Past exam of the particle Astrophysics course of the University of Cambridge / 2016 / Paper 1

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.