COSMOLOGY - Texas A&M Universitypeople.physics.tamu.edu/krisciunas/lecture_cosmology.pdf · 2016....

COSMOLOGYCosmological Principle: The Universe is isotropic and homogeneous,

appearing the same in all directions and at all locations.

COSMOLOGY

Simple Model of the Universe using Newtonian Physics

Consider a Spherical shell in a thin “dust” filled Universe. Dust is everywhere with a uniform density ρ(t).

As the Universe expands, the dust is carried with it. Let

r(t) be the radius at time t of a thin spherical shell containing mass m.

This shell expands with the Universe with recessional

velocity v(t) = dr(t)/dt

Mass m

Dust

r

COSMOLOGYThe mechanical work E of the shell is: K(t) + U(t) = E

The total energy E = constant (conservation of energy !). For convenience we will write the total energy in terms of two constants, k and ϖ

(pronounced “varpi”). k has units of (length)-2 and ϖ has units of length and may be thought of as the present radius of the shell r(t0).

Now E = -(1/2)m kc2 ϖ2 which gives the equation:

Mr is the mass interior to the shell, Mr = (4/3) π r3(t) ρ(t). Note that Mr = constant since no mass is created, the volume expands in lock-step with the

decrease in density. Thus, we can rearrange the eqn above:

COSMOLOGY

The physical nature of the constant k decides the fate of the Universe:

1. if k > 0 then the total energy of the shell is negative and the universe is bounded (closed). The expansion must someday halt and reverse.

2. If k = 0 then the total Energy is exactly zero. The expansion will continue for every and asymptote to zero recessional velocity. The Universe is Flat.

3. If k < 0 the total energy is positive and the universe will is unbounded (open). The Expansion will continue forever.

COSMOLOGY

So far we have dealt only with Newtonian Cosmology, for which spacetime is flat. In reality the mass in the Universe causes spacetime to have curvature. The terms closed, open, flat here describe only the dynamics. Later they

describe the curvature of spacetime (when we use General Relativity).

Cosmological Principle means that the expansion is the same everywhere. We will now write the expansion of any shell as:

Where r(t) is the coordinate distance. ϖ is the comoving distance (does not change with time). R(t) is the scale factor (dimensionless) such that R(t0)

= 1. R is related to the redshift by R = 1/(1+z).

Recall that r3(t) ρ(t) is constant, which means that R3ρ is constant. Thus:

Because R=(1+z)-1 and R3(t0) = 1 this gives:

COSMOLOGYNow we can consider the evolution of our Newtonian Universe. Consider the Hubble Law v(t) = H(t)r(t) = H(t) R(t) ϖ. v(t) is the time derivative of r(t) :

And thus the Hubble Law becomes:

Previously we had:

Inserting v=Hr this then gives:

Inserting our eqn for H above gives:

Multiplying through by R2 gives: RHS is constant, LHS is only

function of t.

COSMOLOGY

When k=0 (the “flat”) case, we can solve for ρ0 and ρC(t):

The present day value (t=t0) is then:

Where H0 = 100 h km/s/Mpc (h is the Hubble parameter). Our current measure is h=0.71 (more on this), which gives:

Or about 6 Hydrogen atoms per cubic meter. Note that the best estimate of baryonic matter density is about 4% of the critical density,

or 2 protons per 2 m x 2 m x 2m box !

COSMOLOGYThe ratio of any density to the critical density is the density parameter, Ω.

The present day cosmic matter density is then

COSMOLOGYGeneral characteristics of our Universe can be determined:

or

We can insert our definition for Ω into the above equations to get:

which for t=t0 becomes

Thus for Ω0 > 1 the Universe is closed.

For Ω0 = 1 the Universe if flat.

For Ω0 < 1 the Universe is open.

Equating the above relations and solving for Ω gives:

This implies that as z→infinity Ω→1. The Universe would be very, very flat. This seemed too perfect; too good to be true to physicists....

COSMOLOGYFor our flat Universe with pressureless dust we can

solve our equations assuming a flat Universe by setting k=0:

Taking the square root and integrating with R=0 at t=0:

Which gives:

valid for Ω0 = 1 and where tH = 1 / H0 is the “Hubble time”, and tH ~ 1010 yrs.

If Ω0 ≠ 1 then the integral above is much more complicated and involves trigonometric and hyberbolic functions. See your book.

COSMIC MICROWAVE BACKGROUND

The idea was flawed because there are no stable isotopes of N=5 or 8 nucleons, leaving He-4 the heaviest element (with trace amounts of Li-7).

At this time there was a bigger problem in that Hubble had published his expansion parameter to be 500 km / s / Mpc, which gave an timescale for the age of the Universe as tH = 1/H0 = 109 yr, much less than the age of the Earth

from Geology.

This led some to argue for a Steady State Model for the Universe.

In 1946 George Gamow made a bold prediction. He was pondering the cosmic abundances. He considered that the early Universe would have been hot enough for nuclear fusion (just like in

the center of the Sun).

Gamow proposed that a very early Universe could explain all the abundances. George Gamow

(1904-1968)

COSMIC MICROWAVE BACKGROUNDSteady State Model for the Universe

In 1946-1949 Hermann Bondi (1915-2005), Thomas Gold (1920-2004), and Fred Hoyle considered a

steady state Universe where Universe is infinitely old, time has no beginning and no end.

The Universe still expands, but new matter is continuously created to fill the void. This changes the interpretation of the Hubble Time, which becomes a

characteristic time for the creating of matter.

Mass-creation rate required is roughly one H atom per cubic meter per billion years.

Fred Hoyle (1915-2001)

Hoyle argued that nucleosynthesis takes place in stars to counter Gamow’s arguments that nucleosynthesis occurred only during the early Universe. In

1957 Hoyle and collaborators published what is still a seminal work on stellar nucleosynthesis that was the basis for this course !


In the 1950s you had two Cosmological Theories that were debated:

1. Steady State Model: headed by Hoyle.

2. “Big Bang” model, headed by Gamow and others.

Steady State model could not explain the Helium abundance, but did explain why the Universe was at least 5 billion years old.

“Big Bang” model (term first coined by Hoyle as a derision) implied the Universe was only a few billion years old at best but could explain the Helium

abundance.

Was there any evidence that such a violent - A “Big Bang” - ever took place ?


Key idea of “Big Bang” was that the early Universe was very hot and dense, like the nucleus of a Star.

Recall that the density evolves like:

For a realistic gas with “pressure”, we need relation between pressure and density. This is the Equation of State. In general we write it as:

For a blackbody (photon) radiation w=1/3 for blackbody radiation. For “pressureless dust” we have w=0.

(If Dark Energy is a constant, then it has w = -1, a big test for cosmology....)

P = w�c2

Reality Check: The “dust” in the Universe is matter, and it has a pressure associated with it (if you smush matter, it pushes back with

some pressure). This matters.

Start with our equation:

Recall from relativity, that mass and energy are the same thing, so we need include both in the density. For nonrelativistic matter, ρ is as above, for but relativistic

matter we must use the equivalent mass density.

Thermodynamics says that dU = dQ - dW, where U is the internal energy, W is the work, and Q is the heat flow.

dQ = 0 because there is no heat flow (Universe in equilibrium). This is called an adiabatic expansion.

Thus:

COSMOLOGY

Substituting V = (4/3)πr3 :

Now, define the energy per unit volume (the energy density): u = U / (4/3 πr3 )

Substituting gives:

We can rewrite u as u = ρ/c2 , which gives:

Finally, using our relation that r = Rϖ, we arrive at the fluid equation:

This gives:

COSMOLOGY

This is the same as before if P=0. Note that P > 0 has the effect of slowing the expansion of the Universe.

This is counterintuitive, but remember that P is the same inside and outside the shell of mass. The motion of particles (our dust) creates the pressure.

However, the assumption that P=0 is valid for most of the Universe’s history, only at the beginning is it different.

The above equation is Birkhoff’s Theorem for G.D. Birkhoff (1884-1944) who showed it in 1923.

There are 3 unknowns in the above equation, but they are not independent. You can relate P, R, and ρ by P = wu = wρc2 where w is a constant.

For pressureless mass, w=0, for blackbody radiation w=(1/3). We get then that:

R3(1+w) ρ = constant = ρ0


in the expansion we can use that for a blackbody the radiation pressure is P = u/3 (where w=1/3 here) and where u is the energy density for a

blackbody: u=aT4.

The energy density evolves in an expanding Universe by :

The energy density today is much, much smaller by a factor of R4. A factor of R3 is due to the change in the volume and another factor of R is due to the

expansion of the wavelength of light.

For our expanding Universe:

COSMIC MICROWAVE BACKGROUNDTransition from the Radiation Era to the Matter Era

Consider that the Pressure was highest during the early Universe when photons were “coupled” to all the baryons. For relativistic particles :

R4 ρrel = ρrel,0

compare this to, for matter:

R3 ρm = ρm,0

Transition point when the density from photons was greater than matter was the transition from a matter era to a radiation era.

COSMIC MICROWAVE BACKGROUNDTransition from the Radiation Era to the Matter Era

WMAP can actually estimate this experimentally, and it gives:

And we can work out what the temperature of the Universe was:


Thus, R4aT4 = aT04 and we find that the blackbody temperature must be related to the temperature at an earlier time as RT = T0.

Consider that Helium fusion requires T~109 K and a density ρb~10-2 kg/m3. Higher temperatures would photodissociate deuterium and lower temperatures

would not allow protons to overcome the Coulomb barrier.

Using the present day value of the baryon density we find that for the

needed value R was :

This gives a present day temperature of: T0 = R T = 3.47 K.


A blackbody with this temperature would emit in the microwave region of the electromagnetic spectrum. From Wien’s law for Blackbody radiation we have:

λmax = 0.00290 m K / T ~ 8.4 x 10-4 m or 360 GHz.

Alpher and Herman made this prediction (they came up with a present-day temperature of 5 K). 16 years later two Princeton physicists, Robert Dicke

and P. J. E. Peebles came up withe the same value, and they wanted to go detect this signal. They were scooped unwittingly by two engineers....

...This gives a present day temperature of: T0 = R T = 3.47 K.


Robert Wilson (left) and Arno Penzias (right) with their 6m Microwave

antennae (horn).

Working for Bell, in 1964 Wilson and Penzias were

building a huge horn antennae to communicate with AT&T’s

Telstar satellite.

Despite all their efforts they detected a continuous “hiss” from all directions of the sky.

The estimated that a blackbody of 3K would produce this hiss, but they did not know the source.

Penzias then learned of Peeble’s prediction of a cosmic microwave background. In 1965 he called Dicke and invited them to Holmdel, NJ to show them their

result and all the pieces fell into place.


Microwave detectors still used in modern communications systems.

Microwaves are good for transmitting information from one place to another because microwave energy can penetrate haze, light rain and snow, clouds, and smoke.

Shorter microwaves are used in remote sensing (including doppler radar used in weather forecasts).

Microwave towers can transmit information like telephone calls and computer data from city-to-city.

Penzias and Wilson had discovered the leftover glow from the Big Bang. It is everywhere and has a peak wavelength of 1.06 mm. This is the

cosmic microwave background (CMB).

This result was a death sentence for the Steady State Model.

In 1991 the COBE satellite measured the full spectrum of the CMB.

Blackbody with T=2.725 K.

Mather et al. 1991, ApJ, 354, L37

Penzias and Wilson received the Nobel Prize in 1978. John Mather received the Nobel Prize in 2006

But, hearing is believing....

Our current measurements of the CMB come from WMAP, the Wilkinson Microwave Anisotropy Probe.

Launched in 2001, it was designed to study the slight fluctuations in left in the CMB (more on this). It has

confirmed small details of the Big Bang.

It also gives us the best measurement of H0:

H0 = 71 (+4/-3) km/s/Mpc

This gives us a current Hubble time of :

tH = 1/H0 = 4.35 x 1017 s = 1.38 x 1010 yr.

The Big Bang happened everywhere. So the CMB comes from everywhere !

BIG BANG NUCLEOSYNTHESISWe can now address the question, why is ~25% of the mass in the Universe

Helium (and the rest Hydrogen?)

At a time t~10-4 s the cosmic temperature was ~1012 K. The Universe was filled with photons, electrons-positrons, neutrinos, and a few protons and neutrons.

There were ~5 p and n for every 1010 photons, therefore the former were constantly being transformed:

The mass difference between a p and n is only: (mp - mn)c2 = 1.293 MeV.

The thermal energy of the Universe is 1012 K is kT ~ 100 MeV. At this energy, the # of neutrons and protons are nearly equal.

BIG BANG NUCLEOSYNTHESISAs the Universe expanded, the temperature fell. At T~1010 K the number of

neutrons to protons was “frozen” at 0.223. There were 223 neutrons for every 1000 protons, and no more neutrons were being created.

The half life of a free neutron is 614 s = 10.2 min.

For p + n to form deuterium requires T ~ 109 K (higher temperatures dissociate deuterium). In this time 176 s passed, so if you started with 223 neutrons, you’d

now have 183, and number of protons increased to 1040.

Now deuterium and Helium fusion can occur at T~109 K.

and

No other elements were made except trace amounts of Li-3. The 183 neutrons and 1040 protons could form 92 He-4 nuclei with 867 protons left over. Because He-4 is

4x more massive than Hydrogen, we should have: [ 4(92)/(867 + 4(92)) ] = 0.299.

Calculated mass abundance of He-4, H-2, He-3, and Li-7. Schramm & Turner 1998, Rev.

mod. Phys, 70, 303.

Calculated mass abundance of H-2, He-3, and Li-7.

Schramm & Turner 1998, Rev. mod. Phys, 70, 303.

Origin of CMB

Final thoughts, from Steven Weinberg in The First Three Minutes:

This is often the way it is in physics - our mistake is not that we take our theories too seriously, but that we do not take them seriously enough. It

is always hard to realize that these numbers and equations ... have something to do with the real world ... The most important thing [of the] discovery of the 3 K radiation background in 1965 was to force us all to

take seriously the idea that there was an early Universe.

The Universe has matter+energy in it. We must use General Relativity to describe how spacetime “curves” in the presence of this matter.

Relativistic Cosmology

Euclidean, Elliptic, and Hyperbolic Geometries


We can describe it with a metric (remember General Relativity?) The cosmological principle: Universe is isotropic (no special directions) and

homogeneous (no spatial places). This leads to curvature in the Universe.

Percy Robertson and Arthur Geoffrey Walker came up with a metric to describe an isotropic, homogeneous universe, the Robertson-Walker metric:

where ϖ is the comoving distance and k is a constant describing the curvature.

Just like for Newtonian cosmology, k > 0 universe is closed, k = 0 universe is flat, k < 0 universe is open.

is the differential proper distance for dt=0.


Aleksandr Friedmann (1888-1925) solved Einstein’s equations for the dynamical evolution of the Universe, which is now known as the Friedmann equation:

Which we derived using only Newton previously. R is the dimensionless scale factor, ρ is the density, everything else is a constant. In 1922 Friedmann

obtained this equation for a nonstatic Universe. It was independently derived in 1927 by a Belgian priest nambed Abbé Georges Lemaître (1894-1966).

Lemaître is considered the first person to propose that the universe evolved from a highly dense

beginning. He is sometimes called the “father of the Big bang”.

Einstein, after hearing him talk about the nonstatic universe, told him “your mathematics is excellent, but

your physics is abominable.” Einstein was wrong.Abbé Georges Lemaître

(1894-1966)

Relativistic CosmologyEinstein had also realized that his equations could not produce a static Universe.

At the time, Hubble’s data had not been published, and everyone thought the Universe had to be static. To make his solution static, Einstein inserted a Cosmological Constant, which is just a constant of integration, into his

solution:

This is our Friedmann equation, except that the “constant” results in an extra potential energy term:

The conservation of energy equation then becomes:

The force due to the new potential is:


When Hubble showed the Universe was expanding, Einstein removed the Cosmological Constant and called it the “biggest blunder of his life”. However,

by the late 1990s the evidence suggested that there is a Cosmological Constant, but with the effect of accelerating the expansion (rather than slowing it down

or keeping it static).

Thus, the Cosmological Constant implies there is some Dark Energy associated with the vacuum, and this is pushing the expansion of the Universe.

Combining this with the fluid equation from before gives us the acceleration equation:


We can then define the mass density of the dark energy:

And, we can define the pressure term due to dark energy:

Which lets us rewrite the acceleration equation as:


We can now rewrite the Friedmann equation as:

where

We can then write the total density parameter as

Which makes the Friedmann equation:

Or, at t=t0 :


We can then rewrite the evolution of Hubble’s constant as:

Where the current values from WMAP are:

This implies that the total density parameter is ~1. The Universe is nearly flat:

Age of the UniverseLook at the behavior of the scale factor to get the age of the Universe

as a function of R.

neglecting the contribution of relativistic particles during the first 55,000 yr (ρrel = 0) then we arrive at an expression :

Plugging in R=1 we get

And WMAP measured:

Redshift-Magnitude Relation using Supernovae Ia

From Perlmutter & Schmidt (2003), data from Perlmutter et

al. ApJ, 517, 565 (1999) and Riess et al. AJ, 116, 1009 (1998).

COSMOLOGY - Texas A&M Universitypeople.physics.tamu.edu/krisciunas/lecture_cosmology.pdf · 2016....

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