Baumann's Cosmology (2)

2 The Expanding Universe

The coarse-grained properties of the universe are remarkably simple. In particular, when averaged over large distances (larger than 100 Mpc) - the size of the observable universe is about 14,000 Mpc - the universe looks isotropic (the same in all directions). Assuming that we don't live at a special point in space - and that nobody else does either - the observed isotropy then implies that the universe is also homogeneous (the same at every point in space). This leads to a simple mathematical description of the universe because the spacetime geometry takes a very simple form.
   Observations of the light from distant galaxies have shown that the universe is expanding. Running this expansion backwards in time, we predict that nearly 14 billion years ago our whole universe was in a hot dense state. The Big Bang theory describes what happened in this fireball., and how it evolved into the universe we see today.

          2.1 Geometry 
Gravity is a manifestation of geometry of spacetime. To describe the evolution of our universe, we therefore stars by determining its spacetime geometry. This geometry is characterized by a spacetime metric. The large degree of symmetry of the homogeneous universe means that its metric will take a rather simple form

          2.1.1 Spacetime and Relativity
The metric is an object that turns coordinate distances into physical distances. In 3-dimensional Euclidean space, the physical distance between two points separated by the infinitesimal coordinate distances 𝑑𝑥, 𝑑𝑦 and 𝑑𝑧
    (2.1)   𝑑𝓁² = 𝑑𝑥² + 𝑑𝑦² + 𝑑𝑧² = ∑³_𝑖,𝑗=1 𝛿_𝑖𝑗𝑑𝑥^𝑖𝑑𝑥^𝑗,     where (𝑥¹, 𝑥², 𝑥³) = (𝑥, 𝑦, 𝑧).  
Here the metric is simply the Kronecker delta 𝛿_𝑖𝑗 = diag(1, 1, 1). If we use polar coordinate,
    (2.2)   𝑑𝓁² = 𝑑𝑟² + 𝑟²𝑑𝜃² + 𝑟²sin²(𝜃)𝑑𝜙² ≡ ∑³_𝑖,𝑗=1 𝑔_𝑖𝑗 𝑑𝑥^𝑖𝑑𝑥^𝑗     where (𝑥¹, 𝑥², 𝑥³) = (𝑟, 𝜃, 𝜙).
Here the metric is, a less trivial form, 𝑔_𝑖𝑗 = diag(1, 𝑟², 𝑟²sin²𝜃). Observers can use different coordinate systems will always agree on the physical distance, 𝑑𝑙 which is an invariant. Hence, the metric turns observer-dependent coordinates into invariants.    A fundamental object in relativity is the spacetime metric. It turns observer-dependent spacetime coordinates 𝑥^𝜇 = (𝑐𝑡, 𝑥^𝑖)  into the invariant line element¹
    (2.3)   𝑑𝑠² = ∑³_𝜇,𝜈=0 𝑔_𝜇𝜈𝑑𝑥^𝜇𝑑𝑥^𝜈 ≡ 𝑔_𝜇𝜈𝑑𝑥^𝜇𝑑𝑥^𝜈.    [𝑥⁰ ≡ 𝑐𝑡 and 𝑥^𝑖, 𝑖 = 1, 2, 3]
In special relativity, the spacetime is Minkowski space, 𝙍^1,3, whose line element is
    (2.4)   𝑑𝑠² = -𝑐²𝑑𝑡 ² + 𝛿_𝑖𝑗 𝑑𝑥^𝑖𝑑𝑥^𝑗,
so that the metric is simply 𝑔_𝜇𝜈 = diag(-1, 1, 1, 1). An important feature of this metric is that the associated spacetime curvature vanishes. In general relativity, on the other hand, the metric depend on the position in spacetime, 𝑔_𝜇𝜈(𝑡, 𝐱), in such a way that the spacetime curvature is nontrivial. This spacetime dependence of the metric incorporates the effects of gravity. How the metric varies throughout spacetime is determined by the distribution of matter and energy in the universe.
Indices  From special relativity the metric is used to raise and lower indices on 4-vectors (and on general tensors),
    (2.5)   𝐴_𝜇 = 𝑔_𝜇𝜈𝐴^𝜈   and     𝐴^𝜇 = 𝑔^𝜇𝜈𝐴_𝜈,
where 𝑔^𝜇𝜈 is the inverse metric defined by [𝑔^𝜇𝜈 [Wikipedia Invertible matrix: 𝐀𝐁 = 𝐁𝐀 = 𝐈_𝑛 where 𝐈_𝑛 denotes 𝑛-by-𝑛 identity matrix.]
    (2.6)   𝑔^𝜇𝜆𝑔_𝜆𝜈 = 𝛿^𝜇_𝜈 = {1   𝜇 = 𝜈,     0   𝜇 ≠ 𝜈.
4-vectors with a lower index, 𝐴_𝜇, are sometimes called covariant to distinguish them from the contravariant 4-vectors, 𝐴^𝜇, with an upper index. Covariant and contravariant vectors can be contracted to produce an invariant scalar,
    (2.7)   𝑆 ≡ 𝐴 ⋅ 𝐵 = 𝐴^𝜇𝐵_𝜇 = 𝑔_𝜇𝜈𝐴^𝜇𝐵^𝜇.
   The spatial homogeneity and isotropy of the universe mean that it can be represented by a time-ordered sequence of 3-dimensional spatial slices, each of which is homogeneous and isotropic (see Fig. 2.1). The 4-dimensional element can then be written as²
    (2.8)   𝑑𝑠² = -𝑐²𝑑𝑡² + 𝑎²(𝑡)𝑑𝓁²,
where 𝑑𝓁² ≡ 𝛾_𝑖𝑗(𝑥^𝑘)𝑑𝑥^𝑖𝑑𝑥^𝑗 is the line element on the spatial slices and 𝑎(𝑡) is the scale factor, which describes the expansion of the universe. [Wikipedia Scale factor: In expanding or contracting universe, 𝑑(𝑡) = 𝑎(𝑡)𝑑₀, where 𝑑(𝑡) is the proper distance at time 𝑡, 𝑑₀ is the distance at the reference time 𝑡₀,𝑑(𝑡₀), usually also referred to as comoving distance, and 𝑎(𝑡) is the scale factor. Thus, by definition, 𝑎(𝑡₀) = 1]

          2.1.2 Symmetric Three-spaces 
Homogeneous and isotropic 3-spaces must have constant intrinsic curvature.Then the curvature can be zero, positive or negative.

• Flat space: The simplest possibility is 3-dimensional Euclidean space 𝐸³. This is the space in which parallel lines don't intersect. Its line element is
    (2.9)   𝑑𝓁² = 𝑑𝐱² = 𝛿_𝑖𝑗𝑑𝑥^𝑖𝑑𝑥^𝑗,
which is clearly invariant under spatial translations 𝑥^𝑖 ↦ 𝑥^𝑖 + 𝑎^𝑖 and rotations 𝑥^𝑖 ↦ 𝑅^𝑖_𝑘𝑥^𝑘, with 𝛿_𝑖𝑗𝑅^𝑖_𝑘𝑅^𝑗_𝑙 = 𝛿_𝑘𝑙.

• Spherical space: The next possibility is a 3-space with constant positive curvature. On such a space, parallel lines will eventually meet. This geometry can be represented as a 3-sphere 𝑆³ embedded in 4-dimensional Euclidean space 𝐸⁴:
    (2.10)   𝑑𝓁² = 𝑑𝐱² + 𝑑𝑢²,     𝐱² + 𝑢² = 𝑅₀², where 𝑅₀ is the radius of the sphere. Homogeneity and isotropy on the surface of the 3-sphere are inherited from the symmetry of the line element under 4-dimensional rotations.  

• Hyperbolic space: We can have a 3-space with constant negative curvature. On such a space parallel lines diverge. This geometry can be represented as a hyperboloid 𝐻³ embedded in 4-dimensional Lorentzian space 𝙍^1,3:
    (2.11)   𝑑𝓁² = 𝑑𝐱² - 𝑑𝑢²,     𝐱² - 𝑢² = -𝑅₀²,
where 𝑅₀² > 0 is a constant determining the curvature of the hyperboloid. Homogeneity and isotropy of the induced geometry on the hyperboloid are inherited from the symmetry of the line element under 4-dimensional pseudo-rotations - i.e. Lorentz transformations, with 𝑢 playing the role of time.

The line elements of the spherical and hyperbolic cases can be combined as
    (2.12)   𝑑𝓁² = 𝑑𝐱² ± 𝑑𝑢²,     𝐱² ± 𝑢² = ±𝑅₀².
The differential of the embedding condition gives 𝑢𝑑𝑢 = ∓𝐱 ⋅ 𝑑𝐱, so that we can eliminate the dependence on the auxiliary coordinate 𝑢 from the line element [𝑑𝑢²/𝑑𝑢 = 𝑑(𝑅₀² -  𝐱²)/𝑑𝐱 𝑑𝐱/𝑑𝑢, 𝑢𝑑𝑢 = -𝐱 𝑑𝐱, 𝑑𝑢² = -(𝐱 𝑑𝐱)²/𝑢²]
    (2.13)   𝑑𝓁² = 𝑑𝐱² ± (𝐱 ⋅ 𝑑𝐱)²/(𝑅₀² ∓ 𝐱²).    
Finally, we can unify (2.13) with (2.9) by writing
    (2.14)   𝑑𝓁² = 𝑑𝐱² + 𝑘 (𝐱 ⋅ 𝑑𝐱)²/(𝑅₀² - 𝑘 𝐱²),    for   𝑘 ≡ {0   𝐸³,     +1   𝑆³,     -1   𝐻³.
To make the symmetries of the space more manifest, it is convenient to write the metric in spherical polar coordinates, (𝑟, 𝜃, 𝜙). Using
    (2.15)   𝑑𝐱² = 𝑑𝑟²+ 𝑟²(𝑑𝜃² + sin²(𝜃)𝑑𝜙),     𝐱 ⋅ 𝑑𝐱 = 𝑟𝑑𝑟,      𝐱² = 𝑟²,
the metric in (2.14) becomes
    (2.16)   𝑑𝓁² = 𝑑𝑟²/(1 - 𝑘𝑟²/𝑅₀²) + 𝑟²𝑑𝛺²,
where 𝑑𝛺² ≡ 𝑑𝜃² + sin²(𝜃)𝑑𝜙 is the metric on the unit 2-spheres.  

         2.1.3 Robertson-Walker Metric
Substituting (2.16) into (2.8), we obtain the Robertson-Walker metric in polar coordinates:
    (2.17)   𝑑𝑠² = -𝑐²𝑑𝑡² + 𝑎²(𝑡)[𝑑𝑟²/(1 - 𝑘𝑟²/𝑅₀²) + 𝑟²𝑑𝛺²],
which is sometimes also called the Friedmann-Robertson-Walker (FRW) metric. [Wikipedia Friedmann-Lemaȋtre-Robertson-Walker metric: in short FLRW or FRW etc.] Notice that the symmetries of the universe have reduced the ten independent components of the spacetime metric 𝑔_𝜇𝜈 to a single function of time, the scale factor 𝑎(𝑡), and a constant, the curvature scale 𝑅₀.

• First of all, the line element (2.17) has a rescaling symmetry
    (2.18)   𝑎 → 𝜆𝑎,    𝑟 → 𝑟/𝜆,     𝑅₀ → 𝑅₀/𝜆
this means that the geometry of the spacetime stays same if we simultaneously rescale  𝑎, 𝑟 and 𝑅₀ by a constant 𝜆 as in (2.18). We can use this freedom to set the scale factor today, at  𝑡 = 𝑡₀, to be unity, 𝑎(𝑡₀) ≡ 1. The scale 𝑅₀ is then the physical curvature scale today, justifying the use of the subscript.

• The coordinate 𝑟 is called a comoving coordinate, which can be changed using the rescaling in (2.18) and hence is not a physical observable. Physical results can only depend on the physical coordinate,  𝑟_phys = 𝑎(𝑡)𝑟 (see Fig. 2.2)
   Consider a galaxy with a trajectory 𝐫(𝑡) in comoving coordinates and 𝐫_phys = 𝑎(𝑡)𝐫 in physical coordinates. The physical velocity of the galaxy
is     (2.19)   𝐯_phys ≡ 𝑑𝐫_phys/𝑑𝑡 = 𝑑𝑎/𝑑𝑡 𝐫 + 𝑎(𝑡) 𝑑𝐫/𝑑𝑡 = 𝐻𝐫_phys + 𝐯_pec,
where we have introduced the Hubble parameter
    (2.20)   𝐻 ≡ 𝑎̇/𝑎.
Here ȧ = 𝑑𝑎/𝑑𝑡. We see that in (2.19) the first term 𝐻𝐫_phys is the Hubble flow, which is the velocity of the galaxy resulting from the expansion of space between the origin and 𝐫_phys(𝑡). the second term 𝐯_pec ≡ 𝑎(𝑡) ṙ, is the peculiar velocity, which is the velocity measured by a "comoving observer" (i.e. an observer who follows the Hubble flow). It describes the motion of galaxy relative to the cosmological rest frame, typically due to the gravitational attraction of other nearby galaxies.

• The complicate 𝑔_𝑟𝑟 component of (2.17) may sometimes be redefined, 𝑑𝜒 ≡ 𝑑𝑟/√(1 - 𝑘𝑟²/𝑅₀²), then
    (2.21)   𝑑𝑠² = -𝑐²𝑑𝑡² + 𝑎²(𝑡)[𝑑𝜒² + 𝑆_𝑘²(𝜒)𝑑𝛺²],
where
    (2.22)   𝑆_𝑘(𝜒) = 𝑅₀ {sinh(𝜒/𝑅₀)  𝑘 = -1,     𝜒/𝑅₀   𝑘 = 0,     sin(𝜒/𝑅₀)   𝑘 = +1.
Note that for for 𝑘 = 0 there is no distinction between 𝑟 and 𝜒. The form of metric in (2.21) will be useful when we define observable distances in section 2.2.3.

• Finally, it is often helpful to introduce conformal time, 𝜂
    (2.23)   𝑑𝜂 = 𝑑𝑡/
so that (2.21) becomes
    (2.24)   𝑑𝑠² = 𝑎²(𝜂)[-𝑐²𝑑𝜂² + [𝑑𝜒² + 𝑆_𝑘²(𝜒)𝑑𝛺²].  
We see that the metric has now factorized into a static part and a time-dependent conformal factor 𝑎(𝜂). This is convenient for studying the propagation of light, for which 𝑑𝑠² = 0. As we will see in section 2.3.6, going to conformal time also will be useful change of variables for certain exact solutions to the Einstein equations.

          2.2 Kinematics   
  Having found the metric of the expanding universe, we now want to determine how particles evolve in the spacetime. It is an essential feature of general relativity that freely falling particles in a curved spacetime moving along geodesics. Some basic facts about geodesic motion in GR will be introduced and then it be applied to FRW spacetime. [RE Appendix A for more detail]

          2.2.1 Geodesics Let us first look at simpler problem of a free particle in 2-dimensional Euclidean space.

          Curvilinear coordinates
In the absence of any forces, Newton's law simply reads 𝑑²𝐱/𝑑𝑡² = 0 [ẍ = 0], which in Cartesian coordinates 𝑥^𝑖 = (𝑥, 𝑦) becomes
    (2.25)   𝑑²𝑥^𝑖/𝑑𝑡² = 0.
In a general coordinate system, however, ẍ = 0 does not have to imply 𝑑²𝑥^𝑖/𝑑𝑡² = 0. For example in the polar coordinate (𝑟, 𝜙), using 𝑥 = 𝑟 cos(𝜙) and 𝑦 = 𝑟 sin(𝜙), the equation of motion in (2.25) imply
    (2.26)   0 = ẍ = 𝑑𝑟̇/𝑑𝑡 cos(𝜙) - 2 sin(𝜙)𝑟̇𝜙̇ - 𝑟 cos(𝜙)𝜙̇² - 𝑟 sin(𝜙) 𝑑𝜙̇/𝑑𝑡,  
                     0 = ÿ = 𝑑𝑟̇/𝑑𝑡 sin(𝜙) + 2 cos (ϕ)𝑟̇𝜙̇ - 𝑟 sin(ϕ)𝜙̇² + 𝑟 cos(ϕ) 𝑑𝜙̇/𝑑𝑡.
Solving this for 𝑑ṙ/𝑑𝑡 and 𝑑𝜙̇/𝑑𝑡, we find
    (2.27)   𝑑𝑟̇/𝑑𝑡 = 𝑟𝜙̇²,     𝑑𝜙̇/𝑑𝑡 = - 2/𝑟 ṙ𝜙̇.
we see that 𝑑𝑟̇/𝑑𝑡 ≠ 0 and 𝑑𝜙̇/𝑑𝑡 ≠ 0. The reason is simply that in polar coordinates the basis vector 𝐫̂ and 𝝓̂ vary in the plane. To keep ẍ = 0, the coordinates must then satisfy (2.27).

Exercise 2.1   Show that the equations (2.27) can also be derive the Lagrangian of the free particle
    (2.28)   𝐿 = 𝑚/2 (𝑟̇² + 𝑟²𝜙̇²),
   where 𝑚 is the mass of the particle.
[Solution]   Using Euler-Lagrange equation for the coordinate 𝑟 is
   (a)   𝑑/𝑑𝑡 (∂𝐿/∂𝑟̇) = ∂𝐿/∂𝑟   ⇒   (∂𝐿/∂𝑟̇) = 𝑑/𝑑𝑡 [𝑚/2 (2𝑟̇ + 0)] = 𝑚 𝑑𝑟̇/𝑑𝑡, ∂𝐿/∂𝑟 = 𝑚/2 (0 + 2 𝑟𝜙̇²) = 𝑚𝑟𝜙̇².   ⇒   𝑑𝑟̇/𝑑𝑡 = 𝑟𝜙̇².
Similarly, for the coordinate 𝜙, we get 
   (b)   𝑑/𝑑𝑡 (∂𝐿/∂𝜙̇) = ∂𝐿/∂𝜙   ⇒   𝑑/𝑑𝑡 (∂𝐿/∂𝜙̇) = 𝑑/𝑑𝑡 [𝑚/2 (0 + 2 𝑟²𝜙̇)] = 𝑚 𝑑/𝑑𝑡 𝑟²𝜙̇ = 𝑚 (𝑑𝑟²/𝑑𝑡 𝜙̇ + 𝑟² 𝑑𝜙̇/𝑑𝑡) = 𝑚 (2𝑟𝑟̇𝜙̇ + 𝑟² 𝑑𝜙̇/𝑑𝑡),
         ∂𝐿/∂ϕ = 𝑚/2 (0 + 0) = 0, 2𝑟𝑟̇𝜙̇ + 𝑟² 𝑑𝜙̇/𝑑𝑡 = 0.   ⇒   𝑑𝜙̇/𝑑𝑡 = -2/𝑟 𝑟̇𝜙̇. ▮

   To derive the equation of motion in an arbitrary coordinate system, which metric 𝑔_𝑖𝑗 ≠ 𝛿_𝑖𝑗, we start from the Lagrangian
    (2.29)   𝐿 = 𝑚/2 𝑔_𝑖𝑗(𝑥^𝑘) ẋ^𝑖 ẋ^𝑗.
Substituting this into the Euler-Lagrangian equation, we find
    (2.30)   𝑑²𝑥^𝑖/𝑑𝑡² = -𝛤^𝑖_𝑎𝑏 𝑑𝑥^𝑎/𝑑𝑡 𝑑𝑥^𝑏/𝑑𝑡,
where we have introduced the Christoffel symbol
    (2.31)   𝛤^𝑖_𝑎𝑏 ≡ 1/2 𝑔^𝑖𝑗(∂_𝑎𝑔_𝑗𝑏 + ∂_𝑏𝑔_𝑗𝑎 - ∂_𝑗𝑔_𝑎𝑏),     with   ∂_𝑗 ≡ ∂/∂𝑥^𝑗.
Derivation   The Euler-Lagrangian equation is
    (2.32)   𝑑/𝑑𝑡 (∂𝐿/∂ẋ^𝑘) = ∂𝐿/∂𝑥^𝑘.
The mass 𝑚 will be cancel on both side. The left-hand side of (2.32) then becomes [∂𝐿/∂ẋ^𝑘 = 1/2 ∂(𝑔_𝑖𝑗ẋ^𝑖𝑗)/∂ẋ^𝑘 = 1/2 (𝑔_𝑖𝑘ẋ^𝑖 + 𝑔_𝑘𝑗ẋ^𝑗) = 𝑔_𝑖𝑘ẋ^𝑖]
    (2.33)   𝑑/𝑑𝑡 (∂𝐿/∂ẋ^𝑘) = 𝑑/𝑑𝑡 (𝑔_𝑖𝑘ẋ^𝑖) = 𝑔_𝑖𝑘ẍ^𝑖 + 𝑑𝑥^𝑗/𝑑𝑡 ∂𝑔_𝑖𝑘/∂𝑥^𝑗 ẋ^𝑖 = 𝑔_𝑖𝑘ẍ^𝑖 + ∂_𝑗𝑔_𝑖𝑘 ẋ^𝑖 ẋ^𝑗 = 𝑔_𝑖𝑘ẍ^𝑖 + 1/2 (∂_𝑖𝑔_𝑗𝑘 + ∂_𝑗𝑔_𝑖𝑘) ẋ^𝑖 ẋ^𝑗,
while the right-hand side is
    (2.34)   ∂𝐿/∂𝑥^𝑘 = 1/2 ∂_𝑘𝑔_𝑖𝑗 ẋ^𝑖 ẋ^𝑗. Combing (2.33) and (2.34) then gives     (2.35)   𝑔_𝑖𝑘ẍ^𝑖 = -1/2 (∂_𝑖𝑔_𝑗𝑘 + ∂_𝑗𝑔_𝑖𝑘 - ∂_𝑘𝑔_𝑖𝑗) ẋ^𝑖 ẋ^𝑗
Multiplying both sides by 𝑔^𝑙𝑘, and using 𝑔^𝑙𝑘 𝑔_𝑘𝑖 = 𝛿^𝑙_𝑖, we get     (2.36)   ẍ^𝑙 = -1/2 𝑔^𝑙𝑘(∂_𝑖𝑔_𝑗𝑘 + ∂_𝑗𝑔_𝑖𝑘 - ∂_𝑘𝑔_𝑖𝑗) ẋ^𝑖 ẋ^𝑗 ≡ -𝛤^𝑙_𝑖𝑗,
which is the desired result (2.30). ▮
   The equation of motion of a massive particle in general relativity will take a similar form as (2.30). However, i this case, the term involving the Christoffel symbol cannot be removed by going to Cartesian coordinate, but is a physical manifestation of the spacetime curvature.

          Curved spacetime
For massive particles, a geodesic is the timelike curve 𝑥^𝜇(𝜏) which extremises the proper time 𝛥𝜏 between two points in the spacetime.³ This extremal path satisfies the following geodesic equation
    (2.37)   𝑑²𝑥^𝜇/𝑑𝜏² = -𝛤^𝑖_𝑎𝑏 𝑑𝑥^𝑎/𝑑𝜏 𝑑𝑥^𝑏/𝑑𝜏,  
where the Christoffel symbol is
    (2.38)   𝛤^𝜇_𝛼𝛽 = 1/2 𝑔^𝜇𝜆(∂_𝛼𝑔_𝛽𝜆 + ∂_𝛽𝑔_𝛼𝜆 - ∂_𝜆𝑔_𝛼𝛽),     with   ∂_𝛼 ≡ ∂/∂𝑥^𝛼.
Notice the similarity between (2.37) and (2.30).
   It will be convenient to write the geodesic equation in terms of the 4-momentum of the particle
    (2.39)   𝛲^𝜇 ≡ 𝑚 𝑑𝑥^𝜇/𝑑𝜏. 
Using the chain rule, we have     (2.40)    𝑑/𝑑𝜏 𝛲^𝜇(𝑥^𝛼(𝜏)) = 𝑑𝑥^𝛼/𝑑𝜏 ∂𝛲^𝜇/∂𝑥^𝛼 = 𝛲^𝛼/𝑚 ∂𝛲^𝜇/∂𝑥^𝛼,  
   so that (2.37) becomes
    (2.41)   [𝑚² 𝑑²𝑥^𝜇/𝑑𝜏² =] 𝛲^𝛼 ∂𝛲^𝜇/∂𝑥^𝛼 = -𝛤^𝜇_𝛼𝛽 𝛲^𝛼𝛲^𝛽.
Rearranging the expression, we can also write
    (2.42)   𝛲^𝛼(∂_𝛼𝛲^𝛼 + 𝛤^𝜇_𝛼𝛽𝛲^𝛽) = 0.
The term in brackets is so-called covariant derivative of 4-vector 𝛲^𝜇 which we denote by ∇_𝛼𝛲^𝜇 ≡ ∂_𝛼𝛲^𝛼 + 𝛤^𝜇_𝛼𝛽𝛲^𝛽. So we can write:
    (2.43)   𝛲^𝛼∇_𝛼𝛲^𝜇 = 0.
   The form of the geodesic equation in (2.43) is particularly convenient because it also applies to massless particles.If we interpret 𝛲^𝜇 ≡ 𝑚 𝑑𝑥^𝜇/𝑑𝜆 as the 4-momentum of a massless particle, where 𝜆 now parameterizes the curve 𝑥^𝜇(𝜆). Accepting that the geodesic equation (2.43) is valid  for both massive and massless particles, we will apply it to particles in the FRW spacetime.

          Free particles in FRW
To evaluate the right-hand side of (2.41), we need the Christoffel symbols for the for the FRW metric,
    (2.44)   𝑑𝑠² = -𝑐²𝑑𝑡² + 𝑎²(𝑡) 𝛾_𝑖𝑗 𝑑𝑥^𝑗𝑑𝑥^𝑖
where the form of the spatial metric 𝛾_𝑖𝑗 depends on our choice of spatial  coordinates and on the curvature of the spatial slices. Substituting 𝑔_𝜇𝜈 = diag(-1, 𝑎²(𝑡) 𝛾_𝑖𝑗) into the definition (2.38), it is straightforward to compute the Christoffel symbols. Note that all Christoffel symbols with two time[-0] indices vanish, i.e. 𝛤^𝜇₀₀ = 𝛤⁰_0𝛽 and 𝛤^𝜇_𝛼𝛽 = 𝛤^𝜇_𝛽𝛼. The only nonzero components are
    (2.45)   𝛤⁰_𝑖𝑗 = 𝑐^-1𝑎ȧ𝛾_𝑖𝑗,    𝛤^𝑖_0𝑗 = 𝑐^-1ȧ/𝑎𝛿^𝑖_𝑗,     𝛤^𝑖_𝑗𝑘 = 1/2 𝛾^𝑖𝑙(∂_𝑗𝛾_𝑘𝑙 + ∂_𝑘𝛾_𝑗𝑙 - ∂_𝑙𝛾_𝑗𝑘).

Example   Let us derive 𝛤⁰_𝛼𝛽 for metric (2.44).
    (2.46)   𝛤⁰_𝛼𝛽 = 1/2 𝑔^0𝜆(∂_𝛼𝑔_𝛽𝜆 + ∂_𝛽𝑔_𝛼𝜆 - ∂_𝜆𝑔_𝛼𝛽).
The factor 𝑔^0𝜆 vanish unless 𝜆 = 0, 𝑔⁰⁰ = -1. Hence we have
    (2.47)   𝛤⁰_𝛼𝛽 = -1/2 (∂_𝛼𝑔_𝛽0 + ∂_𝛽𝑔_𝛼0 - ∂₀𝑔_𝛼𝛽).
The first two terms reduced to derivatives of 𝑔₀₀ and 𝑔₀₀ = -1, so we have
    (2.48)   𝛤⁰_𝛼𝛽 = 1/2 ∂₀𝑔_𝛼𝛽.
∂₀ means ∂/∂𝑥⁰ and 𝑥⁰ ≡ 𝑐𝑡 so ∂₀ = 𝑐^-1∂_𝑡. The derivative is nonzero if 𝑔_𝑖𝑗 = 𝑎²𝛾_𝑖𝑗. In that case we find
    (2.49)   𝛤⁰_𝛼𝛽 = 1/2 𝑐^-1∂/∂𝑡 𝑎²𝛾_𝑖𝑗 = 1/2 𝑐^-1∂/∂𝑎 𝑎²𝛾_𝑖𝑗 ∂𝑎/∂𝑡 = 𝑐^-1𝑎ȧ𝛾_𝑖𝑗. ▮

Exercise 2.2   Derive 𝛤^𝑖_0𝑗 and 𝛤^𝑖_𝑗𝑘 for metric (2.44).
[Solution]   From the definition of the Christoffel symbol, we have
   (a)   𝛤^𝑖_0𝑗 = 1/2 𝑔^𝑖𝜆(∂₀𝑔_𝑗𝜆 + ∂_𝑗𝑔_0𝜆 - ∂_𝜆𝑔_0𝑗),   But 𝑔_0𝑗 = 𝑔_0𝜆 = 0, this becomes 
   (b)   𝛤^𝑖_0𝑗 = 1/2 𝑔^𝑖𝜆∂₀𝑔_𝑗𝜆, 𝑔^𝑖𝜆 = 1/𝑎² 𝛾^𝑖𝜆. And since ∂₀ = 𝑐^-1 ∂_𝑡 and 𝑔_𝑗𝜆 = 𝑎²𝛾_𝑗𝜆, ∂₀𝑔_𝑗𝜆 = 𝑐^-1 2𝑎𝑎̇𝛾^𝑖𝜆.
  ⇒   𝛤^𝑖_0𝑗 = 1/2 1/𝑎² 𝛾^𝑖𝜆 𝑐^-1 2𝑎𝑎̇𝛾^𝑖𝜆 = 𝑐^-1 𝑎̇/𝑎 𝛿^𝑖_𝑗.
Similarly, we have
   (c)   𝛤^𝑖_𝑗𝑘 = 1/2 𝑔^𝑖𝑙(∂_𝑗𝑔_𝑘𝑙 + ∂_𝑘𝑔_𝑗𝑙 - ∂_𝑙𝑔_𝑗𝑘), Since 𝑔^𝑖𝑙 = 1/𝑎² 𝛾^𝑖𝑙 and the derivative is nonzero, if 𝑗, 𝑘 and 𝑙 are spatial indices, 𝑔_𝑗𝑙 = 𝑎² 𝛾_𝑗𝑙, 𝑔_𝑘𝑙 = 𝑎² 𝛾_𝑘𝑙 and 𝑔_𝑗𝑘 = 𝑎² 𝛾_𝑗𝑘 respectively.   ⇒   𝛤^𝑖_𝑗𝑘 = 1/2 1/𝑎² 𝛾^𝑖𝑙(∂_𝑗 𝑎² 𝛾_𝑘𝑙 + ∂_𝑘 𝑎² 𝛾_𝑗𝑙 - ∂_𝑙 𝑎² 𝛾_𝑗𝑘) = 1/2 𝛾^𝑖𝑙(∂_𝑗 𝛾_𝑘𝑙 + ∂_𝑘 𝛾_𝑗𝑙 - ∂_𝑙 𝛾_𝑗𝑘). ▮

   The case of massless particle (like photons) will be specialized. The left-hand side of (2.41) can be written when 𝛼 = 0
    (2.50)   𝛲⁰ ∂𝛲^𝜇/∂𝑥⁰ = 𝛲⁰/𝑐 𝑑𝛲^𝜇/𝑑𝑡, [Notice that 𝑥⁰ ≡ 𝑐𝑡 and ∂₀ = 𝑐^-1∂_𝑡.]
    (2.51)   𝛲⁰/𝑐 𝑑𝛲^𝜇/𝑑𝑡 = -𝛤^𝜇_𝛼𝛽 𝛲^𝛼𝛲^𝛽.
Let us consider the 𝜇 = 0 component of the equation, Then the right term is -𝛤⁰_𝑖𝑗 𝛲^𝑖𝛲^𝑗. Using 𝛲⁰ = 𝐸/𝑐 and 𝛤⁰_𝑖𝑗 = 𝑐^-1𝑎𝑎̇𝛾_𝑖𝑗 then we find,
    (2.52)   𝐸/𝑐³ 𝑑𝐸/𝑑𝑡 = -𝑐^-1𝑎𝑎̇𝛾_𝑖𝑗𝛲^𝑖𝛲^𝑗.
For massless particles, the 4-momentum 𝛲^𝜇 = (𝐸/𝑐, 𝛲^𝑖) obey the constraint
    (2.53)   𝑔_𝜇𝜈𝛲^𝑖𝛲^𝜈 = -𝑐²𝐸² + 𝑎²𝛾_𝑖𝑗𝛲^𝑖𝛲^𝑗 = 0.    [Notice that lightlike 𝑑𝑠² = 0.]
This allow us to write (2.51) as
    (2.54)   1/𝐸 𝑑𝐸/𝑑𝑡 = -𝑎̇/𝑎,     [∫ (1/𝐸 𝑑𝐸/𝑑𝑡) 𝑑𝑡 = log(𝐸) + const; ∫ (-1/𝑎 𝑑𝑎/𝑑𝑡) 𝑑𝑡 = log(𝑎^-1) + const   ⇒   𝐸 ∝𝑎^-1]
which implies that the energy of a massless particle decays with the expansion of the universe, 𝐸 ∝𝑎^-1.

Exercise 2.3   Repeating the analysis for massive particle, with
    (2.55)   𝑔_𝜇𝜈𝑃^𝜇𝑃^𝜈 = -𝑚²𝑐²
   show that the physical 3-momentum, defined as 𝑝² ≡ 𝑔_𝑖𝑗𝑃^𝑖𝑃^𝑗, satisfies 𝑝 ∝𝑎^-1. Show that the momentum can be written as
    (2.56)   𝑝 = 𝑚𝑣/√(1 - 𝑣²/𝑐²).
   where 𝑣² ≡ 𝑔_𝑖𝑗ẋ^𝑖ẋ^𝑗 is the physical peculiar velocity. Since 𝑝 ∝𝑎^-1, freely falling particles will therefore converge onto the Hubble flow.
[Solution]   In the text, we have shown that the geodesic equation of a particle with 4-momentum 𝑃^𝜇 = (𝐸/𝑐, 𝑃^𝑖) can be written as
   (a)   𝐸/𝑐³ 𝑑𝐸/𝑑𝑡 = -𝑐^-1 𝑎̇/𝑎 𝑝²,
   (b)   𝑔_𝜇𝜈𝑃^𝜇𝑃^𝜈 = -𝑐^-2𝐸² + 𝑝² = -𝑚²𝑐²   [Take a time derivative the both side of the equation and arrange them.]
   (c)   𝐸/𝑐³ 𝑑𝐸/𝑑𝑡 = 𝑝/𝑐 𝑑𝑝/𝑑𝑡 = -𝑐^-1 𝑎̇/𝑎 𝑝²,
   (d)   ṗ/𝑝 = -ȧ/𝑎.
The physical 3-momentum of a massive particle therefore decreases as 𝑝 ∝𝑎^-1.
   From the definition of 4-momentum of a massive particle, we get
   (e)   𝑃^𝑖 ≡ 𝑚 𝑑𝑥^𝑖/𝑑𝜏 = 𝑚 𝑑𝑥/𝑑𝜏 ẋ^𝑖 = 𝑚ẋ^𝑖/√(1 - 𝑔_𝑖𝑗ẋ^𝑖ẋ^𝑗/𝑐²) = 𝑚ẋ^𝑖/√(1 - 𝑣²/𝑐²)    
    Now we have (2.56), 𝑝 = 𝑚𝑣/√(1 - 𝑣²/𝑐²).
Since 𝑝 decreases with the expansion of the universe, so will 𝑣, showing that freely falling particles will converge onto the Hubble flow. ▮

          2.2.2 Redshift
The light emitted by a distant galaxy can be interpreted either quantum mechanically as freely-propagating photons or classically as propagating electromagnetic waves. To analyze the observation correctly, we need to take into account that wavelength of light gets stretched (or equivalently, the photons lose energy) by the expansion of the universe.
   Recall wavelength 𝜆 = 𝘩/𝛦, where 𝛦 is photon energy and 𝘩 is Planck's constant. Since the energy of photon evolves as 𝐸 ∝𝑎^-1, the wavelength scales as 𝜆 ∝𝑎. Light emitted at a time 𝑡₁ with 𝜆₁ will be observed at a later time 𝑡₀ with a larger wavelength
    (2.57)   𝜆₁ = 𝑎(𝑡₀)/𝑎(𝑡₁) 𝜆₀.
This increased wavelength is called redshift, since red light has a longer wavelength than blue light.
   The same result can be derived by the classical method. Consider a galaxy at a fixed comoving distance 𝑑. Since the spacetime is isotropic, we can choose coordinates for which light travels in the radial direction with 𝜃 = 𝜙 =const. Then the evolution is determined by a 2-dimensional line element.
    (2.58)   𝑑𝑠² = 𝑎²(𝜂)[-𝑐²𝑑𝜂² + 𝑑𝜒²].
Since photons travel along null geodesics, with 𝑑𝑠² = 0, their path is
    (2.59)   ∆𝜒(𝜂) = ±𝑐𝛥𝜂,
where the plus sign corresponds to outgoing photons and its minus sign to incoming photons. This shows the main benefit of working with conformal time: light rays correspond to straight lines in the 𝜒-𝜂 coordinates. Moreover comoving distance to a source is simply equal to the difference in conformal time between the moments of being emitted and received.
   At a time 𝜂₁, the galaxy emits a signal of a short conformal duration 𝛿𝜂 (see Fig. 2.3).The light arrives at our telescopes at time 𝜂₀ = 𝜂₁ + 𝑑/𝑐. The conformal duration signal measured by the detector is the same as at the source, but the physical time intervals are different at the points of emission and detection.
    (2.60)  𝛿𝑡₁ = 𝑎(𝜂₁)𝛿𝜂   and   𝛿𝑡₀ = 𝑎(𝜂₀)𝛿𝜂.
If 𝛿𝑡 is the period of the light wave, then the light is emitted with wavelength 𝜆₁ = 𝑐𝛿𝑡₁, but is observed with wavelength 𝜆₀ = 𝑐𝛿𝑡₀, so that
    (2.61)  𝜆₀/𝜆₁ = 𝑎(𝜂₀)/𝑎(𝜂₁),
which agree with the result in (2.57).
   It is conventional to define the redshift as the fractional shift in the wavelength:
    (2.62)  𝑧 ≡ (𝜆₀ - 𝜆₁)/𝜆₁.
By comparing the observed wavelengths with those measured in a laboratory on Earth, we determine the redshift. Using (2.61), and setting 𝑎(𝑡₀) ≡ 1, we can write the redshift parameter as
    (2.63)  1 + 𝑧 = 1/𝑎(𝑡₁)
For example, a galaxy at 𝑧 = 1 emitted the observed light when the universe was half its current size. The CMB photons were released at 𝑧 = 1100 and the first galaxies formed around redshift 𝑧 ∼ 10.
   For nearby sources at 𝑧 < 1, we can expand the scale factor in a power series around 𝑡₀:
    (2.64)  𝑎(𝑡₁) = 1 + (𝑡₁ - 𝑡₀)𝐻₀ + ∙ ∙ ∙ ,
where 𝑡₀ - 𝑡₁ is the look-back time and 𝐻₀ is the Hubble constant
    (2.65)  𝐻₀ ≡ 𝑎̇(𝑡₀)/𝑎(𝑡₀).
Equation (2.63) then gives 𝑧 = 𝐻₀(𝑡₀ - 𝑡₁) + ∙ ∙ ∙ , For close objects, 𝑡₀ - 𝑡₁ is simply equal to 𝑑/𝑐, so that the redshift increase linearly with distance. This linear relation is called the Hubble-Lemaitre law. In terms of the recession speed of the object, 𝑣 = 𝑐𝑧, it reads
    (2.66)  𝑣 = 𝑐𝑧 ≈ 𝐻₀𝑑.
Hubble's original measurement of the velocity-distance relations of galaxies are shown in Fig. 2.4. For distant objects (𝑧 > 1), we have to be more careful about the meaning of "distance".

          2.2.3 Distances  
   Measuring distances in cosmology is notoriously difficult. The distances appearing in the metric are not observable. Even the physical distance 𝑑_phs = 𝑎(𝑡)𝜒 cannot be observed because it is the distance between separated events at a fixed time. A more practical definition of  "distance" must take into account that the universe is expanding and that it takes light a finite amount of time to reach us.

          Luminosity distance  
An important way to measure distances in cosmology uses standard candles. These are objects of known intrinsic brightness, so that observed brightnesses can be used to determine their distances.
   Hubble used Cepheids to discover the expansion of the universe. These are stars whose brightness vary periodically. The observed periods were found to be correlated with the intrinsic brightness of stars. By measuring the time variation of Cepheids, astronomers can infer their absolute brightness and then use their observed brightness to infer their distances. To measure larger distances we need brighter sources. These are provided by type Ia supernovae-stellar explosions that arise when a white dwarf accretes too much matter from a companion star. Supernovae are rare (roughly a few per century in a typical galaxy), but outshine all stars in the host galaxy and can therefore be seen out to enormous distances. By observing many galaxies, astronomers can then measure a large number of supernovae.
   The supernovae explosions occur at relatively precise moment - when the mass of the white dwarf exceeds the Chandrasekar limit - and therefore have fixed brightness. Residual variations of their brightness can be corrected for phenomenologically. The use of supernovae as standard candles have been instrumental in the discovery of the acceleration of the universe and they continue to play an important role in observational cosmology.5
   Let us assume that we have identified n astronomical object with a known luminosity 𝐿 (= energy emitted per unit time). The observed flux 𝐹 (= energy per unit time per receiving area) can then be used to infer its (luminosity) distance. Our task is to determine the relation between 𝐿 and 𝐹 in an expanding universe.
   Consider a source at a redshift 𝑧. The comoving distance to the object is
    (2.67)  𝜒(𝑧) = ∫𝑡0𝑡1 𝑑𝑡/𝑎(𝑡) = ∫𝑧0 𝑑𝑧/𝐻(𝑧),
where the redshift evolution of the Hubble parameter, 𝐻(𝑧), depends on the matter content of the universe (see Section 2.3).  We assume That the source emits radiation isotropically (see Fig. 2.5). In a static Euclidean space, the energy would then spread uniformly over a sphere of area 4π𝜒2, and the fraction going through an area 𝐴 (e.g. the collecting area of a telescope) is 𝐴/4π𝜒2. The relation between the absolute luminosity and the observed flux would then be
    (2.68)  𝐹 = 𝐿/4π𝜒2   (static space).
In an expanding spacetime, this result is modified for three reasons:
1. When the light reaches the Earth, at time 𝑡₀, a sphere with radius 𝜒 has and area
    (2.69)  4π𝑎²(𝑡₀)𝑑²_𝑀,
where 𝑑_𝑀 ≡ 𝑆_𝑘 (𝜒) is the"metric distance" defined in (2.22). In a curved space, the metric distance 𝑑_𝑀 differs from the radius 𝜒.
2. The arrive of the photons is smaller than the rate at which they are emitted at the source by a factor of a(𝑡₁)/ a(𝑡₀) = 1/(1 + 𝑧); cf. Fig. 2.3. This reduces the observed flux by the same factor.
3. The energy 𝐸0 = 𝘩𝑓0 of the observed photons is less than the energy 𝐸1 = 𝘩𝑓1 by the same redshift factor 1/(1 + 𝑧). This lowers the observed flux by another factor of 1/(1 + 𝑧).
Hence, the correct formula is
    (2.70)  𝐹 = 𝐿/4π(𝑡0)𝑑2𝑀(1 + 𝑧)2 ≡ 𝐿/4π𝑑2𝐿 .
In the second equality, we have defined the luminosity distance, 𝑑𝐿, so that the relation between luminosity, flux and luminosity instance is the same as in (2.68).
Hence, we find
    (2.71)  𝑑𝐿(𝑧) = (1 + 𝑧)𝑑𝑀(𝑧) ,
where the metric distance for an object with redshift 𝑧 depends on the cosmological parameters. Figure 2.6 shows the luminosity distance as a function of redshift in a universe with and without dark energy. WE see that the luminosity distance out to a fixed redshift is larger in a dark energy-dominated universe than in a matter-only universe.
   For objects at low redshifts (𝑧 <1), we can define perturbative corrections to Hubble's law (2.66). We first extend the Taylor expression of 𝑎(𝑡) in (2.64) to higher order in the look-back time    (2.72)  𝑎(𝑡) = 1 + 𝐻0(𝑡 - 𝑡0) - 1/2 𝑞0𝐻02(𝑡 - 𝑡0)2 + ∙ ∙ ∙ ,
where we have defined
   (2.73)  𝑞0 ≡ -(𝑑𝑎̇/𝑑𝑡)/𝑎𝐻2∣𝑡=𝑡0.
The parameter 𝑞0 was named as the deceleration parameter.
Today, we know that it is negative, 𝑞0 ≈ -0.5, and hence measures the acceleration of the universe. Substituting (2.72) into (2.63), we obtain the redshift as a function of the look-back time
   (2.74)  𝑧 = 1/𝑎(𝑡1) - 1 = 𝐻0(𝑡0 - 𝑡1) +1/2 (2 + 𝑞0)𝐻02(𝑡0 - 𝑡1)2 + ∙ ∙ ∙ .
This can be inverted to give
   (2.75)  𝐻0(𝑡0 = 𝑧 - 1/2(2 + 𝑞0)𝑧2 + ∙ ∙ ∙ ,
where the higher-order terms can be ignored as long as 𝑧 < 1. Using (2.72), we can also write the comoving distance in terms of the look-back time and the redshift    (2.76)  𝜒 = 𝑐 ∫𝑡0𝑡1 𝑑𝑡/𝑎(𝑡) = ∫𝑡0𝑡1 𝑑𝑡[1 + 𝐻0(𝑡 - 𝑡0) + ∙ ∙ ∙ ] = 𝑐(𝑡 - 𝑡0) + 1/2 𝐻0/𝑐 𝑐2(𝑡 - 𝑡0)2 + ∙ ∙ ∙ = 𝑐/𝐻0 [𝑧 -1/2(1 + 𝑞0)𝑧2 + ∙ ∙ ∙ ].
Through (2.71) and (2.22), the determines the luminosity distance as a function of the redshift, 𝑑𝐿(𝑧) . For a flat universe, with 𝑆𝑘(𝜒) = 𝜒, the modified Hubble-Lemaître law reads
   (2.77)  𝑑𝐿 = 𝑐/𝐻0[𝑧 + 1/2(1 - 𝑞0)𝑧2 + ∙ ∙ ∙ ].
The value of 𝐻0 and 𝑞0 can be extracted by fitting the functional form to the observed 𝑑𝐿(𝑧). The value of 𝑞0 will depend on the energy content of the universe. The results of such a measurement in section 2.4 will be presented.
   Measurements of the Hubble constant used to com with very large uncertainties. It therefore become conventional to define  
   (2.78)  𝐻0 ≡ 100𝘩 km s-1Mpc-1,
where the parameter 𝘩 is used to keep track of how uncertainties in 𝐻0 propagate to the inferred value of other cosmological parameters. The latest supernovae measurements have found
   (2.79)  𝘩 = 0.730 ± 0.010    (supernovae),
The Hubble constant cal also be extract from the CMB anisotropy spectrum (see Chapter 7). These observations gives
   (2.80)  𝘩 = 0.674 ± 0.005    (CMB).
As we can see, there is currently a statistically significant discrepancy between the two measurements. It is unclear whether this "Hubble tension" is due to an unidentified observational systematic a breakdown in the standard cosmological model.  

          Angular diameter distance   
  An alternative way to measure distances is to use standard rulers, i.e. objects of known physical size. The observed angular sizes of such objects then depends on their distances. As we will see in Chapter 7, the typical size of hot and cold spots in the CMB can be predicted theoretically and is therefore a standard ruler. The observed angular size of these spots then determines the distance to the CMB's surface of last-scattering.⁶
   Let us assume that an object is at a comoving distance 𝜒 and that the photons emitted at a time 𝑡₁ (see Fig.2.7). The transverse physical size of the object is 𝐷. In static Euclidean space, we would expect its angular size to be
   (2.81)  𝛿𝜃 = 𝐷/𝜒   (static space),
where we assumed that 𝛿𝜃 ≪ 1 (in radian), which is true for all cosmological objects. In an expanding universe, this formula becomes
   (2.82)  𝛿𝜃 = 𝐷/𝑎(𝑡₁)𝑑_𝛭 ≡ 𝑑_𝛢   (expanding space),
where metric distance 𝑑_𝛭 ≡ 𝑆_𝑘(𝜒). Notice that the observed angular size depends on the distance at time 𝑡₁ when the light was emitted. The second equality in (2.820  has defined the angular diameter distance by analogy with Euclidean formula (2.81). In terms of the metric distance 𝑑_𝛭 and redshift 𝑧. the angular diameter distance is
   (2.83)  𝑑_𝛢(𝑧) = 𝑑_𝛭(𝑧)/(1 + 𝑧).
Fig. 2.8 shows the angular diameter as a function of redshift in a flat matter-only universe and in a universe with negative spatial curvature. We see that the angular diameter distance starts to decrease around 𝑧_𝑚 ~ 1.5. At redshift larger than 𝑧_𝑚 objects of a given proper size will appear bigger on the sky with increasing redshift. The effect is due to the expansion of the universe. The spacetime was compressed when the light was emitted and the objected were closer to us. The observered anguar size is therefore larger.

⁵ In the future gravitational waves (GWs) will allow for robust measurement distance using so-called standard sirens. The parameters of source like the masses of the objects in a binary system is determined by GWs signals and GR predicts the emitted GW power. Comparing these data provides a clean measurement of luminosity distance.

          2.3 Dynamics
So far, we have used the symmetries of the universe to fix its geometry and studied the propagation of particles in the expending spacetime. The scale factor 𝑎(𝑡) has remained an unspecified function of time. The evolution of the scale factor follows from the Einstein equation
    (2.84)   𝐺_𝜇𝜈 ( + 𝛬𝑔_𝜇𝜈) = 8π𝐺/𝑐⁴ 𝑇_𝜇𝜈,   [𝛬: cosmological constant, 𝑔_𝜇𝜈:metric tensor]
where 𝐺 = 6.67 × 10^-11 m³ kg^-1 s^-2 is Newton's constant. This relates the Einstein tensor 𝐺_𝜇𝜈 (a measure of the "spacetime curvature" of the universe) to the energy-momentum tensor 𝑇_𝜇𝜈 (a measure of "matter content" of the universe). We will first discuss the possible forms of cosmological energy-momentum tensors, then compute the Einstein tensor for FRW background, and finally put them together to solve for the evolution of the scale factor 𝑎(𝑡) as a function of the matter content.

          2.3.1 Perfect Fluids   
We have seen how the spacetime geometry of the universe is constrained by homogeneity and isotropy. Now, we will determine which types of matter are consistent with these symmetries. We will find that the coarse-grained energy-momentum tensor is required to be that of a perfect fluid.
    (2.85)  𝑇_𝜇𝜈 = (𝜌 + 𝑃/𝑐²)𝑈_𝜇𝑈_𝜈 + 𝑃𝑔_𝜇𝜈,
where 𝜌𝑐² and 𝑃 are the energy density and pressure in the rest frame of the fluid, and 𝑈^𝜇 is its 4-velocity relative to a comoving observer. ]

          Number density 
Let us a simpler object: the number current 4-vector 𝑁^𝜇. 𝑁⁰ measures the number density particles, where a "particle" in cosmology may be   an entire galaxy. 𝑁^𝑖 is the flux of particle in the direction 𝑥^𝑖. We call  𝑁⁰ a three scalar because its value doen't change under a purely spatial coordinate transformation. But 𝑁^𝑖 transforms like a 3-vector under such a transformation. We want to determine which form of the number current is consistent with the homogeneity and isotropy seen by a comoving observer.
   Isotropy requires that the mean value of any 3-vector, such as 𝑁^𝑖, must vanish, and homogeneity require that the mean value of 3-scalar such as 𝑁⁰, is only a function of time. Hence
    (2.86)  𝑁⁰ = 𝑐𝑛(𝑡),     𝑁^𝑖 = 0,
where 𝑛(𝑡) is the number of galaxies per proper volume. A general observer (i.e. an observer in motion relative to the men rest frame of the particles), would measure the following number current 4-vector
    (2.87)  𝑁^𝜇 = 𝑛𝑈^𝜇,
where 𝑈^𝜇 ≡ 𝑑𝑥^𝜇/𝑑𝜏 is relative 4-velocity between particles and the observer. We recover (2.86) for a comoving observer, with 𝑈^𝜇 = (𝑐, 0, 0, 0). Moreover, for 𝑈^𝜇 = 𝛾(𝑐, 𝑣^𝑖, the expression (2.87) gives the correctly boosted result, where 𝛾 is the Lorentz factor for the fact that one of the dimensions is Lorentz contracted.
   How does the number density evolve with time? If the number of particles is conserved, then the rate of of change of number density must equal the divergence of the flux of particles. In Minkowski space, this imply the following continuity equation ∂₀𝑁⁰ = -∂_𝑖𝑁^𝑖, or in relativistic notation,
    (2.88)  ∂_𝜇𝑁^𝜇 = 0.
Equation (2.89) is generalized to curved spacetimes by replacing the partial derivative with a covariant derivative (see Appendix A)
    (2.89)  ∇_𝜇𝑁^𝜇 = 0.
This covariant form of the continuity equation is valid independent of the choice of coordinates.

Covariant derivative     The covariant derivative is an important object in differential geometry and it is fundamental importance in general relativity. But we will not get into these detail here and simply tell us that how the covariant derivatives acts on 4-vectors with upstairs and downstairs indices:
    (2.90-91)   ∇_𝜇𝐴^𝜈 = ∂_𝜇𝐴^𝜈 + 𝑁^𝜇,     ∇_𝜇𝛣_𝜈 = ∂_𝜇𝛣_𝜈 - 𝛤^𝜆_𝜇𝜈𝛣_𝜆.
We already encounter (2.91) in our discussion of the geodesic equation in (2.43),  [Wikipedia Covariant derivative] The action on a general tensor is straightforward generalization of these two results. For example, the covariant derivative of a rank-2 tensor with mixed indices is
    (2.92)    ∇_𝜇𝛵^𝜎_𝜈 = ∂_𝜇𝛵^𝜎_𝜈 + 𝛤^𝜎_𝜇𝜆𝛵^𝜆_𝜈 - 𝛤^𝜆_𝜇𝜈𝛵^𝜎_𝜈.

   Using (2.91) in (2.90, we get
    (2.93)    ∂_𝜇𝑁^𝜇 = -𝛤^𝜇_𝜇𝜆𝑁_𝜆.
Let us evaluate this in the rest frame of fluid. Substituting the components of the number current in (2.86), we obtain
    (2.94)    𝑑𝑛/𝑐 𝑑𝑡 = -𝛤^𝜇_𝜇0 𝑛 = -3/𝑐 𝑎̇/𝑎 𝑛,   [𝑐 𝑑𝑛/𝑐 𝑑𝑡 = -𝛤^𝜇_𝜇0 𝑐 𝑛]
where we have used that 𝛤⁰₀₀ = 0 and 𝛤^𝑖_𝑖0 = -3/𝑐 𝑎̇/𝑎. Hence, we find that
    (2.95)    𝑛̇/𝑛 = -3𝑎̇/𝑎  ⇒   𝑛(𝑡) ∝ 𝑎^-3  [∫ 𝑛̇/𝑛 𝑑𝑡 = ∫ 1/𝑛 𝑑𝑛 = log(𝑛) + const; ∫ -3𝑎̇/𝑎 𝑑𝑡 = ∫ -3/𝑎 𝑑𝑎 = log(1/𝑎³) + const]
As expected, the number density decreases in proportion to the increase of the proper volumne, 𝑉 ∝𝑎³.
          Energy-momentum tensor 
We will now use a similar logic to determine which form of the energy-momentum tensor 𝛵_𝜇𝜈 is consistent with the requirement of homogeneity and isotropy. First, we decompose 𝛵_𝜇𝜈 into a 3-scalar 𝛵₀₀, two 3-vectors  𝛵_𝑖0 and 𝛵_0𝑗 and a 3-tensor 𝛵_𝑖𝑗. The physical meaning of these component is
    (2.96)    𝛵_𝜇𝜈 = (𝛵₀₀, 𝛵_0𝑗, 𝛵_𝑖0, 𝛵_𝑖𝑗) = (energy density, momentum density, energy flex, stress tensor).
In a homogeneous universe, the energy density must be independent of position, but can depend on time, 𝛵₀₀ = 𝜌(𝑡)𝑐². Moreover, isotropy again requires the mean values of the 3-vectors in the comoving frame, 𝛵_𝑖0 = 𝛵_0𝑗 = 0. Finally isotropy around a point 𝐱 = 0 constrains the mean value of 3-tensor, such as  𝛵_𝑖𝑗, to be proportional to 𝛿_𝑖𝑗. Since the metric 𝑔_𝑖𝑗 equals 𝑎²𝛿_𝑖𝑗 at 𝐱 = 0, we have
    (2.97)   𝛵_𝑖𝑗(𝐱 = 0)  ∝ 𝛿_𝑖𝑗 ∝ 𝑔_𝑖𝑗(𝐱 = 0).
Homogeneity requires the proportionality coefficient to be only a function of time. Moreover, Since this is a proportionality between two tensors, 𝛵_𝑖𝑗 and 𝑔_𝑖𝑗, it remains unaffected by transformation of the spatial coordinates, including those transformations that preserve the form of 𝑔_𝑖𝑗 while taking this origin into any other point. Hence homogeneity and isotropy require the component s of energy-momentum tensor everywhere to take the form
    (2.98)   𝛵₀₀ ≡ 𝜌(𝑡)𝑐²,   𝛵_𝑖0 ≡ 𝑐𝜋_𝑖 = 0,   𝛵_𝑖𝑗 = 𝑃(𝑡)𝑔_𝑖𝑗(𝑡, 𝐱).
Raising one of indices, we find ^{[for example?]}
    (2.99)  𝛵^𝜇𝜈 = 𝑔_𝜇𝜈𝛵_𝜇𝜈 = diag(-𝜌𝑐, 𝑃, 𝑃, 𝑃).
which we recognize as the energy-momentum tensor of a perfect fluid in the frame of a comoving observer. More generally, the energy-momentum tensor can be written in the following, explicitly covariant, form [RE (2.84)]
    (2.100)  𝑇_𝜇𝜈 = (𝜌 + 𝑃/𝑐²)𝑈_𝜇𝑈_𝜈 + 𝑃𝑔_𝜇𝜈 where 𝑈^𝜇 is the relative 4-velocity between the fluid and the observer. We can recover (2.98) for a comoving observer, 𝑈^𝜇 = (𝑐, 0, 0, 0). ^{[Why not 𝑈_𝜇?]}

          Continuity equation    
In Minkowski space energy and momentum are conserved. The energy density therefore satisfied the continuity equation ῤ = -∂_𝑖π^𝑖, i.e. the rate of change of the density equals the divergence of the energy flux π. Similarly the evolution of momentum satisfies the Euler equation 𝑑π^𝑖/𝑑𝑡 = ∂_𝑖𝛲. These conservation laws can be combined  into a 4-component conservation equation for the energy-momentum tensor
    (2.101)  ∂_𝜇𝑇^𝜇_𝜈 = 0.
In GR this id promoted to the covariant conservation equation
    (2.102)  ∇_𝜇𝑇^𝜇_𝜈 = 0.
Using (2.92)
    (2.103)  ∇_𝜇𝑇^𝜇_𝜈 = ∂_𝜇𝛵^𝜇_𝜈 + 𝛤^𝜇_𝜇𝜆𝛵^𝜆_𝜈 - 𝛤^𝜆_𝜇𝜈𝛵^𝜇_𝜆 = 0.
This corresponds to four separate equation (𝜈: 0~3).  The evolution of the energy density is determined by the 𝜈 = 0 equation
    (2.104)  ∂_𝜇𝛵^𝜇₀ + 𝛤^𝜇_𝜇𝜆𝛵^𝜆₀ - 𝛤^𝜆_𝜇0𝛵^𝜇_𝜆 = 0.
Since 𝛵^𝑖₀ vanishes by isopropy, this reduces to [∂_𝜇𝛵⁰₀ + 𝛤^𝜇_𝜇𝜆𝛵⁰₀ - 𝛤^𝜆_𝜇0𝛵^𝜇_𝜆 = 0.]
    (2.105)  1/𝑐 𝑑(𝜌𝑐²)/𝑑𝑡 + 𝛤^𝜇_𝜇0(𝜌𝑐²) - 𝛤^𝜆_𝜇0𝛵^𝜇_𝜆 = 0,
where 𝛤^𝜆_𝜇0 vanishes unless 𝜆 and 𝜇 are spatial indices 𝑖 and they are same, i.e. 𝜆 = 𝜇 = 𝑖. So we have 𝛤^𝑖_𝑖0 = 3𝑐^-1 𝑎̇/𝑎. [RE (2.45) and 𝛵^𝜇_𝜆 = 𝛵^𝑖_𝑖 = 𝑃.] The continuity equation (2.105) then becomes
    (2.106)  𝜌̇ + 3 𝑎̇/𝑎 (𝜌 + 𝑃/𝑐²) = 0.   [𝑐 𝑑𝜌/𝑑𝑡 + 3𝑐 𝑎̇/𝑎 (𝜌 + 𝑃/𝑐²) = 0.]
This important equation describes "energy conservation" in the cosmological context. Notice that the usual notion of energy conservation in flat space relies on a symmetry under the translations. This symmetry is broken in an expanding space, so the familiar energy conservation does not have to hold and is replaced by (2.106).

Exercise 2.4   Show that (2.106) can be obtain from the thermodynamic relation 𝑑𝑈 = -𝑃𝑑𝑉, where 𝑈 = (𝜌𝑐²)𝑉 and 𝑉 ∝𝑎³. [Solution]   Using 𝑈 = (𝜌𝑐²)𝑉, we have 𝑑𝑈 = 𝑐²𝑉(𝑑𝜌 + 𝜌 𝑑𝑉/𝑉) and with 𝑑𝑈 = -𝑃𝑑𝑉 becomes the following and then we used  𝑉 ∝𝑎³ in the final equality.
   (a)   𝑑𝜌 = - 𝑑𝑉/𝑉(𝜌 + 𝑃/𝑐²) = -3 𝑑𝑎/𝑎 (𝜌 + 𝑃/𝑐²),
   (b)   (𝜌 + 𝑃/𝑐²) log  𝑉 + const =  (𝜌 + 𝑃/𝑐²) log 𝑎³ + const)
By taking a derivative to time 𝑡 we get the continuity equation,
   (c)   𝑑𝜌/𝑑𝑡 =  -3 𝑎̇/𝑎 (𝜌 + 𝑃/𝑐²).    ⇒    𝜌̇ + 3 𝑎̇/𝑎 (𝜌 + 𝑃/𝑐²) = 0.  ▮

   Most cosmological fluids can be parameterized in terms of a constant equation of state, 𝜔 = 𝛲/(𝜌𝑐²). In that case, we get     (2.107)  𝜌̇/𝜌 = -3(1 + 𝜔) 𝑎̇/𝑎    ⇒    𝜌 ∝ 𝑎^{-3(1 + 𝜔)}     [∫ 𝜌̇/𝜌 𝑑𝑡 = log(𝜌) + const; ∫ [-3(1 + 𝜔) 𝑎̇/𝑎] 𝑑𝑡 = log[𝑎^{-3(1 + 𝜔)}] + const]
which shows how the dilution of the energy density depends on the equation of state.

          2.3.2 Matter and Radiation    
The term matter will be used to refer to a fluid whose pressure is much smaller than its energy density, ∣𝛲∣≪ 𝜌𝑐². As we will see in Chapter 3, this is the case for a gas of non-relativistic particles, for which the energy density is dominated by their rest mass. Setting 𝜔 = 0 in (2.107) gives
    (2.108)  𝜌 ∝ 𝑎^-3.
This dilution of the energy density simply reflects that the fact the volume of a region of space increases as 𝑉 ∝ 𝑎³, while the energy within the region stays constant.

• Baryons   Cosmologists refers to ordinary matter (nuclei and electrons) as baryons.Electrons are leptons, but nuclei are so much heavier than electrons that most of the mass is in this baryons.

• Dark Matter   Most of the matter in the universe is in the form of dark matter. Figure 2.9 shows the orbital speeds of visible stars or gas in the galaxy M33 versus their radial distance from the center of the galaxy. We see that the rotation speeds stay constant for behind the extent of the visible disc. This suggests that there is an invisible halo of dark matter that holds the galaxy together. On larger scale, dark matter has been detected  by gravitational lensing. As the CMB photons travel through the universe they get deflected by the total matter in the universe. The effect can be measured statistically and provides one of the best pieces of evidence for dark matter on a cosmic scale.

   The precise nature of the dark matter is unknown. We usually think of it as a new particle species, but we don't know what it really is. There are many popular dark matter candidates such as WIMPs, axions, MACHOs, and primordial black holes. [Wikipedia Dark matter:] Wile finding a microscopic detail don't affect the large-scale evolution of the universe. On large scales, all dark matter models are described by the same pressureless fluid. This cold dark matter (CDM) is an integral part of the standard model of cosmology.
   The term radiation will be used to denote for which the pressure is on third of the energy density, 𝛲 = 1/3 𝜌𝑐². This is the case for a gas of relativistic particles, for which the energy density to dominated by the kinetic energy. Setting 𝜔 = 1/3 in (2.107) gives
    (2.109)  𝜌 ∝ 𝑎^-4.
The dilution now includes not just the expansion, 𝑉 ∝ 𝑎³, but also the redshifting of the energy of the particles, 𝐸 ∝ 𝑎^-1.

• Light particles   At early times, all particles of the Standard Model acted as relativistic, because the temperature of the universe was larger than the masses of the particles. As the temperature of the universe dropped, the massless of many particles became relevant and they started behave like matter.

• Photons   Being massless, photons are always relativistic. Together with neutrinos, they ere the dominant energy density during Big Bang nucleosynthesis. Today, we observe these photons in the form of the CMB.

• Neutrinos  For most of history of the universe neutrinos behaves like radiation. Only recently have their small masses become relevant, making them act like matter.

• Gravitons  The early universe may also have produced a background of gravitons. The density of these gravitons is predicted to be very small, so they have a negligible effect on the expansion of the universe. Nevertheless, experimental efforts are underway to detect them (see Section 8.4.2).

          2.3.3 Dark Energy 
    We have recently learned that matter and radiation aren't enough to describe the evolution of the universe. Instead the universe today to determined by a mysterious form of dark energy with negative pressure,𝛲 = -𝜌𝑐² and hence constant energy density.
    (2.110)  𝜌 ∝ 𝑎⁰
Since the energy density doesn't dilute, energy has to created on the universe expands. As described above, that doesn't violate the conservation of energy, as long as equation (2.106) is satisfied.

• Vacuum energy   A natural candidate for a constant energy density is the energy associated to empty space itself.As the universe expands, more space is being created and this energy therefore increase in proportion to the volume. In quantum field theory, this so-called "vacuum energy" in actually predicted, leading to an energy-momentum tensor of the form
    (2.111)  𝛵^vac_𝜇𝜈 = -𝜌_vac𝑐²𝑔_𝜇𝜈.
AS expected from Lorentz symmetry, the energy-momentum tensor associated to the vacuum to proportional to the spacetime metric. Comparison with (2.100) shows that this indeed gives 𝛲_vac = -𝜌_vac𝑐². Unfortunately, as it will be described below, quantum field theory also predicts the size of the vacuum energy 𝜌_vac to be much larger than the value inferred from cosmological observations.

• Cosmological constant   The observation of dark energy has also revived the old concept of a "cosmological constant," originally introduced by Einstein to make the universe static. Note that the left-hand side of Einstein equation (2.94) isn't uniquely defined. We can add the term 𝛬𝑔_𝜇𝜈 for some constant 𝛬, without changing the conservation of the energy-momentum tensor, ∇^𝜇𝛵_𝜇𝜈 = 0. To see this, you need to convince yourself that ∇^𝜇𝑔_𝜇𝜈 = 0. In other words, we could have written the Einstein equation as
    (2.112)  𝐺_𝜇𝜈 + 𝛬𝑔_𝜇𝜈 = 8π𝐺/𝑐⁴ 𝑇_𝜇𝜈.
However it has been standard practice to move this extra terms to the right-hand side and treat it as a contribution to the energy-momentum tensor
    (2.113)  𝛵^(𝛬)_𝜇𝜈 = 𝛬𝑐⁴/8π𝐺 𝑔_𝜇𝜈 ≡ -𝜌_𝛬𝑐²𝑔_𝜇𝜈,
which is of the same form as (2.111).

• Dark energy   The therm "dark energy" is often used to describe a more general fluid whose equation of state is not exactly that of a cosmological constant 𝜔 ≈ -1, or may even be varying in time. It is the failure of quantum field theory to explain the size of the observed vacuum energy that has led theorists to consider those more exotic possibilities. Whatever it is, dark energy plays a key role in the standard cosmological model-the 𝛬CDM model (see Section 2.4).

          The cosmological constant problem    
When the cosmological constant was discovered, it came as a relief to observers, but was a shock to theorists. While it reconciled the age of the universe with the age of the oldest starts , its observed value is much smaller than all particle physics scales, making it hard to understand from a more fundamental perspective. This cosmological constant problem is the biggest crisis in modern theoretical physics.
   In Section 2.4. we will see that the observed value of the vacuum energy is
    (2.114)  𝜌_𝛬𝑐² ≈ 6 × 10^-10 J m^-3
To compare this to our expectation from particle physics, we had better to write it in units of high-energy physics, electron volts (eV). Using 1 J = 6.2 × 10¹⁸ eV and ħ𝑐 ≈ 2.0 × 10^-7 eV m, we get (ħ𝑐)³𝜌_𝛬𝑐² = (10^-3 eV)⁴. Written in natural units, with  ħ = 𝑐 ≡ 1, we have then
    (2.115)  𝜌_𝛬 = (10^-3 eV)⁴.
We see that the scale appearing in the vacuum energy, 𝑀^𝛬 ∼ 10^-3 eV, is much smaller than the typical scales relevant to particle physics.    Quantum field theory (QFT) predicts the existence of vacuum energy. However, the expected value of this vacuum energy is vastly too large. To start, recall that that in QFT every particle arises as an expectation of a fundamental field. Moreover, each field can be represented by an infinite number of Fourier modes with frequencies 𝜔_𝜅. These Fourier modes satisfy the equation of a harmonic oscillator and therefore experience the same zero-point fluctuations as a quantum harmonic oscillator (see Section 8.2.1). In other words, the vacuum energy receive contributions of the form 1/2 ℏ𝜔_𝜅 for each mode. The sum over all Fourier modes actually diverges. This divergence arises because we extrapolated our QFT beyond  its regime of validity. Let us therefore assumed that our theory is valid only up to a cutoff scale 𝛭_* and only contributions to the vacuum energy from modes with frequencies below the cutoff. The vacuum energy will now be finite, but depends on the arbitrary cutoff scale. The dependence on the cutoff can be removed by adding a counterterm in the form of a bare cosmological constant 𝛬₀. After this process of renormalization, the remaining vacuum energy is
    (2.116)  𝜌_QFT ∼ 𝛴_𝑖 𝑚⁴_𝑖,
where the sum is over all particles with mass below the cutoff. In the absence of gravity this constant vacuum energy wouldn't affect any dynamics and is therefore  usually discarded. The equivalence principle, however,implies that all forms of energy gravitate and hence affect the curvature of the spacetime. The vacuum energy is no exception.
   The first to worry about the gravitational effects of zero-point energies was Wolfgang Pauli in the 1920s. Considering the contribution from the electron in (2.1116), he estimated that the induced curvature radius of the universe would be 10⁶ km, so that "the world would not even reach to the moon". Including rest of the Standard Model only makes the problem worse:
    (2.117)  𝜌_QFT ≳ (1 TeV)⁴ = 10⁶⁰ 𝜌_𝛬,
where we have added contributions up to the TeV scale. The enormous discrepancy between this estimate and the observed value in (2.114) is the cosmological constant problem.
   But we can add an arbitrary constant to the QFT estimate to match the observed value of this vacuum energy
    (2.118)  𝜌_𝛬 = 𝜌_QFT + 𝜌₀.
To explain the small size of the observed vacuum energy would require two large numbers,𝜌_QFT and  𝜌₀ have 60 digits in common, but differ at the 61st digit,  so that the sum of the two is smaller by 60 orders of magnitude. moreover, each time the universe goes through a phase transition, the value of 𝜌_QFT changes, while the value of 𝜌₀ presumably doesn't. For example, we expect the electroweakland QCD phase transition to induce jumps in the order of ∆_𝜌QFT ∼ (200 GeV)⁴ and (0.3 GeV)⁴ respectively.Any tuning must explain the small value of the cosmological constant today and not at the beginning.
   One way to explain the apparent fine-tuning to the vacuum energy is to appeal to the anthropic principle. Suppose that our universe is part of a much larger multiverse and that the fundamental physical parameters exhibit different values indifferent regions of space. In particular, imagine that the value of the vacuum energy varies across the multiverse. If the vacuum energy were much larger, dark energy would start to dominate before galaxies would have formed and complex structures could not have evolved. In other words, we shouldn't be shocked by the small value of the vacuum energy, since if it was any bigger  w would not be around to ask the question. "We live where we can live" in a vast landscape of possibilities and the observed value of the cosmological constant is simply the accident of "environmental selection."

          2.3.4 Spacetime Curvature  
   Having introduced the relevant types of matter and energy in the universe, we now want to see how they source the dynamic s of the spacetime. To do this, we have to compute the Einstein tensor on the left-hand side of the Einstein equation (2.84)
    (2.119)  𝐺_𝜇𝜈 = 𝑅_𝜇𝜈 - 1/2 𝑅𝑔_𝜇𝜈,
where 𝑅_𝜇𝜈 is the Ricci tensor and 𝑅 = 𝑅^𝜇_𝜇 = 𝑔^𝜇𝜈𝑅_𝜇𝜈 is its trace, the Ricci scalar. In Appendix A, we give the following definition of the Ricci tensor
    (2.120)  𝑅_𝜇𝜈 ≡ ∂_𝜆𝛤^𝜆_𝜇𝜈 - ∂_𝜈𝛤^𝜆_𝜇𝜆 + 𝛤^𝜆_𝜆𝜌𝛤^𝜌_𝜇𝜈 - 𝛤^𝜌_𝜇𝜆𝛤^𝜆_𝜈𝜌.
Now we are going to compute (2.120) and (2.119) for the FRW spacetime. 𝑅_𝑖0 = 𝑅_0𝑖 = 0, because it is a 3-vector and therefore must vanish due to the isotropy of Robertson-Walker metric. The non- vanishing components of Ricci tensor are
    (2.121)  𝑅₀₀ = -3/𝑐² (𝑑𝑎̇/𝑑𝑡)/𝑎,
    (2.122)  𝑅_𝑖𝑗 = 1/𝑐² [(𝑑𝑎̇/𝑑𝑡)/𝑎 + 2 (𝑎̇/𝑎)² + 2 𝑘𝑐²/𝑎²𝑅₀²] 𝑔_𝑖𝑗.

Example   Setting 𝜇 = 𝜈 = 0 in (2.120), we have
    (2.123)  𝑅₀₀ ≡ ∂_𝜆𝛤^𝜆₀₀ - ∂₀𝛤^𝜆_0𝜆 + 𝛤^𝜆_𝜆𝜌𝛤^𝜌₀₀ - 𝛤^𝜌_0𝜆𝛤^𝜆_0𝜌.
Since Christoffel symbols with two time indices-00- vanish, this reduces to
    (2.124)  𝑅₀₀ ≡ -∂₀𝛤^𝑖_0𝑖 - 𝛤^𝑖_0𝑗𝛤^𝑗_0𝑖.
Using 𝛤^𝑖_0𝑗 = 𝑐^-1(𝑎̇/𝑎)𝛿^𝑖_𝑗, {RE (2.45)] we find
    (2.125)  𝑅₀₀ = -1/𝑐² 𝑑/𝑑𝑡 (3 𝑎̇/𝑎) - 3/𝑐² (𝑎̇/𝑎)² = -3/𝑐² (𝑑𝑎̇/𝑑𝑡)/𝑎. ▮
Example   Computing 𝑅_𝑖𝑗 can be greatly simplified using the following trick: we will compute 𝑅_𝑖𝑗 at 𝐱 = 0 where the spatial metric is 𝛾_𝑖𝑗 = 𝛿_𝑖𝑗 and then argue that the result for general 𝐱 must have the form 𝑅_𝑖𝑗 ∝ 𝑔_𝑖𝑗. [RE (2.49) In FRW metric 𝑔_𝑖𝑗 = 𝑎²𝛾_𝑖𝑗]
In Cartesian coordinates the spatial metric is [RE (2.9-14)]
    (2.126)  𝛾_𝑖𝑗 = 𝛿_𝑖𝑗 + 𝑘𝑥_𝑖𝑥_𝑗/[𝑅₀² - 𝑘(𝑥_𝑙𝑥^𝑙)]. [Why? (𝐱 ⋅ 𝑑𝐱)² = 𝑥_𝑖𝑥_𝑗;   𝐱² = 𝑥_𝑙𝑥^𝑙]
The key point is to think ahead and anticipate that we will set 𝐱 = 0 at the end. In this condition the Christoffel symbols contain derivative metric and the Ricci tensor has another derivative, so there will be terms in the final answer with two derivatives acting on the metric from the second term.  We only need to keep terms to quadratic order in 𝑥:  
    (2.127)  𝛾_𝑖𝑗 = 𝛿_𝑖𝑗 + 𝑘𝑥_𝑖𝑥_𝑗/𝑅₀² + 𝑂(𝑥⁴).
Plugging this into the definition of Christoffel symbol gives [RE (2.45) for Christoffel symbol in FRW metric]
    (2.128)  𝛤^𝑖_𝑗𝑘 = 𝑘/𝑅₀² 𝑥^𝑖𝛿_𝑗𝑘 + 𝑂(𝑥³).
This vanishes at 𝐱 = 0, but the derivative does not. From the definition of Ricci tensor, we then have
    (2.129)  𝑅_𝑖𝑗(𝐱 = 0) ≡ ^(𝐴){∂_𝜆𝛤^𝜆_𝑖𝑗 - ∂_𝑗𝛤^𝜆_𝑖𝜆} + ^(𝐵){𝛤^𝜆_𝜆𝜌𝛤^𝜌_𝑖𝑗 - 𝛤^𝜌_𝑖𝜆𝛤^𝜆_𝑗𝜌}.
Let us start with (𝐵). Dropping terms that are zero at 𝐱 = 0, we find
    (2.130)  (𝐵) = 𝛤^𝑙_𝑙0𝛤⁰_𝑖𝑗 - 𝛤⁰_𝑖𝑙𝛤^𝑙_𝑗0 - 𝛤^𝑙_𝑖0𝛤⁰_𝑗𝑙 = 3/𝑐² 𝑎̇/𝑎 𝑎𝑎̇𝛿_𝑖𝑗 - 1/𝑐² 𝑎𝛿_𝑖𝑙 𝑎̇/𝑎 𝑎𝛿^𝑖_𝑗 - 1/ 𝑐² 𝑎̇/𝑎 𝛿^𝑙_𝑖𝑎𝑎̇𝛿_𝑗𝑙 = 𝑎̇²/𝑐² 𝛿_𝑖𝑗.
The two terms labeled (𝐴) can be evaluated by using (2.128),
    (2.131)  (𝐴) = ∂₀𝛤⁰_𝑖𝑗 + ∂_𝑡𝛤^𝑙_𝑖𝑗 - ∂_𝑗𝛤^𝑙_𝑖𝑙 = 𝑐^-2 ∂_𝑡(𝑎𝑎̇)𝛿_𝑖𝑗 + 𝑘/𝑅₀² 𝛿^𝑙_𝑖𝛿_𝑖𝑗 - 𝑘/𝑅₀² 𝛿^𝑙_𝑗𝛿_𝑖𝑙 = 𝑐^-2 (𝑎𝑑𝑎̇/𝑑𝑡 + 𝑎̇² + 2 𝑘𝑐²/𝑅₀²)𝛿_𝑖𝑗.
Hence we get
    (2.132)  𝑅_𝑖𝑗(𝐱 = 0) = (𝐴) + (𝐵) = 1/𝑐²[𝑎𝑎̇ + 2𝑎̇² + 2 𝑘𝑐²/𝑅₀²]𝛿_𝑖𝑗 = 1/𝑐²[𝑎𝑑𝑎̇/𝑑𝑡 + 2𝑎̇² + 2 𝑘𝑐²/𝑅₀²]𝛿_𝑖𝑗 = 1/𝑐²[(𝑑𝑎̇/𝑑𝑡)/𝑎 + 2(𝑎̇/𝑎)² + 2 𝑘𝑐²/𝑅₀²]𝑔_𝑖𝑗(𝐱 = 0).
Since this is a relation between tensors, it holds for general 𝐱, so we get the result quoted in (2.122). ▮

Now we can calculate the Ricci scalar [𝑔^𝜇𝜈 = diag(-1, 1/𝑎² 𝛾^𝑖𝑗)
    (2.133)  𝑅 = 𝑔^𝜇𝜈𝑅_𝜇𝜈 = -𝑅₀₀ - 𝑔^𝑖𝑗𝑅_𝑖𝑗 = 3/𝑐² (𝑑𝑎̇/𝑑𝑡)/𝑎 + 𝑔^𝑖𝑗𝑔_𝑖𝑗 1/𝑐²[(𝑑𝑎̇/𝑑𝑡)/𝑎 + 2(𝑎̇/𝑎)² + 2 𝑘𝑐²/𝑅₀²] = 𝛿^𝑖_𝑖[(𝑑𝑎̇/𝑑𝑡)/𝑎 + 2(𝑎̇/𝑎)² + 2 𝑘𝑐²/𝑅₀²] = 6/𝑐² [(𝑑𝑎̇/𝑑𝑡)/𝑎 + (𝑎̇/𝑎)² + 𝑘𝑐²/𝑅₀²],
and the nonzero components of the Einstein tensor, 𝐺^𝜇_𝜈 ≡ 𝑔^𝜇𝜆𝐺_𝜆𝜈, are
    (2.134)  𝐺⁰₀ = -3/𝑐² [(ȧ/𝑎)² + 𝑘𝑐²/𝑎²𝑅₀²],
Derivation   From (2.119) 𝐺⁰₀ = 𝑔⁰⁰𝐺₀₀ = -𝐺₀₀.   -𝐺₀₀= -𝑅₀₀ + 1/2 𝑅𝑔₀₀ = 3/𝑐² (𝑑𝑎̇/𝑑𝑡)/𝑎 + 1/2 6/𝑐² [(𝑑𝑎̇/𝑑𝑡)/𝑎 + (𝑎̇/𝑎)² + 𝑘𝑐²/𝑅₀²](-1) = -3/𝑐² [(𝑎̇/𝑎)² + 𝑘𝑐²/𝑎²𝑅₀²]. ▮
    (2.135)   𝐺^𝑖_𝑗 = -1/𝑐² [2(𝑑𝑎̇/𝑑𝑡)/𝑎 + (𝑎̇/𝑎)² + 𝑘𝑐²/𝑎²𝑅₀²]𝛿^𝑖_𝑗
Derivation   𝑔^𝑖𝜆𝐺_𝜆𝑗 = 𝐺^𝑖_𝑗.  From (2.122) and (2.133)  𝐺_𝑖𝑗 = 𝑅_𝑖𝑗 -1/2 𝑅𝑔_𝑖𝑗 = 1/𝑐² [(𝑑𝑎̇/𝑑𝑡)/𝑎 + 2 (𝑎̇/𝑎)² + 2 𝑘𝑐²/𝑎²𝑅₀²] 𝑔_𝑖𝑗 - 1/2 6/𝑐² [(𝑑𝑎̇/𝑑𝑡)/𝑎 + (𝑎̇/𝑎)² + 𝑘𝑐²/𝑅₀²]𝑔_𝑖𝑗 = -1/𝑐² [2(𝑑𝑎̇/𝑑𝑡)/𝑎 + (𝑎̇/𝑎)² + 𝑘𝑐²/𝑎²𝑅₀²] 𝑔_𝑖𝑗,  ⇒   𝐺^𝑖_𝑗 = -1/𝑐² [2(𝑑𝑎̇/𝑑𝑡)/𝑎 + (𝑎̇/𝑎)² + 𝑘𝑐²/𝑎²𝑅₀²]𝛿^𝑖_𝑗. ▮

          2.3.5 Friedmann Equations  
   Setting 𝐺⁰₀ equal to (8π𝐺/𝑐⁴)𝛵⁰₀ = 8π𝐺/𝑐² (-𝜌𝑐²) = -(8π𝐺/𝑐²)𝜌, [Re (2.99)] we get Friedmann equation [-3/𝑐² [(𝑎̇/𝑎)² + 𝑘𝑐²/𝑎²𝑅₀²] = -(8π𝐺/𝑐²)𝜌, (𝑎̇/𝑎)² + 𝑘𝑐²/𝑎²𝑅₀² = (8π𝐺/𝑐² 𝑐²/3)𝜌]
    (2.136)   (𝑎̇/𝑎)² = 8π𝐺/3 𝜌 - 𝑘𝑐²/𝑎²𝑅₀² ( + 𝛬𝑐²/3),
where 𝜌 should be understood as the sum of all contributions to the energy density of the universe. We will write 𝜌_𝑟 for the contribution from radiation (with 𝜌_𝛾 for photons and 𝜌_𝜈 for neutrinos), 𝜌_𝑚 for the contribution by matter (with 𝜌_𝑐 for cold dark matter and 𝜌_𝑏 for baryons) and 𝜌_𝛬 for vacuum energy contribution. Equation (2.136) is the fundamental equation describing the evolution of the scale factor. To write it as a closed form equation for 𝑎(𝑡), we have to specify the evolution of the density, 𝜌(𝑎), as in (2.107), 𝜌 ∝ 𝑎^{-3(1 + 𝜔)}.
   The spatial part of the Einstein equation, 𝐺^𝑖_𝑗 = (8π𝐺/𝑐⁴)𝛵^𝑖_𝑗, leads to the second Friedmann equation (also known as the Raychaudhuri equation).
    (2.137)   (𝑑𝑎̇/𝑑𝑡)/𝑎 = -4π𝐺/3 (𝜌 + 3𝑃/𝑐²) ( + 𝛬𝑐²/3), 
Derivation   Taking a time derivative of the first equation (2.136) and using the continuity equation (2.106) ῤ = -3𝑎̇/𝑎 (𝜌 + 𝑃/𝑐²),
       𝑑/𝑑𝑡 (𝑎̇/𝑎)² = 𝑑/𝑑𝑡 (8π𝐺/3 𝜌 - 𝑘𝑐²/𝑎²𝑅₀²),   2/𝑎³ (𝑎𝑎̇ 𝑑𝑎̇/𝑑𝑡 - 𝑎̇³) = 8π𝐺/3 ῤ - (-2𝑎̇/𝑎³ 𝑘𝑐²/𝑅₀²) = 8π𝐺/3 [-3𝑎̇/𝑎 (𝜌 + 𝑃/𝑐²)] + 2𝑎̇/𝑎³ 𝑘𝑐²/𝑅₀²
       (𝑑𝑎̇/𝑑𝑡)/𝑎 = (𝑎̇/𝑎)² + 1/2 8π𝐺/3 [-3(𝜌 + 𝑃/𝑐²)] + 𝑘𝑐²/𝑎²𝑅₀² = 8π𝐺/3 𝜌 - 𝑘𝑐²/𝑎²𝑅₀² + 4π𝐺/3 [-3(𝜌 + 𝑃/𝑐²)] + 𝑘𝑐²/𝑎²𝑅₀² = 4π𝐺/3 (𝜌 + 3𝑃/𝑐²). ▮ 

Newtonian derivation   Some intuition for the form of the Friedmann equations can be developed from a non-relativistic Newtonian analysis. Consider an expanding sphere a matter of uniform mass density 𝜌(𝑡) and radius 𝑅(𝑡) = 𝑎(𝑡)𝑅(0). Let us study the dynamics of a test particle on the surface of the sphere. The acceleration of the particle is
    (2.138)   𝑑Ṙ/𝑑𝑡 = 𝐺𝑀(𝑅)/𝑅²,   𝑀(𝑅) = 4μ/3 𝑅³ 𝜌,
where 𝑀(𝑅) is the mass enclosed in the ball of radius 𝑅. Multiplying this by 𝑅 and integrating, we get
    (2.139)   1/2 Ṙ² - 𝐺𝑀(𝑅)/𝑅 = 𝐸
where the integration constant 𝐸 is the energy per unit mass of the particle. In terms of the scale factor and the density, this becomes
    (2.140)   (𝑎̇/𝑎)² = 8π𝐺/3 𝜌 - 2𝐸/𝑎²𝑅₀².
Identifying 2𝐸 with -𝑘𝑐², this is the form of Friedmann equation for a pressureless fluid. We see that we describe the flatness problem in Chapter 4.

   The first Friedmann equation is often written in terms of Hubble parameter, 𝐻 ≡ 𝑎̇/𝑎,
    (2.141)   𝐻² = 8π𝐺/3 𝜌 - 𝑘𝑐²/𝑎²𝑅₀².
We will use the subscript '0' to denote quantity evaluated today, at 𝑡 = 𝑡₀. A flat universe (𝑘 = 0) corresponds to the following critical density today.
    (2.142)   𝜌_crit,0 = 3𝐻₀²/8π𝐺 = 1.9 × 10^-29 𝘩² g cm^-3 = 2.8 × 10¹¹ 𝘩² 𝛭_⊙ Mpc^-3 = 1.1 × 10^-5 𝘩² protons cm^-3.
It is convenient to measure all densities relative to the critical density and work with following dimensionless density parameters
    (2.143)   𝛺_𝑖,0 ≡ 𝜌_𝑖,0/𝜌_crit,0,   𝑖 = 𝑟, 𝑚, 𝛬, . . .
The density parameter 𝛺_𝑖,0 is often dropped so that 𝛺_𝑖denotes the density today in terms of the critical density today. We will follow the convention. The Friedman equation (2.141) can be then be written as
    (2.144)   𝐻²/𝐻₀² = 𝛺_𝑟𝑎^-4 + 𝛺_𝑚𝑎^-3 + 𝛺_𝑘𝑎^-2 + 𝛺_𝛬,
where we introduces the curvature "density" parameter 𝛺_𝑘, 𝛺_𝑘 ≡ -𝑘𝑐²/(𝑅₀𝐻₀)². Note that 𝛺_𝑘 < 0 for 𝑘 > 0. Evaluating both sides of the equation at the present time, with 𝑎(𝑡₀) ≡ 1, leads to the constraint
    (2.145)   1 = {𝛺_𝑟 + 𝛺_𝑚 + 𝛺_𝛬}(≡𝛺₀) + 𝛺_𝑘
and the curvature parameter can be written as 𝛺_𝑘 = 1 - 𝛺₀. An central task in cosmology is to measure the parameters occurring in the Friedmann equation (2.144) and hence determine the composition of the universe. These measurements will be described in Section 2.4.

          2.3.6 Exact Solutions  
   In general, the Friedmann equation is a complicated nonlinear differential equations for the scale factor 𝑎(𝑡). Some special cases from exact solutions are good approximations to the real universe for certain periods of time

          Single-component universes    
Consider first a flat universe (𝑘 = 0) with just a single fluid. In fact, the different scalings of the densities of radiation (𝑎^-4), matter (𝑎^-3) and vacuum energy (𝑎⁰) imply that for most of its history, the universe was dominated by a single component (first radiation, then matter, then vacuum energy; see Fig. 2.10). Parameterizing this component by its equation of state, 𝜔 = 𝛲/(𝜌𝑐²) captures all cases of interest. The Friedmann equation (2.144) then reduces to
    (2.146)   [𝐻 = 𝑎̇/𝑎 =] 𝑑(ln 𝑎)/𝑑𝑡 ≈ 𝐻₀ √𝛺_𝑖 𝑎^{-3/2 (1+𝜔_𝑖)}.
Derivation  𝛺_𝑖 ≡ 𝜌_𝑖/𝜌_crit, 𝜌 ∝ 𝑎^{-3(1+𝜔_𝑖)} [RE (2.107)], From (2.144) 𝐻²/𝐻₀² = 𝛺_𝑟𝑎^-4 + 𝛺_𝑚𝑎^-3 + 𝛺_𝑘𝑎^-2 + 𝛺_𝛬 ≈ 𝛺_𝑖 𝑎^{-3(1+𝜔_𝑖)}   ⇒   𝐻 ≈ 𝐻₀ √𝛺_𝑖 𝑎^{-3/2 (1+𝜔_𝑖)}.  ▮
Integrating this equation, we obtain the time independence of the scale factor
    (2.147)   𝑎(𝑡) ∝ { 𝑡^{2/3(1 + 𝜔_𝑖)}   𝜔_𝑖 ≠ -1   𝑡^2/3   𝛭𝐷;    𝑡^1/2    𝑅𝐷┃𝑒^{𝐻₀√𝛺_𝛬 𝑡}   𝜔_𝑖 = -1   𝐴𝐷,  
where 𝛭𝐷, 𝑅𝐷 and 𝐴𝐷 stand for matter dominated, radiation dominated and dark energy dominated, respectively.
   In conformal time, the Friedmann equation becomes [RE (2.23) 𝑑𝜂 = 𝑑𝑡/𝑎(𝑡), so 𝑑𝑡 = 𝑑𝜂 𝑎(𝑡), 𝑑(ln 𝑎)/𝑑𝑡 = 𝑑(ln 𝑎)/𝑑𝜂 𝑎 ≈ 𝐻₀ √𝛺_𝑖 𝑎^{-3/2 (1+𝜔_𝑖)}, therefore, as follows]
    (2.148)   𝑑(ln 𝑎)/𝑑𝜂 ≈ 𝐻₀ √𝛺_𝑖 𝑎^{-1/2 (1+3𝜔_𝑖)},
and we get
    (2.149)   𝑎(𝜂) ∝ { 𝜂^{2/(1 + 3𝜔_𝑖)}   𝜔_𝑖 ≠ -1   𝜂²   𝛭𝐷;    𝜂    𝑅𝐷┃(-𝜂)^-1   𝜔_𝑖 = -1   𝐴𝐷,  
which reads 𝜂², 𝜂 and (-𝜂)^-1 for a matter dominated, radiation dominated and dark energy dominated universe, respectively. The special case 𝜔_𝑖 = -1/3 corresponds to a universe dominated by spatial curvature.

• The solution for a pure matter universe is known as the Einstein-de Sitter universe
    (2.150)   𝑎(𝑡) = (𝑡/𝑡₀)^2/3.  
The solutions was first derived by Einstein and de Sitter in an influential two page paper. Equation (2.150) imply following relation between Hubble constant, 𝐻₀, and  the age of the universe, 𝑡₀:   [Time derivative of the both sides of  (2.150)  is: 𝑎̇ = 2/3 1/𝑡₀ and 𝐻₀ = 𝑎̇(𝑡₀)/𝑎(𝑡₀), If we set 𝑡 = 𝑡₀ and 𝑎(𝑡₀ =1, then we have 𝐻₀ = 𝑎̇(𝑡₀)/𝑎(𝑡₀) = 2/3 1/𝑡₀ 1/𝑎(𝑡₀ = 2/3 1/𝑡₀.]
    (2.151)   𝐻₀ = = 2/3 1/𝑡₀
Taking the observed value of the Hubble constant,  𝐻₀ ≈ 70 km s^{-1 -1}, then leads to,
    (2.152)   𝑡₀ = 2/3 𝐻₀^-1 ≈ 9 billion yrs.
This is the famous "age problem": the age of a pure matter universe ia shorter than that of the oldest stars.

•   The above results also apply to a universe without any matter and only a curvature term. The 𝑎^-2 scaling of the curvature contribution implies that we must set 𝜔_𝑘 = - 1/3. In the absence of additional fluid contributions, the Friedmann equation then admits a solution for 𝑘 = -1. In the case we get [In (2.136) if 𝜔_𝑖 =-1/3, (2.144), 𝐻²/𝐻₀² = 𝛺_𝑘𝑎^-2. 𝛺_𝑘 = 𝜌_𝑘/𝜌_crit. In (2.141) if 𝑘 = -1, then 𝜌_𝑘 > 0. In (2.147), if 𝜔_𝑖 =-1/3, then 𝑎(𝑡) ∝ 𝑡.]
       (2.153)   𝑎(𝑡) = 𝑡/𝑡₀,
which is called the Milne universe.

•  The solution for a cosmological constant, 𝜔_𝛬 = - 1, is known de Sitter space. This de Sitter solution turns out to be a good approximation to the evolution of our universe in both far past and the far future. For now it seems that  the time dependence in the de Sitter solution in (2.147) is somewhat a fake. All physical quantities-like the density 𝜌-are independent of time in the exact de Sitter space. To have any cosmological evolution requires that this time-translation invariance of de Sitter background is broken by a time-dependent form of stress-energy.

          A Singularity theorem    
With the exception of the de Sitter solution, the solutions above all have a singularity at 𝑡 = 0. In particular, the scale factor goes to zero and the density diverges as 𝜌 ∝ 𝑡². As famously proven by Hawking and Penrose {RE reference Chapter 2 (12)], this singularity is actually a generic feature of all Big Bang cosmologies.We can get a glimpse of this singularity theorem for the special case of our FRW spacetimes.
   Consider the second Friedmann equation, [RE (2.137)],
    (2.154)   (𝑑𝑎̇/𝑑𝑡)/𝑎 = -4π𝐺/3 (𝜌 + 3𝑃/𝑐²).
We will now show that this implies a singularity in the past, as long as the matter obeys the strong energy condition, 𝜌 + 3𝑃/𝑐² ≥ 0, so that 𝑑𝑎̇/𝑑𝑡 < 0 at all times. This makes sense gravity attractive for all familiar matter sources, so it shows down the expansion. We want to show that 𝑎(𝑡) must have started from zero at some point in the past. It is easiest to prove this graphically (see Fig. 2.11).
   Let us pick up some fiducial time 𝑡 = 𝑡₀ (which may be present time) and specify two conditions on the scale factor at that time
    (2.155)   𝑎(𝑡₀) = 1.     𝑎̇(𝑡₀) = 𝐻₀.   [𝐻 ≡ 𝑎̇/𝑎, 𝐻₀ = 𝑎̇(𝑡₀)/𝑎(𝑡₀) = 𝑎̇(𝑡₀).]
We assume that the universe is expanding at 𝑡₀, so that 𝐻₀ > 0. we use (2.154) to evolve the universe backwards in time. Consider first the special case 𝑑𝑎̇/𝑑𝑡 = 0,. The solution is 𝑎(𝑡) = 1 + 𝐻₀(𝑡 - 𝑡₀). (see the dashed line in Fig 2.11). Now let us compare the special case 𝑑𝑎̇/𝑑𝑡 = 0 to solution with 𝑑𝑎̇/𝑑𝑡 < 0. It has the same tangent at 𝑡₀, but smaller 𝑎(𝑡) at 𝑡 < 𝑡₀. The solid line in the figure shows an example. Any solution with 𝑑𝑎̇/𝑑𝑡 < 0, for all  𝑡 < 𝑡₀, therefore also has a singularity at some 𝑡_𝑖>𝑡₀ - 𝐻₀^-1. [In the figure if 𝑎(𝑡) = 0, then 0 = 1 + 𝐻₀(𝑡 - 𝑡₀), -𝐻₀^-1 = 𝑡 - 𝑡₀, therefore 𝑡 = 𝑡₀ - 𝐻₀^-1.]
   This singularity theorem assumed the FRW solution. The singularity theorem by Hawking and Penrose is more general, but also much more complicated to prove.

          Two-component universes    
It will also be convenient to have solutions for the case of two fluid components. This is relevant in the early universe of a mixture of matter and radiation, in the late universe, where both matter and dark energy are important.
   We often work in conformal time 𝜂 to obtain a solution for the evolution of a scale factor. We will use primes to denote derivative with respect to 𝜂. Using  𝑎̇ = 𝑎ʹ/𝑎 and 𝑑𝑎̇/𝑑𝑡 = 𝑎ʺ/𝑎² - (𝑎ʹ)²/𝑎³. the Friedmann equations (2.136) and (2.137) then are [In (2.23) 𝑑𝜂 = 𝑑𝑡/𝑎, we find 𝑎̇ = 𝑑𝑎/𝑑𝑡 = 𝑑𝑎/𝑎𝑑𝜂 = 𝑎ʹ/𝑎 and we can get 𝑑𝑎̇/𝑑𝑡 from taking a time derivative of the both side. ... more ...]
    (2.156)   (𝑎ʹ)² + 𝑘𝑐²/𝑅₀² 𝑎² = 8π𝐺/3 𝜌𝑎⁴,
    (2.157)   𝑎ʺ + 𝑘𝑐²/𝑅₀² 𝑎 = 4π𝐺/3 (𝜌 - 3𝑃/𝑐²) 𝑎³.

• Matter and radiation 
Consider a flat universe filled with a mixture of matter and radiation. The total density can be written as
    (2.158)   𝜌 ≡ 𝜌_𝑚 + 𝜌_𝑟 = 𝜌_eq/2 [(𝑎_eq/𝑎)³ + (𝑎_eq/𝑎)⁴],
where the subscript 'eq' denotes quantities evaluated at matter-radiation equality. Since 𝜌_𝑚𝑎³ =const = 1/2 𝜌_𝑚𝑎_eq³ and 𝑘 = 0, we can write (2.157) as
    (2.159)   𝑎ʺ = 2π𝐺/3 𝜌_eq𝑎_eq³.  
This equation has the following solution
    (2.160)   𝑎(𝜂) = π𝐺/3 𝜌_eq𝑎_eq³ 𝜂² + 𝐶𝜂 + 𝐷,  
where 𝐶 and 𝐷 are integration constants. Using 𝑎(𝜂=0) ≡ 0, we have 𝐷 = 0. Substituting (2.160) into (2.156) then gives
    (2.161)   𝐶 = (4π𝐺/3 𝜌_eq𝑎_eq⁴)^1/2,
and the solution (2.160) can be written as
    (2.162)   𝑎(𝜂) = 𝑎_eq[(𝜂/𝜂_*)²  + 2 (𝜂/𝜂_*)], 
where
    (2.163)   𝜂_* ≡ (π𝐺/3 𝜌_eq𝑎_eq²)^-1/2 = 𝜂_eq/(√2 - 1).  
For 𝜂 ≪ 𝜂_eq, we recover the radiation-dominated limit, 𝑎 ∝ 𝜂, while for 𝜂 ≫ 𝜂_eq, we agree with the matter-dominated limit, 𝑎 ∝ 𝜂².

• Matter and curvature
A historically important solution is universe containing a mixture of matter and spatial curvature. Before the discovery of dark energy, it was believed that pressureless matter would dominate at late times. In that case, the fate of the universe would be determined by its spatial curvature. A positively-curved universe would recollapse and end in Big Crunch, while a negative-curved universe would expand forever. Let us derive these facts.
   Defining the conformal Hubble parameter as 𝓗 ≡ 𝑎ʹ/𝑎, the two Friedman equations (2.156) and (2.157) can be written as  
[Corollary   𝓗 = 𝐻𝑎.     ⇐     𝐻 = 𝑎̇/𝑎,  𝑑𝜂 = 𝑑𝑡/𝑎(𝑡),  𝓗 ≡ 𝑎ʹ/𝑎 = 𝑑𝑎/𝑑𝜂 (𝑑𝜂/𝑑𝑡 𝑑𝑡/𝑑𝜂) 1/𝑎 = 𝐻𝑎. ▮]
    (2.164)   𝓗² + 𝑘𝑐²/𝑅₀² = 8π𝐺/3 𝜌_𝑚𝑎²,  
    (2.165)   𝑑𝓗/𝑑𝜂 = 4π𝐺/3 𝜌_𝑚𝑎².  
Combing (2.165) and (2.164) allows us to eliminate the dependence on 𝜌_𝑚, and we find
    (2.166)   2 𝑑𝓗/𝑑𝜂 + 𝓗² + 𝑘𝑐²/𝑅₀² = 0.  
introducing the dimensionless time coordinate 𝜃 ≡ 𝑐𝜂/𝑅₀ and the rescaled Hubble parameter 𝓗^* ≡ 𝑅₀𝓗/𝑐, this becomes
    (2.167)   2 𝑑Ȟ/𝑑𝜃 + 𝓗^*2 + 𝑘 = 0.  
The solution of this equation are  
    (2.168)   𝓗^*(𝜃) = 1/𝑎 𝑑𝑎/𝑑𝜃 = { cot(𝜃/2)   𝑘 = +1,    𝜃/2    𝑘 = 0,    coth(𝜃/2)   𝑘 = -1,
which is easily integrated to give the evolution of the scale factor
    (2.169)   𝑎(𝜃) = 𝛢 { sin²(𝜃/2)   𝑘 = +1,     𝜃²    𝑘 = 0,     sinh²(𝜃/2)   𝑘 = -1,
where 𝛢 is an integration constant. At early times, all solutions scale as 𝜃² ∝ 𝜂², as required for a matter-dominated universe. Using the solution for the scale factor, we obtain the following relation between the conformal and physical time coordinates
    (2.170)   𝑡 = 𝑅₀/𝑐 ∫ 𝑎(𝜃) 𝑑𝜃 = 𝑅₀𝛢/2𝑐 { 𝜃 - sin(𝜃)   𝑘 = +1,     𝜃³    𝑘 = 0,     sinh(𝜃) - 𝜃   𝑘 = -1.
Equation (2.169) and (2.170) then determine 𝑎(𝑡) in parametric form. A plot of the solutions is shown in Fig. 2.12. as mentioned, a universe with positive curvature recollapse in the future.
   The maximal expansion of the positive-curvature universe can be found from the Friedmann equation
    (2.171)   𝐻²/𝐻₀² = 𝛺₀/𝑎³ + (1 - 𝛺₀)/𝑎².
At the turn-around point, the expansion rate must vanish, 𝐻(𝑡) = 0, which occurs when
    (2.172)   𝑎_max = 𝛺₀/(𝛺₀ - 1).
This also fixes the integration constant in (2.169) and (2.170) to be 𝛢 = 𝛺₀/∣𝛺₀ - 1∣. Explaining why our universe lives so close to the boundary separating recollapse and eternal expansion, is called the flatness problem (see Chapter 4).

• Cosmological constant and curvature
Next, we consider a universe with cosmological constant and spatial curvature. The Friedman equation is  
    (2.173)   𝐻² = 𝛬𝑐³/3 - 𝑘𝑐²/𝑎²𝑅₀².
For 𝛬 > 0, this has the following solutions   ^{[verification. needed]}
    (2.174)   𝑎(𝑡) = 𝛢 { cosh(𝛼𝑡)   𝑘 = +1,    exp(𝛼𝑡)   𝑘 = 0,     sinh(𝛼𝑡)   𝑘 = -1,
where 𝛼 ≡ √(𝛬𝑐²/3). The normalization 𝛢 equals √(3/𝛬)/𝑅₀ for the 𝑘 = ±1 solutions, and is arbitrary for the 𝑘 = 0 solution. These solutions are shown in Fig. 2.13. We see that the 𝑘 = +1 solution doesn't have a singularity, while the scale factor seems to vanish at 𝑡 = -∞ and 𝑡 = 0, for the 𝑘 = 0 and 𝑘 = -1 solutions, respectively. The latter singularities, however, are coordinate artifacts. In fact the three solutions represent different slicings of the same de Sitter space.⁷
   For 𝛬 < 0, solutions only exist for 𝑘 = -1. The solution of Friedmann equation is
    (2.175)  𝑎(𝑡) = 𝛢 sin(𝛼𝑡),
where  𝛼 ≡ √(-𝛬𝑐²/3). The so called anti-de Sitter space plays an important role as a playground for toy models of quantum gravity [13].

• Matter and cosmological constant
Finally, we look at a flat universe with matter and a positive cosmological constant, which is a good approximation to our universe today. The Friedman equation is
    (2.176)  𝐻²/𝐻₀² = 𝛺_𝑚/𝑎³ + 𝛺_𝛬,
where 𝛺_𝑚 + 𝛺_𝛬 = 1. This has the following solution
    (2.177)  𝑎(𝑡) = (𝛺_𝑚/𝛺_𝛬)^1/3 sinh^2/3 (3/2 𝛼𝑡),  
  where 𝛼 ≡ √𝛺_𝛬 𝐻₀.
At early times, this solution reduces to 𝑎 ~ 𝑡^2/3, as expected for the early matter-dominated period. Similarly, at late times, we recover the scaling of the de Sitter solution in the flat-slicing, 𝑎 ~ exp(𝛼𝑡). Using 𝑎(𝑡₀) ≡ 1, we can solve (2.177) for the age of the universe
    (2.178)  𝑡₀ = 2/(3√ 𝛺_𝛬 𝐻₀) sinh^-1 [√(𝛺_𝛬/𝛺_𝑚)].
Derivation   𝐻₀𝑡 = ∫^𝑎₀ 𝑑𝑎/√(𝛺_𝑚𝑎^-1 + 𝛺_𝛬𝑎²) = 𝑎_𝑚𝛬^3/2/√𝛺_𝑚 ∫^𝑦₀ √𝑦/√(1 + 𝑦³) 𝑑𝑦,   𝑎_𝑚𝛬 ≡ (𝛺_𝑚/𝛺_𝛬)^1/3,   𝑦 ≡ 𝑎/𝑎_𝑚𝛬
                = 2/3 𝑎_𝑚𝛬^3/2/√𝛺_𝑚 sinh^-1(𝑦^3/2) = 2/3 (𝛺_𝑚/𝛺_𝛬)^1/2/√𝛺_𝑚 sinh^-1(𝑎^3/2/ (𝛺_𝑚/𝛺_𝛬)^1/2) = 2/3 1/√𝛺_𝛬 sinh^-1[𝑎^3/2 √(𝛺_𝛬/𝛺_𝑚)]  
          𝑡 = 2/3 1/𝐻₀ 1/√𝛺_𝛬 sinh^-1[𝑎^3/2 √(𝛺_𝛬/𝛺_𝑚)] = 2/(3√𝛺_𝛬 𝐻₀) sinh^-1(𝑎^3/2 √𝛺_𝛬/√𝛺_𝑚).   Since 𝑎(𝑡₀) = 1, we get
          𝑡₀ = 2/(3√ 𝛺_𝛬 𝐻₀) sinh^-1[√(𝛺_𝛬/𝛺_𝑚)].  ▮
Substituting the observed values of the cosmological parameter (𝛺_𝑚 = 0.32 and 𝛺_𝛬 = 0.68), we can find
    (2.179)  𝑡₀ = 0.95 𝐻₀^-1 ≈ 0.95 × 14.4 Gyrs ≈ 13.7 Gyrs. [correction: 0.96→0.95, 14→13.7]
Derivation  1/𝐻₀ ≈ 1/68 s Mpc/km ≈ 1/68 × 3.0857 × 10¹⁹ s ≈ 4.538 × 10¹⁷ s ≈ 4.538/3.154 × 10¹⁰ years ≈ 14.4 Gyrs.
           2/3 √𝛺_𝛬 sinh^-1(√𝛺_𝛬/√𝛺_𝑚) = 2/3 × 1/√0.68 sinh^-1(√0.68/√0.32) ≈ 0.95.  ▮
which solves the age problem of the Einstein-de Sitter universe; cf. (2.152).

⁷ While a matter-dominated universe the spatial curvature is determined by the energy density, for a 𝛬 dominated universe all three types of solutions exist for any given value of 𝜌_𝛬. The different solutions therefore describe the same physical spacetime in different coordinates.