Lecture 7: Origin of CMBR

Question 1

To fully understand the origin and physical properties of the CMBR we will need to think about

Answer 1

the ratio of the number of photons and baryons (this ratio is preserved as the universe expands)

Question 2

the present-day energy density of the CMBR is

Answer 2

ur0 = 4.17 x 10^-14 Jm^-3

Question 3

the typical energy of a CMBR photon is

Answer 3

E__typical = 2.7kBT

=1.02 x 10^-22 J

Question 4

dividing the CMBR energy density by the typical energy of a CMBR photon gives

Answer 4

an estimate of the present-day number density of CMBR

Question 5

We can convert the present-day mass density of baryons to an energy density by

Answer 5

multiplying by c^2 to give ub0

Question 6

how to find number density of baryons

Answer 6

taking the mass of a baryon to equal the proton mass

Question 7

comparing baryon and photon number density, we see

Answer 7

that there are a few billion photons for every baryon in the universe

Question 8

epoch of recombination

Answer 8

The CMBR photons that we observe today come to us from a time when the universe had expanded and cooled sufficiently to allow free electrons to combine with protons and form neutral hydrogen atoms. This
epoch is referred to as recombination

Question 9

Long before the epoch of recombination, when the universe was (say) one millionth of
its present size (which means that its temperature was one million times greater than the
current temperature of the CMBR) it would have been

Answer 9

fully ionised, with photons interacting strongly with free
electrons through Thomson scattering, so the mean free path for any photon was very short

Question 10

mean free path of a photon

Answer 10

the typical distance that a
photon could travel before being scattered by a free electron

Question 11

As the universe expanded and the temperature dropped the interaction rate between photons and electrons also dropped and eventually the mean free path of the photons became

Answer 11

comparable to the size of the universe itself

Question 12

mean free path of e- being approx size of universe means the the photons could

Answer 12

travel unimpeded across the universe, no longer being scattered
(since photons do not interact strongly with neutral hydrogen) for the entire duration of the universe’s subsequent evolution.

Question 13

the point where the scattering of photons stopped occuring is know as

Answer 13

decoupling

Question 14

decoupling marks the point when

Answer 14

the CMBR that we observe today was ‘formed’

Question 15

CMBR is often referred to as the

Answer 15

surface of last scattering

Question 16

The critical physics for understanding the origin of the CMBR is

Answer 16

hydrogen ionisation, related to the binding energy of hydrogen (13.6eV)

Question 17

binding energy of hydrogen

Answer 17

energy needed to ionise from the ground state, a single eectron from the single proton in a hydrogen atom

Question 19

how to get a crude first estimate for the temperature above which neutral hydrogen cannot form

Answer 19

equate the ionisation energy of hydrogen with the mean photon energy of a black body distribution (ie 2.7 kBT = 13.6eV)

Question 21

equating the binding energy of hydrogen with mean photon energy does not take into account that

Answer 21

we have a thermal distribution

Question 22

why do we conclude that the universe must be cooler than 60000K before nucleosynthesis can proceed properly

Answer 22

if we have a photon-to-baryon ration of 10^9 in the universe, there must be enough high energy photons in the tail of the distribution to still ionise hydrogen (and hence prevent its formation)

Question 23

we can make a better estimate of when the CMBR formed by using

Answer 23

the Saha equation

Question 24

the Saha equation allows us to

Answer 24

compute the ionisation fraction X of a gas in thermal equilibrium

Question 25

we define X as

Answer 25

the ratio of the number density of free protons (ionised hydrogen) to the number density of baryons

Question 26

the Saha equation takes into account the

Answer 26

balance between the two relevent processes going on: 1. the formation of hydrogen: the recombination of nuclei and electrons into atoms 2. the ionisation of (newly-formed) hydrogen by the high-energy photons in the tail of the distribution

Question 27

for temperatures much higher than the binding energy of hydrogen, we find that X=

Answer 27

approx 1 ie the universe is fully ionised

Question 28

graph of X against z

Answer 28

starts of horizontal around 0 for small redshift then increases to then horizontal around 1 for high redshift

Question 29

why even this improved estimate of the temperature at decoupling is not quite correct

Answer 29

as the ionisation fraction drops the remaining electrons find it increasingly difficult to combine with the remaining hydrogen nuclei to form atoms because they are both becoming too rare and they do not find each other.

Question 30

the residual level of ionisation can no longer be modelled by the Saha equation since

Answer 30

the universe is no longer in thermal equilibrium (because the interactions between photons and electrons are also becoming too rare)

Question 31

a more sophisticated treatment (beyond scope of course) predicts a residual ionisation fraction of around

Answer 31

10^-3 it also predicts the surface of last scattering to occurs at around 3000K

Question 32

a surface of last scattering temp of 3000K means that

Answer 32

corresponds to a universe one thousandth of its present size and we observe CMBR redshift of around z=1000

Question 33

formally, the surface of last scattering is when the mean free path of the photon equals

Answer 33

the size of the universe

Question 34

from Saha equation and the solution for X as a function of z, we can see that the transition of the universe from fully ionised to fully neutral was

Answer 34

rapid but not instantaneous (**recombination and decoupling did not happen at the same time**)

Question 35

difference between recombination and decoupling

Answer 35

decoupling refers to the point where photons fall out of thermal equilibrium with matter, while recombination refers to the epoch when electrons and protons first become bound as hydrogen nuclei. Strictly speaking, then, decoupling happens after recombination.

Question 36

to work out the age of the universe at the time when the CMBR was formed we need to account for

Answer 36

the fact that the universe was radiation-dominated in its early phase, and modify our expression for E(z) accordingly.

Question 37

We can get a rough idea of the age of the universe at z ≈ 1 000, however, if we assume

Answer 37

that we live in an Einstein de Sitter and matter-dominated universe

Question 38

for z=1000, assuming that we live in an Einstein de Sitter and matter-dominated universe gives

Answer 38

CMBR emitted when universe was a fraction around 3x10^-5 of its current age (400,000 years after Big Bang)

Question 39

omega_rel

Answer 39

general relativistic density parameter that includes the radiation energy density and the neutrinos =omega_r + omega_v

Question 40

taking omega_m=0.3 and h=0.7m we find that z_eq=

Answer 40

3500

Question 41

we find that the epoch of matter-radiation equality occurred around

Answer 41

65,000 years (2x10^12 seconds after Big Bang)

Question 42

what does the epoch of matter-radiation equality occurring at (2x10^12 seconds after Big Bang) mean

Answer 42

the universe was already matter dominated when the CMBR was formed

Question 43

The epoch of nucleosynthesis

Answer 43

universe around 1 second old T= around 10^10K

Question 44

period when t was between 10^-4 and 1s

Answer 44

temperature of the universe cooled to 10^10K the universe comprised a plasma filled with free electrons, protons,neutrons, photons, neutrinos everything strongly interacting with everything else

Question 45

period between t=10^-10 and 10^-4 s

Answer 45

temp cooled to 10^12K universe consisted of a soup of quarks and leptons, strongly interacting quark-hadron transition occurred and protons and neutrons formed

Question 46

the cosmic accelerator (t<10^-10s)

Answer 46

T>10^15K, close to limits of current Earth-bound particle accelerators

Question 47

the epoch of inflation (t=10^-35s)

Answer 47

T=10^27K, universe underwent a short period of accelerated expansion known as cosmological inflation

Question 48

the Planck Scale (t=10^-43s)

Answer 48

classical description of spacetime based on GR breaks down entirely and a quantum description is required

Question 49

how did we obtain 10^-43 s for Planck time

Answer 49

start with planck mass get energy by x by c^2 get time by dividing planck length by c