Unformatted text preview:

11 May 2012 Ay 127 E.S. PhinneyCosmology and Galaxy FormationProblem Set 6 Due in class, Friday 18 May 2012Reading: See the on-line syllabus for lecture-by-lecture readings.Homework Problems:1. There are several important length scales for cosmological flucuations. Here you will estimatethem for yourselves:a) The comoving wavenumber keqof the wave which just fills the horizon at matter-radiationequality. Show that keq= H(zeq)/(c(1 + zeq)), and c alculate numerically (in comovingMpc−1).b) The comoving wavenumber kapof the wave for which kap= π/rs, where rsis the comov-ing distance a sound wave has propagated since the hot big bang be gan. Recall that thedark matter is not coupled to the baryon-radiation fluid, so the sound speed is given bycs=s43pr(ρr+ ρb+ pr/c2)(1)where subscript r indicates radiation, and subscript b indicates baryon. Calculate nu-merically (in comoving Mpc−1).c) The Silk Diffusion wavenumber kD. Because photons have a finite mean free path,cosmological sound waves will be erased on small scales because of the imperfect couplingbetween photons and electrons/protons on scales where relative diffusion occurs. Showthat up to the time just before recombination, photons will have diffused a comovingdistanceL ∼Zt0cdtne(t)σT(1 + z(t))21/2(2)where σT= 0.66 barn is the Thomson (electron scattering) cross-section. EstimatekD∼ 1/L in comoving Mpc−1.2. Magnetic monopoles are generically produced in grand unified theories (GUT) of the strongand electroweak interactions due to spontaneous symmetry breaking as the temperature dropsbelow kTGUT∼ 1015GeV. Since the fields cannot be correlated on scales larger than thehorizon at the time of symmetry breaking, this leads to formation of about one monopole every4 horizon volumes. If the monopole mass M ∼ kTGUT/c2, compute the ratio of monopolenumber density to photon number density (or nearly equivalently, but more exactly, entropydensity). It can be shown that at this density, the monopoles do not significantly annihilatewith antimonopoles, so their number is conserved. Use this to estimate ΩM, the ratio of massdensity in magnetic monopoles to the clos ure density. You should find ΩM∼ 1013, in obviousdisagreement with observation!!This was historically one of the principal arguments for inflation, and is still one of the mainconstraints on the temperature to which the universe is reheated after inflation —to avoidoverproducing monopoles, Treheat< TGUT.13. At times when dark matter and baryons are non-relativistic and decoupled from each other(except via gravity), the dark matter is a pressureless fluid, while the baryons feel the radiationpressure. When an adiabatic perturbation in dark matter, baryons and photons enters thehorizon (i.e. at the t for which its comoving k = H(t)/(c(1 + z (t))))), the baryons andphotons begin acoustic oscillations, leading to zero density and potential fluctuation in thespatial and temporal mean. Thus the dark matter moves in its own self-gravity, but in arapidly expanding background driven mainly by radiation pressure.Starting from the Newtonian equations of dark matter conservation, of motion, and thePoisson equation:∂ρm∂t+ ∇ · (ρmv) = 0(3)∂v∂t+ v · ∇v = −∇φ (4)∇2φ = 4πGρm, (5)linearise the perturbation of these equations about the Hubble flow ρm= ρm0(1 + δ), v =H(t)x + δv, and introduce comoving coordinates q such that the physical x = a(t)q.a) Show that¨δ + 2˙aa˙δ = δ4πGρ0(6)Hints:∂f(x = aq, t)∂tq=∂f∂tx+ Hx · ∇xf (7)The divergence of the 0-order equation of motion is 3˙H + 3H2= −4πGρ, which, usingH = ˙a/a, is just 3 times the Friedmann equation ¨a/a = −4πGρ/3.If you still get stuck, look at Chapter 6 of Mukhanov, or Section 15.2 of Peacock.b) Change the time variable from t to y = a/aeq= ρm/ρr, and note that ( ˙a/a)2= 8πG(ρm+ρr)/3. Show that equation 6 becomes the famous M´esz´aros equation:δ00+2 + 3y2y(1 + y)δ0−32y(1 + y)δ = 0 , (8)where primes indicate differentiation by y.c) Equation 8 is second order linear in δ, so the general solution is a linear combinationof two independent solutions. Show (by inspection or substitution) that one of themis δ1(y) = 1 + 3y/2. As always with second order linear ODEs, this reduces findingthe second solution to an integration: substitute δ2(y) = u(y)δ1(y), and show that theequation for u(y) is first order in u0, i.e. of the form u00+ f(y)u0. Thus show that thegeneral solution to equation 8 isδ(y) = c1(1 + 3y/2) + c2(1 + 3y/2) ln√1 + y + 1√1 + y − 1− 3p1 + y, (9)where c1and c2are integration constants.2For y  1, δ(y) ≈ c1+ c2(−ln(y/4) − 3). The Newtonian equation 6 is only validfor p erturbations inside the horizon. So to determine c1and c2, one must match toa more properly relativistic solution as it enters the horizon (see Dodelson section 7.3for details), but this result gives a flavor for why the dark matter perturbations havelogarithmic growth during the radiation-dominated era.d) Show that for perturbations of fixed comoving wavenumber k, the linearized, Fouriertransformed Poisson equation gives for the perturbed gravitational potentialδφ(k) =4πGρa2δ(k)k2. (10)Show that in the matter-dominated era, when δ ∝ a, δφ(k) becomes independent oftime: even though the density perturbation grows inexorably in the linear regime, thepotential perturbation associated with that comoving scale does not grow. Can youexplain physically why it doesn’t?4. Suppose we model the hot gas in a cluster as a singular isothermal sphere, i.e. the numberdensity scales with radius as n(r) ∝ r−2out to a radius of rmax= 1 Mpc. The temperatureis a constant T = 108K. The total mass in baryons (assumed to be fully ionized hydrogen)is 1014M. To within a factor of 2, what is the radius rcinside of which the cooling timeis less than 1010years? To compute this, assume that the cooling occurs exclusively bybremsstrahlung, Inside this radius, one might expect a cooling flow. Show that if the X-rayluminosity from the cooling region is L(< rc), that the mass of gas cooling per unit time(and, in the absence of heating, expected to form cold gas and thus stars) is˙M(< rc) =2µmp5kTL(< rc) (11)Estimate˙M for this


View Full Document

CALTECH AY 127 - Cosmology and Galaxy Formation

Download Cosmology and Galaxy Formation
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Cosmology and Galaxy Formation and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Cosmology and Galaxy Formation 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?