Cosmology

This is what we seem to see.How to explain it? Where did it come from?

“Cosmologists are often in error but never in doubt.”

L.D. Landau

CosmologyObservation #1: universe is homogeneous and isotropic at large scales

It cannot be stationary! It should expand or contract

Observation #2: universe is expanding (Hubble)

It should have a beginning! Hot or cold??

Observation #3: Cosmic microwave background radiation

Hot Big Bang!

Fate of the universe: depends on mass distribution (or curvature)

Observation #4: Abundance of light elements

Confirms Hot Big BangProves that most matter is dark and non-baryonic

Observation #5: density measurements

Observation #6: Fluctuations of background radiation

Universe is nearly flat; it contains dark matter and “dark energy”;It is accelerating in its expansion!

Observation #7: redshifts of distant Ia supernovae

Problems with standard Big Bang model

Theory of inflation

Formation of structure;Planck scale, Theory of Everything

WHY our universe has the parameters that we observe? Anthropic Principle and beyond

"The Universe must have those properties which allow carbon-based life to develop within it at some stage in its history."

Observation #1: universe is very inhomogeneous and anisotropic at smaller scales …

The Local GroupThere are about 35 galaxies within roughly 1 Mpc of the Milky Way and these have velocities implying they are all

bound to a common center of mass (about 460 kpc in the direction of Andromeda). The most prominent members are the Milky Way (us), and the Andromeda (M31) and Triangulum (M33) galaxies.

Groups clusters superclusters

Virgo Cluster of Galaxies sky.google.com

M84

M86

M87

M58

M88

M89

M90

M91

Central Part of Virgo Cluster

The Virgo Cluster

First recognized by William Herschel where the constellations Virgo and Coma meet. The cluster covers 10 x 10 degrees on the sky (the Full Moon cover 0.5 x 0.5 degrees). The center of

the cluster is ~18 Mpc from Earth.

The Virgo Cluster contains >250 large galaxies and more than 2000 smaller ones contained within an area 3 Mpc across.

The largest galaxies are all ellipticals (M87, M86, M84) and these have sizes equal to the distance between the Milky Way and

Andromeda. These are “giant” Ellipticals (gE).

The Coma Cluster

The Virgo Cluster is small compared to the Coma Cluster.

The Virgo Cluster contains >250 large galaxies and more than 2000 smaller ones contained within an area 3 Mpc across. Most

of the large galaxies are spirals in Virgo.

The Coma Cluster is 15o of Virgo, in the constellation Coma Berenices, and is ~90 Mpc away. It has an angular diameter of ~4o

which at 90 Mpc away is a linear diameter of 6 Mpc.

Coma contains possibly more than 10,000 galaxies. Of the >1000 large galaxies, only 15% are spirals. The majority are

ellipticals (and some S0’s).

The Coma Cluster

The Coma ClusterIn 1933 Fritz Zwicky measured the doppler shift velocities

of galaxies in the Coma Cluster.

He measured the velocity dispersion (average velocity) of cluster galaxies to be σ=977 km/s.

This gives a Virial Mass of M = 5σ2 R /G = 3.3 x 1015 M⊙. Fritz Zwicky1898-1974

Comparing this to all the luminosity from the galaxies in the cluster, Ltot = 5 x 1012 L⊙ gives a mass-to-light ratio of

M/L ≈ 660 M⊙/L⊙.

The Luminous matter in Coma accounts for 1/660 = 0.1% of the mass ! Zwicky argued in 1933 that Dark Matter must dominate clusters. Turns

out it does, but at the time no one believed Zwicky.....

The Coma Cluster

A portion of Zwicky’s “Missing Mass” was discovered in the X-rays. In 1977 the High Energy Astronomical Observatory (HEAO) satellites indicated that clusters contain an intracluster medium (ICM). This includes a hot intracluster gas that is so hot it

emits in the X-rays (by thermal Bremsstrahlung radiation).

Using ne = 300 m-3 and R=1.5 Mpc, the total mass for ionized hydrogen (there is one free proton for every free electron in the gas) we get the total mass of the X-ray emitting gas:

This is 10x higher than the Mass of the galaxies in the cluster (Mgalaxies ~ 1013 M⊙, for L=5 x 1012 L⊙ and M/L ~ 3 M⊙/L⊙).

And, this is still much, much less than the mass from the dynamical measurement, Mtotal = Mgas + Mgalaxies + M??? = 3.3 x 1015 M⊙.

>90% of the mass of the cluster is in the form of some kind of dark matter.

SuperclustersAs the name suggests, superclusters are seen in the distribution of clustering of galaxies

and clustering of clusters. These are structures on scales of ~100 Mpc.

The Milky Way sits at one end of the Local Supercluster, which is ~50 Mpc long.

Distribution of 2175 galaxies out to roughly 50 Mpc from

Tully, 1982, ApJ, 257, 389.

The Milky Way is at the center of the circle and runs in the triangular regions with no

galaxies (can’t see them in the plane of the galaxy)

Redshift SurveysMany galaxy surveys measuring the angular position (RA and Decl) and redshift (the distance) have been carried out. These show strong clustering in all dimensions. The

galaxy distribution is far from random.

v = c z ( = H0 d)

d = 70 Mpc for v=5000 km/s

Angular coordinate on sky (in hours, there are 24 hrs in a

complete circle)

This “slice” is 6 degrees thick (in the page), from

26.50 < δ < 32.50.

Earth

Redshift Surveys

Modern survey: Sloan Digital Sky Survey, probes out to nearly 1000 Mpc.

This spans almost the “whole sky”, except for where the Galaxy blocks

our view....

Earth

400 Mpc

800 Mpc

Courtesy of Michael Blanton.

The universe is homogeneous. This means there is no preferred observing position in the universe. The universe is also isotropic. This means you see no difference in the structure of the universe as you look in different directions.

… but homogeneous and isotropic at large scale(The Cosmological Principle)

The Cosmological PrincipleConsidering the largest scales in the universe, we make the following fundamental assumptions:

1) Homogeneity: On the largest scales, the local universe has the same physical properties throughout the universe.

Every region has the same physical properties (mass density, expansion rate, visible vs. dark matter, etc.)

2) Isotropy: On the largest scales, the local universe looks the same in any direction that one observes.

You should see the same large-scale structure in any direction.

3) Universality: The laws of physics are the same everywhere in the universe.

The universe cannot be stationary!

Force per unit mass:

Energy per unit mass:

Conclusion: the universe should either contract or expand with decreasing speed, because the gravity slows down the expansion

What is in reality?

Hubble’s Law

Distant galaxies are receding from us with a speed proportional to distance

Hubble and Humason 1931: Vrecession = H0 d

The universe expands!

Edwin Hubble

Find z -> find V -> find distance d if H0 is known

The Necessity of a Big Bang

If galaxies are moving away from each other with a speed proportional to distance, there must have been a beginning, when everything was concentrated in one single point:

The Big Bang!

?

The Age of the UniverseKnowing the current rate of expansion of the universe, we can estimate the time it took for galaxies to move as far apart as they are today:

T ≈ d/v = 1/H

Time = distance / velocity

velocity = (Hubble constant) * distance

The Age of the UniverseHubble initially derived a value of H0 = 500 km/s/Mpc.

He could only see Cepheids out to a few Mpc. For more distant galaxies, we assumed that the brightest star he could see was the same luminosity for each galaxy. In most cases the brightest star he could see was instead a Globular Cluster (containing lots

and lots of stars).

He perceived stars being ~100x more luminous intrinsically, thus he thought their distances must be (100)0.5 ~ 10x nearer than they are.

Hubble relation (also called “Hubble Flow”) gives us a way to measure the distance of an object knowing only its redshift:

v = H0 d or d = cz / H0 for z << 1.

To estimate how long all galaxies were in the same place in space and time, calculate the time it would take for a galaxy with a velocity v to have traveled a distance d: t = d / v

= d / (H0 d) = H0-1 = (1.6 x 10-17)-1 s = 1.96 Gyr.

Current value of the Hubble constant H0 = 71 km/s/Mpc

T ~ 14 Gyr

Olbers’s ParadoxWhy is the sky dark at night?

If the universe is infinite, then every line of sight should end on

the surface of a star at some point.

⇒ The night sky should be as bright as the surface of stars!

Solution to Olbers’s Paradox:

If the universe had a beginning, then we can only see light from galaxies that has

had time to travel to us since the beginning of the universe.

⇒ The visible universe is finite!

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

COSMOLOGY

The mechanical energy 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 !

COSMOLOGY

The 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....

COSMOLOGY

For our flat Universe with pressureless dust we can solve our equations 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.

COSMOLOGY

COSMOLOGY

The Lookback Time is defined as how far back in time we are looking when we view an object with redshift z. It is defined as:

tL = t0 - t(z)

where t(z) is given by our previous relation:

Rewriting R = (1+z)-1 we can write the Lookback time as:

for Ω0=1

For the quasar SDSS 1030+0524 at z=6.28 assuming Ω0=1 the Lookback time is

(tL / tH) = (2/3)(1 - 0.0509) =0.633 or tL / t0 = 0.949.

Only 5% of the history of the Universe had unfolded when the light from the quasar left.

COSMOLOGY

Lastly, we can define the deceleration parameter, q(t), which is:

In the 20th century, astronomers were convinced the expansion was slowing down, which is why they came up with this parameter. For pressureless dust, we can solve

that

q(t) = (1/2) Ω(t)

or at the present time q0 = (1/2) Ω0.

This is still used so it is worth knowing what the definition is....

Please submit your course evaluations at

evaluation.tamu.edu

T = 2/3H0 = 9.5 billion yearsBut the age of globular clusters is 13 billion years!

Cosmology and General Relativity

According to the theory of general relativity, gravity is caused by

the curvature of space-time.

The effects of gravity on the largest cosmological scales should be related to the curvature of space-time!

The curvature of space-time, in turn, is determined by the distribution of mass and energy in the universe.

Space-time tells matter how to move;

matter tells space-time how to curve.

General relativistic models

Matter (mass, energy, pressure)

Geometry of space-time

Einstein’s equations

The Expanding UniverseOn large scales, galaxies are moving apart,

with velocity proportional to distance.

It’s not galaxies moving through space.

Space is expanding, carrying the galaxies along!

The galaxies themselves are not expanding!

General relativity picture

Galaxies are at rest in the comoving (expanding) frame

Due to the presence of matter, the universe is non-stationary: all distances change; scale factor R(t) is a function of time

Like any analogy, the balloon analogy has its limits. In the analogy, the balloon expands into the region around it---there is space beyond the balloon. However, with the expanding universe, space itself is expanding in three dimensions---the whole coordinate system is expanding. Our universe is NOT expanding ``into'' anything ``beyond''.

No center and no edge

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

Geometry foundations come from Euclid, around 300 BC. Geometry boils down to 5 postulates that embody self-evident truths, stated without proof:

1. It is possible to draw a straight line from any point to any point.

2. It is possible to [extend] a finite straight line continuously in a straight line.

3. It is possible to describe a circle with any center and [radius].

4. All right angles are equal to one another.

5. If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on the side of which the angles are less than the two angles.

Euclidean, Elliptic, and Hyperbolic Geometries

5. If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on the side of which the angles are less than the two angles.

Huh ?

This is dealing with parallel lines, and was restated by John Playfair (1748-1819) :

5. Given, in a plane, a line L, and a point P not on L, then through P there exists one and only one line parallel to L.

L

P

Euclidean, Elliptic, and Hyperbolic Geometries

19th Century Mathematicians set out to test and disprove Euclid’s 5th postulate, and they managed to do it. It all depends on geometry. Benhard Riemann

(1826-1866) developed an elliptic geometry (like the surface of an ellipsoid, like a sphere). :

Given, in a plane a line L, and a point P not on L, then through P there exists no line parallel to L.

Carl Frederich Gauss (1777-1855), Nikolai Lobachevski (1793-1856) and János Bolyai (1802-1860) developed independently a hyperbolic surface, in which:

Given, in a plane, a line L and a point P not on L, then through P there exist at least two lines parallel to L.

Relativistic Cosmology

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 a special kind of metric (solution to Einstein’s equations);

Friedmann, Lemaitre, Robertson, Walker (FLRW) 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.

Relativistic Cosmology

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 Cosmology

Einstein 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:

Relativistic Cosmology

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 gives us the acceleration equation:

COSMOLOGYIf matter in the Universe was hot, it has a thermal pressure associated with it (if you smush matter, it pushes back with some pressure). This

matters for the rate of expansion.

Start with our equations:

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 mater 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:

From radiation-dominated to matter-dominated era

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.

There are 3 unknowns in the above equation, but they are not independent. You can relate P, R, and ρ by P = w u = w ρc2 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

Relativistic Cosmology

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:

Relativistic Cosmology

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 :

Relativistic Cosmology

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:

Relativistic Cosmology

Look 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:

Relativistic Cosmology

Model Universes on the planeΩm,0 − ΩΛ,0

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).

Cosmological Distances

Recall that proper distance is just the integral of metric, [-(ds)]1/2. Along a radial line from the Earth to a distant object, dθ=dϕ=0, so:

Note that dp,0 = dp(t0) is the proper distance, which is the distance to an object today. It is not the same as the distance between the Earth and the object when

the photon was emitted.

The distance at other times is dp(t) = R(t) dp,0.

Cosmological Distances

The distance to the horizon needs the expression of R(t) for Λ model:

Inserting this into our previous equation gives:

Sadly, this must be solved numerically.... for our WMAP values we find that the distance from t=0 to t=t0 is

d0 = 4.50 x 1026 m = 14,6000 Mpc = 14.6 Gpc

This is the Horizon Distance.

Cosmological Distances

Example: He-4 nuclei were formed when the temperature of the Universe was 109 K at t= 178 s. This early we can assume the Universe was mass+radiation dominated (no Λ) so the scale factor was R(178s) = 2.73 x 10-9. This sets the

“horizon” distance at

d(t) = 2ct = 1.07 x 1011 m = 0.7 AU.

At this point the whole “visible” Universe would fit into the size of the Earth’s orbit.

The “visible” Universe is the “causally connected” Universe. At a time 178s only 0.7 AU regions were causally connected.

At a time t=13.7 Gyr later, this same 0.7 AU region has a present size of d(t) / R(t) = 3.92 x 1019 m = 1.3 kpc.

We can currently see to 14.6 Gpc. The amount of the Universe that is causally connected today is much, much larger than it was at early times.