5.1 Introduction
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M. Pettini: Introduction to Cosmology | Lecture 5 REDSHIFTS AND DISTANCES IN COSMOLOGY 5.1 Introduction As we have seen in the previous lecture, a wide variety of world models are conceivable, depending on the values of the parameters Ωi,ΩΛ, and Ωk. Observational cosmologists are interested in assessing which, if any, of these models is a valid description of the Universe we live in. The measurements on which these tests are based generally involve the redshifts and radiant fluxes of distant objects|modern telescopes now have the capability of reaching to early epochs in the history of the Universe. To accomplish Figure 5.1: The Hubble Ultra-Deep Field|the deepest portrait of the visible Universe ever achieved by mankind|is a two-million-second-long exposure obtained by combining images taken with ultraviolet, optical and infrared cameras on board the Hubble Space Telescope. The area shown is equivalent to that subtended by a grain of sand (1 mm) held at arm's length and yet it contains an estimated 10 000 galaxies, at all distances|from nearby ones to some at redshifts z > 7. 1 this, a connection must be made between the models and the photons that arrive from distant astronomical sources. 5.2 Cosmological Redshifts We first show that the redshift we measure for a distant galaxy (or any other source of photons) is directly related to the scale factor of the Universe at the time the photons were emitted from the source: a 1 + z = 0 e a(t = e) We have already encountered this cosmological redshift in Lecture 1 (eq. 1.8), where we emphasised that it is intrinsically different from kinematic red- shifts. To uncover the origin of cosmological redshifts, we start with the Robertson- Walker metric: 2 2 3 2 2 2 dr 2 2 2 2 (ds) = (c dt) − a (t) 4 + r dθ + sin θ dφ 5 (5.1) 1 − kr2 and exploit the fact that in general relativity the propagation of light is along a null geodesic (ds = 0). With the observer (i.e. us, on Earth) at the origin (r = 0), we choose a radial null geodesic so that dθ = dφ = 0, and eq. 5.1 reduces to: c dt dr = ± (5.2) a(t) (1 − kr2)1=2 where the + sign corresponds to an emitted light ray and the − sign to a received one. Imagine now that one crest of the light wave was emitted at time te at distance re, and received at the origin r0 = 0 at t0 , and that the next wave crest was emitted at te+∆te and received at t0+∆t0 (see Fig. 5.2, where the subscript `1' is used instead of subscript `e' to indicate emission quantities). 2 (t0 +dt 0 , 0) (t0 , 0) (t1 +dt 1 , r1 ) (t1 , r 1 ) O’s world P’s world line line Figure 5.2: Propagation of light rays. These times, which describe how long it takes for successive crests of the light wave to travel to Earth, satisfy the relations: t 0 Z 0 dt 1 Z dr = − p (5.3) a(t) c 1 − kr2 te re for the first crest and: t +∆t 0 0 Z 0 dt 1 Z dr = − p (5.4) a(t) c 1 − kr2 te+∆te re for the next crest. Subtracting these equations produces: t +∆t t 0 Z 0 dt Z 0 dt − = 0 (5.5) a(t) a(t) te+∆te te But t +∆t t t +∆t t +∆t 0 Z 0 dt Z 0 dt 0 Z 0 dt e Z e dt = + − (5.6) a(t) a(t) a(t) a(t) te+∆te te t0 te so that t +∆t t +∆t 0 Z 0 dt e Z e dt = (5.7) a(t) a(t) t0 te Any change in a(t) during the time intervals ∆t0 and ∆te between suc- cessive wave crests can be safely neglected, so that we can treat a(t) as a 3 constant with respect to the time integration. Consequently ∆t ∆t ∆t a(t ) e = 0 ; e = e (5.8) a(te) a(t0) ∆t0 a(t0) The time interval between successive wave crests is the inverse of the fre- quency ν of the light wave, related to its wavelength λ by the relation c = λ · ν (or, if we had stuck to our natural units where c = 1, we could have said that ∆t is the wavelength of the light), so that λ a(t ) 0 = 1 + z = 0 (5.9) λe a(te) Obviously, in a contracting Universe where a(te) > a(t0) [see Figures 4.1 and 4.2] we would measure blueshifts, rather than redshifts, for distant objects. 5.2.1 Time Evolution of the Hubble Parameter Having established this correspondence between redshift and the scale fac- tor of the Universe, we can now derive an expression for the Hubble pa- rameter as a function of redshift, H(z). Recalling the Friedmann equation in the notation of eq. 4.10: 2 2 −1 2 2 a_ = H0 Ωm;0a + H0 ΩΛ;0a (5.10) and the definition of Ωk (eq. 4.7): 2 Ωk ≡ −k=(aH) (5.11) it can be easily seen that: 0 12 H(z) 3 2 @ A = Ωm;0 · (1 + z) + Ωk;0 · (1 + z) + ΩΛ;0 (5.12) H0 The right-hand side of eq. 5.12 is sometimes referred to as E(z), so that: 1=2 H(z) = H0 · E(z) 4 In the simplest case of an Einstein-de Sitter cosmology (Ωm;0 = 1; Ωk;0 = ΩΛ;0 = 0): 3=2 H(z) = H0 · (1 + z) : In today's `consensus' cosmology Ωm;0 = 0:31; Ωk;0 = 0; ΩΛ;0 = 0:69 (see Table 1.1): q 3 H(z) = H0 · Ωm;0 · (1 + z) + ΩΛ;0 : (5.13) A more general version of eq. 5.12 includes the energy density in radiation (i.e. relativistic components of the Universe) which, as we saw in Lecture 2, 4 varies as ρrad / 1=a (eq. 2.23): H(z) q 3 4 2 = Ωm;0(1 + z) + Ωrad;0(1 + z) + Ωk;0(1 + z) + ΩΛ;0 ; (5.14) H0 from which it can be seen that at the highest redshifts it is the relativistic component of the Universe that drives the expansion, Conversely, from z ' 0:3, it is Λ that has progressively become the driving force of cosmic expansion. 5.2.2 Redshift vs. Time We can derive a relationship between time t and redshift z by considering the following: da 1 da dz (1 + z) H(z) ≡ = : (5.15) dt a dz dt a0 But, with: a a a = 0 ; da = − 0 dz ; (5.16) 1 + z (1 + z)2 and therefore: dz dt = − (5.17) H(z) (1 + z) 5 so that: t2 z2 Z 1 Z dz dt = − 1=2 ; (5.18) t1 H0 z1 (1 + z)E(z) from which we can calculate the age of the Universe: t 1 Z 0 1 Z dz t0 = dt = 1=2 : (5.19) 0 H0 0 (1 + z)E(z) In the simplest case of an Einstein-de Sitter cosmology (Ωm;0 = 1; Ωk;0 = ΩΛ;0 = 0) , we recover: 1 0 1 Z dz 2 2 −1 −3=2 −1 t0 = = H0 (1 + z) = H0 (5.20) 5=2 H0 0 (1 + z) 3 1 3 5.3 Cosmological Distances 5.3.1 Proper Distance We saw in Lecture 1 that the cosmological principle implies the existence of a universal time t. Since all fundamental observers see the same sequence of events in the Universe, they can synchronise their clocks by means of these events. This allows us to define a proper distance, as the distance between two events, A and B, in a reference frame for which they occur simultaneously (tA = tB). The proper distance of an object from Earth can be found by starting again from the Robertson-Walker metric: 2 2 3 2 2 2 dr 2 2 2 2 (ds) = (c dt) − a (t) 4 + r dθ + sin θ dφ 5 (5.1) 1 − kr2 with, as before, the Earth at the origin of the radial coordinate, dθ = dφ = 0, but this time setting dt = 0, so that we have: 6 s r Z 0 Z dr s(t) = ds = a(t) 2 1=2 (5.21) 0 0 (1 − kr ) which has three solutions depending on whether k is +ve, 0, or −ve: p 8 p1 −1 > sin (r k) for k > 0 > k > > > s(t) = a(t) · < r for k = 0 (5.22) > > > > 1 −1 q > p sinh (r jkj) for k < 0 : jkj In a flat Universe, the proper distance to an object is just its coordinate distance, s(t) = a(t) · r. However, because the r coordinate was introduced in the context of a flat Newtonian Universe, the coordinate distance will generally not agree with the proper distance. Because sin−1(x) ≥ x and sinh−1(x) ≤ x, in a closed Universe (k > 0) the proper distance to an object is greater than its coordinate distance, while in an open Universe (k < 0) the proper distance to an object is less than its coordinate distance. 5.3.2 The Horizon As the Universe expands and ages, an observer at any point is able to see increasingly distant objects as the light from them has time to arrive. This means that, as time progresses, increasingly larger regions of the Universe come into causal contact with the observer. The proper distance to the furthest observable point|the particle horizon| at time t is the horizon distance, sh(t).