7. Observations and Inflation

Question 1

What are the two most important observational cosmological programmes so far?

Answer 1

• supernovae projects

- CMB fluctutations

Question 2

Supernovae Projects

Answer 2

-mass searches for type Ia supernovae in distant galaxies in the 1990s resulted in large data sets which could provide a way to test our cosmological models making use of the standard candle method

Question 3

CMB Fluctuations

Answer 3

• quantum theory predicts inhomogeneities in the universe with particular characteristic linear scales at the time of decoupling (when the universe becomes transparent to radiation)
• these lead to fluctuations of the CMB which are indeed observed
• thus inhomogeneities can be used as standard rods or bars

Question 4

Fitting Friendmann Models to the Observations

Answer 4

• NONE of the Freidmann models can fit the observations

- contrary to all Friedmann’s models, the expansion of the universe is actually speeding up

Question 5

Deceleration Parameter

Definition

Answer 5

qo = - ao’‘/aoHo²

-must be negative since expansion is actually speeding up

Question 6

The Cosmological Constant

Answer 6

-consider the acceleration equation with a new term in it:
a’‘/a = -4πG/3 [ρ + 3P/c²] + Λ/3
-for both non-relativistic matter and radiation, in models with Λ=0 we have a’‘<0 implying such models are decelerating
-to have an accelerating universe we need Λ>0, thus one possible solution is to consider the Einstein equation with a cosmological constant term

Question 7

Revised Critical Density

Answer 7

-including the cosmological constant in the Friedmann equation:
(a’/a)² + ϰc²/a² = 8πGρ/3 + Λ/3
-where a’/a = H

Question 8

Revised Critical Parameter

Answer 8

Ω = >1, for spatially closed (ϰ>0)
1, for spatially flat (ϰ=0)
<1, for spatially open (ϰ<0)

Question 9

Splitting the Critical Parameter into Components

Answer 9

Ω = Ωm + Ωr + ΩΛ
-where:
Ωm = ρm/ρc
Ωr = ρr/ρc
ΩΛ = Λ/3H²
-we have separated the contributions of the cold or non-relativistic matter, the hot matter or radiation ρr to the total mass-energy density ρ=ρm+ρr and hence their contributions to the critical parameter

Question 10

Standard Cosmological Model

Friedmann’s Equation

Answer 10

(a’/a)² + ϰc²/a² = 8πG/3 (ρr+ρm) + Λ/3

Question 11

Standard Cosmological Model

Fluid Equation for Matter

Answer 11

ρm = ρm,o (a/ao)^(-3)

= ρm,o/A³

Question 12

Standard Cosmological Model

Fluid Equation for Radiation

Answer 12

ρr = ρr,o (a/ao)^(-4)

= ρr,o/A^4

Question 13

Standard Cosmological Model

Normalised Scale Factor

Answer 13

A(t) = a(t)/a(to) = a(t)/ao

• can write Friedmann’s equation in terms of A, ao, ρr,o and ρm,o
• or in terms of A, Ωo, Ωr,o and Ωm,o

Question 14

Standard Cosmological Model

Acceleration Equation

Answer 14

a’‘/a = - 4πG/3 [ρ + 3P/c²] + Λ/3

Question 15

Standard Cosmological Model

Cosmological Observations

Answer 15

-using CMB fluctuation observations and supernovae observations:
Ωm,0~0.27
ΩΛ,0 ~ 0.73
-and Ωr,0 &laquo_space;Ωm,0, based on astronomical observations:
Ωr,0 ~ 8*10^(-5)
-and
Ho ~ 72km/s Mpc^(-1)

Question 16

Standard Cosmological Model
Past of the Universe and Predicting the Future
Curvature Form

Answer 16

-the curvature term 1-Ωo does not vary with A, it is already rather small at present, |1-Ωo|«1, this term is insignificant in the past and future where other terms become much larger thus if the universe is spatially flat it does not matter if ϰ=0,-1,+1 its evolution is almost the same for all these three choices

Question 17

Standard Cosmological Model
Past of the Universe and Predicting the Future
Past

Answer 17

-the matter and radiation terms Ωr,0/A² and Ωm,0/A respectively grow as A->0
thus they dominate in the past when A«1
-since the radiation term grows faster than the matter one as A->0, in the past, there should be a transition frmo the radiation dominated phase to the matter dominated phase
-this happens when:
Ωr,0/A² = Ωm,0/A
=>
Ar,m = Ωr,0 / Ωm,0 = 3*10^(-4)
-this is a much smaller and hence a much younger universe

Question 18

Standard Cosmological Model
Past of the Universe and Predicting the Future
Ar,m

Answer 18

• based on the densities corresponding to Ar,m one can show that the universe must be opaque at this time
• photons cannot propagate freely but get absorbed and emitted again at a very high rate
• matter and radiation are tightly coupled, they decouple only a Adec~10^(-3)
• for A

Question 19

Standard Cosmological Model
Past of the Universe and Predicting the Future
Radiation Dominated Phase

Answer 19

A’ = Ho √Ωr,0/A
-with solution
A² = 2Ho √Ωr,0 t + constant
-this implies existence of a t* such that a(t*)=0 and hence a big bang is still a feature of the standard cosmological model

Question 20

Standard Cosmological Model
Past of the Universe and Predicting the Future
Am,Λ

Answer 20

as the cosmological constant term is ΩΛ,0 A² which grows with S, it will dominate the future of the universe
-the transition from a matter dominated to a Λ-dominated universe occurs when:
Ωm,0/A = ΩΛ,o A²
=>
Am,Λ = [Ωm,0/ΩΛ,0]^(1/3) ~ 0.72
-this is only in the relatively recent past in cosmological terms hence at present the universe is in the transition period to the epoch of Λ-domination

Question 21

Standard Cosmological Model
Past of the Universe and Predicting the Future
Λ-Domination

Answer 21

-ignoring all terms but the cosmological constant one:
A’² = α²A² = Ho² ΩΛ,0
=>
A = K exp(αt)
-hence an exponential expansion is predicted for the future, when A»1

Question 22

The Problems With Standard Cosmology

Answer 22

• there are a number of problems with the standard model cosmology, i.e. cosmology that uses a FRW spacetime and assumes matter is homogeneous and isotropic
• the main issues are the flatness problem and the horizon problem
• other issues are more complex and involve particle physics but they are all solved if initially the expansion of the universe proceeded in a different way to that of the standard model

Question 23

The Flatness Problem

Description

Answer 23

• according to cosmological observations, the critical parameter, Ωo, is very close to one
• unless the universe is exactly flat, ϰ=0, we should have Ωo≠1, this value of ϰ does not seem natural

Question 24

The Flatness Model

Equations

Answer 24

-can express Ω in terms of ϰ and a(t):
|1-Ω| = |ϰ|c²/H²a² = |ϰ|c²/a’c²
-thus the evolution of Ω is determined by the evolution of a’, so regardless of the value of ϰ, if a’->∞ then Ω->1
-however in the standard model a’∝t^(-1/2) during the initial radiation-dominated phase and then as a’∝t^(-1/3) during the matter-dominated phase which continued almost until the current epoch
-so |1-Ω| has been increasing, and hence Ω moving away from 1, the opposite of what we need

Question 25

The Horizon Problem

Description

Answer 25

• the standard cosmological model assumes that the universe is uniform (homogeneous and isotropic) which is in agreement with observations
• for this to be the case and since the highest speed of communication is the speed of light, in the past, the part of the universe which corresponds to the currently observed universe must have been smaller than the contemporary cosmological horizon
• the isotropy of the CMB tells us that the whole of the visible universe was already uniform at the time of decoupling
• hence at t=td, the comoving radius of the cosmological horizon had to exceed the comoving radius of the currently visible universe
• in the standard model, there is no time to establish causal-connectivity across the visible universe and no physical process can erase its initial inhomogeneities

Question 26

The Horizon Problem

Equations

Answer 26

-the metric radius of the cosmological horizon is given by:
rh(t) = c a(t) ∫ du/a(u)
-where the integral is from 0 to t
-for t

Question 27

Inflation

Answer 27

-since for sufficiently small a(t) we enter the conditions not accessible to modern physics, strange things may occur
-suppose that prior to the radiation-dominated phase there was another phase, inflation
-a period when the universe was expanding exponentially:
a(t) = a* e^(αt)
-for 0

Question 28

Solution to the Flatness Problem

Answer 28

-during inflation we have:
a' ∝ e^(αt)
-and hence:
|1-Ω| = c²/a'² ∝ e^(-2αt)
-thus even if initially Ω is very large or very small, it becomes close to one for sufficiently large t, when αt>>1

Question 29

Solution to the Horizon Problem

Answer 29

-

Question 30

Inflation Field

Answer 30

-in inflation, the equation of state is very close to
Pi = -ρi c²
-this leads to very slow changing mass-energy density during inflation
-inflation ends when inflation particles become unstable and decay into other particles which satisfy the equation of state for radiation, P=ρc²/3
-the theory of inflation successfully resolves many other problems of Friedmann’s cosmology