lecture notes in cosmology

7/29/2019 Lecture Notes in Cosmology

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Lecture Notes in Cosmology

Armen Sedrakian

Institute for Theoretical Physics,Frankfurt University,

D-60438, Frankfurt am Main, Germany

February 9, 2011


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Chapter 1

Expanding Universe

Cosmology studies the large-scale nature of the material world around us by methods of natural sciences(i.e., astronomy, physics, mathematics, etc). We can say cosmology studies the universe itself.

We need to realize that the human race occupies a small fraction of the time and space of theuniverse. Therefore, we are able to obtain information only on the part of the universe. Obviously, to

describe the entire universe, with all its parts that are not accessible to us, we need hypothesis whichwill allow us to extrapolate from what we know about our space-time to the entire universe. We willstart with the review of the empirical data that is available on the universe.

1.1 Hubble law

The most important feature of our universe is that, in a first approximation, the universe is homogeneousand isotropic. This statement is known as the cosmological principle. It implies that observations madeon the Earth can be, in fact, used to test cosmological models. We observe about 3000 Mpc of theUniverse. The units of length that we will use are related to each other by

1 Mpc = 3.26 106 light years 3.08 1024 cm.Observations of this segment of our universe suggest that the cosmological principle is true when av-eraging is carried out over the scales in the range 100 l 3000 Mpc. On smaller scales there existlarge inhomogeneities - galaxies, clusters of galaxies, and superclusters of galaxies. Within a galaxy thematter is distributed very inhomogeneously, as is evident from observations of stars, planets, and otherobjects in our galaxy.

The next most important feature of our universe is that it expands. This expansion is accordingto the Hubble law. Let us state this law in the Newtonian theory. Suppose an observer is at rest at

the origin of the system of coordinates K. According to Hubbles law the velocity of any point B withrespect to the observer is given byvB = H(t)rB, (1.1)

where rB is the radius vector of the point B and the parameter H(t) is called the Hubble parameter;sometimes it is called Hubble constant to stress that it is independent of the spatial coordinates; note,however, that it is time-dependent. If we take another point in the same system of coordinates, sayC, then vC = H(t)rC. Consider now a second system of coordinates K

which moves with respect tothe first one with the velocity vB (i.e. in which the particle B is at rest). According to the Galileantransformations

vC = v

C + vB (1.2)

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4 CHAPTER 1. EXPANDING UNIVERSE

where the prime indicates that the quantity is measured in the reference frame K. Then we find

v

C = vC vB = H(t)(rC rB) = H(t)r

C. (1.3)

Thus, we see that the expansion law is independent of the reference frame, i.e., from both referenceframes we see the same expansion according to the Hubble law. A useful way to envision the Hubble

Figure 1.1: Galaxy distribution in the nearby universe (figure from Harvard Smithsonian Center forAstrophysics Redshift survey). Each point is a galaxy. The three dimensional volume which is 900millions light years deep is split into 3 slices. Each slice covers a region 300 300 Mpc.

expansion is the two-dimensional surface of a sphere. Suppose we have a sphere, whose radius a(t) isincreasing with time. The angle between two fixed points A and B on the sphere, which we denote asAB, remains obviously constant. The distance between the two points on the sphere is given by

rAB(t) = a(t)AB. (1.4)

The relative velocity at which the points A and B move is then given by

vAB = rAB(t) = a(t)AB. (1.5)

Eliminating the angle from the last equation using (1.4) we obtain

vAB(t) =a(t)

a(t)rAB(t) = H(t)rAB(t). (1.6)

We see that the Hubble law emerges naturally upon identification H = a/a.


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1.2. DYNAMICS OF DUST IN NEWTONIAN COSMOLOGY 5

Let us go back to the Hubble law in the general 3d case. We see that

v = H(t)r, r = H(t)r,dr

r= H(t)dt, ln r =

H(t)dt + C, (1.7)

where C is the integration constant. Thus we obtain

r = exp H(t)dt eC = a(t) (1.8)where a(t) is called scale factor (which is the analogue of the radius of the 2d sphere in the exampleabove) and = eC is the integration constant which is fixed by the distance between two points at anygiven time. It is called the Lagrangian coordinate (or comoving coordinate of point B in the referenceframe fixed with A).

We want to stress that the Hubble law is valid over the scales where the universe is homogeneous.On smaller scales the motion is dominated by the inhomogeneities in the gravitational field; for examplethe local motion can have orbital nature. The velocity of an object relative to the comoving coordinatesare called peculiar velocity.

How the Hubble constant can be measured? If the peculiar velocities of two objects are small, we

can measure the recession velocity via Doppler shift in spectral lines. If we have also a reliable measureof the distance we can then measure the Hubble constant (more precisely its current value) via theHubble law Eq. (1.1). There are two methods that allow for accurate measurements of the distancesin the space and hence the Hubble constant; these are based on the so called standard candles andstandard rulers.

Standard candles are astrophysical objects which all have about the same luminosity. There aretwo prominent examples: (1) the Cepheid variable stars, which pulse at periodic rate; (2) Type-IAsupernovae, which are bright, exploding stars, which show characteristic spectral pattern. Type-IA arebetter standard candles because they can be observed to much greater distances than Cepheids.

Furthermore, the distance to nearby objects are measured directly by parallax. The inverse squarelaw related the apparent luminosity of distant objects to that of nearby one whose distance is known.

The method based on standard rulers is the same but one identifies a class of objects with the samesize instead of luminosity.

The current value of the Hubble constant is H 65 80 km s1 Mpc1. With this value wecan estimate the age of the Universe if we neglect the effects of gravity and assume that the velocityremains constant over time. Then, if two points are separated by r today, they were coincident att0 = |r|/v = 1/H(t) 1/H time ago, which gives t0 1.5 1010 yr.

1.2 Dynamics of dust in Newtonian cosmology

Consider infinite, expanding, homogeneous and isotropic universe. We shall assume that the universeis filled with dust, which means that its pressure p , where is the energy density. Our initialstudy will be based on Newtonian dynamics, i.e., all the velocities are assumed to be non-relativistic.Consider a sphere within such universe. Its radius changes in time (expands) according to

R(t) = a(t)com, (1.9)

where a(t) is the scale factor, com is the comoving coordinate. The total mass within the sphere isconserved M = const (i.e., is independent of time). Thus we can write

(t) =M

(4/3)R3(t)=

M

(4/3)a3(t)3com=

M

(4/3)a30(t)3com

a30(t)

a3(t)

= 0

a30(t)

a3(t)

, (1.10)


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where 0 is obviously the energy density at t = 0 and we used short-hand notation a(0) = a0. Now wetake the time derivative of Eq. (1.10)

(t) = 30a(t)4a(t) = 30a0

a

3 aa

= 3(t) aa

= 3(t)H(t), (1.11)

i.e.,

(t) + 3(t)H(t) = 0. (1.12)We can compare the last relation with the energy conservation, which quite generally can be written inthe following form

d

dt=

t+ (v) = 0. (1.13)

We note that (r, t) = (t) and that the velocity is given by the Hubble law (1.2). Then the second

term in Eq. (1.13) is transformed as follows: (v) = (v) = H(t)r = 3H(t), which confirms thatEq. (1.12) is nothing else as the conservation of the energy.

Figure 1.2:

1.2.1 Acceleration equation

Gravitational forces among masses are attractive. Therefore, matter masses are self-attractive. As aresult the expansion of matter is slowed down by gravity. Let us return to the example of the 3d spherefrom the previous section. Consider a probe of mass m on the surface of the sphere. Particles outsidethe sphere have no effect on the test mass; only particles inside the sphere are relevant. The equationof motion then reads:

mR = GmMR2

= 43

GmM

(4/3)R3 (t)

R (1.14)

Eliminating the radius in favor of the scale factor we obtain

macom = 43

Gm(t)acom. (1.15)


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1.2. DYNAMICS OF DUST IN NEWTONIAN COSMOLOGY 7

We see that the mass m and the comoving coordinate drop out of this equation, which in combinationwith the energy conservation (1.13) determines the evolution of the system

a = 43

G(t)a. (1.16)

The equations (1.13) and (1.16) completely determine the evolution of the system. These equationsexactly coincide with their counterparts in general relativity. The reason is that the analysis aboveshould be valid for infinitesimally small spheres, in which case the velocities and masses involved aresmall.

Now let us find the solutions of these equations. To eliminate the energy density from the accelerationequation with substitute Eq. (1.10) in Eq. (1.16)

a = 43

G

a0

a(t)

30a. (1.17)

Next we multiply the both sides of this equation by a and integrate; explicitly

aa = 43

Ga300a

a2,

1

2a2 =

4

3aGa300 V(a)

+E, (1.18)

where E is an integration constant. This equation is analogous to the problem which one studies duringthe course of classical mechanics, namely that of the rocket launched from the Earth. This is apparentif we rewrite Eq. (1.18) as

1

2a2 + V(a) = E. (1.19)

The first term is the kinetic energy, the second term the potential energy (gravitational pull of the Earthon the rocket). If the kinetic energy is larger than the potential energy, i.e., E > 0 the rocket leavesthe Earth, otherwise, i.e. when E < 0 the rocket falls back to Earth. In analogy the dust dominatedUniverse expands forever in the case of positive E and recollapses in the case of negative E. Let usrewrite this condition as follows

a2 +8

a

Ga300a

= 2E,

a

a

2

H2

2Ea2

=8G

3

a

30

a3

0

(t)

(1.20)

thus,

H2 2Ea2

=8G

3(t). (1.21)

The fate of the dust dominated universe depends on the sign of E. The following cases are possible:

E > 0, i.e., when the kinetic energy dominates, a/2 > V(a), and we have an expanding universe

E < 0, i.e., when the potential energy dominates, a/2 < V(a), and we have an recollapsinguniverse


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The limiting case E = 0 defines the critical density

cr =3H2

8G. (1.22)

Let us eliminate in Eq.(1.16) the Hubble constant in favor of the critical density. We obtain

8G

3 [cr

(t)] =2E

a2 , (1.23)

or, alternatively,

E =4G

3a2cr [1 (t)] . (1.24)

Here we have introduced a new quantity

(t) =(t)

cr, (1.25)

which is called the cosmological parameter. In general the sign of E is fixed, therefore the quantity1 (t) does not change the sign. This does not mean that the function does not change with time;indeed since the Hubble parameter is time dependent, so is the cosmological parameter.We shall see that the sign ofE determines the spatial geometry of the universe in General Relativity.In a dust dominated universe we have the following cases:

if 0 = 0/cr0 > 1 (the index 0 refers to the current value of the parameter), then E is negativeand the spatial curvature of the universe is positive. This corresponds to closed universe. Duringits expansion the universe reaches some maximal value of the scale factor and then the universerecollapses.

if 0 = 0/cr0 < 1, then E is positive and the spatial curvature of the universe is negative. Thiscorresponds to an open universe, which expands hyperbolically.

if 0 = 1, then E = 0, which corresponds to parabolic expansion of the universe with flat spatialgeometry.

These cases are illustrated in Fig. 1.4. In the cases when universe is flat or open, it expands forever (thescale factor increases forever). In the case of closed universe the scale factor eventually decreases tozero, i.e., the universe returns to the initial singularity. It should be stressed that our example concernsthe dust dominated universe. In general the evolution of the Universe depends on its matter content,so that, for example, we may have a Universe that is closed, but does not recollaps at asymptoticallylarge times.

Consider the special case of flat universe, E = 0. We would like to find an explicit solution to the

scale factor from Eq. (1.18); in that case we have1

2a2 + V(a) = 0,

a2

2 4G0a

30

3a= 0,

aa2

2=

4G0a30

3= Const. (1.26)

The left-hand side of the last relation can be written

aa2 =4

9

da3/2

dt

2= Const. a t2/3. (1.27)

It follows immediately that in this case H(t) = 2/3t and that current age of the universe is t0 = 2/3H0,which differs from our previous estimate on page 5 by a factor of order unity.


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1.3. GEOMETRY OF HOMOGENEOUS AND ISOTROPIC SPACE 9

Figure 1.3: Dependence of the scale factor on time for open, flat and closed universes.

1.3 Geometry of homogeneous and isotropic space

The evolution of our universe in time can be represented by a sequence of three-dimensional spatialhypersurfaces, each of which by assumption are isotropic and homogeneous. Note that isotropy at everypoint implies homogeneity, but homogeneity does not imply isotropy. Homogeneous matter can executemotions that are dependent of directions (e.g. expand in one direction and contract in two others; suchmotions will yield physics which is anisotropic, i.e., depends on directions).

The symmetry group of homogeneous and isotropic spaces include three independent translationsand three rotations. There are three types of such spaces (assuming that the topology of the spaces

are simple): 1) a three dimensional sphere of constant and positive curvature; 2) flat space; 3) a three-dimensional hyperbolic space of constant negative curvature.

There are two well-known examples of embeddings of 2d surfaces in the Euclidean space: flat planeand 2d sphere. These will help us to visualize the isotropic and homogeneous spaces; the extension tohigher spatial dimensions is straightforward. The metric of 3d Euclidean space is given by

dl2 = dx2 + dy2 + dz2. (1.28)

The embedding of a 2d sphere in this 3d space is given by

x2 + y2 + z2 = a2, (1.29)

where a is the radius of the sphere. From this last equation it follows that

dz = xdx + ydyz

= xdx + ydya2 x2 y2 . (1.30)

and, therefore, the metric (1.28) can be written as

dl2 = dx2 + dy2 +(xdx + ydy)2

a2 x2 y2 . (1.31)


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Figure 1.4: Illustration of the Lobachevsky space in two dimensions.

This natural result shows that the distance between two points located on the sphere can be expressedin terms of two independent variables. Let us go over to the spherical (cylindrical) coordinates:

x = cos , y = sin . (1.32)

where

x2 + y2 = 2, xdx + ydy = d. (1.33)

On the other hand

dx2

+ dy2

= d2

+ 2

d2

(1.34)Substituting these in Eq. (1.31) we obtain for the metric

dl2 =

1 +

2

a2 2

d2 + 2d2 =d2

1 2/a2 + 2d2. (1.35)

We can now distinguish three cases: (1) a2 > 0, two-dimensional space with positive curvature; (2)a , which corresponds to a 2d plane, i.e., flat space; (3) a2 < 0, two-dimensional space with negativecurvature; this space is called Lobachevsky space. It cannot be embedded into the Euclidean spacebecause the radius of the sphere is imaginary (sometimes this space is called pseudo-sphere or hyperbolic

space). However, Lobachevsky space arises from the embedding of the surface x2 + y2 z2 = a2 witha >= 0 in the space with metric dl2 = dx2 + dy2 dz2 (see Figure 1.5 for an illustration). One canwrite the metric (1.35) in a different form by introducing = r

|a2|

dl2 = |a2|

dr2

1 kr2 + r2d2

. (1.36)

where k = +1 corresponds to the sphere, k = 1 - pseudo-sphere, and k = 0 to the plane. The meaningof |a| is to quantify the curvature of the space; in the case of the flat space it does not have any physicalmeaning, i.e., when k = 0 it can be absorbed in the definition of r.


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1.3. GEOMETRY OF HOMOGENEOUS AND ISOTROPIC SPACE 11

Similar to the discussion above we can go up in the number of spatial dimensions and obtain the 3dembedding of a sphere (pseudo-sphere) in 4d Euclidean (Lorentzian) space. In that case we obtain

dl2 = |a2|

dr2

1 kr2 + r2(d2 + sin2 d2)

. (1.37)

where as before k = 0, 1. Sometime we will work with the coordinate defined via the relations

r =

sinh , k = 1, k = 0sin , k = +1

and

d2 =dr2

1 kr2 (1.38)Then the metric takes the simple form

dl2 = a2(d2 + 2()d2), d2 = d2 + sin2 d2, (1.39)

and () = [0, ] for k = 0, () = sin [0; ] for k = 1, and () = sinh [0; ] fork = 1.

Next let us consider the embedding of a 2d sphere in a three dimensional space with positivecurvature. The distance element on the sphere of fixed radius is given by

dl2 = a2 sin2 d2. (1.40)

This expression is mathematically identical to that for the sphere in a flat space with radius R =a2 sin2 . The surface area, which is just the integral over the solid angle, follows immediately as

S = 4R2 = 4a2 sin2 . (1.41)

We see that the surface area is non-monotonic as [0; ]. In particular it vanishes at the end pointsof this domain. (A straightforward analogy is the circumference of a circle of constant latitude on ausual sphere. When one moves from the north pole to the equator and then to the south pole; themagnitude of the circumference is non-monotonic.) Because the area remains always bound, we expectthat the volume of the space with positive curvature is also bound. A characteristic feature of the spaceof positive curvature is that the sum of the angles of a triangle formed by geodesics (curves of minimallength) is lager than 180 degree.

The same arguments as above applies to the space of negative curvature k = 1 with metric of thesurface of 2d pseudosphere given by

dl2

= a2

sinh2

d2

, (1.42)which leads to the surface area

S2d = 4a2 sinh2 . (1.43)

The surface area and the volume of such space is unbound. Furthermore, the sum of the angles of atriangle formed by geodesics is smaller than 180 degree.

The volumes of these spaces can be calculated from

dV = S2dad. (1.44)


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Figure 1.5: The surface area of 2d sphere in three cases of open, flat and closed universes.

1.4 Einsteins equations

The general theory of relativity provides a consistent framework for the description of the universe filledin with matter with arbitrary equation of state. In this section we will extend our previous discussionof dynamics of Newtonian universe to general relativity.

We will use Einsteins convention for summation over repeated indices:

gdxdx =

gdxdx. (1.45)

Greek indices will run over 0, 1, 2, 3, while the Latin indices will run over 1, 2, 3.The dynamics of space-time continuum in General Relativity is described by the metric functions

g(x), which obey the Einstein equations:

G = R

1

2R = 8GT . (1.46)

Here R is the Ricci tensor which can be written in terms of the metric tensor and Christoffel symbolsas

R = g

x

x+

, (1.47)

where gg = and the unit tensor is defined as

= 1 if = and 0 otherwise. The Christoffel

symbols are defined entirely in terms of the metric functions

=1

2g

gx

+gx

gx

. (1.48)

Further R = R is the scalar curvature and = const is the cosmological constant. On the left handside of Eq. (1.46) stands the energy-momentum tensor T of matter, which is symmetric in its indecies,i.e.,

T = gT = T. (1.49)


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1.5. FRIEDMAN EQUATIONS 13

The energy momentum tensor is defied by the conservation equation (or equation of motion)

T

x= 0, (1.50)

in Minkowski spacetime. In cured spacetime the equation is modified to

T; = T

x+ T + T = 0. (1.51)

The equation (1.51) follows from the Einstein equations as consequence of the Bianchi identities satisfiedby the Einstein tensor:

G; = 0. (1.52)

For most of the application the matter can be assumed to be perfect fluid with th energy and momentumtensor given by

T = ( +p)uu p , (1.53)

where p = p() is the equation of state of matter, which needs to be specified at each particular case, u

is the four velocity of the fluid, p is the pressure, - the energy density. For ultrarelativistic particlesp = /3; in cosmology we will find often that p = w, where w is a constant.

In relativistic theory the invariant with respect to coordinate transformation is the infinitesimalspacetime interval between events. It is convenient to take the interval in the following form

ds2 = dt2 dl2 = dt2 a(t)2

dr2

1 kr2 + r2d2

gdxdx; (1.54)

the corresponding metric is called the Robertson-Walker metric. Note that the spatial coordinates arecomoving, which means that every object with zero peculiar velocity has constant coordinates r, and. The distance between two comoving observers is

a(t).

1.5 Friedman equations

We have already derived the equations for cosmological evolution in Newtonian theory; these are givenby (1.12), (1.16) and (1.21). Einstein equations determine the evolution of the universe in generalrelativity; we shall discuss the formal derivation somewhat later; for the moment we can heuristicallyderive the general relativistic equation by modifying the Newtonian equations. The main modificationcomes from the fact that we need to relax the assumption that we are dealing with a dust and includethe contribution of pressure. The change in the energy due to the work done by the pressure is given

by (we assume that the entropy of the system is constant, i.e., dissipative processes are negligible)

dE = pdV. (1.55)

Since E = V and V a3, we obtain V d + dV = pdV or d = V1(p + )dV. Then we find thatdV/V = 3a2da/a3 = 3da/a = 3d ln a. Thus,

d = 3( + p)d ln a + 3( + p)H = 0, (1.56)

where we substituted H = a/a. This equation is the relativistic version of Eq. (1.12) and it reflects theenergy conservation in an isotropic and homogeneous universe, T0; = 0. Thus, we see that the general


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relativistic counterparts of the Newtonian equations are obtained by addition of a term arising frompressure. The acceleration equation is now modified from (1.16) to

a = 43

G( + 3p)a. (1.57)

The form of the pressure term arises from the sum of the diagonal terms in the spatial part of the

Einstein equations. Equation (1.56) is the first Friedman equation. To obtain the second Friedmanequation we use the second Eq. (1.56) to eliminate pressure from Eq. (1.57) and multiply the resultingequation by a to bring it to a form suitable for integration. The details are reflected in the followingchain (note that as before = 0a

30/a

3)

aa = 4G3

2

H

aa = 4G

30

2a

30

a3+ 3

a30a

a2H

aa = 4Ga

30

30

a

a3. (1.58)

Now we can integrate this equation to obtain

a2

2=

4Ga303

01

a2+ E. (1.59)

Upon reintroducing the Hubble constant H = a/a and defining k = 2E, simple algebraic manipula-tions lead us to

H2 +k

a2=

8G

3(t). (1.60)

This is the second Friedman equation. The difference to the non-relativistic counterpart (1.21) is thatthis equation applies for arbitrary equation of state. Another important point is that k here has themeaning of the curvature, as introduced above, i.e., k = 0, 1. This identification is possible if wederive the Friedman equations from the 0 0 component of Einsteins equations. Now we can rewrite(1.24) as

k = 2G

3 a2cr [1 (t)] . (1.61)

Thus we see that the value of the cosmological parameter = /cr determines the geometry of theuniverse; specifically, when > 1, the Universe is closed and has a geometry of three-dimensionalsphere k = +1; = 1 corresponds to the flat universe (k = 0); and < 1 corresponds to open universewith hyperbolic geometry (k = 1).

The two Friedman equations (1.57) and (1.60) form a closed system of equations if the equation ofstate p = p() is specified. Note that Eq. (1.57) can be replaced by the second equation of (1.56), i.e.,the conservation law.

1.5.1 Derivation from Einsteins equationsTo derive the Friedman equations from Einsteins equations and to find particular solutions to them itis more convenient to define the conformal time as

dt = a(t)d,

dt

a(t). (1.62)

To be specific let us first consider the universe with positive curvature, i.e., k = +1. In terms ofconformal time the metric is written as

ds2 = a2()[d2 d2 sin2 (d2 + sin2 d2)]. (1.63)


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1.5. FRIEDMAN EQUATIONS 15

Now we would like to write down the Einsteins equations for this metric. For that purpose we identifythe non-vanishing metric functions by comparing Eq. (1.63) with ds2 = gdx

dx. This gives (notethat the coordinates x0, x1, x2, x3 here should be identified with ,,,):

g00 = a2, g11 = a2, g22 = a2 sin2 , g33 = a2 sin2 sin2 . (1.64)

Note that because of isotropy of the space (i.e., equivalence of all directions, the components of metrictensor g0 must be zero in the system of coordinates we have chosen). Now we use the definition ofChristoffel symbols (1.48) to compute the components

000 =a

a, 0 =

a

a ,

00 =

00 = 0, (1.65)

where prime means differentiation with respect to . Now the components of the Ricci tensor can becomputed via Eq. (1.47). We obtain

R00 =3

a4(a

2 aa). (1.66)

From symmetry considerations (isotropy of the space) the components R0 = 0. The remaining com-ponents are

R = 1

a4(2a2 + a

2+ aa). (1.67)

The Ricci scalar is then obtained as

R = R00 + R =

6

a3(a + a). (1.68)

Finally, we need to specify the form of the energy-momentum tensor; in the comoving system of coor-dinates u = 0, and u0 = 1/a, therefore from the definition

Tik = (p + )uiuk pgik, (1.69)

it follows that T00 = . These results should be substituted in the 00 component of Einsteins equations,i.e.,

R00 1

2R = 8GT00 , (1.70)

which gives

a2 + a2

a4=

8G

3. (1.71)

A more convenient form of this equation is obtained upon differentiating it with respect to . Tocompute the derivative note that from the first equation of (1.56) we have

d

d= 3( + p) d

dln a = 3( + p)a

a. (1.72)

Then we find that

2aa + 2kaa =8G

3[a3a( 3p)], a + ka = 4G

3( 3p)a3. (1.73)


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Next we wish to compare the results that we have obtained with those derived heuristically. For thatpurpose, we go over from derivative with respect to the conformal time to the ordinary time (whichinduces an extra factor a). We obtain

1

a2+

a2

a2= H2 +

1

a2=

8G

3, (1.74)

which coincides with the second Friedman equation (1.60) for k = +1. The second equation (whichcloses the system of equations for the unknowns a and ) can be chosen one of those in Eq. (1.56), e.g,

d = 3( + p)d ln a. (1.75)

How the calculations are modified in the case of open universe with negative curvature? The metric inthis case is given by

ds2 = a2()[d2 d2 sinh2 (d2 + sin2 d2)]. (1.76)This can be obtained formally from Eq. (1.63) by replacements i, i, and a ia. Equation(1.75) will not change its form, whereas Eq. (1.74) becomes

H2 1a2

=8G

3, (1.77)

which is consistent with Eq. (1.60).

1.6 Solutions of Friedman equations

To analyze the solutions of the Friedman equations we start with Eq. (1.73)

a + ka =4G

3 ( 3p)a3, (1.78)

which can be solved provided the equation of state p() is given. We will also use Eq. (1.60) which werewrite in terms of conformal time

H2 +k

a2=

8G

3, a2 + k =

8G

3a2, a2 + ka2 = 8G

3a4. (1.79)

In the following we will consider solutions of the Frideman equations for specific equations of state.

1.6.1 Radiation dominated universe, p = /3, k = 0, 1This case is particularly simple since the right hand side of (1.78) vanishes, i.e.

a + ka = 0. (1.80)

The solution of this differential equation is

a() = am

sinh , k = 1, k = 0sin . k = +1


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1.6. SOLUTIONS OF FRIEDMAN EQUATIONS 17

where am is one of the integration constants, the other is fixed from the condition a( = 0) = 0. Thephysical time is the integral of a()d. Integration in Eq. (1.81) gives

t = am

cosh 1, k = 12/2, k = 01 cos . k = +1

One conclusion that can be drawn from these equations is that in the case of flat universe a t1/2,form which it follows that H = 1/2t. We can also find the energy density from Eq. (1.60) with k = 0:

=3

32Gt2 a4. (1.81)

The same can be found from the Eq. (1.56)

d = 4d ln a d ln = d ln a4 + C a4. (1.82)

This means that the energies of particles (whose number is conserved) is reduced proportional to a1.

1.6.2 Dust dominated universe p = 0, k = 0, 1First we note that Eq. (1.56) in this case implies

d = 3d ln a d ln = d ln a4 + C a3. (1.83)Therefore a3 = const and

a + ka = C. (1.84)

The solutions of this equation read

a() = am

cosh 1, k = 12, k = 0

1 cos , k = +1and the physical time is given via integration

t = am

sinh , k = 13/3, k = 0 sin . k = +1

Since the scale factor varies in the range 0

a

we see that the conformal time varies in the

range 0 in the cases of open or flat universes independent whether it is radiation or matterdominated. In a closed universe the conformal time is bounded and assumes values in the range [0; ]for radiation dominated universe and [0;2] for matter dominated universe.

1.6.3 Strongly interacting universe p We consider a concrete case of flat universe k = 0. We already saw that the energy density for radiationscales as R a4 and for matter scales as M a3. The total energy density is the sum of both,which we write as

= R + M = CRa4 + CMa

3, (1.85)


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where CM and CR are constants. It will be convenient to work with the energy of universe when thesetwo contributions are equal; we denote the scale factor and the total energy density in these cases as aand . By definition

= CRa4 + CMa

3, CRa4 = CMa

3 CRa1 = CM (1.86)

Substituting the last relation in Eq. (1.86) in the first one we find

= CRa4 + CRa

4, CR = 2

a4, CM =

2a3. (1.87)

Thus the total energy density is given by

=

2

a4a4 + a3a3

. (1.88)

Taking a derivative of Eq. (1.79) we obtain

a =2G

3 a3. (1.89)

Because the right hand side is a constant, we can straightforwardly integrate this equation to obtain

a() =G

3a32 + C1 + C2. (1.90)

From the condition a( = 0) = 0 it follows that C2 = 0. To obtain the second integration constant wesubstitute in Eq. (1.79) written for the case k = 0

a2 =8G

3a4, (1.91)

the solution (1.90): G

3a3 + C1

2=

4G

3

a4 + aa3

, (1.92)

where in the right hand side we substituted Eq. (1.88). Note that the first term in the right hand sideis due to radiation and the second due to the matter. Substituting the initial condition a( = 0) = 0once more we obtain

C21 =4G

3a4, C1 = +

4G

3a4

1/2

. (1.93)

We see that the term C1 is controlled by the radiation contribution, since the matter contributiondropped out upon imposing the initial conditions. The plus sign in the last relation follows from therequirement of positivity of the scale factor. Then the solution (1.90) can be written in an elegant form

a() = a

2

2+ 2

, (1.94)

where

1 =

G

3a21/2

. (1.95)


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1.7. MILNE UNIVERSE 19

In Eq. (1.94) the first term can be identified with the matter contribution, while the second term withthe radiation contribution. When a = a the conformal time satisfies the equation

2

2+ 2

= 1, (1.96)

whose solution (which we will denote by ) is given by

= (2 1). (1.97)

Note, however, that for the purpose of order of magnitude estimates we can use both s, since .We see that if the linear term, representing the radiation contribution, in Eq. (1.94) dominates.The scale factor grows linearly with . Conversely, when the matter contribution dominates anda() 2.

1.7 Milne universe

An interesting special case is the case of Milne universe, which is defined as an open universe (k = 1)with zero matter energy density. In that case from Eq. (1.79) we obtain

a2(t) = 1, a(t) = t. (1.98)Then the Robertson-Walker (RW) metric becomes

ds2 = dt2 a(t)2

dr2

1 kr2 + r2d2

= dt2 t2

dr2

1 + r2+ r2d2

= dt2 t2 d2 + sinh2d2 ,

(1.99)

where the coordinate is defined via Eq. (1.38). Since the Milne universe does not have any mattercontent we can expect that its metric should be related to the Minkowski metric which corresponds tothe metric of empty space. It can be written

ds2 = d2 dr2 r2d2. (1.100)(The Minkowski coordinate r should not be confused with the one appearing in the RW-metric). Tomap the Minkowski metric (1.100) onto the RW metric we define new coordinates

= t cosh, r = t sinh. (1.101)

We can compute further the differentials

d = dt cosh + t sinhd, dr = dt sinh + t coshd, (1.102)

from which we see thatd2 dr2 = dt2 t2d2. (1.103)

We conclude that (1.99) and (1.100) are equivalent. However, the Milne coordinates do not coverthe entire Minkowski space. To see this consider a particle which has fixed comoving coordinates; itsvelocity is equal

|v| = r

= tanh < 1. (1.104)


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Figure 1.6: The constant t hyperboloids in the Minkowski space cover the Milne universe.

The proper time of the particle is given by 1 |v|2 = t. (1.105)

Note that on the left hand side of these equation are the Minkowski coordinates, on the right handsides the Milne coordinates. Now we find the hypersurfaces of constant time t which in the Minkowskicoordinates are given expressed

t =

2 r2 = Const. (1.106)These can be plotted in the Minkowski space (i.e. as a function of the coordinates and r). The result

is shown in figure Fig. (1.6). We see that the hypersurface t = 0 is simply the light cone; the surfacesof constant t > are hyperboloids in the Minkowski coordinates which are within (half of) the forwardlight cone. Thus the Milne universe covers only 1/4 of the entire Minkowski space (the uncovered partsinclude the backward light-cone, a half of the forward light cone). Some remarks are in order

The Milne universe has a center, which is apparent from its coverage shown in the figure. The Milne universe is an example of a universe which is flat in the 4d space, but has a curvature

in the 3d space (since k = 1). In the universes with energy content the correct foliation of the space is given by the constant

energy hypersurfaces. Obviously in the Milne universe we have a translational symmetry in time- all the hypersurface have the same zero energy.

1.8 De Sitter universe

The de Sitter universe is a space-time void of matter and radiation with positive constant 4-curvaturewhich is isotropic and homogeneous. Since it is a space-time, it can be derived from purely geometricalconsiderations. (We will return to more physical derivation from Friedman equations later on). We haveseen that in the case of purely spatial spaces we were able to represent curved spaces as embeddings ofvarious (hyper)surfaces in higher dimensional flat spaces. To make the discussion intuitively accessible


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1.8. DE SITTER UNIVERSE 21

Figure 1.7: de Sitter space-time in two dimensions.

we will consider only two spatial dimensions. Addition of third dimension will be trivial. Considerthree-dimensional Minkowski metric given by

ds2 = dz2 dx2 dy2. (only spatial) (1.107)In this space we consider an embedding (hyperboloid) defined as

H2 = z2 x2 y2. (1.108)

The hyperboloid will be parameterized in terms of the coordinates x and y. Since zdz = xdx + ydy andz =

x2 + y2 H2

1/2the metric can be expressed entirely in terms of the variable x and y

ds2 =(xdx + ydy)2

x2 + y2 H2 dx2 dy2. (1.109)

where according to (1.107) x2 +y2 > H2. This metric describes the de Sitter space in two-dimensions.

By coordinate transformation this metric can be brought in a more compact form (which also makeapparent its symmetries) by, for example, transformation

x = H1cosh(Ht)cos , y = H

1cosh(Ht)sin . (1.110)

Where t and [0;2]. Then, the metric takes the formds2 = dt2 H2cosh2(Ht)d2. (1.111)

This space-time is described by the hyperboloid shown in Fig. 1.7. In four dimensions this metriccorresponds to closed universe with positive curvature.

Now we consider the four-dimensional de Sitter universe, which is described by the equation ofstate p = . The index indicates that the pressure originates from cosmological constant in theEinsteins equations. The general equation (1.56) in this case shows that

d = 3( +p)d ln a d = 3( + p)d ln a = 0, = Const. (1.112)


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Figure 1.8: Scale factor in de Sitter universe as a function of time. Note the exponential expansion fort .

Then the acceleration equation (1.57) reads

a = 43

G( + 3p)a a 83

Ga = 0, a H2a = 0, (1.113)

where H = [8G/3]1/2. The solution of Eq. (1.113) is given by

a(t) = C1 exp(Ht) + C2 exp(Ht), (1.114)

with C1 and C2 being the integration constants. To fix the integration constants we go back to thesecond Friedman equation (1.60) and write it as

H2 +k

a2=

8G

3 H2. (1.115)

Inserting H = a/a in this equation, we rewrite it as

H2

a2

a2 = k. (1.116)

We substitute the solution (1.114) in this equation to obtain

C1C2 =k

4H2. (1.117)

Now let us consider separately the following cases:

Flat universe k = 0; either C1 = 0 or C2 = 0. If C2 = 0 we obtain exponentially expandinguniverse.


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1.8. DE SITTER UNIVERSE 23

For closed universe k = 1 and we can choose |C1| = |C2| to obtain C1 = C2 = (2H)1. In thiscase the scale factor becomes

a(t) = H1

exp(Ht) + exp(Ht)

2

= H1 sinh(Ht). (1.118)

For open universe k =

1 and we can choose

|C1

|=

|C2

|, i.e., C1 =

C2 = (2H)

1. In this case

the scale factor reads

a(t) = H1

exp(Ht) exp(Ht)

2

= H1 cosh(Ht). (1.119)

The metric for these three cases can be written (upon substituting the scale factor a(t)) as

ds2 = dt2 H2

sinh2(Ht)exp(2Ht)

cosh2(Ht)

d2 +

sinh22

sin2

d2

.


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Chapter 2

Light propagation in expanding universe

Most of the universe is seen by us in the electromagnetic spectrum. Although the existence of thegravitational waves has been indirectly proven in the binary systems involving two compact stars, theirdirect detection has not been achieved yet. Thus vast astronomical information reaches us in formof photons of different wave-lengths. (Neutrinos and high energy cosmic rays are other sources that

complement our information on the universe).The fact (known from the special relativity) that the speed of any particle is limited to that of the

light limits the amount of information that can be transmitted. Indeed since no information can travelfaster than light their exist horizons beyond which we can not obtain information about the universe.

2.1 Light trajectories

Consider a massless particle propagating in a flat universe at the speed of light (e.g. photon). Itstrajectory is given by the geodesics that satisfies the condition

ds2 = 0. (2.1)

We know that in the curved universes we can always choose a local frame where physics corresponds toa flat universe so that the condition (2.1) remains intact.

Consider isotropic universe, with light propagating radially in a coordinate system where observeris at the origin. To describe the trajectories we use the conformal time

=

dt

a(t). (2.2)

The metric is written as

ds2 = a2() d2 d2 2()(d2 + sin2 d2) , (2.3)where

2() =

sinh2 k = 12 k = 0

sin2 k = +1

.

Since we have spherical symmetry the radial trajectories with , = const are geodesics (i.e. d = 0 =d). Then, from the requirement ds2 = 0 it follows

d2 d2 = 0. (2.4)

25


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26 CHAPTER 2. LIGHT PROPAGATION IN EXPANDING UNIVERSE

The solution of this equation is() = + C. (2.5)

We see that the trajectories are simply straight lines at 45 degree in the plane.

2.2 Horizons

The information that we receive about the universe around us is limited by the speed of light and thefact that the universe has a finite age. We can receive only information about parts of the universefrom which the light could have traveled within the time equal or less than the age of the universe.The volume from which we can obtain information is the bounded; its boundary is called the particlehorizon.

The age of the universe is 1.5 1010 yr, so that the size of particle horizon is about 1.5 1010 lightyr. Let us examine now the distance that light can propagate in an expanding universe. According to(2.5)

p() = i = t

t1

dt

a. (2.6)

where t1 is the initial time (start of the universe). Therefore, at time an event with > p() isinaccessible to the observer located at the origin of the coordinate system = 0. Note that we cannotalways put i = 0 = ti; this is only valid for the universes that are singular at the beginning. There areexample of other (e.g. de Sitter) universes where the initial state is non-singular. The physical distanceis obtained upon multiplying the the comoving coordinate with the scale factor

dp(t) = a(t)p = a(t)

tt

dt

a. (2.7)

Now we need to take into account the fact that the early universe was dense and photons where in termalequilibrium with matter. Their decoupling from matter occured at the time of hydrogen recombination.At that time the universe was about 1000 time smaller than now. Therefore, it is easy to see thatthe information that we obtain is, in fact, limited by the distance that the light can travel since therecombination time. Effectively the initial time is replaced by the recombination time

dopt(t) = a(t)p = a(t)

tt

dt

a, (2.8)

where t is the time of recombination. This horizon is clearly less distant that the particle horizon. Itis called optical horizon. Only primordial neutrinos or gravitational wave can allow us to see beyondthe optical horizon (but only up to the particle horizon).

For any concrete model of the universe for which the scale factor is know we can compute thehorizons. As an example let us take a flat universeand assume that radiation dominates; then a(t) t1/2and we find that dp = 2t (note that c = 1); in the case of matter dominated universe a(t) t2/3 anddp = 3t.

There is another scale in the universe that sometimes is called horizon. Evidently we have thelength scale 1/H (again c = 1). This is called curvature horizon and Hubble horizon. In some modelscurvature horizoncould be of the same order as the particle horizon. However there are counterexamples.One such counterexample is the flat de Sitter space where a(t) exp(Ht) and therefore

dp(t) = exp(Ht)

tt

exp(Ht)dt = H1 {exp[H(t t)] 1} . (2.9)


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2.3. REDSHIFT 27

For times satisfying tt H1, dp grows exponentially, i.e., the particle horizon is very large comparedwith the Hubble horizon H1 which is constant.

One can also define an event horizon. The event horizon is defined with respect to a given time (interms of conformal time ). It is the boundary of the volume from which no signal can be obtained bythe observer in the future. The coordinates of these points are defined

> e() = max

d = max . (2.10)

where max refers to the final time. The physical location of the horizon at the time t is given by

de(t) = a(t)

tmaxt

dt

a, max =

tmaxt

dt

a(t). (2.11)

If universe expands forever then by definition the time tmax . The value ofmax and de(t) obviouslydepends on the convergence of the integral defining these quantities. Let is consider several examples.

Open or flat universe; since t

maxand

maxare infinite there are two possibilities:

the integrand in Eq. (2.11) is convergent, i.e., there is a finite event horizon

this integrand is divergent, i.e., there is no event horizon.

The convergence of the integrand depends on the function a(t).

For linear or slower growing a(t) the integrand is divergent (in the linear case logarithmically).This corresponds to decelerating universe. Mathematically, this means that a 0.

If a(t) grows faster than linear the integrand is convergent. This corresponds to acceleratinguniverse (a > 0) (gebremstes Universum).

de Sitter universe. For the event horizon we obtain

de(t) = exp(Ht)

t

exp(Ht)dt = H1. (2.12)

This means that the event horizon is equal to the curvature scale. All the events that occur at agiven moment at a distance greater than 1/H have no effect on an observer because the universeis expanding too fast. In closed and decelerating universes the tmax is finite and therefore therealways exists an event horizon and a particle horizon.

In the closed universe there is always an event horizon because the integration range is limitedfrom above.

2.3 Redshift

Imagine a photon emitted from a distance galaxy. If we (observers) were static with respect to thatgalaxy, then the wave-length of the photon will be the same for us as it was at the moment of emission.This is not true in an expanding universe. The wave-length of the photon will undergo a redshiftbecausethe universe is expanding.


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Figure 2.1: Illustration of light propagation from the source to the observer in an expanding universe.

Consider a source or radiation at em and tem. Suppose it emits a signal with a duration . Theequation defining the light trajectory is then [see Eq. (2.5), where we take the sign ]

() = + C. (2.13)The integration constant is fixed from the initial condition (em) = em + C or C = em + em.Substituting this constant in the above equation we obtain

() = em

(

em), (2.14)

which defines the trajectory of a photon. The photon reaches the observer at the location obs = 0 attime obs = em + em. The physical intervals for the emission of light at the point of emission anddetection are different and are given by

tem = a(em) emission point

, tobs = a(obs) observation point

. (2.15)

The wave-length and the period of the photonic wave are related simply by (c = 1) em = tem If tis the period of the light wave, the light is emitted with the wavelength em = tem; similar relationholds for the observed wavelength. Their ratio is then given by

obsem

=tobstem

=aobsaem

. (2.16)

We see that the expansion of the universe leads to the scaling (t) a(t). Then the frequency (t) 1/(t) 1/a(t).

If all the photons have their frequencies shifted by /a the shape of the Planck distributiondoes not change. The temperature, which is determined from the maximum of the Planck distributionis therefore shifted by the amount 1/a. Therefore the energy radiation (photon luminosity T4) scalesas 1/a4. Because the number density of photons scales as n T3 a3, while the volume scales asV a3, the total number photons N = nV is conserved.


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2.4. DOPPLER SHIFT 29

2.4 Doppler shift

The redshift of photons can be interpreted as a Doppler shift due to the relative expansion of twogalaxies. Let us denote the observed frequency as o and source frequency as s. The relativisticDoppler effect states that

o = (1

)s, =

v

c, =

11 2 . (2.17)If the velocities are non-relativistic 1, then = 1 and we can write the Doppler law

s o = s . (2.18)Suppose we have two galaxies (1 and 2) at a distance l H1. At these small scales we can identify alocal inertial frame in which the effects of general relativity can be neglected and space can be assumedto be flat (i.e., we can use the Special Theory of Relativity). Hubble law tells us that the galaxies aremoving apart with the velocity v = Hl.

Consider now a photon which has a frequency 1 in (source) galaxy 1 at the moment t1. Its frequency

is measured at a later time t2 when it reaches the (observation) galaxy 2. The shift in the frequency isgiven then by the Doppler law (2.18)

= (t1) (t2) (t1)v = (t1)H(t)l. (2.19)Since t = t2 t1 = l we can rewrite this as

t= = H(t) (2.20)

which has the solution

1a

. (2.21)

Applying this argument to small pieces of photon trajectories we conclude that our solution is valid ingeneral relativistic case as well.

2.5 Peculiar velocity induced redshifts

Motions that are superimposed on the general expansion of the universe were defined as peculiar ve-locities. We can expect redshifts originating from these velocities. Suppose we have two observers (1and 2). The observer 1 is located in the lab frame; the observer 2 is located at the origin of a framewhich moves with respect to the lab frame according to the Hubble law with the velocity v = H(t)l,where l is the distance between the origins of the two reference frames. The velocity measured bythe observer 1 is denoted by w1(t1) and that by the observer 2 by w2(t2). Then applying the Galileantransformation to connect the velocities w(t) in both frames we obtain

w(t1) w(t2) v = H(t)l. (2.22)We assume that the times t1 and t2 are infinitesimally close; in fact we will take the limit t1 t2 shortly.The time that will take a particle to move from observer 1 to observer 2 is t = t2 t1 = l/w. Thenfrom Eq. (2.22) w(t1) w(t2) = H(t)wt. Taking the limit t 0 we obtain

w = H(t)w w 1a

. (2.23)


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We see that the peculiar velocity of a particle decays inversely as the scale factor increases. We concludethat the overall expansion of the universe leads to a damping of the peculiar motions in the comoving

frame!

Consider as an example now a non-relativistic non-interacting gas. Its temperature is proportionalto their kinetic energy of particles, i.e., their peculiar velocity squared

Tnonrel.gas

w2

a2. (2.24)

This behavior differs from that for photons (radiation) for which T a1. We see that if radiationand (non-relativistic) matter are decoupled, then the gas cools faster than the radiation as the universeexpands. As in the case of radiation we can extend our argument to the case of General Relativity byconsidering piece-wise tracks with locally inertial systems attached to them.

2.5.1 Redshift as a measure of time

Next we define the redshift parameter (or simply redshift) through the following relation

z =

obs

em

em , (2.25)

where obs is the wave-length of the photon observed currently on the Earth, em is the wave-length ofthe photon emitted by a distant galaxy. However, according to Eq. (2.16) obs/em = aobs/aem and wecan write

1 + z =aobs

a(tem). (2.26)

We will take further the observation time tobs = t0, i.e., the contemporary time, and therefore a(tobs) =a0.

Equation (2.26) establishes a one-to-one correspondence between the time and the redshift. There-fore, we conclude that redshift can be used to parameterize the history of the universe, instead of time,

i.e. all the functions of time can be rewritten in terms of z. Applying Eq. (2.26) to the current time weobtain

1 + z =a0

a(tem). (2.27)

This relation tells us that the universe was 1 + z times smaller at redshift z than now.It is useful to rewrite some of our previous expressions in terms of the redshift. For example,

Eq. (1.56), which reads d = 3( + p)d ln a can be integrated to obtain(z)0

d

+ p()= 3 ln a(z)

a0,

(z)0

d

+p()= 3 ln(1 + z), (2.28)

where we inserted (2.26) for the ratio scale factor under the logarithm. Next, let us obtain the expressionfor the Hubble parameter H in terms of z. We modify the second Friedman equation (1.74) as

H2 +k

a2 H2 + k

a20(1 + z)2,

8G

3 =

8G

3cr

H20

0cr0

(z)

0, (2.29)

which gives us finally

H2(z) +k

a20(1 + z)2 = 0H

20

(z)

0. (2.30)


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2.5. PECULIAR VELOCITY INDUCED REDSHIFTS 31

Note that z = 0 point corresponds to the current time (i.e. all the quantities have an index 0) andz = corresponds to the initial time. If z = 0 then H(z) = H0 and Eq. (2.30) reduces to

k

a20= (0 1)H20 . (2.31)

This is an interesting relation, since it expresses the current value of the scale factor a0 in terms of

current values of Hubble constant H0 and cosmological parameter 0 provided the universe is not flat,i.e., k = 1. Now let us substitute (2.31) back into (2.30) to eliminate the factor k/a20; then

H(z) = H0

(1 0)(1 + z)2 + 0 (z)

0

1/2. (2.32)

This is an alternative to Eq. (2.30) which expresses the time dependence of the Hubble parameter interms of 0, 0 and (z).

Next we would like to express the time in terms of the redshift. Expression (2.26) 1 + z = a0/a(t)provides an indirect relation. To obtain a direct expression let us differentiate it

dz = a0a2(t)

a(t)dt = (1 + z)H(t)dt, (2.33)

which, after integrating, gives

t =

z

dz

H(z)(1 + z). (2.34)

Thus given (z) we can obtain from Eq. (2.30) H(z) after which we can integrate (2.34) to obtain thetime. Note that the upper limit z = corresponds to the initial time (t = 0).

2.5.2 Redshift as a measure of distance

If we measure the redshift of light from a distant galaxy we can determine the distance to that galaxy,i.e., the redshift can be used effectively to measure not only time but also distances. The comovingdistance is generally given by

= 0 em =t0tem

dt

a(t)=

1

a0

t0tem

dt(1 + z) (2.35)

where we have substituted a(t) = a0/(1 + z). According to Eq. (2.33) (1 + z)dt = dz/H(t) and wefind

(z) =1

a0 z

0

dz

H(z). (2.36)

If the universe is curved k = 0 then according to (2.31)1

a0=

|0 1|H0, (z) =

|0 1|H0z

0

dz

H(z). (2.37)

Now let us give an example, where we know the function (z). Such example is provided by thedust-dominated universe where (z) = 0(1 + z)

3. Substituting this in Eq. (2.32)

H(z) = H0(1 + z)1 + 0z. (2.38)


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Figure 2.2: Illustration of geometry of light emission by a standard ruler.

Further assume that the universe is flat 0 = 1. Then Eqs. (2.34) and (2.36) valid for this case can beintegrated analytically upon substituting (2.38) and we find

t(z) =2

3H0(1 + z)3/2, (z) =

2

a0H0

1 1

1 + z

. (2.39)

In the limit z 0 (current time) we obtain = 0 and t = 2/3H0, as expected.

2.6 Determining cosmological parameters

We already mentioned that there are two ways to determine the cosmological parameters. First, one canmeasure the angular size of certain objects and find the evolution of the angular size with the redshift.If these objects have about the same size (standard rulers) we can make conclusion about the dynamicsof the universe. Similarly, by using objects which have same total brightness (standard candles) wecan determine their apparent luminosities at different redshifts to fix the cosmological parameters. Byusing such observation one can study the recent expansion of the universe with different matter contentand distinguish between different models.

2.6.1 Standard rulers

We start by considering an object with the size l at distance em, where it emits light. Without lossof generality we can assume that the light propagates along the geodesics, and therefore the ends ofthis object have angular coordinates 0, 0 and 0, 0 + , (see Fig. 4.2) We are interested in photonsemitted by the endpoints of the standard ruler. The observer is located at comoving coordinate 0 = 0and observes at t = t0. The size of the object is given by the interval

l =

s2 = a(tem)(em), (2.40)


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2.6. DETERMINING COSMOLOGICAL PARAMETERS 33

where we used the metric (2.3)

ds2 = a2()

d2 d2 2()(d2 + sin2 d2) , (2.41)with = const for fixed conformal time and comoving coordinate. The angle is then given by

=l

a(tem)(em)

=l

a(0 em)(em)(2.42)

where we have replaced tem 0 em. Consider limiting cases: the object is close to us em 0 [i.e. the emission time em 0], therefore

a(0 em) a(0), (em) em, = la(0)em

=l

d, (2.43)

where d is the distance to the object and the second relation assumes approximately flat universe.The last expression is the one that we would expect from the Euclidean geometry.

the object is located far away (close to the particle horizon) em = 0

em

0, then

a(em) a0, (em) (p) = Const, (2.44)and the general formula for this case reads

la(em)

. (2.45)

We see that as a 0 the angle increases to infinity and the object covers the whole horizon.However, the luminosity of the object drops drastically with the distance (otherwise it will outshineall the objects in the vicinity of the observer).

To compare with concrete predictions of cosmological models we first express the angular size interms of redshift instead of emission time:

(z) = (1 + z)l

a0(em(z)), em(z) =

1

a0

z0

dz

H(z), (2.46)

where we used that (1 + z)/a0 = 1/a(tem) and (2.36) for em.This first formula can be applied to concrete cosmological models. Let us take the example of flat,

dust dominated universe, in which case em is given by Eq. (2.39) and Eq. (2.4) implies (em) = em.Then substituting em from Eq. (2.36) we obtain

(z) = lH0

2(1 + z)3/2

(1 + z)1/2 1 . (2.47)

The behavior of this curve is as follows: z 1, (z) z1, it has a minimum at 5/4 and is zfor z 1. If we could experimentally determine the angular diameter for several redshifts, we couldtest the cosmological models. However, such measurements are seldom because of the lack of standardrulers. The only successful example is the measurement of the cosmic microwave background (i.e.just one standard ruler). The measurements of acoustic peaks in the power spectrum of temperatureautocorrelation function allows to determine the cosmological parameter in an almost model independentway from the first acoustic peak. We will return to this problem in a later lecture.


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2.6.2 Standard candles

Consider a source located at the comoving coordinate em with total luminosity L. The energy outputat time tem within the conformal interval is given by

Eem = Ltem = La(tem). (2.48)

We see that the energy of emitted photons depends on the scale factor, however the conformal widthof the shell from which the photons are emitted = const. Therefore, at the moment of observationthe energy output will be

Eobs = Eema(tem)

a0= L

a2(tem)

a0. (2.49)

Generalizing Eqs. (1.41) and (1.43) for surface area of spheres in expanding universe we can write

S(t0) = 4a20

2(em), (2.50)

since em = obs = Const. The time interval needed for the shell to pass over the observers position ist = l, where l is the width of the shell is given by

l = a0 = a0. (2.51)

The measured energy per unit area per unit time is equal to

F =Eobs

S(t0)t=

L

S(t0)t

a2(tem)

a0=

L

42(em)t

a2(tem)

a30=

L

42(em)

a2(tem)

a40. (2.52)

But the ratio of the scale factors can be expressed via the redshift according to Eq. (2.26), so that

F =L

4a

2

02

[em(z)](1 + z)2

, (z) =1

a0 z

0

dza(z)2

a(z

)2

. (2.53)

In astronomy we measure the bolometric magnitude

mbol = 2.5log10 F = 2.5log10 L + 2.5log10(4a202[em(z)](1 + z)2)= 5 log10(1 + z) + 5 log10([em(z)]) + 2.5log10(4a

20) 2.5log10 L. (2.54)

The last two terms are constants independent of z. Now let us expand this function in the limit z 1(contemporary time). Since to leading order in small we have sin and sinh , thenindependent of the curvature of the universe we can substitute [em(z)] = em(z). The expansion toleading order gives

mbol(z) = 5 log10 z+2.5

ln10(1 q0)z+ O(z2), q = a

aH2. (2.55)

where q0 is the deceleration parameter defined as

q0 =1

20

1 + 3

p

0

. (2.56)

By determining the function mbol(z) experimentally from observations and fitting it by some choice ofparameters entering q0 we can determine these parameters, i.e., 0, p0 and 0.

Type IA supernovae are suitable standard candles. Their observation led to the following conclusions:


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Figure 2.3: Fitting the supernova data with models with various fractional contributions from matterand cosmological constant.

The expansion of the Universe is accelerating. From Friedman equation

a = 43

G( + 3p)a (2.57)

we see that the universe will be accelerating if a > 0, which means that the right hand side ofthis equation must be positive, which in turn means that a substantial fraction of the total energy

density is dark energy with negative pressure (e.g. p = w, with w being negative).

What is the origin of the dark energy? Cosmological constant, i.e., w = 1. Dynamical field, e.g., time-dependent scalar field (called quintessence).

Following problems of modern cosmology are related to the acceleration of the universe: Why the dark energy is dominating the universe at the current time (much later than the

birth of the universe)

The long term future of the universe can not be predicted since we can not predict thebehavior of the dark matter (e.g. the scalar field can dissipate by creating matter andradiation)

The expansion (2.55) is not valid for typical observed supernovae, since their redshifts are of order ofunity. One then needs to carry out a numerical study by using different models of the universe. Forexample, assume that the universe is flat and consists of cold dark matter and cosmological constant,i.e., 0 = + m = 1. Then

[em(z)] = em(z) =1

H0a0

z0

dz

m(1 + z)3 + (1 m)

. (2.58)


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With this assumption we can compute the exact expression (2.55) for bolometric luminosity and compareto observations. The data is best described by m = 0.3.

2.6.3 Further methods

Let us calculate the number of a galaxies at a given redshift. We will need to make some simplifications;

we assume that the density is uniform and is given by n(z). The differential number of the galaxies isgiven by

N = n(z)V(z) (2.59)

where V(z) is an element of volume at a distance z. To calculate the latter quantity we note that asurface element can be written

dS(z) = a(z)2((z))2d, (2.60)

where d is a element of a solid angle. This quantity should be multiplied with the third (physical)dimension d = a(z), so that we obtain

V = a(z)

3

2

()(z). (2.61)

Next we use that z = aH0 and 1 + z = a0/a(z) [Eq. (2.26)] to obtain

V = (1 + z)3a302()(a0H)

1z = (1 + z)3a202()H1(z)z. (2.62)

To show a concrete example, we substitute the following expression for 2(),

((z)) =2|0 1|20(1 + z)

{0z+ (0 2)[(1 + 0z)1/2 1]}, (2.63)

which is valid for dust dominated universe (obtained in an Exercise)

N

z=

4n(z)

H30 40

0z+ (0 2)[(1 + 0z)1/2 1]

2(1 + z)6(1 + 0z)1/2

. (2.64)

Given the function n(z) measurements of N/z can be used to test cosmological models. Thismethod does not take into account the fact that the galaxies undergo some evolution (e.g. merge andinteract); however this is not a real problem if we can compute the galaxy dynamics.

An additional, higher order effect arises from the evolution of the redshift. Consider again theexample of light emitted at the conformal time em at a distance which is observed at the current

conformal time 0. Then,z(0) =

a0ae

=a(0)

a( 0 emission t

). (2.65)

Now let us look at the time derivative of the redshift

dz(t)

dt=

1

a(0)

dz()

d0=

1

a(0)

d

d0

a(0)

a(0 ) =1

a(0)

a(0)

a(0 ) a(0)

a(0 )2 a(0 )

=

a0ae

aeae

(2.66)


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where prime denotes derivative with respect to conformal time and dot denotes derivative with respectto physical time (dt = ad). Further,

z =a0ae

aeae

=a0ae

1+z

a0a0

aeae

= (1 + z)H0 H(z). (2.67)

Consider a concrete example of universe which consists of a mixture of matter and vacuum density suchthat its energy density is given by

(z) = cr0 [ + m(1 + z)3]. (2.68)

We use the expression for the Hubble parameter (2.32)

H(z) = H0

(1 0)(1 + z)2 + 0 (z)

0

1/2. (2.69)

Substituting we obtain

z = (1 + z)H0

1 (1 0 + m(1 + z) + (1 + z)21/2 . (2.70)

Suppose the universe is flat: 0 = 1 and = 1m. The change in the redshift is u = z/(1 +z) =zt/(1 + z) and thus

u = tH0

(m(1 + z) + (1 m)(1 + z)21/2 1 . (2.71)

We see that when m 1, u > 0 and when m 0, u < 0. Suppose m = 1 and = 0 and wemake an observation with an interval 1 yr; then

v 2(1 + z 1) cm s1. (2.72)

Currently it is possible to measure shifts that are by several orders of magnitude larger than those thatare required. However, it is conceivable that in the future decades such a shift can be measured. Suchmeasurements will provide a direct probe of acceleration of the universe and should be a complementto other tests discussed above.


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Chapter 3

Thermal evolution of the universe

This chapter deals with the evolution of the universe at energies below few MeV. Compared to theearlier (higher-energy) stages the physics at this scales is well understood.

The main elements of the composition of the universe at this stage are:

Primordial radiation. This is the cosmic microwave radiation at temperature T = 2.73 K. We canscale back the temperature (T a1) to see that it was very high in the early universes. Thecurrent energy content of this radiation is negligibly small ( = 10

34 g cm3, which is only 105

fraction of the total energy density).

Baryonic matter. This is the component of matter from which ordinary objects are made. Thecontribution of the baryonic matter to the total energy density is small, of the order of severalpercent.

Dark matter. Experiments indicate that a large fraction of matter and energy of the universeis dark, i.e., invisible in ordinary light. The total of experimental data indicates that the dark

matter is cold and can contribute to the gravitational instability and structure formation in theearly universe; cold dark matter contributes about 1/4 of the critical density. A good candidatefor such matter are weakly interacting massive particles (WIMPs) predicted by supersymmetrictheories.

Dark energy is non-clustered, i.e., it does not participate in the gravitational instability forma-tion and has negative pressure, which is equivalent to a cosmological constant (p = ). Itconstitutes the significant rest 3/4 of the energy density of the universe.

Primordial neutrinos are relicts of the early stage of the evolution of the universe. They havenegligible contribution to the energy density of the universe and their temperature (in the case of

only three neutrino families) is 1.9 K.

The energy densities of these components (actually, those that contribute substantially to the totalenergy density) scale with the redshift as

D = cr0 D(1 + z)

3(1+w), m = cr0 m(1 + z)

3, = 0m(1 + z)4, (3.1)

where cr0 = 3H20 /8G is the current value of the critical density, m is the contribution of the baryonic

and cold dark matter to the cosmological parameter, D is the same for dark energy with w 1/3(currently). From the large z asymptotics we see that initially dominated the radiation, then the

39


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40 CHAPTER 3. THERMAL EVOLUTION OF THE UNIVERSE

Figure 3.1: The thermal evolution of the universe. This chapter deals with the stages below the fewMeV temperatures, i.e., nucleosynthesis and recombination.

baryonic matter, and finally the dark energy. The transition from baryonic to dark energy dominateduniverse occurs at modern times.

Comparing the redshifts at which the energy densities of these components become comparable weobtain three basic epochs of evolution in terms of redshift:

radiation dominated epoch zR > 104, ultra-relativistic matter p = /3 and a

t.

matter dominated epoch 104

zM 0.33 1.33, p = 0 component (dark matter) determines theevolution with a t2/3.

dark energy dominated epoch z < zM, the equation of state is p = w and a t2/3(1 + w), wherew is typically negative.

Let us take a closer look at the evolution of the universe and the main events that occur. If we startfrom cold universe and go back in time to the hot universe, there are roughly the following stages intimes/energies:

1016 1017 s. Galaxy formation through gravitational instability.


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3.1. PHYSICS AT LEPTON ERA 41

1012 1013 s. Free electrons and protons recombine to form neutral hydrogen. Last scatteringsurface forms for photons - imprints are seen in the CMB (Hintergrundstrahlung) temperaturefluctuations.

1011 s (T eV). Matter - radiation equipartition.

200

300 s (T

0.05 MeV). Light elements are formed from free neutrons and protons through

the nuclear reactions. We can measure the abundances of the light elements from primordialnucleosynthesis.

1 s (T 0.5 MeV) The temperature is of the order of the electron mass - electron-positronannihilation takes place leaving behind photons. (Only a small fraction of electrons survives).

0.2 s (T 12 MeV) Weak interactions are out of equilibrium and primordial neutrinos decouplefrom the rest of the matter. The ratio of neutrons to protons remains fixed as there is no chemicalequilibrium among them.

105 s/200 MeV. The quark-gluon phase transition takes place.

1010 1014 s/100 GeV-10 TeV. The electroweak symmetry is restored and gauge bosons aremassless.

1014 1043 s/10 TeV-1019 GeV. Up to these times/energies the classical general relativityremains likely intact. Matter composition is unclear. Possibly supersymmetry plays a role byrequiring that the supersymmetric partners of the particles in the standard model be present.

1043 s/1019 GeV (the Planckian scale). Classical gravity is not valid and should be replaced byquantum gravity. Several orders below this energy scale one expects a Grand Unification of allforces in Nature.

3.1 Physics at lepton era

In this section we consider matter below the phase transition from free quarks to baryons, i.e., thetemperature T 100 MeV and the time t > 104 s. At these energies the matter composition is givenby

n, p e+, e e, , . (3.2)

Other components are

, , . . . (, negligible) (3.3)

where . . . stand for other mesons. The leptons, i.e., electrons, muons, tauons and neutrinos play acrucial role in the reactions taking place at these times, therefore one calls this epoch lepton era.

For each conserved quantity we need one chemical potential; in the lepton era the conserved quan-tities are electric charge, baryon number, and lepton number. Suppose the particles pi, i = 1, 2, 3, 4 arein equilibrium via a reaction

p1 + p2 = p3 + p4. (3.4)

Then the equilibriummeans the equality of their chemical potentials

1 + 2 = 3 + 4. (3.5)


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Thus, in general we need to establish the equilibrium reactions among the particles, form which wedetermine the chemical potentials for each conserved quantity:

Q, B, l, (3.6)

i.e. we need three equations. The chemical potentials of anti-particles are equal to that of the particlein magnitude but have an opposite sign.

Let us outline how this works on an example where we have the following constituents of matter

n,p, (baryons) (3.7)

0, , (mesons) (3.8)

e,,,e, , (leptons). (3.9)

The reactions taking place in such a mixture and relation between the chemical potentials that followfrom application of the rules (3.5) are as follows:

n

p + e + e,

n = p + e + e. (3.10)

The life time of the pion is short (in free space) so that

0 , 0 = 0 = (3.11)

Heavy muon decays into light particles

e + e + e, = e e . (3.12)

The same applies for lepton

e + e + , = e e . (3.13)

Finally

e + e, = e + e. (3.14)Conservation of baryon number implies that the number of baryons minus the number of anti-baryonsscales as a3; if there is no dissipation in matter then the entropy scales also as a3 and the entropyper baryon remains constant:

B =1

s(np np + nn nn) = const. (3.15)

Conservation of the total electric charge is

Q =1

s

i=p,e,,

(ni ni). (3.16)

Finally the lepton number conservation for each lepton family f = e,, is given by

Li =1

s

f=e,,

(nf nf). (3.17)


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The equilibrium distribution functions for fermionic and bosonic constituents are (we omit here thedegeneracy factors that take into account their internal degrees of freedom, but will recover them inthe following section)

f =1

expET

+ 1

, g =1

expET

1 , (3.18)so that the densities defined as

nf =

d3p

(2)3f(E(p), , T ), ng =

d3p

(2)3g(E(p), , T ). (3.19)

The entropy conservation impliesd

dt

sa3

= 0; (3.20)

note that s is the entropy density and should be multiplied with the volume to yield the entropy ofthe system; since V a3, we obtain (3.20). The conservation laws above determine the six unknownfunctions

T, e, n, and f (f = e,,). (3.21)

The entire universe is electrically neutral, therefore

Q = 0. (3.22)

Baryon per entropy is established from observations to be in the range

B 1010 109. (3.23)

The lepton to entropy ratio is not well established. From the arguments that will become clear in thelater chapters, the numbers of leptons and baryons can not be very different; this implies that

|Li| < 109. (3.24)

When the temperature is much larger than the mass m of the particles - many particle and anti-particlepairs are created from vacuum, i.e., the number of pairs are of the order of the number of photons, whichscales as n T3. What happens when T < m? We might expect that the pairs annihilate until asmall number of particles are left. We will need to recover some basic relations from thermodynamicsin order to quantify the particle number excess.

3.1.1 A recourse in thermodynamics

Let us recall some basics thermodynamics. We have pointed out above that the equilibrium distributionfunctions for fermions and bosons are given in terms of the Fermi-Dirac and Bose-Einstein distributionfunctions as

f =1

expET

+ 1

, g =1

expET

1 . (3.25)The density of the Fermi/Bose particles is then given as

n = g

0

d3p

(2)31

exp ET 1

, (3.26)


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where the upper sign corresponds to fermions and the lower sign to bosons, g is the degeneracy factor(g = 2s +1, where s is the value of the spin of particles). Suppose now the particle spectrum is given byE =

p2 + m2, which corresponds to an ideal, i.e., non-interacting gas of relativistic particles. Then

we have

EdE = pdp, 0 p , m E . (3.27)Changing the integration in Eq. (3.3) from momentum to energy we obtain

n =g

22

m

dE E

E2 m2

expET

1 , (3.28)where we have carried out the angular integration over the phases d = sin dd = 4, since thespectrum and, therefore, the integrand are independent of direction. The energy of the system can bewritten as

= g

0

d3p

(2)3E

exp

ET

1 =g

22

m

dE E2

E2 m2exp

ET

1 . (3.29)

The expression for the pressure is then given by1

p =

3 m

2g

62

m

E2 m2

expET

1 . (3.30)Adding the contribution from anti-particles amounts to adding identical integrals, but with .Thus, we see that to compute the thermodynamics quantities of non-interacting relativistic gas we needintegrals of the type

I() (, ) =

(x2 2)/2ex

1

+ ( ), (3.31)

where we have introduced non-dimensional quantities = m/T, = /T and the variable x = E/T.Using these definitions we can write the previous expression in a compact form. The energy becomes

+ =gT4

22

I(3) +

2I(1)

. (3.32)

For the pressure we obtain

p =gT4

62I

(3) . (3.33)

Finally, the excess of particles over the anti-particles can be written as

n n = gT3

62

I(3) . (3.34)

While the integrals (3.31) are easily calculated numerically it is instructive to consider several limitingcases.

1We will accept this expression without proof. See any text on thermodynamics for further detail.


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High temperature expansion of the integrals

Consider the limit where T m, we have , 1. and the corresponding expansions for fermionsand bosons

I(1) =

3

3 2

2

2 2 1

22 + O(4, 22),

2

6+

2

2+

2

2 + O(4, 22),

(3.35)

where CF = 0.577 is the Euler number and

=

4+ CE 1

2

. (3.36)

Further

I(3) =

24

15+

2

2(22 2) + (2 2)3/2 + O(6, 42),

74

60+

2

4(22 2) + 3

4(ln 2)4 + O(4, 42),

(3.37)

where

=1

8 24 622 34 ln

CE

4e3/4 . (3.38)Low temperature expansion of the integrals

The low temperature limit is given by the conditions = m/T 1 and 1. In this limit weobtain

I(1) =

2e cosh

1 +

3

8+ O(2)

, (3.39)

and

I(3) =

183e cosh

1 +

15

8+ O(2)

. (3.40)

With this formulae we can compute the thermodynamic quantities of of non-relativistic particles in thelimit 1. It is clear that in this case the Fermi and Bose distribution functions reduce to theBoltzmann distribution function, i.e., the thermodynamic parameters are the same for both statistics.

3.1.2 Thermodynamics of ultra-relativistic and non-relativistic gases

Now using the general formulae we can obtain explicit results for some specific examples that are ofinterest in cosmology.

Ultrarelativistic particles: bosons

The form of the Bose distribution function implies that in order the distribution to be positive definitethe chemical potential of a boson can not exceed its mass m. We take the high-temperatureexpansion for bosons (3.37) and evaluate the particle-anti-particle asymmetry from (3.34). The resultis

n n = gT3

3T, ( < m). (3.41)

Note that this equation does not take into account the possibility of Bose-Einstein condensation, inwhich case a macroscopically large number of particles can occupy the ground state with

= limp0

p2 + m2 = m. (3.42)


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For ultra-relativistic particles p; setting = 0 in Eq. (3.26) we obtain the estimate

n (3)gT3

2. (3.43)

Note that Eqs. (3.41) and (3.43) can not be compared to each other since the latter is taken in the limitof Bose-Einstein condensation while the former excludes that limit. We stress again that The number

of particles that can be accommodated by Bose condensate in its ground state can be macroscopicallylarge.

However, in the absence of Bose-Einstein condensation then nn n and particle and anti-particlescontribute equally to the thermodynamic quantities - energy density, pressure, and entropy

=1

2( + )

2gT4

30, (3.44)

p =

3, (3.45)

s =4

3T=

24

45(3)n. (3.46)

Ultrarelativistic particles: fermions

For fermions the chemical potential is not bounded from above. We take the ultrarelativistic limit byformally putting m = 0, i.e., = 0 in Eq. (3.37) to obtain

+ =72

120gT4

1 +

302

72+

154

74

, (3.47)

which is valid at arbitrary chemical potentials for ultrarelativistic gas. As before we have p = /3.Using Eq. (3.34) we obtain for particle-antiparticle asymmetry

n n = gT3

6

1 +

2

2

. (3.48)

The entropy can be computed from a thermodynamical equation s = T1( + p n):

s + s =72

90gT3

1 +

152

72

. (3.49)

As noted Eqs. (3.1.3)-(3.49) are valid at any . Let us consider now some limiting cases.Large . This constitutes the degenerate Fermi gas, where T

. In that case the rightmost terms

in these equations are dominant. The energy density and entropy of a degenerate Fermi gas to leadingorder in /T is then given by

+ =g4

82+

gT22

4, (3.50)

s + s =g2T

4. (3.51)

We see that to leading order the energy of degenerate gas is independent of temperature and is entirelydue to the degeneracy pressure (Pauli principle). The next to leading correction arises from the narrow


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shell around the Fermi surface. The entropy again is linear in temperature and vanishes in the zerotemperature limit (as it should).

Small . This corresponds to /T = 1, i.e., we can neglect the chemical potentials in (3.26).The integral is the straightforward and gives

n =3(3)gT3

42. (3.52)

At the same time from the general equation (3.48) we obtain in this limit

n n gT3

6 n, (3.53)

i.e., the particle over anti-particle densities are parametrically suppressed by compared to the particledensity. For the remaining parameters one finds

=72

240gT4, p =

3, s =

4

3T. (3.54)

If we have simultaneously bosons and fermions with the same number of internal degrees for freedom(i.e. g is the same) then their entropies are related by

sfermions =7

8sbosons. (3.55)

Non-relativistic particles

Here we use the results obtained in the case of low-temperature expansion. We saw that it is valid inthe limit = (m )/T 1. In this case the exponential factor is larger than the unity and wehave Boltzmann distribution function. For this case we use again Eq. (3.34) to obtain

n n 2gT m

2

3/2exp

m

T

sinh

T

1 +

15T

8m

. (3.56)

The number density of the particle is given by

n g

T m

2

3/2exp

m

T

sinh

T

1 +

15T

8m

. (3.57)

The number density of anti-parti

lecture notes in cosmology

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