1. Cosmology 2002 20031 The Models and The Cosmological Parameters Guido Chincarini Here we derive the observable as a function of different cosmological parameters. The goal is to derive a fairly complete set of equations useful for data reduction. See also the following set discussing the relation to geometry and some applications.

2. Cosmology 2002 20032 The Metric and The Dynamics The RW metric tell us where in a 3 dimension space is an event and at which proper time. The coordinates of the event do not change as a function of time but the coordinate system will change as a function of the proper time because of the expansion factor a(t). The geometry is determined by the value of k = -1, 0, +1.

3. Cosmology 2002 20033 Einstein Equation – Einstein 1916 Ricci’s Tensor Ricci’s scalar Cosmological Constant Energy momentum Tensor simplified in a Robertson Walker Metric Metric – The Universe is homogeneous and isotropic. In this equation figure both the geometry and the Energy Stress Tensor

4. Cosmology 2002 20034 A Newtonian Model V=H 0 r m 00 r0r0 We consider a sphere exdpanding with a velocity v = H r. vWe assume also that the energy and the mass are conserved Critical density

5. Cosmology 2002 20035

6. 6 E  0

7. Cosmology 2002 20037 And I can write:

8. Cosmology 2002 20038 The Equation to be solved

9. Cosmology 2002 20039  0 >1 E<0

10. Cosmology 2002 200310  0 0

11. Cosmology 2002 200311 Obviously (as an example  >1):

12. Cosmology 2002 200312 Friedmann This is very similar to the equation we derived in the Newtonian Toy so that the solution has been already derived for k=1, 0, -1. Here as you have seen the difference is the constant in the Equations. Furthermore I can write – see for  >, <, = 1:

13. Cosmology 2002 200313 Adiabatic Expansion The two Friedmann equations are not indipendent and are related by the adiabatic equation: And in case I have   0 I keep the same equation using:

14. Cosmology 2002 200314 Radiation and Matter

15. Cosmology 2002 200315

16. Cosmology 2002 200316 And

17. Cosmology 2002 200317 Dust Universe to make it easy

18. Cosmology 2002 200318

19. Cosmology 2002 200319

20. Cosmology 2002 200320 Angular separation From Mag Notes: ds 2 = -r 1 2 a 2 (t 1 )  2 = -d 2

21. Cosmology 2002 200321 Integration, Mattig solution see transparencies

22. Cosmology 2002 200322 Generalities As we have seen we can integrate for a dust Universe. We can derive a compact equation also when we account for a Matter + Radiation Universe. It is impossible to integrate if we account also for the cosmological constant or Vacuum Energy. In this case we must integrate numerically. On the other hand we must also remember, as we will see shortly that the present Universe is a Dust Universe so that what is needed to the observer are the equations we are able to integrate. Recent results, however, both on the Hubble relation using Supernovae and on the MWB, with Boomeramg and WMAP, have shown that the cosmological constant is dominant and  tot =1 so that we should use such relations.

23. Cosmology 2002 200323 Compare with toy Model:  m =1,   =0 & The real thing  m =0.1,   =0.9  m =1.0,   =0.0  m =0.3,   =0.0  m =0.3,   =0.7  m =0.3,   =0.9

24. Cosmology 2002 200324 The look back time

25. Cosmology 2002 200325 Distance – In a different way I have a standard source emitting a flash of light at the time t em. If the photons emitted are N em a telescope of Area A would receive, disregarding expansion and considering the physical distance is a(t em ) r, A / 4  (a(t em ) r) 2 photons. However the Universe is expanding and at the time of detection the area of the sphere where the photons arrive as expanded to 4  (a(t obs ) r) 2. That is N em /N detected = A / (a(t obs ) r) 2 We have to take into account of two more factors: 1.Photons are redshifted so that their energy changes by a factor 1=z=a(t obs )/a(t em ) 2.The time between flashes  t at the source increases by a factor (1+z). This also corresponds to a loss of Energy per Unit Area of the telescope. The Luminosity distance can be defined as:

26. Cosmology 2002 200326 Distance

27. Cosmology 2002 200327 And with 

28. Cosmology 2002 200328 And for  k < 0

29. Cosmology 2002 200329 And for  k > 0 - Open

30. Cosmology 2002 200330 Green   = 0.0,  m =0.1 Light blue   = 0.0,  m =0.3 Blue   = 0.0,  m =1.0 Red   = 0.3,  m =0.7 Dark Red   = 0.2,  m =0.9 Dark Blue   = 0.1,  m =0.5

31. Cosmology 2002 200331 Magnitudes

32. Cosmology 2002 200332 Volume - Proper

33. Cosmology 2002 200333 Volume - Comoving

34. Cosmology 2002 200334 Comoving - Dust Universe

35. Cosmology 2002 200335 03, 07 01,05 02,09 02,00 m m 