Transcript Document

Distribution of objects, Fluxes etc
The Geometry and q0
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.
Cosmology 2002/2003
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Counts -  = 0.0
If the density of objects is n  z 
dN  n  z  dVd  dz and the counts, for n  z   const go as Vol.
In general n  z   n0 1  z 
1.4 10
1.2 10
1 10
8 10
6 10
4 10
2 10
Non realistic at high z because
3
of evolution.
m=0.1
9
9
9
m=0.2
8
8
m =0.4
8
8
0
0
2
4
6
Cosmology 2002/2003
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2
And adding 
2 10
1.5 10
1 10
5 10
9
m=0.3 m=0.7
9
9
8
0
0
2
4
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8
3
Radiation by background objects
Definitions:
j => emissivity in erg cm-3 s-1 Hz-1
z1 z2 => 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 = L0 - 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.
Cosmology 2002/2003
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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+dp3. Using the
phase distribution function f(x,p,t) we can write dN=f dV dp3.
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.
Cosmology 2002/2003
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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 dp3  p2dpd  2 d d and dx3  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:
f  t,x, p  
dN
3
dx dp
3

dN
c dte dAe  
2
e
d e d  e 

dN
c dtr dAr   r2 d r d  r 
dN
 in var iant
dt dA  2 d d 
Cosmology 2002/2003
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From this it follows:
dE  h  dN
and the int ensity is defined
as
dE
h  dN  3
1
2
1
1
I

erg
s
cm
Hz
sr
dt dA d d  dt dA d d   3
I
dN

 in var iant
3
2

dt dA  d d 
 4
Density of Energy u  
 c
3

3
3
I



1

z

a
 


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in a time dt j  z  dt which at z  0 means du  
1 z
j  z  dt
1  z 
3
from sources in the reshift range z1 z2 I have
u  z 0   u 0  
t1
t2
j  0  1  z  ,z  dt dz
z2 j  0  1  z  ,z 
dz

3
3
dz z
H  z  1  z 
1  z 
1  z 
 c
I    
 4
c
I  obs  
4

z2
z
j  0  1  z  ,z  dz
c

4
H  z  4
1

z
 

z2
z

 u

j  0  1  z  ,z 
1  z 
dz
0 z  1
H0 1  z 
4
for H 2  z   H 0 2  1  z   0 z  1 Dust Universe and
2
j  0  1  z  ,z  
I  obs  
dL  em 
;
d em
c
4 H 0

z2
z
class of sources n  z   n0 1  z 
dL  em 
1
d em  1  z 2
Cosmology 2002/2003
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dz
0 z  1
8
Olber’s Paradox
L
f 
4 r 2
f tot  

0

L
2
n 4 r dr   dr  
2
0
4 r
Density of Objects
The flux I receive from each star or galaxy
But the Night Sky is Dark
Cosmology 2002/2003
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I  obs  
c n0
4 H 0

z2
z1
 and
dL  em 
1
d em  1  z 2
for 0  1 
dL  em 


  for
 L0  
d em
0 z  1 

c n0 L0
4 H 0
dz
z  em 
z2
1

dz
1  z 
5
2
 min     max  min   obs  1  z    max  z2  1 
 max
 obs
3
 

2
z
d
1

z
c
n
L





   0 0  obs  1   obs
cn L 2

I  obs   0 0   obs 1  z 


5
0


4 H 0
1  z  2 4 H 0  3       max  
2

3
 

dL  obs     obs  2 
c n0
I  obs  
1
  for   1 and  max   

3
 d obs    max  
4 H 0    


2

dL  obs 
c n0
I  obs  
3
 d obs
4 H 0    
2


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X ray Background - Preliminaries
• The discovery goes back to Giacconi et al. (1962)
Cosmology 2002/2003
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The X ray Background
see also Peacock Page 358 and Steeve Holt – See Math X_Bck
1eV  1.6022 10 12 erg 1keV  1.1605 107 K
 E 

3keV


0.71
 E
exp  
 kT
 I  2.7 10 11 

2
1
 W m sr

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.
1 
 E 

3keV


 I  1.11011 
 E 
exp    W m2 sr 1
 kT 
Cosmology 2002/2003
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The Plots – (See Math X_Bck)
6 10
5 10
4 10
3 10
2 10
1 10
-11
Fit of the Background
-11
-11
-11
-11
-11
0
0
20
40
Cosmology 2002/2003
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80
100
13
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
Cosmology 2002/2003
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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 To = 3.6 keV and
gash2 ~ 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 n2.
Cosmology 2002/2003
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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.
Cosmology 2002/2003
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Numbers now

1
2
  6.8 10 32 T ne2 e
for h
kT

h
kT
G  ,T  in W m 3 Hz 1 ; 1W  107 erg / s
 kT 
1  log 
 to 20% accuracy
 h 
kT
y    T ne
dl
2
me c
G
T  To  1  z 
2
Plasma He  25% by mass;
3 parts H 1 part He; Number Density

ne  0.88  g
 mH
H
 12 that is
He

3
2
3

9.84

h
1

z
m



g

 T 
y  1.2 10 4  g h 2  0  
 keV  0
zmax
Cosmology 2002/2003
1  z 
3
1  z
dz
17
Conclusions
• Using the limit given by COBE we derive zmax <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 = <ne2>/<ne>2 and we should have f >104. 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.
Cosmology 2002/2003
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HI – 21 cm
The emission of the neutral Hydrogen could be written as
j()=(3/4) A nH h  (-H).
A=2.85 10-15 s-1
nH = n0 (1+z)3 [1-x(z)] ; x(z) fraction of ionized gas at z.
n0 present number density of hydrogen atoms + ions.
In the expanding Universe we will:
a) Observe the flus at all frequencies with 0 < H and no Flux at 0
> H
b) The discontinuity is given by [possibly derive this]:
F 
3ch
3ch
An
1

x
z

0

AnH 0 


 0 

16 H0
16 H0
Cosmology 2002/2003
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Expansion and Absorption
Question: What is the Optical depth due to matter in the redshift range 0 to z
dt
dz
dz
z   0  1  z   n  z 
d     n  z  cdt     n  z  c
  0 ,z       n  z  c
z
0
dt
c
dz 
dz
H 0 0
3A
 Absorption cross sec tion    
16 
3A
  0 ,z  
32
 h

 k BTspin
 h

 2 k BTspin
1  z 
2
1  z
and for 21 cm
  c 2
      H  ; A Transition Pr obability
  
  c  nH  0   1  zc 2
1  x  zc   ;
  

H
1  z
0
 H 
3
 1  zc  
H
0
Absorption will take place anywhere between 0 and zsource and the
flux will be diminished at all frequencies in the band  H/(1+z) and
H. The discontinuity will be given by:
3A  h   c  nH  0 
 
1  x  0  

  
32  k BTspin    H  H 0
3
Cosmology 2002/2003
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The q0 Dust Universe
8
4 G 
p
2
a k c 
G a a  
   3 2  a and for p  0
3
3 
c 
4
4
a
G  a  2aa  2
G  a 2 and
3
3
a 2 2aa kc 2 a0 2 2a0 a0
2
2
a  2aa  kc  2  2  2  2  2 
a0
a0
a0
a0
a0
2
2

2a0 a0 a0 2 
2
 H 1 2

H
0  1  2q0 
2 
a0 a0 

aa
q0   0 2 0  0  2q0
a0
2
0
Cosmology 2002/2003
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0
 
 c2
3H 0 2
3H 02  c 2
1  c2
  

2
8 G 3H 0 8 G 1
8
8
 c2a2
2
2
2
2
a k c 
G   m     a  a  k c 
G m a 
3
3
3
a03 2  c 2 a 2
8
2
2
a  k c 
G 0 3 a 
3
a
3
a03  c 2 a 2
8
2
2
a  k c 
G 0

3
a
3
2
2
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 acrit = Max size.
Cosmology 2002/2003
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>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 d2a/dt2 =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.
Cosmology 2002/2003
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<0
=0
k=-1
k= 0
k=+1
t
Cosmology 2002/2003
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>0
k=-1
k= 0
c >  > 0
 > c
k=+1
 = c
Cosmology 2002/2003
t
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