Massachusetts Institute of Technology Department of Physics 8.962 Spring 2006 Problem Set 9 Post date: Thursday, April 27th Due date: Thursday, May 4th 1. Time scale for variation in microlensing Consider the microlensing scenario described in the April 20 lecture. (“Microlensing” is the limit in which the images cannot be separated; one just examines the total intensity produced by the sum of the two images.) Imagine that the source is a star in the Large Magellanic Cloud (about 100 kiloparsecs away; 1 pc = 3.26 light years = 3.1 × 1013 km); imagine that the lens is an unseen object of about one solar mass in the galaxy, roughly 10 kiloparsecs away. If the lens is orbiting in the galactic p otential with a typical galactic orbit speed vorb ∼ 200 km/sec, what is the timescale over which you expect the light curve to vary? In other words, roughly how long will it take for the light curve to brighten and then dim? For students who could not attend th at lecture, see Hartle’s textbook, Chapter 11. 2. Proper motion distance Using the definition of proper motion distance given in Carroll [Eq. (8.124)], compute dM(z). Your final result should be similar in form to Eq. (8.123) of Carroll and should confirm the rule that dL(z) = (1 + z)dM(z) = (1 + z)2dA(z) , where dL(z) is luminosity distance and dA(z) is angular diameter distance. 3. Particle horizons in the early universe [Note: Carroll Sec. 8.8 provides important b ackground to this and the next two prob-lems. A very readable and even more comprehensive discussion is given in Chapter 3 of Cosmological Inflation and Large Scale Structure, by Andrew R. Liddle and David H. Lyth (Cambridge University Press, 2000), and in Chapter 13 of An Introduction to Modern Cosmology, by Andrew R. Liddle (Wiley, 2003).] The early universe was so hot and dense that all matter existed in the form of a plasma. As such, photons did not stream freely, but rather scattered repeatedly off the ionized charges in the plasma. “Recombination” is the name given to the point at which the universe cooled sufficiently that electrons and protons could bind into atoms; at this point, photons can stream freely. The cosmic microwave background (CMB) dates from the recombination epoch, which is now known to have occurred at a redshift zr � 1200.Consider a k = 0 FRW cosmology which is radiation dominated prior to recombination and is matter dominated afterwards1 . Let rH(tr) be the maximum coordinate distance that a photon can travel from the big bang to the epoch of recombination; this is the “comoving horizon radius” at recombination. (More properly, it is the comoving particle horizon radius, to be contrasted with the event horizon we will define in about two weeks.) Only points that are within each other’s particle horizons are in causal contact. Let robs(tr) be the coordinate distance from recombination to an observer on earth; thus 2 × robs(tr) is the coordinate separation between patches of the CMB on opposite parts of the sky. (a) Compute rH(tr) and robs(tr). The CMB is extremely isotropic; its temp erature varies by roughly a part in 105 over the sky. (b) Given the isotropy of the CMB, why is the smallness of the ratio rH/robs so puz-zling? (If you find that this ratio is not small, you might want to revise your answer to part a!) This is known as the “horizon problem”. (c) Compute the angular scale θc such that sources separated by θ < θc as seen today were in causal contact prior to t = tr, but sources separated by θ > θc were not. Some background: For the remaining two problems, we repeatedly need to refer to temperature and energy scales commonly used in early universe cosmology. Since this subject grew out of particle physics, the units of particle physics describe its relevant quantities — e.g., temperatures are usually given in MeV or GeV. This means that temperatures are really being reported in the combination kBT , where kB is Boltzmann’s constant. The following conversions and definitions will prove useful in problems 4 and 5: T0 ≡ Mean temperature of universe today (CMB temperature) = 2.73 K kB ≡ Boltzmann’s constant = 8.6 × 10−11 MeV/K h¯ = 6.6 × 10−22 MeV sec π2k4 σ ≡ Stefan-Boltzmann constant = B . 60¯h3 c2 The last constant comes in handy when converting between temperature and energy density for radiation: A gas of photons2 at temperature T has an energy density ρ = 4(σ/c)T4 . Finally, don’t forget that the temperature of radiation redshifts: Radiation whose temperature is T today was at temperature a(t)T/a(0) = (1 + z)T at an earlier time t. 1In fact, the universe became matter dominated somewhat prior to recombination: The redshift at which matter and radiation contribute equal to the mean density of the universe is zeq � 3000. The error we make in approximating zeq � zr is small enough for the purposes of this and the next two problems that we may safely consider the universe to be radiation dominated prior to recombination, and matter dominated afterwards. 2If you read the texts by Liddle, you will discover that we are neglecting neutrinos and other relativistic particles in this analysis. Our conclusions will thus be off by a factor of order unity; fine for our purposes.4. The horizon problem resolved: The inflationary universe Suppose that at some very early epoch, t1 ≤ t ≤ t2 (where t1 � t2 � tr), the universe resides in a “false vacuum” state: A scalar field φ fills spacetime and provides an effective vacuum energy ρvac � V (φ). (For more details about how a scalar field of this type is characterized and evolves, see Carroll pp. 369 – 370.) This vacuum energy acts like a cosmological constant Λ = 8πρvac. Up to factors of order unity, we may write ρvac � (4σ/c)T4 ,