Chapter 23

Cosmology(the study of the big and the small)

Hadrons – interact via strong force and have internal structure (made of quarks)

Baryons – massive (ex. proton and neutron)composite fermionsMesons – less massivecomposite bosons

Quark combination for:Proton = uudNeutron = udd

Leptons – do not interact by strong force but obey Pauli exclusion (ex. electron and neutrino)

Fundamental Particles Primer (the small)

Quark properties

Photons – massless with energy E = hc/λ

The Cosmological Principle (the big) : the universe is homogeneous and isotropic on sufficiently large scales

The universe looks pretty much like this everywhere – “walls” and “voids” are present but no larger structures are seen….

It follows that the Universe has no “edge” or center.

But is the Universe the same at all times?

Recall universal expansion:

recession velocity = Ho x distance(Hubble Law)

The cosmological principle does not imply that the Universe is constant at all times.

This was once thought to be the case - Steady State Universe

Universal expansion points to a beginning of the Universe and implies that the Universe is changing over time…

The Hot Big Bang model – Universe has expanded from an initial hot and dense state to its current cooler and lower-density state

First Cosmological Observation:The night sky is dark! (Olbers’ Paradox)

Every line of sight would eventually hit a star in an infinite and eternal Universe and the sky would be always bright

Number of stars increases by r2 for each shellbrightness decreases by r2 for each shell

Brightness per shell is a constant!

The fact that the sky is not uniformly bright indicates that either

• the Universe has a finite size

and/or

• the Universe has a finite age

Second Cosmological Observation:Hubble Law

Indicates homogeneous, isotropic expansion where

r(t) = a(t) ro

ro = r(to) is the separation between two points at the current time to

a(t) is a dimensionless scale factor

Distance between two points increases with velocity

Which takes form of the Hubble law v(t) = H(t) r(t) where

H(t) is the Hubble parameter and Ho = H(to) is the Hubble ConstantThus, the primary resolution to Olbers’ paradox is that the Universe has a finite age stars beyond the horizon distance ro ~ c/Ho are invisible to us because their light hasn’t had time to reach us.

Third Cosmological Observation:Cosmic Microwave Background

•At the time of the Big Bang, the Universe was very HOT – emitting gamma-rays – the hot dense universe radiates as a blackbody.

•This radiation has been redshifted by the expansion of the Universe so that the peak of the radiation is now at radio/microwave wavelengths.

•Current temperature is To = 2.727 K

Third Cosmological Observation:Cosmic Microwave Background

The Universe was opaque to light at the time of the Big Bang (T>104 K) due to high particle density, ionization, and scattering

When cooled to ~3000 K during expansion, recombination of atoms allowed photons to pass without as much scattering or absorbing.

At current temp (To = 2.7 K), CMB is cooled by a factor of 1100.

T ~ 1/a(t) temperature is inversely proportional to the scale factorλ ~ a(t) wavelength is directly proportional to the scale factor

Discovery of the CMB

• First observed (inadvertently) in 1965 by Arno Penzias and Robert Wilson at the Bell Telephone Laboratories in Murray Hill, NJ

• Detected excess noise in a radio receiver peak emission at BB temp = 3 degrees

• In parallel, researchers at Princeton were preparing an experiment to find the CMB.

• When they heard about the Bell Labs result they immediately realized that the CMB had been found

•The result was a pair of papers in the Physical Review: one by Penzias and Wilson detailing the observations, and one by Dicke, Peebles, Roll, and Wilkinson giving the cosmological interpretation. •Penzias and Wilson shared the 1978 Nobel prize in physics

•Almost immediately after its detection, the Steady State theory was dead

These three cosmological observations underpin the concept of the Hot Big Bang model – Universe has expanded from an initial hot and dense state to its current cooler and lower-density state

…but how exactly does the Universe expand?

Newtonian Gravity in CosmologyWhat can we learn about the evolution of an expanding Universe by applying Newtonian gravity?

Assume Isotropy Universe is spherically symmetric from any point – spherical volume evolves under its own influence and homogeneity (r) = constant

For a test mass moving on the surface of a sphere with mass M at position r

(d2r/dt2) = -GM/r(t)2 (23.27)

Multiply each side by dr/dt and integrate over time to get

½ (dr/dt)2 = GM/r(t) + k (23.28)

Where k is an integration constant. Now let the sphere have any old mass and radius

M = (4π/3) ρ(t) r(t)3 & use r(t) = a(t) ro

Divide each side by (roa)2/2 to get

(23.31)

(23.32)

Friedmann Equation

H(t)2 =

k = 0 The Universe expands at an ever decreasing rate (dr/dt) 0 as t infinityBorderline Universe or Marginally Bound

k > 0Right hand side of equation is always positiveLeft hand side is always greater than zeroExpansion continues forever!Open or Unbound Universe

½ (dr/dt)2 = GM/r(t) + k (23.28)

k < 0Right hand side is zero when r reaches some maximum rmax = GM/(k)After this, the Universe starts to collapseClosed or Bound Universe

The future of a self-gravitating sphere depends on the sign of k

For a given value of H(t) there is a critical mass density for which k=0, where gravity is just sufficient to halt the expansion

ρc = 3H(t)2/(8πG)

At the present time:

Ho = 70 km/s/Mpc, then

ρc,o ≈ 1.4 x 1011 Msun/Mpc3 (about one H atom in 200 L volume of space)

H(t)2 = (23.32)

The fate of the Universe

depends on its density…

High density = enough matter to gravitationally halt expansion and cause gravitational collapse

Low density = not enough gravitational attraction to stop expansion…it goes on forever

Except in the case of a positive Cosmological Constant – introduced by Einstein in context of General Relativity

Einstein’s principle of equivalence led to the understanding that space-time is curved in the presence of a gravitational field. The mass/energy density of the Universe determines the geometry of space-time over large scales in the Universe.

Cosmology and General Relativity

κ = 0 Flat/Marginally Bound/Critical UniversePlane infinite area, no edge/boundary

κ = -1 Open UniverseHyperboloid infinite area, no edge/boundary

Describe curvature of space (in 2-d) in terms of angles/areas of a triangle.If the angles of a triangle are α, β, γ then

α + β + γ = 180° + (κA/rc2)

where κ is the curvature constant, A is the areas of the triangle and rc is the radius of curvature

κ = +1 Bound/Closed UniverseSphere finite area, no edge/boundary

α

β

γ

How is space curved?

Deviations of α + β + γ from 180 degrees are tiny unless the area of the triangle is comparable to r2

c,o. Some simple observations show that if the Universe is curved, the radius of curvature must be close to the Hubble distance, c/Ho ~ 4300 Mpc. To see why, consider looking at a galaxy with diameter D a distance d away.

If κ = 0, α = D/d (small angle formula)

If κ > 0, α > D/d and mass-energy content acts like a magnifying lenswhen d = π rc,o, galaxy size would fill the sky – not observed!galaxy would also be seen at d+Co, d+2Co, etc. where Co = 2πrc,o

If κ < 0, α < D/d and Universe acts like demagnifying lensObjects at d much greater than rc will be exponentially tiny

Since galaxies are resolved in angular size to distances comparable to Hubble distance, and the above effects are not seen, we conclude radius of curvature is comparable to or larger than Hubble distance. Universe consistent with flat.

180

Distances in 3-d space are given by the metric (omitting parentheses)

Including time and switching to spherical coordinates gives Minkowski Metric used in SR

For an expanding, flat Universe we use the Robertson-Walker Metric

Where a is the scale factor of the Universe and r, θ, φ are the comoving coordinates of a point in space. If the expansion of the universe is homogeneous and isotropic, comoving coordinates are constant with time. The photons from distant galaxies follow null geodesics (where dl=0).

Metrics of Space-timeCompute distances between 2 events or objects in 4-d space-time

Proper distance: length of geodesic between two points when the time and scale factor are fixed (assume flat Universe with θ and φ constant)

Since proper distance is impossible to measure, we measure photons from the galaxy emitted at time te < to, remembering that photons follow null geodesics.

Relating this to something we can observe…Redshift – which tells us scale factor at te

Typo in book! Should be λo/λe = a(to)/a(te)

In GR, functional form of a(t), curvature constant κ and rc,o are determined by the Field Equations. These equations link curvature to energy density and pressure at each point in space-time through the relativistic Friedmann Equation

Main differences with Newtonian form:

mass density ρ energy density u (photons have energy and contribute as well as massive particles)

k κ (positively/negatively curved space rather than open/closed Universe)

Addition of Λ/3 term on right – Cosmological Constant (with units 1/time2)

This term adds a constant (with time) energy density to the Universe

Newtonian

1) The Universe is infinite in volume and mass (keeping it from gravitational collapse).

2) The Universe is expanding fast enough to overcome the attractive force of gravity.

3) The Universe is collapsing (either #2 or #3 imply a beginning of the Universe).

Since Newton believed the universe was eternal and unchanging, he believed that the universe was therefore infinite.

However, once Hubble’s observations of an expanding Universe were discovered in 1920, Einstein realized a non-zero Λ was unnecessary (#2 above) and called it his “greatest blunder”.

In fact, both Hubble and Einstein were correct – the Universe is expanding and changing which does point to a beginning (BB), but this does not exclude the possibility of a non-zero cosmological constant and, in fact, current estimates predict that it is non-zero.

History of the Cosmological Constant

Newton understood the nature of gravity on a Universe that contains matter. He postulated that either

Solving Einstein’s field equations resulted in the constant of integration Λ. This energy density could balance the mass energy density and allow for a static Universe that contains matter – seemed a great finding to support the idea of a Steady-State Universe!

ur = radiation density (from relativistic particles like photons)um = matter density (non-relativistic like protons, electrons, WIMPS)uΛ = lambda density (vacuum density from cosmological constant)

Flatness of Universe implies ur + um + uΛ = uc (critical density)

Rewrite Friedmann equation in terms of energy density components

Components of Universe often given in terms of a dimensionless Density Parameter Ω(t) = u(t)/uc(t)

where Ω<1 is a negatively curved Universe and Ω>1 is positively curved.Thus, knowing how the Universe expands with time requires knowing how much energy density is in radiation, matter and Λ today and how radiation and matter density evolve with time.