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Part I The Homogeneous Universe
2 The Expanding Universe
Our goal in this chapter is to derive, and then solve, the equation governing the evolution of the entire universe.
Fortunately, the coarsegrained 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 3dimensional Euclidean space, the physical distance between two points separated by the infinitesimal coordinate distances 𝑑𝑥, 𝑑𝑦 and 𝑑𝑧 is
(2.1) 𝑑𝓁^{2} = 𝑑𝑥^{2} + 𝑑𝑦^{2} + 𝑑𝑧^{2} = ¢²^{3}_{𝑖,𝑗=1} 𝛿_{𝑖𝑗}𝑑𝑥^{𝑖}𝑑𝑥^{𝑗}, where (𝑥^{1}, 𝑥^{2}, 𝑥^{3}) = (𝑥, 𝑦, 𝑧).
Here the metric is simply the Kronecker delta 𝛿_{𝑖𝑗} = diag(1, 1, 1). If we use polar coordinate,
(2.2) 𝑑𝓁^{2} = 𝑑𝑟^{2} + 𝑟^{2}𝑑𝜃^{2} + 𝑟^{2}sin^{2}𝜃𝑑𝜙^{2} ¡Õ ¢²^{3}_{𝑖,𝑗=1} 𝑔_{𝑖𝑗} 𝑑𝑥^{𝑖}𝑑𝑥^{𝑗} where (𝑥^{1}, 𝑥^{2}, 𝑥^{3}) = (𝑟, 𝜃, 𝜙).
Here the metric is, a less trivial form, 𝑔_{𝑖𝑗} = diag(1, 𝑟^{2}, 𝑟^{2}sin^{2}𝜃). Observers can use different coordinate systems will always agree on the physical distance, 𝑑𝑙 which is an invariant. Hence, the metric turns observerdependent coordinates into invariants.
A fundamental object in relativity is the spacetime metric. It turns observerdependent spacetime coordinates 𝑥^{𝜇} = (𝑐𝑡, 𝑥^{𝑖}) into the invariant line element^{1}
(2.3) 𝑑𝑠^{2} = ¢²^{3}_{𝜇,𝜈=0} 𝑔_{𝜇𝜈}𝑑𝑥^{𝜇}𝑑𝑥^{𝜈} ¡Õ 𝑔_{𝜇𝜈}𝑑𝑥^{𝜇}𝑑𝑥^{𝜈}. [𝑥^{0} ¡Õ 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) 𝑑𝑠^{2} = 𝑐^{2}𝑑𝑡 ^{2} + 𝛿_{𝑖𝑗} 𝑑𝑥^{𝑖}𝑑𝑥^{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 4vectors (and on general tensors),
(2.5) 𝐴_{𝜇} = 𝑔_{𝜇𝜈}𝐴^{𝜈} and 𝐴^{𝜇} = 𝑔^{𝜇𝜈}𝐴_{𝜈},
where 𝑔^{𝜇𝜈} is the inverse metric defined by [𝑔^{𝜇𝜈} [RE wikipedia Invertible matrix : 𝐀𝐁 = 𝐁𝐀 = 𝐈_{𝑛} where 𝐈_{𝑛} denotes 𝑛by𝑛 identity matrix.]
(2.6) 𝑔^{𝜇𝜆}𝑔_{𝜆𝜈} = 𝛿^{𝜇}_{𝜈} = {1 𝜇 = 𝜈, 0 𝜇 ¡Á 𝜈.
4vectors with a lower index, 𝐴_{𝜇}, are sometimes called covariant to distinguish them from the contravariant 4vectors, 𝐴^{𝜇}, with an upper index. Covariant and contravariant vectors can be contracted to produce an invariant scalar,
(2.7) 𝑆 ¡Õ 𝐴 ⋅ 𝐵 = 𝐴^{𝜇}𝐵_{𝜇} = 𝑔_{𝜇𝜈}𝐴^{𝜇}𝐵^{𝜇}.
The spatial homogeneity and isotropy of the universe mean that it can be represented by a timeordered sequence of 3dimensional spatial slices, each of which is homogeneous and isotropic (see Fig. 2.1). The 4dimensional element can then be written as^{2}
(2.8) 𝑑𝑠^{2} = 𝑐^{2}𝑑𝑡^{2} + 𝑎^{2}(𝑡)𝑑𝓁^{2},
where 𝑑𝓁^{2} ¡Õ 𝛾_{𝑖𝑗}(𝑥^{𝑘})𝑑𝑥^{𝑖}𝑑𝑥^{𝑗} is the line element on the spatial slices and 𝑎(𝑡) is the scale factor, which describes the expansion of the universe. [RE wikipedia Scale factor : In expanding or contracting universe, 𝑑(𝑡) = 𝑎(𝑡)𝑑_{0}, where 𝑑(𝑡) is the proper distance at time 𝑡, 𝑑_{0} is the distance at the reference time 𝑡_{0},𝑑(𝑡_{0}), usually also referred to as comoving distance, and 𝑎(𝑡) is the scale factor. Thus, by definition, 𝑎(𝑡_{0}) = 1]
2.1.2 Symmetric Threespaces _{}^{}
Homogeneous and isotropic 3spaces must have constant intrinsic curvature.Then the curvature can be zero, positive or negative.
• Flat space: The simplest possibility is 3dimensional Euclidean space 𝐸^{3}. This is the space in which parallel lines don't intersect. Its line element is
(2.9) 𝑑𝓁^{2} = 𝑑𝐱^{2} = 𝛿_{𝑖𝑗}𝑑𝑥^{𝑖}𝑑𝑥^{𝑗},
which is clearly invariant under spatial translations 𝑥^{𝑖} ↦ 𝑥^{𝑖} + 𝑎^{𝑖} and rotations 𝑥^{𝑖} ↦ 𝑅^{𝑖}_{𝑘}𝑥^{𝑘}, with 𝛿_{𝑖𝑗}𝑅^{𝑖}_{𝑘}𝑅^{𝑗}_{𝑙} = 𝛿_{𝑘𝑙}.^{[how?]}
• Spherical space: The next possibility is a 3space with constant positive curvature. On such a space, parallel lines will eventually meet. This geometry can be represented as a 3sphere 𝑆^{3} embedded in 4dimensional Euclidean space 𝐸^{4}:
(2.10) 𝑑𝓁^{2} = 𝑑𝐱^{2} + 𝑑𝑢^{2}, 𝐱^{2} + 𝑢^{2} = 𝑅_{0}^{2},
where 𝑅_{0} is the radius of the sphere. Homogeneity and isotropy on the surface of the 3sphere are inherited from the symmetry of the line element under 4dimensional rotations.
• Hyperbolic space: We can have a 3space with constant negative curvature. On such a space parallel lines diverge. This geometry can be represented as a hyperboloid 𝐻^{3} embedded in 4dimensional Lorentzian space 𝓡^{1,3}:
(2.11) 𝑑𝓁^{2} = 𝑑𝐱^{2}  𝑑𝑢^{2}, 𝐱^{2}  𝑢^{2} = 𝑅_{0}^{2},
where 𝑅_{0}^{2} > 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 4dimensional pseudorotations  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) 𝑑𝓁^{2} = 𝑑𝐱^{2} ¡¾ 𝑑𝑢^{2}, 𝐱^{2} ¡¾ 𝑢^{2} = ¡¾𝑅_{0}^{2}.
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 [𝑑𝑢^{2}/𝑑𝑢 = 𝑑(𝑅_{0}^{2}  𝐱^{2})/𝑑𝐱 𝑑𝐱/𝑑𝑢, 𝑢𝑑𝑢 = 𝐱 𝑑𝐱, 𝑑𝑢^{2} = (𝐱 𝑑𝐱)^{2}/𝑢^{2}]
(2.13) 𝑑𝓁^{2} = 𝑑𝐱^{2} ¡¾ (𝐱 ⋅ 𝑑𝐱)^{2}/(𝑅_{0}^{2} ∓ 𝐱^{2}).
Finally, we can unify (2.13) with (2.9) by writing
(2.14) 𝑑𝓁^{2} = 𝑑𝐱^{2} + 𝑘 (𝐱 ⋅ 𝑑𝐱)^{2}/(𝑅_{0}^{2}  𝑘 𝐱^{2}), for 𝑘 ¡Õ {0 𝐸^{3}, +1 𝑆^{3}, 1 𝐻^{3}.
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) 𝑑𝐱^{2} = 𝑑𝑟^{2}+ 𝑟^{2}(𝑑𝜃^{2} + sin^{2}𝜃 𝑑𝜙), 𝐱 ⋅ 𝑑𝐱 = 𝑟𝑑𝑟, 𝐱^{2} = 𝑟^{2},
the metric in (2.14) becomes
(2.16) 𝑑𝓁^{2} = 𝑑𝑟^{2}/(1  𝑘𝑟^{2}/𝑅_{0}^{2}) + 𝑟^{2}𝑑𝛀^{2},
where 𝑑𝛀^{2} ¡Õ 𝑑𝜃^{2} + sin^{2}𝜃 𝑑𝜙 is the metric on the unit 2spheres.
2.1.3 RobertsonWalker Metric _{}^{}
Substituting (2.16) into (2.8), we obtain the RobertsonWalker metric in polar coordinates:
(2.17) 𝑑𝑠^{2} = 𝑐^{2}𝑑𝑡^{2} + 𝑎^{2}(𝑡)[𝑑𝑟^{2}/(1  𝑘𝑟^{2}/𝑅_{0}^{2}) + 𝑟^{2}𝑑𝛀^{2}],
which is sometimes also called the FriedmannRobertsonWalker (FRW) metric. [RE wikipedia FriedmannLemaȋtreRobertsonWalker 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 𝑅_{0}.
• First of all, the line element (2.17) has a rescaling symmetry
(2.18) 𝑎 ¡æ 𝜆𝑎, 𝑟 ¡æ 𝑟/𝜆, 𝑅_{0} ¡æ 𝑅_{0}/𝜆
this means that the geometry of the spacetime stays same if we simultaneously rescale 𝑎, 𝑟 and 𝑅_{0} by a constant 𝜆 as in (2.18). We can use this freedom to set the scale factor today, at 𝑡 = 𝑡_{0}, to be unity, 𝑎(𝑡_{0}) ¡Õ 1. The scale 𝑅_{0} 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} ¡Õ 𝑑𝐫_{phys}/𝑑𝑡 = 𝑑𝑎/𝑑𝑡 𝐫 + 𝑎(𝑡) 𝑑𝐫/𝑑𝑡 = 𝐻𝐫_{phys} + 𝐯_{pec},
where we have introduced the Hubble parameter
(2.20) 𝐻 ¡Õ 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} ¡Õ 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 ¡Õ 𝑑𝑟/¡î(1  𝑘𝑟^{2}/𝑅_{0}^{2}), then
(2.21) 𝑑𝑠^{2} = 𝑐^{2}𝑑𝑡^{2} + 𝑎^{2}(𝑡)[𝑑𝜒^{2} + 𝑆_{𝑘}^{2}(𝜒)𝑑𝛀^{2}],
where
(2.22) 𝑆_{𝑘}(𝜒) = 𝑅_{0} {sinh(𝜒/𝑅_{0}) 𝑘 = 1, 𝜒/𝑅_{0} 𝑘 = 0, sin(𝜒/𝑅_{0}) 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 [RE wikipedia Particle horizon : The conformal time is the amount time that light travel from where we located to the furthest observable distance. 𝜂_{0} (of today) is not yet actually passed, but it is conceptually meaningful with particle horizon.]
(2.23) 𝑑𝜂 = 𝑑𝑡/𝑎(𝑡) [RE wikipedia Particle horizon : 𝜂 = ¡ò^{𝑡}_{0} 𝑑𝑡'/𝑎(𝑡')]
so that (2.21) becomes
(2.24) 𝑑𝑠^{2} = 𝑎^{2}(𝜂)[𝑐^{2}𝑑𝜂^{2} + [𝑑𝜒^{2} + 𝑆_{𝑘}^{2}(𝜒)𝑑𝛀^{2}].
We see that the metric has now factorized into a static part and a timedependent conformal factor 𝑎(𝜂). This is convenient for studying the propagation of light, for which 𝑑𝑠^{2} = 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 2dimensional Euclidean space.
Curvilinear coordinates _{}^{}
In the absence of any forces, Newton's law simply reads 𝑑^{2}𝐱/𝑑𝑡^{2} = 0 [ẍ = 0], which in Cartesian coordinates 𝑥^{𝑖} = (𝑥, 𝑦) becomes
(2.25) 𝑑^{2}𝑥^{𝑖}/𝑑𝑡^{2} = 0.
In a general coordinate system, however, ẍ = 0 does not have to imply 𝑑^{2}𝑥^{𝑖}/𝑑𝑡^{2} = 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 = ϔ cos ϕ  2 sin ϕ ṙ ᾠ  𝑟 cos ϕ ᾠ^{2}  𝑟 sin ϕ ᾥ, 0 = ÿ = ϔ sin ϕ + 2 cos ϕ ṙ ᾠ  𝑟 sin ϕ ᾠ^{2} + 𝑟 cos ϕ ᾥ.
Solving this for ϔ and ᾥ, we find
(2.27) ϔ = 𝑟 ᾠ^{2}, ᾥ =  2/𝑟 ṙ ᾠ.
we see that ϔ ¡Á 0 and ᾢ ¡Á 0. The reason is simply that in polar coordinates the basis vector ȓ and ῷ 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 (ṙ^{2} + 𝑟^{2}ᾡ^{2}). Using EulerLagrange equation, 𝑑/𝑑𝑡 (¡Ó𝐿/¡Óẋ^{𝑘}) = ¡Ó𝐿/¡Ó𝑥^{𝑘}
[Solution] Let us start, 𝑑/𝑑𝑡 (¡Ó𝐿/¡Óṙ) = 𝑑/𝑑𝑡 𝑚/2 (2 ṙ + 0) = 𝑚ϔ, ¡Ó𝐿/¡Ó𝑟 = 𝑚/2 (0 + 2 𝑟ᾡ^{2}) = 𝑚𝑟ᾡ^{2}. ¢¡ ϔ = 𝑟ᾡ^{2}. Similarly, 𝑑/𝑑𝑡 (¡Ó𝐿/¡Óᾡ) = 𝑑/𝑑𝑡 𝑚/2 (0 + 2 𝑟^{2}ᾡ) = 𝑚 𝑑/𝑑𝑡 𝑟^{2}ᾡ = 𝑚 (𝑑𝑟^{2}/𝑑𝑡 ᾡ + 𝑟^{2}𝑑ᾡ/𝑑𝑡) = 𝑚 (2𝑟 ṙ ᾡ + 𝑟^{2} ᾥ), ¡Ó𝐿/¡Óϕ = 𝑚/2 (0 + 0) = 0, 2𝑟 ṙ ᾡ + 𝑟^{2} ᾥ = 0. ¢¡ ᾥ = 2/𝑟 ṙ ᾡ. ▮
To derive the equation of motion in an arbitrary coordinate system, which metric 𝑔_{𝑖𝑗} ¡Á 𝛿_{𝑖𝑗}, we start from the Lagrangian [RE wikipedia Lagrangian
mechanics: 𝐿 = 𝑇  𝑉, where 𝑇 and 𝑉 are kinetic and potential energy of the system, respectively.]
(2.29) 𝐿 = 𝑚/2 𝑔_{𝑖𝑗}(𝑥^{𝑘}) ẋ^{𝑖} ẋ^{𝑗}.
Substituting this into the EulerLagrangian equation, we find
(2.30) 𝑑^{2}𝑥^{𝑖}/𝑑𝑡^{2} = 𝛤^{𝑖}_{𝑎𝑏} 𝑑𝑥^{𝑎}/𝑑𝑡 𝑑𝑥^{𝑏}/𝑑𝑡,
where we have introduced the Christoffel symbol
(2.31) 𝛤^{𝑖}_{𝑎𝑏} ¡Õ 1/2 𝑔^{𝑖𝑗}(¡Ó_{𝑎}𝑔_{𝑖𝑏} + ¡Ó_{𝑏}𝑔_{𝑗𝑎}  ¡Ó_{𝑗}𝑔_{𝑎𝑏}), with ¡Ó_{𝑗} ¡Õ ¡Ó/¡Ó𝑥^{𝑗}.
Derivation The EulerLagrangian equation is
(2.32) 𝑑/𝑑𝑡 (¡Ó𝐿/¡Óẋ^{𝑘}) = ¡Ó𝐿/¡Ó𝑥^{𝑘}.
The mass 𝑚 will be cancel on both side. The lefthand side of (2.32) then becomes [¡Ó𝐿/¡Óẋ^{𝑘} = 1/2 ¡Ó(𝑔_{𝑖𝑗}ẋ^{𝑖}^{𝑗})/¡Óẋ^{𝑘} = 1/2 (𝑔_{𝑖𝑘}ẋ^{𝑖} + 𝑔_{𝑘𝑗}ẋ^{𝑗}) = 𝑔_{𝑖𝑘}ẋ^{𝑖}]
(2.33) 𝑑/𝑑𝑡 (¡Ó𝐿/¡Óẋ^{𝑘}) = 𝑑/𝑑𝑡 (𝑔_{𝑖𝑘}ẋ^{𝑖}) = 𝑔_{𝑖𝑘}ẍ^{𝑖} + 𝑑𝑥^{𝑗}/𝑑𝑡 ¡Ó𝑔_{𝑖𝑘}/¡Ó𝑥^{𝑗} ẋ^{𝑖} = 𝑔_{𝑖𝑘}ẍ^{𝑖} + ¡Ó_{𝑗}𝑔_{𝑖𝑘} ẋ^{𝑖} ẋ^{𝑗} = 𝑔_{𝑖𝑘}ẍ^{𝑖} + 1/2 (¡Ó_{𝑖}𝑔_{𝑗𝑘} + ¡Ó_{𝑗}𝑔_{𝑖𝑘}) ẋ^{𝑖} ẋ^{𝑗},
while the righthand side is
(2.34) ¡Ó𝐿/¡Ó𝑥^{𝑘} = 1/2 ¡Ó_{𝑘}𝑔_{𝑖𝑗} ẋ^{𝑖} ẋ^{𝑗}.
Combing (2.33) and (2.34) then gives
(2.35) 𝑔_{𝑖𝑘}ẍ^{𝑖} = 1/2 (¡Ó_{𝑖}𝑔_{𝑗𝑘} + ¡Ó_{𝑗}𝑔_{𝑖𝑘}  ¡Ó_{𝑘}𝑔_{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}(¡Ó_{𝑖}𝑔_{𝑗𝑘} + ¡Ó_{𝑗}𝑔_{𝑖𝑘}  ¡Ó_{𝑘}𝑔_{𝑖𝑗}) ẋ^{𝑖} ẋ^{𝑗} ¡Õ 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 [RE Appendix A]
(2.37) 𝑑^{2}𝑥^{𝜇}/𝑑𝜏^{2} = 𝛤^{𝑖}_{𝑎𝑏} 𝑑𝑥^{𝑎}/𝑑𝜏 𝑑𝑥^{𝑏}/𝑑𝜏, [RE wikipedia Fourvelocity : 𝑑𝑡 = 𝛾(𝑢)𝑑𝜏, 𝛾(𝑢) = 1/¡î(1  𝑢^{2}/𝑐^{2}), 𝑢 = ∣𝐮∣ = ¡î[(𝑢^{1})^{2} + (𝑢^{2})^{2} + (𝑢^{3})^{2}], 𝐔 = (𝑈^{0}, 𝑈^{1}, 𝑈^{2}, 𝑈^{3}), 𝑥^{0} = 𝑐𝑡, 𝑈^{0} = 𝑑𝑥^{0}/𝑑𝜏 = 𝑐 𝑑𝑡/𝑑𝜏 = 𝑐𝛾(𝑢) 𝑈^{𝑖} = 𝑑𝑥^{𝑖}/𝑑𝜏 = 𝑑𝑥^{𝑖}/𝑑𝑡 𝑑𝑡/𝑑𝜏 = 𝛾(𝑢)𝑢^{𝑖}.]
where the Christoffel symbol is
(2.38) 𝛤^{𝜇}_{𝛼𝛽} = 1/2 𝑔^{𝜇𝜆}(¡Ó_{𝛼}𝑔_{𝛽𝜆} + ¡Ó_{𝛽}𝑔_{𝛼𝜆}  ¡Ó_{𝜆}𝑔_{𝛼𝛽}), with ¡Ó_{𝛼} ¡Õ ¡Ó/¡Ó𝑥^{𝛼}.
Notice the similarity between (2.37) and (2.30).
It will be convenient to write the geodesic equation in terms of the 4momentum of the particle
(2.39) 𝛲^{𝜇} ¡Õ 𝑚 𝑑𝑥^{𝜇}/𝑑𝜏. [RE wikipedia Four momentum : 𝑝^{𝜇} = 𝑚𝑢^{𝜇}]
Using the chain rule, we have
(2.40) [𝑚 𝑑^{2}𝑥^{𝜇}/𝑑𝜏^{2} =] 𝑑/𝑑𝜏 𝛲^{𝜇}(𝑥^{𝛼}(𝜏)) = 𝑑𝑥^{𝛼}/𝑑𝜏 ¡Ó𝛲^{𝜇}/¡Ó𝑥^{𝛼} = 𝛲^{𝛼}/𝑚 ¡Ó𝛲^{𝜇}/¡Ó𝑥^{𝛼},
so that (2.37) becomes
(2.41) [𝑚^{2} 𝑑^{2}𝑥^{𝜇}/𝑑𝜏^{2} =] 𝛲^{𝛼} ¡Ó𝛲^{𝜇}/¡Ó𝑥^{𝛼} = 𝛤^{𝜇}_{𝛼𝛽} 𝛲^{𝛼}𝛲^{𝛽}.
Rearranging the expression, we can also write
(2.42) 𝛲^{𝛼}(¡Ó_{𝛼}𝛲^{𝛼} + 𝛤^{𝜇}_{𝛼𝛽}𝛲^{𝛽}) = 0.
The term in brackets is socalled covariant derivative of 4vector 𝛲^{𝜇} which we denote by 𝛻_{𝛼}𝛲^{𝜇} ¡Õ ¡Ó_{𝛼}𝛲^{𝛼} + 𝛤^{𝜇}_{𝛼𝛽}𝛲^{𝛽}. So we can write:
(2.43) 𝛲^{𝛼}𝛻_{𝛼}𝛲^{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} ¡Õ 𝑚 𝑑𝑥^{𝜇}/𝑑𝜆 as the 4momentum 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 righthand side of (2.41), we need the Christoffel symbols for the for the FRW metric,
(2.44) 𝑑𝑠^{2} = 𝑐^{2}𝑑𝑡^{2} + 𝑎^{2}(𝑡) 𝛾_{𝑖𝑗} 𝑑𝑥^{𝑗}𝑑𝑥^{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, 𝑎^{2}(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^{𝜇}_{00} = 𝛤^{0}_{0𝛽} and 𝛤^{𝜇}_{𝛼𝛽} = 𝛤^{𝜇}_{𝛽𝛼}. The only nonzero components are
(2.45) 𝛤^{0}_{𝑖𝑗} = 𝑐^{1}𝑎ȧ𝛾_{𝑖𝑗}, 𝛤^{𝑖}_{0𝑗} = 𝑐^{1}ȧ/𝑎𝛿^{𝑖}_{𝑗}, 𝛤^{𝑖}_{𝑗𝑘} = 1/2 𝛾^{𝑖𝑙}(¡Ó_{𝑗}𝛾_{𝑘𝑙} + ¡Ó_{𝑘}𝛾_{𝑗𝑙}  ¡Ó_{𝑙}𝛾_{𝑗𝑘}).
Example Let us derive 𝛤^{0}_{𝛼𝛽} for metric (2.44).
(2.46) 𝛤^{0}_{𝛼𝛽} = 1/2 𝑔^{0𝜆}(¡Ó_{𝛼}𝑔_{𝛽𝜆} + ¡Ó_{𝛽}𝑔_{𝛼𝜆}  ¡Ó_{𝜆}𝑔_{𝛼𝛽}).
The factor 𝑔^{0𝜆} vanish unless 𝜆 = 0, 𝑔^{00} = 1. Hence we have
(2.47) 𝛤^{0}_{𝛼𝛽} = 1/2 (¡Ó_{𝛼}𝑔_{𝛽0} + ¡Ó_{𝛽}𝑔_{𝛼0}  ¡Ó_{0}𝑔_{𝛼𝛽}).
The first two terms reduced to derivatives of 𝑔_{00} and 𝑔_{00} = 1, so we have
(2.48) 𝛤^{0}_{𝛼𝛽} = 1/2 ¡Ó_{0}𝑔_{𝛼𝛽}.
¡Ó_{0} means ¡Ó/¡Ó𝑥^{0} and 𝑥^{0} ¡Õ 𝑐𝑡 so ¡Ó_{0} = 𝑐^{1}¡Ó_{𝑡}. The derivative is nonzero if 𝑔_{𝑖𝑗} = 𝑎^{2}𝛾_{𝑖𝑗}. In that case we find
(2.49) 𝛤^{0}_{𝛼𝛽} = 1/2 𝑐^{1}¡Ó/¡Ó𝑡 𝑎^{2}𝛾_{𝑖𝑗} = 1/2 𝑐^{1}¡Ó/¡Ó𝑎 𝑎^{2}𝛾_{𝑖𝑗} ¡Ó𝑎/¡Ó𝑡 = 𝑐^{1}𝑎ȧ𝛾_{𝑖𝑗}. ▮
Exercise 2.2 Derive 𝛤^{𝑖}_{0𝑗} and 𝛤^{𝑖}_{𝑗𝑘} for metric (2.44).
[Solution] 𝛤^{𝑖}_{0𝑗} = 1/2 𝑔^{𝑖𝜆}(¡Ó_{0}𝑔_{𝑗𝜆} + ¡Ó_{𝑗}𝑔_{0𝜆}  ¡Ó_{𝜆}𝑔_{0𝑗}), But 𝑔_{0𝑗} = 𝑔_{0𝜆} = 0, 𝛤^{𝑖}_{0𝑗} = 1/2 𝑔^{𝑖𝜆}¡Ó_{0}𝑔_{𝑗𝜆},
In FRW metric, 𝑔^{𝑖𝜆} = 1/𝑎^{2} 𝛾^{𝑖𝜆}. And since ¡Ó_{0} = 𝑐^{1} ¡Ó_{𝑡} and 𝑔_{𝑗𝜆} = 𝑎^{2}𝛾_{𝑗𝜆}, ¡Ó_{0}𝑔_{𝑗𝜆} = 𝑐^{1} 2𝑎 ȧ 𝛾^{𝑖𝜆}. So we have
𝛤^{𝑖}_{0𝑗} = 1/2 1/𝑎^{2} 𝛾^{𝑖𝜆} 𝑐^{1} 2𝑎 ȧ 𝛾^{𝑖𝜆} = 𝑐^{1} ȧ/𝑎 𝛿^{𝑖}_{𝑗}. ▮
𝛤^{𝑖}_{𝑗𝑘} = 1/2 𝑔^{𝑖𝑙}(¡Ó_{𝑗}𝑔_{𝑘𝑙} + ¡Ó_{𝑘}𝑔_{𝑗𝑙}  ¡Ó_{𝑙}𝑔_{𝑗𝑘}),
Since 𝑔^{𝑖𝑙} = 1/𝑎^{2} 𝛾^{𝑖𝑙} and the derivative is nonzero, if 𝑗, 𝑘 and 𝑙 are spatial indices, 𝑔_{𝑗𝑙} = 𝑎^{2} 𝛾_{𝑗𝑙}, 𝑔_{𝑘𝑙} = 𝑎^{2} 𝛾_{𝑘𝑙} and 𝑔_{𝑗𝑘} = 𝑎^{2} 𝛾_{𝑗𝑘} respectively. So we have
𝛤^{𝑖}_{𝑗𝑘} = 1/2 1/𝑎^{2} 𝛾^{𝑖𝑙}(¡Ó_{𝑗} 𝑎^{2} 𝛾_{𝑘𝑙} + ¡Ó_{𝑘} 𝑎^{2} 𝛾_{𝑗𝑙}  ¡Ó_{𝑙} 𝑎^{2} 𝛾_{𝑗𝑘}) = 1/2 𝛾^{𝑖𝑙}(¡Ó_{𝑗} 𝛾_{𝑘𝑙} + ¡Ó_{𝑘} 𝛾_{𝑗𝑙}  ¡Ó_{𝑙} 𝛾_{𝑗𝑘}). ▮
The case of massless particle (like photons) will be specialized. The lefthand side of (2.41) can be written when 𝛼 = 0
(2.50) 𝛲^{0} ¡Ó𝛲^{𝜇}/¡Ó𝑥^{0} = 𝛲^{0}/𝑐 𝑑𝛲^{𝜇}/𝑑𝑡, [Notice that
𝑥^{0} ¡Õ 𝑐𝑡 and ¡Ó_{0} = 𝑐^{1}¡Ó_{𝑡}.]
(2.51) 𝛲^{0}/𝑐 𝑑𝛲^{𝜇}/𝑑𝑡 = 𝛤^{𝜇}_{𝛼𝛽} 𝛲^{𝛼}𝛲^{𝛽}.
Let us consider the 𝜇 = 0 component of the equation, Then the right term is 𝛤^{0}_{𝑖𝑗} 𝛲^{𝑖}𝛲^{𝑗}. Using 𝛲^{0} = 𝐸/𝑐 and 𝛤^{0}_{𝑖𝑗} = 𝑐^{1}𝑎ȧ𝛾_{𝑖𝑗} then we find, [RE wjkipedia photon : 𝐸^{2} = 𝑝^{2}𝑐^{2} + 𝑚^{2}𝑐^{4}, if 𝑚 = 0, then 𝐸 = 𝑝𝑐; Fourmomentum : In special relativity, 𝑝^{𝜇} = (𝐸/𝑐, 𝑝_{𝑥}, 𝑝_{𝑦}, 𝑝_{𝑧}),]
(2.52) 𝐸/𝑐^{3} 𝑑𝐸/𝑑𝑡 = 𝑐^{1}𝑎ȧ𝛾_{𝑖𝑗}𝛲^{𝑖}𝛲^{𝑗}.
For massless particles, the 4momentum 𝛲^{𝜇} = (𝐸/𝑐, 𝛲^{𝑖}) obey the constraint
(2.53) 𝑔_{𝜇𝜈}𝛲^{𝑖}𝛲^{𝜈} = 𝑐^{2}𝐸^{2} + 𝑎^{2}𝛾_{𝑖𝑗}𝛲^{𝑖}𝛲^{𝑗} = 0. [Notice that lightlike 𝑑𝑠^{2} = 0.]
This allow us to write (2.51) as
(2.54) 1/𝐸 𝑑𝐸/𝑑𝑡 = ȧ/𝑎, [¡ò (1/𝐸 𝑑𝐸/𝑑𝑡) 𝑑𝑡 = log(𝐸) + const; ¡ò (1/𝑎 𝑑𝑎/𝑑𝑡) 𝑑𝑡 = log(𝑎^{1}) + const ¢¡ 𝐸 ¡ð𝑎^{1}]
which implies that the energy of a massless particle decays with the expansion of the universe, 𝐸 ¡ð𝑎^{1}.
Exercise 2.3 Repeating the analysis for massive particle, with
(2.55) 𝑔_{𝜇𝜈}𝑃^{𝜇}𝑃^{𝜈} = 𝑚^{2}𝑐^{2}
show that the physical 3momentum, defined as 𝑝^{2} ¡Õ 𝑔_{𝑖𝑗}𝑃^{𝑖}𝑃^{𝑗}, satisfies 𝑝 ¡ð𝑎^{1}. Show that the momentum can be written as
(2.56) 𝑝 = 𝑚𝑣/¡î(1  𝑣^{2}/𝑐^{2}).
where 𝑣^{2} ¡Õ 𝑔_{𝑖𝑗}ẋ^{𝑖}ẋ^{𝑗} is the physical peculiar velocity. Since 𝑝 ¡ð𝑎^{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 4momentum 𝑃^{𝜇} = (𝐸/𝑐, 𝑃^{𝑖}) can be written as
𝐸/𝑐^{3} 𝑑𝐸/𝑑𝑡 = 𝑐^{1} ȧ/𝑎 𝑝^{2},
𝑔_{𝜇𝜈}𝑃^{𝜇}𝑃^{𝜈} = 𝑐^{2}𝐸^{2} + 𝑝^{2} = 𝑚^{2}𝑐^{2} [Take a time derivative the both side of the equation and arrange them.]
𝐸/𝑐^{3} 𝑑𝐸/𝑑𝑡 = 𝑝/𝑐 𝑑𝑝/𝑑𝑡 = 𝑐^{1} ȧ/𝑎 𝑝^{2},
ṗ/𝑝 = ȧ/𝑎.
The physical 3momentum of a massive particle therefore decreases as 𝑝 ¡ð𝑎^{1}.
From the definition of 4momentum of a massive particle, we get
𝑃^{𝑖} ¡Õ 𝑚 𝑑𝑥^{𝑖}/𝑑𝜏 = 𝑚 𝑑𝑥/𝑑𝜏 ẋ^{𝑖} = 𝑚ẋ^{𝑖}/¡î(1  𝑔_{𝑖𝑗}ẋ^{𝑖}ẋ^{𝑗}/𝑐^{2}) = 𝑚ẋ^{𝑖}/¡î(1  𝑣^{2}/𝑐^{2})
Now we have (2.56), 𝑝 = 𝑚𝑣/¡î(1  𝑣^{2}/𝑐^{2}).
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 freelypropagating 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 ¡ð𝑎^{1}, the wavelength scales as 𝜆 ¡ð𝑎. Light emitted at a time 𝑡_{1} with 𝜆_{1} will be observed at a later time 𝑡_{0} with a larger wavelength
(2.57) 𝜆_{1} = 𝑎(𝑡_{0})/𝑎(𝑡_{1}) 𝜆_{0}.
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 2dimensional line element.
(2.57) 𝑑𝑠^{2} = 𝑎^{2}(𝜂)[𝑐^{2}𝑑𝜂^{2} + 𝑑𝜒^{2}].
Since photons travel along null geodesics, with 𝑑𝑠^{2} = 0, their path is
(2.58) 𝛥𝜒(𝜂) = ¡¾𝑐𝛥𝜂,
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 𝜒𝜂 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 𝜂_{1}, the galaxy emits a signal of a short conformal duration 𝛿𝜂 (see Fig. 2.3).The light arrives at our telescopes at time 𝜂_{0} = 𝜂_{1} + 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) 𝛿𝑡_{1} = 𝑎(𝜂_{1})𝛿𝜂 and 𝛿𝑡_{0} = 𝑎(𝜂_{0})𝛿𝜂.
If 𝛿𝑡 is the period of the light wave, then the light is emitted with wavelength 𝜆_{1} = 𝑐𝛿𝑡_{1}, but is observed with wavelength 𝜆_{0} = 𝑐𝛿𝑡_{0}, so that
(2.61) 𝜆_{0}/𝜆_{1} = 𝑎(𝜂_{0})/𝑎(𝜂_{1}),
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 ¡Õ (𝜆_{0}  𝜆_{1})/𝜆_{1}.
By comparing the observed wavelengths with those measured in a laboratory on Earth, we determine the redshift. Using (2.61), and setting 𝑎(𝑡_{0}) ¡Õ 1, we can write the redshift parameter as
(2.63) 1 + 𝑧 = 1/𝑎(𝑡_{1})
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 𝑡_{0}:
(2.64) 𝑎(𝑡_{1}) = 1 + (𝑡_{1}  𝑡_{0})𝐻_{0} + ∙ ∙ ∙ ,
where 𝑡_{0}  𝑡_{1} is the lookback time and 𝐻_{0} is the Hubble constant
(2.65) 𝐻_{0} ¡Õ ȧ(𝑡_{0})/𝑎(𝑡_{0}).
Equation (2.63) then gives 𝑧 = 𝐻_{0}(𝑡_{0}  𝑡_{1}) + ∙ ∙ ∙ , For close objects, 𝑡_{0}  𝑡_{1} is simply equal to 𝑑/𝑐, so that the redshift increase linearly with distance. This linear relation is called the HubbleLemaitre law. In terms of the recession speed of the object, 𝑣 = 𝑐𝑧, it reads
(2.66) 𝑣 = 𝑐𝑧 ≈ 𝐻_{0}𝑑.
Hubble's original measurement of the velocitydistance 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 tke 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 supernovaestellar 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) = ¡ò^{𝑡0}_{𝑡1} 𝑑𝑡/𝑎(𝑡) = ¡ò^{𝑧}_{0} 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 4¥ð𝜒^{2}, and the fraction going through an area 𝐴 (e.g. the collecting area of a telescope) is 𝐴/4¥ð𝜒^{2}. The relation between the absolute luminosity and the observed flux would then be
(2.68) 𝐹 = 𝐿/4¥ð𝜒^{2} (static space).
In an expanding spacetime, this result is modified for three reasons:
1. When the light reaches the Earth, at time 𝑡_{0}, a sphere with radius 𝜒 has and area
(2.69) 4¥ð𝑎^{2}(𝑡_{0})𝑑^{2}_{𝑀},
where 𝑑_{𝑀} ¡Õ 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(𝑡_{1})/
a(𝑡_{0}) = 1/(1 + 𝑧); cf. Fig. 2.3. This reduces the observed flux by the same factor.
3. The energy 𝐸_{0} = 𝘩𝑓_{0} of the observed photons is less than the energy 𝐸_{1} = 𝘩𝑓_{1} 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/4¥ð(𝑡_{0})𝑑^{2}_{𝑀}(1 + 𝑧)^{2} ¡Õ 𝐿/4¥ð𝑑^{2}_{𝐿} .
In the second equality, we have defined the luminosity distance, 𝑑_{𝐿}, so that the relation between luminosity, flux and luminosity istance is the same as in (2.68).
Hence, we find
(2.71) 𝑑_{𝐿}(𝑧) = (1 + 𝑧)𝑑_{𝑀}(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 energydominated universe than in a matteronly 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 lookback time
(2.72) 𝑎(𝑡) = 1 + 𝐻_{0}(𝑡  𝑡_{0})  1/2 𝑞_{0}𝐻_{0}^{2}(𝑡  𝑡_{0})^{2} + ∙ ∙ ∙ ,
where we have defined
(2.73) 𝑞_{0} ¡Õ ä/𝑎𝐻^{2}∣_{𝑡=𝑡0}.
The parameter 𝑞_{0} was named as the deceleration parameter. [Re wikipedia Accelerating expansion of the universe: It was before the accelerated expansion of the universe was discovered in 1998.] Today, we know that it is negative, 𝑞_{0} ≈ 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 lookback time
(2.74) 𝑧 = 1/𝑎(𝑡_{1})  1 = 𝐻_{0}(𝑡_{0}  𝑡_{1}) +1/2 (2 + 𝑞_{0})𝐻_{0}^{2}(𝑡_{0}  𝑡_{1})^{2} + ∙ ∙ ∙ .
This can be inverted to give
(2.75) 𝐻_{0}(𝑡_{0} = 𝑧  1/2(2 + 𝑞_{0})𝑧^{2} + ∙ ∙ ∙ ,
where the higherorder terms can be ignored as long as 𝑧 < 1. Using (2.72), we can also write the comoving distance in terms of the lookback time and the redshift
(2.76) 𝜒 = 𝑐 ¡ò^{𝑡0}_{𝑡1} 𝑑𝑡/𝑎(𝑡) = ¡ò^{𝑡0}_{𝑡1} 𝑑𝑡[1 + 𝐻_{0}(𝑡  𝑡_{0}) + ∙ ∙ ∙ ] = 𝑐(𝑡  𝑡_{0}) + 1/2 𝐻_{0}/𝑐 𝑐^{2}(𝑡  𝑡_{0})^{2} + ∙ ∙ ∙ = 𝑐/𝐻_{0} [𝑧 1/2(1 + 𝑞_{0})𝑧^{2} + ∙ ∙ ∙ ].
Through (2.71) and (2.22), the determines the luminosity distance as a function of the redshift, 𝑑_{𝐿}(𝑧) . For a flat universe, with 𝑆_{𝑘}(𝜒) = 𝜒, the modified HubbleLemaître law reads
(2.77) 𝑑_{𝐿} = 𝑐/𝐻_{0}[𝑧 + 1/2(1  𝑞_{0})𝑧^{2} + ∙ ∙ ∙ ].
The value of 𝐻_{0} and 𝑞_{0} can be extracted by fitting the functional form to the observed 𝑑_{𝐿}(𝑧). The value of 𝑞_{0} 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) 𝐻_{0} ¡Õ 100𝘩 km s^{1}Mpc^{1},
where the parameter 𝘩 is used to keep track of how uncertainties in 𝐻_{0} propagate to the inferred value of other cosmological parameters. The latest supernovae measurements have found
(2.79) 𝘩 = 0.730 ¡¾ 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 ¡¾ 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 lastscattering.^{6}
Let us assume that an object is at a comoving distance 𝜒 and that the photons emitted at a time 𝑡_{1} (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 ¡ì 1 (in radian), which is true for all cosmological objects. In an expanding universe, this formula becomes
(2.82) 𝛿𝜃 = 𝐷/𝑎(𝑡_{1})𝑑_{𝛭} ¡Õ 𝑑_{𝛢} (expanding space),
where metric distance 𝑑_{𝛭} ¡Õ 𝑆_{𝑘}(𝜒). Notice that the observed angular size, 𝑑_{𝛢}, depends on distance at the time 𝑡_{1} when the light was emitted. The second equality in (2.82) has defined the angular diameter distance. In terms of the metric distance 𝑑_{𝛭} and 𝑧, the angular diameter distance is
(2.83) 𝑑_{𝛢}(𝑧) = 𝑑_{𝛭}(𝑧)/(1 + 𝑧).
Fig 2.8 shows the shows that angular diameter distance as a function of redshift in a flat matteronly universe and in a universe with spatial curvature. We see that the angular distance diameter starts decrease around 𝑧_{𝑚} ¡ 1.5. The spacetime was compressed when the light was emitted and the objects were closer to us than are today. The observed angular size therefore larger.
^{1} we use Einstein's convention where repeated indices are summed over. Our metric signature will be (, +, +, +). In this chapter he speed of light will be kept explicit, but in the rest of the book natural units with 𝑐 ¡Õ 1 will be used.
^{2} Component 𝑔_{00} is absorbed into the time coordinate, 𝑑𝑡' ¡Õ ¡î𝑔_{00}𝑑𝑡. [It's very confusing comment.]
^{3} Proper time is defined by the relation 𝑐^{2}𝑑𝜏^{2} = 𝑑𝑠^{2}. The relativistic action of a massive particle is 𝑆 = 𝑚𝑐^{2} ¡ò 𝑑𝜏, so extremising the proper time between the events is equivalent to extremising the action.
^{4} We have used the parametrization (2.24) for the radial coordinate 𝜒, so that (2.57) is conformal to a 2dimensional Minkowski space and the curvature scale 𝑅_{0} of the 3dimensional spatial slices is absorbed into the definition of 𝜒.
^{5} In the future, gravitational waves (GWs) will allow for robust measurements of luminosity distances using socalled standard sirens. The observed waveforms of GW signals determine the parameters of sources (like the masses of the objects in a binary system) and general relativity then predicts the emitted GW power. Comparing this to the observed GW amplitude provides a clean measurement of the luminosity distance.
^{6}We observed the CMB at a fixed moment in t 

