Evolution of Density Perturbations in Expanding Universe - Prof. Michael Miller, Study notes of Astronomy

Download Evolution of Density Perturbations in Expanding Universe - Prof. Michael Miller and more Study notes Astronomy in PDF only on Docsity! Structure Formation in an Expanding Universe Last time we talked about how extra-dense regions can collapse when the background is static. However, we know that the universe is expanding. The effects of expansion are the topic of this lecture. The Evolution of Density Perturbations Consider a spherical Newtonian explosion in vacuum. When the sphere has radius r, a particle on the edge of the explosion has speed v(r), and the mass of the material interior to the particle is M = Mcrit(1+ δ), where Mcrit is the mass that would make v(r) exactly equal to the escape speed at r and δ ¿ 1. How does δ depend on r? To determine this, consider the specific energy of the particle, which is its energy per unit mass. This is 1 2 v2 − GM/r = C , (1) where the total specific energy C is a constant. If M = Mcrit, then the total energy is zero (by definition of Mcrit), so we can rewrite this as 1 2 v2 − GMcrit(1 + δ)/r = 1 2 v2 − GMcrit/r − GMcritδ/r = −GMcritδ/r = C . (2) Since C is a constant and so is Mcrit for δ ¿ 1, it follows that δ ∝ r. Let’s take stock of what this really means. It says that if a patch of space currently is slightly overdense, say by 1%, then the scale factor of the universe must double so that the patch is now overdense by 2% (note that this is entirely independent of the size of the patch). But because spacetime is expanding, the critical density is decreasing. This means that the actual density of the patch, measured say in kg m−3 is decreasing during this period. That’s rather different than what we explored in the last class, where the density would just go up. However, the density will not decrease indefinitely. In fact, a useful perspective is as follows: as Newton showed, matter inside a spherical shell feels no force from the shell. The extension in general relativity, called Birkhoff’s theorem, then means that we can treat a slightly overdense spherical region of space as if it is a universe in its own right, with its own values of Ω, Ωk, and ΩΛ. Suppose we ignore ΩΛ, which is okay for the redshifts z > 10 when structure first formed. Then a patch of space slightly denser than normal evolves as if it had positive curvature, meaning that the expansion slows, stops, and turns around. In more detail, we can do a simplified (yet still realistic) treatment to determine the overdensity at the moment of turnaround. Suppose that the average density of the universe is exactly the critical density, and that a spherical region of mass M has a constant slight overdensity. We will assume that the matter in this region is cold dark matter: the “cold” means pressureless, so that in practice the matter just feels gravity but doesn’t collide. Ignore dark energy, and consider the total energy of a particle of mass m at the edge of the region (of radius r). Energy is conserved, and m is conserved, thus so is the specific energy: U = 1 2 ṙ2 − GM/r < 0 . (3) As a result, ṙ2 = 2U + 2GM/r. Ask class: how do they expect the turnaround time to depend on U? Suppose that the region starts with a radius rmin, and expands to a maximum radius rmax, at which point ṙ = 0 and the region stalls before turning around. The time T that this takes is T = ∫ rmax rmin dr/ṙ = ∫ rmax rmin dr √ 2U + 2GM/r . (4) We will now assume that rmin ¿ rmax, so that we can set rmin → 0 in the integral. Since rmax is given by the condition that ṙ = 0, it means that rmax = GM/(−U) (recall that U < 0, so this is positive as it should be!). We can then write the time as T = ∫ −GM/U 0 dr √ −2U √ −GM/U r − 1 . (5) We change variables to x ≡ r/(−GM/U), leading to T = GM −U √ −2U ∫ 1 0 √ x 1 − x dx . (6) Doing the integral then yields T = πGM −2U √ −2U . (7) To determine the overdensity, though, we need to know the radius r0 to which the background would have expanded, then the density ratio will be (r0/rmax) 3. Because the background is at exactly the critical density, it means the energy is zero, so that ṙ2 = 2GM/r. Therefore, T = ∫ r0 0 dr √ 2GM/r . (8) Solving gives r 3/2 0 = (3π/4)(−GM/U)3/2. The density contrast is then ρ/ρcrit = (r0/rmax) 3 = (3π/4)2 . (9) Wow! What a miracle! The energy U does not enter the calculation at all, so this result applies for all scales. leading to filamentary structure. Finally, collapse occurs along a to get to a roughly spherical distribution (which could be modified if there is rotation present). Note, in fact, that the figure shown in the last lecture demonstrates at least the filamen- tary nature of large-scale structure, but the spherical nature of smaller-scale structure. The reason, as we shall discuss shortly, is that initial large-scale perturbations are smaller than small-scale perturbations, so small scales can collapse and become spherical more quickly. In fact, clusters of galaxies are the largest spherical virialized gravitationally-bound systems in the universe (see Figure 1, which shows X-ray and optical images of a cluster). Hierarchical Structure Formation What, then, is the overall picture we have about how matter got together? Since cold (i.e., pressureless) dark matter dominates the mass, most simulations have ignored baryons although a few are starting to be more comprehensive. From the standpoint of dark matter, the issue is that small structure forms first because the initial perturbations were greatest, then as time went on (and the scale factor of the universe went up), larger structure formed. However, you should not think of these as isolated incidents, where in one place small structure formed and in a distinct place large structure appeared. Instead, it is a very dynamic dance indeed. In a large overdense region there are small extra-overdense regions that form structure. Then, those structures got together to form yet larger structures. The picture is then of dark matter blobs smashing together all the time to form bigger and bigger things. Nice movies of this are at http://star-www.dur.ac.uk/∼moore/movies.html. One aspect of all these collisions that appeals to me is that because many dark matter collections are thought to have black holes at their centers, collisions and dynamical friction (settling of heavy stuff) probably means that there are many inspirals and mergers of supermassive black holes with each other. Future space-based gravitational wave detectors will be able to see this, and thus give us a unique view of this crucial period in the history of the universe. Intuition Builder We described collisions of dark matter balls (and their associated baryons) as central to structure formation. Is that happening now? In particular, consider a galaxy cluster which currently has 1000 galaxies in a region of space 1 Mpc across. If you checked back in a trillion years, how would you expect it to look? Fig. 1.— X-ray (bottom) and optical (top) images of the galaxy cluster Abell 2029. Hot gas contains several times more mass than exists in the cluster stars. From http://www.sr.bham.ac.uk/exgal/images/99150main abell2029 comp m.jpg