Addendum 14: Early Universe: Chapter 23

First define some astronomical and physics constants and dimensional units that may be needed below

1. Critical Density. The critical mass density is the density required to just stop the observed Hubble expansion (after an infinite time). We can estimate this density by considering a sphere of radius R and enclosed mass M, with a small test mass m on the periphery of the sphere. The test mass will be receding from the center of the sphere at a speed given by the Hubble expansion:

The mass will be slow down due to the attractive gravitational force exerted by the mass M. If the gravitational and kinetic energies are equal the mass will stop at infinity (conservation of total energy, same as the escape speed calculation from earlier in semester). This condition is:

The test mass m cancels, and we can rewrite the mass M in terms of the density r:

and R terms cancel. Solving for r:

This is numerically:

2. Temperature vs. cosmic time. The temperature of the CBR radiation (and matter, when radiation and matter were coupled, before 300,000 yrs), can be calculated by using the critical density and ne additional formula: the energy density of a radiation field. We can obtain this easily be considering a thermal body and using the Stephan-Boltzmann law. Consider a small volume above the surface of a thermal body, so the radiant energy emitted per unit area per second is:

Now consider a thin annulus around the sphere: photons will stream through the annulus at the speed of light c, so the radiative energy density u_rad will be

Now let's write the critical density equation (section 1 above) as a function of time (since as the Universe exapnds the Hubble 'constant' will change (unless the Universe is empty - which it isn't!)

The radiation energy density (which will dominate in the radiative era) can be written as an equivalent mass density:

or

Finally, we can write the 'age' of the Universe as the inverse Hubble time

Solving for T(t), we obtain:

numerically:

This expression is (roughly) correct for the radiation-dominated era (before t < 10⁵ yr) but isn't quite correct for later times (the exponent for matter-dominated times is -2/3, not -1/2).

Example: What was the age of the Universe at the temperature 'freezing out' of leptons (e.g. electrons) ? This happened at a temperature T ~ 6*10⁹ K (see next section)

solving for t above:

3. Relationship between 'freezing out' temperatures and rest masses. The minimum temperature required to allow a particle with rest mass m to freely exchange energy with the vacuum is given by equating its thermal energy with its rest mass equivalent energy (mc²⁾. Below this temperature the particle 'freezes out', i.e. becomes stable.

Example: At what temperature and age do baryons freeze-out (become stable particles)?

A baryon (e.g. proton,neutrons)

4. Number densities of photons and baryons, and the present asymmetry of matter over anti-matter. The number density of photons and baryons in the present Universe are very different, a fact which is important is understanding why matter dominates over anti-matter.

a. Number density of photons. The total radiation field is completely dominated by the CMB radiation if averaged over Gpc scales. We can estimate the number density by starting with the energy density of radiation (section 2 above) and dividing by the average energy per photon:

where l_max = 1 cm is the wavelength of maximum intensity of the CMB radiation and T (= 2.7K) is the present temperature of the CMB.

b. Number density of baryons. This is an observed quantity (doesnt include dark matter). We can form a crude estimate by summing the mass in a cluster by ite volume, and dividing by the massof a proton (Universe being mostly hydrogen remember)

Hence, the ratio of photons to baryons is enormous, 10⁹ to 1. Here's the key point: Before baryons were frozen-out (t~ 10^-4 sec), the number of photons and baryons/anti-baryon pairs must have been nearly equal, since they were constantly exchanging (via pair production/annilhilation). Indeed, the matter-antimatter pairs must also have been equal, i.e. the Universe was equal parts matter and anit-mattter. At freeze-out, there must have been a slight excess of matter over anti-matter (1 part in 10⁹), so the present Universe is dominated by matter.

5. Space-time at the smallest scales: Planck length and Planck time. At the smallest scales (much smaller than anything probed in the lab, or even though possible in the future), the fabric of space and time are thought to be quantized by gravity. The idea is the that this is the length for which a particle's Schwarzschild radius is equal to the size (wavelength) of photon that would be generated if it were annihilated (into the vacuum). The Planck length can be obtained by:

a. Considering the wavelength of a photon whose energy is equal to the rest mass energy of the particle. This is known as the Compton wavelength:

b. Considering the Schwarschild radius (event horizon) of that particle:

c. Equating the two lengths and solving for the mass (called the Planck mass), and dropping the numerical factor of 2 (unimportant):

d. Putting the Planck mass back into either (a) or (b) and solving for the length (Planck length):

This is a remarkable result: It combines the fundamental constants of gravity (G), quantum mechanics (h), and relativity (c) in a unique manner. Planck first noticed this without deriving as we have done above: He simply asserted that this length must be significant becvause it is inly way to combing these constants to obtain a length. similarly, the only combination which results in a time unit is:

or

This is the Planck time. It is the time a photon takes to move one Planck length.

Some people assert that this time is the shortest interval in which events can occurs: it's the granularity or quantum unit of time. It would mean that time doesn't 'flow' smoothly, but happens in discrete units of the Planck time.

The painting below, by John Walte (2000), is titled: Not Bound by Planck Time, The Gods Prepare Hydrogen for the Inflationary Epoch