PHYS4003 Introduction to Cosmology Notes PDF

PHYS4003 Introduction to Cosmology Prof. Anne Green CAPT B110 [email protected] Contents 2 Observing the Universe...

PHYS4003 Introduction to Cosmology Prof. Anne Green CAPT B110 [email protected] Contents 2 Observing the Universe 1 2.1 Expansion rate and age............................... 1 2.1.1 Present day expansion rate......................... 1 2.1.2 Rough estimate of age............................ 2 2.1.3 Observational limits on age......................... 3 2.1.4 Accurate calculation of age......................... 3 2.2 Light travel and horizons.............................. 4 2.2.1 Robertson-Walker metric.......................... 4 2.2.2 Redshift revisited............................... 5 2.2.3 Cosmological horizon distance........................ 6 2.2.4 Cosmological event horizon......................... 6 2.3 Distances....................................... 7 2.3.1 Proper distance................................ 7 2.3.2 Luminosity distance............................. 7 2.3.3 Angular diameter distance.......................... 9 2 Observing the Universe 2.1 Expansion rate and age 2.1.1 Present day expansion rate The Hubble parameter is the (time dependent) constant of proportionality in the Hubble law: v = Hr , (1) and its present day value (often known as the Hubble constant) is written as H0 = 100h km s−1 Mpc−1 , (2) with h parametrizing the uncertainty in its value. The Hubble constant feeds into many other cosmological parameters, and hence the uncertainty in its value leads to uncertainties in the values of these other parameters (we’ll see the factor of h cropping up in various equations). 1  In principle to measure H0 we just need to measure the speeds of galaxies at different distances. In practise there are complications in measuring both the speeds and distances. As recently as the 1990s there was a factor of 2 uncertainty in the value of the Hubble constant (and the value measured by Hubble himself was too large by an order of magnitude, due to errors in his distance measurements). The velocity of a galaxy is actually the sum of its recession velocity and its peculiar velocity: v = H0 r + vpec. (3) However if we observe galaxies at sufficiently large distances ( & 10Mpc) then we can ensure that vpec  H0 r (see problem sheet 1). Distances are hard to measure. In astronomy the distances of nearby objects can be mea- sured using parallax (the apparent movement of an object due to the Earth’s orbit), however for a galaxies at distances larger than 1 Mpc the parallax is smaller than a micro arcsecond, and hence far too small to be measured. Instead we have to use standard candles, objects that are assumed to always have the same absolute brightness, so their apparent brightness can be used to calculate their distance. There are a number of different standard candles, for instance Cepheid variable stars (whose period is related to their luminosity), type 1a SNe and the brightest galaxy in a cluster (see Sec. 1.4 of Roos for more information). However these methods typically give relative, rather than absolute, distances. They can be used to confirm the Hubble law, v ∝ r, but not to measure the constant of proportionality (i.e. the Hubble parameter). To measure absolute distances, and hence H0 , we need to know not just that the objects are standard candles, but what their absolute brightness is. This is done by using a chain of different standard candles to measure the distance of distant objects (this is known as the cosmic distance ladder). For instance parallax can be used to measure the distance to nearby Cepheids, with Cepheids in distant galaxies then being used to calibrate other standard candles. Observations using this method typically find results close to h = 0.73 ± 0.02. The anisotropies in the cosmic microwave background radiation can, with some assumptions, also be used to determine the Hubble constant (see Sec. 3.3 ‘Thermal Universe: cosmic microwave background’). In this case the values obtained are somewhat (compared to the size of the error bars) smaller: h = 0.68 ± 0.01. This discrepancy, known as the Hubble tension, is one of the hottest topics in cosmology at the moment see e.g. https://arxiv.org/abs/2001.03624. 2.1.2 Rough estimate of age A very rough estimate of the age of the Universe, t0 , can be obtained from the Hubble law r 1 t0 ∼ =. (4) v H0 Converting the units into time gives 1 3.1 × 1022 t0 ∼ = s = 3.1 × 1017 h−1 s = 9.8 × 109 yr ∼ 10 Gyr , (5) 100h kms−1 Mpc−1 100h × 103 where we’ve used h ∼ 1 in the final steps. This estimate ignores the fact that v changes with time, therefore we don’t expect it to be very accurate. 2  For most of its evolution the Universe is matter dominated, therefore we can obtain a better estimate for the age of a flat, k = 0, Universe by using the expression for the Hubble parameter which we derived in the ‘Expanding Universe’ section: H = 2/(3t). This gives 2 t0 = H0−1 ≈ 7 Gyr , (6) 3 i.e. a factor of 2/3 smaller than the very rough estimate. 2.1.3 Observational limits on age The Universe must be older than the objects in it. This allows us to use measurements of the age of various objects to place lower limits on the age of the Universe: From geological data the age of the Earth is of order 5 Gyr. Uranium isotopes produced in SNe decay at different rates. Their ratios can be used to measure the age of the Milky Way, tMW ∼ 7 Gyr. Studies of the cooling of old white dwarfs find tWD ∼ 10 Gyr. Studies of the evolution of stars within globular clusters (clumps of stars that are believed to be among the oldest objects in the Universe) find tGC ∼ 10 Gyr In each case we need to add the time after the Big Bang at which these objects form (roughly 1 Gyr for white dwarfs and globular clusters). The fact that the observational limits on the age of the Universe are roughly comparable with the estimates from the Big Bang model is reassuring (and to some extent a vindication of the Big Bang). However the theoretical estimate for a flat matter dominated Universe was somewhat smaller that the observational limits. To check that the Big Bang does indeed produce a Universe which is older than the objects in it we need a more accurate theoreti- cal calculation of its age, which drops the simplifying assumption of a flat matter-dominated Universe. 2.1.4 Accurate calculation of age The exact value of the age of the Universe can be expressed as Z t0 Z a0 =1 Z 1 da da t0 = dt = =. (7) 0 0 ȧ 0 aH Since redshift is defined as 1 + z = 1/a this can be rewritten, using da dz =− , (8) a 1+z as Z 0 Z ∞ dz dz t0 = − = , (9) ∞ (z + 1)H(z) 0 (z + 1)H(z) 3 where we showed in the Expanding Universe section that, allowing for non-zero curvature and cosmological constant H 2 = H02 Ωr,0 (1 + z)4 + Ωm,0 (1 + z)3 + Ωk,0 (1 + z)2 + ΩΛ,0.   (10) In general this has to be solved numerically, but there are some special cases where solutions can be found analytically An open matter-dominated universe (i.e. with no cosmological constant) and a flat universe with a cosmological constant (i.e. Ωm + ΩΛ = 1) are both older than a matter-dominated flat universe (with the same value of H0 ). For the matter-dominated open universe this is simply because there is less matter to slow the expansion down (and the opposite is true for a matter- dominated closed universe). A non-zero cosmological constant increases the expansion rate at late times, therefore for a fixed present day expansion rate, it is older than a matter-dominated flat universe. 2.2 Light travel and horizons In this part we’ll answer the question ‘How big is the Universe?’ (or, more accurately, ‘How far has light travelled since the Big Bang?’). To do this we’ll use some results from general relativity, and in the process derive the red-shifting of wavelength properly. 2.2.1 Robertson-Walker metric In general relativity the metric is a fundamental quantity which describes the geometry of space- time, giving the distance between neighbouring points. A proper derivation of this requires general relativity, so instead we’re just going to motivate its form in a more qualitative way. If you find this confusing, don’t worry. You just need to be able to use the Robertson-Walker metric, Eq. (15), in the calculations we do subsequently. On the other hand if you’d like to understand more about where the form for the spatial part of the metric comes from, see section 2.2 of Roos. On a flat 2d surface the distance, ∆s between 2 points with coordinates (Xa1 , Xb1 ) and (Xa2 , Xb2 ) is given by ∆s2 = ∆Xa2 + ∆Xb2 , (11) where ∆Xa = Xa2 − Xa1 and ∆Xb = Xb2 − Xb1. This is just Pythagoras’ rule. In comoving coordinates, (xa , xb ), this becomes ∆s2 = a2 (t)(∆x2a + ∆x2b ). (12) In 4d space-time in general X ds2 = gµν dxµ dxν , (13) µ,ν where the indices µ and ν take values 0, 1, 2, 3. x0 = ct is the time coordinate, and x1 , x2 , x3 are the 3 spatial coordinates. gµν is the metric and it has 16 components (4 columns and 4 rows). 4  The cosmological principle states that the Universe has no preferred locations, which tells us that the spatial part of the metric, ds23 , has constant curvature. The most general spatial metric for which this is true is, in spherical polar coordinates, dr2 ds23 = + r2 (dθ2 + sin2 θdφ2 ) , (14) 1 − kr2 where k is the curvature (n.b. here r is a comoving coordinate). We need to take into account, using the scale factor, that space can expand or contract, which gives us the Robertson- Walker metric: dr2   2 2 2 2 2 2 2 2 2 2 2 2 ds = −c dt + a (t)ds3 = −c dt + a (t) + r (dθ + sin θdφ ). (15) 1 − kr2 2.2.2 Redshift revisited We can use the Robertson-Walker metric to derive the redshift of radiation properly. Light obeys ds = 0 so therefore for a light ray which travels radially (dθ = dφ = 0) c dt dr =√. (16) a 1 − kr2 Consider a light ray which travels radially from r = 0 at t = te to r = r0 at t = tr : Z tr Z r0 c dt dr = √. (17) te a 0 1 − kr2 A light ray emitted a short time later from the same galaxy (hence with the same comoving coordinates) satisfies: Z tr +dtr Z r0 c dt dr = √. (18) te +dte a 0 1 − kr2 The RHSs of these expressions must be equal, therefore Z tr Z tr +dtr c dt c dt =. (19) te a te +dte a Dividing these integrals up: Z te +dte Z tr Z tr Z tr +dtr c dt c dt c dt c dt + = + , (20) te a te +dte a te +dte a tr a the overlapping regions cancel so that Z te +dte Z tr +dtr c dt c dt =. (21) te a tr a The time intervals dte and dtr are small, so the integrals are given by the value of the integrand times the width of the region and hence: c dte c dtr =. (22) a(te ) a(tr ) If we consider 2 successive crests of a single wave λ ∝ dt ∝ a(t) and hence λr a(tr ) =. (23) λe a(te ) 5 2.2.3 Cosmological horizon distance The cosmological horizon distance is the maximum distance light has travelled since the Big Bang (at t = 0). In comoving units it is given by, rH where Z rH Z t dr c dt̃ √ =. (24) 0 1 − kr2 0 a(t̃) In a flat Universe, k = 0, this simplifies to Z rH Z t dt̃ dr = c , 0 0 a(t̃) Z t dt̃ rH = c. (25) 0 a(t̃) To get the physical distance we need to multiply the comoving distance by the scale factor: Z t dt̃ RH = a(t)rH = a(t)c. (26) 0 a(t̃) Today a(t0 ) = 1 and hence physical and comoving distances coincide: RH (t0 ) = rH (t0 ). If the cosmological constant is zero, Λ = 0, then for most of its evolution the Universe is matter dominated with a(t) = (t/t0 )2/3 and hence Z t0 h it0 2/3 2/3 rH (t0 ) ≈ ct0 t−2/3 dt = ct0 3t1/3 = 3ct0. (27) 0 0 and RH (t0 ) = rH (t0 ) = 3ct0. Light from galaxies further away than r0 won’t have had time to reach us. Note that rH (t0 ) > ct0 as the Universe expands as light travels across it. The cos- mological horizon distance is sometimes also referred to as the particle horizon (since particles have v ≤ c it also serves as an upper limit on the distance particles can have travelled since the Big Bang). 2.2.4 Cosmological event horizon The cosmological event horizon, rev , is the radius within which signals emitted at time t can be observed by time tmax. The comoving cosmological event horizon is given by Z rev Z tmax dr c dt̃ √ =. (28) 1 − kr 2 a(t̃) 0 t In a flat k = 0 Universe this simplifies to Z tmax dt̃ rev (t) = c. (29) t a(t̃) The physical size of this region is Z tmax dt̃ Rev (t) = a(t)rev (t) = ca(t). (30) t a(t̃) In an open universe, with Λ = 0, tmax → ∞ and Rev (t) → ∞, while in a closed Λ = 0 universe tmax is finite, and hence so is Rev. 6 2.3 Distances In this section we’ll see how redshift affects how the properties of objects, such as luminosity and diameter, appear to us. 2.3.1 Proper distance R The proper distance is the length of the spatial geodesic at some time t i.e dp = ds. The proper distance to a galaxy with comoving coordinates (r0 , 0, 0) is given by Z Z r0 dr dp (t) = a(t) ds = a(t) √. (31) 0 1 − kr2 We’re usually interested in the proper distance today dp ≡ dp (t0 ). We know that for light emitted at time te and observed at t0 Z r0 Z t0 dr dt̃ √ =c , (32) 2 0 1 − kr te a(t̃) therefore the proper distance to an object today, at time t0 , can also be written as Z t0 dt̃ dp = c. (33) te a(t̃) In a flat universe with k = 0, Z r0 dp = dr = r0. (34) 0 √ In a closed universe with k > 0 if we make the substitution u = kr Eq. (31) becomes √ Z kr0 1 du dp = √ √ , (35) k 0 1 − u2 and making a further substitution u = sin y gives √ √ Z sin−1 ( kr0 ) Z sin−1 ( kr0 ) √ 1 cos y dy 1 1 dp = √ p =√ dy = √ sin−1 ( kr0 ). (36) k 2 k k 0 1 − sin y 0 While for an open universe a similar calculation (see problem sheet 2) gives 1 dp = p sinh−1 ( |k|r0 ). p (37) |k| 2.3.2 Luminosity distance The luminosity distance, dlum , is defined as the distance an object appears to have, assuming the inverse square law holds. This is not the actual distance as the Universe is expanding and the geometry is not necessarily flat. 7  If a flux of photons, f , is measured from an object with luminosity, L, then the luminosity distance is defined as L 1/2   dlum =. (38) 4πf The photons emitted at time te are today spread out over a sphere of radius r0 and surface area 4πr02. The expansion of the Universe also affects the flux of photons received. Firstly, the energy of each photon is decreased, E ∝ λ−1 −1 −1 r ∝ λe (1 + z). Secondly, the rate at which photons arrive is also decreased by a factor of (1 + z). [If the time difference between photons is initially δte , then the distance between then is c δte. When the photons are detected at t0 this distance has increased to c δte (1 + z), and hence the time interval between photons is δt0 = (1 + z)δte.] Therefore for an object at r0. L f= , (39) 4πr02 (1 + z)2 and inserting this in Eq. (38) we find: dlum = r0 (1 + z) , (40) i.e. distant objects appear further away than they actually are since redshift reduces their apparent luminosity. In a flat k = 0 universe dp = r0 so that dlum = dp (1 + z). For nearby objects, z  1, dlum ≈ dp. In a closed k > 0 universe from Eq. (36) 1 √ r0 = √ sin ( kdp ) , (41) k and hence 1+z √ dlum = √ sin ( kdp ). (42) k so the extent to which distant objects appear further away is reduced (the opposite is true in an open universe). For nearby objects (z  1) dp is small and hence 1+z √ dlum ≈ √ ( kdp ) ≈ dp. (43) k This is also the case in an open universe. The luminosity distance can be used to measure cosmological parameters if we have objects, known as ‘standard candles’, with known luminosity. We’ll see in the ‘Dark Side of the Universe’ section how type 1a SNe can be used as standard candles and provide evidence for accelerated expansion and dark energy. 8 2.3.3 Angular diameter distance The angular diameter distance, ddiam , is defined as the distance an object of known physical extent (i.e. a ‘standard ruler’) appears to be at assuming Euclidean geometry. In other words it is a measure of how large objects appear. If an object with physical extent l is perpendicular of the line of sight and subtends an angle dθ then its angular diameter distance is l ddiam =. (44) dθ For an observer at r = 0 and an object at r = r0 aligned in the θ direction ds2 = a2 r02 d2 θ and hence l = ds = r0 a(te )dθ , (45) where te is the time at which light is emitted from the object. Light rays propagate radially so the angular extent of the object is preserved and we perceive an angular extent l l(1 + z) dθ = = , (46) r0 a(te ) r0 since a(te ) = 1/(1 + z), and hence l r0 ddiam = = , (47) l(1 + z)/r0 1+z For nearby objects, z  1, ddiam ≈ r0 and in a flat universe ddiam ≈ dp once more. Note that in the real Universe we don’t get objects with the same physical extent existing at all z (i.e. galaxies aren’t the same at low and high redshift). However there is a feature in the clustering of galaxies, known as Baryon Acoustic Oscillations (BAO), which acts as a standard ruler and has been used in recent years to measure the angular diameter as a function of red-shift and hence constrain cosmological parameters. 9