We must continue the previous discussion further. One may use (14.36) (14.39) equations conveniently. Also equation (14.39) can be useful alternative forms as follows

𝑑(𝜌𝑅

And the density and the pressure of a fluid are related by its equation of state. In cosmology we assume that the cosmological fluid follows

𝑝 = 𝑤𝜌𝑐

where the

𝑑(𝜌𝑅

This equation has the immediate solution, which we can use from now on

(15.0 ← 14.42 & 14.44)

**15.1 Components of cosmological fluid** _{.}

We assume that the cosmological fluid consists of *three* components, matter, radiation and the vacuum, each with a different equation of state. The total equivalent mass density is simply the sum of each one,

(15.1) 𝜌(𝑡) = 𝜌_{𝑚}(𝑡) + 𝜌_{𝑟}(𝑡) + 𝜌_{𝛬}(𝑡),

where 𝑡 is the cosmic time and we shall assume that these are non-interacting. although matter and radiation did interact in the early universe. So each component has an equation of state of the form

𝑝_{𝑖} = 𝑤_{𝑖}𝜌_{𝑖}𝑐^{2},

where the equation-of-state parameter 𝑤_{𝑖}, is a constant and i labels the component. In particular, 𝑤_{𝑖} = 0 for pressureless 'dust', 𝑤_{𝑖} = 1/3 for radiation and 𝑤_{𝑖} = -1 for the vacuum. In general, if 𝑤_{𝑖} is a constant then, requiring that the weak energy condition is satisfied and that the local sound speed^{**} (𝑑𝑝/𝑑𝜌)^{1/2} is less than 𝑐, one finds that 𝜔_{𝑖} must lie in -1 ≤ 𝑤_{𝑖} ≤ 1. We now discuss each components in turn and conclude with a description of total density as the universe evolve.

[Matter]

In general, matter in the universe may come in several different forms. Indeed, observations of the large-scale structure in the universe suggest the most of he matter is in the form of non-baryonic *dark matter*, which interact electromagnetically only very weakly and hence invisible or 'dark'. Moreover, dark matter may itself come in different forms, such as *cold dark matter* (CDM) and *hot dark matter* (HDM), the naming of which is connected to whether the typical energy of the particle is non-relativistic or relativistic. So we have

𝜌_{𝑚}(𝑡) = 𝜌_{𝑏}(𝑡) + 𝜌_{𝑑𝑚}(𝑡).

But in the following discussion, we will deal only the total matter density that determines how the scale factor 𝑅(𝑡) evolves with cosmic time 𝑡. We shall also commonly assume that the matter particles have thermal energy that is much less than their rest mass energy, and the matter can be considered to be pressureless, i.e. *dust*. Thus, from (15.0), if a present-day proper density 𝜌_{𝑚}(𝑡_{o}) ≡ 𝜌_{𝑚,0}, its density of some other cosmic time 𝑡 is given by

__𝜌 _{𝑚}(𝑡) = 𝜌_{𝑚,0}[𝑅_{o}/𝑅(𝑡)]^{3} or 𝜌_{𝑚}(𝑧) = 𝜌_{𝑚,0}(1 + 𝑧)^{3},__

These expressions concur with our expectation for the behavior of the space density of dust particles in an expanding universe.

[Radiation]

Radiation includes not only photons but also other species with very small or zero rest mass, such as a neutrino, which is a small non-zero rest mass. The total equivalent mass density of radiation in the universe at some cosmic time 𝑡 may be written as follows

𝜌

. However, only the total energy density determines the behavior of the scale factor. For radiation, in general, we have 𝑤 = 1/3 and from (15.0), if the total radiation in the universe has a present-day energy density of 𝜌

It is worth nothing that, to a very good approximation, the dominant contribution to the radiation energy density of the universe is due to the photons of the cosmic microwave background (CMB). The radiation is to a very high degree of accuracy uniformly distributed throughout the universe and has a

(15.2) 𝑛(𝜈, 𝑇)𝑑𝜈 = 8π𝜈

where 𝑇 is the 'temperature' 0f the radiation. since the energy per unit frequency is simply 𝑢(𝜈, 𝑇) = 𝑛(𝜈, 𝑇)𝘩𝜈, the total equivalent mass density of the radiation is

𝜌

where 𝑎 = 4π

(15.3)

[Vacuum]

As the vacuum can be modeled as a perfect fluid having as equation of state 𝑝 = -𝜌𝑐

[Relative contributions of the components]

On combining the above results, we find

(15.4) 𝜌(𝑡) = 𝜌

Typically one wold expect radiation to dominate the total density when 𝑅(𝑡) is small. As the universe expands, the radiation energy density dies away the most quickly and matter becomes the dominent component. Finally, if the universe continues to expand then the matter density also dies away and the vacuum ultimately dominate the energy density. We conclude by that cosmologists often define the

𝑎(𝑡) ≡ 𝑅(𝑡)/𝑅

in terms of which the above results are more compactly written, since 𝑎

**15.2 Cosmological parameters** _{.}

In our very simplified model of the universe, its entire history is determined by only a few *cosmological parameters*. Indeed the three values of the equivalent mass densities , 𝜌_{𝑚}(𝑡), 𝜌_{𝑟}(𝑡) and 𝜌_{𝛬} and the Hubble parameter 𝐻(𝑡) are sufficient to determine the scale factor 𝑅(𝑡) at all cosmic time using (14.19) (14.36). So the cosmological model is entirely fixed by specifying the four quantity

𝐻_{0}, 𝜌_{𝑚,0}, 𝜌_{𝑟,0}, 𝜌_{𝛬,0}.

However, it is common to work in terms of alternative *dimensionless* quantities, called *density parameters* or simply *density* defined by

(15.5) __𝛺 _{𝑖}(𝑡) = 8π𝐺/3𝐻^{2}(𝑡) 𝜌_{𝑖}(𝑡)__.

In terms of these new dimensionless parameters, the cosmological model may thus be fixed by following the four present-day quantities

(15.6) 𝐻

A major goal of obsevational cosmology is therefore to determine these quantities for our universe. With significant advances, about 2005, we know them to an accuracy of a few percent. They are

(15.7)

where 1 Mpc = 10

We also note that cosmologists define further these density for the individual contributions to the matter and the radiation. For examples, 𝛺

(15.8) 𝛺

where we see the familiar baryonic matter density is one-sixth of the total matter density. Moreover, the majority of the baryonic matter seems not to reside in ordinary (hydrogen-burning) stars; the contribution of such stars is only 𝛺

The reason for defining densities (15.5) becomes clear when we rewrite the second of the cosmological field equations (14.36) in terms of them. Dividing this equation through by 𝑅

(15.9) 1 = 𝛺

where, for simplicity, we omitted the explicit time dependence of the variables. It is also common practice to define the

(15.10)

so that all the cosmic times 𝑡, we have the elegant relation

(15.11) 𝛺

From (15.9), we see that 𝛺

𝛺

𝛺

𝛺

The fact that the sum of the density parameters is unity means that the universe cannot evolve from one form of the FRW geometry to another. Cosmologists often also define

(15.12)

where from (15.1) and (15.50 we see that 𝛺 = 8π𝐺𝜌/(3𝐻

Finally, for flat cosmological model one require 𝛺 = 1 and it is common to describe the corresponding total equivalent mass density as the

(15.12a)

Also it is worth noting that 𝜌

**15.3 The cosmological field equations** _{.}

Recalling that 𝐻 = Ṙ/𝑅, the second cosmological field equations of (14.36) may be written

𝐻^{2} = 8π𝐺/3(𝛴_{𝑖}𝜌_{i}) - 𝑐^{2}𝑘/𝑅^{2},

where the label 𝑖 includes matter, radiation and the vacuum. From (15.4), (15.5) and (15.10), we therefore find

(15.13) __𝐻 ^{2} = 𝐻_{0}^{2}(𝛺_{𝑚,0} 𝑎^{-3} + 𝛺_{𝑟,0} 𝑎^{-4} + 𝛺_{𝛬} + 𝛺_{𝑘,0} 𝑎^{-2}).__

where 𝑎 = 𝑅/𝑅

We now turn to the first equation in (14.36). multiplying the equation through by 𝑅/Ṙ

𝑅Ȑ/Ṙ

where left-hand side is equal to minus the deaccecleration parameter 𝑞 defined in (14.19) as 𝑞(𝑡) ≡ -𝑅(𝑡)Ȑ(𝑡)/Ṙ(𝑡)

(15.14)

Also one can write this equation in terms of present-day values by using the result (15.13)

𝛺

**15.4 General dynamical behavior of the universe** _{.}

Because 𝛺_{𝑟,0} is significantly smaller than 𝛺_{𝑚,0} or 𝛺_{𝛬,0} according to the observations (15.7), it is a reasonable approximation to neglect it and parameterize a universe in terms of just 𝛺_{𝑚,0} and 𝛺_{𝛬,0}. **Figure 15.1** is a marvelous summary of the properties of FRW universe dominated by matter and vacuum energy (known as Lemaitre models). In the figure the 'open-close' line comes from (15.11) in case of zero spatial curvature

𝛺_{𝛬,0} = 1 - 𝛺_{𝑚,0}.

The 'accelerating-decelerating' line is obtained by setting 𝑞_{0} = 0 in (15.14) for 𝑡 = 𝑡_{0}, which gives

𝛺_{𝛬,0} = 1/2 𝛺_{𝑚,0}.

When we set 𝑡 = 𝑡_{*} for 𝐻(𝑡_{*}) = 0 in (15.13), we can rearrange as follows

(15.15) 𝑓(𝑎) ≡ 𝛺_{𝛬,0}𝑎^{3} + (1 - 𝛺_{𝑚,0} - 𝛺_{𝛬,0})𝑎 + 𝛺_{𝑚,0} = 0.

In the figure for the case 𝛺_{𝛬,0} < 0 from (15.14) we may deduced that Ȑ or a is always negative and Ṙ > 0 (because we observe redshifts), hence the 𝑎(𝑡) graph must be *convex* for all values of 𝑡. This means that 𝑎(𝑡) must have equaled zero at some point in the past, usually in 𝑡 = 0 and the universe must eventually recollapse. As the universe expands, the vacuum energy eventually dominates and and so we need only consider the 𝛺_{𝛬} on the right-hand side of (15.14), which will not tend to zero as the scale factor increases. Thus a cannot tend to zero and so 𝑎 → 0 at some finite cosmic time in the future.

In case of 𝛺_{𝛬,0} = 0, the equation (15.15) has the single solution 𝑎_{*} = 𝛺_{𝑚,0}/(𝛺_{𝑚,0} - 1), which is negative in the range 0 ≤ 𝛺_{𝑚,0} ≤ 1, indicating that there is no turning point. Therefore over this range, the 'expand-forever-recollapse' line is simply given by 𝛺_{𝛬,0} = 0.

Finally in case of 𝛺_{𝛬,0} > 0, 𝑓(𝑎) → ∓∞ as 𝑎 → ∓∞ and 𝑓(0) = 𝛺_{𝑚,0} ≈ 0.3. Thus 𝑓(𝑎) has a positive root, it must have a turning point in the region 𝑎 > 0. On evaluating the derivatives 𝑓'(𝑎) and 𝑓"(𝑎) with respect to 𝑎, 𝑓(𝑎) must have the general form illustrated in **Figure 15.2**. Thus we requires 𝑓(𝑎_{*}) = 𝑓'(𝑎) = 0, which yields

(15.16) 𝑎_{*} = (𝛺_{𝑚,0}/2𝛺_{𝛬,0})^{1/3}.

On substituting this expression back into (15.5) one can a equations for 𝛺_{𝛬,0} > 0, given by

(15.17) 4(1 - 𝛺_{𝑚,0} - 𝛺_{𝛬,0})^{3} + 27𝛺_{𝑚,0}^{2}𝛺_{𝛬,0} = 0

By introducing the variable 𝑥 = [𝛺_{𝛬,0}/4𝛺_{𝑚,0})]^{1/3}, we have

𝑥^{3} - 3𝑥/4 + (𝛺_{𝑚,0} - 1)/4𝛺_{𝑚,0} = 0,

Which is convenient form to finding the solutions. If we write in terms of 𝛺_{𝛬,0}, one finds the following three cases:

• 0 < 𝛺_{𝑚,0} ≤ 1/2, one positive root at

(15.18) 𝛺_{𝛬,0} = 4𝛺_{𝑚,0} cosh^{3}[1/3 cosh^{-1} (1 - 𝛺_{𝑚,0})/𝛺_{𝑚,0}];

• 0 < 𝛺_{𝑚,0} ≤ 1/2, one positive root at

(15.19) 𝛺_{𝛬,0} = 4𝛺_{𝑚,0} cosh^{3}[1/3 cos^{-1} (1 - 𝛺_{𝑚,0})/𝛺_{𝑚,0}];

• 𝛺_{𝑚,0} > 1, two positive root, larger given by (15.19) and smaller by

(15.20) 𝛺_{𝛬,0} = 4𝛺_{𝑚,0} cos^{3}[1/3 cosh^{-1} (1 - 𝛺_{𝑚,0})/𝛺_{𝑚,0}].

Moreover, from (15.16), we finds that 𝑎_{*} < 1 for (15.18, 15.19), whereas 𝑎_{*} > 1 for (15.20). the universe is expanding, if 𝑎_{*} < 1, there is a turning pint in the past and no big bang, whereas 𝑎_{*} > 1, there is a turning point in the future (i.e. recollapse).

The resulting lines of **Figure 15.1** shows some interesting features. In particular, when 𝛺_{𝛬,0} = 0, any combination of spatial geometry and eventual fate is possible. It is worth noting that the red colored region is consistent with the recent cosmological observations of (0.3, 0.7) in 𝑘 = 0, i.e. flat spatial model, These shows that the expansion of the universe to be accelerating and during 1998 the accelerating expansion was astonishingly discovered! They also require the universe to have started at a big bang at some finite cosmic time in the past and to expand forever in the future.

**15.5 Evolution of the scale factor** _{.}

We now discusshow to find the form of the 𝑎(𝑡) curve at all cosmic times, for a given st of present-day cosmological parameters. From (15.13), remembering 𝐻 = ά/𝑎 and multiplying 𝑎^{2}, we obtain

(15.21) (𝑑𝑎/𝑑𝑡)^{2} = 𝐻_{0}^{2}(𝛺_{𝑚,0} 𝑎^{-1} + 𝛺_{𝑟,0} 𝑎^{-2} + 𝛺_{𝛬} 𝑎^{2} + 1 - 𝛺_{𝑚,0} - 𝛺_{𝑟,0} - 𝛺_{𝛬,0}).

It is more convenient to introduce the new dimensionless variable

(15.22) 𝙩 = 𝐻_{0}(𝑡 - 𝑡_{0}),

which measures cosmic time relative to the present epoch in the units of the 'Hubble time' 𝐻_{0}^{-1}. in terms of 𝙩, (15.21) becomes

(15.23) __(𝑑𝑎/𝑑𝙩) ^{2} = 𝛺_{𝑚,0} 𝑎^{-1} + 𝛺_{𝑟,0} 𝑎^{-2} + 𝛺_{𝛬} 𝑎^{2} + 1 - 𝛺_{𝑚,0} - 𝛺_{𝑟,0} - 𝛺_{𝛬,0}.__

When 𝙩 = 0 (the present epoch) , for which 𝑎

(15.24) 𝑎

where 𝛥𝙩 is the small step size. The coefficient of 𝛥𝙩 and (𝛥𝙩)

In the bottom panel we have 𝛺

It is worth to discuss the general case in the limit 𝑎 → 0. We can assume that the energy density of the universe is dominated by one kind of source. in this case, (15.23) can be written

(15.25) (𝑑𝑎/𝑑𝙩)

(15.26)

where 𝙩

**15.6 Analytical cosmological models** _{.}

Although in the general case the evolution of the (normalized) scale factor 𝑎(𝑡) must be solved explicitly, there exist a number of special cases, corresponding to the particular values of cosmological parameters 𝛺_{𝑚,0}, 𝛺_{𝑟,0} and 𝛺_{𝛬,0}, for which equation (15. 21) can be solved analytically. We now discuss some of these analytical models in terms of cosmic time 𝑡 directly.

[The Friedmann models]

Cosmological models with a zero cosmological constant is known as the *Friedmann models*, after the mathematician A. A. Friedmann (1888-1925). **Figure 15.4** shows some information without solving the equations (14.36) and (15.21) explicitly. We see that the 𝑎(𝑡)-curve is everywhere convex and it crosses the point (𝑡_{0}, 𝑎_{0}) (note that 𝑎_{0} = 1) and the tangent of the point reaches the 𝑡-axis if the expansion had been uniform, that is, ά = constant and ἄ = 0. The time elapsed from that point to the present epoch is simply ά(𝑡_{0})/𝑎(𝑡_{0}) = 𝐻_{0}^{-1} i.e. the Hubble time. Thus

__𝑡 _{0} < 𝐻_{0}^{-1}.__

After the big-bang, the evolution depends critically on the curvature density parameter 𝛺

𝛺

𝛺

𝛺

The three case above are illustrated in

(15.27) ά

• For 𝛺

(15.28a)

• For 𝛺

(15.28b)

• For 𝛺

(15.28c)

(15.29a) ά

• For 𝛺

(15.29b)

• For 𝛺

(15.29c)

(15.30a) ά

which may be integrating by substituting 𝑦 = 𝛺

(15.30b)

Thogh this expression cannot be easily inverted to give 𝑎(𝑡), it becomes 2/3 𝑎

[The Lemaitre models]

The Lemaitre models are a generalization of the Friedmann models in which the cosmological constant is non-zero. In particular we will focus on matter-only models.

(15.31) ά

Though the equation is very complicate to solve for 𝑎(𝑡), we see that for small 𝑎 the first term on the right-hand side dominates and for large 𝑎 the second term does. Thus, the 𝑎(𝑡)-curve at first increases as, after starting from a big-bang origin at 𝑡 = 0, then the matter energy density decreases and the vacuum energy eventually dominates.

From the above behavior the universe must make a transition from a decelerating to an accelerating phase. This occurs when ἄ = 0 where has a point of inflection. Differentiating (15.31), we find that

(15.32) ἄ = 1/2 𝐻

From this result, we may verify that when 𝑎 is small, we have ἄ < 0 and so the expansion is decelerating and the deceleration gradually decrease until ἄ changes sign, after which the expansion accelerates ever more rapidly. We see that ἄ = 0 occurs is given by

(15.33) 𝑎

To obtain an approximate analytical expression for 𝑎(𝑡) in the vicinity of the point of inflection, denoting the cosmic time of inflection by 𝑡

𝑎 ≈ 𝑎

(15.34) ά

Differentiating (15.32) to obtain the third derivative of 𝑎(𝑡), 𝑑

ά

This equation can now integrated analytically and has the solution

(15.35)

An interesting property of the model is that in case of 𝑘 = 1, there is a 'coasting period' in the vicinity where ἄ = 0 during which the value of 𝑎(𝑡) remains almost equals to 𝑎

ά

This integral can be made tractable by the substitution 𝑦

𝐻

where the plus sign corresponds to the case 𝛺

(15.36)

From (15.36) we may obtain analytic expressions for 𝐻(𝑡), 𝜌

[The de Sitter model]

The de Sitter model is a particular special case of a Lemaitre model defined by the cosmological parameters 𝛺

For the de Sitter model, the cosmological field equation (15.13) reads

(ά/𝑎)

which tells us that the Hubble parameter 𝐻(𝑡) is a constant the normalized scale factor increase exponentially as

where the second equality is expressed in terms of the cosmological constant 𝛬. Thus, the de Sitter model has no big-bang singularity at a finite time in the past.

[Einstein's static universe]

Einstein derive his field equations well before the discovery of the expansion of the universe and he was worried that he could not find static cosmological solutions. He therefore introduced the cosmological constant to constructing static solution.

For 𝛬 > 0 and static universe, i.e. ἄ = ά = 0, the Hubble parameter 𝐻 is zero always, and from (14.36) we see that we require

In fact the first equality can be succinctly written as 𝜌

How well did Einstein's static universe fit with cosmological observations of the time? For the mean matter density, recent cosmological observations suggest that

𝜌

Adopting the above value of 𝜌

However, it has the theoretically undesirable feature of being

** local sound speed: ?

*** Whether such coincidences have some deeper significance is the the subject of current cosmological research.

^{※} attention: some rigorous derivation might be required