7. Observations and Inflation Flashcards
What are the two most important observational cosmological programmes so far?
- supernovae projects
- CMB fluctutations
Supernovae Projects
-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
CMB Fluctuations
- 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
Fitting Friendmann Models to the Observations
- NONE of the Freidmann models can fit the observations
- contrary to all Friedmann’s models, the expansion of the universe is actually speeding up
Deceleration Parameter
Definition
qo = - ao’‘/aoHo²
-must be negative since expansion is actually speeding up
The Cosmological Constant
-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
Revised Critical Density
-including the cosmological constant in the Friedmann equation:
(a’/a)² + ϰc²/a² = 8πGρ/3 + Λ/3
-where a’/a = H
Revised Critical Parameter
Ω = >1, for spatially closed (ϰ>0)
1, for spatially flat (ϰ=0)
<1, for spatially open (ϰ<0)
Splitting the Critical Parameter into Components
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
Standard Cosmological Model
Friedmann’s Equation
(a’/a)² + ϰc²/a² = 8πG/3 (ρr+ρm) + Λ/3
Standard Cosmological Model
Fluid Equation for Matter
ρm = ρm,o (a/ao)^(-3)
= ρm,o/A³
Standard Cosmological Model
Fluid Equation for Radiation
ρr = ρr,o (a/ao)^(-4)
= ρr,o/A^4
Standard Cosmological Model
Normalised Scale Factor
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
Standard Cosmological Model
Acceleration Equation
a’‘/a = - 4πG/3 [ρ + 3P/c²] + Λ/3
Standard Cosmological Model
Cosmological Observations
-using CMB fluctuation observations and supernovae observations:
Ωm,0~0.27
ΩΛ,0 ~ 0.73
-and Ωr,0 «_space;Ωm,0, based on astronomical observations:
Ωr,0 ~ 8*10^(-5)
-and
Ho ~ 72km/s Mpc^(-1)
Standard Cosmological Model
Past of the Universe and Predicting the Future
Curvature Form
-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
Standard Cosmological Model
Past of the Universe and Predicting the Future
Past
-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
Standard Cosmological Model
Past of the Universe and Predicting the Future
Ar,m
- 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
Standard Cosmological Model
Past of the Universe and Predicting the Future
Radiation Dominated Phase
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
Standard Cosmological Model
Past of the Universe and Predicting the Future
Am,Λ
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
Standard Cosmological Model
Past of the Universe and Predicting the Future
Λ-Domination
-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
The Problems With Standard Cosmology
- 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
The Flatness Problem
Description
- 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
The Flatness Model
Equations
-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
