Chapter 2

FRW (Friedmann-Robertson-Walker)

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 - 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 in SDSS appears to be homogeneous on scales larger than 70 Mpc/h. ' In this chapter, we study the kinematics and of the FRW universe.

2.1 FRW

The 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 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- -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 and 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 (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 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 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 of the local . 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 dϕ . (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 ) | | and with the spherical coordinate, we get  2  2 2 dr 2 2 2 2 dsK<0 = a + r dθ + sin θ dϕ . (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 dϕ2 , (2.16) 3D = i j + 1 K x 2 = 1 K r2 + + − | | − 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 dϕ2 . = + ( ) i j + 2 = + ( ) 2 + + − 1 K x − 1 K r − | | − (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 − | |  2 1  (1 K r )− 0 0 − 2 g˜i j =  0 r 0  (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 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 , the free particles move µ along the 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. (2.24) as p dp a˙ • dp a˙ ˜ p2 p p 0, (2.26) p0 dλ + a = d t + a = whose solution is p 1/a. Therefore, in an expanding universe, amplitude of the spatial momentum decreases proportional∝ to the scale factor. We call this phenomenon cosmological redshift. For the massless (relativistic) particles, the change in momentum is directly connected to the change of : 1 E = p . (2.27) ∝ a For a photon of ν emitted when the scale factor is a(t) and observed by an observer at present a(t0) a0, the obeserved frequency and can be calculated from ≡ Eobs νobs λ a(t) 1 = = = . (2.28) E ν λobs a0 ≡ 1 + z(t) the last equality defines cosmological redshift parameter z(t). For the massive particles, the -shell condition

µ 0 2 2 2 pµ p = (p ) + p = m , (2.29) − − relates the energy to the momentum as

 2 1/2 2 Æ p p E p0 m2 p2 m 1 m v4/c4 . (2.30) = = + = + 2 + + ( ) m ' 2m O The cosmological redshift for the massive particles can be easily understood when we consider what the means. Velocity of a particle, or its momentum, can be only defined reference to a frame of observers. In the case of the FRW coordinate system, the reference observers are the comoving observers 2.3. 5 standing still at their space coordinate. Let us consider a particle moving with speed v(t)xˆ at time t and passing through a space coordinate x. Of course, v(t)xˆ is the velocity of the particle measured by a comoving observer at x. After an infinitesimal time δt later, the particle pass through the comoving observer at x + v(t)δtxˆ who is moving away from the comoving observer at x by v = H(t)v(t)δtxˆ, and therefore measures the velocity of the particle as

v(t + δt) = v(t) H(t)v(t)δt, (2.31) − from which we set up the differential equation: dv Hv 0, (2.32) d t + = which reads the cosmological redshift as v 1/a. ∝

2.3 Friedmann equations

Let us consider the dynamics of the FRW universe that determines time evolution fo the scale factor a(t). The dynamics of cosmic expansion is determined by its energy contents, or the energy-momentum tensor. In a isotropic universe, any vector vµ and tensor tµν must satisfy (A) vi = 0, (B) t0i = 0 (C) ti j gi j in order to avoid defining a preferred direction. Also, in a homogeneous universe, the ∝ proportionality constant must only depends on time: ti j = f (t)gi j. Therefore, the energy-momentum tensor of the contents of the FRW universe must be the form of the perfect fluid

Tµν = (ρ + P)uµuν + P gµν (2.33) with ρ = ρ(t), P = P(t) depend only on time. Once we know the form of the and the energy-momentum tensor, the Einstein equation will supply the equation of for the metric. In your homework, you have done this by yourself to find out the following two Friedmann equations: K 3H2 3 8πGρ, (2.34) + a2 = a¨ 4πG = (ρ + 3P). (2.35) a − 3 The second Friedmann equation is sometimes also called . Here, 1 da a˙ H(t) = (2.36) ≡ a d t a is the Hubble parameter. The hubble constant is the current value of the Hubble parameter H0 H(t0). Note that here K = 1, 0, 1 and a(t) in the open and closed universe carries, as a curvature≡ radius, dimension of length. One− can, of course, replace K κ K/a2 as a spatial curvature, the Friedmann equation then becomes → ≡ 2 3H + 3κ = 8πGρ. (2.37) Note that in this case we have a freedom to redefine the scale factor reference to some epoch t so that ∗ a(t ) = 1. The usual choice is setting the scale factor unity at present: a(t0) = a0 = 1. ∗ µ You have also worked out in your homework that the energy-momentum conservation eqution µ Tν = 0 reads ∇ ρ˙ + 3H(ρ + P) = 0, (2.38) 6 CHAPTER 2. FRW (FRIEDMANN-ROBERTSON-WALKER) UNIVERSE and show that Eq. (2.38) does not give any more information to Eqs. (2.34)–(2.35). But, Eq. (2.38) is a useful form when calculating the time evolution of energy density. ————————————————————————————————————————————– Now that you know how to derive the Friedmann equations in the right way, I will tell you how people usually cheat on this. Consider a test particle at the distance r from the center (where?), and comoving 5 with the expansion velocity v = Hr. The Newton’s raw of says that this test particle gets a gravitational attraction only from the material inside of the radius r (yes, from the center!). The equation of motion reads 4 a¨ = πGρa. (2.39) −3 Comparing to Eq. (2.35), we miss 3P because plays no role in the Newtonian gravity! The effective Poisson equation in the presence of pressure is

2 φ = 4πG(ρ + 3P), (2.40) ∇ then we replace ρ to ρ + 3P to reproduce Eq. (2.35). The second law of hermodynamics says

dQ = dU + PdV. (2.41)

As there is no energy exchange from/to the Universe (what’s outside of the Unvierse?), the expansion is an adiabatic process (dQ = 0). With total internal energy U = ρV , the second law becomes

0 = dρV + (ρ + P)dV ρ˙ + 3H(ρ + P) = 0 (2.42) → because dV /V = 3da/a. Yes, we reproduce Eq. (2.38) Are you convinced? If not? what do you think is the flaw in the argument above? ————————————————————————————————————————————–

2.3.1 Density parameters

What are these equations telling us about the expansion of the Universe? The first Friedmann equation, Eq. (2.34), essentially says that the Hubble parameter H(t) is entirely determined by the energy density and the curvature of the Universe. In literature, we define the critical density as

3H2 ρ (2.43) crit = 8πG and rewrite the energy density with respect to the critical density as ρ 8πG Ω ρ, (2.44) = 2 ≡ ρcrit 3H and define the curvature parameter as κ Ω , (2.45) k = 2 − H so that Ω + Ωk = 1. (2.46) When the Universe consists of multiple constituents (with index X ), so that total energy density is given P by ρ = X ρX , we also define the density parameter for each component as ΩX ρX /ρcrit. For the purpose of Friedmann equation, we categorize the contents of the Universe by three≡ kinds:

5This holds in GR as well with Birkoff’s theorem. 2.3. FRIEDMANN EQUATIONS 7

Matter, or non-relativistic particles (ΩM) • All particle moves slower than the speed of falls in this category: dark , , 2 massive neutrino with mν ¦ Tν 0.168(1 + z) meV. Because P (v/c) 1, we neglect the pressure for these particles: ' ∝  P = 0. (2.47)

Radiation, or relativistic particles (ΩR) • The relativistic particles such as photon, massless neutrinos, and massive neutrinos with mν ® Tν. For relativistic particles, the equation of state is ρ P . (2.48) = 3

Cosmological constant, or vacuum energy (ΩΛ) • The energy density of this component stays constant throughout the cosmic history:

P = ρ. (2.49) − Topological defects • Symmetry breaking at early epoch can generate topological defects such as domain wall, cosmic string and monopole (depending on the internal symmetry that is broken), whose equation of state is wdw = 2/3, ws = 1/3, and wm = 0—this is to keep to energy density inside of the defect to be constant—.− But− there is no significant observational evidence of these defects, and we will not discuss them further in the class.

Using the density parameters, Friedmann equation now becomes

ΩM + ΩR + ΩΛ + Ωk = 1. (2.50)

For a general component X , we parameterize the pressure by its equation-of-state parameter wX :

PX = wX ρX . (2.51)

Once the equation-of-state is given, the continuity equation, Eq. (2.38), tells how the energy density evolves in an expanding universe (provided that the expansion conserves the particle number density). We transform Eq. (2.38) to ρ˙X a˙ + 3 (1 + wX ) = 0, (2.52) ρX a and find the solution, when w is constant in time6, as

3(1+wX ) ρX a− . (2.56) ∝ 6When w is not constant in time, one can rewrite the equation above as

d ln ρX + 3(1 + wX )d ln a = 0, (2.53) which can be integrated as – a ™ Z da  a ‹ 3[1+weff(a)] ρ a ρ exp 3 1 w a ρ − . (2.54) X ( ) = 0 a ( + X ( )) 0 a − a0 ≡ 0 with the effective equation of state of given by a 1 Z w a d ln a w a . (2.55) eff( ) = ln a/a 0 ( 0) ( 0) a0 8 CHAPTER 2. FRW (FRIEDMANN-ROBERTSON-WALKER) UNIVERSE

That is, the energy density of matter, radiation and cosmological constants evolve with scale factor as

3 4 ρM a− , ρR a− , ρΛ constant. (2.57) ∝ ∝ ≡ What does that mean? For the non-relativistic particles (), the rest mass energy dominates over the kinetic energy, and the total energy density scales as the number density: ρX = mX nX . As total number of particles in the comoving volume stays constant, the physical number density decreases as 3 n a− ; thus, energy density drops with the same amount. For the relativistic particles, on the other hand,∝ the energy and momentum of each particles decrease as 1/a(t) (this is due to cosmic redshift. See the next section.), on top of the dilution of the number density due to expansion. That accounts for an additional factor of a. stays constant, as it is defined to be. Now, let us denote the quantities at present with the subscript 0. Then,

 ‹ 3  ‹ 4 a − 3 a − 4 ρM(a) = ρM0 = ρM0 (1 + z) , ρR(a) = ρR0 = ρM0 (1 + z) , ρΛ(a) = ρΛ0. (2.58) a0 a0

Here, we use the redshift 1 + z = a0/a defined in Eq. (2.28). Then, the Friedmann equation becomes

2  3 4  2 3H = 8πG ρM0 (1 + z) + ρR0 (1 + z) + ρΛ0 3κ0 (1 + z) , (2.59) − 2 2 with κ0 K/a0 is the spatial curvature at present. Dividing the both side with 3H0 and using ≡ 2 29 2 3 ρcrit0 = 3H0 /(8πG) = 1.878 10− h g/cm , (2.60) × we find probably the single most frequently used equation in this cosmology lecture note:

2 2  3 4 2 H (z) =H0 ΩM0 (1 + z) + ΩR0 (1 + z) + ΩΛ0 + Ωk0 (1 + z) 2  3 4 2 =H0 ΩM0 (1 + z) + ΩR0 (1 + z) + ΩΛ0 + (1 ΩM0 ΩR0 ΩΛ0)(1 + z) . (2.61) − − − This is the first order differential equation for a(t), or rather z(t) but they are really the same, which can be easily solved once the present values of Hubble parameter H0 and density parameters are all known. The current best measurement of the present value of the density parameters are (maximum likelihood values from “+WP+highL+BAO” column of the Planck 2013 paper)

ΩM = 0.3086, ΩΛ = 0.6914, h = 0.6777, (2.62)

5 2 and ΩR = 4.175 10− h− based on the CMB Tγ = 2.726 K and neutrino temperature × Tν = 1.946 K. Although Eq. (2.34) and Eq. (2.38) are enough to solve for the dynamics of expansion a(t), Eq. (2.35) provides a useful information about the acceleration of the Universe. From Eq. (2.35), one can see that the expansion of the Universe accelrates (a¨ > 0) only if ρ + 3P < 0. This is so called strong energy condition, and cannot be satisfied using matter and radiation. This is exactly the reason why we need a new energy component called dark energy to accelate the expansion.

Another words of convention. In this section, I used ΩX 0 for the density parameter at present and ΩX for an arbitrary time to emphasize that the Friedmann equation is satisfied at all . But, for most of applications, we only need the density parameter at present time—anyway, that’s what we measure from Planck, for example. Therefore, I will omit the subscript 0 hereafter. Instead, when I refer the density parameter at different epoch, I will refer it as ΩX (z) to distinguish from the present value ΩX . 2.3. FRIEDMANN EQUATIONS 9

102 Ω (a) Ω (a) Ω (a) 1.0 R M Λ 101 Λ-dominated era

100 0.8

) Matter-dominated era a ( 0 X a 10 1

/ − Ω ) t (

0.6 a 2 NOW 10− a ∝ t2/3

3 1/2 0.4 10− a ∝ t scale factor density parameter 4 Radiation-dominated era 10− CMB last 0.2 5 10−

6 0.0 6 5 4 3 2 1 0 1 2 10− 6 5 4 3 2 1 0 1 2 3 4 10− 10− 10− 10− 10− 10− 10 10 10 10− 10− 10− 10− 10− 10− 10 10 10 10 10 scale factor a/a0 cosmic time t (Myrs)

Figure 2.1: Time evolution of the density parameter as a function of scale factor (left column), and the scale factor as a function of cosmic time (right column, in the units of Mega-years).

2.3.2 Evolution of scale factor

Let us solve our master equation for the expansion of the Universe, Eq. (2.61) with the best fitting density parameters listed above. It is easier to solve it with the scale factor a as a time variable instead of the redshift z, as a runs from 0 to 1 while z runs from to 0. We rewrite Eq. (2.61) in terms of ∞ x = a/a0:  ‹2 1 d x 2  3 4 2 = H0 ΩM x− + ΩR x− + ΩΛ + (1 ΩM ΩR ΩΛ) x− . (2.63) x d t − − −

Remember the new convention that ΩX refers to the density parameter at present. Rearranging it a bit, we find the integral form of the solution as

t a/a Z 1 Z 0 d x d t = . (2.64) H p 2 2 0 0 0 ΩM /x + ΩR/x + ΩΛ x + (1 ΩM ΩR ΩΛ) − − − The result of the numerical integration is shown in the right panel of Fig. 2.1. This figure shows the evolution of the scale factor as a function of cosmic time t in the unit of Mega years. Let us digest this figure more in depth. Although the Unvierse is dominated by the dark energy (cosmo- logical constant) now, the dominant energy component has been changing from the past. It is because each energy component evolves differently in time, or scale factor, as we have discussed in Eq. (2.58). For example, the energy density of the matter component is smaller than the cosmological constant by 3 a factor of 2.24 now, but back in the early time the matter density grows as a− while the cosmological 3 constant density stays the same so that the matter density gets dominated for a < p2.24. Similar thing happens for matter and radiation, although it takes more time for radiation to catchup with matter, as 4 the radiation density grows as a− , it will eventually dominate the energy budget of the Universe in the very early Universe. The left hand side of Fig. 2.1 shows the density parameter ΩX (a) at earlier times. From the early times, the Universe has been dominated by, in order, radiation, matter, and cosmological constant. Do mind how big the radiation fraction is at, say, a = 0.01 (z = 99). It is easy to overlook the radiation component at that redshift, but the radiation fraction is not that small! The reason is that the relative importance of radiatio compared to matter decreases only with a, while matter to cosmological constant decreases by a3. 10 CHAPTER 2. FRW (FRIEDMANN-ROBERTSON-WALKER) UNIVERSE

When the Universe is dominated by a single component, we can ignore the contribution from the other components, and estimate the evolution of the scale factor analytically as following:

1) Radiation-dominated era • The integration Eq. (2.64) becomes

a/a 1 Z 0 xd x 1  a ‹2 t p = p , (2.65) ' H0 0 ΩR 2H0 ΩR a0 which gives v a Ç 1/2 t t 2H0ΩR t 0.001150 . (2.66) a0 ' ' Myr 2) Matter-dominated era • Now we turn off all but matter density parameter to have

a/a 1 Z 0 x1/2d x 2  a ‹3/2 t t , (2.67) H p 3H p a − ∗ ' 0 a /a0 ΩM ' 0 ΩM 0 ∗ which gives  ‹2/3  ‹2/3 a 3 p t H0 ΩM t 0.001494 . (2.68) a0 ' 2 ' Myr Here, I approximate that the scale factor (a ) and cosmic tiem (t ) at the end of the radiation epoch are small compared to the corresponding∗ values in deeply matter-dominated∗ era.

3) Cosmological-constant-dominated era • For this case, a/a 1 Z 0 d x 1  a ‹ t t? = = ln , (2.69) H pΩ x H pΩ a − 0 a?/a0 Λ 0 Λ ? or  p  a = a? exp H0 ΩΛ(t t?) . (2.70) − That is, once the Universe is dominated by the cosmological constant, it will exponentiate the expansion. This is the reason why we have a sharp up turn at the end of the right hand side of Fig. 2.1.

2.4 The cosmological constant problem

In [1], Einstein has applied his general to the Universe only to find out that the theory only predicts a non-. He then introduced a minimal modification to the theory by introducing a constant λ (called cosmological constant):

Gµν λgµν = 8πGTµν. (2.71) − Note that gµν also satisfies gµν;ν = 0 like Gµν;ν = 0 (Bianchi identity) and Tµν;ν = 0 (energy-momentum conservation). Then, as you have done in the homework, Einstein can manage to find a static solution (although, the solution is unstable, as you have proven in the homework). 2.4. THE COSMOLOGICAL CONSTANT PROBLEM 11

Then, as we all know, Hubble and Lemaître7 came along and discovered that the Universe is in fact expanding, and Einstein probably did not say that it is his “greatest blunder” (see, Mario Livio’s book ‘Brilliant Blunders’ for this). It seemed that, arguably even back then, cosmological constant has no use. 8 In 1998, Search Team [3] and Supernova Cosmology Project [4] had discovered that the expansion of the Universe is in fact accelerating now. If attributed the entirety of current acceleration to the cosmological constant, then its energy density occupies 70% of the total energy budget, corresponds to 47 2 4 123 ρΛ = ΩΛρcrit 5.67 10− h GeV 1.17 10− , (2.72) ' × ' × in the natural unit and the Planck unit, respectively. What could be the cosmological constant? In the , there is one candidate which stays the same everywhere with the same value for a long time: the vacuum energy. Vacuum energy is the energy associated with the quantum ground state, where particle and anti-particle pairs are continuously created/annihiliated. A direct analogy of this energy is the zero-point energy 1/2ħhω of quantum harmonic oscillator, but the harmonic oscillator is everywhere in the space (called field). We can sum over the zero-point energy of the quantum fields in the momentum space as

Z Λ 3 Z Λ 3 4 d k 1 d k 1p Λ ρ ω k2 m2 , (2.73) V = 3 = 3 + 2 〈 〉 0 (2π) 2 0 (2π) 2 ' 16π where we introduce a cutoff Λ below which it is uncertain if we can apply the physics that we know now. The most natural choice for Λ is, of course, the Planck scale. So, the vacuum energy is in planck unit, 1 ρ 0.00633 5.41 10120ρ ! (2.74) V 2 = Λ 〈 〉' 16π ' × This is 10120 times greater than the value that we need to accelerate the expansion of the Universe! Of course, there are many types of fields in , and some fields (bosonic) contribute positively to the vacuum energy and some fields (fermionic) contribute negatively, with each contribution of order ρV . But, it is highly unlikely that summing up the whole contribution leads to such a tiny little number!〈 This〉 is called cosmological constant problem. For a detailed review, see, e.g. [5].

7 [2] contains interesting story about the English translation of the Lemaître’s article. His measurement of the Hubble parameter only appeared in the French version. 8For Adam Riess’s personal recollection, see http://www.stsci.edu/ ariess/documents/Shaw%20Prize%20Lecture_web.pdf The figure in page 7 is particu- larly interesting! 12 CHAPTER 2. FRW (FRIEDMANN-ROBERTSON-WALKER) UNIVERSE Bibliography

[1] A. Einstein, Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften (Berlin), Seite 142-152. 142 (1917).

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