In this section using the four present-day cosmological parameters 𝐻

Remembering (14.26) of Chapter 14, if a comoving galaxy emmited a photon at cosmic time 𝑡, 'look-back time' 𝑡

(15.37) 𝑡

From the cosmological field equation (15.13), on noting that 𝑎 = 𝑅/𝑅

(15.38) 𝐻

A more convenient integral is obtained by substituting 𝑥 = (1 + 𝑧)

(15.39)

Assuming 𝛺

In cosmological model, the

𝑡

where 𝑓 is the value of the integral, which is typically a number of order unity. But it is not possible to perform the integral analytically and so one has to resort to numerical integration.

𝑡

Clearly, one requires 𝑡

Simply by setting 𝑡 = 𝑡

𝑡

**15.8 The distance-redshift relation** _{.}

We may also obtain a general expression for the comoving 𝜒-coordinate of a galaxy emitting a photon at time 𝑡 that is recieved at time 𝑡_{0} with redshift 𝑧. This is given by from (14.27)

𝜒 = ∫_{𝑡}^{𝑡0} 𝑐𝑑𝑡/𝑅(𝑡) = 𝑐/𝑅_{0} ∫_{0}^{𝑧} 𝑑𝑧/𝐻(𝑧)

.
We may substitute for 𝐻(𝑧) using the expression (15.38). Thus the 𝜒-coordinate of a comoving object with redshift is given by

(15.40) 𝜒(𝑧) = 𝑐/𝑅_{0}𝐻_{0} ∫_{0}^{𝑧} 𝑑𝑧/√{𝛺_{𝑚,0} (1 + 𝑧)^{3} + 𝛺_{𝑟,0} (1 + 𝑧)^{4} + 𝛺_{𝛬,0} + 𝛺_{𝑘,0} (1 + 𝑧)^{2}}.

As before a simpler form for the integral is obtained by making the substitution 𝑥 = (1 + 𝑧)^{-1}, which yields

(15.41) __𝜒(𝑧) = 𝑐/𝑅 _{0}𝐻_{0} ∫_{(1 + 𝑧)-1}^{1} 𝑑𝑥/√(𝛺_{𝑚,0}𝑥 + 𝛺_{𝑟,0} + 𝛺_{𝛬,0}𝑥^{4} + 𝛺_{𝑘,0}𝑥^{2})__.

From (14.29) and (14.31), the corresponding luminosity distance 𝑑

𝑑

where 𝑆(𝜒) is given by (14.12), whereas the proper distance to the object is simply 𝑑(𝑧) = 𝑅

𝑅

𝑅

which allows simple direct evaluation of 𝑑

As was the case in the previous section, for general value of 𝛺

The integral (15.41) can be evaluated in some simple cases. As an example, consider Einstein-de-sitter(EdS) model (𝛺

𝜒(𝑧) = 𝑐/𝑅

Thus, the luminosity distance in EdS model is given by

𝑑

and the angluardiameter distance by

𝑑

We note that 𝑑

What we need to use the above relation are a standard candle and a standard ruler for them to fix the values of 𝛺

**15.9 The volume-redshift relation** _{.}

We found in Section 14.12 that the proper volume in the range 𝜒 → 𝜒 + 𝑑𝜒 and subtending an infinitesimal solid angle 𝑑𝛺 = sin 𝜃 𝑑𝜃 𝑑𝜙 at the observer is

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

where a redshift 𝑧 is given by 𝑑𝑉(𝑧) = 𝑑𝑉_{0}/(1 + 𝑧)^{3}. We may now express 𝑑𝑉_{0} in terms of 𝐻_{0}, 𝛺_{𝑚,0}, 𝛺_{𝑟,0} and 𝛺_{𝛬,0}. Using (15.40), (15.38) and (15.10) for 𝜒(𝑧), 𝐻(𝑧) and 𝛺_{𝑘} respectlvely, we find that

(15.43) __𝑑𝑉 _{0} = (𝑐𝐻_{0}^{-1})^{3}/𝘩(𝑧) ∣𝛺_{𝑘,0}∣^{-1}𝑆^{2}√ ∣𝛺_{𝑘,0}∣ 𝐸(𝑧) for 𝛺_{𝑘,0} ≠ 0,__

𝘩(𝑧) ≡ 𝐻(𝑧)/𝐻

One must once again resort to numerical integration to obtain 𝑑𝑉

**15.10 Evolution of the density parameters** _{.}

It is possible to investigate of the evolution of these densities 𝛺_{𝑚}, 𝛺_{𝑟} and 𝛺_{𝛬}) as the universe expands.

From (15.5) we have the equation of 𝛺_{𝑖}(𝑡) and using Quotient rule we obtain

(15.44) 𝛺_{𝑖}(𝑡) = 8π𝐺/3𝐻^{2}(𝑡) 𝜌_{𝑖}(𝑡) ⇒ 𝑑𝛺_{𝑖}/𝑑𝑡 = 8π𝐺/3𝐻^{2}[ῤ_{𝑖} - (2Ḣ/𝐻)𝜌_{𝑖}],

where the label 𝑖 denotes '𝑚', '𝑟' or '𝛬' and dots denotes differentiation with respect to cosmic time 𝑡 as usual. From the equation of motion for the cosmological fluid (14.39) we have

ῤ_{𝑖} = -3(1 + 𝑤_{𝑖})𝐻𝜌_{𝑖},

where 𝐻 = Ṙ/𝑅 and the equation-of state parameter 𝑤_{𝑖} = 𝑝_{𝑖}/(𝜌_{𝑖}𝑐^{2}). Thus {15.44) becomes

(15.44) 𝑑𝛺_{𝑖}/𝑑𝑡 = -𝛺_{𝑖}𝐻[3(1 + 𝑤_{𝑖}) + 2Ḣ/𝐻^{2}].

We now find an expression for Ḣ in 𝑅, Ṙ or Ȑ and the deceleration parameter 𝑞

Ḣ = 𝑑/𝑑𝑡 (Ṙ/𝑅) = Ȑ/𝑅 - (Ṙ/𝑅)^{2} = Ȑ/𝑅 - 𝐻^{2}, Ḣ/𝐻^{2} = 𝑅Ȑ/Ṙ^{2} - 1 = -(𝑞 + 1).

Substituting this result into (14.45) and using (15.14), we finally obtain the neat relation

𝑑𝛺_{𝑖}/𝑑𝑡 = 𝛺_{𝑖}𝐻(𝛺_{𝑚} + 2𝛺_{𝑟} - 2𝛺_{𝛬} - 1 - 3𝑤_{𝑖}),

Setting 𝑤_{𝑖} = 0, 1/3 and -1 respectively for matter(dust), radiation and the vacuum, we thus obtain

(15.46) __𝑑𝛺 _{𝑚}/𝑑𝑡 = 𝛺_{𝑚}𝐻[(𝛺_{𝑚} -1) + 2𝛺_{𝑟} - 2𝛺_{𝛬}],__

By dividing these equations by one another, we may remove the dependence on 𝐻 and 𝑡 and hence obtain a set of copled first-order differentiable equations. Therefore, these equations define a unique trajectory that passes through some points. As an illustration, let us consider the case 𝛺

𝑑𝛺

which define a set of trajectories or 'flow line' in the (𝛺

It is worth noting that the profound effect of a non-zero cosmological constant on the evolution of the density parameters. In case of 𝛬 = 0, any slight deviation from 𝛺

**15.11 Evolution of the spatial curvature** _{.}

We may investigate directly the behavior of the spatial curvature from (15.10) as follows

(15.47) 𝛺_{𝑘} = 1 - 𝛺_{𝑚} - 𝛺_{𝑟} - 𝛺_{𝛬} = - -𝑐^{2}𝑘/(𝐻^{2}𝑅^{2}).

Differentiating the right-hand side with respect to 𝑡 and combining the derivatives (15.46), one finds that

(15.48) __𝑑𝛺 _{𝑘}/𝑑𝑡 = 2𝛺_{𝑘}𝐻𝑞 = 𝛺_{𝑘}𝐻(𝛺_{𝑚} + 2𝛺_{𝑟} - 2𝛺_{𝛬}), __

where 𝑞 is the deceleration parameter. If 𝛺

We may obtain an analytical expression for the 𝑘 as a function of redshift 𝑧. Substituting for 𝑐

𝛺

Using (15.38) for 𝐻(𝑧) then gives

From the above expression we find that even if 𝛺

**15.12 The particle horizon, event horizon and Hubble distance** _{.}

It is interesting to consider the extent of the region 'accessible' to some comoving observer at a given cosmic time 𝑡.

[Particle horizon]

Let us consider a comoving observer 𝑂 at comoving coordinates 𝜒 and a emitter 𝐸 at 𝜒_{1} which emits a photon at 𝑡_{1} that 𝑂 receives by the time 𝑡 in the condition that 𝜒 < 𝜒_{1}.

Since along the photon path 𝑑𝑠 = 𝑑𝜃 = 𝑑𝜙 = 0, from (14.11), the comoving coordinates 𝜒_{1} of the emitter 𝐸 is determined by

(15.49) 𝜒_{1} = 𝑐 ∫_{t1}^{t} 𝑑𝑡'/𝑅(𝑡').

If the integral on the right-hand side diverges as 𝑡_{1} → 0, then 𝜒_{1} can be made as large as we please by taking 𝑡_{1} sufficiently small. In this case, it is possible to receive signals emitted at sufficiently early epochs from *any* comoving particle. But if the integral converges as 𝑡_{1} → 0, then 𝜒_{1} can never exceed a certain value for given 𝑡. In this case our vision of the universe is limited by a *particle horizon*. The 𝜒-coordinate of the particle horizon is given by

(15.50) __𝜒 _{𝑝𝘩}(𝑡) = 𝑐 ∫_{0}^{t} 𝑑𝑡'/𝑅(𝑡')__

the corresponding proper distance to the particle horizon is 𝑑

On differentiating (15.50) with respect to 𝑡, we have 𝑑𝜒

We can obtain explicit expression for the paricle horizon in some cosmological models. For example, a matter-dominated model at early epoch obeys 𝑅(𝑡)/𝑅

𝑑

These proper distances are karger than 𝑐𝑡 because the universe has expanded while the photon has been travelling. Alternatively, if one has an analytical expression for 𝜒(𝑧) for some cosmological model, then 𝜒

The particle horizon for common cosmological models illustrates the

[event horizon]

Although our particle horizon grows as the cosmic time increase, in some cosmological models there could be events that we may

So 𝜒

In a similar way to define the Hubble time 𝐻

which provide a characteristic length scale for the universe. We may also define the

(15.51) 𝜒

The Hubble distance is the length scale at which general-relativistic effects become important; indeed, on length scales much less than 𝑑

𝑑

Thus, the Hubble distance is of same order as the particle horizon and so is often described simply as the 'horizon'. But in

* According to this table we should take the value 13.5 Gyr which is very close to the known value 13.8 Gyr (as of 2015).

** We can solve max{1/(1 + 𝑧) [1- (1 + 𝑧)^{-1/2}]} = 4/27 at 𝑧 = 5/4, (e.g. by using WolframAlpha with iPad).

*** In practice, our view of the universe is not by our particle horizon but by the epoch of recombination occured at 𝑧 ≈ 1500.
^{※} attention: some rigorous derivation might be required