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M. Pettini: Introduction to Cosmology EXERCISES: Set 4 of 4 Q1: (You will need a ruler and a calculator) The attached figure shows sections of the Lyman alpha forest in three quasars. Three lines are marked. (a) Indicate on which portion of the Curve-of-Growth each line falls. (b) Measure the equivalent widths of lines (1) and (2). Why is it not possible to measure the equivalent width of line (3) accurately? (c) Assuming that lines (1) and (2) are Lyα lines, deduce the corresponding column densities of neutral hydrogen, N (H i), if it is possible to do so. If it is not possible to do so, state what additional information you require. How would you measure N (H i) in cloud (3)? (d) Does cloud (1) correspond to an over- or under-density of matter? What is its likely linear size? And its density (in units cm−3 ) of neutral gas? What is the angle subtended by this Lyα cloud on the sky? (You may assume an Einstein-de Sitter universe). 1 Figure 4.1: Portions of the Lyα forest. The x-axis is Observed Wavelength, while the y-axis is Relative Intensity. 2 Q2(a): Using the Friedmann equation for a pressureless cosmological model ȧ2 kc2 8πG Λc2 + = ρ + , a2 a2 3 3 show that for a population of absorbing gas clouds with constant number density per comoving volume and with constant physical cross-section, the probability that the line of sight to a distant source intersects such a cloud per unit redshift at redshift z is proportional to: (1 + z)2 [Ωm,0 (1 + z)3 + Ωk,0 (1 + z)2 + ΩΛ,0 ]1/2 where Ωm,0 + ΩΛ,0 + Ωk,0 = 1. Q2(b): Surveys have shown that, on average, the spectrum of a quasar at redshift zem = 2 shows five sets of absorption lines from elements heavier than helium (i.e. five metal line systems) at redshifts zabs < zem . If such systems are produced by intervening galaxies, randomly distributed along the line of sight to the quasar, estimate the gaseous size of galaxies required to satisfy the absorption line statistics in a ΩΛ,0 = 0, Ωm,0 = 0.3 cosmology. (Hint: Use the average density of galaxies implied by the galaxy luminosity function). Comment on the implications of your answer. Q3(a): Derive the functional form of the rotation curve expected for the outer regions of our Galaxy, where the density of stars is negligible, in the absence of dark matter. Q3(b): The rotation curve of the Milky Way is found to remain flat beyond r ∼ 10 kpc, where v ' 220 km s−1 . Deduce M (r) in the outer regions of the Galaxy, including the constant of proportionality in your answer. Q3(c): Instead of appealing to dark matter as the explanation for the flat rotation curves of spiral galaxies, it has been proposed that the law of gravity should be modified on large scales. If the gravitational force is given by F = GM m/rγ , what value of γ is required to remove the need for dark matter? 3 Q3(d): Derive the density ρ(r) in the outer regions of the Milky Way corresponding to the value of M (r) found in part (b). Include the constant of proportionality in your answer. Q3(e): Under the assumption that the Milky Way has a constant density in its inner regions, derive an expression for the rotation curve close to the centre. Q4: The nature of dark matter has yet to be established. Suppose that all the DM in the halo of our Galaxy consisted of black holes of mass m = 10−8 M . (a) How far away would you expect the nearest such black hole to be? (b) How frequently would you expect such a black hole to pass within 1 AU of the Sun? (c) Repeat both (a) and (b) for the case where the dark matter in the Galactic halo consists of Jupiter-mass planets, with m = 10−3 M . (d) Can the above considerations help exclude either candidate for Galactic DM? Note: approximate estimates will suffice. You can assume that the average density of matter near the Sun is ρm = 0.06 M pc−3 . Q5(a): A light ray just grazes the surface of the Earth. Ignoring atmospheric refraction, through what angle α is the light bent by gravitational lensing? Repeat your calculation for a solar-mass white dwarf and for a neutron star, and comment on the results. Q5(b): In lecture 16.4.1 we saw that the bending of the images of stars whose light rays just graze the surface of the Sun observed during the total solar eclipse of 29 May 1919 confirmed one of the predictions of General Relativity. It was also stated that the first empirical confirmation of strong lensing was the realisation, in 1979, that the two quasars, 0957+561A and 0957+561B are in fact two images of the same object. Does gravitational lensing by the Sun cause two images? If so, why was only one (displaced) image of the background stars seen during the total solar eclipse of May 1919? 4 Q6: The Nobel Prize in Physics 2015 was awarded jointly to Takaaki Kajita and Arthur B. McDonald “for the discovery of neutrino oscillations, which shows that neutrinos have mass”. An oscillation is the transmutation of one flavour of neutrino into another. The rate at which two neutrino flavours oscillate is proportional to the difference of the squares of their masses. Current data indicate that: m(νµ )2 − m(νe )2 c4 = 5 × 10−5 eV2 and m(ντ )2 − m(νµ )2 c4 = 3 × 10−3 eV2 What values of m(νe ), m(νµ ), and m(ντ ) minimize the sum m(νe )+m(νµ )+ m(ντ ), given the above constraints? Q7: The pair of quasars 1146+111B,C are separated by 157 arcseconds on the sky. Both quasars have magnitude mV = 18.5, and their spectra exhibit broad Mg ii λ2798 emission lines at the same redshift z = 1.012±0.001 and with the same width FWHM = 64 ± 4 Å. Discuss the alternative possibilities that this pair is: (a) the same quasar gravitationally lensed into two images, or (b) two separate quasars. Propose a set of observations that would distinguish between the two possibilities. 5 Note: Q8 is at an advanced level. However, try your best: even partial attempts will be useful ahead of the supervision. Q8(a): Give the definition and the physical interpretation of the power spectrum P (k) and of the correlation function ξ(r) of the density fluctuation field δ. Q8(b): Write an expression for the rms mass fluctuation σM within a sphere of radius R using the power spectrum P (k). Assuming a top-hat filtering of density fluctuations, recover the expression for the window function in Fourier space W (kR). [N.B. you should use the expansion in spherical harmonics of the quantity exp[ik · x] or alternatively you can Fourier transform the top-hat filter]. Q8(c): Sketch the function W (kR) and find its behaviour for small values of x = kR and large values of x. Q8(d): Let us assume that P (k) = Ak n and that W (kR) has the two asymptotic behaviours found in 7(c) for k < km and for k > km . Find out a power law expression, dependent on km only, which relates σM to R and M. Calculate the ratio of rms mass fluctuations between the particle horizon now and the fluctuations at R = 1 Mpc [You can assume an EdS cosmology here] adopting n = 1 (a scale invariant power spectrum). Q8(e): The correlation function of star forming galaxies at redshift z = 3 can be well approximated by ξ(r) = (r/r0 )−1.6 with r0 = 4 Mpc. Write down the probability of finding a pair of galaxies at a generic separation r. Let us assume that there is another set of cosmological objects—Lyman alpha clouds—with a uniform distribution, which sample the same volume and the same redshifts, and whose number density is twice of that of the galaxies. At which r is the probability of finding two galaxies in the infinitesimal volumes dV1 and dV2 the same as that of finding two Lymanalpha clouds in the same volumes? 6