Exploring Simple Cosmological Models: Understanding the Evolution of the Universe - Prof. , Study notes of Astronomy

Download Exploring Simple Cosmological Models: Understanding the Evolution of the Universe - Prof. and more Study notes Astronomy in PDF only on Docsity! Simple Cosmological Models Following Liddle’s Chapter 5, we will now explore some simple solutions to the Fried- mann equation ( ȧ a )2 = 8πG 3 ρ − k a2 (1) and the fluid equation ρ̇ + 3 ȧ a ( ρ + p/c2 ) = 0 . (2) A straw poll of cosmologists roughly a decade ago would have found most of them agreeing that these equations describe the universe pretty well. However, since that time strong evidence has emerged that the universe is accelerating in its expansion. This requires an additional ingredient, called dark energy. For now, though, let’s ignore that and go back to the blissful days before dark energy. What is the left hand side in the Friedmann equation? Note that if we consider two objects that are currently at a distance r from each other and are moving with the universal expan- sion, then at any given time their separation is proportional to the scale factor: r ∝ a, hence ṙ ∝ ȧ. Therefore, ṙ/r = ȧ/a. We note from the Friedmann equation that as a result, ṙ/r is independent of r. This is just what Hubble’s Law tells us: the apparent recession speed is proportional to the distance. We can define H ≡ ȧ/a, and H0 as the value right now. Note that, indeed, the units of the Hubble “constant” are one over time: H0 = 72 km s −1 Mpc−1 (distance per time per distance). We therefore find an evolution equation for H = H(t): H2 = 8πG 3 ρ − k a2 . (3) Let’s consider what this means. We know that ρ decreases as a increases. If k = 0 then as a increases H therefore also decreases, so the expansion slows down with time. If k < 0 then both terms are positive and both terms decrease as a decreases, so again H decreases with increasing a and thus the expansion slows down. If k > 0 then since H > 0 now we know the first term on the right hand side is larger than the second. However, for ρ ∝ 1/a3 (for nonrelativistic matter) or ρ ∝ 1/a4 (for relativistic matter), the first term decreases more rapidly with increasing a than the second. As a result, when k > 0 there will come a time when H = 0, at which point the expansion has stopped and the universe will recontract. Even in this case, though, since H > 0 now we expect the expansion to slow down in the future. As a result, without something else happening, an accelerating expansion is not expected. Expansion and Redshift What happens to a photon as the universe expands? The answer, in short, is that the wavelength of the photon is proportional to the scale factor a (see Figure 1). As Liddle shows, the straightforward way to motivate this is to consider two nearby galaxies, with separation dr. Assuming that they both move with the universal expansion, this tells us that their apparent relative speed is dv = (ȧ/a)dr. Light is emitted from one at wavelength λe, and the Doppler shift is therefore given by dλ/λe = dv/c . (4) We also know that it takes a time dt = dr/c to travel the distance, so that dλ λe = ȧ a dr c = ȧ a dt = da a . (5) Integrating gives λ ∝ a. You can then imagine a large number of such infinitesimal motions, calculus-style, and conclude that λ ∝ a is valid for any distance traveled. The redshift z is usually defined as 1 + z = λr λe = a(tr) a(te) (6) where a photon emitted at wavelength λe is received at wavelength λr, and the global time of emission and reception are te and tr, respectively. This definition implies z = 0 at the present time. The redshift (or scale factor) is the most relevant single quantity to refer to cosmological epochs. Unfortunately, the popular press likes to talk about times instead: “the most distant galaxy was reported yesterday, having emitted the light we see 13 billion years ago.” The problem is that quoting a time like that requires a specific cosmological model, whereas the redshift is a statement of observational fact! Solving the Equations The fluid equation involves both the mass-energy density ρ and the pressure p. As a result, to follow the evolution of the universe we need to know the relation p(ρ) (or more generally when the temperature is important, p(ρ, T )). This is called the equation of state. This comes up in many contexts in physics. For example, consider water. When it is water vapor, a small change in pressure leads to a significant change in density. In contrast, when it is liquid, a very large change in pressure is needed for even a small change in density. This tells us that in the most general circumstances we need to worry about phase changes, where a substance alters its character due to a change in density, temperature, or something else. These can have importance in the early universe, where, for example, it is thought that there was a shift from a very high-density high-temperature environment in which quarks is also redshifted as E ∝ a−1, giving a mass-energy density ∝ a−4. We then find a(t) = (t/t0) 1/2 ; ρ(t) = ρ0(t0/t) 2 . (10) This also implies that in a radiation-dominated epoch, H = 1/(2t). Note that the universe expands more slowly than during the matter-dominated epoch. Pressure is a form of energy, and like any energy it gravitates, so again one should be cautious not to think of pressure as blowing up the universe. The current universe is overwhelmingly matter-dominated, but has that always been the case? The answer is no, as can be seen by ρrad ∝ a−4 but ρmatter ∝ a−3. There was a stage, at a redshift of a few thousand and above, when radiation dominated. Therefore, the early part of the expansion was driven by radiation, but more recently has been governed by matter. What About Curvature? We obtained the previous solutions by assuming k = 0, but perhaps the universe doesn’t work like that. What if k 6= 0? Consider again the Friedmann equation: ( ȧ a )2 = 8πG 3 ρ − k a2 . (11) If k 6= 0 we might worry that the whole nature of the solutions changes. However, we can obtain substantial insight by assuming that either the first term or the curvature term (the one with k) dominates. Note that for matter ρ ∝ a−3 and for radiation ρ ∝ a−4, whereas the curvature term is ∝ a−2. This tells us right away that when a is small enough, the first term is more important. In such early phases, we really can ignore the curvature term for our solutions, and since measurements indicate that the curvature term is currently small (and possibly zero), the entire history of the universe thus far has had minimal influence from curvature. What about later, though? If k < 0 then the universe will expand forever, meaning that a will grow arbitrarily large. It is therefore inevitable that the curvature term will dominate, at which stage the Friedmann equation becomes (ȧ/a)2 ∝ 1/a2, or ȧ ∝const, meaning a ∝ t. The universe thus enters a coasting phase. If k > 0 then eventually the right hand side becomes zero, the universe turns around, and recollapses. We therefore conclude that unless k = 0, curvature will eventually dominate. Note, though, that this does not apply if there is dark energy around! Intuition Builder Suppose the main component of the universe is some sort of field with an equation of state p = wρ (with c ≡ 1). What are the requirements on w for this field to be more important than curvature at very late times (i.e., large a)?