Theoretical Astrophysics and Cosmology Master Degree in Astronomy and Erasmus-Mundus A.A. 2016/17 Alberto Franceschini Cosmology Course

Transcription

1 Theoretical Astrophysics and Cosology Master Degree in Astronoy and Erasus-Mundus A.A. 16/17 Alberto Franceschini Cosology Course Hoogeneous Friedan Universe.1

2 PROGRAMME FOR THE COSMOLOGY COURSE. The Hoogeneous and Isotropic (Friedann) Universe Hubble law. The Cosological Principle. Isotropic curved spaces. The Robertson-Walker etric. Geoetrical properties of the space-tie. Cosic dynaics, the Newtonian and general-relativistic approach. Cosological odels and paraeters. Fundaental observables. The redshift. Luinosity and angular diaeter distances. Tie-redshift relations. Hubble diagras. Generalized dynaical equations. The cosological constant. Observational evidences [4] Good knowledge of everything 1. The Large Scale Structure of the Universe. Local properties General and structural properties of the universe. Large scale distribution of galaxies. Angular and spatial correlation functions. Higher order correlations. Liber relation. Power-spectru of the cosic structures. Relationship of the power-spectru and ξ(r). Observational data on the large scale structure. The initial power-spectru of the perturbations. 3D apping of galaxies, clusters, AGNs. Counts-in-cells. Outline of fractal and topological analyses of the large-scale structure of the universe. [] Chap. all, Cap. 14.1, 14., 14.4, 14.6 (see below bibliographic references) [1] Chap. 13.1, 13., 13.8, 16.1, 16., 16.3, 16.4, 16.6, 16.7, 16.8, 16.11, Deviations fro hoogeneity. Gravitational lensing Point-like lenses and isotheral spherical distributions. Lens potentials. Einstein radius. Lensing crosssections. Lensing effects on tie lags. Caustics. Observations of the gravitational lensing and cosological applications. Estiate of the total galaxy cluster ass. Estiates of H. Effects of a cosological constant in the lensing statistics. Microlensing and weak-lensing. Mapping of the ass distribution. [1] Cap. 19.1, 19., 19.3, 19.4, 19.5 [] Cap. 4.6 Hoogeneous Friedan Universe.

3 Further reading: [3] Cap. 4., 4.3, 4.4, Cosological evolution of perturbations in the cosic fluid. Cosological evolution of perturbations in the thero-dynaical paraeters of the various coponents of the cosic fluid. General equations in a static universe and in an expanding one. Evolution in a atter doinated universe. Hubble drag. Relationship of perturbations and velocity fields. 4. Perturbations in an expanding universe. Peculiar otions of galaxies and structures. Deviations fro the Hubble flow, peculiar velocity fields in the coso. Observations of peculiar velocity fields. The cosic viral theore. Origin of the large scale otions. Constraints on the cosological paraeters fro the large scale otions. [] Chap. 11.1, 11., 11.3, [3] Chap Brief theral history of the Universe The atter and radiation content of the Universe. Energy densities and their evolution. Radiationdoinated universes. The epoch of recobination and equivalence. Tie-scales of cosic evolution. Cosic entropy per baryon. Priordial nucleo-synthesis and its consequencies. [1] Chap..1,.,.6,.7,.8 6. The Cosic Microwave Background Discovery of the CMB. Observations fro ground and fro space. COBE, WMAP & Planck. Origin of the CMB. Spatial properties, isotropy of the CMB. Statistical description of the angular structure. Origin of the CMB angular fluctuations. Physical processes in operation on the large scales. Fluctuations on interediate angular scales. Contributions of sources to the anisotropies on sall scales. Cosological re-ionization and its ipact on CMB. Constraints of CMB observations on the cosological paraeters. The CMB spectru. Spectral distorsions. The Sunyaev-Zeldovich effect. Observational liits on the spectral distorsions and their iplications. Polarization properties of CMB photons, observations and iplications. [] CHAP..1. Cap. 15.1, 15., 15.3, 15.4, 15.5 solo cenni, 15.6, 15.7, 15.8 solo cenni, 15.9, Cap Hoogeneous Friedan Universe.3

4 [1] Chap. 17.1, 17., 17.3, The Priordial Universe, Big Bang, phase transitions, cosological inflation The proble of the cosological horizons: propagation of the inforation and visibility of the universe. Big Bang singularity. Planck tie. Brief overview of the standard odel of eleentary particles. Fundaental interactions. Cosological phase transitions and their epochs. Open questions about the standard Big Bang odel. The horizon proble. The flatness proble. Cosological inflation and solutions to the probles. The Anthropic Principle. Current interpretation of the origin of the perturbation field and of the cosic large-scale structure. [1] Chap. 6.1, 6., 6.3, 7.1, 7., 7.3, 7.4, 7.5, 7.8, 7.9, 7.1, 7.11, 7.1, Origin and Evolution of the Cosological Structure Generation of the perturbation field (the Priordial Power Spectru). Fluctuations fro inflation. General coposition of the cosic fluid: the Dark Matter coponent. Properties of the Cold and Hot Dark Matter fluids. Scales and asses involved within the Cosological Horizon. Scale-invariant priordial spectru and the horizon entrance of perturbations of different scales Free-streaing and daping in the Dark Matter fluid. HDM versus CDM. Stagnation of the Dark Matter perturbation before the equivalence epoch. Evolution of the DM cosological perturbation field. The transfer function in the linear regie. CDM cosology. Effects of non-linear evolution in the DM. Evolution of the cosic fluid after recobination: the fate of the baryon gas. Collapse of DM halos. The Press-Schechter theory. Galaxy foration. 9. The Post-Recobination Universe Intergalactic diffuse gas. Absorption-lines in quasar spectra, Lyan-alpha clouds. The issing baryon proble. Evolutionary history of star foration and the stellar ass function. Production of heavy eleents, constraints fro the background radiations. Cosological evolution of Black-Hole accretion in Active Galactic Nuclei and quasars and relationship with galaxy evolution. [] Chap. 17, 18, 19 Hoogeneous Friedan Universe.4

5 Main bibliographic references [1] Coles & Lucchin: Cosology, Second Edition, J Wiley, [] Longair: Galaxy Foration, Second Edition. Springer, 8 [3] Peacock, Cosological Physics, Cabridge University Press, 5 [4] Rowan-Robinson Cosology, Clarendon Press, Oxford and: Franceschini, Lecture Notes in Friedann Cosology, 15, see Note: We will refer in this course to Sections, as the 8 listed above. Chapters will instead ean chapters inside these Sections. These notes can be found in: Hoogeneous Friedan Universe.5

6 Section THE HOMOGENEOUS FRIEDMANN UNIVERSE.1 Basic concepts about the fundaental cosological etrics It is useful here to suarize soe basic concepts about the hoogeneous and isotropic Friedann Universe. These concepts are the basis for our understanding and description of the Universe in its present structure and evolution. As it is discussed in soe good detail in Sect. 1, while strongly structured on scales saller than a few to several tens of Mpc, all observations show excellent evidence for isotropy and hoogeneity on larger scales, and particularly alost perfect hoogeneity on scales sufficiently larger than few hundreds Mpc. The above is the first fundaental evidence about the global properties of the Universe, and akes the content of the Cosological Principle: the Universe appears as hoogeneous and isotropic and in fact looks to share the sae properties to any fundaental observer at a given cosic tie. This is key to any our attept to describe the general properties of the Universe based on what we observe in our past light-cone. Global size of the Universe proportional to the scale factor R(t) function only of tie R(t) The second ingredient for our foral description of it is the Hubble law, the recession velocities of galaxies are proportional R(t) to their distances, d d v = Hd. These two Hoogeneous Friedan Universe.6

7 general properties, and the syetry properties that they anifest, can be viewed euristically with the analogy of the inflating balloon, see figure. This can be very effectively foralized into a etrics, called the di Robertson Walker (RW), aking the general cosological etrics: = dt a c ( t) dr 1 kr + r ( dϑ + sin ϑdϕ ) ds (.1) where a( t) R( t) R and where Rt ( ) is the scale factor detailing the expansion law of the Universe, R the scale factor at the present tie. This etrics involves only the syetry properties and nothing else, like the precise law of gravitation (like General Relativity). k is the curvature constant, does not vary with tie, and can be either, -1 or +1 (corresponding to flat Euclidean, open hyperbolic of closed spherical odels). the radial r and angular coordinates are in the so-called cooving units, that is they ake a tie invariant coordinate grid to identify cosic objects at a given tie. Proper (or physical) coordinates are siply scaled fro the cooving ones as d = a( t) r. In these ters, the Hubble constant is siply: H ( ) ( ) at = at t= t Concerning the geoetry of the Universe, the three cases of positive, negative and zero constant k to correspond positive, negative and Hoogeneous Friedan Universe.7

8 flat geoetries. These Universes are called closed, open and flat, respectively. Surfaces and volues are finite in the first cases and infinite in the others. In the case of a closed universe a light signal, or a space ship, traveling in an arbitrary direction, will travel straight away, but will be deflected by gravity onto a circular orbit all around the Universe.. The cosological redshift The first application of the Robertson-Walker etrics is a fundaental generalization of the Hubble law v = cz = Hr and the concept of the relation between recession velocity and distance. Let us consider a photon travelling fro a distant source and a fundaental observer (an observer cooving with the Hubble flow). Then ds = in eq.(.1) for these photons. Now, by separating the radial (spatial) and teporal coponents in (.1), we have r dr e = (.1) te c 1 kr () t dt at and also, independently of tie: c dr 1 kr dt e = = = at ( e) at ( ) a( te) a( t ) dt, where at ( ) 1; so that e = = that is, in ters of frequency: e dt dt n = n at ( ) e n n e and because 1 ( ne n)/ n = ne / n 1 z= at ( e) 1 1 we finally get (1 + z) = at ( e) = ne / n Hoogeneous Friedan Universe.8

9 a relation generalizing the concept of redshift (in the classical specially relativistic liit this is just v = c z) to sources at any distances in space-tie. The conclusion is that the cosological redshift z is a easure of how uch the scale factor has changed fro the epoch when the signal has been sent to that it has been received. This introduces a substantial odification of the standard Doppler redshift. For large distances, the redshift is a very fundaental easure of distance, and expresses the ratio of the scale factor at the tie the photon is eitted to that when it is received..3 Cosic dynaics and the expansion history of the Universe Now, if we want to define the x O O: observer (coordinates t=t, r=) Schee of Newtonian treatent of dynaics. Shell of particles at proper distance x to observer O. appropriate values of k and the function R(t) we need to insert a specific odel to treat the gravitational effects in the Universe and how these and the universal geoetry are affected by the various ass-energy coponents (nonrelativistic atter, relativistic particles like photons and neutrinos in particular). We can get a first insight into this potentially coplicated proble by considering an heuristic approach of using a siple Newtonian treatent, that is illustrated in the figure. Let us iagine an observer at arbitrary position O and an expanding shell of particles at distance x. Assuing the origin at the position O (this is very arbitrary), these will feel the gravitational pull of the atter content inside x, but, because of the Gauss theore, will not be affected by all the rest of the surrounding universe. Then we have, in ters of the universal scale factor: Hoogeneous Friedan Universe.9

10 GrV x 4πGrx 4 πgr() t a() t r F = x = = and in cooving coord.: a ( t) r = 3 x 3 3 t at 4 πgr() t at () 4π at () G r = = at () Now, considering the ass conservation r() = r (), An integral of this one is the faous Friedann law: 8π a () t = Gρ kc 3 at () This approach, however, is not coplete and not copletely self-consistent, for any different reasons. The ost iportant is that it cannot include into the analysis the relativistic particle coponents of the Universe (e.g. photons) that can be relevant during soe epochs. Another aspect not resolved by the classical treatent is the fact that in the classical approach the gravitational inforation travels at infinite speed, so that the local dynaics ay be influenced by events at infinite distances, while a relativistic approach and the finite speed of light would naturally liit this sphere of influence to a relatively sall environent of any rando point. For these and other reasons, we need to refer to the ore general fraework provided us by the General Relativity and its field equations: 1 8π G R g R = T, ik ik 4 ik c with R ik = R the Ricci tensor given by the contraction of the Rieann tensor l ilk l R ijk, and with ik R= g R ik a scalar quantity, the curvature scalar, and where the quantity' g ik is the etric tensor that is synthetically given by eq. (.1) in its diagonal for. The quantity T p c U U pg = ( + ρ ) ik i k ik is the energy-oentu tensor for a perfect fluid, where p and ρ are the pressure and energy density of atter and Ui the four-velocity of the cosic fluid. An even ore general for of the field Hoogeneous Friedan Universe.1

11 equations has also been considered by Einstein that is still generally co-variant (it is the ost general co-variant for of the equations): 1 8π G R g R g = T, ik ik ik 4 ik c where is an absolute constant, a scalar that is fully independent on position and tie, called by Einstein the cosological constant, in relation with his cosological static odel of the Universe. The 16 field equations becoe iediately 4 equations if we consider the diagonal for of the etrics g ik in (.1). Once applied to our cosological situation and to the RW etrics brings to the following two dynaical laws (the second is the faous Friedann equation, fro which the universal odels based on these eqs. are called Friedann universes): 4p G 3 pt () 1 a = at t+ + at 3 c 3 8pGρ( t) a = a ( t) kc + a ( t) 3 3 ( ) ρ() ( ) (.) with the cosological constant and k the curvature constant, that is the sae k constant appearing in the etrics (.1) and defining the universal geoetry. Note that these equations, without the relativistis ters p and, are perfectly consistent with what we previously found by siple application of the Newtonian gravity theory. Often in the first equation of the (.) we assue p = to treat the so-called dust universes, or atter-doinated universes, or universes without relativisti coponents. A third and fourth equations that can be inferred fro the field equations are the ass-energy conservation rules (the first has already been entioned previously): r () t = r at () ; r () t = r at () atter 3 4 radiation It is custoary to use the following paraeters to treat the cosological dynaics: Hoogeneous Friedan Universe.11

12 that are the Hubble and density paraeters. These are called the cosological paraeters. As for the ters in the above equations, the cosological constant, this is an hypothetical ter constant in tie introduced by Einstein for consistency with his assued static Universe solution of the field equations that he tried to odel. The paraeter needs to be independent on cosic tie to aintain the general covariance (general independence fro the reference syste) of the field equations (.). This constant has been recently re-introduced into the equations ruling the cosic dynaics based on the observations of high-redshift Type-1A supernovae (that are the best standard candles for cosology), whose Hubble diagra showed very faint agnitudes for their given redshift. To explain such faint observed fluxes (large values of the apparent agnitudes in the figure) based on standard cosology would require negative ass density (!), corresponding to an accelerating Hubble expansion, instead of a naturally decelerating one under the effect of self-gravity. Hoogeneous Friedan Universe.1

13 To achieve an accelerating expansion, as needed by the observations, the constant ust be positive. Fro a ore physical point of view, this constant can be interpreted as a structural property of the cosic geoetry (like interpreted by Einstein). Alternatively, it is possible to interpret it as an energy density of the vacuu ρ V by assuing that: = 8π Gρ V (.3) with associated density paraeter, siilarly to what we defined for the atter density: ( t ) 8π Gρ = = 3H 3 V H, (.4a) hence contributing to the dynaics of the expansion at the current tie t, together with the density paraeter of gravitating atter 8π G = ρ 3H ( t ) =,. (.4b) In this way the first dynaical equation in (.) calculated at the present tie becoes ( t ) 8π Gρ V 4π Gρ a ( t ) =. 3 3 (.5) After the above definitions and using the cosological paraeters, the dynaical equations can be expressed as ( ) at = H + a, ( t) ( ) H at H, a ( t) = kc + H a ( t). a ( t) (.6) Putting these two equations together and resolving for the ter kc, and calculating the at the present tie (k does not vary with tie, as we have seen) for a = 1 and a = H, we get the following fundaental relation between the curvature constant and the density paraeters Hoogeneous Friedan Universe.13

14 [ ] ck= H + 1 (.7) A further cosological paraeter easuring the deceleration or the acceleration of the cosological expansion is q aa = a t (.8) that is naturally positive (to indicate deceleration under self-gravity), but ay becoe negative (acceleration) for a sufficiently large cosological constant. We see fro eqs. (.7) and (.8) the profound relationship between the geoetrical and dynaical properties of the Universe, and their relation with its kineatical properties. A flat Universe can be obtained when the su of the density paraeters + = 1, corresponding to a sort of balance between the kinetic energy of the expansion, the self gravitational energy, and the repulsion ter. Siilar relations are reported in Galaxy Foration, (MS Longair), where the constant c ( ) 1/ R= R t = k is the curvature radius of the RW etrics at the present tie. The sae radius at generic tie t will be given by R c ( t) R a( t) = hence: ( t) a ( t) = k R = R c 1 (.9) c R = kc. (.1) Using (.7) in the second of (.6), the Friedann equation can be written quite effectively as a a ( t) H a = or, collecting a Hoogeneous Friedan Universe.14

15 a + 1 = H ( t) = H 3 + a a a (.11) and passing to the redshift z: da dt ( t) d( 1 + z) H ( 1+ z) [( 1+ z) ( z + 1) z ( + ) ] a ( t) = z. 1 dz [ ] 1 1 = = dt H ( ) ( ) ( ) = 1 + z ( z 1 ) z ( z ) z dt 1 + z (.1) (.13) and eventually dz dt ( ) ( ) ( ) ( ) 1 = at ( )(1 + z) = H 1 + z 1 + z z+ 1 z + z (.14) Then the tie redshift relation is dz dz dt = = + z z + z + z ( ) ( ) ( ) ( ) ( ) 1 1, z at () H 1+ z, t and ( t, ) with = ( ) =. (.15) We can further express the relations between tie, redshift and the cooving radial coordinate dr in the Robertson Walker etrics using dr c dt = a ( t) dr = 1 + z (.16) dr being the radial distance increent. Setting to dx = dr ( 1 + z) the proper radial increent, we get fro (.15) dt dx = c dz = dz H c [ ] 1 dz ( ) ( z) 1 + ( z ) z z ( z) (.17) Hoogeneous Friedan Universe.15

16 ( 1 + z) ( z + 1) z ( + z) = ( z + z + 1)( z + 1) z ( 1 )( z) +. (.18) By referring to a flat universal geoetry + = 1 we get 1 for the cooving radial coordinate dr = c H dz 3 [ ( + z) + ] 1 1 (.19) Note that, particularly for treating precisely the epochs before the Recobination (see Sect. 5 later), eqs. (.17) and (.19) have to be copleted with the contributions of relativistic particles and radiation to the cosic dynaics. We can easily see (Sect. 5) that (.11) is substituted in such a ore general case by: γ ah () t a( t) = H + + a 1 γ + a a (.) where,, γ are the density paraeters of the gravitating atter, the ter, and radiation calculated at the present cosic tie (for siplicity we have oitted the undersigned ). Note that the different factors containing and γ coe fro the different dependences of the atter and radiation densities with redshift: ρ at () 3, and 4 ρ γ at (). As well known, at the present tie we have: 1 For a flat Universe geoetry + = 1 we have ( 1 + z) ( z + 1) z ( + z) = ( z + z + 1)( z + 1) z ( 1 )( z) + By developing all expressions and products, and all siplifications, this becoes 3 ( 3 3 ) 1 ( 1 ) 3 z + z + z + = + z +, where we have taken + = 1. allowing us to write for the radial coordinate dr = c H dz 3 [ ( + z) + ] 1 1 Hoogeneous Friedan Universe.16

17 γ (.1a) 3 1 We often use the following notation in ters of a curvature paraeter : k 1. (.1b) k γ Then we are in a Universe with flat geoetry if k =. Then the fundaental relation allowing us to calculate all relevant quantities, like ties, distances, redshift, horizons, ecc., is γ a ( t) = H + + a + a a (.) k For exaple, if we want to calculate the cosic tie corresponding to a given redshift (tie fro Big Bang to that corresponding to a given redshift z), using (.15) we have dz tz ( ) = = z 1 at () ( + z) dz 3 4 = z k + H ( ) ( 1 z ) ( 1 ) ( 1 ) z γ z 1+ z 1 (..b) a relation easy to solve nuerically. The cosic look-back tie is instead the copleentary quantity: t look ( z) = back z dz. 1 at () ( + z) Hoogeneous Friedan Universe.17

18 Cosic look-back tie versus redshift, in a liited redshift interval, and for standard values of the cosological paraeters (see Sect..6 below). Extrapolating to the Big Bang ( ) we obtain the present tie (tie lapsed fro Big Bang to now):. Hoogeneous Friedan Universe.18

19 .4 Distances in cosology Distances in a curved space-tie are not so siply defined as in a flat Euclidean space. The ere integral of the cooving distance r or of the proper distance is not related to any observation we can iagine. So, depending on the way you ake the observation, you have a different distance easure. There are at least two kinds of distances obviously related to observations: that relating the observed angular scale and the intrinsic size (the angular size distance), and that relating observed flux and intrinsic luinosity (the luinosity distance). In what follows we illustrate the approach of Carroll et al. (199). We define as the angulare diaeter distance the quantity d A D = dϑ (.3) where D is the intrinsic size of the source and dϑ its apparent angular size. Instead we define as luinosity distance : 1 (.4) d L = π L 4 S where L is the source luinosity and S the flux. Let us first define as a theoretical easure of distance the radial cooving coordinate dm = r (Distance Measure) (.5) 1 where r 1 is the cooving radial distance to the observer, that is the integral of the eleent in (.16) and (.19). The distance in (.5) has purely a theoretical significance, because it cannot be easured with observations. This latter can be re-written for exaple for a flat Universe + = 1, fro integration of (.19), Hoogeneous Friedan Universe.19

20 z( r) z c 3 1 dm = r1 = dr ( 1 ) z = + + dz H that for z>>1 and finite, is easily integrated and gives d M c = 1 z H 1/ 1/ (.6) Then using the RW etrics, it is easy to find the following expressions for the distances: for the angular diaeter distance, the proper source size in a reference syste coherent with the observer is c ds = a t r d θ = D, so that ( ) 1/ ( ) dt= e d A = D dm rat ( e) dϑ = = 1 + z (.7) This graph here illustrates the situation with respect to the Robertson-Walker etrics and coordinate syste: (r, θ+d θ, φ, t e.) D dθ Observer in r=, t=t (r, θ, φ, t e ) Hoogeneous Friedan Universe.

21 Effects of the angular diaeter distance on the apparent angular sizes of cosic sources as a function of redshift. observer with coordinates: (r=r, θ, φ, t ) O d dθ dθ For the luinosity distance, the photon collecting area of the telescope as seen by the source is: Q source with coordinates (, θ, φ, t e ) da = r dθ r sinθdφ = r d. The observed flux, which is the energy flown per unit area, will correspond to the fraction of the eitted radiant energy within d with Hoogeneous Friedan Universe.1

22 da = 1, d= 1 r : S = L L 1 4πd = 4 πr (1 + z) L where the (1 ) + z factor coes fro consideration that both the rate of arrival and the energy are decreased by 1+z. So for the luinosity distance, having set r = r 1 in (.5), we have L M A ( ) d = r (1 + z) = d (1 + z) = d 1 + z. (.8) Again fro the RW etrics dr 1 kr c dt = = c(1 + z) dt a t ( ) This last equation is easily integrated for a flat Universe, that is + = 1 and k =. Fro (.14) and (.15) we easily obtain r1 z c 1 dr = r = d = ( 1+ z) ( z + 1) z ( + z) dz H M (.8) fro which d A dm = 1 + z = H c ( 1 + z) z 1 [( 1 + z) ( z + 1) z ( + z) ] dz. (.9) and analogous expression fro the luinosity distance. This integral can be easily solved nuerically, for exaple using the Cosology Calculator by Edward Wright, University of California, Los Angeles ( Note that, to first approxiation, for >> 1 (e.g. during Recobination, see Sects. 5 and 6), and z neglecting the ter in the denoinator, the angular-diaeter, distance-easure integral relation (.9) siplifies to Hoogeneous Friedan Universe.

23 d. c H z (.3) A and correspondingly d M c/h 1/ Gpc at the last-scattering surface at z=1,.5 although a ore precise deterination requires the full integration of the (.8) and (.9), see Sect. 6. Note that the angular diaeter and luinosity distances at z=1, for exaple, differ by a factror 1 6. Figure.1 Illustration of the behavior of angular diaeters and cooving distances as a function of redshift, for cosological odels with and without curvatures and the cosological constant ter. [Figure taken fro Coles & Lucchin ]. A nice illustration of the effects of the various cosological paraeters in shaping the cooving radial and angular size distances is offered in Figure.1 Concerning the radial distance, the cosological constant has the effect to stretch it, for a given redshift, copared to the case of a flat atter doinated Universe. Siilar stretching is produced by an open Universe. A flat Labda- Hoogeneous Friedan Universe.3

24 doinated Universe shows an interediate-level of radial stretching, but a very large one in the direction orthogonal to the line-of-sight, such that a given angular size subtends a very large propersize source..5 Siple dynaical cosological odels 1- The Milne vacuu odel, Rt () k=-1, ρ = t - The Einstein de-sitter odel k=, q =.5 R c t = ± Rt () R t /3 3- The open odel k=-1, q <.5 Hoogeneous Friedan Universe.4

25 R(t) 4- The closed odel k=-1, q >.5 t As an iportant note, fro eq. (.6) a ( t),h = kc + H a ( t) we have that, back in the a ( t) past, when a(t) is sufficiently sall, clearly while the second addendu keeps constant and the third decreases fast because of a, the first addendu gets larger and larger, and doinates the other two. The consequence is that during ost of the Hubble tie the Einstein de Sitter odel = 1, q =.5, k = ) applies with good precision. (.6 The current standard dynaical odel of the Universe We can anticipate here that a vast variety of high-precision cosological observations, that will be discussed in good datails during later parts of the Course, have deterined the cosological paraeters to assue the following values: H. 7 K / sec/ Mpc..3 γ Based on this, it is iediate to plot the evolution of the universal scale factor as a function of tie, as illustrated in the figure. Hoogeneous Friedan Universe.5

26 Cosological scale factor on a linear scale for the standard odel.... on a log scale... Hoogeneous Friedan Universe.6