Cosmology Winter School 5/12/2011! Jean-Philippe UZAN!

Transcription

1 Cosmology Winter School 5/12/2011! Lecture 1:! Cosmological models! Jean-Philippe UZAN! Cosmological models! We work in the framework of general relativity so that the Universe is described by a spacetime M and a metric g µ!.! The geometry and the matter distribution are then related by the Einstein equations.! Cosmological program:! Which solution of the Einstein equations describes our! universe?! or more modestly! Which solution of the Einstein equtions is a good model of! our universe?! Such questions raise a issue of scale which is not defined.! Let us first see the difficulties that appear to achieve this program.!

2 Cosmological models! Einstein equations! Matter sector! This can be derived simply from an action! What are the matter constituents?! What is their stress-energy tensor?! Solving this set of equations! The possibility to solve this set of equations depends on the quantity of astrophysical data! Even with ideal data, we have to face (at least) 2 limitations:! - we have only 1 universe! Unique object that cannot be compared to other similar objects to! discuss it genericity.! - we observe our universe from 1 spacetime position! This position cannot be chosen.!

3 Consequences! Geophysical / astrophysical! -!Astrophysical data are located on our past light cone.! -! Geophysical data ( ) up to a redshift of order 0.4! -! Interpretation of data is not independent of the hypotheses on the structure of the spacetime! -! There exist many spacetimes with same observational signature on the past light cone! -! We need to distinguish observable universe & universe! Solving this set of equations! The possibility to solve this set of equations depends on the quantity of astrophysical data! Even with ideal data, we have to face (at least) 2 limitations:! - we have only 1 universe! Unique object that cannot be compared to other similar objects to! discuss it genericity.! - we observe our universe from 1 spacetime position! This position cannot be chosen.! There is a historical dimension! We need to reconstruct the history of the universe compatible with! the data and what we know of physics! We will learn about initial conditions in the early universe.! We need to make hypotheses on the structure of the universe.!

4 Principle of uniformity! Data indicate that the Universe looks isotropic around us.! We can restrict to this class of models.! Principles of uniformity!

5 Principles of uniformity! Copernican principle and its consequences! Copernican principle: We do not leave in a particular place in the universe.! Cosmological principle: The universe is spatially isotropic and homogeneous.! Note: the notions of isotropy and homogeneity have to be considered in a statistical sens. They involve implicitely a smoothing scale.! Spacetime can be foliated in homogeneous space.!

6 Cosmological metric! Notion of fundamental observer with 4-velocity u µ.! It allows to define the cosmic time, i.e. the propoer time of this family of observers! The induce metric on " t is then related to the spacetime metric by! so that the 4-dimensional metric takes the form! The geometry of the 3-dimensional space is the one of a homogeneous and isotropic space. They are known and there are 3 types of possible geometries.! Cosmological metric! In coordinates, the metric takes the form with It is convenient to use the conformal time defined by

7 Description of the matter! The next step is to describe matter. The symmetries (CP) imply that it has to be of the form! It follows that the Einstein equations reduce to the Friedmann equations! and the conservation equation takes the form! Set of 2 (3-1) independent equations for 3 variables (a,#,p) of the cosmic time. Not closed. Need to specify an equation of state.! - dust/radiation/cosmological constant! Exercise: Check this set of equations. Come and see me if any problem here. Implications (kinematics)! Hubble law! Propagtion of light! - cosmological redshift! - redshift drift! Geodesic of test particles! - align to the Hubble flow (i.e. to worldlines of fundamental observers)!

8 The standard cosmological models! a 0!! Eq. state! Scaling Scale factor! radiation! w=1/3! a -4! t 1/2! Matter (dust)! w=0! a -3! t 2/3!!" w=-1! a 0! e Ht! Early universe always dominated by radiation (see lecture 2)! Dynamics! The Friedman equations can be rewritten in a reduced form by introducing! The conservation equation implies that! and the Friedman equation takes the forme Full dynamics depends on 5 numbers!

9 Dynamics! Universe is dominated by radiation in the past [see lecture 2] RDU MDU (structure can form)!du

10 If w>-1/3 Saddle point Repulsor Attractor Exercise: rewrite the Friedmann equations as a dynamical system using p=ln a as a time variable. Observable universe! Light cone! transparent! Galaxy that can be observed in the futre.! opaque! Only a finite part of the universe can be observed! (c) L. Haddad & G. Duprat

11 Typical value of the parameters! The Hubble constant is decomposed as which gives the typical spatial & time scales of the universe Conformal time representation! X

12 Hypotheses! Gravitation is well described by General relativity Many extensions motivated by the fact that GR is not renormalizable and difficulty to construct a quantum theory of gravity [see lectures by K. Koyama] Many tests: local / constants / large scale structure [see lecture 5] Matter composition Need for dark matter [see lectures by L. Covi] Need for dark energy Fluid description On what scale is it a good description? Copernican principle Can it be tested? On what scale is it a good hypothesis Topology Testable of the size of the observable universe A good model! Many successes: - Hubble law - primordial hot phase & thermal history that leads to * a relic radiation density with Planck spectrum T~(1+z) is constrained * relic abundances of light elements - formation of structure: * distribution of galaxies * CMB anisotropies

13 But still an incomplete model! Why is FL a good description of the universe? flatness, horizon problems [see lecture 3] Dark sector [see lectures by L. Covi for dark matter] Origin of structures - what are the seeds of density inhomogeneity? - it seems that they have super-hubble initial correlations. [see Lectures by J. Dunkley] - Is there a new horizon problem? Beyond FL! There exist a large number of cosmological solutions of Einstein equations. They are characterised by their Killing vectors (that are related to the symmetries of the spacetime) How many Killing vectors for Minkowski? Friedmann-Lemaître?

14 Beyond FL! There exist a large number of cosmological solutions of Einstein equations. They are characterised by their Killing vectors (that are related to the symmetries of the spacetime) The Killing vectors have the property that their commutator is also a KV One can define the group of transitivity dimension = s the group of isotropy [fixed point] dimension = q the group of isometries dimension = r Beyond FL! In d=4 dimensions:

15 Non-FL models: example! General dynamics of cosmological spacetimes! «Popular» universe model: Lemaître-Tolman-Bondi - spherically symetric but inhomogeneous spacetime - i.e. spherical symetry around one worldline only : center Two expansion rates, a priori different [for an off-center observer, the universe does not look isotropic] The solution depends on 2 arbitrary functions of r FL limit Observationally

16 To keep in mind! To construct a cosmological model, we need: - hypotheses on the fundamental law of physics [action] - hypotheses on the structure of spacetime Need to test for this hypothesis difficult because of the light-cone degeneracy, but there are some ways Standard cosmological model: - FL type + CDM + cosmological constant - in agreement with all data There exist many othe cosmological solutions that may be usefull to interpret the data and construct counter-example Data cannot be interpreted independently of a model.