Chapter 18 Friedmann Cosmologies

Transcription

1 Chapter 18 Friedmann Cosmologies Let us now consider solutions of the Einstein equations that may be relevant for describing the large-scale structure of the Universe and its evolution following the big bang. To do so, we must first make a choice for the form of the spacetime metric governing our Universe. 487

2 488 CHAPTER 18. FRIEDMANN COSMOLOGIES 18.1 The Cosmological Principle Possible forms for the metric of spacetime are strongly constrained by the cosmological principle: The Universe on large scales is both homogeneous and isotropic. This implies a proper time such that at any instant the 3-dimensional spatial line element of the Universe, dl 2 = γ i j dx 1 dx j (i, j = 1,2,3) (where γ i j is the spatial part of the metric tensor) is the same in all places and all directions, with ds 2 = dt 2 + a(t) 2 dl 2 = dt 2 + a(t) 2 γ i j dx i dx j. The scale parameter a(t) describes expansion or contraction of the spatial metric. Cosmological principle no reason for time to pass at different rates for different locations in an isotropic and homogeneous Universe. (If time depended on spatial coordinates, measurement of time could distinguish one place from another contradiction.) Thus the time term is simply dt and not a more complicated expression as in the Schwarzschild metric.

3 18.1. THE COSMOLOGICAL PRINCIPLE 489 Thus, we must investigate 3D curved spaces that are both homogeneous and isotropic. Let us first consider this question in two dimensions, where visualization is easier, and then generalize to three spatial dimensions.

4 490 CHAPTER 18. FRIEDMANN COSMOLOGIES Open Flat Closed Figure 18.1: Isotropic, homogeneous 2-dimensional spaces Homogeneous, Isotropic 2-Dimensional Spaces In two dimensions there are three independent possibilities for homogeneous, isotropic spaces: 1. Flat Euclidean space. 2. A sphere of constant (positive) curvature. 3. An hyperboloid of constant (negative) curvature. These three possibilities are illustrated in Fig Constant negative curvature surfaces can t be embedded in 3-D Euclidean space. The saddle-like open surface only approximates constant negative curvature near its center. In each case the corresponding space has neither a special point nor a special direction. Thus, these are 2- dimensional spaces with underlying metrics consistent with the cosmological principle.

5 18.2. HOMOGENEOUS, ISOTROPIC 2-DIMENSIONAL SPACES 491 Let us examine the 2-sphere as a representative example. From We are used to thinking of 2-spheres in terms of a 2-dimensional surface embedded in a 3-dimensional space. But a metric defines intrinsic properties of a space that should be independent of any additional embedding dimensions (recall Gaussian curvature). Therefore, it should be possible to express the metric of the 2-sphere in terms of only two coordinates. x 2 + y 2 + z 2 = S 2, we may deduce that (Exercise) dz 2 = (xdx+ydy)2 S 2 x 2 y 2 Thus, we may write the metric for the 2-sphere in the form dl 2 = dx 2 + dy 2 + dz 2 = dx 2 + dy 2 + (xdx+ydy)2 S 2 x 2 y 2 This metric describes distances on a 2D surface and depends on only two coordinates (S is a constant). Distances are specified entirely by coordinates intrinsic to the 2D surface, independent of 3rd embedding dimension.

6 492 CHAPTER 18. FRIEDMANN COSMOLOGIES 18.3 Homogeneous, Isotropic 3-Dimensional Spaces Let us now generalize the discussion of the preceding section. Our spacetime appears to have three rather than two spatial dimensions; thus we consider the embedding of 3D spaces in four Euclidean dimensions.

7 18.3. HOMOGENEOUS, ISOTROPIC 3-DIMENSIONAL SPACES Constant Positive Curvature For a 3-sphere the generalization is obvious: x 2 + y 2 + z 2 + w 2 = S 2, where w is now a fourth Euclidean dimension. We may expect that the metric of the 3-sphere is independent of the embedding and therefore expressible in terms of only three of these coordinates. By analogy with the 2-sphere, the metric may be written as dl 2 = dx 2 + dy 2 + dz 2 + (xdx+ydy+zdz)2 1 x 2 y 2 z 2, where we now specialize to a unit 3-sphere (S = 1) because the overall spatial scale will be set by the expansion factor a(t). Introduce spherical polar coordinates (r,θ,ϕ) through the relations x = r sinθ cosϕ y = r sinθ sinϕ z = r cosθ. The metric of the unit 3-sphere then takes the form where we have defined dl 2 = dr2 1 r 2 + r2 dθ 2 + r 2 sin 2 θdϕ 2 = dr2 1 r 2 + r2 dω 2 dω 2 dθ 2 + sin 2 θdϕ 2

8 494 CHAPTER 18. FRIEDMANN COSMOLOGIES The 3-sphere, by analogy with the 2-sphere, Corresponds to a homogeneous, isotropic space that is closed and bounded. It has great circles as geodesics. It is a space of constant positive curvature. An ant dropped onto the surface of an otherwise featureless 3-sphere would find that No point or direction appears any different from any other. The shortest distance between any two points corresponds to a segment of a great circle. A sufficiently long journey in a fixed direction would return one to the starting point. The total area of the sphere was finite.

9 18.3. HOMOGENEOUS, ISOTROPIC 3-DIMENSIONAL SPACES Constant Negative Curvature The generalization of the 2-hyperboloid to three dimensions is given by the equation x 2 + y 2 + z 2 + w 2 = S 2. By a similar argument as above, the metric in this case can be expressed as dl 2 = dr2 1+r 2 + r2 dω 2, By analogy with the 2D case: This describes a space that is homogeneous and isotropic, but now unbounded and infinite. It has constant negative curvature, with hyperbolas as geodesics. Our hypothetical ant would find No preferred direction or location. The shortest distances between any two points would now be segments of hyperbolas. The ant would never return to the starting point by continuing an infinite distance in a fixed direction. The ant would find that the volume of the space is infinite.

10 496 CHAPTER 18. FRIEDMANN COSMOLOGIES Zero Curvature Finally, for a Euclidean 3-space the metric may be expressed in the form This space is dl 2 = dr 2 + r 2 dω 2, Homogeneous and isotropic. Of infinite extent, with straight lines as geodesics. Obviously, this space corresponds to the limit of no spatial curvature. The volume is infinite, and a straight path in one direction will never return to the starting point.

11 18.4. THE ROBERTSON WALKER METRIC The Robertson Walker Metric We may combine the results of the preceding section and write the most general spatial metric in three dimensions that incorporates the isotropy and homogeneity constraints as dl 2 = dr2 1 kr 2 + r2 dω 2, where the parameter k determines the nature of the curvature: +1 hypersphere of positive curvature k = 0 flat Euclidean space 1 hyperboloid of negative curvature In fact, k can have other normalizations but one is free to rescale the equations so that it takes only these three values. Finally, combining dl 2 = dr2 1 kr 2 + r2 dω 2 ds 2 = dt 2 + a(t) 2 dl 2 we arrive at the most general metric for 4-dimensional spacetime that is consistent with the homogeneity and isotropy required by the cosmological principle, ( ds 2 = dt 2 + a(t) 2 dr 2 ) 1 kr 2 + r2 dθ 2 + r 2 sin 2 θdϕ 2, where k = 0,±1.

12 498 CHAPTER 18. FRIEDMANN COSMOLOGIES The metric ( dr ds 2 = dt 2 + a(t) 2 2 ) 1 kr 2 + r2 dθ 2 + r 2 sin 2 θdϕ 2, is commonly called the Robertson Walker (RW) metric, and is the starting point for any description of our Universe on scales sufficiently large that the cosmological principle applies. The time variable t appearing in the Robertson Walker metric is the time that would be measured by an observer who sees uniform expansion of the surrounding Universe; it is termed the cosmological proper time or the cosmic time.

13 18.4. THE ROBERTSON WALKER METRIC 499 The Robertson Walker metric may be expressed in an alternative form (Exercise) by introducing the 4-dimensional generalization of polar angles w = cos χ x = sin χ sinθ cosϕ y = sin χ sinθ sinϕ z = sin χ cosθ (with ranges 0 ϕ 2π, 0 θ π, and 0 χ π) into the equations for spherical geometry, and these variables with the substitutions w iw χ iχ S is, into the equations for hyperbolic geometry (and choosing S = 1 in both cases). Then the Robertson Walker metric may be written as dχ 2 + sin 2 χ(dθ 2 + sin 2 θdϕ 2 ) (closed) ds 2 = dt 2 + a(t) 2 dχ 2 + χ 2 (dθ 2 + sin 2 θdϕ 2 ) (flat) dχ 2 + sinh 2 χ(dθ 2 + sin 2 θdϕ 2 ) (open) which is related to ( dr ds 2 = dt 2 + a(t) 2 2 ) 1 kr 2 + r2 dθ 2 + r 2 sin 2 θdϕ 2, by the change of variables sin χ r = χ sinh χ (closed) (flat) (open)

14 500 CHAPTER 18. FRIEDMANN COSMOLOGIES Notice that The derivation of the Robertson Walker metric was purely geometrical, subject to the constraints of isotropy and homogeneity. No dynamical considerations enter explicitly into its formulation. Of course, dynamics are implicit, to the extent that the overall dynamical structure of the Universe must be consistent with the cosmological principle that was used to construct the metric.

15 18.4. THE ROBERTSON WALKER METRIC 501 The Robertson Walker metric may be used to express the line element for homogeneous, isotropic spacetime in matrix form: ds 2 = g µν dx µ dx ν = (dt dr dθ dϕ) a 2 1 kr a 2 r a 2 r 2 sin 2 θ dt dr dθ dϕ. Thus, the RW metric is diagonal, with g 00 = 1 g 11 = a2 1 kr 2 g 22 = a 2 r 2 g 33 = a 2 r 2 sin 2 θ as its non-zero covariant components, and the corresponding contravariant components are g 00 = 1 g 11 = 1 kr2 a 2 g 22 = 1 a 2 r 2 g 33 = since the metric tensor is diagonal in the RW metric and g µα g αν = δ µ ν. 1 a 2 r 2 sin 2 θ,

16 502 CHAPTER 18. FRIEDMANN COSMOLOGIES R Figure 18.2: Hubble expansion in two spatial dimensions Comoving Coordinates Homogeneous, isotropic expansion of the Universe can be exemplified in 2D by placing dots on a balloon and blowing it up (Fig. 18.2). The spherical coordinates (θ,ϕ) remain the same but the distance between points changes with the scale factor R(t). Example: 2 positions on the surface of the Earth, defined by latitude and longitude. Expand the size of the globe by a factor of 2. The actual distance between the two cities also increases by a factor of 2, but the coordinates of the two cities are unchanged. Termed comoving coordinates or a comoving frame. An observer attached to a comoving coordinate (comoving or fundamental observer) sees all other points receding from him and sees a homogeneous, isotropic universe. An observer not comoving does not see an isotropic universe. The receding points maintain their comoving coordinates.

17 18.5. COMOVING COORDINATES 503 Although the balloon analogy is useful, one must guard against misconceptions that it can generate, particularly in popular-level discussions. First, the surface that is expanding is 2-dimensional; the center of the balloon is in the third dimension and is not part of the surface, which has no center. Second, the Universe is not being expanded by a pressure. For that matter, neither is the balloon. The expansion of the balloon is generated by a difference in pressure. But in a homogeneous, isotropic universe there can be no pressure differences on global scales. Furthermore, because pressure couples to gravity in the Einstein equation, addition of (positive) pressure to the Universe would slow, not increase, the expansion rate Later GR Result: ä = 4πG a(ε + 3P). 3 Finally, if the dots on the balloon represent galaxies, they too will expand. But real galaxies don t partake of the general Hubble expansion because they are gravitationally bound. A better analogy is to glue solid objects to the surface of the balloon to represent galaxies, so that they don t expand when the balloon expands.

18 504 CHAPTER 18. FRIEDMANN COSMOLOGIES If we now generalize this idea to three spatial dimensions, The coordinates (r,θ,ϕ) of the Robertson Walker metric are comoving coordinates. In the RW metric, as the Universe expands the galaxies (to the degree that peculiar motion relative to the Hubble flow can be ignored) keep the same coordinates (r,θ,ϕ), and only the RW scale factor a(t) changes with time. Just as in the 2D analogy, the galaxies recede from us but, if we are comoving observers, the receding galaxies maintain their comoving coordinates and the recession is described entirely by the time dependence of the scale factor a(t). Peculiar velocities will change the comoving coordinates, but these are small on the large scales where the cosmological principle and therefore the RW metric is valid.

19 18.6. PROPER DISTANCES Proper Distances Let us consider the measurement of distances between galaxies in the RW metric. To simplify considerations, we take one galaxy to be at comoving coordinates (r,θ,ϕ) = (0,0,0) and the other to be at (r,0,0), at fixed time t. Thus, the portions of the line element depending on time and the angles θ and ϕ make no contribution and we obtain from ( dr ds 2 = dt 2 + a(t) 2 2 ) 1 kr 2 + r2 dθ 2 + r 2 sin 2 θdϕ 2 the proper distance ds 2 = a(t) 2 ( dr 2 l = a(t) r 0 1 kr 2 dr 1 kr 2. )

20 506 CHAPTER 18. FRIEDMANN COSMOLOGIES We may evaluate this for the three generic curvature parameters. 1. Flat Euclidean space (k = 0): the integral is then trivial and we obtain r l = a(t) dr r = l 0 1 kr 2 a Thus, r increases without limit as l increases at fixed a(t), implying that the space is unbounded. 2. Positive curvature (k = +1): solution for k = 1 gives r l = a 0 dr 1 r 2 = asin 1 r r = sin ( ) l. a In a universe of constant positive spatial curvature, r returns to the origin whenever l = πa and the space is bounded and closed. 3. Negative curvature (k = 1): Solution gives r l = a 0 dr 1+r 2 = asinh 1 r r = sinh ( ) l. a Therefore, r grows without bound in a Universe of constant negative curvature as l is increased at fixed a(t), implying an unbounded and open geometry.

21 18.6. PROPER DISTANCES 507 The proper distance is the rulers end to end distance that would be measured by a set of observers with rulers distributed between the two objects. Although this notion of distance is conceptually well-grounded, it is impractical to implement in an astrophysics context where instead essentially all distance information comes from data carried by signals propagating on null geodesics (light). Therefore, we shall later have to consider more extensively the meaning of distance and how to specify and measure it practically in observational astronomy.

22 508 CHAPTER 18. FRIEDMANN COSMOLOGIES 18.7 The Hubble Law and the RW Metric For a galaxy participating in the Hubble flow r is a comoving coordinate and so is constant in time. Therefore, for k = 0 r l = a(t) dr 0 1 kr 2 r l = ȧ(t) dr = l ȧ(t) 0 a(t), where l = a(t)r from above. This may be recognized as a generalized form of Hubble s law, with v l = Hl H = ȧ(t) a(t). Similar results follow from the equations for positive and negative curvature at small r. The RW metric, and the cosmological principle upon which it rests, imply the Hubble law. Conversely, the observation of a Hubble law is an indicator of homogeneous, isotropic spacetime.

23 18.8. PARTICLE AND EVENT HORIZONS Particle and Event Horizons An important consequence of the metric structure of spacetime and the finite speed of light is the possibility that regions of spacetime may be intrinsically unknowable for a fixed observer, either at the present time, or perhaps for all time. Such limitations are termed horizons. We shall distinguish two related concepts: A particle horizon An event horizon. The former is of particular importance in cosmology and one often finds the generic term horizon used to mean particle horizon. We shall, for the present discussion, keep the two terminologies distinct.

24 510 CHAPTER 18. FRIEDMANN COSMOLOGIES Particle Horizons in the RW Metric A particle horizon is the largest distance from which a light signal could have reached us at time t if it were emitted at time t = 0. Imagine a spherical light wave emitted by us at the time of the big bang. Over time it sweeps out over more and more galaxies. By symmetry, at the same instant that those galaxies can see us we can see them. So this spherical light front divides the galaxies into two groups: 1. Those inside our particle horizon, for which their light has had time to reach us since the big bang. 2. Those outside our particle horizon, for which their light has not yet had time to reach us since the bang.

25 18.8. PARTICLE AND EVENT HORIZONS 511 In the RW metric, light travels along a geodesic ds 2 = 0. Choosing the direction θ = ϕ = 0, from ds = 0 and ( dr ds 2 = dt 2 + a(t) 2 2 ) 1 kr 2 + r2 dθ 2 + r 2 sin 2 θdϕ 2 we have dt = adr 1 kr 2. Integrating both sides of this expression gives t 0 dt rp a(t ) = dr 1 kr 2, 0 where r p is the comoving distance to the particle horizon. The proper distance to the particle horizon is l h = a(t) rp 0 dr 1 kr 2. Combining the previous two expressions we obtain l h = a(t) t 0 dt a(t ). Whether a particle horizon exists depends on the behavior of the integral. Convergence at the lower limit is not ensured because the scale factor in the denominator generally tends to zero at the lower limit in RW cosmologies.

26 512 CHAPTER 18. FRIEDMANN COSMOLOGIES Example of Particle Horizon: Flat, Static Universe Flat, Static Universe As a simple example of a particle horizon, assume a flat, static Universe. Then the RW metric reduces to the Minkowski metric expressed in spherical coordinates with a equal to a constant and t l h = a(t) 0 dt a(t ) l h = a ct a = ct cτ H = r H where r H is the Hubble radius introduced earlier with heuristic arguments and is equal to the radius that light could reach in a time τ H = 1/H in a flat, static universe.

27 18.8. PARTICLE AND EVENT HORIZONS Conformal Time and Horizons It is sometimes useful to introduce a new time coordinate η called the conformal time through dt = a(t)dη. The flat Robertson Walker metric then may be expressed as ds 2 = dt 2 + a 2 (dr 2 + r 2 dθ 2 + r 2 sin 2 θdϕ 2 ) = a 2 dη 2 + a 2 (dr 2 + r 2 dθ 2 + r 2 sin 2 θdϕ 2 ) = a 2 ( dη 2 + dr 2 + r 2 dθ 2 + r 2 sin 2 θdϕ 2 ), This is the same form as the metric for a uniformly expanding Minkowski space. For radial light rays dθ = dϕ = ds = 0, so this becomes a(t) 2 (dη 2 + dr 2 ) = 0 dη = ±dr. Thus, in the η r plane light rays move at 45 degree angles at all times, which simplifies discussion of horizon and causality issues.

28 514 CHAPTER 18. FRIEDMANN COSMOLOGIES Conformal time is a special case of a conformal transformation, which is a transformation on the metric of the form g µν f g µν, where f is an arbitrary spacetime function. Generally, if g µν is a solution of the Einstein equation, f g µν is not except for the trivial case where f is a constant. However, null geodesics are conformally invariant, so conformal transformations have the useful feature that they preserve the light cone structure of the metric.

29 18.8. PARTICLE AND EVENT HORIZONS 515 η η 2 η 0 η 1 Big bang singularity Current horizon r Figure 18.3: Particle horizons in conformal time. Figure 18.3 illustrates the behavior of particle horizons in conformal time. The horizon at the present conformal time η 0, t0 dt r h = a(t) 0 is indicated. Clearly the horizon was smaller at the earlier time η 1, and will be larger at the future time η 2. As we saw, the k = 0 Robertson Walker universe discussed above is related to Minkowski spacetime by a conformal transformation. It is called flat for this reason, even though its spacetime is not flat (spacelike slices of its spacetime are flat). Generally, the RW metric is said to be conformally flat for all k, meaning that, for all k, coordinate transformations exist that permit the metric to be cast in the Minkowski form.

30 516 CHAPTER 18. FRIEDMANN COSMOLOGIES Event Horizons An event horizon is the most distant present event from which a world line could ever reach our world line. Proceeding in analogy with our discussion of particle horizons, the comoving distance r e to an event horizon can be defined through the integral tmax dt re t 0 a(t ) = dr. 1 kr 2 Then the proper distance to the event horizon is re dr tmax l e (t) = a(t) = a(t) dt 0 1 kr 2 t 0 a(t ), which differs from the expression for l h only in the limits of the integral. Similar to the case of particle horizons, whether an event horizon exists depends on the behavior of the integral on the right side of this equation. If this integral converges as t max (which depends on the detailed behavior of a(t) in this limit), an event horizon exists. 0

31 18.8. PARTICLE AND EVENT HORIZONS 517 Some metrics have event horizons and some don t. The expanding balloon analogy illustrates this qualitatively. Suppose inhabitants of galaxies on the surface of the balloon can exchange signals of constant local speed. Since the physical distance between galaxies is increasing with time, the exchanged signals must cover a greater distance in going from one galaxy to the next than in a static spacetime. If the space is expanding fast enough, the distance to distant galaxies may increase so rapidly that the signal will never reach those distant galaxies, even after an infinite amount of time (treadmill analogy). If the expansion is sufficiently rapid, a sphere centered on each galaxy divides other galaxies into two groups: 1. Those that have already been reached by a signal sent from the galaxy, or will be reached by the signal at some point in the finite future. 2. Those galaxies that will never be reached by the signal, even after infinite time has elapsed. This radius, if it exists, defines an event horizon for the observer, for by symmetry no signals from galaxies beyond this radius will ever reach the observer.

32 518 CHAPTER 18. FRIEDMANN COSMOLOGIES Such event horizons, if they exist, are similar to event horizons associated with black holes. One important difference is that cosmological event horizons are generally defined relative to an observer. Each observer in a universe containing event horizons has her own event horizon.. The event horizon associated with a Schwarzschild or Kerr black hole, on the other hand, is associated with a particular region of spacetime.

33 18.8. PARTICLE AND EVENT HORIZONS 519 Particle horizons and event horizons are defined by the same integrals, but with different limits. A particle horizon represents the largest distance from which light could have reached us today, if it had traveled since the beginning of time. An event horizon is the largest distance from which light emitted today could reach us at any future time. Thus particle and event horizons are distinct: Cosmological event horizons, as for black hole event horizons, separate regions of spacetime according to the causal properties of the spacetime. Particle horizons separate spacetime events according to whether objects in the spacetime can be seen by a particular observer at a particular time and place. Hence the meaning of particle horizons is similar to our normal meaning of horizons on the Earth. Horizons on the Earth also illustrate clearly the dependence on location of the observer.

34 520 CHAPTER 18. FRIEDMANN COSMOLOGIES t t max Future Timelike Spacelike Now y x Spacelike Timelike Past Event horizon Particle horizon t min Figure 18.4: Schematic representation of particle and event horizons in cosmology. The maximum possible time coordinate is t max. The relationship between particle and event horizons is illustrated in Fig From this we see that an event horizon, if it exists, may be interpreted as the ultimate particle horizon.

35 18.9. THE EINSTEIN EQUATIONS FOR THE RW METRIC The Einstein Equations for the RW Metric Let s now solve the Einstein equations that may be relevant for a description of our Universe. A simple set of such solutions Use the Robertson Walker metric. Assume the homogeneously and isotropically distributed matter of the Universe to be perfect fluid characterized by an energy density ε and pressure P. The corresponding cosmologies are commonly termed Friedmann Cosmologies.

36 522 CHAPTER 18. FRIEDMANN COSMOLOGIES The Stress Energy Tensor The most general form of the stress energy tensor is T µν = (ε + P)u µ u ν + Pg µν. However, for a comoving observer in the Robertson Walker metric the space time, time space, and nondiagonal space space terms are identically zero, T 0i = T i0 = T i j (i j) = 0 (RW comoving) and the stress energy tensor can be chosen diagonal. The most general form for a comoving Robertson Walker observer is T µµ = (ε + P)u µ u µ + Pg µµ, with explicit non-vanishing components T 00 = ε T 11 = Pa2 1 kr 2 T 22 = Pr 2 a 2 T 33 = Pr 2 a 2 sin 2 θ, since for comoving observers (u 0 ) 2 = 1 and (u i ) 2 = 0.

37 18.9. THE EINSTEIN EQUATIONS FOR THE RW METRIC The Connection Coefficients The required connection coefficients are given by inserting the metric tensor components g 00 = 1 g 11 = a2 1 kr 2 g 22 = a 2 r 2 g 33 = a 2 r 2 sin 2 θ g 00 = 1 into g 11 = 1 kr2 a 2 g 22 = 1 a 2 r 2 g 33 = ( Γλ σ µ = 1 gµν 2 gνσ x λ + g λν x µ g ) µλ x ν For example ( Γ 2 12 = 1 g20 2 g02 x 1 + g 10 x 2 g ) 21 x 0 ( g21 g12 x 1 + g 11 x 2 g 21 x 1 ( g22 g22 x 1 + g 12 x 2 g 21 x 2 ( g23 g32 x 1 + g 13 x 2 g 21 x 3 ) ( = 1 g22 2 g22 x 1 + g 12 x 2 + g 21 x 2 = 1 ( ) 1 ( 2 r 2 a 2 r 2 a 2) = 1 r r. ) ) ) 1 a 2 r 2 sin 2 θ (ν = 0 term) (ν = 1 term) (ν = 2 term) (ν = 3 term) ( ) = 1 g22 2 g22 x 1

38 524 CHAPTER 18. FRIEDMANN COSMOLOGIES Table 18.1: Non-vanishing Friedmann connection coefficients Γ 0 11 = aȧ/(1 kr2 ) Γ 0 22 = r2 aȧ Γ 0 33 = r2 sin 2 θaȧ Γ 1 01 = ȧ/a Γ1 11 = kr/(1 kr2 ) Γ 1 22 = r(1 kr2 ) Γ 1 33 = r(1 kr 2 )sin 2 θ Γ 2 02 = ȧ/a Γ 2 12 = 1/r Γ 2 33 = sinθ cosθ Γ3 03 = ȧ/a Γ3 13 = 1/r Γ3 23 = cotθ The coefficients are symmetric in the lower indices: Γ µ αβ = Γµ β α The nonvanishing connections coefficients are summarized in Table 18.1.

39 18.9. THE EINSTEIN EQUATIONS FOR THE RW METRIC The Ricci Tensor and Ricci Scalar The Ricci tensor may now be constructed from the connection coefficients R µν = Γ λ µν,λ Γλ µλ,ν + Γλ µν Γσ λσ Γσ µλ Γλ νσ. Utilizing the connection coefficients from Table 18.1, the non-vanishing components of the Ricci tensor are R 00 = 3ä a R 11 = (aä+2ȧ 2 + 2c 2 k) (1 kr 2 ) R 22 = r 2 (aä+2ȧ 2 + 2c 2 k) R 33 = R 22 sin 2 θ, and the Ricci scalar is obtained by contraction with the metric tensor, R = g µν R µν = 6(aä+ȧ2 + c 2 k) a 2.

40 526 CHAPTER 18. FRIEDMANN COSMOLOGIES The Friedmann Equations We now have the necessary ingredients to construct the Einstein equations R µν 1 2 g µνr = 8πGT µν. From the 00 and 11 components, utilizing the previous results for R µν, R, g µν and T µν, 3ȧ 2 + 3k = 8πGρa 2 2aä+ȧ 2 + k = 8πGPa 2. These are termed the Friedmann equations, and they represent the solution of the covariant gravitational equations with the conditions that we have imposed (the 22 and 33 components don t give any new results).

41 18.9. THE EINSTEIN EQUATIONS FOR THE RW METRIC Static Solutions and the Cosmological Term Let us first ask if the Friedmann equations have a static solution (corresponding to a scale factor constant in time). Setting ä = ȧ = 0, the Friedmann equations become 3k = 8πGρa 2 0 k = 8πGP 0 a 2 0, from which we find that k a 2 0 = 8πG 3 ρ = 8πGP 0. But from this we conclude that 1. For the present mass density ρ to be positive, we must have k = If ρ > 0, the pressure must be negative, P 0 < 0! Thus, we find that the Friedmann universe cannot be static: it is unstable against either expansion or contraction. When Einstein first realized this, Hubble had not yet discovered the expansion of the Universe and the natural assumption was that a correct cosmology should give a static solution. Thus, Einstein was led to make what he reportedly confided to Born was the greatest mistake of his career.

42 528 CHAPTER 18. FRIEDMANN COSMOLOGIES Einstein modified the field equations by subtracting Λg µν from the left side, where Λ is a scalar constant carrying dimensions of inverse length squared (in c = 1 units): R µν 1 2 g µνr Λg µν = 8πGT µν. Unlike the other terms on the left side, the Λ term does not vanish in the limit of vanishing mass and curvature. The cosmological term Λg µν is a rank-2 tensor since Λ is a scalar, and it has vanishing covariant divergence since D α g µν = g µν;α = 0, Thus it satisfies all the properties that we expect for a term in the Einstein equations. The corresponding Friedmann equations are ȧ 2 a 2 + k a 2 Λ 3 = 8π 3 Gρ 2ä a + ȧ2 a 2 + k a 2 Λ = 8πGP. A positive value of Λ becomes a repulsive force that counteracts gravity and a negative value becomes an attractive force that adds to the gravitational force. By proper adjustment of Λ, it is then possible to obtain a static Friedmann universe.

43 18.9. THE EINSTEIN EQUATIONS FOR THE RW METRIC 529 When Hubble discovered the expansion of the Universe, Einstein realized the opportunity that had been missed. Had he more confidence in his original field equations, he could have predicted that the Universe had to be either expanding or contracting. Once Hubble demonstrated that the Universe was expanding, Einstein discarded the cosmological term. But in modern cosmology there may still be a need for the cosmological term, although for reasons very different from Einstein s original motivation. Because the data require Λ to be very small if it exists, it can play a role only over volumes of space that are cosmological in dimension; this is why it is commonly termed the cosmological constant. Because it is equivalent to an energy density associated with the ground state of the Universe, Λ is also often termed the vacuum energy density.

44 530 CHAPTER 18. FRIEDMANN COSMOLOGIES Because of the interpretation of Λ as a vacuum energy density, it is convenient in modern applications to absorb the effect of the cosmological constant on the left side of R µν 1 2 g µνr Λg µν = 8πGT µν. into the terms arising from the stress energy tensor on the right side. Therefore, in the following development we shall omit the explicit Λ terms from the left sides of the Friedmann equations and include the possibility of a finite vacuum energy by a redefinition of the density and pressure variables appearing in T µν on the right side.

45 RESOLUTION OF DIFFICULTIES WITH NEWTONIAN VIEW Resolution of Difficulties with Newtonian View Finally, let us comment that the covariant theory of gravitation implies conceptual differences relative to Newtonian cosmology. 1. Expanding space in general relativity alleviates inconsistencies associated with apparent recessional velocities that would exceed the speed of light at large distances. Since the recessional velocities are generated by the expansion of space itself, not by motion within space, there is no conceptual difficulty with recessional velocities larger than light velocity. Recall from the discussion of parallel transport that defining relative velocities between two different spacetime points requires parallel transport of a vector on a curved surface, which depends on the path taken. Thus, relative velocity between objects located at two different spacetime points is ambiguous. 2. Because of the finite speed of (massless) gravitons implied by Lorentz invariance, gravitation is no longer felt instantaneously over large distances. It propagates at the speed of light. 3. In general relativity spacetime is generated by matter (and energy and pressure), so the idea of a boundary between a homogeneous universe and an empty space outside does not arise. 4. We shall see that the cosmological constant term in the Friedmann equations suggests a way to deal with dark energy.

46 532 CHAPTER 18. FRIEDMANN COSMOLOGIES Therefore, general relativity solves, at least in principle, several conceptual difficulties with the Newtonian approach to cosmology, in addition to providing a more solid quantitative basis.