The Expanding Universe

Allday: 12.1.1; Silk: Chap 5, Math notes 1-3 (p 409-411)

Olber’s Paradox

Why is the night sky so dark?✦ Consider a very large universe filled uniformly with stars, then eventually every line of sight will intercept a star → the night sky should be as bright as the Sun!More rigourously: assume the density of stars is N0, and each star has luminosity L, then the flux Δf (received at the origin) due to all the stars in a spherical shell with radius r, is Δf=NoL4πr2dr/r2 → i.e., each shell in the sphere contributesthe same flux (i.e.,when it is twice as far away, the stars are 4times fainter, but the volume in the shell is 4 times larger).

See Silk, pages 55-58

- if the universe is infinite, the sky should be infinitely bright!- stars actually have size, and not a point, so light from stars farther away may be intercepted by closer stars → this means that the sky should be as bright as the sun.

- This problem was known by Kepler’s time, and popularized by Olber in the 19th century.

Possible solutions to Olber’s Paradox* The Universe is finite -- difficult to have a Universe with edges (Newton surmised that such a universe would collapse under its gravity!)

* There is some absorbing material out there -- but this presents the same problem: the absorbing material needs to re-radiate the energy as it heats up. * The Universe is expanding, and light is redshifted into undetectability -- this works some, but not the dominant effect.* The Universe is young -- we can see only so far; distant light hasn’t reach us yet (The “horizon effect”). This is the dominant reason why the sky is dark.

The Expanding Universe and the Scale Factor R(t)

The expanding universe shouldbe thought of as the stretching of space,not that galaxies are receding fromeach other.

The expandinguniverse + thecosmologicalprinciple, automatically lead to theHubble law: v=H0d

:,

- The scale factor R(t) relates the size at one epoch to the size at another: R(t)/R(to)=r/ro (Also often written as a(t), as in Allday.)

If at time to, the scale factor R(to)=1, and the radiusof the balloon is ro, then, if at time t, the scale factoris R(t), the radius of the balloon is R(t)ro.→ all the co-ordinates are scaled by R(t), i.e., thedistance between two objects would be scaled by R(t).

Note again that: the size of a galaxy (or any gravitationally bound entity) does not scale with R(t), just the distances between galaxies (that are not boundtogether by gravity).

We normally take R(now)=R(to)=1, so that in the past, R(t) for earlier time is always less than 1 (i.e., the universe was “smaller” in the past).

Redshift and the Scale Factor

Redshift is actually due to the stretching of the wavelength of light (due to the stretching of space).

R(t) R(to)

(1+z) ≡ ν(rest)/ν(obs) = λ(obs)/λ(rest) = R(to)/R(t), E.g., at redshift of 1, the universe is a factor of 2 smaller.

Cosmology with Newtonian Gravity

✦ cosmological models can only be derived self-consistently using General Relativity; but with a bit of “cheating”, we can use Newtonian gravity to derive how an expanding universe would behave. This provides some physical insight, but is not a correct interpretation of the universe.- the “fix” is to apply what is called the “Birkoff’s rule” to approximate the solution from GR: a test particle of mass m is affected only by gravity from mass within r in a spherical co-ordinate system.

ρor

m

Assume the test particle m isexpanding from the center, andthe expansion is opposed by the mass M interior to r.

Is the particle bound to the mass?Or, equivalent to asking: Is the Universe bound?

M

The mass interior to r: ≡ constant as the sphere expands Total energy of the system:

if E0, the system will expand forever E=0, the system is just bound (still expands forever!)

For a “just bound” universe:Substitute v=Hor for an expanding universe, with E=0: then: define ρoc as the “critical density” at which the system is just balance →

For ρ>ρoc, the system is decelerating sufficiently fast and will eventually collapse; ρ

How large a density is needed to “close” the Universe? (i.e., make it collapse eventually)For H0~70km/s/Mpc → ρoc~1x10-29 gm/cm3 Or about 5 hydrogen atoms per cubic meter

Define the (matter) density parameter: Ωm=ρ/ρc Ωm > 1 → closed universe (there is enough matter for to slow down the expansion for the universe to collapse eventually). Ωm = 1 → “critical universe” (still expands forever) Ωm < 1 → open universe (expands forever).

(Note: this is a VERY simple-minded scenario in which we deal with only one type of matter/energy in the universe.)

The Equation of Motion of the Universe and the Scale Factor

We can write a more general eqn of motion andrelate it to the scale factor:

substitute: M(r)= ⁴⁄₃ πr3ρ

→

Now introduce the scale factor R(t) such that R(t)=r(t)/r(t0), substitute and get:

→ an equation describing how the scale factor of the universe behave with time

Some important points about R(t)

✦ we already showed: (1+z) = R(to)/R(t) ✦ R3(t)ρ(t)=R3(t0)ρ(t0) , the matter density of the universe increases as (1+z)3 as we look back in time.

✦ The Hubble constant and R(t): v(r)=H(t)r , or → dr/dt =H(t)r ; Substitute r=R(t) r0 → r0 dR(t)/dt = H(t) R(t) r0 → H(t)=1/R(t) dR/dt

For t=t0 (now): H0= dR/dt |t=t0 i.e., the Hubble constant is just the rate of change of the scale factor.

Writing the eqn of motion in R(t):

Substitute ρ=ρo/R3 →

This is a second order differential equation in R(t), with the solution (see next page for derivation):

Solving the equation of the Scale Factor:

(1)

Solutions to R(t)

1. k=0: “critical universe”: rewrite the solution (eqn 1, last page) as: R½dR = (⁸⁄³ π Gρo)½ dt

Integrate both sides: ³⁄₂ R ³⁄₂ = (⁸⁄₃ πGρo)½ t

→ R(t) ∝ t ⅔

The universe always expands, but the rate gets slower and slower (dR/dt ∝ t -⅓ )

For the other two cases, the solutions are complicated, but we can decipher how R(t) behaves.

2. k>0: dR/dt eventually approaches zero, and changes sign → the universe collapses. (very roughly, for Ω=2, this occurs at ~30 Gyr)

3. k

The Age of the k=0 Universe(I.e., the “critical universe”, the “flat universe”; ρ=ρc; or Ω=1)

k>0

k=0k

The Big Bounce

"Some say the world will end in fire,Some say in ice.From what I've tasted of desireI hold with those who favor fire.But if it had to perish twice,I think I know enough of hateTo say that for destruction iceIs also greatAnd would suffice."

-- Robert Frost

The Fate of the Universe(Big Empty)

Radiation Dominated Universe

- What we have derived so far is for a matter dominated universe (i.e., mass density ∝ R(t)-3)

- But the universe also has lots of radiation (i.e., the CMB) as its mass-energy content

- currently, the matter mass density dominates over radiation energy density (by comparing the average matter mass density to CMB photon energy density) -- But this has not always been the case, because radiation density changes differently with R(t).

The Scale Factor and Radiation

- The CMB (cosmic microwave background) radiation is a blackbody radiation of certain temperature T, (i.e., with a blackbody (bbd) Planck spectrum), reflecting the temperature of the universe.

- As the universe expands, it can be shown straight- forwardly that the bbd spectrum is shifted in such a way that the temperature cools as 1/R(t). i.e., T(t) =T(t0) R(t0)/R(t) = T(t0)/R(t)

Example: currently CMB: T=2.70K; at redshift 10, R=1/11, so T(z=10)=2.7*11=29.7oK

Recall Stefan’s law relating the temperature and energy of blackbody radiation: E=σT4

As the universe expands and cools, the energy density drops as T4

Since T∝R-1 → ρrad=ρrad0 /R4

(compare this with matter density: ρm=ρm0 /R3 )

✦ Radiation mass density drops faster than matter mass density as the universe expands

log R(t), or log 1/z

How matter and radiationdensity, ρm and ρrad change with R(t) (i.e., time)

rad

At some redshift in the past,the universe was radiationdominated (question in problem set 3).

This has an important effectin understanding the historyof the universe, and we willcome back to this in the next section.

Recall the eqn of motion of R(t) for matter dominated:

How does R(t) behave when the universe is radiation dominated?

and we substituted ρ=ρo/R3 → to get

providing an integrated-once solution of

We’ll do the same for radiation dominated, but use ρ=ρo/R4

(for k=0)

Substitute ρ=ρo/R4 , and get:

By following the same derivation (integrating once) as for the matter dominated case, we find for the case of k=0 (i.e., critical universe): R(t) dR = constant dt , which has the solution: R2∝t, or R(t)∝t1/2Compare this with matter dominated case for k=0: R(t) ∝ t ⅔ i.e., the universe expand slower when it was radiation dominated

The Friedmann Equation

The Friedmann Eqn is a solution to Einstein’s general relativity field equation.

Compare this to the Newtonian gravity derivation:

The kc2/A2 term can be interpreted as the curvatureof space associated with the mass density of the universe,where A is the curvature (units of 1/radius), and k,the curvature sign.

✦ The second term (Λ) can be thought of as the curvature of the vacuum (energy), i.e., theenergy density of empty space (!).This arises naturally as an integration constant, butEinstein decided to include it only as a way to balanceout the first constant term to “prevent” an expandinguniverse. (He called this his “biggest mistake”, when,in the 1930’s, astronomers found that the universe is in fact expanding!)

The Friedmann Equation

InterpretingSpace Curvature(as 3-D creatures, we can onlyillustrate (i.e., visualize) curved space for 2-D space, curving into the 3rd dimension.)

Parallel lines indifferentkinds of space

The area of a circle of radiusr is different for spaces ofdifferent curvatures.A=πR2 for flat space,but A>πR2 for negative curv.,and A

How do we determine the cosmological model for our Universe?Two main classes of methods: 1) measure the mass density, or 2) measure the geometry.

1. Measure the average mass density of the universe and compare it to the critical density ρc, i.e., measure Ω = ρ/ρc. - measure the luminosity density (i.e., counting galaxies) and transform to mass density using M/L (also measured). ~ measure the motion of galaxies (e.g., “bulk flow” of galaxies), and model the mass density consistent with the gravitational field.

While these are simple in principle, in practice, they arevery complex and difficult tasks.

2. Measure the geometry of the universe (i.e., curvature), by measuring how volume or length changes with distance from us. - measure how volume changes with redshift; e.g., by counting galaxies or galaxy clusters per unit volume at different distances (tricky: galaxies and galaxy clusters evolve with time.) - measure how distance changes with redshift: (see details in later lectures) 1. using standard candles: e.g., Supernova Ia 11. using standard rulers: e.g., ripples on the CMB 3. A related method is the growth of structure: how quickly

galaxy clusters and clustering grow with time. This is akin to a combination of the dependence on mass density and geometry.

We will come back to this in a few lectures to see what astronomers measure