Modern Cosmology Solutions 4: LCDM Universe

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1 Modern Cosmology Solutions 4: LCDM Universe Max Camenzind October 29, 200. LCDM Models The ansatz solves the Friedmann equation, since ȧ = C cosh() Ωm sinh /3 H 0 () () ȧ 2 = C 2 cosh2 () sinh 2/3 () ( Ω m)h0 2 ( ) 2/3 Ωm = H 2 +sinh 2 () 0 Ω m sinh 2/3 () ( Ω m) ( ) /3 ( ) = H0 2 Ωm 2/3 Ωm Ω m Ω m sinh 2/3 () +H2 0 sinh 4/3 ()( Ω m ) Ω m = H 2 0[ Ωm a +( Ω m)a 2]. (2) Age of the Universe t(z) follows from the inversion with a(t) = /(+z). Define x(z) [/C(+z)] 3/2, the inversion gives t(z) = 2 Ωm arsinh(x(z)) 2 Ωm ln(x+ x 2 +). (3) Here we use the identity arsinh(x) = ln(x+ x 2 +). The age t 0 of the Universe follows from this for z = 0, i.e. for x 0 = /C 3/2 = [( Ω m )/Ω m ] /2 =.644 for Ω m = 0.27, and therefore t 0 = 2 Ωm ln(x 0 + Light travel time distance d T (z) follows from this by x ) =.272 = 0.99 = 3.7Gyrs. (4) H 0 H 0 d T (z) = c(t 0 t(z)). (5) The age t of the galaxy at z = 8.6 follows for z = 8.6, i.e. for x g = /[C( + z g )] 3/2 = [( Ω m )/Ω m ] /2 /(+z g ) 3/2 = for Ω m = 0.27, and therefore t(z g ) = 2 ln(x g + x 3H 2 g +) = = = 600Mioyrs. 0 Ωm H 0 H 0 (6)

2 Light travel time distance for the galaxy at redshift 8.6 follows from this by Ω m = 0.27, /H 0 = 3.7 Gyrs and C = 0.4 with t 0 = 3.7 and t(8.6) = 600 Mio yrs d T (z = 8.6) = c(t 0 t(z = 8.6)) = 3.Glyrs = 4.25Gpc. (7) Neutrinos decoupled at the temperature of 0.8 MeV, shortly before electron-positron pairs annihilated. Therefore, the neutrino temperature is lower than the photon temperature, T γ /T ν = (/4) /3 =.40, T ν =.93 K. If electron neutrinos have a mass, they are non-relativistic today with a mass-density The number of electron neutrinos is then ρ ν = m ν [n ν (T ν )+n ν (T ν )]. (8) n ν = n ν = n γ = 3 22 n γ = 53cm 3. (9) ev = J = erg. So the mass density follows ρ ν ν = /c m ν [ev]gcm 3. (0) This has to be compared with the critical density and therefore ρ crit = h 2 gcm 3 () Ω ν = ρ ν ν mν ρ crit ev h 2. (2) Neutrinos cannot be important in the present evolution of the Universe! 2. Cosmological Distances and Fundamental Plane Observable Universe In Big Bang cosmology, the observable Universe consists of the galaxies and other matter that we can in principle observe from Earth in the present day, because light (or other signals) from those objects has had time to reach us since the beginning of the cosmological expansion. Assuming the Universe is isotropic, the distance to the edge of the observable universe is roughly the same in every direction that is, the observable universe is a spherical volume centered on the observer, regardless of the shape of the Universe as a whole (Fig. ). The observable universe appears from our perspective to be spherical. Every location in the Universe has its own observable universe which may or may not overlap with the one centered around the Earth. In practice, we can see objects only as far as the surface of last scattering, which is when particles were first able to emit photons that were not quickly re-absorbed by other particles, before which the Universe was filled with a plasma opaque to photons. The photons emitted at the surface of last scattering are the ones we detect today as the cosmic microwave background radiation (CMB). The current comoving distance to the particles which emitted the CMB, representing the radius of the visible universe, is calculated to be about 4.0 Gigaparsecs (45.7 2

3 Figure : The observable Universe as seen from the Milky Way in comoving distances. The Universe is spherically symmetric around us. A representation in light travel distances would give a wrong impression on the matter distribution. We are surrounded by galaxies, followed at higher distances by Quasars (dots), the Dark Age and finally the last scattering surface (at a distance of 4 Gpc), which has a wavy structure due to the perturbations in the density and in the temperature distribution. Close to this surface we find the Particle Horizon (at a distance of 4.3 Gpc), which represents the Big Bang and therefore the ultimate edge of the observable Universe. billion light years), while the current comoving distance to the edge of the observable Universe is calculated to be 4.3 Gigaparsecs (46.6 billion light years), about 2% larger. While Special Relativity constrains objects in the Universe from moving faster than the speed of light with respect to each other, there is no such constraint when space itself is expanding. This means that the size of the observable universe could be smaller than the entire Universe; there are some parts of the Universe which might never be close enough for the light to overcome the speed of the expansion of space, in order to be observed on Earth. Some parts of the Universe, which are currently observable, may later be unobservable due to ongoing expansion. The figures quoted above are distances now (in cosmological time), not distances at the time the light was emitted. For example, the cosmic microwave background radiation that we see right now was emitted at the time of recombination, 38,000 years after the Big Bang, which occurred around 3.7 billion years ago. This radiation was emitted by matter that has, in the intervening time, mostly condensed into galaxies, and those galaxies are now calculated to be about 46 billion light-years from us. To 3

4 estimate the distance to that matter at the time the light was emitted, a mathematical model of the expansion must be chosen and the scale factor, a(t), calculated for the selected time t since the Big Bang. For the observationally-favoured Lambda- CDM model, using data from the WMAP spacecraft, such a calculation yields a scale factor change of approximately,292. This means the Universe has expanded to,292 times the size it was when the CMB photons were released. Hence, the most distant matter that is observable at present, 46 billion light-years away, was only 36 million light-years away from the matter that would eventually become Earth when the microwaves we are currently receiving were emitted. Parameters for the homogeneous FLRW Universe: Figure 2: The fundamental parameters of the Friedmann Universe, as following from WMAP and Supernovae observations. The inhomogeneous Universe has a lot more parameters. Fundamental Plane of Cosmology Besides the Hubble constant H 0, the density parameters Ω m, Ω Λ and Ω k are the fundamental parameters of a FLRW universe. In addition, we need an equation of state for the Dark Energy. Since the sum of all density parameters is one in a Friedmann Universe, essentially only the density parameter Ω m and the DE parameter Ω Λ are independent. These two parameters build the fundamental plane of Cosmology (FP, FigureFP bottom). Each dot in the FP represents a possible model of our Universe. Various data give constraints on these models (Supernova data, CMB data, Dark Matter observations in clusters of galaxies, gravitational lensing etc.) 4

5 Figure 3: Top: Density evolution in FLRW models in terms of the density parameters Ω i (z) for i = m,r,λ. Bottom: The Fundamental Plane of Cosmology. Special dots represent a pure vacuum universe (desitter), an empty universe (k = ), the SCDM universe (Einstein-deSitter) and the present LCDM (green). 5