Past exam of the particle Astrophysics course of the University of Cambridge / 2015 / Paper 1 / 3

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.