**14.1 The cosmological principle** _{.}

We have good evidence that the universe is *isoptropic* on the very larges scales, to high accuarcy. If the universe has no prefered center then *isotropy* also implies *homogeneity*. We assume the *cosmological principle*, which states that *at any particular time, the universe the same from all positions in space and all directions in space at any point are equivalent.*

**14.2 Slicing and threading spacetime** _{.}

In general relativity the concept of a 'moment of time' is ambiguous and is replaced the notion of a *three-dimensional spacelike hypersurface*. To define a 'time' parameter we slice 'slice up' spacetime by introducing a series of non-intersecting spacelike hypersurfaces that are labelled by some parameter *𝑡*. This parameter then defines a universal time in that 'particular time' means a given hypersurface.

It is useful at this point to introduce the idealized concept of *fundamental observers*, who are assumed to have no motion relative to the overall *cosmological fluid* associated with the 'smeared-out' motion of all the galaxies and other matter in the universe. Whe we adopt *Weyl's postulate (1923)* asumming that the worldlines of galaxies are a bundle or *congruence* of geodesics in spacetime diverging from a point in the finite or infinitely distant past or future or both, the worldline means the timelike worldline of the fundamental observer. The set of worldlines is sometimes described as providing *threading* for the spacetime.

The hypersurfaces 𝑡 = constant may now be naturally constructed in such a way that the 4-velocity of any fundamental observer is orthogonal to the hypersurface. Thus, the surface of simultaneity of the local Lorentz frame of any such observer coincides locally with the hypersurface (see **Figure 14.1**). Each hypersurface may therefore be considered as the 'meshing together' of all the local Lorentz frames of the fundamental observers.

**14.3 Syncronous comoving coordinates** _{.}

The spacelike hypersurfaces are labelled by a parameter 𝑡, which may be taken to be the *proper time* along the worldline. The parameter 𝑡 is the called the *synchronous time coordiniate or cosmic time*. In addition, we may introduce spatial coordinates (𝑥^{1}, 𝑥^{2}, 𝑥^{3}) that are *constant* along the worldline. Thus each fundamental observer has fixed (𝑥^{1}, 𝑥^{2}, 𝑥^{3}) coordinates, and is called *comoving coordinates*. The line element takes the form

__(14.1) 𝑑𝑠 ^{2} = 𝑐^{2}𝑑𝑡^{2} - 𝑔_{𝑖𝑗}𝑑𝑥^{𝑖}𝑑𝑥^{𝑗} (𝑖,𝑗 = 1,2,3),__

where the 𝑔

Let 𝑥

(14.2) 𝑥

Since 𝑑𝑥

(14.3) [𝑢

𝑔

Using (3.3), we see that we require only that 𝛤

𝛤

which is easily shown to be zero by using the fact that 𝑔

**14.4 Homogeneity and isotropy of the universe** _{.}

The metric (3.1) does *not* yet incorporate the property that space is homogeneous and isotropic.

Let us now incorporate the postulates of homogenitey and isotropy. The (squared) spatial separation on the same hypersurface 𝑡 = constant of two nearby galaxies at coordinates (𝑥^{1}, 𝑥^{2}, 𝑥^{3}) and (𝑥^{1} + 𝛥𝑥^{1}, 𝑥^{2} + 𝛥𝑥^{2}, 𝑥^{3} + 𝛥𝑥^{3}) is

𝑑𝜎^{2} = 𝑔_{𝑖𝑗}𝛥𝑥^{𝑖}𝛥𝑥^{𝑗}.

If we consider the triangle formed by three nearby galaxies at some particular time 𝑡, then isotropy requires that the triangle at some later time must be similar to the original triangle. Moreover the homogeneity requires that he magnification factor must be *indepenent* of the position of the triangle in the 3-space.It follows that time 𝑡 can enter the 𝑔_{𝑖𝑗} only trough a common factor, so that the ratios of small distances are the same at all times, Hence the metric must take the form

__(14.4) 𝑑𝑠 ^{2} = 𝑐^{2}𝑑𝑡^{2} - 𝑆^{2}(𝑡)𝘩_{𝑖𝑗}𝑑𝑥^{𝑖}𝑑𝑥^{𝑗},__

where 𝑆(𝑡) is a time-dependent

**14.5 The maximally symmetric 3-space** _{.}

We clearly require the 3-space spanned by the spacelike coordinates (𝑥^{1}, 𝑥^{2}, 𝑥^{3}) to be homogeneous and isotropic. This leads us to study the *maximally symmetic 3-space*. A *maximally symmetric space* is specified by the *curvature K*, which is independent of the coordinates. Such *constant curvature* spaces must clearly be *homogeneous and isotropic*.

The simplest expression that satisfies the various symmetry properties and identities of 𝑅_{𝑖𝑗𝑘𝑙} and contains just 𝐾 and the metric tensor is given by

(14.5) 𝑅_{𝑖𝑗𝑘𝑙} = 𝐾(𝑔_{𝑖𝑘}𝑔_{𝑗𝑙} - 𝑔_{𝑖𝑙}𝑔_{𝑗𝑘}),
which is the definition of a maximally symmetric space.

The Ricci tensor and Ricci scalar are given as follows

𝑅_{𝑗𝑘} = 𝑔^{𝑖𝑙}𝑅_{𝑖𝑗𝑘𝑙} = 𝐾𝑔^{𝑖𝑙}(𝑔_{𝑖𝑘}𝑔_{𝑗𝑙} - 𝑔_{𝑖𝑙}𝑔_{𝑗𝑘}) = 𝐾(𝛿^{𝑙}_{𝑘}𝑔_{𝑗𝑙} - 𝛿^{𝑙}_{𝑙}𝑔_{𝑗𝑙}𝑔_{𝑗𝑘}) = 𝐾(𝑔_{𝑗𝑘} - 3𝑔_{𝑗𝑙}𝑔_{𝑗𝑘}) = -2𝐾𝑔_{𝑗𝑘}.

𝑅 = 𝑅^{𝑘}_{𝑘} = -2𝐾𝛿^{𝑘}_{𝑘} = -6𝐾.
The metric of an isotropic 3-space must depend only on the rotational invariants

𝐱 ⋅ 𝐱 ≡ 𝑟^{2}, 𝑑𝐱 ⋅ 𝑑𝐱, 𝐱 ⋅ 𝑑𝐱,
and in spherical polar coordinates (𝑟, 𝜃, 𝜙) it must take the form

𝑑𝜎^{2} = 𝐶(𝑟)(𝐱 ⋅ 𝑑𝐱)^{2} + 𝐷(𝑟)(𝑑𝐱 ⋅ 𝑑𝐱)^{2} = 𝐶(𝑟)𝑟^{2}𝑑𝑟^{2} + 𝐷(𝑟)(𝑑𝑟^{2} + 𝑟^{2}𝑑𝜃^{2} + 𝑟^{2}sin^{2}𝜃 𝑑𝜙^{2}) = 𝐵(𝑟)𝑑𝑟^{2} + 𝑟^{2}𝑑𝜃^{2} + 𝑟^{2}sin^{2}𝜃 𝑑𝜙^{2},
where 𝐵(𝑟) is an arbitrary function of 𝑟.

The above line element is identical to the space part of Schwartzschild metric. The only non-zero connection coefficients are

𝛤^{𝑟}_{𝑟𝑟} = 1/2𝐵(𝑟) 𝑑𝐵(𝑟)/𝑑𝑟, 𝛤^{𝑟}_{𝜃𝜃} = -𝑟/𝐵(𝑟), 𝛤^{𝑟}_{𝜙𝜙} = -𝑟sin^{2}𝜃/𝐵(𝑟), 𝛤^{𝜃}_{𝑟𝜃} = 𝛤^{𝜙}_{𝑟𝜙} = 1/𝑟, 𝛤^{𝜃}_{𝜙𝜙} = -sin 𝜃 cos 𝜃, 𝛤^{𝜙}_{𝜙𝜃} = cot 𝜃,

The Ricci tensor in terms of connection coefficients and non-zero components are

𝑅_{𝑖𝑗} = 𝛤^{𝑘}_{𝑖𝑘,𝑗} - 𝛤^{𝑘}_{𝑖𝑗,𝑘} + 𝛤^{𝑘}_{𝑖𝑘} 𝛤^{𝑘}_{𝑖𝑗} - 𝛤^{𝑙}_{𝑖𝑗} 𝛤^{𝑘}_{𝑙𝑘},

𝑅_{𝑟𝑟} = -1/𝑟𝐵 𝑑𝐵/𝑑𝑟, 𝑅_{𝜃𝜃} = 1/𝐵 - 1 - 𝑟/2𝐵^{2} 𝑑𝐵/𝑑𝑟, 𝑅_{𝜙𝜙} = 𝑅_{𝜃𝜃} sin^{2}𝜃

Since 𝑅_{𝑖𝑗} = -2𝐾𝑔_{𝑖𝑗}, so we get following equations

(14.6) 1/𝑟𝐵 𝑑𝐵/𝑑𝑟 = 2𝐾𝐵(𝑟), ∴ 𝐵(𝑟) = 1/(𝛢 - 𝐾𝑟^{2})

(14.7) 1+ 𝑟/2𝐵^{2} 𝑑𝐵/𝑑𝑟 - 1/𝐵 = 2𝐾𝑟^{2}. 1 - 𝛢 + 𝐾𝑟^{2} = 𝐾𝑟^{2}, ∴ 𝛢 = 1

__(14.8) 𝑑𝜎 ^{2} = 𝑑𝑟^{2}/(1 - 𝐾𝑟^{2}) + 𝑟^{2}𝑑𝜃^{2} + 𝑟^{2}sin^{2}𝜃 𝑑𝜙^{2},__

which is the line element for the maximally symmetric 3-space with one number, 𝐾, the

**14.6 The Friedmann-Robertson-Walker metric** _{.}

Combining the line element (3.4) for cosmological principle and Weyl's postulate and equation (3.8) we obtain

(14.9) 𝑑𝑠^{2} = 𝑐^{2}𝑑𝑡^{2} - 𝑆^{2}(𝑡)[𝑑𝑟^{2}/(1 - 𝐾𝑟^{2}) + 𝑟^{2}(𝑑𝜃^{2} + sin^{2}𝜃 𝑑𝜙^{2})].

If we define 𝑘 = 𝐾/∣𝐾∣ and introduce the rescaled coordinates 𝑟̄ = ∣𝐾∣^{1/2}𝑟, then (3.9) becomes

𝑑𝑠^{2} = 𝑐^{2}𝑑𝑡^{2} - 𝑆^{2}(𝑡)/∣𝐾∣[𝑑𝑟^{2}/(1 - 𝑘𝑟^{2}) + 𝑟^{2}(𝑑𝜃^{2} + sin^{2}𝜃 𝑑𝜙^{2})].

Finally, we define a rescaled scale function 𝑅(𝑡) by

𝑅(𝑡) = {𝑆(𝑡)/∣𝐾∣^{1/2} if 𝐾 ≠ 0, 𝑆(𝑡) if 𝐾 = 0}

Then, dropping the bars on the radial coorrdinates, we obtain the standard form for the *Friedmann-Robertson-Walker* (FRW) line elements,

__(14.10) 𝑑𝑠 ^{2} = 𝑐^{2}𝑑𝑡^{2} - 𝑅^{2}(𝑡)[𝑑𝑟^{2}/(1 - 𝑘𝑟^{2}) + 𝑟^{2}(𝑑𝜃^{2} + sin^{2}𝜃 𝑑𝜙^{2})],__

where 𝑘 takes the values -1, 0, or 1 depending on whether the spatial section has negative, zero or positive curvature respectively.

**14.7 Geometric properties of the FRW metric** _{.}

[Positive spatial curvature: 𝑘 = 1]

In this case, because the coefficient of 𝑑𝑟 in FRW metric becomes singular as 𝑟 → 1, we introduce a new radial coordinates 𝜒 as follows

𝑟 = sin 𝜒 ⇒ 𝑑𝑟 = cos 𝜒 𝑑𝜒 = (1 - 𝑟^{2})^{1/2}𝑑𝜒

𝑑𝜎^{2} = 𝑅^{2}[𝑑𝜒^{2} + sin^{2}𝜒 (𝑑𝜃^{2} + sin^{2}𝜃 𝑑𝜙^{2})],

Consider the 3-space as *embedded* in a four-dimensional Euclidean space with coordinates (𝑤, 𝑥, 𝑦, 𝑧) given by

𝑤 = 𝑅 cos 𝜒, 𝑥 = 𝑅 sin 𝜒 sin 𝜃 cos 𝜙, 𝑦 = 𝑅 sin 𝜒 sin 𝜃 sin 𝜙, 𝑧 = 𝑅 sin 𝜒 cos 𝜃;

𝑤^{2} + 𝑥^{2} + 𝑦^{2} + 𝑧^{2} = 𝑅^{2},

𝑑𝜎^{2} = 𝑑𝑤^{2} + 𝑑𝑥^{2} + 𝑑𝑦^{2} + 𝑑𝑧^{2} = 𝑅^{2}[𝑑𝜒^{2} + sin^{2}𝜒 (𝑑𝜃^{2} + sin^{2}𝜃 𝑑𝜙^{2})].

Consider again at a particular time 𝑡 the 2-surface given by 𝜒 = constant, whence we get 𝑤 = 𝑅 cos 𝜒, and

𝑥^{2} + 𝑦^{2} + 𝑧^{2} = 𝑅^{2} sin^{2} 𝜒

This behaviour is similar to what happen on a 2-sphere in a three-dimensional Euclidean space. When 𝜒 = constant, The surface area 𝛢 and a *finite* total volume 𝑉 are given as follows,

𝛢 = ∫^{π}_{𝜃=0}∫^{2π}_{𝜙=0} (𝑅 sin 𝜒 𝑑𝜃)(𝑅 sin 𝜒 sin 𝜃 𝑑𝜙) = 4π𝑅^{2} sin^{2}𝜒,

𝑉 = ∫^{π}_{𝜒=0}∫^{π}_{𝜃=0}∫^{2π}_{𝜙=0} (𝑅 𝑑𝜒)(𝑅 sin 𝜒 𝑑𝜃)(𝑅 sin 𝜒 sin 𝜃 𝑑𝜙) = 2π^{2}𝑅^{3},

where 𝑅 is often referred to as the 'radius of the universe'.

[Zero spatial curvature: 𝑘 = 0]

In this case, if we set 𝑟 = 𝜒, the 3-space line element is simply the ordinary three-dimensional Euclidean space as follows,

𝑑𝜎^{2} = 𝑅^{2}[𝑑𝜒^{2} + 𝜒^{2} (𝑑𝜃^{2} + sin^{2}𝜃 𝑑𝜙^{2})],

𝑥 = 𝑅𝜒 sin 𝜃 cos 𝜙, 𝑦 = 𝑅𝜒 sin 𝜃 sin 𝜙, 𝑧 = 𝑅𝜒 cos 𝜃;

𝑑𝜎^{2} = 𝑥^{2} + 𝑦^{2} + 𝑧^{2}.

[Negative spatial curvature: 𝑘 = -1]

In this case, we introduce a new radial coordinates 𝜒 given by

𝑟 = sinh 𝜒 ⇒ 𝑑𝑟 = cosh 𝜒 𝑑𝜒 = (1 + 𝑟^{2})^{1/2}𝑑𝜒

𝑑𝜎^{2} = 𝑅^{2}[𝑑𝜒^{2} + sinh^{2}𝜒 (𝑑𝜃^{2} + sin^{2}𝜃 𝑑𝜙^{2})],

which is embedded in a four-dimensonal Minkowski space with coordinates (𝑤, 𝑥, 𝑦, 𝑧) given by

𝑤 = 𝑅 cosh 𝜒, 𝑥 = 𝑅 sinh 𝜒 sin 𝜃 cos 𝜙, 𝑦 = 𝑅 sinh 𝜒 sin 𝜃 sin 𝜙, 𝑧 = 𝑅 sinh 𝜒 cos 𝜃;

𝑑𝜎^{2} = 𝑑𝑤^{2} - 𝑑𝑥^{2} - 𝑑𝑦^{2} - 𝑑𝑧^{2}

𝑤^{2} - 𝑥^{2} - 𝑦^{2} - 𝑧^{2} = 𝑅^{2},

which shows that the 3-space can be represented as a three-dimensional hyperboloid in the four-dimensional Minkowski space. The hypersurface is defined by the coordinate rnges,

0 ≤ 𝜒 ≤ ∞, 0 ≤ 𝜃 ≤ π, 0 ≤ 𝜙 ≤ 2π

The 2-surfaces 𝜒 = constant are 2-spheres with surface area

𝛢 = 4π𝑅^{2} sinh^{2}𝜒 '

which increases indefinitely as 𝜒 increases. The proper radius of such a 2-sphere is 𝑅𝜒, and so the surface area is larger than that in Euclidean space. The total volume of the space is infinite.

We can summarize the above discussion as follows

__(14.11) 𝑑𝑠 ^{2} = 𝑐^{2}𝑑𝑡^{2} - 𝑅^{2}(𝑡)[𝑑𝜒^{2} + 𝑆^{2}(𝜒)(𝑑𝜃^{2} + sin^{2}𝜃 𝑑𝜙^{2})],__

**14.8 Geodesics in the FRW metric** _{.}

In the comoving coordinate systems the 'cosmological fluid' is at rest and we now consider the motion of particles travelling with respect to the frame. It is free in the sense that it is affected only by the 'background' cosmological gravitatinal field and no other fields. This could be a projectile shot out of a galaxy or a light wave (photon) travelling through intergalactic space.

It is convenient to the express the FRW metric in the form (3.11) and write [𝑥^{𝜇}] = (𝑡, 𝜒, 𝜃, 𝜙], so that

𝑔_{00} = 𝑐^{2}, 𝑔_{11} = -𝑅^{2}(𝑡), 𝑔_{22} = -𝑅^{2}(𝑡)𝑆^{2}(𝜒), 𝑔_{33} = -𝑅^{2}(𝑡)𝑆^{2}(𝜒) sin^{2}𝜃.

The path of a particle is given by the geodesic equation. (Refer to[Islam 2002 p.46] and [Hobson Efstathiou Lasenby 2006] as follows together!)

𝑑𝑢^{𝜇}/𝑑𝜆 + 𝛤^{𝜇}_{𝜈𝜎}𝑢^{𝜈}𝑢^{𝜎},

where 𝑢^{𝜇} = 𝑑𝑥^{𝜇}/𝑑𝜆. For our present purpose it will be more useful to use rewrite the geodesic equation in the form [Hobson Efstathiou Lasenby 2006, p.81]

𝑑𝑢^{𝜇}/𝑑𝜆 = 1/2 𝑔_{𝜈𝜎,𝜇} 𝑢^{𝜈}𝑢^{𝜎},

which shows if the metric 𝑔_{𝜈𝜎} is independent of a particular coordinate 𝑥^{𝜆} then 𝑢_{𝜆} = constant, i.e. it is conserved along the geodesic.

Let us suppose that the geodesic passes through some spatial point 𝑃 of which 𝜒 = 0 and consider 𝜙-component 𝑢^{3}, 𝜃-component 𝑢^{2}, 𝑟-component 𝑢^{1} in turn.

For the 𝜙-component, since the metric is independent of 𝜙, we have 𝑑𝜇_{𝜇}/𝑑𝜆 = 0 so that 𝑢_{3} is constant along the geodesic.

𝑢_{3} = 𝑔_{33}𝑢^{3} = -𝑅^{2}(𝑡)𝑆^{2}(𝜒) sin^{2}𝜃 𝑢^{3}

so that 𝑢_{3} = 0 at the point 𝑃 where 𝜒 = 0, Thus 𝑢_{3} = 0 along the path and also we have 𝑢^{3} = 𝑑𝜙/𝑑𝜆 = 0 as well. So 𝜙 is constant along the along the geodesic,

__𝜙 = constant.__

For the 𝜃-component, we have,

(14.13) 𝑑𝑢_{2}/𝑑𝜆 = -1/2 𝑔_{𝜈𝜎,2} 𝑢^{𝜈}𝑢^{𝜎} = 0

The only component of 𝑔_{𝜈𝜎} which depends on 𝑥^{2} = 𝜃 is 𝑔_{33} and since 𝑢^{3} = 0, 𝑑𝑢_{2}/𝑑𝜆 vanishes. So 𝑢_{2} is constant along the geodesic. Again,

𝑢_{2} = 𝑔_{22}𝑢^{2} = -𝑅^{2}(𝑡)𝑆^{2}(𝜒) 𝑢^{2},

which vanishes at 𝑃(𝜒 = 0), and so 𝑢_{2} is zero along the geodesic, as 𝑢^{2}, so that

__𝜃 = constant.__

For the 𝜒-component, we have 𝑢^{2} = 𝑢^{3} = 0, while 𝑔_{00} and 𝑔_{11} are independent of 𝜒.

Thus

(14.14) 𝑑𝑢_{1}/𝑑𝜆 = -1/2 𝑔_{𝜈𝜎,1} 𝑢^{𝜈}𝑢^{𝜎}.

We have 𝑑𝑢_{1}/𝑑𝜆 = 0, so that 𝑢_{1} is constant along the geodesic. so 𝑢_{1} = 𝑔_{11}𝑢^{1} must be constant.Thus we have

(14.15) __𝑅 ^{2}(𝑡) 𝑑𝜒/𝑑𝑠 = constant.__

where we have taken the parameter 𝜆 to be the propertime 𝑠. [Islam 2002 p.47]

Finally, 𝑢

**14.9 The cosmological redshift** _{.}

Suppose that a photon is emitted at cosmic time 𝑡_{𝐸} by a comoving observer with fixed spatial coordinates (𝜒_{𝐸}, 𝜃_{𝐸}, 𝜙_{𝐸}) and received by another observer of (𝑡_{𝐸}, 𝜒_{𝐸}, 𝜃_{𝐸}, 𝜙_{𝐸}). We may take the latter to be at the origin of our spatial coordinate system.

For a photon one can choose an affine parameter such that the 4-momentum is 𝑝^{𝜇} = 𝑑𝑥^{𝜇}/𝑑𝜆. Since 𝑑𝜃 = 𝑑𝜙 = 0 along the photon geodesic or equivalently 𝑝^{2} = 𝑝^{3} = 0 and 𝑝_{1} is constant along the geodesic and the photon momentum is null, we require 𝑔^{𝜇𝜈}𝑝_{𝜇}𝑝_{𝜈} = 0, which reduces to

1/𝑐^{2} (𝑝_{0})^{2} - 1/𝑅^{2}(𝑡) (𝑝_{1})^{2} = 0,

from which we find 𝑝_{0} = 𝑐𝑝_{1}/𝑅(𝑡).

According to [Hobson Efstathiou Lasenby 2006 Appendix 9A], for an emitter and recevier with fixed spatial coordinate, the frequency shift of the photon is given, in general, by

(14.16) 𝜈_{𝑅}/𝜈_{𝐸} = 𝑝_{0}(𝑅)/𝑝_{0}(𝐸) [𝑔_{00}(𝐸)/𝑔_{00}(𝑅)]^{1/2}

For the FRW metric we have 𝑔_{00} = 𝑐^{2}, and so we find immediately that

(14.17) __1 + 𝑧 ≡ 𝜈 _{𝐸}/𝜈_{𝑅} = 𝑅(𝑡_{𝑅})/𝑅(𝑡_{𝐸}),__

where 𝑧 = (𝜆

(14.17a) ∫

Now, if the emmitter sends a second light purse at time 𝑡

∫

from which we see immediatly [Ohanian Ruffini 1994 p.567 Fig. 9.18] that

∫

Asumming 𝛿𝑡

𝛿𝑡

1 + 𝑧 ≡ 𝜈

**14.10 The Hubble and decceleration parameters** _{.}

In a common notation we denote the present cosmic time as 𝑡_{0}. If a nearby galaxy emits a photo at cosmic time 𝑡, we can write 𝑡 = 𝑡_{0} - 𝛿𝑡, where 𝛿𝑡 ≪ 𝑡_{0}. Let us expand the scale factor 𝑅(𝑡) as a power series about the present epoch 𝑡_{0} to obtain

(14.18) 𝑅(𝑡) = 𝑅[𝑡_{0} - (𝑡_{0} - 𝑡)]

= 𝑅(𝑡_{0}) - (𝑡_{0} - 𝑡)Ṙ(𝑡_{0}) + 1/2 (𝑡_{0} - 𝑡)^{2}Ȑ(𝑡_{0}) - ∙∙∙

= 𝑅(𝑡_{0})[1 - (𝑡_{0} - 𝑡)𝐻(𝑡_{0}) - 1/2 (𝑡_{0} - 𝑡)^{2}𝑞(𝑡_{0})𝐻^{2}(𝑡_{0}) - ∙∙∙ ],

where we have introduced the *Hubble parameter* 𝐻(𝑡) and the *decceleration parameter* 𝑞(𝑡). These are given by

(14.19) __𝐻(𝑡) ≡ Ṙ(𝑡)/𝑅(𝑡), 𝑞(𝑡) ≡ - Ȑ(𝑡)𝑅(𝑡)/Ṙ ^{2}(𝑡)__

where the dot corresponds to differentiation with respect to cosmic time 𝑡. The present-day values of them are usually denoted by 𝐻

𝑧 = 𝑅(𝑡

and assuming that 𝑡

(14.20) 𝑧 = (𝑡

(14.21)

Using the Taylor expansion (3.18), we can also obtain an approximate 𝜒-coordinate of the emitting galaxy from (3.17a),

𝜒 = ∫

Assuming once more that 𝑡

(14.22) 𝜒 = 𝑐𝑅

(14.23)

From the FRW metric, we see that the proper distance 𝑑 to the emitting galaxy at 𝑡

(14.24)

which is approximately valid for small 𝑧. This is

(14.25)

So far, we have been considering the low-𝑧 limit. In general, we have

𝑑𝑧 = 𝑑(1+ 𝑧) = 𝑑(𝑅

where we used quotient rule and get a useful relation between an interval in redshift and the corresponding interval in cosmic time. Thus we can write the look-back time as

(14.26)

and galaxiy's 𝜒-coordinate is given by

(14.27)

In order to evaluate these integrals we must know 𝐻(𝑧) which requires knowledge of the evolution of the scale factor 𝑅(𝑡).

**14.11 Distances in the FRW geometry** _{.}

Distance measures in an expanding universe can be confusing. What do you mean by the 'distance' to a galaxy? From the FRW metric

𝑑𝑠^{2} = 𝑐^{2}𝑑𝑡^{2} - 𝑅^{2}(𝑡)[𝑑𝜒^{2} + 𝑆^{2}(𝜒)(𝑑𝜃^{2} + sin^{2}𝜃 𝑑𝜙^{2})],

The parameter 𝜒 is sometimes referred to as the *coordinate distance*, whereas the *proper distance* to an object at some cosmic time is 𝑑 = 𝑅(𝑡)𝜒, but this cannot be measured in practice. The two most important *operationally defined* distance measures are the *luminosity distance* and the *angular diameter distance*. These distance measures form the basis for observation of the universe.

[Luminosity distance]

In an ordinary static Euclidean universe, if a source of *absolute lumnosity* 𝐿 (in 𝑊 = 𝐽/𝑠) is a distance 𝑑 then the *flux* that we received (in 𝑊/𝑚^{2}) is 𝐹 = 𝐿/(4π𝑑^{2}). In the FRW geometry, if we know the luminosity 𝐿 and we observe a flux 𝐹, then the quantity

(14.28) __𝑑 _{𝐿} = (𝐿/4π𝐹)^{1/2},__

is called the

Consider an emitting souce 𝐸 with a fixed comoving coordinate 𝜒 relative to an observer 𝑂 (the emitter would assign the

𝛢 = 4π𝑅

However, each photon received by 𝑂 is redshifted in frequency, so that

𝜈

and the arrival rate of the photons is also reduced by the sme factor. Thus, the

𝐹(𝑡

Thus the luminosity distance defined above is now

(14.29)

[Angular diameter distance]

In Euclidean spaceit were at a distance 𝑑 it would subtend an angular diameter 𝛥𝜃 = 𝐷/𝑑. in an FRW geometry, we thus define the

(14.30)

This is again an operational definition. Suppose we have two radial null geodesics (light path) meeting at the observer at time 𝑡

𝓁 = 𝑅(𝑡

(14.31)

Of course, we need to know the time history of 𝑅(𝑡) to evaluate 𝑑

**14.12 Volumes and number densities in the FRW geometry** _{.}

The interpretation of cosmological observation often requires one to determine the volume of some three-dimensional region of the FRW geometry. Consider a comoving observer at the origin 𝜒 = 0 and we see that at cosmic time 𝑡_{0}, the proper volume of the region of space lying in the infinitesimal coordinate range 𝜒 → 𝜒 + 𝑑𝜒 and subtending an infinitesimal solid angle 𝑑𝛺 = sin 𝜃 𝑑𝜃 𝑑𝜙 at the obse

rver is
𝑑𝑉_{0} = (𝑅_{0} 𝑑𝜒)[𝑅_{0}^{2}𝑆^{2}(𝜒) 𝑑𝛺] = 𝑅_{0}^{3}𝑆^{2}(𝜒) 𝑑𝜒 𝑑𝛺

.
For the interval 𝜒 → 𝜒 + 𝑑𝜒 in the radial comoving coordinate there exists a corresponding interval 𝑧 → 𝑧 + 𝑑𝑧 and 𝑡 → 𝑡 + 𝑑𝑡 within which the light observed by 𝑂 at 𝑡 = 𝑡_{0} was emitted. We may therefore write the volume element as

𝑑𝑉_{0} = 𝑅_{0}^{3}𝑆^{2}(𝜒) (𝑑𝜒/𝑑𝑧) 𝑑𝑧 𝑑𝛺.

From (14.27) we can find the equivalent of 𝑑𝜒/𝑑𝑧 and so

𝑑𝜒/𝑑𝑧 = 𝑐/𝑅_{0}𝐻(𝑧), 𝑑𝑉_{0} = 𝑐𝑅_{0}^{2}𝑆^{2}(𝜒(𝑧))/𝐻(𝑧) 𝑑𝑧 𝑑𝛺,

where we have a function of 𝑧. This volume element is illustrated in **Figure 14.5**. Finally using (14.17) we have

(14.32) __𝑑𝑉(𝑧) = 𝑑𝑉 _{0}/(1+ 𝑧)^{3} = 𝑐𝑅_{0}^{2}𝑆^{2}(𝜒(𝑧))/[(1+ 𝑧)^{3}𝐻(𝑧)] 𝑑𝑧 𝑑𝛺.__

The main use of (14.32) is in predicting the numbers of galaxies. Suppose the proper number density of galaxies at a redshift 𝑧 is given by 𝑛(𝑧) and the total number 𝑑𝛮 of such objects is

(14.33) 𝑑𝛮 = 𝑛(𝑧)𝑑𝑉(𝑧) = 𝑐𝑅

**14.13 The cosmological field equation** _{.}

The *dynamics* of spacetime geometry is characterized entirely by the scale factor 𝑅(𝑡), we must solve the gravitational field equations in the presence of matter.

The standard Einstein field equations are

𝑅_{𝜇𝜈} - 1/2 𝑔_{𝜇𝜈}𝑅 = -𝜅𝑇_{𝜇𝜈}, 𝜅 = 8π𝐺/𝑐^{4},

where in the right-hand side we may use minus sign for consistency with Newtonian theory. In fact Einstein proposed a modification known as the *cosmological term*. Because we can add *any constant multiple* of 𝑔_{𝜇𝜈} to the left-hand side of the above equation and still obtain a *consistent* set of field equations. It is usual to denote this multiple by 𝛬. So the field equation become [Hobson Efstathiou Lasenby 2006 p.185]

(14.34) 𝑅_{𝜇𝜈} = -𝜅(𝑇_{𝜇𝜈} - 1/2 𝑇𝑔_{𝜇𝜈}) + 𝛬𝑔_{𝜇𝜈}, 𝑇 = 𝑇^{𝜇}_{𝜇}.

For simplicity, we assume a *perfect flluid*, which is characterized at each point by its proper density 𝜌 and the pressure 𝑝 in the instantaneous rest frame. The energy-momentum tensor is given by [Hobson Efstathiou Lasenby 2006 p.179]

(14.35) 𝑇^{𝜇𝜈} = (𝜌 + 𝑝/𝑐^{2})𝑢^{𝜇}𝑢^{𝜈} - 𝑝𝑔^{𝜇𝜈}.
Since we are seeking solutions for a homogeneous and isotropic universe, the density 𝜌 and the pressure 𝑝 *must be funtions of cosmic time t alone*.

We may perform the calculation by adopting the comoving coordinates [𝑥^{𝜇}] = (𝑡, 𝑟, 𝜃, 𝜙) in which the FRW metric is from (3.10)

𝑑𝑠^{2} = 𝑐^{2}𝑑𝑡^{2} - 𝑅^{2}(𝑡)[𝑑𝑟^{2}/(1 - 𝑘𝑟^{2}) + 𝑟^{2}(𝑑𝜃^{2} + sin^{2}𝜃 𝑑𝜙^{2})].

Thus the covariant components 𝑔_{𝜇𝜈} of the metric are

𝑔_{00} = 𝑐^{2}, 𝑔_{11} = -𝑅^{2}(𝑡)/(1 - 𝑘𝑟^{2}), 𝑔_{22} = -𝑅^{2}(𝑡)𝑟^{2}, 𝑔_{33} = -𝑅^{2}(𝑡)𝑟^{2}sin^{2}𝜃.
Since the metric is diagonal, the contravariant components 𝑔^{𝜇𝜈} are simply the reciprocals of them.

The connection is given in terms of the metric by the following equation and from which we have only non-zero coefficients are

𝛤^{𝜎}_{𝜇𝜈} = 1/2 𝑔^{𝜎𝜌}(𝑔_{𝜇𝜈}(𝑔_{𝜌𝜇,𝜈} + 𝑔_{𝜌𝜈,𝜇 } - 𝑔_{𝜇𝜈,𝜌},

𝛤^{0}_{11} = 𝑅Ṙ/𝑐^{2}(1 - 𝑘𝑟^{2}), 𝛤^{0}_{22} = 𝑅Ṙ𝑟^{2}/𝑐^{2}, 𝛤^{0}_{33} = 𝑅Ṙ𝑟^{2} sin^{2}𝜃/𝑐^{2}

𝛤^{1}_{01} = Ṙ/𝑅, 𝛤^{1}_{11} = 𝑘𝑟/(1 - 𝑘𝑟^{2}), 𝛤^{1}_{22} = -𝑟(1 - 𝑘𝑟^{2}), 𝛤^{1}_{33} = -𝑟(1 - 𝑘𝑟^{2}) sin^{2}𝜃,

𝛤^{2}_{02} = 𝛤^{3}_{03} = Ṙ/𝑅, 𝛤^{2}_{12} = 𝛤^{3}_{13} = 1/𝑟, 𝛤^{2}_{33} = - sin 𝜃 cos 𝜃, 𝛤^{3}_{23} = cot 𝜃,

where the dot denote differentiation with respect to cosmic time 𝑡. We next substitute these expression for the Ricci tensor,

𝑅_{𝜇𝜈} = 𝛤^{𝜎}_{𝜇𝜎,𝜈} - 𝛤^{𝜎}_{𝜇𝜈,𝜎} + 𝛤^{𝜌}_{𝜇𝜎}𝛤^{𝜎}_{𝜌𝜈} - 𝛤^{𝜌}_{𝜇𝜈}𝛤^{𝜎}_{𝜌𝜎}.

𝑅_{00} = 3Ȑ/𝑅, 𝑅_{11} = -(𝑅Ȑ + 2Ṙ + 2𝑐^{2}𝑘)𝑐^{-2}/(1 - 𝑘𝑟^{2}), 𝑅_{22} = -(𝑅Ȑ + 2Ṙ + 2𝑐^{2}𝑘)𝑐^{-2}𝑟^{2}, 𝑅_{33} = -(𝑅Ȑ + 2Ṙ + 2𝑐^{2}𝑘)𝑐^{-2}𝑟^{2} sin^{2}𝜃,

In our comoving coordinate system (𝑡, 𝑟, 𝜃, 𝜙), the 4-velocity of the fluid simpy

[𝑢^{𝜇}] = (1, 0, 0, 0),

which we can write 𝑢^{𝜇} = 𝛿^{𝜇}_{0}. Thus the covariant components of the 4-velocity and the energy-momentum tensor are

𝑢_{𝜇} = 𝑔_{𝜇𝜈}𝛿^{𝜈}_{0} = 𝑔_{𝜇0} = 𝑐^{2}𝛿^{0}_{𝜇},

𝑇_{𝜇𝜈} = (𝜌𝑐^{2} + 𝑝)𝑐^{2}𝛿^{0}_{𝜇}𝛿^{0}_{𝜈} - 𝑝𝑔_{𝜇𝜈}.

𝑇 = 𝑇^{𝜇}_{𝜇} = (𝜌 + 𝑝/𝑐^{2})𝑐^{2} - 𝑝 - 𝑝𝛿^{𝜇}_{𝜇} = 𝑝𝑐^{2} - 3𝑝.

𝑇_{𝜇𝜈} - 1/2 𝑇𝑔_{𝜇𝜈} = (𝜌𝑐^{2} + 𝑝)𝑐^{2}𝛿^{0}_{𝜇}𝛿^{0}_{𝜈} - 1/2 (𝑝𝑐^{2} - 𝑝)𝑔_{𝜇𝜈}.

Including the cosmological-constant term, the field equations (3.24) vanish for 𝜇 ≠ 𝜈. The non-zero components reads

𝑅_{00} = -𝜅(𝑇_{00} - 1/2 𝑇𝑔_{00}) + 𝛬𝑔_{00} = -1/2 𝜅(𝜌𝑐^{2} + 3𝑝)𝑐^{2} + 𝛬𝑐^{2},

𝑅_{11} = -𝜅(𝑇_{11} - 1/2 𝑇𝑔_{11}) + 𝛬𝑔_{11} = -1/2 [𝜅(𝜌𝑐^{2} - 𝑝) + 𝛬]𝑅^{2}/(1 - 𝑘𝑟^{2}),

𝑅_{22} = -𝜅(𝑇_{22} - 1/2 𝑇𝑔_{22}) + 𝛬𝑔_{22} = -1/2 [𝜅(𝜌𝑐^{2} - 𝑝) + 𝛬]𝑅^{2}𝑟^{2},

𝑅_{33} = -𝜅(𝑇_{33} - 1/2 𝑇𝑔_{33}) + 𝛬𝑔_{33} = -1/2 [𝜅(𝜌𝑐^{2} - 𝑝) + 𝛬]𝑅^{2}𝑟^{2} sin^{2}𝜃,

Combining these expression with those of Ricci tensor, the field equation yield just the two independent equations

3Ȑ/𝑅 = -1/2 𝜅(𝜌𝑐^{2} + 3𝑝)𝑐^{2} + 𝛬𝑐^{2},

𝑅Ȑ + 2Ṙ + 2𝑐^{2}𝑘 = [1/2 𝜅(𝜌𝑐^{2} - 𝑝) + 𝛬]𝑐^{2}𝑅^{2}.

Finally, we arrive at the *cosmological field equation* by eliminating Ȑ from the second equation and remembering 𝜅 = 8π𝐺/𝑐^{4}

__Ȑ = -4π𝐺3 (𝜌 + 3𝑝/𝑐 ^{2})𝑅 + 1/3 𝛬𝑐^{2}𝑅.__

(14.36)

These two equations are known as the

**14.14 Equation of motion for the cosmological fluid** _{.}

We can derive one more important equation which is often useful in shortening calculations) from the fact that energy-momentum conservation requires [Islam 2002 p.52; with the value adjusted of 𝑇^{00} because of using 𝑥^{0} = 𝑐𝑡 instead of 𝑥^{0} = 𝑡]

𝑇^{𝜇𝜈}_{;𝜈} = 𝑇^{𝜇𝜈}_{,𝜈} + 𝛤^{𝜇}_{𝜈𝜎}𝑇^{𝜎𝜈} + 𝛤^{𝜈}_{𝜈𝜎}𝑇^{𝜇𝜎} = 0.

𝑇^{00} = 𝜌𝑐^{2}, 𝑇^{11} = 𝑝(1 - 𝑘𝑟^{2})/𝑅^{2}, 𝑇^{22} = 𝑝/(𝑟^{2}𝑅^{2}), 𝑇^{33} = 𝑝/(𝑟^{2} sin^{2}𝜃 𝑅^{2}),

(14.39) __ῤ + (𝜌 + 𝑝/𝑐 ^{2})3Ṙ/𝑅 = 0.__

^{※} attention notes when some rigorous derivation might be required