4 Friedmann–Robertson–Walker Universehkurkisu/cosmology/Cosmo4.pdf · 4 FRIEDMANN–ROBERTSON–WALKER UNIVERSE 37 • “Matter” (called “matter” in cosmology, but “dust”

4 Friedmann–Robertson–Walker Universe

We shall now apply the Einstein equation to the homogeneous and isotropic case,which leads to Friedmann–Robertson–Walker (FRW) cosmology. The metric is nowthe Robertson–Walker metric,

ds2 = −dt2 + a2(t)

[dr2

1 −Kr2+ r2dϑ2 + r2 sin2 ϑ dϕ2

]

. (1)

Calculating the Einstein tensor from this metric gives

G00 =3

a2(a2 +K) (2)

G11 = − 1

a2(2aa+ a2 +K) = G22 = G33. (3)

We use here the orthonormal basis (signified by theˆover the index).We assume the perfect fluid form for the energy tensor

T µν = (ρ+ p)uµuν + pgµν . (4)

(For the present universe this implies a very large-scale viewpoint, where individualgalaxies are “microscopic fluid particles”, too small to be resolved). Isotropy impliesthat the fluid is at rest in the Robertson–Walker coordinates, so that uµ = (1, 0, 0, 0)and (remember, gµν = ηµν = diag(−1, 1, 1, 1))

T µν =

ρ 0 0 00 p 0 00 0 p 00 0 0 p

. (5)

Homogeneity implies that ρ = ρ(t), p = p(t).The Einstein equation becomes now

3

a2(a2 +K) = 8πGρ (6)

−2a

a−

(a

a

)2

− K

a2= 8πGp . (7)

Let us rearrange this pair of equations to

(a

a

)2

+K

a2=

8πG

3ρ (8)

a

a= −4πG

3(ρ+ 3p) . (9)

These are the Friedmann equations. (“Friedmann equation” in singular refers toEq. (8).)

The general relativity version of energy and momentum conservation, energy-momentum continuity, follows from the Einstein equation. In the present case thisbecomes the energy continuity equation

ρ = −3(ρ+ p)a

a. (10)

35

4 FRIEDMANN–ROBERTSON–WALKER UNIVERSE 36

Since the fluid is at rest, there is no equation for the momentum. (Exercise: Derivethis from the Friedmann equations!)

We define H ≡ a/a. This quantity H = H(t) gives the expansion rate ofthe universe, and it is called the Hubble parameter. Its present value H0 is theHubble constant. The dimension of H is 1/time (or velocity/distance). In time dt adistance gets stretched by a factor 1 +Hdt (a distance L grows with velocity HL).The Friedmann equation (8) connects the three quantities, the density ρ, the spacecurvature K/a2, and the expansion rate H of the universe,

ρ =3

8πG

(

H2 +K

a2

)

= ρc +3K

8πGa2. (11)

(Note that the curvature quantity K/a2 is invariant under the r coordinate scalingwe discussed earlier.) Here we have defined the critical density

ρc(t) ≡3H2

8πG, (12)

corresponding to a given value of the Hubble parameter.1 Defined this way, thecritical density changes as the Hubble parameter evolves. Usually, by critical densitywe mean its present value, given by the value of the Hubble constant,

ρc ≡ ρc(t0) ≡3H2

0

8πG. (13)

The nature of the curvature then depends on the density ρ :

ρ < ρc ⇒ K < 0 (14)

ρ = ρc ⇒ K = 0 (15)

ρ > ρc ⇒ K > 0 . (16)

(17)

The density parameter Ω(t) is defined

Ω(t) ≡ ρ(t)

ρc(t). (18)

Thus Ω = 1 implies a flat universe, Ω < 1 an open universe, and Ω > 1 a closeduniverse. The Friedmann equation can now be written as

Ω(t) = 1 +K

H2a2, (19)

a very useful relation. Here K is a constant, and the other quantities are functionsof time Ω(t), H(t), and a(t). Note that if Ω < 1 (or > 1), it will stay that way. Andif Ω = 1, it will stay constant, Ω = Ω0 = 1. Observations suggest that the densityof the universe today is close to critical, Ω0 ≈ 1.

To solve the Friedmann equations, we need the equation of state p(ρ). Thesimplest cases are

1We could also define likewise a critical Hubble parameter Hc corresponding to a given density ρ,but since, of the above three quantities, the Hubble constant has usually been the best determinedobservationally, it has been better to refer other quantities to it.


• “Matter” (called “matter” in cosmology, but “dust” in general relativity),meaning nonrelativistic matter (particle velocities v ≪ 1), for which p ≪ ρ,so that we can forget the pressure, and approximate p = 0. From Eq. (10),d(ρa3)/dt = 0, or ρ ∝ a−3.

• “Radiation”, meaning ultrarelativistic matter (where particle energies are ≫their rest masses, which is always true for massless particles like photons), forwhich p = ρ/3. From Eq. (10), d(ρa4)/dt = 0, or ρ ∝ a−4.

• Vacuum energy (or the cosmological constant), for which ρ = const. FromEq. (10) follows the equation of state for vacuum energy: p = −ρ. Thusa positive vacuum energy corresponds to a negative vacuum pressure. Youmay be used to pressure being positive, but there is nothing unphysical aboutnegative pressure. In other contexts it is often called (positive) “tension”instead of (negative) “pressure”.2

We know that the universe contains ordinary, nonrelativistic matter. We alsoknow that there is radiation, especially the cosmic microwave background. In Chap-ters 5 and 6 we shall discuss how the different known particle species behave asradiation in the early universe when it is very hot, but as the universe cools, themassive particles change form ultrarelativistic (radiation) to nonrelativistic (mat-ter). During the transition period the pressure due to that particle species fallsfrom p = ρ/3 to p ∼ 0. We shall discuss these transition periods in Chapter 6. Inthis chapter we focus on the later evolution of the universe (after big bang nucleosyn-thesis). Then the known forms of matter and energy in the universe can be dividedinto these two classes: matter (p ≈ 0) and radiation (p ≈ ρ/3). Except that we donot know the small masses of neutrinos. Depending on the values of these masses,neutrinos may make this radiation-to-matter transition sometime during this “laterevolution”.

We already revealed in Chapter 1 that the present observational data cannotbe explained in terms of known forms of particles and energy using known laws ofphysics, and therefore we believe that there are other, unknown forms of energy inthe universe, called “dark matter” and “dark energy”. Dark matter has by definitionnegligible pressure, so that we can ignore its pressure in the Friedmann equations.However, to explain the observed expansion history of the universe, an energy com-ponent with negative pressure is needed. This we call dark energy. We do not knowits equation of state. The simplest possibility for dark energy is just the cosmo-logical constant (vacuum energy), which fits the data perfectly. Therefore we shallcarry on our discussion assuming three energy components: matter, radiation, andvacuum energy. We shall later comment on how much current observations actuallyconstrain the equation of state for dark energy.

If the universe contains these three energy components, we can arrange Eq. (8)

2In Chapter 5 we derive formulae for the pressures of different particle species in thermal equilib-rium. These always give a positive pressure. The point is that there we ignore interparticle forces.To make the pressure from particles negative, would require strong attractive forces between parti-cles. But the vacuum pressure is not from particles, its from the vacuum. If the dark energy is notjust vacuum energy, it is usually thought to be some kind of field. For fields, a negative pressurecomes out more naturally than for particles.


into the form (a

a

)2

= α2a−4 + β2a−3 −Ka−2 +1

3Λ, (20)

where α, β, K, and Λ are constants.3 The four terms on the right hand side are dueto radiation, matter, curvature, and vacuum energy, in that order. As the universeexpands (a grows), different components on the right hand side become importantat different times. Early on, when a was very small, the universe was radiation-dominated. If the universe keeps expanding without limit, eventually the vacuumenergy will become dominant (it may already be the largest term). In the middlewe may have matter-dominated and curvature-dominated eras.

We know that the radiation component is insignificant at present, and we canforget it in Eq. (20), if we exclude the first few million years of the universe fromdiscussion.

In the “inflationary scenario”, there was something resembling a very large vac-uum energy density in the very early universe (during the first small fraction ofthe first second), which then disappeared. So there may have been a very early“vacuum-dominated” era (inflation).

Let us now solve the Friedmann equation for the case where one of the four termsdominates. The equation has the form

(a

a

)2

= α2a−n or an

2−1 da = α dt. (21)

Integration gives2

na

n2 = αt, (22)

where we chose t = 0, so that a(t = 0) = 0. We get the three cases:

n = 4 radiation dominated a ∝ t1/2

n = 3 matter dominated a ∝ t2/3

n = 2 curvature dominated (K < 0) a ∝ tThe cases K > 0 and vacuum energy have to be treated differently (exercise).

Example: Age of the flat universe with Λ = 0. Consider the simplest case, Ω = 1(K = 0) and Λ = 0. The first couple of million years when radiation can not be ignored,makes an insignificant contribution to the present age of the universe, so we can ignoreradiation also. We have now the matter-dominated case. For the density we have

ρ = ρ0

(a0

a

)3

= Ω0ρc

(a0

a

)3

= ρc

(a0

a

)3

. (23)

The Friedmann equation is now

(a

a

)2

=8πG

3ρc

︸︷︷︸

H2

0

(a0

a

)3

⇒ a1/2da = H0a3/20dt

⇒∫ a2

a1

a1/2da = H0a3/20

∫ t2

t1

dt ⇒ 2

3(a

3/22

− a3/21

) = H0a3/20

(t2 − t1).

3Here we ignore any transfer of energy between these components. This kind of transfer isimportant only in the early universe, before big bang nucleosynthesis. In Chapter 6 we return tothe question of the early universe to discuss its expansion law more exactly.


Thus we get

t2 − t1 =2

3H−1

0

[(a2

a0

) 3

2

−(a1

a0

) 3

2

]

=2

3H−1

0

[1

(1 + z2)3/2− 1

(1 + z1)3/2

]

(24)

where z is the redshift.

• Let t2 = t0 be the present time (z = 0). The time elapsed since t = t1 correspondingto redshift z is

t0 − t =2

3H−1

0

[

1 −(a1

a0

) 3

2

]

=2

3H−1

0

[

1 − 1

(1 + z)3/2

]

. (25)

• Let t1 = 0 and t2 = t(z) be the time corresponding to redshift z. The age of theuniverse corresponding to z is

t =2

3H−1

0

(a2

a0

) 3

2

=2

3H−1

0

1

(1 + z)3/2= t(z). (26)

This is the age-redshift relationship. For the present (z = 0) age of the universe weget

t0 =2

3H−1

0. (27)

The Hubble constant is H0 ≡ h·100 km/s/Mpc = h/(9.78 × 109 yr), or H−1

0= h−1 · 9.78 ×

109 yr. Thus

t0 = h−1 · 6.52 × 109yr =

9.3 × 109 yr h=0.7

13.0 × 109 yr h=0.5(28)

The ages of the oldest stars appear to be at least about 12 × 109 years. Considering the

HST value for the Hubble constant (h = 0.72 ± 0.08), this model has an age problem.

Cosmological parameters

If the universe does not have the critical density, then we have to include the spatialcurvature term. Also, the vacuum energy may be nonzero and important today. Wedivide the density into its matter, radiation, and vacuum components ρ = ρm +ρr +ρvac, and likewise for the density parameter, Ω = Ωm +Ωr +ΩΛ, where Ωm ≡ ρm/ρc,Ωr ≡ ρr/ρc, and ΩΛ ≡ ρvac/ρc ≡ Λ/3H2. Ωm, Ωr, and ΩΛ are functions of time(although ρvac is constant, ρc(t) is not). In standard notation, Ωm, Ωr, and ΩΛ

denote the present values of these density parameters, and we write Ωm(t), Ωr(t),and ΩΛ(t), if we want to refer to their values at other times. Thus we write

Ω0 ≡ Ωm + Ωr + ΩΛ. (29)

The present radiation density is relatively small, Ωr ∼ 10−4 (we shall calculate it inChapter 6). So we usually write just

Ω0 = Ωm + ΩΛ. (30)

The radiation density is also known very accurately from the temperature of thecosmic microwave background, and therefore Ωr is not usually considered as a cos-mological parameter (in the sense of an inaccurately known number that we try to


fit with observations). The FRW cosmological model is thus defined by giving thepresent values of the three cosmological parameters, H0, Ωm, and ΩΛ.

Observations favor the values h ∼ 0.7, Ωm ∼ 0.3, and ΩΛ ∼ 0.7. (We havealready discussed the observational determination of H0. We shall discuss the ob-servational determination of Ωm and ΩΛ both in this Chapter and later.)

Since the critical density is ∝ h2, it is often useful to use instead the “physical”or “reduced” density parameters, ωm ≡ Ωmh

2, ωr ≡ Ωrh2, which are directly pro-

portional to the actual densities in kg/m3. (An ωΛ turns out not be so useful andis not used.)

Age of the FRW universe

Consider now the general case with arbitrary values of Ωm and ΩΛ. Now the Fried-mann equation has four terms on the right hand side,

(a

a

)2

=8πG

3Ωrρc

︸︷︷︸

ΩrH20

(a0

a

)4

+8πG

3Ωmρc

︸︷︷︸

ΩmH20

(a0

a

)3

+ ΩΛH20 − K

a2

⇒ da

dt= H0a0

√

Ωra20a

−2 + Ωma0a−1 + ΩΛa2a−20 −KH−2

0 a−20 .

Defining

x ≡ a

a0

=1

1 + z, (31)

we get1

a0

da

dt≡ dx

dt= H0

√

Ωrx−2 + Ωmx−1 + ΩΛx2 + (1 − Ω0), (32)

where we have used Eq. (19). We shall later have much use for this convenientform of the Friedmann equation. Now we integrate from it the time it takes for theuniverse to expand from a1 to a2, or from redshift z1 to z2,

∫ t2

t1

dt = H−10

∫ x2

x1

dx√

(1 − Ω0) + Ωrx−2 + Ωmx−1 + ΩΛx2(33)

This is integrable to an elementary function if two of the four terms under the rootsign are absent.

From this we get the age-redshift relationship

t(z) =

∫ t

0

dt = H−10

∫ 1

1+z

0

dx√

(1 − Ω0) + Ωrx−2 + Ωmx−1 + ΩΛx2. (34)

(This gives t(z), that is, t(a). Inverting this function gives us a(t), the scale factoras a function of time. Now a(t) is not necessarily an elementary function, even ift(a) is. Sometimes one can get a parameter representation a(ψ), t(ψ) in terms ofelementary functions.)

In Fig. 1 we have integrated Eq. (32) from the initial conditions a = a0, a = H0a0,both backwards and forwards from the present time t = t0 to find a(t) as a functionof time.


-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5time (Hubble units)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

a/a 0

Matter only

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5time (Hubble units)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

a/a 0

Flat universe

Figure 1: The expansion of the universe a(t) for a) the matter-only universe ΩΛ = 0,Ωm = 0, 0.2,. . . ,1.8 (from top to bottom) b) the flat universe Ω0 = 1 (ΩΛ = 1 − Ωm),Ωm = 0, 0.05, 0.2, 0.4, 0.6, 0.8, 1.0, 1.05 (from top to bottom). The time axis givesH0(t− t0), i.e, 0.0 corresponds to the present time.

For the present age of the universe we get

t0 =

∫ t0

0

dt = H−10

∫ 1

0

dx√

(1 − Ω0) + Ωrx−2 + Ωmx−1 + ΩΛx2. (35)

where ignoring the Ωr term causes negligible error.

Example: Age of the open universe. Consider now the case of the open universe (K < 0or Ω0 < 1), but without vacuum energy (ΩΛ = 0), and approximating Ωr ≈ 0. IntegratingEq. (35) (e.g., with substitution x = Ωm

1−Ωm

sinh2 ψ2) gives for the age of the open universe

t0 = H−1

0

∫ 1

0

dx√1 − Ωm + Ωmx−1

= H−1

0

[1

1 − Ωm− Ωm

2(1 − Ωm)3/2arcosh

(2

Ωm− 1

)]

. (36)

A special case of the open universe is the empty, or curvature-dominated, universe(Ωm = 0 and ΩΛ = 0). Now the Friedmann equation says dx/dt = H0, or a = a0H0t, andt0 = H−1

0.

From the cases considered so far we get the following table for the age of the universe:

Ωm ΩΛ t00 0 H−1

0

0.1 0 0.90H−1

0

0.3 0 0.81H−1

0

0.5 0 0.75H−1

0

1 0 (2/3)H−1

0

There are many ways of estimating the matter density Ωm of the universe, some of which

are discussed in Chapter 8. These estimates give Ωm ∼ 0.3. With Ωm = 0.3, ΩΛ = 0 (no

dark energy), and h = 0.72, we get t0 = 12.2 × 109 years. This is about the same as the

lowest estimates for the ages of the oldest stars. Since it should take hundreds of millions of


years for the first stars to form, the open universe (or in general, a no-dark-energy universe,

ΩΛ = 0) seems also to have an age problem.

The cases (Ωm > 1, ΩΛ = 0) and (Ω0 = Ωm + ΩΛ = 1, ΩΛ > 0) are left asexercises. The more general case (Ω0 6= 1, ΩΛ 6= 0) leads to elliptic functions.

Distance-redshift relationship and the horizon

In cosmology, the typical velocities of observers (with respect to the comoving coor-dinates) are small, v <1000 km/s, so that we do not have to worry about Lorentzcontraction. The expansion of the universe brings, however, other complications tothe concept of distance. Do we mean by the distance of a galaxy how far it is now(longer), how far it was when the observed light left the galaxy (shorter), or thedistance the light has traveled (intermediate)? By comoving distance we mean theproper distance scaled to the present value of the scale factor (or sometimes to someother special time we choose as the reference time). If the objects have no peculiarvelocity their comoving distance at any time is the same as their distance today.

From the Robertson-Walker metric,

ds2 = −dt2 + a2(t)

[dr2

1 −Kr2+ r2dϑ2 + r2 sin2 ϑ dϕ2

]

, (37)

we already got that the proper distance from the origin to the coordinate value r ata given time t is (I have changed the notation from s to d)

d(t) =

∫ r

0

a(t)dr√

1 −Kr2=

a(t)K−1/2 arcsin(K1/2r) ,K > 0

a(t) r ,K = 0

a(t)|K|−1/2arsinh(|K|1/2r) ,K < 0

(38)

To facilitate handling all three cases simultaneously, we define the “generalizedsine”,

Sk(x) ≡

sin(x) , k = 1

x , k = 0

sinh(x) , k = −1

(39)

and its slightly more generalized version

SK(x) ≡

K−1/2 sin(K1/2x) , K > 0

x , K = 0

|K|−1/2 sinh(|K|1/2x) . K < 0

(40)

We write S−1

k and S−1K for their inverse functions. The functions SK and S−1

K convertbetween the two natural “unscaled” (i.e., you still need to multiply this distance bythe scale factor a) radial distance definitions for the RW metric: d/a, the properdistance measured along the radial line, and r which is related to the length of thecircle and area of the sphere at this distance with the familiar 2πr and 4πr2. Thuswe have

d(t) = a(t)S−1K (r) = a(t)|K|−1/2S−1

k (|K|1/2r) (41)

for the proper distance.


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1

-0.5

0

0.5

1

1.5

2

open --- closed

accelerating --- decelerating

no big bang

recollapses eventually

Ωm

ΩΛ

Age of the universe / H0-1

0.55

0.6

0.650.7

0.75

0.8

0.70.

750.8

0.9

11.2

1.5

2

Fig. by E. Sihvola

Figure 2: The age of the universe as a function of Ωm and ΩΛ.


Figure 3: Calculation of the distance-redshift relationship.

As the universe expands, this distance grows,

d(t) = a(t)S−1K (r) =

a0

1 + zS−1

K (r) =d0

1 + z, (42)

where d0 is the present proper distance to r, or the comoving distance to r.Neither the distance d, nor the coordinate r of a galaxy are directly observable.

Observable quantities are, e.g., the redshift z, the angular diameter, and the appar-ent luminosity. We want to use the FRW model to relate these observable quantitiesto the coordinates and actual distances.

Let us first derive the redshift-distance relationship. We see a galaxy with redshiftz; how far is it? (We assume z is entirely due to the Hubble expansion, 1 + z =a/a0, i.e., we ignore the contribution from the peculiar velocity of the galaxy or theobserver).

Since for light,

ds2 = −dt2 + a2(t)dr2

1 −Kr2= 0, (43)

we have

dt = −a(t) dr√1 −Kr2

⇒∫ t0

t1

dt

a(t)=

∫ r

0

dr√1 −Kr2

=d

a=d0

a0

. (44)

The comoving distance to redshift z is thus

d0(z) = a0

∫ t0

t1

dt

a(t)=

∫dt

x=

∫dx

x

1

dx/dt, (45)


0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0redshift

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

dist

ance

(H0-1

)

Matter only

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0redshift

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

dist

ance

(H0-1

)

Flat universe

Figure 4: The distance-redshift relationship, Eq. (46), for a) the matter-only universeΩΛ = 0, Ωm = 0, 0.2,. . . ,1.8 (from top to bottom) b) the flat universe Ω0 = 1 (ΩΛ = 1−Ωm),Ωm = 0, 0.05, 0.2, 0.4, 0.6, 0.8, 1.0, 1.05 (from top to bottom). The thick line in both casesis the Ωm = 1, ΩΛ = 0 model.

where x ≡ a/a0 = 1/(1 + z). We have dx/dt from Eq. (32), giving

d0(z) =

∫dx

x

1

H0

√

Ωmx−1 + Ωrx−2 + ΩΛx2 + (1 − Ω0)

= H−10

∫ 1

1

1+z

dx√

ΩΛx4 + (1 − Ω0)x2 + Ωmx+ Ωr

≈ H−10

∫ 1

1

1+z

dx√

ΩΛx4 + (1 − Ω0)x2 + Ωmx

= H−10

∫ 1

1

1+z

dx√

Ω0(x− x2) − ΩΛ(x− x4) + x2, (46)

where we have dropped the Ωr term, which has negligible effect, and utilized Ωm =Ω0 − ΩΛ to get the last form. This is the distance-redshift relationship. We seethat it depends on three independent cosmological parameters, for which we havetaken H0, Ω0, and ΩΛ. In this parametrization, the distance at a given redshift isproportional to the Hubble distance, H−1

0 . If we give the distance in units of H−10 ,

then it depends only on the two remaining parameters, Ω0 and ΩΛ.If we increase Ω0 keeping ΩΛ constant (meaning that we increase Ωm), the dis-

tance corresponding to a given redshift decreases. This is because the universe hasexpanded faster in the past (see Fig. 1), so that there is less time between a givena = a0/(1+z) and the present. The distance to the galaxy with redshift z is shorter,because photons have had less time to travel. Whereas if we increase ΩΛ with a fixedΩ0 (meaning that we decrease Ωm), we have the opposite situation and the distanceincreases. (Note that (x− x2) and (x− x4) are always positive in Eq. 46).

If the galaxy has stayed at the same coordinate value r, i.e., it has no peculiarvelocity, then the comoving distance is equal to its present distance. The actual


distance to the galaxy at the time t1 the light left the galaxy is

d1(z) =d0(z)

1 + z. (47)

We encounter the beginning of time, t = 0, at a = 0 or z = ∞. Thus thecomoving distance light has traveled during the entire age of the universe is

dhor = H−10

∫ 1

0

dx√

ΩΛx4 + (1 − Ω0)x2 + Ωmx+ Ωr

. (48)

This distance (or the sphere with radius dhor, centered on the observer) is calledthe horizon, since it represent the maximum distance we can see, or receive anyinformation from.

There are actually several different concepts in cosmology called the horizon. Tobe exact, the one defined above is the particle horizon. Another horizon concept isthe event horizon, which is related to how far the light can travel in the future. TheHubble distance H−1 is also often referred to as the horizon (especially when onetalks about subhorizon and superhorizon distance scales).

Distance and redshift in the flat matter-dominated universe

Let us look at the simplest case, (Ωm,ΩΛ) = (1, 0) (with Ωr ≈ 0), in more detail.Now Eq. (46) is just

d0(z) = H−10

∫ 1

1

1+z

dx

x1/2= 2H−1

0

(

1 − 1√1 + z

)

. (49)

Expanding 1/√

1 + z = 1 − 12z + 3

8z2 − 5

16z3 · · · we get

d0(z) = H−10 (z − 3

4z2 +

5

8z3 − · · · ) (50)

so that for small redshifts, z ≪ 1 we get the Hubble law, z = H0d0. At the timewhen the light we see left the galaxy, its distance was

d1(z) =1

1 + zd0(z) = a(t)r = 2H−1

0

(1

1 + z− 1

(1 + z)3/2

)

(51)

= H−10 (z − 7

4z2 +

19

8z3 − · · · ) (52)

so the Hubble law is valid for small z independent of our definition of distance.The distance d(t) = a(t)r to the galaxy grows with the velocity d = ra = raH,

so that today d = ra0H0 = d0H0 = 2(1 − 1/√

1 + z). This equals 1 (the speed oflight) at z = 3.

We note that d1(z) has a maximum d1(z) = 827H−1

0 at z = 54

(1 + z = 94). This

corresponds to the comoving distance d0(z) = 2

3H−1

0 . See Fig. 6. Galaxies that arefurther out were thus closer when the light left, since the universe was then so muchsmaller.

The distance to the horizon in this simplest case is

dhor ≡ d0(z = ∞) = 2H−10 = 3t0. (53)


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1

-0.5

0

0.5

1

1.5

2

open --- closed

accelerating --- decelerating

no big bang

recollapses eventually

Ωm

ΩΛ

Horizon / H0-1

1.6

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1.8

1.92

2.4

3

22.2

2.4

2.7

3

3.5

4

5

67

Fig. by E. Sihvola

Figure 5: The horizon as a function of Ωm and ΩΛ.


Figure 6: Spacetime diagrams for a flat universe giving a) the actual distance b) thecomoving distance from origin as a function of cosmic time.

Just like any planar map of the surface of the Earth must be distorted, so is thatof the curved spacetime. Although this simplest case has flat space, the spacetimeis curved due to the expansion. Thus any spacetime diagram is a distortion ofthe true situation. In Figs. 6 and 7 there are three different ways of drawing thesame spacetime diagram. In the first one the vertical distance is proportional to thecosmic time t, the horizontal distance to the actual distance at that time, d1. Thesecond one is in the comoving coordinates (t, r), so that the horizontal distance isproportional to the comoving distance d0 (Note that for Ω = 1, i.e., K = 0, we haved0 = a0r, see Eq. (38)). The third one is in the conformal coordinates (η, r), withnormalization a0 = 1. The last one has the advantage that light cones are always ata 45 angle. This is thus the “Mercator projection” of spacetime.

Angular diameter distance

The distance-redshift relationship obtained above would be nice if we already knewthe values of the cosmological parameters H0, Ω0, and ΩΛ. We can turn the situationaround and use an observed distance-redshift relationship, to determine the cosmo-logical parameters. But for that we need a different distance-redshift relationship,one where the “distance” is replaced by some directly observable quantity.

Astronomers employ various “fake”, or auxiliary, distance concepts, like the an-

gular diameter distance or the luminosity distance. These would be equal to the truedistance in Euclidean non-expanding space.

To answer the question: “what is the physical size of an object, whom we seeat redshift z, subtending an angle ϑ on the sky?” we need the concept of angular


Figure 7: Spacetime diagram for a flat universe in conformal coordinates.

Figure 8: Defining the angular diameter distance.

diameter distance dA.In Euclidean geometry,

s = ϑd or d =s

ϑ. (54)

Accordingly, we define

dA ≡ s

ϑ, (55)

where s is the actual diameter of the object (the diameter it had when the light wesee left it), and ϑ is the observed angle. For large-scale structures, which expandwith the universe, we use the comoving angular diameter distance dc

A ≡ s0/ϑ, wheres0 = (1 + z)s is the comoving diameter of the structure. Thus dc

A = (1 + z)dA.Suppose we have a set of standard rulers, objects that we know are all the

same size s, observed at different redshifts. Their observed angular sizes ϑ(z) thengive us the observed angular diameter distance as dA(z) = s/ϑ(z). This can thenbe compared to the theoretical dA(z) for the FRW universe to find the parametervalues which give the best fit between observation and theory.

From the RW metric, the proper distance corresponding to an angle ϑ is, fromds2 = a2(t)r2dϑ2 ⇒ s = a(t)rϑ. Thus

dA = a(t)r =a0

1 + zr. (56)


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gula

rdi

amet

erdi

stan

ce(H

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1.4

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1.8

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angu

lar

diam

eter

dist

ance

(H0-1

)

Flat universe

Figure 9: The angular diameter distance -redshift relationship, Eq. (58), for a) the matter-only universe ΩΛ = 0, Ωm = 0, 0.2,. . . ,1.8 (from top to bottom) b) the flat universe Ω0 = 1(ΩΛ = 1 − Ωm), Ωm = 0, 0.05, 0.2, 0.4, 0.6, 0.8, 1.0, 1.05 (from top to bottom). The thickline in both cases is the Ωm = 1, ΩΛ = 0 model. Note how the angular diameter distancedecreases for large redshifts, meaning that the object that is farther away may appear largeron the sky. In the flat case, this is an expansion effect, an object with a given size occupiesa larger comoving volume in the earlier, smaller universe. In the matter-only case, the effectis enhanced by space curvature effects for the closed (Ωm > 1) models.

We know (see Eq. 38) that the coordinate r is related to the comoving distance d0

byr = SK(d0/a0), (57)

and using the distance-redshift relationship, Eq. (46), we have the final result

dA(z) =a0

1 + zSK

[1

a0H0

∫ 1

1

1+z

dx√

Ω0(x− x2) − ΩΛ(x− x4) + x2

]

, (58)

for the angular diameter distance, and

dcA(z) = a0SK

[1

a0H0

∫ 1

1

1+z

dx√

Ω0(x− x2) − ΩΛ(x− x4) + x2

]

, (59)

for the comoving angular diameter distance.For a flat universe the comoving angular diameter distance is equal to the co-

moving distance,

dcA(z) = d0(z) = H−1

0

∫ 1

1

1+z

dx√

Ω0(x− x2) − ΩΛ(x− x4) + x2. (60)

For the open/closed case, we can write Eq. (59) as

dcA(z) = H−1

0

√

K

Ω0 − 1SK

[√

Ω0 − 1

K

∫ 1

1

1+z

dx√

Ω0(x− x2) − ΩΛ(x− x4) + x2

]

=H−1

0√

|Ω0 − 1|Sk

[√

|Ω0 − 1|∫ 1

1

1+z

dx√

Ω0(x− x2) − ΩΛ(x− x4) + x2

]

, (61)

eliminating the apparent dependence on the normalization-dependent quantity a0.


0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0redshift

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mov

ing

angu

lar

diam

eter

dist

ance

(H0-1

)Matter only

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1.6

1.8

2.0

com

ovin

gan

gula

rdi

amet

erdi

stan

ce(H

0-1)

Flat universe

Figure 10: Same as Fig. 9, bur for the comoving angular diameter distance. Now theexpansion effect is eliminated. For the closed models (for Ωm > 1 in the case of ΩΛ = 0)even the comoving angular diameter distance may begin to decrease at large enough redshifts.This happens when we are looking beyond χ = π/2, where the universe “begins to closeup”. The figure does not go to high enough z to show this for the parameters used. Notehow for the flat universe the comoving angular diameter distance is equal to the comovingdistance (see Fig. 4).

We shall later (in Cosmology II) use the angular diameter distance to relate theobserved anisotropies of the cosmic microwave background to the physical lengthscale of the density fluctuations they represent. Since this length scale can be calcu-lated from theory, their observed angular size gives us information of the cosmologicalparameters.

Luminosity distance

In transparent Euclidean space, an object whose distance is d and whose absoluteluminosity (radiated power) is L would have an apparent luminosity l = L/4πd2.Thus we define the luminosity distance of an object as

dL ≡√

L

4πl. (62)

Consider the situation in the FRW universe. The absolute luminosity can beexpressed as:

L =number of photons emitted

time× their average energy =

NγEem

tem. (63)

If the observer is at a coordinate distance r from the source, the photons haveat that distance spread over an area

A = 4πa20r

2 . (64)

The apparent luminosity can be expressed as:

l =number of photons observed

area · time× their average energy =

NγEobs

tobsA. (65)


0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0redshift

0

1

2

3

4

5

6

7

8

9

10lu

min

osity

dist

ance

(H0-1

)Matter only

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0

1

2

3

4

5

6

7

8

9

10

lum

inos

itydi

stan

ce(H

0-1)

Flat universe

Figure 11: Same as Fig. 9, bur for the luminosity distance. Note how the vertical scale nowextends to 10 Hubble distances instead of just 2, to have room for the much more rapidlyincreasing luminosity distance.

The number of photons Nγ is conserved, but their energy is redshifted, Eobs =Eem/(1 + z). Also, if the source is at redshift z, it takes a factor 1 + z longer toreceive the photons ⇒ tobs = (1 + z)tem. Thus,

l =NγEobs

tobsA=

NγEem

tem

1

(1 + z)21

4πa20r

2. (66)

From Eq. (62),

dL ≡√

L

4πl= (1 + z)a0r

= (1 + z)H−10

√

K

Ω0 − 1SK

[√

Ω0 − 1

K

∫ 1

1

1+z

dx√

Ω0(x− x2) − ΩΛ(x− x4) + x2

]

= (1 + z)2dA(z) = (1 + z)dcA(z) . (67)

Compared to the angular diameter distance, dA(z), we have two extra redshiftfactors 1 + z, one from the redshift of photon energy, one from the cosmologicaltime dilation in receiving the emitted radiation, both contributing to making large-redshift objects dimmer.

Suppose now that we have a set of standard candles, objects that we know allhave the same L. From there observed redshifts and apparent luminosities we thenget an observed luminosity-distance-redshift relationship dL(z) =

√

L/4πl, whichcan then be compared to the theoretical one to find the values of the cosmologicalparameters which give the best fit between theory and observations.

As we discussed in Chapter 2, astronomers have the habit of giving luminositiesas magnitudes. From the definitions of the absolute and apparent magnitude,

M ≡ −2.5 lgL

L0

, m ≡ −2.5 lgl

l0, (68)

and Eq. (62), we have that the distance modulus m−M is given by the luminositydistance as

m−M = −2.5 lgl

L

L0

l0= 5 lg dL + 2.5 lg 4π

l0L0

= −5 + 5 lg dL(pc) . (69)


0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0redshift

-5

-4

-3

-2

-1

0

1

2

3

4

5m

agni

tude

Matter only

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-5

-4

-3

-2

-1

0

1

2

3

4

5

mag

nitu

de

Flat universe

Figure 12: Same as Fig. 9, bur for the magnitude-redshift relationship. The constantM − 5 − 5 lgH0 in Eq. (70), which is different for different classes of standard candles, hasbeen arbitrarily set to 0.

(As explained in Chapter 2, the constants L0 and l0 are chosen so as to give thevalue −5 for the constant term, when dL is given in parsecs.) For a set of standardcandles, all having the same absolute magnitude M , we find that their apparentmagnitudes m should be related to their redshift z as

m(z) = M − 5 + 5 lg dL(pc)

= M − 5 − 5 lgH0 + 5 lg

(1 + z)

√

K

Ω0 − 1×

×SK

[√

Ω0 − 1

K

∫ 1

1

1+z

dx√

Ω0(x− x2) − ΩΛ(x− x4) + x2

]

(70)

We find that the Hubble constant H0 contributes only to a constant term in thismagnitude-redshift relationship. If we just know that all the objects have the sameM , but do not know the value of M , we cannot use the observed m(z) to determineH0, since both M and H0 contribute to this constant term. On the other hand, theshape of the m(z) curve depends only on the two parameters Ω0 and ΩΛ.

Type Ia supernovae (SNIa) are fairly good standard candles. Two groups, theSupernova Cosmology Project4 and the High-Z Supernova Search Team 5 have beenusing observations of such supernovae up to redshifts z ∼ 1 to try to determine thevalues of the cosmological parameters Ω0 and ΩΛ.

In 1998 they announced [1, 2] that their observations are inconsistent with amatter-dominated universe, i.e., with ΩΛ = 0. In fact their observations requiredthat the expansion of the universe is accelerating. This result was named the “Break-through of the Year” by the Science magazine [3]. Later more accurate observationsby these and other groups have confirmed this result. This SNIa data is one of themain arguments for the existence of dark energy in the universe.6 See Fig. 14 for

4http://supernova.lbl.gov/5http://cfa-www.harvard.edu/cfa/oir/Research/supernova/HighZ.html,

http://www.nu.to.infn.it/exp/all/hzsnst/6The other main argument comes from combining CMB anisotropy and large-scale-structure

data, and will be discussed in Cosmology II.

REFERENCES 54

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0redshift

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

m

relative difference in m(z)

Figure 13: The difference between the magnitude-redshift relationship of the differentmodels in Fig. 12 from the reference model Ωm = 1, ΩΛ = 0 (which appears as the horizontalthick line). The red (solid) lines are for the matter-only (ΩΛ = 0) models and the blue(dashed) lines are for the flat (Ω0 = 1) models.

SNIa data from 2004, and Fig. 15 for a determination of Ωm and ΩΛ from this data.As you can see, the present data is not good enough for a simultaneous accuratedetermination of both Ωm and ΩΛ. But by assuming a flat universe, Ω0 = 1, Riesset al. [4] find ΩΛ = 0.71+0.03

−0.05 (⇒ Ωm = 0.29+0.05−0.03). (The main evidence for a flat

universe, Ω0 ≈ 1 comes from the CMB anisotropy, which we shall discuss later.)We have in the preceding assumed that the mysterious dark energy component

of the universe is vacuum energy, or indistinguishable from a cosmological constant,so that pde = −ρde. Making the assumption7 that the equation-of-state parameterwde ≡ pde/ρde for dark energy is a constant, but not necessarily equal to −1, Riesset al. [4] find the limits −1.48 < wde < −0.72, when they assume a flat universe,and use an independent limit on Ωm from other cosmological observations.

References

[1] A.G. Riess et al., Astron. J. 116, 1009 (1998)

[2] S. Perlmutter et al., Astrophys. J. 517, 565 (1999)

[3] J. Glantz, Science 282, 2156 (18 Dec 1998)

[4] A.G. Riess et al., Astrophys. J. 607, 665 (2004), astro-ph/0402512

7There is no particular theoretical justification for this assumption. It is just done for simplicitysince the present data is not good enough for determining a larger number of free parameters inthe dark-energy equation of state to a meaningful accuracy.

REFERENCES 55

-1.0

-0.5

0.0

0.5

1.0

∆(m

-M)

(mag

)

HST DiscoveredGround Discovered

0.0 0.5 1.0 1.5 2.0z

-0.5

0.0

0.5

∆(m

-M)

(mag

)

ΩM=1.0, ΩΛ=0.0

high-z gray dust (+ΩM=1.0)Evolution ~ z, (+ΩM=1.0)

Empty (Ω=0)ΩM=0.27, ΩΛ=0.73"replenishing" gray Dust

Figure 14: The Supernova Ia luminosity-redshift data. The top panel shows all supernovaof the data set. The bottom panel show the averages from different redshift bins. Thecurves corresponds to three different FRW cosmologies, and some alternative explanations:“dust” refers to the possibility that the universe is not transparent, but some photons getabsorbed on the way; “evolution” to the possibility that the SNIa are not standard candles,but were different in the younger universe, so that M = M(z). This Figure is from Riess etal., astro-ph/0402512 [4].

REFERENCES 56

0.0 0.5 1.0 1.5 2.0 2.5ΩM

-1

0

1

2

3

ΩΛ

68.3

%

95.4

%

99.7

%

No Big

Bang

Ωtot =1

Expands to Infinity

Recollapses ΩΛ=0

Open

Closed

Accelerating

Decelerating

q0=0

q0=-0.5

q0=0.5

^

Figure 15: Ωm and ΩΛ determined from the Supernova Ia data. The dotted contours arethe old 1998 results[1]. This Figure is from Riess et al., astro-ph/0402512 [4].

Documents

4 Friedmann–Robertson–Walker Universehkurkisu/cosmology/Cosmo4.pdf · 4 FRIEDMANN–ROBERTSON–WALKER UNIVERSE 37 • “Matter” (called “matter” in cosmology, but “dust”