Formation and evolution of the cosmic web - Aurora Simionescu...Charles Messier catalogued...

Formation and evolution of the cosmic web

Aurora Simionescu!ISAS/JAXA

aurorasimionescu.wordpress.com/teaching

http://aurorasimionescu.wordpress.com/teaching

Course Evaluation

One short essay (around 2-4 pages) exploring in more detail one of the topics presented in class.

Suggested essay themes will be provided at the end of each lecture.

Choose only ONE topic; essays due after the end of the course (due date TBA).

Your goal: tell me something I did not know before reading your essay.

(70%) !

Active participation in class! (30%)

Textbooks: “Extragalactic Astronomy and Cosmology”, Peter Schneider

“High Energy Astrophysics”, Malcolm Longair

Lecture I The expanding Universe

v=H*d H=500 km/s/Mpc, Hubble 1929 (current value ~70 km/s/Mpc)

Discovery of the cosmic expansion

vc c

c c source is static: observer sees wavelength λ

source is moving: observer sees wavelength λ’= λ*(1+v/c) if v<<c

Doppler Effect

Vesto Melvin Slipher was the first to observe the shift of spectral lines of galaxies in 1912

How do we measure the velocity?

Charles Messier catalogued catalogued all extended nebulae producing a list of 103 objects by 1784

(final list today has 110) !

But for ~140 years (!!!) it was not known how far the Messier objects were. They could be gas clouds in our own galaxy, as well as other very

distant “copies” of the Milky Way

M108M20

How do we measure the distance?

Henrietta Swan Leavitt (1908)

one of the women human "computers" hired by Edward Charles Pickering to measure and catalog the brightness of stars as they appeared in the

Harvard College observatory's photographic plate collection !

discovered that for a certain class of variable stars called Cepheids there is a relation between the luminosity and the period

Absolute distance measurements using Cepheids

brightness=luminosity/(4π*distance2)

luminosity inferred based on period

—> knowing luminosity and brightness, we can measure distance

1913: Ejnar Hertzsprung uses Cepheids to measure distance to the Small Magellanic Cloud as 37,000 light years (current value 199,000 ly)

1923: Edwin Hubble discovers variable star in Andromeda

Andromeda “nebula” using the 100-inch Hooker telescope at

Mount Wilson Observatory

1924: Edwin Hubble determines distances to Andromeda and the Triangulum “nebulae” as 900,000 and 850,000 light years: much too far

to be a part of the Milky Way! These must be other galaxies! (current distance values 2.5 million and 2.7 million light years)

It has been less than 100 years since we discovered that there are other galaxies apart from the Milky Way.!

!The general theory of relativity was published before we were sure that

other galaxies exist. !

Although astronomy is one of the oldest professions, extragalactic astronomy is one of the youngest fields of science.

performed a systematic study of how fast galaxies are moving away from us

The fascinating life of Milton Humason

• dropped out of school at the age of 14 !

• became a "mule skinner" taking materials and equipment up the

mountain while Mount Wilson Observatory was being built

!• became a janitor at the observatory in

1917; volunteered as a night assistant !

• in 1919, George Ellery Hale made him a Mt. Wilson staff member recognising his

dedication and technical skill !

• got his PhD in 1950 from Lund University; retired in 1957

v=H*d H=500 km/s/Mpc, Hubble 1929 (current value ~70 km/s/Mpc)

Discovery of the cosmic expansion

Progress in measuring cosmic expansion

https://www.cfa.harvard.edu/~dfabricant/huchra/hubble/

Hubble Space Telescope Key Project (2001): 72+/-7 WMAP (2011) 70.2+/-1.4

Planck (2015) 67.74 +/- 0.46

https://www.cfa.harvard.edu/~dfabricant/huchra/hubble/

What does cosmic expansion mean?

Imagine an elastic band:

Now, someone stretched it out to double its initial size:L L L L L

2L 2L 2L 2L 2L

d=L v= L/t

d= 2L v= 2L/t

d=3L v=3L/t

d= 4L v= 4L/t

We recover v∝d!

But the Universe is NOT expanding away from any “privileged” point that remains fixed in space!

A B CXY

If you live in galaxy A

A B CXY

If you live in galaxy B

v2v

2v v v3v

v 2v

3D intuitive equivalent to cosmic expansion (‘expanding raisin bread model’)

But, in the Universe, there is no dough… the space itself is growing

Relativistic Doppler effect

vc c

λ’= (c+v)T with v positive for receding source [all quantities measured in observer’s frame]

in relativity we must now in addition consider time dilation:

�0 = (c+v)T0p1�v2/c2

=p

1+�p1��

�0

�0 =p

1+�p1��

�0

1

β=v/c

The concept of “redshift”

Redshift (z) is defined as !!

*recall that in the non-relativistic limit, λobs=λ0(1+v/c) therefore z=v/c !

The recession velocity v of a galaxy has two components: (1) “Hubble flow” (velocity due to the expansion of space)

(2) Peculiar velocity (galaxy is moving through space) !

the expansion velocity v=H*d, thus if d is small, the expansion velocity is small and peculiar velocities may dominate

!in the distant universe, the Hubble flow velocity becomes large and peculiar velocities are insignificant in comparison; therefore “redshift” is often used to indicate the distance of an

object rather than quoting it in Mpc !

Note that z can become (much) larger than 1

�0 = (c+v)T0p1�v2/c2

=p

1+�p1��

�0

�0 = (1 + z)�0 =p

1+�p1��

�0

1

Mathematical description of expanding universe

to account for expansion with time, it is useful to transform to comoving coordinates

r=a(t)x !!

We are free to choose, for convenience, a=1 at the present (i.e. units of x is Mpc in

local Universe)

physical distance (e.g. km or Mpc)

coordinate distance (any arbitrary units;

e.g. size on a paper in cm)

a(t) is called the “scale factor”

Mathematical description of expanding universe

If nothing is moving with respect to our coordinate grid (x does not change with time) r=a(t)x → v=dr/dt=(ȧ/a)r!

!Hubble’s law: v= Hr, thus H=ȧ/a

!Of course things are moving through space, so in reality

v=dr/dt=(ȧ/a)r+u!with u being the “peculiar velocity” and (ȧ/a)r the “Hubble flow”

Redshift is due to the fact that the photon wavelength expands as the space expands:

λ(a)=aλobs; Remembering the definition of redshift:

λobs=λemitted*(1+z) Therefore, a=1/(1+z)

The homogeneous and isotropic Universe

Over 2 million galaxies in a region 100 degrees across centered toward the Milky Way Galaxy's south pole.

Locally, the Universe clearly has structure, but when averaged over very large spatial scales, it becomes well approximated as uniform: isotropic (the same in all directions) and

homogeneous (the same at all places).

Now, consider the evolution of a spherical volume of radius L(t)=L0a(t) (if we choose small L then GR corrections become small and we can use Newtonian dynamics). Expansion will

slow down because of gravitational pull of matter inside sphere:

�0=

(c+v)T0p1�v2/c2

=

p1+�p1��

�0

�0= (1 + z)�0 =

p1+�p1��

�0

Let n be the average number density of galaxies. The probability of finding

a galaxy in the volume element dV is:

P1 = ndV

The probability of finding a galaxy in the volume element dV at location x and

another galaxy at location y is:

P2 = (ndV )

2[1 + ⇠(x,y)]

If the galaxies are uncorrelated, P2 = (P1)2and ⇠ = 0. For a statistically

homogeneous and isotropic universe, ⇠ can only depend on r = |x� y|.�(x) = ⌃akcos(x · k)

✏↵⌫ =

32⇡Z2e6neni

3mec3

r2⇡

3kBTmee�h⌫/kBT gff (T, ⌫)

✏↵⌫ is flat for h⌫ << kBT and exponentially decreasing for h⌫ > kBT .�TSZTCMB

= f(⌫)yc

⇠(r) =⇣

rr0

⌘��

@⇢@t +r · (⇢v) = 0

@v@t + (v ·r)v = �rP

⇢ �r�

r2� = 4⇡G⇢� ⇤

d2L

dt2= �GM

L2

V = 4⇡L3

3

⇢ = ⇢m +

u

c2

1

so, in a matter dominated Universe, we obtain

(deceleration must be non-zero!)

�0=

(c+v)T0p1�v2/c2

=

p1+�p1��

�0

�0= (1 + z)�0 =

p1+�p1��

�0



P1 = ndV



P2 = (ndV )

2[1 + ⇠(x,y)]



✏↵⌫ =

32⇡Z2e6neni

3mec3

r2⇡



= f(⌫)yc

⇠(r) =⇣

rr0

⌘��

@⇢@t +r · (⇢v) = 0

@v@t + (v ·r)v = �rP

⇢ �r�

r2� = 4⇡G⇢� ⇤

d2L

dt2= �GM

L2

V = 4⇡L3

3

⇢ = ⇢m +

u

c2

a = � 4⇡G⇢m

3 a⇢m(t) = ⇢0a30/a(t)

3

1

with respect to a point in time where we know the density to be ρ0

�0=

(c+v)T0p1�v2/c2

=

p1+�p1��

�0

�0= (1 + z)�0 =

p1+�p1��

�0



P1 = ndV



P2 = (ndV )

2[1 + ⇠(x,y)]



✏↵⌫ =

32⇡Z2e6neni

3mec3

r2⇡



= f(⌫)yc

⇠(r) =⇣

rr0

⌘��

@⇢@t +r · (⇢v) = 0

@v@t + (v ·r)v = �rP

⇢ �r�

r2� = 4⇡G⇢� ⇤

d2L

dt2= �GM

L2

V = 4⇡L3

3

⇢ = ⇢m +

u

c2

a = � 4⇡G⇢m

3 a⇢m(t) = ⇢0a30/a(t)

3

a = � 4⇡G⇢0a30

31a2

2aa = d(a2)/dt = � 4⇡G⇢0a30

32aa2 = � 8⇡G⇢0a

30

3 d(�1/a)/dt

a2 =

8⇡G⇢0a30

31a � kc2

1

Equations of motion of uniform matter-dominated universe

Integrating once with respect to time (multiply both sides by 2ȧ):

�0=

(c+v)T0p1�v2/c2

=

p1+�p1��

�0

�0= (1 + z)�0 =

p1+�p1��

�0



P1 = ndV



P2 = (ndV )

2[1 + ⇠(x,y)]



✏↵⌫ =

32⇡Z2e6neni

3mec3

r2⇡



= f(⌫)yc

⇠(r) =⇣

rr0

⌘��

@⇢@t +r · (⇢v) = 0

@v@t + (v ·r)v = �rP

⇢ �r�

r2� = 4⇡G⇢� ⇤

d2L

dt2= �GM

L2

V = 4⇡L3

3

⇢ = ⇢m +

u

c2

a = � 4⇡G⇢m

3 a⇢m(t) = ⇢0a30/a(t)

3

a = � 4⇡G⇢0a30

31a2

2aa = d(a2)/dt = � 4⇡G⇢0a30

32aa2 = � 8⇡G⇢0a

30

3 d(�1/a)/dt

a2 =

8⇡G⇢0a30

31a � kc2

1

integration constant

The meaning of k: • RHS must be greater than zero. If k > 0, then there is a maximum value of a at which the expansion turns around and collapse begins. This is a closed Universe. • If k < 0, expansion rate eventually tends to a constant determined by the value of k - an open Universe. • If k = 0, expansion slows but never reverses - a flat Universe.

scal

e fa

ctor

a

Einstein - de Sitter Universe:!For k=0, a∝t2/3!

And we can define the critical density (such that k=0) and remembering that H=ȧ/a

�0=

(c+v)T0p1�v2/c2

=

p1+�p1��

�0

�0= (1 + z)�0 =

p1+�p1��

�0



P1 = ndV



P2 = (ndV )

2[1 + ⇠(x,y)]



✏↵⌫ =

32⇡Z2e6neni

3mec3

r2⇡



= f(⌫)yc

⇠(r) =⇣

rr0

⌘��

@⇢@t +r · (⇢v) = 0

@v@t + (v ·r)v = �rP

⇢ �r�

r2� = 4⇡G⇢� ⇤

d2L

dt2= �GM

L2

V = 4⇡L3

3

⇢ = ⇢m +

u

c2

a = � 4⇡G⇢m

3 a⇢m(t) = ⇢0a30/a(t)

3

a = � 4⇡G⇢0a30

31a2

2aa = d(a2)/dt = � 4⇡G⇢0a30

32aa2 = � 8⇡G⇢0a

30

3 d(�1/a)/dt

a2 =

8⇡G⇢0a30

31a � kc2

1

Equations of motion of uniform matter-dominated universe

Note! The Hubble “constant” is not a constant

In-class exercise:

The Hubble constant at the present time is 70 km/s/Mpc !

(1) If the Universe is empty, what is its age? !

(2) In an Einstein - de Sitter (matter dominated, critical density) Universe, what is the time since the Big Bang?

2/3H0=9.2Gyr1/H0=14Gyr

[The answer to #2 is true of any kind of matter that gravitates, both baryons and dark matter]

�0=

(c+v)T0p1�v2/c2

=

p1+�p1��

�0

�0= (1 + z)�0 =

p1+�p1��

�0



P1 = ndV



P2 = (ndV )

2[1 + ⇠(x,y)]



✏↵⌫ =

32⇡Z2e6neni

3mec3

r2⇡



= f(⌫)yc

⇠(r) =⇣

rr0

⌘��

@⇢@t +r · (⇢v) = 0

@v@t + (v ·r)v = �rP

⇢ �r�

r2� = 4⇡G⇢� ⇤

d2L

dt2= �GM

L2

V = 4⇡L3

3

⇢ = ⇢m +

u

c2

a = � 4⇡G⇢m

3 a⇢m(t) = ⇢0a30/a(t)

3

a = � 4⇡G⇢0a30

31a2

2aa = d(a2)/dt = � 4⇡G⇢0a30

32aa2 = � 8⇡G⇢0a

30

3 d(�1/a)/dt

a2 =

8⇡G⇢0a30

31a � kc2

1

a2 =

8⇡G⇢m

3 a2 � kc2

�aa

�2= H2

=

8⇡G(⇢m+⇢r+⇢⇤)3 � kc2

a2

�aa

�2= H2

=

8⇡G⇢⇤

3⇢r = ur/c

2and ⇢⇤ = ⇤/8⇡G

⌦m =

⇢m,0

⇢cr,0, ⌦r =

⇢r,0

⇢cr,0,

⌦⇤ =

⇢⇤,0

⇢cr,0and ⌦0 = ⌦m + ⌦r + ⌦⇤

H2(t) = H2

0

h⌦r

a4(t) +⌦m

a3(t) +kc2

H20a

2(t)+ ⌦⇤

i

⇢c,0 =

3H20

8⇡G

2

All of the stars in a globular cluster formed at roughly the same time, thus they can serve as cosmic clocks. The older the cluster, the less massive the biggest hydrogen burning star in it

can be. The oldest globular clusters contain only stars less massive than 0.7 solar masses,

suggesting that they are between 11 and 18 billion years old.

Equations of motion of uniform expanding universe with !a cosmological constant

for matter dominated Universe; adding radiation:

�0=

(c+v)T0p1�v

2/c

2=

p1+�p1��

�0

�0= (1 + z)�0 =

p1+�p1��

�0



P1 = ndV



P2 = (ndV )

2[1 + ⇠(x,y)]



✏↵⌫

=

32⇡Z2e6ne

ni

3me

c3

r2⇡

3kB

Tme

e�h⌫/kBT gff

(T, ⌫)

✏↵⌫

is flat for h⌫ << kB

T and exponentially decreasing for h⌫ > kB

T .�TSZ

TCMB= f(⌫)y

c

⇠(r) =⇣

r

r0

⌘��

@⇢

@t

+r · (⇢v) = 0

@v@t

+ (v ·r)v = �rP

⇢

�r�

r2� = 4⇡G⇢� ⇤

d2L

dt2= �GM

L2

V = 4⇡L3

3

⇢ = ⇢m

+

u

c2

a = � 4⇡G⇢m

3 a⇢m

(t) = ⇢0a30/a(t)3

a = � 4⇡G⇢0a30

31a

2

a

a

= � 4⇡G3 (⇢

m

+ ur

/c2) + ⇤/3

2aa = d(a2)/dt = � 4⇡G⇢0a30

32aa

2 = � 8⇡G⇢0a30

3 d(�1/a)/dt

a2 =

8⇡G⇢0a30

31a

� kc2

1

�0=

(c+v)T0p1�v

2/c

2=

p1+�p1��

�0

�0= (1 + z)�0 =

p1+�p1��

�0



P1 = ndV



P2 = (ndV )

2[1 + ⇠(x,y)]



✏↵⌫

=

32⇡Z2e6ne

ni

3me

c3

r2⇡

3kB

Tme

e�h⌫/kBT gff

(T, ⌫)

✏↵⌫



T .�TSZ

TCMB= f(⌫)y

c

⇠(r) =⇣

r

r0

⌘��

@⇢

@t

+r · (⇢v) = 0

@v@t

+ (v ·r)v = �rP

⇢

�r�

r2� = 4⇡G⇢� ⇤

d2L

dt2= �GM

L2

V = 4⇡L3

3

⇢ = ⇢m

+

u

c2

a = � 4⇡G⇢m

3 a⇢m

(t) = ⇢0a30/a(t)3

a = � 4⇡G⇢0a30

31a

2

a

a

= � 4⇡G3 (⇢

m

+ ur

/c2) + ⇤/3

2aa = d(a2)/dt = � 4⇡G⇢0a30

32aa

2 = � 8⇡G⇢0a30

3 d(�1/a)/dt

a2 =

8⇡G⇢0a30

31a

� kc2

1

radiation energy density: number of photons decreases as a-3, but

wavelength also increases as a so that this term decreases as a-4

matter density (baryons+dark matter):

proportional to a-3

In 1964, Arno Penzias and Robert Wilson accidentally discover the signal from this background radiation.

!George Gamow (see also work of Ralph Alpher and Robert Herman) had predicted in

1953 that radiation left over from the Big Bang should fill the whole Universe with a temperature of 7K.

!Its discovery supports the model in which the Universe started with a hot Big Bang, and

provides further evidence for an isotropic Universe.

The radiation-dominated era

400 photons per cubic centimeter. In this room! A few percent of TV static.

There are 1 billion photons for every baryon in the Universe.

Cosmic microwave background basics

We see the “surface of last scattering” when Universe

became transparent to light.

CMB temperature is the same within +/-0.0035K; those variations are due to the absolute velocity of the Earth through the Universe, v = 627 +/- 22 km/s (Kogut et al. 1993, COBE). Once that is taken out, the temperature is the same within 1 in 100,000.

If we do not live in a special place and the Universe is isotropic to all other observers then it must be homogeneous

When matter and radiation were compressed at very high densities, matter was first in the form of a “quark soup” (quark-gluon plasma);

the radiation density was much higher than the matter density, so the dynamics of the expansion was driven by radiation (photons).

!As expansion continued and temperatures dropped, quarks

combined to protons and neutrons and the universe became filled with hydrogen plasma. The Universe was still opaque to light and as

the universe expanded further, both the plasma and the radiation filling it grew cooler.

!When the universe cooled enough, protons and electrons combined to form neutral hydrogen atoms (“epoch of recombination”). These

atoms could no longer absorb the thermal radiation, and so the universe became transparent (~380,000 years after Big Bang).

!After recombination, matter and radiation evolve independently. The radiation left over from this epoch follows a black-body distribution

whose effective temperature drops as the universe expands.

The radiation-dominated era

Evolution of the temperature of the Universe

4.3 Consequences of the Friedmann expansion 187

where D is the distance of a source with redshift z. Thiscorresponds to a light travel time of!t D D=c. On the otherhand, due to the definition of the Hubble parameter, we have!a D .1 ! a/ " H0!t , where a is the scale factor at timet0!!t , and we used a.t0/ D 1 andH.t0/ D H0. This impliesD D .1!a/c=H0. Utilizing (4.42), we then find z D 1!a, ora D 1 ! z, which agrees with (4.41) in linear approximationsince .1 C z/!1 D 1 ! z C O.z2/. Hence we conclude thatthe general relation (4.41) contains the local Hubble law as aspecial case.

Energy density in radiation. A further consequenceof (4.41) is the dependence of the energy density of radiationon the scale parameter. As mentioned previously, the numberdensity of photons is / a!3 if we assume that photons areneither created nor destroyed. In other words, the numberof photons in a comoving volume element is conserved.According to (4.41), the frequency " of a photon changesdue to cosmic expansion. Since the energy of a photon is/ ",E# D hP" / 1=a, the energy density of photons decreases,$r / nE# / a!4. Therefore (4.41) implies (4.24).

Cosmic microwave background. Assuming that, at sometime t1, the universe contained a blackbody radiation oftemperature T1, we can examine the evolution of this photonpopulation in time by means of relation (4.41). We shouldrecall that the Planck function B" (A.13) specifies the radia-tion energy of blackbody radiation that passes through a unitarea per unit time, per unit frequency interval, and per unitsolid angle. Using this definition, the number density dN" ofphotons in the frequency interval between " and " C d" isobtained as

dN"d"

D 4% B"

c hP"D 8%"2

c31

exp!hP"kBT1

"! 1

: (4.43)

At a later time t2 > t1, the universe has expanded by afactor a.t2/=a.t1/. An observer at t2 therefore observes thephotons redshifted by a factor .1 C z/ D a.t2/=a.t1/, i.e., aphoton with frequency " at t1 will then be measured to havefrequency "0 D "=.1C z/. The original frequency interval istransformed accordingly as d"0 D d"=.1 C z/. The numberdensity of photons decreases with the third power of thescale factor, so that dN 0

"0 D dN"=.1C z/3. Combining theserelations, we obtain for the number density dN 0

"0 of photonsin the frequency interval between "0 and "0 C d"0

dN 0"0

d"0D dN"=.1C z/3

d"=.1C z/

D 1

.1C z/28%.1C z/2"02

c31

exp!hP.1Cz/"0kBT1

"! 1

D 8%"02

c31

exp!hP"0kBT2

"! 1

; (4.44)

where we used T2 D T1=.1 C z/ in the last step. Thedistribution (4.44) has the same form as (4.43) except thatthe temperature is reduced by a factor .1C z/!1. If a Planckdistribution of photons had been established at an earliertime, it will persist during cosmic expansion. As we haveseen above, the CMB is such a blackbody radiation, with acurrent temperature of T0 D TCMB " 2:73K. We will showin Sect. 4.4 that this radiation originates in the early phase ofthe cosmos. Thus it is meaningful to consider the temperatureof the CMB as the ‘temperature of our Universe’ which is afunction of redshift,

T .z/ D T0.1C z/ D T0 a!1 ; (4.45)

i.e., the Universe was hotter in the past than it is today. Theenergy density of the Planck spectrum is given by (4.26), i.e.,proportional to T 4, so that $r behaves like .1C z/4 D a!4 inaccordance with (4.24).7

Finally, it should be stressed again that (4.41) allows allrelations to be expressed as functions of a as well as of z.For example, the age of the Universe as a function of z isobtained by replacing the upper integration limit, a ! .1Cz/!1, in (4.36).

Interpretation of cosmological redshift. The redshiftresults from the fact that during the expansion of theuniverse, the energy of the photons decreases in proportion to1=a, which is the reason, together with the decreasing propernumber density, that $r.a/ / a!4. Our considerations inthis section have derived this 1=a-dependence of the photonenergy.

But maybe this is puzzling anyway—if photons loseenergy during cosmic expansion, then, having in mind theconcept of energy conservation, one might be tempted to ask:Where does this energy go to?

To answer this question, we start with pointing out thatenergy conservation in cosmology is expressed by the ‘firstlaw of thermodynamics’ (4.17), which has as one of itsconsequences the 1=a-behavior of photon energy. Thus, thereis no reason to lose sleep about this issue.

But it may be useful to be more explicit here. We firstpoint out that ‘the energy’ of a photon, or any other particle,

7Generally, it can be shown that the specific intensity I" changes due toredshift according to

I"

"3D I 0"0."0/3

: (4.46)

Here, I" is the specific intensity today at frequency " and I 0"0 is thespecific intensity at redshift z at frequency "0 D .1C z/".

The Planck function Bν specifies the radiation energy of blackbody radiation that passes through a unit area per unit time, per unit frequency interval, and per unit solid angle. The number density dNν of photons in the frequency interval between ν and ν+dν is:

At two different points in time (t2 later than t1), an observer measures a different photon frequency due to the expansion of space: ν’= ν/(1+z), thus dν’ is changed accordingly, and dN’ν’ has diminished by (1+z)3, where 1+z=a(t2)/a(t1).





dN"d"

D 4% B"

c hP"D 8%"2

c31

exp!hP"kBT1

"! 1

: (4.43)




dN 0"0

d"0D dN"=.1C z/3

d"=.1C z/

D 1

.1C z/28%.1C z/2"02

c31

exp!hP.1Cz/"0kBT1

"! 1

D 8%"02

c31

exp!hP"0kBT2

"! 1

; (4.44)


T .z/ D T0.1C z/ D T0 a!1 ; (4.45)








I"

"3D I 0"0."0/3

: (4.46)






dN"d"

D 4% B"

c hP"D 8%"2

c31

exp!hP"kBT1

"! 1

: (4.43)




dN 0"0

d"0D dN"=.1C z/3

d"=.1C z/

D 1

.1C z/28%.1C z/2"02

c31

exp!hP.1Cz/"0kBT1

"! 1

D 8%"02

c31

exp!hP"0kBT2

"! 1

; (4.44)


T .z/ D T0.1C z/ D T0 a!1 ; (4.45)








I"

"3D I 0"0."0/3

: (4.46)


Once established, a black body distribution is conserved by the expansion of space, with an evolving temperature T(z)=T0(1+z). At recombination, z=1100, so T~3000K.

CMB density

182 4 Cosmology I: Homogeneous isotropic world models

where !mCr combines the density in matter and radiation. Inthe second equation, the pressureless nature of matter, Pm D0, was used so that PmCr D Pr. By inserting the first of theseequations into (4.14), we indeed obtain the first Friedmannequation (4.18) if the density ! there is identified with !mCr

(the density in ‘normal matter’), and if

!v D"

8#G: (4.23)

Furthermore, we insert the above decomposition of densityand pressure into the equation of motion (4.22) and imme-diately obtain (4.19) if we identify ! and P with !mCr andPmCr D Pr, respectively. Hence, this approach yields bothFriedmann equations; the density and the pressure in theFriedmann equations refer to normal matter, i.e., all matterexcept the contribution by ". Alternatively, the "-termsin the Friedmann equations may be discarded if insteadthe vacuum energy density and its pressure are explicitlyincluded in P and !.

4.2.7 Discussion of the expansion equations

Following the ‘derivation’ of the expansion equations, wewill now discuss their consequences. First we consider thedensity evolution of the various cosmic components resultingfrom (4.17). For pressure-free matter, we immediately obtain!m / a!3 which is in agreement with (4.11). Inserting theequation of state (4.20) for radiation into (4.17) yields thebehavior !r / a!4; the vacuum energy density is a constantin time. Hence

!m.t/ D !m;0 a!3.t/ I !r.t/ D !r;0 a

!4.t/ I

!v.t/ D !v D const: ; (4.24)

where the index ‘0’ indicates the current time, t D t0.The physical origin of the a!4 dependence of the radiationdensity is seen as follows: as for matter, the number densityof photons changes / a!3 because the number of photonsin a comoving volume is unchanged. However, photons areredshifted by the cosmic expansion. Their wavelength $changes proportional to a (see Sect. 4.3.2). Since the energyof a photon isE D hP % and % D c=$, the energy of a photonchanges as a!1 due to cosmic expansion so that the photonenergy density changes/ a!4.

Analogous to (4.16), we define the dimensionless densityparameters for matter, radiation, and vacuum,

˝m D !m;0

!crI ˝r D

!r;0

!crI ˝" D !v

!crD "

3H20

;

(4.25)

so that˝0 D ˝m C˝r C˝".5

By now we know the current composition of our Universequite well. The matter density of galaxies (including theirdark halos) corresponds to ˝m & 0:02, depending on the—largely unknown—extent of their dark halos. This valuetherefore provides a lower limit for ˝m. Studies of galaxyclusters, which will be discussed in Chap. 6, yield a lowerlimit of ˝m & 0:1. Finally, we will show in Chap. 8 that˝m ! 0:3.

In comparison to matter, the radiation energy densitytoday is much smaller. The energy density of the photons inthe Universe is dominated by that of the cosmic backgroundradiation. This is even more so the case in the early Universebefore the first stars have produced additional radiation.Since the CMB has a Planck spectrum of temperature 2:73K,we know its energy density from the Stefan–Boltzmann law,

!CMB D aSB T4 "

!#2 k4B15„3 c3

"T 4

' 4:5 # 10!34!

T

2:73K

"4 gcm3

; (4.26)

where in the final step we inserted the CMB temperature;here, „ D hP=.2#/ is the reduced Planck constant. Thisenergy density corresponds to a density parameter of

˝CMB ' 2:4 # 10!5h!2 : (4.27)

As will be explained below, the photons are not the onlycontributors to the radiation energy density. In addition, thereare neutrinos from the early cosmic epoch which add to thedensity parameter of radiation, which then becomes

˝r ' 1:68˝CMB ! 4:2 # 10!5h!2 ; (4.28)

so that today, the energy density of radiation in the Universecan be neglected when compared to that of matter. However,(4.24) reveal that the ratio between matter and radiationdensity was different at earlier epochs since !r evolves fasterwith a than !m,

!r.t/

!m.t/D !r;0

!m;0

1

a.t/D ˝r

˝m

1

a.t/: (4.29)

Thus radiation and dust had the same energy density at anepoch when the scale factor was

5In the literature, different definitions for ˝0 are used. Often thenotation ˝0 is used for ˝m.

From Stefan-Boltzmann law:




!v D"

8#G: (4.23)




!m.t/ D !m;0 a!3.t/ I !r.t/ D !r;0 a

!4.t/ I

!v.t/ D !v D const: ; (4.24)



˝m D !m;0

!crI ˝r D

!r;0

!crI ˝" D !v

!crD "

3H20

;

(4.25)




!CMB D aSB T4 "

!#2 k4B15„3 c3

"T 4

' 4:5 # 10!34!

T

2:73K

"4 gcm3

; (4.26)


˝CMB ' 2:4 # 10!5h!2 : (4.27)


˝r ' 1:68˝CMB ! 4:2 # 10!5h!2 ; (4.28)


!r.t/

!m.t/D !r;0

!m;0

1

a.t/D ˝r

˝m

1

a.t/: (4.29)



a2 =

8⇡G⇢m

3 a2 � kc2

�aa

�2= H2

=

8⇡G(⇢m+⇢r+⇢⇤)3 � kc2

a2

�aa

�2= H2

=

8⇡G⇢⇤

3⇢r = ur/c

2and ⇢⇤ = ⇤/8⇡G

⌦m =

⇢m,0

⇢cr,0, ⌦r =

⇢r,0

⇢cr,0,

⌦⇤ =

⇢⇤,0

⇢cr,0and ⌦0 = ⌦m + ⌦r + ⌦⇤

H2(t) = H2

0

h⌦r

a4(t) +⌦m

a3(t) +kc2

H20a

2(t)+ ⌦⇤

i

⇢c,0 =

3H20

8⇡G

2

Compared to the critical density:




!v D"

8#G: (4.23)




!m.t/ D !m;0 a!3.t/ I !r.t/ D !r;0 a

!4.t/ I

!v.t/ D !v D const: ; (4.24)



˝m D !m;0

!crI ˝r D

!r;0

!crI ˝" D !v

!crD "

3H20

;

(4.25)




!CMB D aSB T4 "

!#2 k4B15„3 c3

"T 4

' 4:5 # 10!34!

T

2:73K

"4 gcm3

; (4.26)


˝CMB ' 2:4 # 10!5h!2 : (4.27)


˝r ' 1:68˝CMB ! 4:2 # 10!5h!2 ; (4.28)


!r.t/

!m.t/D !r;0

!m;0

1

a.t/D ˝r

˝m

1

a.t/: (4.29)



and including neutrinos in addition to photons,




!v D"

8#G: (4.23)




!m.t/ D !m;0 a!3.t/ I !r.t/ D !r;0 a

!4.t/ I

!v.t/ D !v D const: ; (4.24)



˝m D !m;0

!crI ˝r D

!r;0

!crI ˝" D !v

!crD "

3H20

;

(4.25)




!CMB D aSB T4 "

!#2 k4B15„3 c3

"T 4

' 4:5 # 10!34!

T

2:73K

"4 gcm3

; (4.26)


˝CMB ' 2:4 # 10!5h!2 : (4.27)


˝r ' 1:68˝CMB ! 4:2 # 10!5h!2 ; (4.28)


!r.t/

!m.t/D !r;0

!m;0

1

a.t/D ˝r

˝m

1

a.t/: (4.29)



In class exercise: what was the scale factor of the Universe, a, at matter-radiation equality, as a function of Ωm?

a2 =

8⇡G⇢m

3 a2 � kc2

�a

a

�2= H2

=

8⇡G(⇢m+⇢r+⇢⇤)3 � kc

2

a

2

�a

a

�2= H2

=

8⇡G⇢⇤

3⇢r

= ur

/c2 and ⇢⇤ = ⇤/8⇡G⌦

m

=

⇢m,0

⇢cr,0, ⌦

r

=

⇢r,0

⇢cr,0,

⌦⇤ =

⇢⇤,0

⇢cr,0and ⌦0 = ⌦

m

+ ⌦

r

+ ⌦⇤

2

we define


deceleration must be non-zero!

�0=

(c+v)T0p1�v

2/c

2=

p1+�p1��

�0

�0= (1 + z)�0 =

p1+�p1��

�0



P1 = ndV



P2 = (ndV )

2[1 + ⇠(x,y)]



✏↵⌫

=

32⇡Z2e6ne

ni

3me

c3

r2⇡

3kB

Tme

e�h⌫/kBT gff

(T, ⌫)

✏↵⌫



T .�TSZ

TCMB= f(⌫)y

c

⇠(r) =⇣

r

r0

⌘��

@⇢

@t

+r · (⇢v) = 0

@v@t

+ (v ·r)v = �rP

⇢

�r�

r2� = 4⇡G⇢� ⇤

d2L

dt2= �GM

L2

V = 4⇡L3

3

⇢ = ⇢m

+

u

c2

a = � 4⇡G⇢m

3 a⇢m

(t) = ⇢0a30/a(t)3

a = � 4⇡G⇢0a30

31a

2

a

a

= � 4⇡G3 (⇢

m

+ ur

/c2) + ⇤/3

2aa = d(a2)/dt = � 4⇡G⇢0a30

32aa

2 = � 8⇡G⇢0a30

3 d(�1/a)/dt

a2 =

8⇡G⇢0a30

31a

� kc2

1

Einstein added a cosmological constant term to force ä to be 0 when it was believed that the Universe was static (before discovery

of expansion)

General relativity allows for integration constant Λ in


�0=

(c+v)T0p1�v

2/c

2=

p1+�p1��

�0

�0= (1 + z)�0 =

p1+�p1��

�0



P1 = ndV



P2 = (ndV )

2[1 + ⇠(x,y)]



✏↵⌫

=

32⇡Z2e6ne

ni

3me

c3

r2⇡

3kB

Tme

e�h⌫/kBT gff

(T, ⌫)

✏↵⌫



T .�TSZ

TCMB= f(⌫)y

c

⇠(r) =⇣

r

r0

⌘��

@⇢

@t

+r · (⇢v) = 0

@v@t

+ (v ·r)v = �rP

⇢

�r�

r2� = 4⇡G⇢� ⇤

d2L

dt2= �GM

L2

V = 4⇡L3

3

⇢ = ⇢m

+

u

c2

a = � 4⇡G⇢m

3 a⇢m

(t) = ⇢0a30/a(t)3

a = � 4⇡G⇢0a30

31a

2

a

a

= � 4⇡G3 (⇢

m

+ ur

/c2) + ⇤/3

2aa = d(a2)/dt = � 4⇡G⇢0a30

32aa

2 = � 8⇡G⇢0a30

3 d(�1/a)/dt

a2 =

8⇡G⇢0a30

31a

� kc2

1

radiation energy density: number of photons decreases as a-3, but

wavelength also increases as a so that this term decreases as a-4

matter density (baryons+dark matter):

proportional to a-3

cosmological constant: does not depend on a at all

Useful to define a radiation density and a “cosmological constant density” as

a2 =

8⇡G⇢m

3 a2 � kc2�a

a

�2= H(t)2 =

8⇡G⇢m

3 a2 � kc

2

a

2

⇢r

= ur

/c2 and ⇢⇤ = ⇤/8⇡G

2

�0=

(c+v)T0p1�v

2/c

2=

p1+�p1��

�0

�0= (1 + z)�0 =

p1+�p1��

�0



P1 = ndV



P2 = (ndV )

2[1 + ⇠(x,y)]



✏↵⌫

=

32⇡Z2e6ne

ni

3me

c3

r2⇡

3kB

Tme

e�h⌫/kBT gff

(T, ⌫)

✏↵⌫



T .�TSZ

TCMB= f(⌫)y

c

⇠(r) =⇣

r

r0

⌘��

@⇢

@t

+r · (⇢v) = 0

@v@t

+ (v ·r)v = �rP

⇢

�r�

r2� = 4⇡G⇢� ⇤

d2L

dt2= �GM

L2

V = 4⇡L3

3

⇢ = ⇢m

+

u

c2

a = � 4⇡G⇢m

3 a⇢m

(t) = ⇢0a30/a(t)3

a = � 4⇡G⇢0a30

31a

2

a

a

= � 4⇡G3 (⇢

m

+ ur

/c2) + ⇤/3

2aa = d(a2)/dt = � 4⇡G⇢0a30

32aa

2 = � 8⇡G⇢0a30

3 d(�1/a)/dt

a2 =

8⇡G⇢0a30

31a

� kc2

a2 =

8⇡G⇢m

3 a2 � kc2

1

then more generally becomes:a2 =

8⇡G⇢m

3 a2 � kc2

�a

a

�2= H2

=

8⇡G(⇢m+⇢r+⇢⇤)3 � kc

2

a

2

⇢r

= ur

/c2 and ⇢⇤ = ⇤/8⇡G

2

“Friedmann equations”

As a increases, ρr decreases the fastest (as the 4th power of a). At some point, ρr becomes smaller than ρm which

decreases only as the 3rd power of a.

Early universe (small a): “radiation dominated”

As a increases further, ρm decreases as the third power

of a while ρΛ does not change. No matter how small ρΛ is

initially, because it does not decrease with a, it will become

dominant for large a.

Three stages of the Universe

scale factor a

Expansion dominated by matter vs. cosmological

constant

(any kind of) matter

cosmological constant

1 8

a2 =

8⇡G⇢m

3 a2 � kc2�a

a

�2= H(t)2 =

8⇡G⇢m

3 a2 � kc

2

a

2

⇢r

= ur

/c2 and ⇢⇤ = ⇤/8⇡G

2

�0=

(c+v)T0p1�v2/c2

=

p1+�p1��

�0

�0= (1 + z)�0 =

p1+�p1��

�0



P1 = ndV



P2 = (ndV )

2[1 + ⇠(x,y)]



✏↵⌫ =

32⇡Z2e6neni

3mec3

r2⇡



= f(⌫)yc

⇠(r) =⇣

rr0

⌘��

@⇢@t +r · (⇢v) = 0

@v@t + (v ·r)v = �rP

⇢ �r�

r2� = 4⇡G⇢� ⇤

d2L

dt2= �GM

L2

V = 4⇡L3

3

⇢ = ⇢m +

u

c2

a = � 4⇡G⇢m

3 a⇢m(t) = ⇢0a30/a(t)

3

1

a2 =

8⇡G⇢m

3 a2 � kc2

�a

a

�2= H2

=

8⇡G(⇢m+⇢r+⇢⇤)3 � kc

2

a

2

�a

a

�2= H2

=

8⇡G⇢⇤

3⇢r

= ur

/c2 and ⇢⇤ = ⇤/8⇡G

2

exponential increase!

a2 =

8⇡G⇢m

3 a2 � kc2

�a

a

�2= H2

=

8⇡G(⇢m+⇢r+⇢⇤)3 � kc

2

a

2

⇢r

= ur

/c2 and ⇢⇤ = ⇤/8⇡G

2

Effect of cosmological constant on age of Universea2 =

8⇡G⇢m

3 a2 � kc2

�a

a

�2= H2

=

8⇡G(⇢m+⇢r+⇢⇤)3 � kc

2

a

2

�a

a

�2= H2

=

8⇡G⇢⇤

3⇢r

= ur

/c2 and ⇢⇤ = ⇤/8⇡G⌦

m

=

⇢m,0

⇢cr,0, ⌦

r

=

⇢r,0

⇢cr,0,

⌦⇤ =

⇢⇤,0

⇢cr,0and ⌦0 = ⌦

m

+ ⌦

r

+ ⌦⇤

2

a2 =

8⇡G⇢m

3 a2 � kc2

�a

a

�2= H2

=

8⇡G(⇢m+⇢r+⇢⇤)3 � kc

2

a

2

⇢r

= ur

/c2 and ⇢⇤ = ⇤/8⇡G

2

From numerical solutions to:

Solves “problem” of age of Universe

compared to oldest globular star clusters

(13.8 Gyr)

The homogeneous Universe: take-home points

On large scales (hundreds of Mpc), the Universe is uniform (isotropic&homogeneous)

The rate of expansion and age of the Universe depend on the contents of the Universe. The scale factor a=1/(1+z) evolves as (Friedmann eqns):a2 =

8⇡G⇢m

3 a2 � kc2

�a

a

�2= H2

=

8⇡G(⇢m+⇢r+⇢⇤)3 � kc

2

a

2

⇢r

= ur

/c2 and ⇢⇤ = ⇤/8⇡G

2

“baby” universe: radiation dominated“teenager” universe: matter dominated

“adult/senior” universe: cosmological constant dominated

�0=

(c+v)T0p1�v

2/c

2=

p1+�p1��

�0

�0= (1 + z)�0 =

p1+�p1��

�0



P1 = ndV



P2 = (ndV )

2[1 + ⇠(x,y)]



✏↵⌫

=

32⇡Z2e6ne

ni

3me

c3

r2⇡

3kB

Tme

e�h⌫/kBT gff

(T, ⌫)

✏↵⌫



T .�TSZ

TCMB= f(⌫)y

c

⇠(r) =⇣

r

r0

⌘��

@⇢

@t

+r · (⇢v) = 0

@v@t

+ (v ·r)v = �rP

⇢

�r�

r2� = 4⇡G⇢� ⇤

d2L

dt2= �GM

L2

V = 4⇡L3

3

⇢ = ⇢m

+

u

c2

a = � 4⇡G⇢m

3 a⇢m

(t) = ⇢0a30/a(t)3

a = � 4⇡G⇢0a30

31a

2

a

a

= � 4⇡G3 (⇢

m

+ ur

/c2) + ⇤/3

2aa = d(a2)/dt = � 4⇡G⇢0a30

32aa

2 = � 8⇡G⇢0a30

3 d(�1/a)/dt

a2 =

8⇡G⇢0a30

31a

� kc2

1

The Universe is filled with a microwave “background” radiation left over from the last time matter interacted with light. It follows a Planck distribution with T=2.73K and increasing in the past T(z)=T(1+z).

The Universe is expanding. Expansion velocity measured via Doppler effect as a “redshift” (z) of the spectra:

�0 = (c+v)T0p1�v2/c2

=p

1+�p1��

�0

�0 = (1 + z)�0 =p

1+�p1��

�0

1

Suggested essay topics

1) Explain in more detail how Cepheids are used for distance determination. What are the different types of Cepheids? How were Hubble’s measurements of distances to Andromeda and the Triangulum galaxy revised, and what drove the changes in

the measurements of the Hubble constant over time? !

2) Explain the role of neutrinos during the radiation-dominated phase of the Universe. At what temperature does neutrino decoupling happen, and what was then the age and scale

factor of the Universe? How many neutrinos per cubic centimeter of space are there from the cosmic neutrino

background?

Formation and evolution of the cosmic web - Aurora Simionescu...Charles Messier catalogued...

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Transcript of Formation and evolution of the cosmic web - Aurora Simionescu...Charles Messier catalogued...