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FRW (Friedmann-Robertson-Walker) Universe Chapter 2 FRW (Friedmann-Robertson-Walker) universe We start our discussion from the evolution of the homogeneous and isotropic expanding universe, of- ten called Friedmann-Robertson-Walker (FRW) world model. This somewhat idealized world model is consistent with the observations thus far: Hubble diagrams (which shows the distance-redshift relation) have been continuously updating to confirm that the Universe is expanding, observed cosmic microwave background radiation is isotropic from all lines of sight, and distribution of galaxies in SDSS appears to be homogeneous on scales larger than 70 Mpc=h. ' In this chapter, we study the kinematics and dynamics of the FRW universe. 2.1 FRW metric The spacetime of spatially homogeneous and isotropic, expanding universe can be best thought of as a contiguous bundle of homogeneous and isotropic spatial (constant-time) hypersurfaces whose length scale evolves in time. The metric in such a spacetime can be written as1 2 µ ν 2 2 i j ds = gµνd x d x = d t + a (t)g˜i j(x)d x d x , (2.1) − where g˜i j(x) is a time-independent spatial metric defined on the constant-time hypersurface, and a(t) is the scale factor that relates the proper (physical) distance to the comoving (coordinate) distance. As we shall show below, except for the three maximally symmetric cases2—Minkowski, de-Sitter, and anti de-Sitter spacetime, defining the space-like hypersurface can be naturally done with, e.g. constant- density-hypersurface, in the FRW world model. A word about the convention. Throughout the course, we shall use the Greek for the spacetime in- dices µ, ν, = 0, 1, 2, 3 with the index 0 being the time index, and the Latin for the space indices ··· 1 The following form of metric is uniquely dictated from spatial homogeneity and isotropy as we should not have any g0i component which would introduce a preferred direction nor the space dependence in g00 as that breaks homogeneity. Then, we rescale the time coordinate so that g00 = 1. As we shall see below, there are only three choices for homogeneous and isotropic spatial metric. Then, the scale factor− describes the homogeneous time evolution of the spatial coordinate. 2For the maximally symmetric cases—everywhere in the spacetime is the same—, the Riemann tensor is given by Rµνρσ = κ gµρ gνσ gµσ gνρ , (2.2) − where κ = 0 for Minkowski spacetime, κ > 0 for de-Sitter spacetime and κ < 0 for anti de-Sitter spacetime. 1 2 CHAPTER 2. FRW (FRIEDMANN-ROBERTSON-WALKER) UNIVERSE i, j, k, = 1, 2, 3 that refere to the three spatial dimensions. From here, we also use Einstein’s sum- mation··· convention that summing over the indices repeated up-and-down is understood. For example, compare Eq. (2.1) to the same equation in the introdution. 2.1.1 Spatial metric The spatial metric g˜i j(x) is a metric for a homogeneous and isotropic 3-space (called constant-time- hypersurface). As an illustrative example of the homogeneous and isotropic space (but in a two- 2 dimensional space that we can easily imagine), let us consider the metric on the two-sphere (S ) em- 3 bedded in the three-dimensional Euclidean space (R ) with coordinate (x, y, z). The two-sphere in the three-dimension is defined by 2 2 2 2 x + y + z = a , (2.3) with a being a radius of the sphere. The metric in the 3D space is given as 2 2 2 2 ds3D = d x + d y + dz , (2.4) but we only need two coordinate to describe the two-sphere. Let us use (x, y) for that. Then, zdz = xd x yd y yields − − 2 2 2 2 2 xd x yd y 2 2 (xd x + yd y) ds2D = d x + d y + − − = d x + d y + (2.5) z a2 (x2 + y2) − 2 2 2 Now we use the polar coordinate system that x = r sin θ and y = r cos θ (r = x + y ) to transform the metric as r2dr2 a2dr2 ds2 dr2 r2dθ 2 r2dθ 2. (2.6) 2D = + + a2 r2 = a2 r2 + Finally, changing the radial coordinate to r ar, we− find − ! 2 2 2 2 2 2 (xd x + yd y) 2 dr 2 2 ds2D = a d x + d y + = a + r dθ . (2.7) 1 (x2 + y2) 1 r2 − − 2 That is, the overall factor a is the radius of the S . Let’s go back to the FRW metric in Eq. (2.1). There, 2 the scale factor a(t) depends on time, and it describes the situation that the radius of S evolves in time. As the radius changes in time, distance between any given two points on the sphere also changes with the same rate. This is the essence of the FRW world model! Now all we have to do is to extend the analysis above to three-dimension. There are three spaces that 3 3 3 are homogeneous and isotropic: R (Euclidean space), S (sphere), and H (hyperspherical surface) where the spatial curvatures are K = 0, K > 0 and K < 0 respectively, everywhere the same in the space. 3 3 3 3 These three spaces are usually referred as flat (R ), closed (S ) and open (H ) in the literature . The 3 metric in R is trivially: ds2 a2 d x2 d y2 dz2 a2 dr2 r2 dθ 2 sin2 θ dφ2 , (2.8) K=0 = + + = + + where we use the Cartesian coordinate and the spherical coordinate, respectively. 3 The metric for the three-dimensional sphere S with radius a can be found by extending the 2D calcu- lation above. Consider the three-sphere embedded in the four-dimensional space (x, w) 2 2 2 x + w = a (2.9) 3This is the case when we assumes the homogeneity and isotropy to the entire space. But, of course, in the real universe we can only know the curvature of the local observable Universe. 2.1. FRW METRIC 3 with the four dimensional metric of 2 2 2 ds4D = dx3D + dw . (2.10) Then, we can work out that the metric will be: 2 2 2 2 2 2 2 (xd x + yd y + zdz) 2 dr 2 2 2 2 dsK>0 = a d x + d y + dz + = a + r dθ + sin θ d' . (2.11) 1 (x2 + y2 + z2) 1 r2 − 3 − 4 The three-dimensional hyperspherical surface H can be embedded in the pseudo-Euclidean space with line element 2 2 2 ds4D = dx3D dw , (2.12) − as 2 2 2 w x = a . (2.13) − Differentiating the Eq. (2.13) and plugging into Eq. (2.12), we have x dx 2 xd x yd y zdz 2 ds2 dx2 ( ) a2 d x2 d y2 dz2 ( + + ) , (2.14) K<0 = 3D 2 · 2 = + + 2 2 2 − a + x − 1 + (x + y + z ) j j and with the spherical coordinate, we get 2 2 2 dr 2 2 2 2 dsK<0 = a + r dθ + sin θ d' . (2.15) 1 + r2 2.1.2 FRW metric We can combine all three cases we consider above as x i x j dr2 ds2 a2 δ K d x i d x j a2 r2 dθ 2 sin2 θ d'2 , (2.16) 3D = i j + 1 K x 2 = 1 K r2 + + − j j − where K = 0 for flat, K = 1 for closed and K = 1 for open universe. Inserting the spatial metric into Eq. (2.1), we have arrived at the FRW metric: − x i x j dr2 ds2 d t2 a2 t δ K d x i d x j d t2 a2 t r2 dθ 2 sin2 θ d'2 . = + ( ) i j + 2 = + ( ) 2 + + − 1 K x − 1 K r − j j − (2.17) That is, in Eq. (2.1), x i x j g˜ δ K (2.18) i j = i j + 1 K x 2 for the Cartesian coordinate, and − j j 0 2 1 1 (1 K r )− 0 0 − 2 g˜i j = @ 0 r 0 A (2.19) 0 0 r2 sin2 θ for spherical coordinate. Yes, metric is nothing mysterious but a matrix once you define a basis! The inverse metric in Cartesian coordinate is 1 1 g00 1, gi j g˜i j δi j K x i x j , (2.20) = 2 = 2 − ≡ a (t) a (t) − and corresponding Christoffel symbols are a˙ Γ 0 Γ 0 Γ i 0, Γ 0 aa˙ g˜ , Γ i δ , Γ i K g˜ x i. (2.21) 00 = 0i = 00 = i j = i j j0 = a i j jk = jk 4Note one minus sign in the 4D metric, that’s why it is called ‘pseudo’. 4 CHAPTER 2. FRW (FRIEDMANN-ROBERTSON-WALKER) UNIVERSE 2.2 Cosmological redshift How would ‘free’ particles move in the FRW universe? In General Relativity, the free particles move µ along the geodesic x (λ) in the curved spacetime following the geodesic equation: d2 xµ d xα d xβ Γ µ 0, (2.22) dλ2 + αβ dλ dλ = where λ is a parameter along the geodesic curve (called affine parameter). We can define the affine parameter λ so that d xµ pµ . (2.23) = dλ Then, 0-component of the geodesic equation becomes dp0 dp0 a2H g˜ pi p j Hp2 0, (2.24) dλ + i j = dλ + = 2 i j 2 i j µ where we define p = gi j p p = a g˜i j p p . For the relativistic particles, pµ p = 0, and for non- µ 2 relativistic particles, pµ p = m , both of which leads − 0 0 p dp = pdp, (2.25) that transform Eq.
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