Chapter 24

Or, how the scale factor grew in the past and will change in the future….

# density (photons) ~ 1/a(t)3

But, the cosmological redshifting of the photons means that energy per photon ~ 1/a(t). Thus,

ur (energy density of radiation) ~ 1/a(t)4

Energy density of radiation drops more quickly than matter as a(t) increases

Radiation to Matter Dominated Universe

Some time before the surface of last scattering (CMB), the Universe went from radiation to matter dominated.

um (energy density of matter) ~ ρmc2 (recall E=mc2)

ρm ~ 1/a(t)3

If Universe has a non-zero cosmological constant, it eventually becomes “lambda dominated” since the cosmological constant does not depend on the scale factor

Consensus Model Based on the apparent flatness of the Universe, Ω(t) = u(t)/uc(t) ~ 1, how is energy density distributed among components such that Ω(t)=1?

Radiation •Photons mostly from Big Bang (CMB)•Small amount of photons from stars•relativistic neutrinos

Photons and Neutrinos contribute only a small fraction of Ω today!

Matter •total matter determined dynamically from galaxy cluster motions + estimate of mass in voids•baryonic matter (protons, neutrons, electrons) determined from BBN models (more on this later)

Most of the mass density must be in nonbaryonic dark matter (e.g. WIMPS)

Number density of photons is much greater than that of baryonic matternbary = ρbary/mp = 0.22 m-3

Non-zero cosmological constant!

Consensus Model Friedmann equation

Evolution of the scale factor as a function of density parameters and Hubble parameter

Three components of energy density have different dependencies on time. Currently, we are lambda dominated. At an earlier time (smaller scale factor) matter and lambda densities were equal.

Even earlier, radiation dominated the energy density budget of the Universe

Note error h

ere!

arm = 0.00028

In the early, radiation dominated Universe (a < 0.00028), Friedmann equation is

such that a(t) ~ t1/2

Universe expansion slowing due to gravity acting on photons and relativistic particles

At scale factors 0.00028 < a < 0.75, Universe is matter dominated

such that a(t) ~ t2/3

Universe expansion slowing during matter dominated Universe

In lambda dominated Universe

Universe expansion speeding up Dark Energy!

Integration of Friedmann Equation over time for the Consensus Model

trm = 3.3 x 10-6 Ho-1 = 47,000 yr

tmΛ = 0.70 Ho-1 = 9.8 Gyr

to = 0.964 Ho-1 = 13.5 Gyr

Computing proper distances in the Consensus Model

As z ∞ , proper distance to horizon (horizon distance) is 3.24c/Ho =14,000 Mpc

Expressing Distances in an Expanding Universe

The geometry and expansion rate of the Universe effects angular sizes and distances measured.

DH = c/Ho Hubble Distance = 4300 Mpc

DA = L(size)/θ(angular size) Angular Distance

DL = sqrt (Luminosity/4πFlux) Luminosity Distance

DL = (1+z)2 DA

See Ned Wright’s Javascript Cosmology Calculator to calculate these distances for various cosmological parameters (density of the Universe, Hubble parameters, and value of the cosmological constant)http://www.astro.ucla.edu/~wright/CosmoCalc.html

θ

DA

L

(from Hogg 2000 astro-ph 9905116)

At high z, angular diameter distance is such that 1 arcsec is about 5 kpc.

Angular size vs z (plotting DA/DH where DA=L/θ)

DH=c/Ho= 4300 Mpc

Luminosity distance vs z (plotting DL/DH)

flat, Λ=0 – solidopen, Λ=0 – dottedflat, non-zero Λ - dashed

The Accelerating UniverseThe Consensus Model indicates a currently accelerating Universe.

Model confirmed by observations of Type 1 supernovae (standard candles)

Galaxies were receding slower in the past!

Models showing decelerating universe – more distant galaxies are receding faster than nearby galaxies.

What values of Ωm,o and ΩΛ,o give the best fits to the Type 1a SN data?

Assume radiation density negligible

Flatness criterion – dashed line

a = 0 – solid diagonal

Big Crunch below solid horizontal

..

SN data do not constrain curvature - curvature based on angular sizes of distant objects (discussed earlier) and measurement of CMB

Refresher – what is the Cosmic Microwave Background?

• Time when Universe went from opaque to transparent – photons travel freely through the Universe.

• Photons underwent their last scattering from free electrons.

Last scattering surface – surface of glowing, opaque ionized gas that filled the early Universe.

Every observer in the Universe is surrounded by a spherical last scattering surface emitting photons from the Big Bang.

Observations of the CMB

Cosmic Background Explorer Satellite (COBE) launched in 1989 and revealed precise spectrum of CMB – best fit blackbody peaks at 2.725K

Scale factor at time of CMB (last scattering)als = To/Tls = 2.725 K / 3000 K = 9.1 x 10-4

z = 1/als -1 = 1100 and tls ~ 400,000 yrs with Consensus Model energy density values

All sky plot of CMB radiation with bright regions (yellow) being hotter and dark regions being cooler caused by Doppler shift of photons due to our motion.

Correcting for solar system motion and Galactic rotation speed, local group moving towards Hydra at 630 km/s.

CMB should be generally isotropic but anisotropies on many scales of are observed

Major source of anisotropy is Earth’s motion through the Universe – Dipole Anisotropy

Observations of the CMBWilkinson Microwave Anisotropy Probe (WMAP)

Launched in 2001First all-sky maps released in 2003Last data release Jan 2011

WMAP orbits at the L2 lagrange point

Comparison of WMAP and COBE results minus dipole anisoptropy

Small scale fluctuations in the CMB map are ~10-5 the strength of the radiation itself.

Observations of the CMBPlanck

The Planck mission released their map of the CMB in March 2013

Power spectrum reveals relative intensities of fluctuations on different angular scales

The dominant angular scale fluctuation is the angle subtended by the sonic horizon at the surface of last scattering. In a flat universe, where light will move in a straight line, this scale is roughly one degree.

Negatively curved Universe: photons move on diverging paths. Our ruler would appear to have a smaller angular size - location of the first peak would appear at smaller angular scales (grey line)

Positively curved Universe: Angle would appear larger (first peak shifted to the left)

Flat Universe: A flat universe – undistorted (red line)

http://map.gsfc.nasa.gov/mission/sgoals_parameters_geom.html for movie!

Other CMB results

• dark matter and atoms become less dense as the universe expands• photon and neutrino particles lose energy as the universe expands, so their energy density decreases faster than the matter. • dark energy density does not appear to decrease• it now dominates the universe even though it was a tiny contributor 13.7 billion years ago

Planck 2013 CMB results

Big Bang Nucleosynthesis

In the first three minutes, the Universe was hot enough for nuclear reactions to take place.

Protons and neutrons formed 2H (deuterium or D), 3He and 4He.

4He is most stable and, within 3 minutes, made up 25% of the Universe.

What determined the abundance of 4He? need to know Universal conditions (density, relative number of neutrons and protons) at T=109 K (about t ~ 200 to 300s after Big Bang) when the temperature was hot enough for nuclear fusion.

No stable nuclei with atomic number 5, but small amounts of 6Li and 7Li can be made by fusing 4He + D and 4He + 3H.

Final mixture of elements from BBN depends on density of baryons when reactions started.

Protons have lower rest energy than neutrons and, at t~100s after BB, become more numerous than neutronsby a factor of 7:1.

Group of 2 n and 14 p 4He + 12 p

When Universal temperature dropped too low for BBN, we have

X (H mass fraction) = 12/16 ~ 75%Y (He mass fraction) = 4/16 ~ 25%

(and small amounts of other elements)

Reactions affecting neutron, proton abundances:

Density of baryons

Best density estimate from abundance data

Determining abundance of D important!Fits to models from abundance data, baryons make up ~5% of ucrit.

This is almost exactly what we observe in interstellar regions that have not been largely affected by stellar enrichment.

CMB Anisotropies – The Flatness Problem

The small scale fluctuations of the CMB indicate the Universe is flat. But if Universe is close to flat now, it must have been very close to flat in the past. During matter and radiation dominated epochs, deviations of Ω from 1 grew at rates proportional to the rate of growth of the scale factor. At time of BBN (t ~ 3 min)

When D and 4He were forming, Universe was extremely close to flat. If it had not been, Universe would have either collapsed in Big Crunch or expanded to low-density Big Chill after only a few years. In both scenarios, there would not have been enough time for galaxies, stars, planets, UF students to form…

Coincidence? Or is there a mechanism for flattening the early Universe?

CMB Isotropy – The Horizon Problem

Is the background radiation too isotropic and smooth?Conditions should only be identical at different locations if they have some way of communicating with each other. Two objects separated by a distance greater than that which light can traverse cannot affect each other – they are not in causal contact.

θo = t/ato is the maximum current separation between 2 points that could have been causally connected before decoupling

What is the maximum angular separation for causality if the Universe is 13.7 Gyr old and was 400,000 years old at decoupling?

Coincidence? Or is there a mechanism for homogenizing the early Universe?

a/ao = To/T = 3K/3000K = 10-3 = aθo = t/ato = 400000 yr/[(10-3)(13.7 Gyr)] = 0.03 radians or 1.7 degrees

Particle carriers:

QED theory - photons carry EM force (can be real or virtual photons)

Massless graviton thought to carry gravity (so far undetected)

Strong force carried by pion and its mass is determined by the force range

Weak force carried by massive W and Z particles (80 and 90 x proton mass)

Inflationary Theory - a possible solution to these problems. Describes an early period when the acceleration of Universal expansion was positive and thus dominated by a cosmological constant. But first, let’s look at some events in early history and the role of the 4 fundamental forces…

Unification of Forces – are all forces a manifestation of one larger force?

Maxwell unified electricity and magnetic forces

Nobel prize for physics in 1979

Electroweak force - photon (massless) and W (or Z) particle (massive) are the same. This can only happen when particle energy is greater than the difference in mass (nature is symmetric as long as there is enough energy).

Energies must be even greater to unite the electroweak and strong forces

Predictions of Grand Unified Theory (GUT):Decay of proton and magnetic monopole (not observed yet)

At higher energies, forces become more unified

Now imagine a Universe expanding and cooling over time…..

Big Bang

Inflationary Theory Induced by a phase transition in the vacuum at the end of GUT.

The vacuum behaves like a supercooled liquid – below freezing but temporarily remains in a liquid state until something triggers a phase change. When freezing occurs, latent heat warms surrounding – adds energy to the Universe (Cosmological Constant).

Resulted in extremely rapid expansion. Scale factor underwent exponential growth (1043 scale factor change in 10-32 s).

Solves flatness problem – an exponentially, quickly expanding Universe leads to

Where N~60 during GUT, such that even if Ω were strongly curved (close to 0) initially, it ends up close to 1 at the end of inflation – Universe flattens out!

Solves horizon problem – everything within horizon was closer together in past and temperatures could “even out”.

Proper “size” before inflation:

Horizon distance before inflation

Assuming inflation began at ti ~ 10-35 s

•Begin at Planck Time – light travel time across Planck length (set by equating Rs to particle wavelength)

•High temps allow for Grand Unified Theory of forces. Particles, anti-particles are created and annihilated producing photons.

•As the Universe cools we are left with a slight excess of matter over antimatter.

•One excess particle for every 1010 particle-antiparticle pairs produced.

•Explains 1010 photons for every baryon in the Universe

Brief History of Early Universe

• Inflation period occurs during/near end of GUT.

• Universe cools (at 10-35 s after BB) to temp where strong and electroweak forces separate (end of GUT). Baryon number now conserved.

•By 10-12 s, Universe cools to temp allowing for separation of EM and weak forces (average energy is comparable to mass of W particle).

•Hot Universe allows quarks to move as in a fluid until 10-6 s. Then quarks are confined to hadrons.

•Weak force continues to weaken w.r.t. EM force. At 1s it is weak enough that neutrinos are rarely absorbed by matter (matter-neutrino decoupling- CNB).

• After 3 minutes, BBN has set the Hydrogen, Helium primordial abundances

• After ~100,000 yrs all forces have separated with their particle carriers.

• ~380,000 yrs after BB, decoupling of matter-radiation – CMB produced!

Brief History of Early Universe

History of the Universe – Cosmic Timeline…

http://www.nationalgeographic.com/cosmic-dawn/questions-index.html