Past exam of the relativistic Astrophysics and Cosmology course of the University of Cambridge / 2014 / Paper 1 / 3

An active galactic nucleus is powered by accretion onto a central supermassive black hole. Binding-energy release gives $L\sim \eta \dot{M}{c}^2$, so a compact source can outshine its host without converting its entire mass into radiation. A disc produces a thermal continuum, a hot corona can Comptonize disc photons into X-rays, and surrounding gas produces broad and narrow emission lines. Dust can obscure and reradiate the central emission. Orientation explains part of the diversity between unobscured and obscured nuclei, but accretion state and intrinsic variability also matter.

relativistic jets channel energy and momentum into collimated magnetized outflows. Disc rotation and, in some models, black-hole rotation supply the energy; magnetic stresses accelerate the plasma and connect the engine to the outflow. synchrotron radiation traces relativistic electrons and magnetic fields, while inverse Compton scattering can produce higher-energy emission. Bright knots, terminal hotspots and large radio lobes show that the deposited energy is not confined to the unresolved nucleus.

For bulk speed $\beta c$ at viewing angle $i$, the Doppler factor is $\mathcal{D}=[\Gamma (1-\beta \cos i){]}^{-1}$, and an approaching feature hasLight-travel delay can therefore give apparent superluminal motion without any material or signal travelling locally faster than light. Beaming also makes approaching jets appear brighter and more rapidly variable; a one-sided image is not proof of a physically one-sided outflow. Spectral, polarization and resolved-motion measurements constrain the geometry and emission mechanisms. Jets can inflate cavities and heat surrounding gas, linking black-hole accretion to galaxy and cluster feedback.

Write the dimensionless scale factor as $a=R/R_0$ and the density contrast as $\delta =(\rho -\bar{\rho })/\bar{\rho }$. Gravity amplifies overdensities while expansion and pressure oppose growth. In a sub-Hubble Newtonian single-fluid approximation, a comoving Fourier mode obeysFor pressureless matter in an Einstein-de Sitter background, the growing solution is $\delta \propto t^{2/3}\propto a$ and the decaying solution is $t^{-1}$. Jeans support suppresses growth when the pressure term dominates. The equation is not a substitute for relativistic perturbation theory on horizon-sized scales.

cold dark matter clusters without baryonic pressure support. Sub-horizon growth is much weaker during radiation domination, and baryons are initially tied to the photon fluid, producing acoustic oscillations rather than unrestricted collapse. Horizon entry, radiation-matter equality, acoustic evolution and diffusion therefore give a scale-dependent transfer function. The primordial curvature spectrum and the later matter-density spectrum are not the same function: evolution and the gravitational constraints relate them. Late dark-energy domination reduces the growth rate.

When $|\delta |$ approaches unity, linear theory fails. Collapse, virialization, mergers and tidal interactions form a hierarchy of haloes. Gas must cool and lose angular momentum to make dense stellar systems; feedback then changes its supply and distribution. Simulations combine collisionless gravitational dynamics with increasingly complex gas physics rather than extending the linear growing mode indefinitely. Galaxy clustering measures biased tracers of matter; gravitational lensing and CMB constraints provide complementary tests. Approximately adiabatic, nearly scale-invariant initial perturbations explain the coherent acoustic pattern and supply seeds for this later growth.^[ref]

Without feedback, efficiently cooling gas would turn into stars too readily. Young stars return radiation, winds and supernova energy and momentum to their surroundings. This can heat gas, disrupt dense clouds and drive outflows, with the strongest fractional effect in shallow potential wells. The outflow mass-loading factor ${\eta }_w={\dot{M}}_{\mathrm{out}}/{\dot{M}}_{\ast }$ is useful, but its dependence on escape speed differs between energy-limited and momentum-limited models. Some expelled gas later recycles; feedback also transports metals and changes subsequent cooling.

active-galactic-nucleus feedback is important in massive galaxies and hot haloes. Radiative or mechanical power from black-hole accretion can suppress the supply of cold gas. Collimated jets inflate radio/X-ray cavities, while broader winds can carry significant momentum and kinetic energy. For a relativistic cavity fluid with adiabatic index $4/3$, its enthalpy is approximately $4pV$, so $4pV/t_{\mathrm{age}}$ estimates mechanical power. Comparing it with the gas cooling luminosity tests whether that energy budget is sufficient, rather than merely showing that a radio source is present.

Deep Chandra observations of Perseus resolved bubbles, a density jump and weak surface-brightness ripples consistent with repeated mechanical disturbance of the intracluster gas. Their interpretation as heating requires a dissipation mechanism and a coupling history, not just a large instantaneous power.^[ref]

Observed outflow geometry, filling factors, cavity ages and cooling corrections create substantial uncertainties. Feedback can also compress gas and locally promote star formation. Simulations must track radiative losses and the scale on which energy or momentum couples: depositing unresolved thermal energy is not automatically equivalent to driving a resolved wind. A successful model should explain stellar masses, gas fractions, metallicity and circumgalactic or intracluster thermodynamics together.

The weak equivalence principle is universality of free fall for sufficiently small test bodies, independent of composition, when non-gravitational forces and self-gravity are negligible. Equality of inertial and passive gravitational mass makes this possible in the Newtonian limit. The Einstein equivalence principle adds local Lorentz invariance and local position invariance for nongravitational experiments: a small freely falling laboratory has the local physics of special relativity.

This motivates a universal spacetime metric. Freely falling unforced test particles follow geodesics, $u^{\nu }{\nabla }_{\nu }{u}^{\mu }=0$. At a point one can choose local inertial coordinates with vanishing connection, but curvature and tidal relative acceleration remain measurable over a finite region. Gravity cannot be transformed away globally. A uniformly accelerating small laboratory and a local gravitational field can have the same nongravitational experimental behaviour without having identical global geometry.

Clock comparisons predict a fractional rate difference of order $g\Delta h/c^2$ in a weak static field. The Pound-Rebka experiment tested the associated photon redshift through resonant gamma-ray emission and absorption at different heights, using controlled Doppler shifts to match the tiny energy displacement. Such a result tests the metric redshift prediction, not uniquely every component of the gravitational field equations.^[ref]

general relativity adds the dynamical Einstein field equations,The equivalence principle alone does not derive this unique equation; assumptions about covariance, dynamics and the Newtonian limit are also needed. The strong equivalence principle extends the universality claim to self-gravitating systems and gravitational experiments. Alternative metric theories can satisfy the weaker local principles while differing in gravitational dynamics, so composition tests, clock/redshift tests and self-gravity tests probe distinct claims.

core-collapse supernovae follow loss of pressure support in the evolved core of a massive star. electron capture and nuclear dissociation contribute to collapse; formation of a hot compact object launches a shock that can stall as dissociation and neutrino losses drain it. neutrino heating and multidimensional fluid motions can revive the shock, but some collapses fail to produce a normal bright explosion. The remnant may be a neutron star or black hole. The neutrino burst detected from SN 1987A provides direct evidence for the core-collapse energy channel: the water-Cherenkov signal is a particle-interaction measurement, requiring detector-response and accidental-background checks rather than an optical light-curve inference.^[ref]

Type Ia supernovae instead involve thermonuclear destruction of a degenerate carbon-oxygen white dwarf. They leave expanding ejecta without an ordinary surviving neutron-star core. The ignition channel need not have a unique total mass or binary history. Radioactive nickel/cobalt decay powers much of the later luminosity, and empirical light-curve corrections make these events useful distance indicators. Spectral type is not identical to explosion mechanism: hydrogen-rich Type II events are core collapse, but stripped-envelope Ib/c events can also be core collapse despite lacking conspicuous hydrogen.

The ejecta first expand nearly freely while swept-up mass is small. A strong shock heats ambient gas, and once swept-up material dominates, an approximately adiabatic remnant approaches the Sedov-Taylor scalingSubsequent radiative cooling makes a dense shell and ends that energy-conserving approximation. Thermal X-ray lines probe shocked gas and nucleosynthetic ejecta; radio and nonthermal X-rays trace relativistic particles and magnetic fields. shock acceleration makes remnants plausible contributors to Galactic cosmic rays. Expansion, ambient-density structure, cooling and cosmic-ray energy losses complicate energy estimates; a bright synchrotron shell alone is not proof of a specific accelerated-particle composition. Supernovae return metals and mechanical energy to the interstellar medium and therefore connect stellar death to the feedback cycle.