1. Cosmology 2002/20031 Distribution of objects, Fluxes etc The Geometry and q 0 Prof. Guido Chincarini The idea of this chapter that will be upgraded as soon as possible is to give to the student the capability to understand the cosmic background at different frequencies of the electromagnetic spectrum and how known objects and populations yet to be detected contribuite to the observed spectrum and brightness. We will also discuss the effects of the IGM and absorption and how this matter will interact with photons.

2. Cosmology 2002/20032 Counts -  = 0.0  m =0.1  m =0.2  m =0.4

3. Cosmology 2002/20033 And adding    m =0.3  m =0.7

4. Cosmology 2002/20034 Radiation by background objects Definitions: j => emissivity in erg cm -3 s -1 Hz -1 z 1 z 2 => redshift interval over which the sources are dsitributed. U => density of energy in erg cm -3 Hz -1 H(z) => Hubble constant at the epoch z I => Intensity in units ergs cm -3 s -1 Hz -1 sr -1 dL/d = L 0 -  spectral distribution between min and max for a sample of sources Also we should recall that the Volume goes as (1+z) 3. Better we will show that I/ 3 is an invariant.

5. Cosmology 2002/20035 Conservation Consider a stream of particle propagating freely in the space time. A comoving observer at the time t finds dN particle in the comoving volume dV. These have momentum in the range p p+dp 3. Using the phase distribution function f(x,p,t) we can write dN=f dV dp 3. At a later time, t+dt, the proper volume is increased by a factor [a(t+dt)/a(t)] 3 and the volume in the momentum space {we have shown a few lectures ago velocity (or momentum)  a -1 See the Chapter the Hubble expansion and the cosmic redshift} goes as [a(t)/a(t+dt)] 3 so that the phase volume occupied by the particles does not change during the free propagation. Since the number of particles is also conserved it follows that the phase space distribution function is conserved along the streamline. We can use a similar reasoning for photons where we also showed that the frequency changes (redshift) as a function of the expansion.

6. Cosmology 2002/20036 We can reason using photons For photons we can demonstrate an important invariant in a similar way. In this case we can write the momentum space volume as dp 3  p 2 dpd   2 d d  and dx 3  cdt dA {dA is the area normal to the direction of propagation}. Using the subscript e for the emission and the subscript r to indicate the photons received by the surface we can write for the conserved number of photons per unit phase space volume:

7. Cosmology 2002/20037 From this it follows:

8. Cosmology 2002/20038

9. 9 Olber’s Paradox The flux I receive from each star or galaxy Density of Objects But the Night Sky is Dark

10. Cosmology 2002/200310

11. Cosmology 2002/200311 X ray Background - Preliminaries The discovery goes back to Giacconi et al. (1962)

12. Cosmology 2002/200312 The X ray Background see also Peacock Page 358 and Steeve Holt – See Math X_Bck Here kT = 40 keV. If we subtract a source density of a density of about 400 sources deg -2 (2.5 10 -18 W m -2 ) that accounts for 60% of the Background we have the relation below with 0 <  < 0.2 [TBChecked] and kT about 23 – 30- keV.

13. Cosmology 2002/200313 The Plots – ( See Math X_Bck) Fit of the Background

14. Cosmology 2002/200314 Home work The class or the student developes all this part according to the latest observations related to the Chandra deep field etc. References: –Hasinger

15. Cosmology 2002/200315 How could we explain an X ray background anyhow The Universe after recombination must go through a reheating process. The main reason for this is that the Universe is completely transparent to radiation and HI is not detected. On the other hand the structure formation process can not have been 100% efficient so that HI should have been left around. Recombination occurs at z ~ 1000 and the gas Temperature, for a gas we can use  = 5/3 and for Radiation  = 4/3, according to the adiabatic expansion T  R -3 (  -1) and therefore T  a -2. Guilbert and Fabian (1986, MNRAS 220, 439) estimate T o = 3.6 keV and  gas h 2 ~ 0.24 assuming reheating at redshift z=6 [This also should be revisited for reheating may be at higher redshift]. Note that if the gas is clumped the density is higher in the emitting region since Bremsstrahlung goes as n 2.

16. Cosmology 2002/200316 The y parameter If we have a gas at very high energy the Compton inverse effect will be very strong. The electron hits a photon and convert a low energy photon to a high energy photon by a factor of the order  2 in the case of relativistic electrons. This type of scattering would cause a distortion in the Microwave Spectrum (see also the Sunayev Zeldovich effect) by depopulating the Rayleigh Jeans regime in favour of photons in the Wien tail. On the other hand the observations of the Microwave Background is thermal to an extremely high degree of accuracy (COBE and WMAP) so that we can rule out models with too much hot gas. According to Fixsen et al. 1996 Ap J 473, 576 COBE sets a limit of y<1.5 10 -5. It is very important to estimate if there is any background at all. If present we may have too look into the emission by a population of low luminosity active galaxies. Something it is worth searching for.

17. Cosmology 2002/200317 Numbers now

18. Cosmology 2002/200318 Conclusions Using the limit given by COBE we derive z max <0.1 in contraddiction with the fact that reheating is at high redshift, larger than z. This to account fot the X ray Background. We might have a rather clumped gas. The X ray emission scale with the parameter f = / 2 and we should have f >10 4. In this case however we must make a model for the evolution of the temperature in the clumps. If the background is made out of discrete sources then need to have a density of about 5000 sources per squaredegree [TBC]. We do not have that many clusters of galaxies and that is why we are considering active low luminosity galaxies. It is very clear that the next X ray observations must give a detailed number counts of sources. To do that we need a fairly good resolution.

19. Cosmology 2002/200319 HI – 21 cm The emission of the neutral Hydrogen could be written as j( )=(3/4) A n H h  ( - H ). A=2.85 10 -15 s -1 n H = n 0 (1+z) 3 [1-x(z)] ; x(z) fraction of ionized gas at z. n 0 present number density of hydrogen atoms + ions. In the expanding Universe we will: a)Observe the flus at all frequencies with 0 H b)The discontinuity is given by [possibly derive this]:

20. Cosmology 2002/200320 Expansion and Absorption Question: What is the Optical depth due to matter in the redshift range 0 to z Absorption will take place anywhere between 0 and z source and the flux will be diminished at all frequencies in the band H /(1+z) and H. The discontinuity will be given by:

21. Cosmology 2002/200321 The q 0 Dust Universe

22. Cosmology 2002/200322 00 For small a(t) the term in  can be disregarded, it will dominate for large values of a. And this becomes a puzzle, we are in the right epoch to be capable of measuring  because it is now the dominating term. If  <0 a can not become extremely large since da/dt must be real. For this value of a crit = Max size.

23. Cosmology 2002/200323  > 0 K=0 or K=-1 –For a(t) large the model enter a phase of exponential expansion – The equation becomes K=+1 We need a fine tuning among the different terms. –We could fine tune  to have da/dt=0 and d 2 a/dt 2 =0 (static model) –For  larger the repulsive force dominates and the Universe will expand forever –For smaller  we find a range with a<0. These values are therefore forbidden. –Etc The student practice.

24. Cosmology 2002/200324 t  < 0  = 0 k= 0 k=+1 k=-1

25. Cosmology 2002/200325 t  > 0 k= 0 k=+1 k=-1  c >  > 0  =  c  >  c