Growth of Perturbations! density ! and pressure p velocity distribution is v the gravitational potential " at any point is given by: Conservation of mass d ! ! ! = #!($ % v) eqn of continuity dt Euler’s eqn ! dv 1 ! ! eqn of motion of fluid element = # $p # $& Lecture Seven: dt Poisson’s eqn ! Gravitational potential $2& = 4"G! Co-moving coordinates, x = a(t)r, where r is co-moving coordinate Galaxy Formation! distance and a(t) is the scale factor, so that #x = #[a(t)r] = r#a(t) + a(t) #r. ...fluctuations to galaxies…! Perturb these equations by a very small amount Eliminate the peculiar velocities by taking the co-moving divergence of Euler‘s equation and the time derivative of the continuity equation, combining these & remembering Poisson’s eqn, get a wave equation 2 adiabatic perturbations are related via the sound speed to pressure and density ! p = cs !" and given $ = #!/! Longair, chapter 11 - 13, 16 Seek wave solutions for $, where kc is the wave-vector in co-moving coords See also chapters on galaxy formation in: Peacock, Physical Cosmology Wave equation for $ Binney & Tremaine, 2nd edition Galaxy Dynamics Monday 1st March The Jeans Instability! The Jeans Instability! Virial Theorem: 2K + U = 0 Wave equation for $ Describes the condition of equilibrium for a stable, gravitationally bound system Jeans criterion: we have an oscillating solution where the pressure gradient is sufficient to support the region gravitational potential energy of a spherical cloud of constant density if the right hand side is positive. If it is negative we have an exponentially growing (or decaying) solution and the gravitational attraction is stronger than the pressure. dispersion relation 2 2 cloud’s internal kinetic energy, no. of particles, 4!G"e > cs k unstable modes 1/2 In terms of wavelength: 2! $ ! % " > "J = = cs & ' kJ ( G#e ) Physical basis is very simple – the instability is driven by the self-gravity of the region and the tendency to collapse is resisted by internal pressure gradient. It is also possible to derive Jeans instability criterion by initial mass considering the pressure support of a region, pressure p; density !; and radius r. condition for collapse density of cloud dp/dr ~ -p/r 3 2 M ~ ! r and since cs ~ p/! Jeans Mass region becomes unstable if hydrostatic support Jeans Length write in terms of sound waves, p=A!& from ideal gas law (&=5/3)% Thus the Jeans Length is the scale which is just stable against gravitational collapse Note: the Jeans length is the distance a sound wave travels in a collapse time Jeans Mass !% $=#!/! rJ Evolution of Jeans Mass! Evolution of Jeans Mass! The temperature dependence of the Jeans mass changed dramatically at the time of recombination RADIATION DOMINATED UNIVERSE MATTER DOMINATED UNIVERSE Bender, IMPRS Astrophysics Introductory Course Bender, IMPRS Astrophysics Introductory Course Bender, IMPRS Astrophysics Introductory Course Bender, IMPRS Astrophysics Introductory Course Bender, IMPRS Astrophysics Introductory Course Oscillations & Damping! The temperature dependence of the Jeans mass changed dramatically Theory vs. Observation! at the time of recombination Temperature fluctuation maps of the CMB from WMAP give $T/T ~ 10-5 at z~1000, as the earliest evidence for inhomogeneities. CMB fluctuations can at best (in a critical density universe) grow by a factor 1/(1+z) = R ~ 1000 by today, and thus we should expect for the amplitude of baryon fluctuations today: ! B (t = t0 ) < 0.1 Effect of dark In sharp contrast to the large Therefore, in a purely baryonic universe we cannot matter inhomogenieties observed in understand how galaxies and clusters could form! the local universe! Only temperature dependence is sound speed SILK DAMPING DARK MATTER Rate of growth of density DRIVES GALAXY perturbations EVOLUTION Bender, IMPRS Astrophysics Introductory Course Dark Matter & Baryons! Coupled Perturbations: Two collapse scenarios: Evolution of perturbations that contain distinct components (e.g., baryons & dark matter) Initial collapse Hierarchical merging Time dependence of two different modes means that baryons can fall into dark potential wells and quickly match the dark matter perturbations. This means that Universes containing dark matter can (bottom up) produce small anisotropies in the microwave background: radiation drag allows dark matter to (top down) undergo growth between matter-radiation equality and recombination, while the baryons cannot. ! ! ' ' Fragmentation ! ! ' ' ! ! Merging ' ' Baryons follow Dark Matter Cold dark matter •" Devised to explain rotation curves and missing mass in clusters. •" Assumed non-interacting except via gravity. •" Also required to explain large-scale structure and CMB. •" Numerical simulations on cosmological scales •" Power law of initial fluctuations set at CMB surface. •" Growth via gravity alone. •" Robust prediction of Large Scale Structure. •" Halo build-up via hierarchical merging. •" Testable under assumption light traces matter •" Numerical simulations can now predict dark matter distributions very well (20 million particles+gravity) Bender, IMPRS Astrophysics Introductory Course Post-Recombination Era ( Standard AfterCDM two its principle model constituents, vacuum I. Between epoch of recombination & reionisation ( z ~ 1000 - 7) energy and cold dark matter also known as the DARK AGES Exactly when reionisation took place is a key issue for contemporary cosmology II. The Observable Universe (z ~ 7 - 0) •" On large scales the Universe is homogeneous & isotropic (Friedman- Populations of galaxies and quasars have been observed to evolve dramatically over this period. Robertson-Walker metric) •" Geometry of Universe is flat, as predicted by inflation -2 •" Present densities of baryonic matter, )b0 = (0.0455 ±0.0015)h7 , dark matter, )m0 = (0.237 ±0.034) and vacuum energy )(0 = 1 - )m0 = 0.763 ± 0.034, where h7 = 1.05 ± 0.05. •" Dark matter is cold (random velocity at decoupling ~50km/s) •" Initial density fluctuations in density were small (|#|<<1) and described by a random gaussian field. •" Initial power spectrum of density fluctuations was approximately a Harrison- Zeldovich spectrum, P(k) * kn, where n=1 Bringing together the wealth of observational data into a convincing and coherent picture of galaxy formation and evolution is an important modern goal. The intrinsically non-linear nature of the processes involved makes the subject an ideal challenge for large scale computer simulations - which can be used to try to understand the underlying physical processes. Non-linear collapse of density Biased Galaxy Formation:" Peaks and Patches! perturbations Galaxies form at peaks The density of a luminous galaxy at a radius of a few kpc is ~105 times larger than 11 -2 -3 in the density field. critical density, !c (=1.3599x10 h7 M!Mpc ). Threshold decreases with time leading to more Thus, galaxy formation involves highly non-linear density fluctuations, and and bigger galaxies. our linear formulism must be supplemented by approximate analytic arguments and numerical simulations to follow structure formation into the non-linear regime. Although full development of gravitational instability cannot be solved exactly without N-body techniques there are some very useful special cases and approximations that help to understand the general case, Spherical (Top-Hat) Collapse Isotropic Top-Hat Collapse A calculation that can be carried out exactly is the collapse Can now work out the density of the perturbation at maximum scale factor !max of a uniform spherical density perturbation in an relative to that of the background !0 which we assume to be critical model, otherwise uniform Universe )0 = 1, )( = 0. The density within the perturbation was times that of the background model The dynamics of such a regions are precisely the )0 same as those of a closed universe with )0 > 1 The variation of the `scale factor’, or relative size, of the perturbation ap is cycloidal and given by the parametric solution to the Friedman model (matter dominated Universe) Where the scale factor of the background model has been evaluated at cosmic time tmax. Thus, by the time the perturbed sphere has stopped expanding, its density was already 5.55 times greater than the background density. The perturbation had no internal pressure and so it will collapse to infinite density at time t = 2+D, twice the time it took to reach maximum expansion. As we showed before a * t 2/3 and so the relation between redshift of maximum expansion, zmax, and redshift of collapse, zc is: The perturbation reached maximum size at `turnaround’, when it stopped expanding, at = 0 at '=+ and so has separated out of the expanding background. This occurred at scale factor: This means that the collapse of the perturbation occured very rapidly once it separated out from the background, e.g., if zmax = 20 then zc = 12; zmax = 10 then zc = 6 etc. turn-around '=+% Violent Relaxation '=2+% Interpreted literally, the spherical perturbed region would collapse to a black hole, but in practise the presence of dark matter density sub-perturbations and tidal effects of linear non-linear collapse: violent neighbouring perturbations means fragmentation into sub-units which then reach relaxation dynamical equilibrium (virial equilibrium) under the influence of large scale bound structure gravitational potential gradients: VIOLENT RELAXATION (Lynden-Bell 1967) We can work out the final dimensions of the virialised dark matter halo: At zmax, the sphere was stationary and all energy of the system in the form of gravitational energy. For a 2 uniform sphere of radius rmax, the gravitational PE is -3GM /5rmax 2 If the system did not lose mass and collapsed to half its radius, gravitational PE would be -3GM /(5rmax/2) 8 density 5.5 Thus, by collapsing by a factor of 2 in radius from its maximum radius of expansion, the kinetic energy of collapse became half the negative gravitational potential energy. Once this energy was randomised by the process of violent relaxation, the condition for dynamical equilibrium according to virial theorem is satisfied. Thus the density of perturbation is increased by further factor 8, while the -3 background density continues to decrease.