can change the same small-scale mass distribution. There are intrinsic inversion ambiguities: the mass-sheet transformationleaves reduced gravitational shear unchanged. Source-redshift information, absolute magnification or other data can restrict such degeneracies. Lensing establishes where gravitating matter is and tests its distribution and interactions; identification of its particle species requires complementary evidence.

Extra spatial dimensions offer mechanisms for unification and for the large separation between weak and gravitational scales. They are hypotheses about geometry and fields, not observations established by the hierarchy problem itself.

In a Kaluza-Klein theory construction, a compact extra coordinate can turn components of a higher-dimensional metric into four-dimensional gauge and scalar fields. For a field periodic on a circle of radius $R$, momentum in that direction is quantized. In units $\hslash =c=1$,The resulting tower explains why a small compact dimension can be hard to see at low energy: nonzero modes need energy of order $1/R$. A realistic model must also explain its matter representations, chirality, internal geometry and stabilization of Lorentz scalar moduli; a compact coordinate alone does not reproduce the Standard Model.

Large-extra-dimension models instead use volume to weaken effective gravity. With a fixed unwarped compact space of volume $V_d$, integrating the higher-dimensional Einstein-Hilbert action gives, in a consistent reduced-Planck-mass convention,The Standard Model can be localized on a brane while gravity propagates in the bulk. A large $V_d$ makes the observed four-dimensional gravitational scale large even if the fundamental higher-dimensional scale is much smaller. The proposal identifies gravitational dilution as a possible hierarchy mechanism, with deviations from ordinary gravity at sufficiently short distance.^[ref] For approximately flat dimensions of common size $R$, gravity at distances much smaller than $R$ spreads through more directions and the force scales as $r^{-(2+d)}$, returning to $r^{-2}$ at large distance.

Warped models use a different mechanism. A five-dimensional metricredshifts a mass on a distant brane by $m_{\mathrm{IR}}=e^{-kL}{m}_{\mathrm{UV}}$. An exponential hierarchy can therefore arise without a large flat compact volume. The original Randall–Sundrum construction makes this distinction explicit and predicts characteristic massive spin-two excitations.^[ref] It still needs a consistent stabilization and phenomenology; the flat-volume formula and the warped redshift should not be treated as interchangeable explanations.

Extra dimensions also occur in string-theory constructions, where consistency and compactification can organize gravitational and gauge degrees of freedom. That is a framework motivation, not proof that a particular compactification describes nature. Laboratory inverse-square-law tests, collider missing-momentum channels from escaping bulk modes, massive-mode resonances, and astrophysical or cosmological energy-loss constraints probe different parameter ranges. Background estimates and dependence on field localization matter: a brane model with invisible bulk emission need not share the signatures of a model with visible resonant decays. The useful claim is a testable mechanism for a specified hierarchy or unification problem, with compactification and stabilization assumptions stated.

A topological defect can remain after spontaneous symmetry breaking when the field cannot be continuously put into the vacuum everywhere. If the vacuum manifold is $\mathcal{M}=G/H$, its topology classifies possible obstructions. Causally separated regions can settle into different local order-parameter configurations; joining them during a phase transition can trap defects. This is a formation mechanism, not an assertion that every transition creates every possible defect.

In three spatial dimensions the homotopy classification of cosmological defects gives areDisconnected vacua can meet across a wall; winding around a line can obstruct filling its core; a sphere around a point can carry monopole charge. Textures differ from permanent localized cores: they can collapse and unwind when the field leaves the vacuum manifold. Local gauge and global symmetry breaking also differ in their long-range fields and radiation. The original cosmological analysis connects vacuum topology with domain and string formation and their subsequent evolution.^[ref]

The energy budget determines cosmological importance. Nonrelativistic monopoles dilute approximately as $a^{-3}$, so a large primordial abundance can become unacceptable as radiation falls faster. Inflation can dilute an earlier population, provided the same defect-producing transition is not repeated afterwards. A slowly evolving wall network dilutes especially slowly and can dominate the energy budget or create excessive anisotropy.

Strings can lose energy by intercommutation and loop formation, with loops radiating gravitational waves or field quanta. A horizon-scaling cosmic string network whose characteristic separation is of order the horizon can have density ${\rho }_{\mathrm{string}}\sim \mu /t^2$ in natural units, keeping a limited fraction of the total rather than accumulating indefinitely. This differs from an ideal frozen noninteracting network's simple $a^{-2}$ dilution law. Similarly, a horizon-scaling domain wall network has ${\rho }_{\mathrm{wall}}\sim \sigma /t$, while a frozen wall network follows $a^{-1}$. The evolutionary assumptions must accompany any quoted scaling.

Defects can seed density wakes and other inhomogeneities, perturb the microwave background and generate non-Gaussian signatures. A sufficiently straight ideal cosmic string has conical spacetime and can create doubled background images; a cosmic-string network can generate gravitational-wave bursts and a stochastic background. Textures can produce localized microwave-background temperature features while collapsing. These signatures constrain defect abundance and symmetry-breaking scale; an isolated suggestive feature is not a unique defect detection.

The predictions depend on network evolution, reconnection, loop sizes, radiation channels and the relation between symmetry breaking and inflation. topology identifies which configurations are possible; formation and evolution determine whether they survive and whether their gravitational or radiative effects are observable. A defect's unseen energy does not automatically make it the dominant cold dark matter.

Original solutions for the shared Physics and Astrophysics paper. ^[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.

The primordial-element abundances from Big Bang nucleosynthesis, together with the cosmic microwave background, constrain the total baryonic density. massive compact halo objects such as ordinary faint stars, brown dwarfs and stellar remnants are baryonic whether or not they shine. With the stipulated density fractions, the entire baryonic budget is only one fifth of the required dark-matter density, and much of it is already accounted for by ordinary gas and stars.

A second test does not require a compact object to emit light: gravitational microlensing temporarily magnifies a background star as the object crosses the line of sight. Monitoring many Magellanic-Cloud stars constrains the microlensing optical depth and event-duration distribution. Surveys such as EROS-2 found too few suitable events for a standard halo predominantly composed of compact objects over their tested mass range.^[ref] Artificial-event injection, cadence and blending corrections are needed because very short or long events are easily missed; lensing by stars within the Clouds must also be separated from halo lensing. These are mass-range and halo-model constraints, not a proof excluding every compact-object mass. Primordial black holes are not limited by the ordinary-baryon budget in the same way as stellar remnants; their viability requires their own observational bounds.

Standard Model active neutrinos decouple while relativistic and retain a thermal momentum distribution. They therefore constitute hot dark matter for the light masses compatible with the standard cosmological picture. Their collisionless free streaming erases linear cosmological density perturbations below a characteristic scale. Making them the dominant gravitating matter would delay small-scale structure formation and suppress the observed matter power spectrum.

For standard thermal abundances and nonrelativistic neutrinos today,Increasing the mass sum enough to provide all the stipulated dark matter conflicts with the successful cold dark matter description of cosmological structure. Independently, the Tremaine-Gunn bound on the fermionic phase-space occupation limits the density light neutrinos can supply in small galactic haloes.^[ref] Oscillations establish nonzero neutrino mass differences, but do not make the relic neutrinos cold. Active neutrinos can contribute some matter; they cannot supply the dominant cold component. A heavier sterile neutrino candidate would be a different hypothesis, not an ordinary Standard Model relic.

Particle candidate. A stable, electrically neutral weakly interacting massive particle can become nonrelativistic before cosmological particle freeze-out. Its annihilation rate per particle $\Gamma =n⟨\sigma v⟩$ eventually falls below the Hubble parameter; the surviving comoving number then remains approximately fixed. A weak-scale annihilation cross section can give a cosmologically significant thermal relic. This is a mechanism and a parameter-scale motivation, not a guarantee that every weakly interacting particle has the right abundance.

supersymmetry supplies neutralinos, mixtures of neutral electroweak gauginos and higgsinos. If a neutralino is the lightest supersymmetric particle and R-parity is conserved, it cannot decay solely into Standard Model particles and is stable. Its electroweak interactions can make it a WIMP. Conservation of R-parity and a neutral lightest state are additional assumptions; R-parity violation, coannihilation, composition and resonant annihilation can substantially change the conclusion.

An underground test. XENON100 at Gran Sasso used a liquid-xenon target in a two-phase time projection chamber. A WIMP scattering elastically from a xenon nucleus could produce a low-energy nuclear recoil. The interaction makes prompt particle scintillation, S1, and ionization electrons. An electric field drifts the electrons upward; their extraction into xenon gas produces delayed electroluminescence, S2. The delay measures depth, while the S2 light pattern determines the transverse position. An inner fiducial volume suppresses surface backgrounds through xenon self-shielding; underground overburden reduces cosmic-ray muons, and shielding and material selection reduce radioactivity.^[ref]

The S2/S1 ratio separates most electron recoils from nuclear recoils. Gamma-source calibration maps the electron-recoil leakage; an americium–beryllium neutron source calibrates nuclear-recoil response and cut acceptance. Purity and position corrections must be measured, and selection fixed before unblinding the search region. neutrons can themselves produce single nuclear recoils, so discrimination is not perfect; material assays and transport simulations constrain this background. In the 224.6-live-day, 34-kg analysis, two candidate events were compatible with the expected $1.0\pm 0.2$ background events. The result constrained scattering cross sections rather than establishing a discovery.^[ref]

A predicted count also depends on the local dark matter halo density, distribution of velocities, recoil threshold and nuclear response. Thus a null result constrains a particle model under stated astrophysical assumptions; it does not exclude all dark matter. A collider missing-momentum signal would likewise need independent evidence for a cosmological lifetime.

Relic energy density in the stipulated freeze-out approximation. Let $M=m{c}^2=1900\mathrm{GeV}$ and $\theta =k_{B}T$. Writing the given distribution in energy units givesBy sharp-freeze-out relic dilution, after cosmological particle freeze-out conserve particle number, $n{a}^3=\mathrm{constant}$. In the approximation $aT=\mathrm{constant}$ for the background bath,Inserting the printed inputs yieldsand henceThis is approximately $93\%$ of the stipulated dark-matter energy density, $0.2\times 5=1\mathrm{GeV}{\mathrm{m}}^{-3}$, so the toy calculation gives the right order and is close to the target. At fixed $P$ the cancellation of $M$ in $n_0$ is exact for the given prefactor, while the present energy density remains proportional to $M$.

The equilibrium exponential must be evaluated at freeze-out, not at today's temperature: equilibrium no longer maintains the particle number afterward. More accurately, comoving entropy conservation gives $a^3{g}_{\ast s}{T}^3=\mathrm{constant}$ and multiplies the displayed $n_0$ by $g_{\ast s,0}/g_{\ast s,f}$, when both counts use the same bath-temperature convention. TODO for a refined numerical abundance: the question supplies no entropy degrees of freedom or annihilation history for that refinement. The value $0.185$ is therefore the intended constant-$g_{\ast s}$ estimate, not a precision relic-abundance prediction; realistic entropy release, gradual freeze-out and the model's annihilation channels matter.

When a cosmological phase transition breaks a symmetry $G$ to a subgroup $H$, the degenerate vacuum configurations form a vacuum manifold $\mathcal{M}=G/H$. Causally separated regions need not choose the same vacuum orientation. The Kibble mechanism explains how joining those choices can leave a field configuration that cannot be continuously deformed to a uniform vacuum without leaving $\mathcal{M}$. The field then stores energy in a defect core, gradients or gauge fields.^[ref]

The relevant homotopy classification of cosmological defects uses the vacuum map on a surface surrounding a proposed defect:A domain wall separates different disconnected vacua. A cosmic string has nonzero winding number on a loop around its core. A magnetic monopole has a nontrivial vacuum map on an enclosing sphere. Nontrivial ${\pi }_3(\mathcal{M})$ can instead support cosmological texture configurations extending over a volume; cosmological textures can unwind dynamically by leaving the vacuum manifold and need not possess a permanently localized stable core. Nontrivial topology permits a defect; formation efficiency, interactions and the cosmological history still determine its abundance.

For example, a complex scalar with a circular set of minima can wind by $2\pi n$ around a cosmic string. Continuity forces the field away from its vacuum value near the core. A local gauge symmetry allows gauge fields to screen much of the exterior gradient and trap flux, whereas a global symmetry leaves long-range Goldstone-field gradients. Thus local and global cosmic strings have different energy-loss channels; they should not be treated as identical networks.

Relic defects can be tightly constrained because their energy dilutes differently from radiation. Nonrelativistic magnetic monopoles approximately dilute as $a^{-3}$, so excessive production causes the cosmological monopole problem. A roughly frozen wall network has energy density proportional to $a^{-1}$, even slower dilution. An evolving cosmic string network can approach a scaling regime with characteristic separation proportional to cosmic time and density of order $\mu /t^2$ in $c=1$ units, losing energy into loops and radiation. That is a dynamical network statement, not the dilution law of a fixed collection of cosmic strings.

cosmic inflation can dilute defects made before it; reheating and later symmetry breaking must not regenerate an unacceptable abundance. Observational tests include microwave-background anisotropy, gravitational lensing by cosmic strings, gravitational radiation from string loops, and searches for magnetic monopoles. Each is sensitive to model-dependent tensions, couplings and evolution. Defects are possible relics of symmetry breaking, not an inevitable dominant dark-matter population. Their absence constrains a transition and its subsequent history rather than every theory possessing a nontrivial vacuum topology.

supersymmetry predicts partners with spins differing by one half. At a proton collider, strongly interacting squarks and gluinos can be pair produced if kinematically accessible. With conserved R-parity, their cascade decays end in a stable lightest supersymmetric particle, often a neutralino. The generic signature is energetic particle jets, possibly leptons or heavy-flavour jets, together with missing transverse momentum from invisible particles. Longitudinal missing momentum is not an equally clean observable because the initial parton momentum fractions are unknown.

A practical inclusive search defines collider signal regions using jet multiplicity, transverse momenta, missing transverse momentum and an event scale such asLeptonic and heavy-flavour selections probe other decay chains. For a compressed supersymmetric spectrum, the daughter particles can be soft, making a hard initial-state radiation jet useful for triggering and recoil. Direct chargino or neutralino production instead motivates multilepton searches. Decays with R-parity violation can yield high jet or lepton multiplicities with little missing momentum, and long-lived particles motivate displaced vertices, disappearing tracks or anomalous ionization. No single selection covers all these cases.

The difficult issue is background inference. Invisible $Z$ decay with jets is a real missing-momentum background: a photon-plus-jets sample or reconstructed $Z$ decays to charged leptons constrains its normalization and recoil shape through modeled transfer factors. A single-lepton collider control region constrains $W$ and top backgrounds whose charged lepton is missed in the collider signal region. Jet mismeasurement in ordinary multijet events gives instrumental missing momentum; angular cuts, detector-quality checks and control samples test that tail. Validation regions check these extrapolations without tuning on the final signal count.

detector acceptance, trigger efficiency, jet calibration, collider luminosity, theoretical production rates and transfer factors enter a likelihood as nuisance parameters. One compares the observed signal-region distribution with background-only and signal-plus-background hypotheses, using a profile likelihood and reporting exclusions for specified masses, branching fractions and spectra. As a pre-examination example, the 2014 ATLAS zero-lepton jets-plus-missing-momentum search used $20.3\mathrm{f}{\mathrm{b}}^{-1}$ at $8\mathrm{TeV}$ and found no significant excess. Its interpretation depended on R-parity conservation and a neutralino lightest state; different simplified models gave different mass exclusions.^[ref]

A null search excludes the tested parameter region, not supersymmetry as a whole. Conversely, a missing-momentum excess alone would not establish supersymmetry: one would need consistent masses, decay chains, couplings and eventually spin information. A stable particle on detector flight-time scales is not thereby proved stable for the age of the universe.

The inference of dark matter uses a gravitational law to turn observed motions and lensing into a mass distribution. modified gravity asks whether some discrepancies instead reveal a breakdown of that law. The strongest motivation at galactic scales is the regular relation between baryonic structure and galaxy rotation curves, including approximately flat outer rotation speeds and the baryonic Tully-Fisher relation.

In modified Newtonian dynamics, the low-acceleration spherical relation is $g\mu (g/a_0)=g_N$, with $\mu (x)\to 1$ at high acceleration and $\mu (x)\to x$ at low acceleration. For an isolated baryonic mass $M_b$ in the latter regime,Thus an asymptotically flat rotation curve and a fourth-power mass–speed relation arise from a common acceleration scale, without assigning an independent halo profile to every galaxy. This is a substantive phenomenological motivation. Distances, stellar mass-to-light ratios, gas distributions and the external gravitational environment must still be controlled.^[ref]

A nonrelativistic acceleration prescription does not itself specify light bending or cosmology. A relativistic completion must determine the metric followed by photons and matter. tensor-vector-scalar gravity illustrates the need for additional dynamical fields and a specified physical metric; lensing predictions then have to be calculated within that theory rather than imported unchanged from general relativity.^[ref]

Several observations challenge a simple replacement of dark matter by a single galactic acceleration law. galaxy clusters retain substantial mass discrepancies even after such a modification. In the bullet Cluster dark-matter separation, the dominant X-ray-emitting baryonic gas is displaced from the galaxies, whereas weak-lensing mass concentrations lie nearer the galaxies. Collisionless dark matter accompanying the galaxies provides a natural account of this separation. The inference uses a lensing model and does not constitute a theorem excluding every conceivable modified-gravity theory, but a successful alternative must reproduce the displaced lensing signal quantitatively.^[ref]

The cosmic microwave background, acoustic structure, expansion history and growth of cosmic structure provide additional simultaneous tests. A theory that fits galaxy rotation curves must also supply the appropriate cosmological perturbation evolution and lensing. Adding extra clustering fields can help, but may reintroduce an effective unseen gravitating component. Solar-system and binary-pulsar agreement with general relativity constrain alternatives in their tested regimes, although a modification designed to act only at very low acceleration is not automatically ruled out by a high-acceleration test.

A discriminating programme compares dynamics, which probes the potential accelerating nonrelativistic matter, with lensing, which probes the sum of the two scalar metric potentials. Scale-dependent discrepancies, gravitational slip and environmental screening can then be tested with controlled galaxy bias and selection effects. Galaxy-scale tensions in a dark-matter model must likewise be separated from uncertain baryonic astrophysics. Regular galactic scaling relations motivate modified gravity; clusters and the combined cosmological evidence strongly challenge its simplest dark-matter-free forms. Failure to detect a particular WIMP is not positive evidence for a particular modified-gravity theory.

For a fixed comoving cell, take physical volume $V=R^3$ and energy density $u=\rho c^2$. With negligible chemical potentials and fixed relativistic species, $u=A{T}^4$ and $P=u/3$. The first law of thermodynamics givesChoosing zero entropy at zero temperature yields $S=4R^{3}u/(3T)$. Reversible expansion conserves this entropy; when species disappear, the effective entropy degrees of freedom must be changed rather than keeping $A$ fixed.

Before electron-positron annihilation, photons contribute two states, electrons plus positrons four, and three neutrino/antineutrino species six. ThusAfter neutrinos decouple, the electromagnetic entropy alone is conserved through annihilation: $(2+7\times 4/8)T_i^3{R}_i^3=2T_{\gamma }^3{R}_f^3$, while $T_{\nu }{R}_f=T_i{R}_i$. ConsequentlyThis assumes instantaneous decoupling, negligible neutrino masses/chemical potentials and complete electromagnetic thermalization. Small noninstantaneous corrections are outside this approximation.

In the Pati-Salam model, right-handed quarks and leptons fill a $(4,1,2)$ multiplet. The neutrino component has no colour, $T_L=0$ and hypercharge $Y=T_{3R}+(B-L)/2=1/2-1/2=0$. Hence ${\nu }_R$ is a Standard Model gauge singlet neutrino, with no ordinary weak or electromagnetic gauge coupling. It is sterile under the low-energy Standard Model, not a singlet of the full unbroken Pati-Salam group. A heavy Majorana mass can arise after symmetry breaking. The representation alone fixes neither its mass nor the stability, mixing or relic abundance needed for dark matter.^[ref]

Space example: AMS-02, a magnetic spectrometer on the International Space Station. Tracker curvature determines rigidity and charge sign; transition-radiation detection and an electromagnetic calorimeter distinguish electrons/positrons from the much larger proton background. It measures charged cosmic-ray spectra, including positrons that could arise from halo-particle annihilation. This is an indirect dark-matter search, despite directly recording the charged products: it does not measure scattering of neutral halo WIMPs, and astrophysical sources can mimic a positron excess. Calling it direct WIMP detection would misstate the technique.^[ref]

Underground example: XENON100, a dual-phase xenon detector. A candidate WIMP scatters from a xenon nucleus, producing prompt liquid scintillation S1 and ionization electrons. Drift and extraction into gas produce delayed proportional scintillation S2. Their delay and light patterns reconstruct the interaction position; S2/S1 discriminates nuclear recoils from electron-recoil backgrounds. Underground shielding, fiducial cuts and calibration reduce backgrounds, though neutrons can imitate nuclear recoils. This is direct recoil detection; a null excess constrains cross sections for an assumed halo and interaction model.^[ref][ref]

For successive radial null rays from a comoving source, ${\int }_{t_1}^{t_0}cdt/R(t)$ equals the same spatial integral. Subtracting two nearby wavecrests gives $\delta t_0/R_0=\delta t_1/R_1$, hence the cosmological redshiftFor transparent propagation, photon energies are reduced by $1+z$ and arrival intervals increased by $1+z$. The observer-area sphere is $4\pi (R_{0}r{)}^2$, so the observed bolometric flux isThus the requested $D_L=(1+z)D$ uses $D=R_{0}r$, the present transverse proper/area distance. In a curved metric this is not the radial proper distance $R_0{\int }_0^{r}d{r}^{\prime }/\sqrt{1-K{r}^{\prime 2}}$; the two coincide for $K=0$. The source's unqualified “proper distance” needs this distinction.

Type Ia supernovae are standardizable candles: light-curve shape and colour calibrate their luminosity. Distant events are fainter, hence have larger inferred luminosity distances, than appropriate matter-only decelerating models predict. Their redshift–distance relation favours recent accelerated expansion. In general relativity, sufficiently negative pressure, $\rho +3P/c^2<0$, drives acceleration. Dust, evolution, calibration and selection are checked as possible systematics; a single faint supernova is not the argument.^[ref]

An independent complementary test combines the cosmic microwave background acoustic distance/near-flat geometry with matter-density and baryon acoustic oscillation constraints. Matter alone does not supply the required late-time expansion history; a dark-energy component fits the joint data. CMB geometry alone has curvature/expansion degeneracies and is not an assumption-free detection of a cosmological constant.^[ref]

In natural units $c=\hslash =1$, a regulated zero-point energy estimate isThe integral is ultraviolet divergent and requires renormalization; it is not a finite prediction from counting “virtual pairs.” An electroweak scale estimate gives $|{\rho }_{\mathrm{vac}}|\sim m_t^4=(174{)}^4\mathrm{Ge}{\mathrm{V}}^4\simeq 9.2\times 10^8\mathrm{Ge}{\mathrm{V}}^4$, or somewhat smaller with loop factors. Compared with $10^{-47}\mathrm{Ge}{\mathrm{V}}^4$ this is roughly $10^{56}$ too large.

The source's units are inconsistent: $10^{-47}\mathrm{Ge}{\mathrm{V}}^4$ is a vacuum energy density, whereas $8\pi G{\rho }_{\mathrm{vac}}$ has units $\mathrm{Ge}{\mathrm{V}}^2$. The comparison above consistently uses densities. Bare and quantum contributions must cancel to extraordinary accuracy if this naive scale estimate is representative: the cosmological constant problem. Its severity is not a measured value of an individual cutoff integral.

Supersymmetry relates bosons and fermions whose spins differ by one-half. Broken supersymmetry can soften Higgs-mass sensitivity and improve gauge-coupling unification in suitable spectra; neither feature establishes that nature realizes it. Collider searches look for squarks/gluinos or electroweak superpartners through jets/leptons, missing transverse momentum and appropriate cascade-decay kinematics. Null searches constrain specified masses and branching fractions, with weaker reach for compressed or nonstandard spectra.

With conserved R-parity, superpartners are pair-produced and the lightest supersymmetric particle is stable. A neutral weakly coupled lightest state, such as a neutralino, can be a dark-matter candidate. Its thermal abundance follows annihilation freeze-out when $n_{\chi }⟨\sigma v⟩\sim H$; later expansion dilutes number density while leaving the comoving relic abundance approximately fixed. Coannihilation and resonances can change the result. Recoil searches and cosmic-ray/photon searches test complementary couplings.

Cosmology also constrains unstable long-lived relics through nucleosynthesis and reheating; not every supersymmetric spectrum gives the observed dark matter. R-parity is an additional assumption: supersymmetry by itself does not guarantee a stable neutral relic. Laboratory discovery and a matching cosmological abundance would both be needed.

Cosmic inflation is accelerated expansion with decreasing comoving Hubble radius, $\ddot{R}>0$ or ${ϵ}_H=-\dot{H}/H^2<1$. It can make the observed region causally connected before inflation and dilute spatial curvature or unwanted relics. A slowly rolling scalar has energy and pressure ${\rho }_{\varphi }={\dot{\varphi }}^2/2+V$, $P_{\varphi }={\dot{\varphi }}^2/2-V$ in natural units. Potential domination gives near-exponential expansion; ending inflation and reheating must create the hot radiation era.

For a canonical single scalar with reduced Planck mass $M_{\mathrm{Pl}}$,Quantum fluctuations stretched to cosmological scales seed density perturbations. Tests include the scalar spectral tilt, acoustic coherence, spatial curvature, non-Gaussianity and tensor-induced CMB B-mode polarization. Foregrounds and lensing must be separated from primordial tensors. A tensor non-detection does not exclude all inflation; tilt and tensor consistency predictions depend on the model. Approximate flatness and near-scale-invariant fluctuations support the framework but do not uniquely determine a potential or prove an exact de Sitter phase.

Baryogenesis dynamically creates a baryon–antibaryon imbalance rather than assuming it as an initial condition. Under the usual local relativistic assumptions, the Sakharov conditions are baryon-number violation, C and CP violation, and departure from thermal equilibrium. Without the last condition, inverse reactions and equilibrium constraints cancel a net asymmetry; CP-conjugate rates must differ in a channel with the required number violation.

In heavy-boson decay baryogenesis, heavy particles decay through baryon-violating channels with CP-asymmetric branching. Expansion makes decays/inverse decays fall out of equilibrium; washout reactions must then become ineffective. Inflation would dilute an earlier asymmetry, so the generation/reheating history matters.

In leptogenesis, out-of-equilibrium Majorana-neutrino decays first produce lepton number. Electroweak sphalerons violate $B+L$ while preserving $B-L$ and convert part of the surviving asymmetry to baryon number. In the minimal Standard Model chemical-equilibrium limit, $B=28(B-L)/79$; the coefficient depends on particle content and equilibrated interactions. Electroweak baryogenesis is another possibility if CP violation and a sufficiently strong nonequilibrium phase transition are supplied. The Standard Model's observed parameters do not by themselves furnish an adequate conventional electroweak scenario. Each proposal must quantitatively match the relic asymmetry and avoid washout.

For an isolated approximately spherical mass distribution outside its luminous extent, retain the gravitational force $GMm/r^2$ and use the specified low-acceleration inertial law. circular motion then givesThe radius cancels when enclosed luminous mass is effectively constant. This is the asymptotic modified Newtonian dynamics result associated with the baryonic Tully-Fisher relation; an arbitrary extended disk cannot everywhere be replaced exactly by a point mass.

At the disk edge,Using $G=6.67\times 10^{-11}{\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$ givesThis is an order-of-magnitude isolated, low-acceleration estimate. The edge acceleration is below $a_0$, but not extremely small compared with it; a full interpolation law, disk geometry and an external gravitational field can affect a more precise mass fit.

For the spherical mass profile for a flat rotation curve, assume spherical symmetry where the halo dominates. A flat rotation curve gives $M(<r)=v^{2}r/G$, and differentiating $M(<r)=4\pi {\int }_0^r\rho (s)s^{2}ds$ yieldsHere $\rho$ is the density of the spherical gravitating model; it is not an exact deprojection of a thin luminous disk or a proof that the central density must diverge. At the outermost stated measured radius,This is enclosed mass within $2R$, not a measured global total. Treating it as the entire galaxy-plus-halo mass additionally assumes truncation there. Extending $\rho \propto r^{-2}$ to infinity would give divergent mass; no outer cutoff is provided. TODO: A global total requires the unprovided outer halo extent or a continuation of its mass profile. Subtracting the MOND-inferred luminous mass to obtain a Newtonian halo mass would mix independent model-dependent estimates without an independently measured luminous mass.