Past exam of the particle Astrophysics course of the University of Cambridge / 2012 / Paper 1 / 3

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

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

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

A cosmological constant has fixed density and equation of state $w=p/(\rho c^2)=-1$. A separately conserved dynamical component obeysIf $w$ departs from $-1$, its history changes the Hubble parameter, luminosity distances and angular-diameter distances. Type Ia supernovae probe a relative distance-redshift relation, baryon acoustic oscillations provide standard-ruler distances and expansion information, and the cosmic microwave background fixes early-time scales and geometric combinations. Combining these breaks some degeneracies with matter density, curvature and calibration. Distance observables integrate the expansion history, so they smooth rapid variations in $w$ and cannot determine an arbitrary function exactly.

Structure-growth measurements add a dynamical test. Galaxy clustering, redshift-space distortions and weak gravitational lensing constrain the growth of matter perturbations and metric potentials. The late Integrated Sachs-Wolfe effect probes potential evolution through CMB-matter correlations. A scalar-field dark-energy model can have perturbations and an effective sound speed different from other models with a similar background expansion. Comparisons of geometry and growth can therefore distinguish some models and test gravity assumptions.

A statistically significant, robust departure from $w=-1$ or from constant dark-energy density would disfavor a pure cosmological constant within the stated framework. Consistency with $w=-1$ over a finite redshift range does not prove exact constancy: a slowly rolling field can mimic it. Supernova calibration/evolution, galaxy bias, photometric redshift errors, intrinsic galaxy alignments and foregrounds must be controlled; modified gravity can also change growth. Parameterizations such as $w(a)=w_0+w_a(1-a)$ are useful finite-dimensional tests rather than a complete classification of all dark-energy histories.

The baryon excess requires a mechanism that produces different net baryon numbers from an initially symmetric ensemble. In the usual local, CPT-invariant framework, the Sakharov conditions are baryon-number violation, C and CP violation, and departure from thermal equilibrium. Baryon conservation would preserve zero net baryon number; exact conjugation symmetries equate the conjugate production processes; equilibrium inverse reactions and detailed balance erase a would-be asymmetry. These necessary conditions do not determine its magnitude.^[ref]

For example, let equal numbers of a heavy particle $X$ and its antiparticle decay into channels $i$ with baryon numbers $B_i$. Their mean net yield per pair isIt can be nonzero even though CPT requires equal total widths: the branching fractions differ among channels carrying different baryon numbers. A nonzero decay-rate asymmetry requires interference with both a CP-odd weak phase and an absorptive phase. If the expansion rate overtakes the inverse-reaction rate, decays can occur out of equilibrium; later baryon-violating washout must then be weak enough for the asymmetry to survive.

In electroweak baryogenesis, CP-asymmetric transport near a first-order phase-transition boundary can bias baryon-violating sphaleron transitions. CP violation alone is insufficient: the transition and washout dynamics also matter. In leptogenesis, out-of-equilibrium decays of heavy Majorana neutrinos can generate lepton asymmetry, part of which sphalerons convert into baryon asymmetry while conserving $B-L$. The CKM phase demonstrates CP violation in the Standard Model but does not by itself supply a successful cosmological mechanism. The key quantity is a surviving net charge yield, not merely the existence of a CP-violating reaction.

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

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

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