Chapter 1

The Smooth Universe

In this chapter we will study the background dynamics of the standard cosmological model. We will assume a homogeneous and isotropic universe, an assumption known as the Cosmological Principle. We will also assume it is filled with unperturbed, ideal fluids such as dust, dark matter and radiation.

This assumption is valid only on scales larger than about 100 Mpc but it gives a very good description of the evolution in the model. Any observer (galaxy) will be treated as a test (massless) particle. 1

2

Particle Cosmology

1.1

Newtonian picture

In the classical Big Bang picture the universe is expanding and the only expansion law consistent with the Cosmological Principle is the Hubble law which states that two (test) observers will observe each other receding away from each other at a velocity proportional to the distance between them !v = H(t)!r

(1.1)

where H(t) is the Hubble rate and !r is the vector connecting the two observers. It is sometimes useful to make the analogy of a balloon being blown up with two points on the surface r(t)

χ

a(t)

Figure 1.1: The balloon analogy...not to be taken too far...

3

The smooth universe In this case the distance between the two points scales as the radius r(t) = a(t)χ

(1.2)

and taking the derivative v(t) = r(t) ˙ = a˙ χ =

a˙ r. a

(1.3)

In the cosmological context a(t) is the scale factor which scales distances. We can describe the distance between observers in a homogeneous isotropic cosmology using physical distance r or comoving (Lagrangian) distances χ. Called comoving because the coordinate system follows the Hubble expansion. Comoving distances remain constant. Note that the Hubble rate (sometimes called the Hubble constant) is not constant but changes with time. It’s measurement is one of the classic cosmological observations. Today we know that H0 ∼ 72 km s−1 Mpc−1 .

(1.4)

The inverse of the Hubble rate has units of time and it can be use to estimate the age of the universe since to observers separated by a distance r today would have been at the same point in space at a time t0 =

1 r = , v H0

(1.5)

in the past. Using 1 Mpc= 3.0856 × 1019 km we have that t0 = 13.6 Gyrs.

(1.6)

This actually turns out to be quite accurate! Note that in reality all real objects will have a peculiar velocity with respect to the Hubble expansion due to their own dynamics e.g. galaxies have few 100 km s−1 peculiar velocities and we are not truly comoving observers. However at large distances these become small compared to the Hubble flow. !v = H(t)!r .

(1.7)

4

Particle Cosmology We will assume the universe is homogeneous, isotropic and filled with dust i.e.

pressure p = 0 with energy density ρ. The universe is expanding and we also assume the dust is non-relativistic i.e. the speed of the particles of dust is much less than the speed of light. Also assume we are in the weak gravity regime. Consider an expanding sphere of radius a(t). The sphere contains a total mass M which is conserved, therefore the energy density of the mass is M ρ(t) = = ρ0 (4π/3)a3(t)

!

a0 a(t)

"3

,

(1.8)

where ρ0 is a reference density at with radius a0 . Taking time derivative we have ρ(t) ˙ = −3ρ0

!

a0 a(t)

"3

a˙ = −3 H(t) ρ(t). a

(1.9)

This is the non-relativistic continuity equation ∂ρ = −∇ · (ρ!v ). ∂t

(1.10)

Note that using the continuity equation and assuming ρ(!x, t) = ρ(t) gives a unique solution for the homogeneous expansion i.e. the Hubble law We need the second derivative to determine the full evolution of the sphere of dust since gravity acts on the matter to slow down the expansion. Consider a test particle of mass m at a distance a(t) from the centre of the sphere. The particles outside the radius do not contribute to the acceleration of the test particle so we have ma ¨=−

GmM 4π M = − Gm a, 2 a 3 (4π/3)a3

(1.11)

which gives the acceleration at the surface of the sphere as a ¨ 4π = − Gρ. a 3 This is the acceleration equation.

(1.12)

5

The smooth universe

We can now obtain some simple Newtonian solutions to this system which will be useful later since we can generalise them to the fully relativistic case. Substituting the expression for ρ into the acceleration equation we have a ¨=−

a3 4π Gρ0 02 , 3 a

(1.13)

multiply by a˙ and integrate #

4π a ¨ a˙ dt = − Gρ0 a30 3

#

a˙ dt. a2

(1.14)

Easy to show that d 2 a˙ = 2¨aa, ˙ dt

(1.15)

d1 a˙ = − 2, dt a a

(1.16)

and

we have 1 1 2 4π a˙ − Gρ0 a30 = E 2 3 a 1 2 a˙ + V (a) = E. 2

(1.17)

This is just the conservation equation for a projectile thrown from the Earth at a velocity a˙ and climbing to a distance a. Depending on the sign of the integration constant E we have escape - positive, orbit - zero, fallback - negative. We can re-write the conservation equation in a more cosmological form H2 =

E 8πG ρ + 2 2. 3 a

(1.18)

Setting E = 0 in the above we obtain the definition of the critical energy density ρcrit =

3H 2 . 8πG

Note that this decreases in time for this example since H is decreasing.

(1.19)

6

Particle Cosmology The Hubble parameter today is often expressed in terms of a dimensionless con-

stant h as H0 = 100h Km s−1 Mpc−1 = h 3.247 × 10−18 s−1

(1.20)

putting this in we find the value of the critical energy density of matter today ρcrit =

3H02 , 8πG

(1.21)

using G = 6.67 × 10−8 cm3 g−1 s−1 we get ρcrit = 1.88h2 × 10−29 g cm−3 ≡ 2.775h2 × 1011 M" Mpc−3 . a(t) E > 0, open

E = 0, flat

initial singularity a → 0 final singularity

E < 0, closed

t

Figure 1.2: Fate of the scale factor for various energies. The open and flat solutions expand forever whereas the the closed one has both initial and final singularities. Energy densities in the universe are usually expressed in units of the critical density e.g. Ω(t) =

ρ(t) . ρcrit (t)

(1.22)

Expressing the value of E in terms of the critical energy density we have E=

4πG 2 a ρcrit (1 − Ω) , 3

(1.23)

7

The smooth universe

so since a and ρcrit don’t change sign if we determine Ω at any point in the evolution it tells us the sign of the integrations constant E. e.g. we determine today’s energy density Ω0 . E is connected to the spatial geometry of the universe in the relativistic picture. Ω0 > 1 E < 0 closed, positive curvature Ω0 < 1 E > 0 open, negative curvature Ω0 = 1 E = 0

critical, flat

(1.24)

8

Particle Cosmology

1.2 1.2.1

Relativistic cosmology Homogeneous, isotropic 3-geometries

The evolution of a homogeneous and isotropic universe can be represented by the a sequence of three-dimensional, space-like hypersurfaces. Homogeneity and isotropy impose the greatest symmetry on the hypersurfaces; the universe is the same at every spatial point and looks the same in every direction from any one point (6-degrees of freedom). There are only three possible types of hypersurfaces with maximal symmetry and simple topology; 1. 3-dim flat space 2. 3-dim spheres of constant positive curvature 3. 3-dim hyperbolic space of constant negative curvature To derive the correct infinitesimal line element of the spaces it is useful to embed the space in four dimensional Euclidean space covered by the Cartesian coordinates (w, x, y, z). (This can be done formally only for the plane and 3-sphere, hyperbolic case has imaginary radius). e.g. the equation for the 3-sphere is w 2 + x2 + y 2 + z 2 = a2 ,

(1.25)

where a is a constant. Then we can differentiate to get dw 2 =

(xdx + ydy + zdz)2 (a2 − x2 − y 2 − z 2 )

(1.26)

and substitute into the Euclidean 4-dimensional metric dl2 = dw 2 + dx2 + dy 2 + dz 2 , =

(xdx + ydy + zdz)2 + dx2 + dy 2 + dz 2 . (a2 − x2 − y 2 − z 2 )

(1.27)

9

The smooth universe

such that the distance between two points is described purely as a function of x, y, and z and is bounded by a. We can then change to polar coordinates x = r sin θ cos ϕ, y = r sin θ sin ϕ, z = r cos θ,

(1.28)

x2 + y 2 + z 2 = r 2 ,

(1.29)

with

and 3-metric $ % dx2 + dy 2 + dz 2 = dr 2 + r 2 dΩ2 = dr 2 + r 2 dθ2 + sin2 θdϕ2 .

(1.30)

Since xdx + ydy + zdz = rdr we then have

r 2 dr 2 + dr 2 + r 2 dΩ2 , a2 − r 2 dr 2 2 2 = & 2 ' + r dΩ . 1 − ar 2

dl2 =

(1.31)

This line element also describes infinitesimal distances on hyperbolic (negative, constant curvature) 3-spaces if we assume a2 < 0 i.e. the radius of the 3-sphere is imaginary. The flat case corresponds to the limit a → ∞. that if the curvature is positive (closed) then the space is truly bounded but unbounded for the flat and negative (open) curvature cases. ( By making the substitution R = r/ |a|2 we can write the line element in the

standard form

2

dl = |a|

2

where K = |a|2 /a2 i.e.

!

" dR2 2 2 + R dΩ , 1 − KR2

(1.32)

K = +1 positive curvature; K=0

flat;

(1.33)

K = −1 negative curvature. It is sometimes useful to change to the variable χ using the transformations    sinh(χ) K = −1 (open)    R ≡ SK (χ) = , (1.34) χ K = 0 (flat)      sin(χ) K = +1 (closed)

10

Particle Cosmology

such that dχ2 =

dR2 . 1 − KR2

(1.35)

In this case the 3-metric can be written as & ' 2 dl2 = |a|2 dχ2 + SK (χ)dΩ2 ,

(1.36)

with χ spanning 0 to +∞ for flat and open spaces, and 0 ≤ χ ≤ π in closed spaces. The function a is related to a radius of curvature in both positive and negatively curved spaces. i.e. for positive curvature space (sphere) we can define L = a sin χ,

(1.37)

with e.g. the surface area of the sphere being S = 4πL2 = 4πa2 sin2 χ.

(1.38)

Note however that χ here goes from 0 to π, so the volume and surface area reach a maximal value at π/2 and go back to zero at π. Thus the positively curved space has a finite volume. In positive curvature space triangles defined by geodesics between three points in space have angles which add up to more than 180 degrees. For the negative curvature case we have L = a sinh χ,

(1.39)

with χ from 0 to ∞ so the volume in this case is infinite, i.e. space is unbounded. In negative curvature space triangles defined by geodesics between three points in space have angles which add up to less than 180 degrees.

The smooth universe

1.2.2

11

Spacetime metric

We now bring in the time coordinate (the proper time measured by comoving observers) in the description of the spacetime interval such that at each separate time we describe the space as a 3 dimensional hypersurface introduced above. As usual the space-time interval is described by the metric gµν

ds2 = −dt2 + dl2 = gµν dxµ dxν µ, ν = 0, 3.

(1.40)

this tells us how to calculate the invariant distance between points with different spacetime coordinates. We adopt a signature convention (−, +, +, +) with greek indices running over all four space-time and roman ones only over the spatial dimensions. Indices are raised and lowered using the metric tensor e.g. Aµ = g µν Aν with summation over repeated indices implicit in the contraction. Also the contraction of the metric tensor gives a delta g µν gνγ = δγµ . Following the Cosmological Principle we have . & 2 ' dr 2 2 2 2 2 2 2 ds = −dt + a (t) + r dΩ = −dt + a (t) dχ + S (χ)dΩ . (1.41) K 1 − Kr 2 2

2

2

-

This is known as the Friedmann–Robertson–Walker (FRW) metric and describes an isotropic, homogeneous expanding space–time. Note that now we have normalized the scale factor a(t) and assume it is positive. We will often use the FRW form defined using the comoving radial coordinate (comoving distance) χ. Most of the time we will be looking at the flat case since we know the universe is very close to flat today. However it is important to look at the overall dynamics for non-zero curvature.

12

Particle Cosmology

1.2.3

Geodesic Equation

The geodesic equation in general relativity describes the motion of a particle in a space-time with no external (non-gravitational) force applied i.e. it is the equivalent of Newton’s Law with zero force d!x2 = 0. dt2

(1.42)

This is trivial in a Minkowski spacetime but when time and spatial coordinates are non-trivially connected i.e. dxi /dt )= 0 then the form is more complicated α β d2 xµ µ dx dx = −Γ , αβ dλ2 dλ dλ

(1.43)

where we have introduced λ, the affine parameter which increases monotonically along the spacetime path since the time coordinate also depends on the spacetime metric, and the Christoffel symbol or affine connection which describes the “connections” between the coordinates Γµαβ

. g µν ∂gαν ∂gβν ∂gαβ + − . = 2 ∂xβ ∂xα ∂xν

(1.44)

Evaluation of the connection components for the flat FRW cosmology is left as an exercise. Here we simply list the results. Γ000 = Γ00i = Γ0i0 = 0 Γ0ij = δij aa ˙ = δij Ha2 Γi0j = Γij0 = δji

a˙ = δji H. a

(1.45)

13

The smooth universe

1.2.4

The Friedmann Equations

The spacetime is a dynamic quantity in general relativity and the expressions relating the spacetime geometry (or gravitational field) to the matter are known as the Einstein equations Gµν = 8πGTµν ,

(1.46)

where Tµν is the stress-energy tensor describing the energy and pressure (both isotropic and anisotropic) of the matter and the Einstein tensor is given by 1 Gµν = Rµν − gµν R − gµν Λ, 2

(1.47)

where Rµν is the Ricci tensor and its contraction R = Rµµ , the Ricci scalar. The Λ term is the cosmological constant introduced as an integration constant in the derivation of the equations. The Ricci tensor is given by Rµν = Γαµν,α − Γαµα,ν + Γαβα Γβµν − Γαβν Γβµα ,

(1.48)

We will use commas to denote partial derivatives while semi-colons to denote covariant derivatives e.g. uµ;ν =

∂uµ + uγ Γµγν . ∂xν

(1.49)

The Einstein equations can be derived by varyng the Einstein-Hilbert action # √ 1 IG = −gRd4 x, (1.50) 16πG and the right hand side is obtained by the formal definition of the stress-energy tensor through the variation of the matter action wrt the metric # √ 1 d4 x −gT µν δgµν . δIM = 2

(1.51)

Their derivation is left as an optional exercise. In the flat FRW case only the {0, 0} and {i, i} components of the Ricci tensor are non vanishing. The time-time; R00 = Γα00,α − Γα0α,0 + Γαβα Γβ00 − Γαβ0 Γβ0α ,

(1.52)

14

Particle Cosmology

using the fact that Γα00 = 0 for the FRW metric we get −δii

R00 =

d dt

! " ! "2 a˙ a˙ − δij δ ij . a a

(1.53)

recall that for 3 spatial dimensions δii = 3 so ! "2 a˙ a ¨ −3 = −3 . a a

(1.54)

Rij = Γαij,α − Γαiα,j + Γαβα Γβij − Γαβj Γβiα ,

(1.55)

R00 = −3

!

a¨ a˙ 2 − a a2

"

For the spatial components;

spatial derivatives disappear to give Rij = Γ0ij,0 + Γαβα Γβij − Γαβj Γβiα ,

(1.56)

Rij = Γ0ij,0 + Γ0β0 Γβij + Γkβk Γβij −Γ0βj Γβi0 − Γkβj Γβik , -

. d 2 2 Rij = δij (aa) ˙ + 3a˙ − 2a˙ , dt $ % Rij = δij a ¨a + 2a˙ 2 .

(1.57) (1.58) (1.59)

The Ricci scalar can be calculated as

R = g µν Rµν = −R00 + to give

/

a ¨ R=6 + a

1 Rii , a2

! "2 0 a˙ . a

(1.60)

(1.61)

This defines the left hand side of the Einstein equations, now we define the right hand side by describing the matter in the universe as a perfect fluid. A perfect fluid is described purely by its energy density ρ and isotropic pressure p i.e. there is no shear, viscosity (or heat conduction). Its stress-energy takes the form T µν = (ρ + p)U µ U ν − pg µν ,

(1.62)

15

The smooth universe

where U µ is the four-velocity of the fluid. In the comoving frame of reference, the fluid is at rest wrt expansion so that the velocity is unit normalized U µ = (1, 0, 0, 0) with U µ Uµ = −1. This gives the stress-energy tensor of the perfect fluid as    −ρ 0 0 0     0 p 0 0    T µν =  (1.63) .  0 0 p 0      0 0 0 p

Perfect fluids can be described conveniently through their equation of state

which describes their pressure as a (linear) function of their energy density p = wρ,

(1.64)

where w is the equation of state parameter. Note this can change in time. Putting this into the time-time component of the Einstein equations then gives us ! "2 a˙ 8πG H = ρ, = a 3 2

(1.65)

which is the (first) Friedmann equation. The space-space component gives 1 Rij − gij R = 8πGTij , 2 which is δij

7

/! " 08 2 a ˙ a ¨ 2a˙ 2 + a¨a − 3a2 + = 8πGa2 pδij , a a

(1.66)

(1.67)

which, using first Friedmann equation gives the second (some avoid second and call this the acceleration equation) 4πG a ¨ =− (ρ + 3p). a 3

(1.68)

we have seen this before in the Newtonian picture (except we were assuming zero pressure for the dust).

16

Particle Cosmology

General Friedmann equations The generalized Friedmann equations are ! "2 a˙ 8πG Λ K H = = ρ + − 2, a 3 3 a

(1.69)

4πG Λ a ¨ =− (ρ + 3p) + . a 3 3

(1.70)

2

and

17

The smooth universe

1.2.5

Stress-energy conservation

In Minkowski spacetime we can re-write energy and momentum conservation as the equations ∂T µν = 0, ∂xν

(1.71)

from which we can derive the equations of motion. For curved spacetimes these generalize to T µν;ν = 0.

(1.72)

The Christoffel symbols appearing in the equations account for the presence of a gravitational field. Note that these equations can be obtained by the fact that the Einstein tensor satisfies the Bianchi Identities Gµν;µ = 0.

(1.73)

Taking the time-time conservation equation and assuming homogeneity and isotropy we have −

dρ − Γ0 00 ρ − Γi dt

0i ρ

− Γα 00 T 0 α − Γα 0i T i α = 0,

(1.74)

but Γα 0µ = 0 for α = µ = 0 else = δij (a/a) ˙ so we get dρ a˙ + 3 (ρ + p) = 0, dt a

(1.75)

This is the relativistic generalization of the continuity equation that we have already seen in the Newtonian limit. Re-writing the above in terms of the equation state parameter w = p/ρ we have dρ a˙ = −3ρ (1 + w), dt a

(1.76)

ρ dρ = −3 (1 + w), da a

(1.77)

getting rid of time variable

which is obviously satisfied by a power law solution of the type ρ(a) = ρ0 (a/a0 )n . Substituting this in we equate n = −3(1 + w).

(1.78)

18

Particle Cosmology

This gives the evolution of the energy densities with respect to the scale factor for the various fluids • relativistic matter e.g. photons (w = 1/3) ργ ∼ Ωγ ∝ a−4 ,

(1.79)

(think of thermodynamics of particles with momentum moving in 3 spatial dimensions...) • non-relativistic e.g. dust-like, pressureless matter (w = 0) ρm ∼ Ωm ∝ a−3 ,

(1.80)

(pressure is zero by definition in dust) • Cosmological constant type matter e.g. vacuum energy (w = −1) ρΛ ∼ ΩΛ ∝ const.

(1.81)

(by definition a cosmological constant has constant energy density ρΛ so stressenergy conservation implies pΛ = −1ρΛ ) We can also immediately infer that the curvature “density” term ΩK = ρK 8πG/3H 2 scales as ρK ∼ ΩK ∝ a−2 ,

(1.82)

i.e. scales the slower than matter and radiation energy densities. The first Friedmann equation can then be recast in a useful form with respect to energy densities in units of ρcrit ! "2 $ % a˙ = H02 ΩΛ (a0 ) + Ωm (a0 )a−3 + Ωγ (a0 )a−4 − ΩK (a0 )a−2 . H (a) = a 2

(1.83)

The smooth universe

19

Figure 1.3: Evolution of the energy densities ΩX (in critical units) for the various components in a typical ΛCDM model; radiation, non-relativistic matter and cosmological constant.

20

Particle Cosmology

1.2.6

Solutions to the Friedmann equations

To examine different solutions to the Friedmann equations it is convenient to introduce the concept of conformal time. We can factorise out the scale factor from the metric so that you can write the metric as a conformal transformation of the Minkowski metric gµν = σ(t, !x)ηµν ds2 = −dt2 + a2 (t)d!x2 → a2 (η)(−dη 2 + d!x2 ),

(1.84)

where we have defined the conformal time variable as η=

t

#

0

- # t . cdt# dt# = . # a(t# ) 0 a(t )

(1.85)

We can change from physical time to conformal in the Friedmann equations using dt = a(η)dη with a˙ = and

a# a

(1.86)

/ ! # "2 0 1 a## a a¨ = − a a a

(1.87)

2

! # "2 8πG 2 a = ρa − K, H = a 3

(1.88)

a## 4πG 2 = a (ρ − 3p) − K. a 3

(1.89)

to get

and

Dust filled universe Consider a flat universe filled only with dust. This is known as the Einstein-de Sitter solution. Using the first Friedmann equation we have a#2 =

8πG 8πG 4 ρa → ρ0 a. 3 3

(1.90)

Integrating we have 1/2

a

=

!

8πG ρ0 3

"1/2

η + C.

(1.91)

21

The smooth universe Setting a(η = 0) = 0 the integration constant disappears and we have a = a0 η 2 .

(1.92)

Integrating the #

t

#

η

a0 η #2 dη # ,

(1.93)

t ∼ η 3 and a ∼ t2/3 .

(1.94)

0

#

dt =

0

we get

Putting this back into the Friedmann equation we get a relation between the time and Hubble rate t=

2 1 , 3H

(1.95)

thus the estimate we used to get the age of the universe in the simple Newtonian picture was nearly correct (assuming the universe can be described as dust filled for most of its history.) Radiation filled universe Now consider a universe filled only with an ultra-relativistic fluid such as photons. Easiest to use the second Friedmann equation since for a flat model we have a## = 0.

(1.96)

Integrating and setting the same initial conditions we find a ∼ η,

(1.97)

t ∼ η 2 and a ∼ t1/2 .

(1.98)

which in turn gives

Again we have a relation between the time and Hubble rate t=

11 2H

(1.99)

We now look at two further solutions as examples with extra (time) symmetries

22

Particle Cosmology

The empty universe The first one is an empty open model. This is known as the Milne universe. With ρ = 0 and K = −1 the Friedmann equation gives a = t,

(1.100)

ds2 = −dt2 + t2 (dχ2 + sinh2 χdΩ2 ).

(1.101)

such that we have the metric

Since the Milne universe is empty i.e. it does not have 4-curvature you would expect to be able to describe it with a Minkowski metric. This can be done by changing the coordinate system as τ = t cosh χ, r = t sinh χ.

(1.102)

ds2 = −dτ 2 + dr 2 + r 2 dΩ2 .

(1.103)

This gives the metric

Thus the Milne universe can be described using special relativity only. It’s coordinate system covers only one quadrant of the full Minkowski coordinates, the forward light cone. i.e. the Milne universe actually has an outside. It describes a cosmological bubble expanding into nothing. The surfaces of constant time in the Milne model are hyperboloids with t2 = τ 2 − r 2 . and the velocity of a comoving test particle is v=

r = tanh χ < 1. τ

(1.104)

Thus in Milne coordinates the velocity never exceeds the speed of light and it’s proper time is related to coordinate time τ as t=

√

1 − v 2 τ.

(1.105)

23

The smooth universe

τ

t=

0,

χ

=

∞

χ =const hype-surfaces

t =const hypersurfaces

r

Figure 1.4: The empty universe is one with zero 4-curvature i.e. a special relativity one which can be described by a Minkowski metric for a suitable choice of coordinates. However the Minkowski coordinates only cover one quadrant of an infinite Minkowski spacetime - the forward light cone. This is the one case where the intuitive notion of inside and outside applies! The light cone is expanding into nothing and I will never see observers entering the horizon e.g. the entire, past and future universe is contained in the observer’s light cone.

24

Particle Cosmology Note however that if we were to define a recession velocity as the Minkowski

coordinate divided by the proper time we would have u=

r = sinh χ, t

(1.106)

and this would be greater than the speed of light for χ > 1. But this definition has no invariant meaning. Any well defined speed has to be defined using the same inertial coordinate system. The Hubble velocity in the Milne universe can also be greater than 1 since vH = aχ ˙ = χ,

(1.107)

but again this has no invariant meaning. Only for χ , 1 do the three speeds agree. In a curved (non-zero 4-curvature R) Friedmann cosmology the definition of a classical Hubble velocity has even less meaning on large scales since an inertial coordinate system can only be defined locally on scales less than 1/H. Thus the common misconception that a very distant object would be receding from us at a speed greater than the speed of light is not valid.

25

The smooth universe Vacuum universe

This is known as the de Sitter solution and is the maximally symmetric homogeneous, isotropic solution as it has time-translational symmetry. Consider a universe filled only with the vacuum. This is not an empty universe since the vacuum actually has an energy density. In quantum field theory this is just the energy density of virtual particles created out of spontaneous particle/antiparticle creation in empty space. From a GR point of view it can just be added as an integration constant when deriving the Einstein equations. We can think of the vacuum energy as a fluid itself but with constant energy density ρΛ =

Λ . 8πG

(1.108)

Taking the continuity equation dρΛ ρΛ = −3 (1 + w) = 0, da a

(1.109)

then the equation of state for the vacuum energy must have w = −1. Consider the acceleration equation, for this case we have a ¨ 8πG = ρΛ . a 3

(1.110)

The de Sitter universe is one where the expansion is accelerating. We can define a (truly constant) Hubble rate ! "1/2 9 Λ 8πG HΛ = ρΛ = . 3 3

(1.111)

Integrating we have a = C1 eHΛ t + C2 e−HΛ t .

(1.112)

The integration constants can be constrained using the Friedmann equation which in this case we can write a˙ 2 = HΛ2 a2 − K

(1.113)

26

Particle Cosmology

to give K = 4HΛ2 C1 C2

(1.114)

which shows that in de Sitter the sign of K is set by choices of integration constants, not by the content of the universe i.e. we can have open, closed and flat universes for any value of the cosmological constant. For K = 0 we can chose one of the constant to be zero. e.g. C1 = 1/2HΛ and C2 = 0. For K = +1 or −1 we can chose |C1 | = |C2 | at t = 0 to get C1 = C2 =

1 closed, 2HΛ

(1.115)

1 open. 2HΛ

(1.116)

and C1 = −C2 =

So for any choice of ρΛ we can have  2  sinh (HΛ t) 1  ds2 = −dt2 + 2  exp(2HΛ t) HΛ   cosh2 (HΛ t)



  2  (dχ2 + SK (χ)dΩ2 ).  

(1.117)

The three choices all describe the same physical spacetime but with different coordinate systems. This is a consequence of the time-translational symmetry of de Sitter space. So in the pure de Sitter solution we have the scale factor behaving in different ways depending on the coordinate system, but can switch between spatial curvatures and describe the same solution. Note that the t = 0 point for open models is purely a coordinate singularity (al three curves are the same physical solution). At late times t > 1/HΛ the solution looks the same for all three coordinate choices a(t) ∼ eHΛ t .

(1.118)

27

The smooth universe

4

3

K=+1

2

1

K=0

0 -2

-1

K=-1

0

1

2

Figure 1.5: The de Sitter solution in the three different coordinates. The vanishing scale factor at t = 0 for the open case is a coordinate singularity which can be avoided by switching coordinates. The scale factor approaches the same exponential solution in all coordinate systems for t > 1/HΛ .

28

Particle Cosmology

1.3

Cosmological horizons

We now introduce the concept of horizons which are of fundamental importance in relativistic cosmology. Since particle speeds are limited by the speed of light it is clear that we can only see out to a certain distance if the universe has a finite age. The nature of horizons is related to the definition of conformal time η=

#

dt . a(t)

(1.119)

Consider a light ray propagating through the universe. This must be a geodesic and therefore along the path we have 2 ds2 = 0 = a2 (η)[−dη 2 + dχ2 + SK (χ)dΩ2 ].

(1.120)

Since θ, φ = const by symmetry then we have χ(η) = ±η + C,

(1.121)

i.e. 45◦ lines in the χ, η plane. Note that we are equating a distance with time simply because we are working in units of c = 1 otherwise χ(η) = ±cη + C.

(1.122)

The Particle Horizon is the boundary created by the maximum distance a photon can have traveled throughout the life of the universe dp (η0 ) = a(η0 )χ(η0 ) = a(η0 )η0 = a(t0 )

#

t0

ti

dt , a(t)

(1.123)

where η0 and t0 are times at a particular point in the expansion, say today and ηi and ti are initial times. i.e. if there is an initial singularity beyond which the time coordinate is not defined the ti = ηi = 0. This is not the case for all models though. We can’t receive any signals from distances farther than the particle horizon. However note that in practice the distance we can see using light is limited to a

29

The smooth universe

20

15

10

5

0 0

10

20

30

40

Figure 1.6: Different solutions to the matter dominated Friedman equations. These are the same solutions you would find for the spherical collapse of matter. i.e. any self contained region of the universe can be treated as an isolated Friedman model (Birkhoff’s theorem)

30

Particle Cosmology

Figure 1.7: “Phase” diagram of full ΛCDM cosmology. As an exercise derive the 3

loitering curve where ΩΛ = 4Ωm {F (1/3 F −1[1/Ωm − 1])} where F (x) = cosh x if Ωm < 0.5 and F (x) = cos x otherwise.

31

The smooth universe

slightly smaller radius since photons were tightly coupled with matter until recombination of electrons and Hydrogen nuclei. This is known as the optical horizon dopt = a(η)(η − ηr ).

(1.124)

The particle horizon is usually close to the Hubble scale 1/H and finite for a universe where the strong energy condition has never been violated ρ + 3p > 0

(1.125)

We can see this by changing the integration variable to a in the expression for the horizon a √0 dp = H0 ΩX

#

a0 0

da a2 anX /2

,

(1.126)

which is finite for any form of matter that decreases more rapidly than a−2 . However if we have a de Sitter model with nX = 0 then the integrand blows up as we go towards a → 0. Note that in a pure de Sitter model there is no initial singularity and t → −∞ as a → 0. The Hubble scale (also called the Hubble Horizon) remains constant however in the de Sitter model at 1/HΛ so the particle horizon and Hubble scale diverge. As we shall see later on the Hubble scale is an important concept in perturbation theory, describing the scale at which the expansion dominates the equation of motion of perturbations. The divergence of these two scales will play a very important role that allows inflation to overcome the horizon problem of big bang cosmology. The future horizon or event horizon is defined as the maximum distance that a photon can travel from today to a time tmax χe (η0 ) = ηmax − η0 .

(1.127)

Observers at distances larger than χe will never receive the signal. The physical size of the event horizon is now a √0 de = H0 ΩX

#

amax

a0

da a2 anX /2

,

(1.128)

32

Particle Cosmology

with amax → ∞ if the universe expands forever in a decelerating (matter or radiation) universe. i.e. there is no future horizon in a radiation or Einstein-de Sitter model and the event horizon is infinitely large However amax is finite for a closed universe so there is both a past (particle) and future (event) horizon. The integral converges for a de Sitter universe following the same argument as above. Thus a de Sitter universe has a future (event) horizon but no past (particle) horizon. In summary;

Past (Particle)

Future (Event)

Flat, Open

yes

no

Closed

yes

yes

de Sitter

no

yes

33

The smooth universe

1.4

Conformal Diagrams - FRW Solutions

Conformal diagrams are useful for understanding the causal structure of different solutions, particularly if the solutions are isotropic such that the spacetime can be represented on a 2d coordinate system (time-radial coord). The first step is conformal mapping to a minkowski metric. This is simple for the FRW case where we just need to use conformal time ds2 = a2 (η)(dη 2 − dχ2 ) .

(1.129)

Then light propagation (geodesics) is on x = y lines in the χ η space χ(η) = ±η + C ,

(1.130)

where C is a constant describing the initial conditions for the propagation. The second step is to check if the η and χ coordinates span a finite range. If not we must introduce another transformation which maps the infinite boundaries into a finite range.

1.4.1

Closed Universe

The metric in this case is ds2 = a2 (η)(dη 2 − dχ2 − sin2 χdΩ2 ) ,

(1.131)

with K = +1. The solutions for the scale factor are a(η) = a0 sin(η) , w =

1 , π ≥ χ ≥ 0, π > η > 0, 3

a(η) = a0 (1 − cos(η)) , w = 0 , π ≥ χ ≥ 0 , 2π > η > 0 .

(1.132) (1.133) (1.134)

Draw Diagrams Note that the dust filled universe lives twice as long as the radiation filled one before recollapse (final singularity). Geodesics can travel around the

34

Particle Cosmology

whole spacetime twice before recollapse in the dust case (out and back to starting point). While in radiation there is only time to just cover the entire spacetime before recollapse. Coordinates are finite so there is no need for second transformation.

1.4.2

Minkowski

Metric is Minkowski by definition but coordinates span an infinite range −∞ ≤ t ≤ ∞ and 0 ≤ r ≤ inf ty. Map to new coordinates η and χ with tanh t =

sin η sin χ , tanh r = , cos χ cos η

(1.135)

with ranges −π/2 ≤ η ≤ π/2 and 0 ≤ χ ≤ π/2. In these coordinates the Minkowski metric is ds2 =

cos2

& ' 1 −dη 2 + dχ2 + Φ2 (η, χ)dΩ2 , 2 χ − sin η

(1.136)

wehre the function Φ(η, χ) describes the transformation of the solid angle which is not relevant since we are still considering isotropic spaces. Draw diagram + Milne as quadrant of Minkowski

1.4.3

Flat FRW

In this case we have ds2 = a2 (˜ η )(−d˜ η 2 + dχ˜2 + χ˜2 dΩ2 ) .

(1.137)

Can use similar transformation for the infinities tanh η˜ =

sin η sin χ , tanh χ˜ = , cos χ cos η

with ranges 0 ≤ η ≤ π/2 and 0 ≤ χ ≤ π/2. Draw diagram

(1.138)

35

The smooth universe

1.5

Cosmological Redshift

A consequence of an expanding spacetime is that radiation is redshifted. We can see this by considering the emission of a light signal of conformal period ∆η. The conformal period is invariant but the physical duration ∆t (proper time interval measured by observer) will depend on the time of emission and observation ∆t(te ) = a(ηe )∆η ∆t(to ) = a(η0 )∆η.

(1.139)

Thus we have that the wavelength of the signal changes as a(ηe ) λe = . λ0 a(η0 )

(1.140)

This is known as redshifting since the frequency decreases as 1/a and wavelength increases as a i.e. blue to red in an expanding universe. Redshifting can be interpreted as a Doppler effect on scales smaller than 1/H. i.e. can think of it as due to the relative motion of galaxies, however since relative distance has no invariant meaning this can’t be extended to larger scales. The cosmological redshift z is defined as z=

λ0 − λe a0 = − 1. λe ae

(1.141)

Thus the use of scale factor a and redshift z is interchangeable and we can define cosmic age in terms of a redshift integral t0 =

#

0

∞

dz , H(z)(1 + z)

(1.142)

and comoving distance to a certain redshift in the past as 1 χ(z) = a0

#

0

z

dz . H(z)

(1.143)

36

Particle Cosmology

1.6

Distance measures in cosmology

Having defined most of the concepts with which we describe a cosmological model (dynamics, content, scales and ages etc.) we now look at some of the classical measures of distances used to constrain the expansion history of the universe.

1.6.1

Angular diameter distance

Consider an extended object of proper size l at some comoving distance χ at a conformal time η. figure The proper size of the object is related to the radial distance and angle subtended as l = a(η)SK (χ)∆θ.

(1.144)

So by observing objects whose proper size can be estimated kinematically (e.g. galaxy or globular cluster) at various redshifts, we can use their apparent angular sizes to constrain the expansion history and/or the geometry of the universe along the line of sight. To understand how this works consider an object which is close by. The metric looks nearly flat such that SK (χ) ≈ χ and a(η) ≈ a0 then ∆θ ≈

l = , a0 χ d l

(1.145)

where d is the physical distance to the object. This is the same as in Euclidean space and simple to understand. However when the object is much further away i.e. close to the particle horizon, the comoving distance has nearly converged to a constant and a(η) , a0 such that the apparent size grows as ∆θ ∼

1 . a(η)

(1.146)

37

The smooth universe

This is quite strange since an object at the horizon would cover the whole sky! however it becomes very faint so we can’t see it... In analogy with the euclidean case we define the angular diameter distance of an object as dA =

l a0 SK (χ) = a(η)SK (χ) = . ∆θ (1 + z)

(1.147)

38

Particle Cosmology

1.6.2

Luminosity distance

One quantity that we can measure about objects in the universe is the flux (Energy/unit time/unit area) of incoming radiation. This is measured directly by a telescope measuring the intensity of radiation collected over a certain aperture. If we could somehow predict the intrinsic luminosity of certain objects in the sky then by checking the observed flux we could calculate their distance. Comparing this to the redshift of the object we would then be able to map the expansion history of the universe. Consider an object at a comoving distance χ emitting radiation with a certain luminosity (energy/time) L. The energy emitted in a time interval ∆t = a(η)∆η is just ∆E = La(η)∆η

(1.148)

and is travelling outwards along geodesics in a spherical shell of thickness ∆χ = ∆η. figure An observer will see the photons in this shell redshifted by the cosmological expansion that occurred as the radiation was travelling ∆E(η0 ) = ∆E

a(η) a2 (η) =L ∆η. a0 a0

(1.149)

But the surface area of the shell has also grown to 2 A(η0 ) = 4πd2 = 4πa20 SK (χ),

(1.150)

since the physical distance is related to the comoving distance as d = aSK (χ) in a general curved model. The time it takes for the shell to pass the observer is ∆t(η0 ) = a0 ∆η.

(1.151)

Thus the flux reaching the observer is F =

∆E(η0 ) L a2 (η) L = . = 2 4 2 A(η0 )∆t(η0 ) 4πSK (χ) a0 4πSK (χ)a20 (1 + z)2

(1.152)

39

The smooth universe

Again, in analogy with the Euclidean case where we would simply define F = L/4πd2 we define a luminosity distance to an object as dL =

!

L 4πF

"1/2

.

(1.153)

The common way the flux is used in astronomy is to re-write it as an apparent magnitude m(z) = −2.5 log10 F = 5 log10 (1 + z) + 5 log10 SK (χ(z)).

(1.154)

This is how modern Hubble diagrams are plotted, usually using Type 1a supernovae as observations. The intrinsic luminosity of these types of supernovae explosions can be estimated very accurately if the light curve is measured during the explosion. Thus they are an ideal candidate to use as a distance measure. They are also bright so we can see them going off in distant galaxies. These type of observations led to the discovery that the expansion is apparently accelerating today and we appear to be entering a de sitter like epoch.

40

Particle Cosmology

Figure 1.8: Plot of distances for two (flat) models; Einstein-de Sitter and ΛCDM for H0 = 70 Kms−1 Mpc−1 . Comoving distance to horizon is ∼ 8.5 Gpc and 14.0 Gpc for the two models respectively

41

The smooth universe

HST Discovered Ground Discovered

45 ust

Binned Gold data

40

-z

high

35

Δ(m-M) (mag)

µ

0.5

tion ~

Evolu

z

pure acceleration: q(z)=-0.5

w=-1.2, d

w/dz=-0.5

0.0

ΩM =1

.0, Ω

Λ

-0.5

30

d gray

w=-0.8

=0.0

Empty (Ω=0) ΩM=0.29, ΩΛ=0.71

0.0

0.5

0.5

1.0 z

1.0 z

, dw/dz

~ pur

e dec

elera

tion:

1.5

1.5

=+0.5

q(z)=

0.5

2.0

2.0

Figure 1.9: The latest (Riess et al. 2007) SN1a magnitude diagram. Notice how the pure Einstein-de Sitter solution is strongly ruled out. The most distant supernovae are now probing a time when the universe was still matter dominated (and decelerating).

42

Particle Cosmology

1.7

The Big Bang Puzzles

The standard Big Bang picture has some fundamental problems associated with it.

Flatness Problem Recall how we can write the Hubble equation in terms of the critical energy density at any time as H 2 = H 2Ω −

K a2

(1.155)

where Ω contains contribution from matter or radiation. We can write this as ΩK ≡ Ω − 1 =

K a2 H 2

(1.156)

So if Ω = 1 perfectly it will remain so but if Ω−1 is not zero it will evolve away from it since in general a˙ is decreasing in time. For example in the matter dominated universe we had a ∼ t2/3

(1.157)

a ∼ t1/2

(1.158)

ΩK ≡ |Ω − 1| ∼ t2/3 or ∼ t

(1.159)

and in the radiation epoch

such that

so if the universe today is close to flat then it had to be much closer to unity in the past. Assuming for example that t0 ≈ 1017 s, and teq ≈ 1011 s and the initial conditions were set at close to the Planck time tpl ≈ 10−43 s we can estimate that ΩK (tpl ) < 10−61 , if ΩK (t0 ) < 0.02.

(1.160)

43

The smooth universe Horizon Problem

The most striking example of the horizon problem is exposed in the CMB. Recall the definition of the comoving particle horizon # t0 dt η= a 0

(1.161)

i.e. it is the conformal distance that a particle could have traveled from an initial time to today. At recombination the particle horizon is roughly η& ∼ 200Mpc

(1.162)

since η0 is about 14 Gpc, the surface of last scattering we see today spans lots of regions that were causally disconnected at recombination. However we know that the CMB is homogeneous to one part in 104 even though no causal physical process could have made regions so homogeneous. The problem is even more severe if we consider the size of a causal region at the Planck time. We can estimate that dp (tpl ) ≈ 10−26 . dp (t0 )

(1.163)

Thus the universe we see today should be made up of about 1078 regions that were causally disconnected at the Planck time and yet the distribution of matter was very smooth over this whole region. One assumption we are making here is that the expansion of the universe does not destroy inhomogeneity, we will confirm this when looking at perturbations.

1.8

Inflation - Solution of the Cosmological Puzzles

We saw earlier how the standard (matter or radiation dominated) models suffer from a number of problems namely, the flatness, horizon, homogeneity problems etc.

44

Particle Cosmology Recall how we can write the Hubble equation in terms of the critical energy

density at any time as H 2 = H 2Ω −

K a2

(1.164)

where Ω contains contribution from matter or radiation. We can write this as Ω−1 =

K a2 H 2

(1.165)

So if Ω = 1 perfectly it will remain so but if Ω−1 is not zero it will evolve away from it since in general a˙ is decreasing in time. For example in the matter dominated universe we had a ∼ t2/3

(1.166)

a ∼ t1/2

(1.167)

|Ω − 1| ∼ t2/3 or ∼ t

(1.168)

and in the radiation epoch

such that

so if the universe today is close to flat then it had to be much closer in the past. The most striking example of the horizon problem is exposed in the CMB. Recall the definition of the comoving particle horizon # t0 dt η= a 0

(1.169)

i.e. it is the conformal distance that a particle could have traveled from an initial time to today. At recombination the particle horizon is roughly η& ∼ 200Mpc

(1.170)

since η0 is about 14 Gpc, the surface of last scattering we see today spans lots of regions that were causally disconnected at recombination. So why is the CMB so homogeneous? In addition we see the same phase of the acoustic oscillations in all directions in the CMB, this means that the initial conditions for the oscillators (i.e. the

45

The smooth universe

phase) were in all causal patches. The initial conditions are just the super-horizon modes entering the horizon, so how could these have been set up without violating causality? A useful way to re-write the comoving particle horizon is η=

#

0

a0

da 1 = a aH

#

a0

d ln a

0

1 aH

(1.171)

i.e. the logarithmic integral over the history of the comoving Hubble radius. Recall that the particle horizon tells us whether two points were ever in causal contact. The Hubble radius tells us whether two points are in causal contact at time/scale factor a. The trick is to realize that if the comoving Hubble radius was larger in the past and then decreased in size the current Hubble radius could be smaller than in the past but the comoving particle horizon could still be very large in particular η0 >>

1 a0 H0

(1.172)

with most of it’s contribution coming from early times. If the comoving Hubble radius is decreasing then regions of space that are in causal contact eventually fall out of contact. i.e. wavelengths that are initial sub-horizon can exit the horizon and become super-horizon modes What is the condition required to get the comoving Hubble radius to decrease? we simply need a form of energy density which gives a˙ increasing with time i.e. an accelerating scale factor. What kind of matter can do this? The second Friedman equation is 4πG a ¨ =− (ρ + 3p) a 3

(1.173)

so for a ¨ > 0 we must have p 0 for a period of time and then the normal Big Bang scenario takes over.

47

The smooth universe

1.9

Scalar Field Cosmology

We can achieve this scenario with a scalar field φ(!x, t) which we will assume to be homogeneous for now i.e. φ(!x, t) → φ(t)

(1.175)

1 Lφ = − g µν ∂µ φ∂ν φ − V (φ) 2

(1.176)

Its Lagrangian density is given by

with the energy tensor given by the variation of the action . α αν α 1 µν Tβ = g ∂β φ∂ν φ − gβ g ∂µ φ∂ν φ + V (φ) . 2

(1.177)

For a homogeneous field in the rest frame the space derivatives vanish and we can identify T00 = −ρ and T0i = p to get 1 ρφ = φ˙ 2 + V (φ) 2

(1.178)

1 pφ = φ˙ 2 − V (φ). 2

(1.179)

and

Note the negative sign in the equation for the pressure. If we arrange the scalar field such that the potential is (positive) and greater than the kinetic energy we have negative pressure 1 V (φ) > φ˙ 2 2

(1.180)

If this condition is valid we have inflation. The first models of inflation (Guth 1981) used a scalar field trapped in a false vacuum show figure. Since φ˙ is small in the local minima then the scalar field is potential dominated with a constant value of the potential and the a ¨ > 0. In this case we have a constant Hubble rate with the scale factor growing exponentially a ∝ eHt

(1.181)

48

Particle Cosmology

1.9.1

How many e-foldings?

Let’s consider the comoving Hubble radius at the end of inflation vs today assuming the universe was radiation dominated (dominant contribution to 1/aH is from radiation period) then H∼

1 a0 H0 a0 and = 2 a ae He ae

(1.182)

where ae is the scale factor at the end of inflation. The comoving Hubble radius is then related to the one today as ae 1 T0 1 1 = = ae He a0 a0 H0 Tinf a0 H0

(1.183)

since in the radiation era a ∼ 1/T and Tinf is the energy scale of inflation which we match to the temperature of the universe at the start of the Big Bang picture. Using T0 = 2.725K = 2.348 × 10−4eV (kB = 1 units) and assuming that inflation happens at energies Tinf > 1015 GeV

(1.184)

1 1 ≈ 1028 ae He a0 H0

(1.185)

then we have

So the visible universe was 28 orders of magnitude smaller at the end of inflation. The minimal requirement to solve the horizon problem is that all the largest scales we see today were sub-horizon at the start of inflation (i.e. they were inflated out of the horizon and are just now re-entering) So the comoving Hubble radius had to decrease by at least 28 orders of magnitude during inflation. Since H is constant we need the scale factor to grow by 28 orders of magnitude during inflation a(ts ) = ae eH(ts −te ) ts < te

(1.186)

so the number of e-foldings of the scale factor during inflation is ln(1028 ) ∼ 64.

(1.187)

49

The smooth universe

1.9.2

Slowly Rolling Scalar Field

The simple case of a scalar field trapped in a false vacuum suffered from the problem of how to end inflation. Guth and Weinberg 1983 and Hawking, Moss and Stewart 1982 proved inflation could not end nicely. Since the only way to stop inflation was for the field to tunnel to the true vacuum regions that would have done so already would have to coalesce so that the whole visible universe to be at the true vacuum. They showed that this could not happen because the regions would be inflated away by the bubbles still in the false vacuum. An alternative scenario was proposed by Linde 1982 and Albrecht and Steinhardt 1982 in which the field is rolling towards the true vacuum and isn’t trapped in a local minimum. Under specific conditions the universe would be inflating. To understand how this work we need an equation of motion for the scalar field. Taking the time derivative of the first Friedman equation we have / ! "2 0 a ¨ a˙ 8πG @ ˙ ¨ ˙ A 2H − = φφ + φV,φ a a 3

(1.188)

substituting the equation for a ¨/a the lhs becomes −8πGH(ρ + p) = −8πGH φ˙ 2

(1.189)

φ¨ + 3H φ˙ + V,φ = 0

(1.190)

φ## + 2Hφ´ + a2 V,φ φ = 0

(1.191)

so that we have

or in conformal time

Slow roll is defined as the case where φ## and φ# are small and hence H is nearly constant. In this case we can write η=

#

a

ae

da 1 ≈ Ha2 H

#

a

ae

da 1 ≈ − a2 aH

(1.192)

since the scale factor at the start of inflation is much smaller than at the end.

50

Particle Cosmology Slow roll is generally quantified in term of the slow roll parameters their meaning

follows from the condition of inflation, we can write a ¨ = H˙ + H 2 > 0 a

(1.193)

such that −

H˙ ≡0