Cosmo Exam

Question 1

What is a comoving coordinate system?

Answer 1

Coordinate system where the grid lines expand with the objects of the universe.

Question 2

What is a peculiar velocity

Answer 2

The small movement objects may have with respect to the comoving coordinate system. However, due to the homogeneity and isotropy of the universe, on average galaxies are at rest wrt comoving coordinate system.

Question 3

What are comoving observers

Answer 3

Hypothetical observers expanding along with the coordinates.

Question 4

What does isotropic mean

Answer 4

Same in all directions

Question 5

What does homogeneity mean

Answer 5

Same at all points in space

Question 6

What is the cosmological principle

Answer 6

The universe is both isotropic and homogenous on large scales

Question 7

What is the scale factor

Answer 7

Describes the evolution of the universe

Question 8

What is the expansion rate

Answer 8

The Hubble parameter å/a

Question 9

Over what spatial scales in our universe is the cosmological principle approximately satisfied

Answer 9

Satisfied in the universe over spatial scales >~ 100Mpc

Question 10

What is freeze out

Answer 10

When the abundance of any species(particle or radiation) departs from its value in thermal equilibrium. Freeze out occurs when the rate of interactions that keeps the species in thermal equilibrium decreases below the expansion rate of the universe. After freeze out, the comoving number density of the species remains constant, in the absence of any other subsequent interactions.

Question 11

What is a hot relic

Answer 11

A species is said to be a hot relic if it freezes out when it is still relativistic

Question 12

What is a cold relic

Answer 12

A species is said to be a cold relic if it freezes out when it is non relativistic

Question 13

State the Sakharov conditions ( for matter to dominate in abundance over antimatter) with explanations

Answer 13

1) there must be reactions in which baryon number is not conserved in order to produce a matter/ anti matter imbalance in the first place
2) there must be reactions that violate charge parity symmetry. If CP violation does not occur, then any reactions that yield a net change in baryon number will be exactly offset by reactions that yield a net change in anti baryon number, producing equal quantities of matter and anti-matter.
3) there must be a departure from thermal equilibrium. If this does not happen, then by the principle of detailed balance in thermal equilibrium, any reaction that produces an excess of baryons will be exactly offset by its opposite reaction that reduces the number of baryons, thereby producing no net increase in the baryon number

Question 14

Explain the term standard candle

Answer 14

A standard candle is an object that has a known luminosity, so that observations of the flux received from it can be used to deduce its distance

Question 15

What is critical density

Answer 15

The critical density is the density required today to produce a flat universe with no cosmological constant.

Question 16

What is the dependence of ρ_r on Z?

Answer 16

Proportional to (1+z)^4

Question 17

What is the dependence of ρ_m on Z?

Answer 17

Proportional to (1+z)^3

Question 18

Define angular diameter distance

Answer 18

d_A = D /θ

Where D is the physical size, diameter, of the object and θ is it’s observed angular size diameter in the sky.

Question 19

Formula for redshift

Answer 19

z = (ν_e / ν_ο) - 1


Or

a_o / a = (1+z)

Question 20

Why does recombination become efficient when kT ~ 0.3 eV instead of 13.6eV

Answer 20

Hydrogen will remain almost completely ionised as long as there are sufficient number of ionising photons (i.e. E> 13.6eV). However, since the number density of baryons is vastly smaller than the number density of photons, there are enough high energy photons in the exponentially declining tail of the Plank distribution to keep the hydrogen ionised even when the average photon energy falls much below 13.6 eV. It is only when the photon energy falls to 0.3eV that the number of high energy photons in the tail of the distribution becomes too small to keep most of the hydrogen ionised and recombination becomes efficient.

Question 21

Describe the cosmological constant problem.
What does inflation have to say about this problem

Answer 21

The cosmological constant problem expresses the problem with the small value of the observed cosmological constant when compared to its natural value. The natural scale is set by the plank mass.

Inflation has nothing to say about the cosmological constant problem.

(In an exam need to use numbers):

In natural units, the cosmological constant has an observed energy density of ΩΛ x ρcrit = 0.7 x 10^ -11 (eV)^4

The natural scale is set by the plank mass Epl ~ 10^28 eV -> the expected density is Epl^4 ~ 10^112 (eV)^4. Thus the ratio of the observed to expected value of the cosmological constant is ~ 10^-123.

Question 22

Describe the flatness problem in the standard Big Bang cosmological model

Answer 22

The flatness problem describes the evolution of the curvature parameter with scale factor. For a radiation dominated case, we have Ω_k proportional to a^2, and for matter domination it is proportional to a. Both of these are increasing functions of a, hence if Ω_k is non zero, it will get larger at late times. Hence the curvature parameter must have been even smaller in the past. The flatness problem is how such a finely tuned initial value could have arisen.

Question 23

What is w for radiation domination

Answer 23

w = 1/3

Question 24

What is w for dust

Answer 24

0

Question 25

What is the formula for the sound speed

Answer 25

C^2 = dp / dρ = p/ρ = wc^2

Question 26

What 2D shapes satisfy homogeneity and isotropy

Answer 26

Flat space,
Sphere,
Hyperbolic paraboloid

Question 27

How does the geometry depend on the factor k

Answer 27

k>0 three sphere at any given time, universe has positive curvature. (Triangles have more than 180 degrees, parallel lines will converge). Universe has finite volume at any one time.
k=0, flat manifold (Minkowksi with the extra factor of a^2 on the spatial part)
k < 0 negative curvature. 3D saddle shape. Triangle has less than 180 and parallel likes may cross and will diverge. Universe is infinitely large

Question 28

In our FRW metric, what does the coordinate r represent

Answer 28

The radius in comoving length of a sphere centred at r = 0

Question 29

What path does light travel in curved space

Answer 29

Light travels along geodesics of the manifold which are null, meaning ds=0

Question 30

What is the definition of redshift

Answer 30

The wavelength of freely propagating photons increases with the expansion of the universe, proportional to the scale factor.

(The expansion is very much like climbing out of a potential well, so it is plausible that photons lose energy as they propagate)

Question 31

How does the energy density scale with a for PHOTONs

What else does this apply to

Answer 31

Density decreases proportionally to L^3 ie a^-3

The energy per particle is proportional to the frequency. this scales as wavelength^-1 hence the total energy density scales as

a^-4

If the first term dominated in E^2 = p^2c^2 + m^2c^4, the particles behave like photons (radiation). Also applies to high speed particles

Question 32

What is the dependence of a on z

Answer 32

a/a_o = 1/(1+z)

Question 33

What is the density parameter

Answer 33

The ratio of actual density to the critical density

Question 34

How does the density of non relativistic matter scale with a

Answer 34

a^-3

Question 35

How does the density of radiation scale with a

Answer 35

a^-4

Question 36

How do we define the total density parameter

Answer 36

Ω_tot = Ωm + Ωr + ΩΛ = 1 -Ωk

Question 37

Describe which density terms dominated at each stage in the history and future of the universe

Answer 37

Due to the different powers in (1+z) higher powers dominate further back in time.

Further and further back in time Ωr and Ωm dominates over ΩΛ and Ωk hence the universe will look geometrically flat with no cosmological constant.

The furtherest back is radiation dominated. Then we move to matter domination. We expect k and Λ to dominate eventually if they are nonzero. (We think Ωk is zero).

We are experiencing the change from matter domination to Λ domination now.

Question 38

What is an open universe

Answer 38

k<0
Ωk > 0

Question 39

What is an empty universe

Answer 39

k<0,
no matter, no radiation, no cosmological constant.

Milne universe.

We find a scales as t^-1.

This corresponds to the far future evolution of any open universe, where the matter and radiation densities have diluted so much as to become negligible. As long as there is no cosmological constant.

Question 40

What does a flat matter dominated universe correspond to

Answer 40

Early times for a universe with non relativistic matter, but not so early that radiation dominates.
This is the Einstein de Sitter Universe.

Question 41

What is a closed universe

Answer 41

k>0
Ωk < 0

Question 42

What can we learn from our cosmological models

Answer 42

All universes start out looking like a flat universe, but diverge depending upon the total density. (I.e. the sign of k).
k = 0 case does not asymptote to a constant value of H, but it grows more slowly than the open case. If you normalise the 3 cases to the same expansion rate at some particular time, the open universe is older than the flat universe, which is older than the closed universe.

Question 43

Describe the principle of causality

Answer 43

Information cannot travel faster than the speed of light. Hence causality cannot act faster than the speed of light. An event can only be causally connected to something within its lightcone.

Question 44

How do you define χ_H

What is r_H

Answer 44

This is the comoving distance to the horizon at comoving coordinate value r_H. The comoving distance to the sphere along t=constant surface, defined by coordinate r_H.

r_H does not correspond to a distance along the manifold. It is a coordinate.

χ_H = integral (c/a(t)) wrt dt from 0 to t

Question 45

How do we convert a comoving coordinate to a proper distance

Answer 45

We multiply by the scale factor a(t)

Question 46

Define luminosity distance

Answer 46

Flux F = L / (4π (d_L)^2)


dL can only depend on redshift once we have fixed the spacetime

Question 47

Define the equation of state parameter w

Answer 47

p/ (ρc^2)

Relation between the density and pressure

Question 48

What is the relationship between temperature and redshift

Answer 48

The effect of redshifting upon a black body is a blackbody at the new temperature

T(z) = (1+z) To

Question 49

What is the number of degrees of freedom for fermions

Answer 49

2

Question 50

What are interactions between particle species governed by

Answer 50

- Interaction rates, which may depend on the temperature, particle masses and cross sections
- particle masses which determine the equilibrium abundances

Question 51

When is something in temperature equilibrium with photons

Answer 51

When kT >> mc^2, the particle acts as radiation. If the interaction with photons is sufficiently strong, it can interconvert with photons and will have the same temperature.

Question 52

What is the mean free path

Answer 52

The mean free path is the mean distance traveled before an interaction

λ ~ v/Γ ~ c / Γ

Question 53

What is the mean time between interactions

Answer 53

1/Γ

Question 54

What is the maximum distance between particles in causal contact

Answer 54

The calculation of the horizon distance ~ ct ~ cH

Question 55

When are interactions unable to maintain thermal equilibrium

Answer 55

Since the distance where particles are in causal contact is ~ c/H we expect when λ >> d_H we expect that interactions will be unable to maintain thermal equilibrium

If H >> Γ it will not remain in thermal equilibrium

Question 56

What is some evidence that interactions are no longer in equilibrium

Answer 56

• different particle species have different temperatures
• there is considerably more matter than antimatter
•there are nucleons rather than quarks
• there are atoms rather than ions
• there is structure rather than a smooth featureless gas at a single temperature

Question 57

Why does the CMB have a higher temperature than the neutrino background

Answer 57

Neutrinos interact very weakly and hence came out of equilibrium far before the electrons and positrons, before they could annihilate into other light particles. Hence the total neutrino plus antineutrino density is higher than that of electrons plus positrons. However, when the electron/ positron pairs annihilated, they produced photons. Hence there was considerable extra energy dumped onto the photon background.

Question 58

What is a wimp

Answer 58

Weakly interacting massive particle, which might be dark matter. They are massive, stable particles that interact weakly with normal matter. They behave like cold relics.

Question 59

Roughly how did the universe go from being an ionised plasma to a neutral gas of mostly hydrogen

Answer 59

The early universe was hot enough kT >> 13.6eV to keep the universe ionised, and therefore for the thermal bath of photon to remain tightly coupled to the ions because of Thomson scattering. When the universe cooled sufficiently electrons and photons could recombine to form neutral hydrogen, which also frees the photons from their interactions with matter. - CMB.

Question 60

Describe the last scattering surface

Answer 60

A consequence of the interactions freezing out is that photons, once tightly coupled to the ionised plasma (which is opaque due to Thompson scattering) are able to stream freely through the now neutral, hence transparent, hydrogen gas. Thus we see the photons as if frees from a cloud, this is the last scattering surface, and if forms the cosmic microwave background. Further away, higher redshift, it is opaque, nearer is transparent.

Question 61

What are the 3 distinct hydrogen events

Answer 61

•Recombination: the equilibrium ionisation fraction X goes from nearly one to nearly zero
• last scatterings the freeze out of e^- γ Thompson scattering, the decoupling of photons from the baryons
• the freeze in of residual ionisation at a higher value than equilibrium i.e. the freeze out of the recombination reaction

Question 62

What is Big Bang by nucelosynthesis

Answer 62

Synthesis of light nuclei from the initial primordial mixture of protons and neutrinos.

At early times, protons, neutrons and light nuclei such as deuterium, tritium and Helium are in equilibrium. As the universe expands and cools, different nuclear reactions freeze out, leaving us with relic abundances of the stable nuclei.

Question 63

BBN complications

Answer 63

- we have to track the abundance of not just one end product, but several different nuclei
- neutrons are unstable when not in a nucleus, with a half life of about 11 minutes.
- several of the possible end states have binding energies that are small or comparable to kT, hence freeze out can be delayed.

Question 64

Why do we expect most nucleons to end up in helium equilibrium.

Answer 64

4He is the most strongly bound light nucleus. There is a lack of stable nuclei at all mass 5 or 8 makes it very difficult to get higher-mass nuclei beyond helium.

Question 65

Why is it difficult to make deuterium

How does this lead to the deuterium bottleneck

Answer 65

The equilibrium deuterium fraction is controlled by an equation similar to the Saha equation, with factors of (kT/mpc^2)^3/2 η << 1

The large baryon to photon ratio keeps the reaction pn-> dγ from proceeding forward. (I.e. The photons dissociate the deuterium) well below the binding energy. The deuterium fraction remains small until kT ~ 0.1MeV.


Even at this point the reaction rate / H > 1 for kT > 0.05 MeV.

Only a very small number of deuterium nuclei are formed from the baryons therefore there’s a very low rate of the dd interactions. We never make it to equilibrium abundances of helium.

Question 66

What happens after 0.05 MeV to protons

Answer 66

We can form deutrium and both of the pathways proceeds. There is a rapid burning of almost all of the remaining protons and neutrons into helium.

Question 67

What is the Coulomb barrier

Answer 67

There are no stable elements with A=5 and A=8 so we cannot synthesise elements beyond helium by adding just a single nucleon. To create heavier elements, we must fuse several light nuclei together, but each would have a positive charge, hence the reaction rates are strongly suppressed by the Coulomb barrier.

Question 68

What is the dependence on lithium abundance with η

Answer 68

Lithium is very easy to destroy via collisions with protons. So increasing η actually decreases final lithium abundance, until the beryillium pathway opens up, and then the lithium abundance increases with η since beryillium inverse beta decays via electron capture to 7Li.

Question 69

How can we aim to resolve the cosmological constant problem

Answer 69

• first - Λ =0 but there is another physical mechanism, dark energy, which can provide an energy density with w = -1, not a true vacuum energy. A scalar field can provide this.

Inflation is thought to be driven by a scalar field however the energy scales of inflation. And dark energy are so different that nobody can come up with a single mechanism for producing both epochs of w=-1

• another possibility is that the properties of the vacuum depend on the details of the fundamental theory, e.g. string theory. There are a huge number of ways to compacting down to the 3+1 dimensions and the cosmological constant could be different in each of them. We then use the anthropic argument to find the cosmological constant. With greater values the universe would look much greater than it does today. I.e no structure formation. The universe would have started exhibiting accelerated expansion before the structures would form. -> we couldn’t observe the cosmological constant if it were much larger -> we should observe the largest possible value of Λ consistent with structure formation

Question 70

Describe the horizon problem

Answer 70

The largest distance that physics ought to be able to act is the horizon size. Any two patches of the CMB more than a couple of degrees apart should not have been in causal contact. Why are they then the same temperature? Why is the matter density roughly similar?

The physical scale of a galaxy grows along with the scale factor, scales as t^2/3 (MD) or t^1/2 (RD). But the horizon scale grows as t. Today the scale is inside the horizon. However at some point, we must have had the size being bigger than the horizon. The time of equality is horizon crossing. Due to causality, we expect that the structures can only grow when they are inside the horizon.

Question 71

What is the relic particle problem?

Answer 71

GUT (grand unified theory) combines the strong and electroweak forces, the theory has a very massive electromagnetic monopole. We expect the monopole to be formed with a number density of about one monopole per horizon volume at some temp T_GUT. Hence the physical number density of monopoles scales as H^3_GUT = T^2_GUT /m_pl

If we calculate the present day mass density parameter of monopoles, we find it is much bigger than 1, since Ω ~ 1, this is not possible.

It is said that the relics would over close the universe, they would cause the universe to close and collapse again on very short timescales.

Question 72

Describe the idea of inflation

Answer 72

A period of accelerated expansion takes a very small volume of the early universe and blows it up so much and so quickly that any in homogeneities or curvature in this volume are smoothed out and the density of non relativistic particles is diluted. At the same time any quantum fluctuations are blown up to macroscopic size, providing the seeds for large scale structure.

Question 73

How does inflation solve various problems?

Answer 73

The accelerating expansion means that the Hubble scale remains constant, but comoving scales increase much more rapidly. The true horizon scale is now much larger than the Hubble length, thus accelerating expansion should be able to solve the horizon problem.


It can solve the flatness problem as it means the value of Ω_k gets driven closer and closer to zero while acceleration is occurring.

It solves the relic monopole problem, it dilutes the number of massive relic particles in a given physical volume much faster than ordinary decelerating expansion.

Question 74

What is reheating

Answer 74

Expanding the universe cools the universe down proportional to the scale factor. Therefore we must find a mechanism for reheating. We need to find a way to make the universe radiation dominated after the period of accelerated expansion.

Question 75

Timeline for inflation

Answer 75

At very early times the universe is radiation dominated. At late time the universe is Λ dominated.

At the very early universe, we are inflation dominated. At the GUT scale t ~ 10^-34, the transition causes inflation to start. Lasting till 10^-32s. During which H is approximately constant, causing exponential expansion. We then need to reheat the universe to a high temperature which converts the entire energy density into radiation particles, vastly increasing the energy density of the universe.

Question 76

What is the slow roll regime

Answer 76

When the scalar field is sitting on the approximately flat part of the potential, we say it is in the slow roll regime.

Question 77

What are the values in the concordance universe

Answer 77

Ω_k = 0
Ω_m ~ 0.3
Ω_r ~ 10^-5
Ω_Λ ~ 0.7

Question 78

Define the meaning of particle horizon

Answer 78

Proper distance to the comoving position at that time of a point on there light cone at time t->0. The furthest point from which a light ray could have reached an observer at time t, measured in present day proper units.

Question 79

What is the approximate number of baryons in the universe for every photon? What are the important measurements and theoretical ideas that have gone into the determination of these two numbers

Answer 79

There are roughly one billion photons for every baryon . The photon density comes from the temperature of the CMB and the measurement that it is a black body. The baryon density comes from measurements of the primordial abundance and the theory of the Big Bang nucleosynthesis.

Question 80

Explain the link between H recombination and Thomson scattering of photons off electrons. Why does the H recombination reaction freeze out at the same time as the CMBR is formed

Answer 80

Due to Thompson scattering, the thermal bath of photons remains tightly coupled to the ions. Hence, when CMBR is formed, the photons are not tightly coupled to the ionised plasma, they stream freely. This is the surface of last scattering.

Question 81

Open
Closed
Flat

Compare the evolution of the scale factor in these three cases

Answer 81

The scale factor for a closed universe rises to a maximum value and then symmetrically falls back to zero.

The flat universe acts as the critical case and rises smoothly heading to infinity asymptotically.
An open universe scale factor rises more rapidly again, heading to infinity.

Question 82

How have type 1a supernova been used to constrain the properties of dark energy

Answer 82

They act as standardisable candles that with a single free parameter describing brightness-decay time can be used as standard candles
Given a population of standard candles distributed at different redshifts, one could measure the distance redshift relation dL
From dL one could in principle measure cosmological parameters Ho ΩΛ Ωm and w for dark energy

Question 83

How does H depend on T for a radiation dom universe flat

Answer 83

H ~ T^2/mp

Question 84

What is the weak interaction scale

Answer 84

σ ~ G_F^2 T^2

G_F = α_W/M_W

Question 85

What is a standard ruler

Answer 85

An object of known physical size, by measuring angular size we can determine it’s distance from Earth

Question 86

What is the copernican principle

Answer 86

we do not live in a special place or time

Question 87

Plank units

Answer 87

G^-1/2 = Mpl

tpl = l_pl = G^1/2

In natural units

Question 88

How is string theory used to solve the cosmological constant problem

Answer 88

In string theory, there are a huge number of ways to compactify down to 3+1 dimensions, and the cosmological constant could be different in each of them. We use the anthropic argument to find the cosmological constant. The theory predicts greater values of Λ than observed. However, with these greater values, it would be difficult to form galaxies, the universe would start exhibiting accelerated expansion before the structures could form. Hence we wouldn’t be here to observe the cosmological constant if it were much larger. Thus the prediction is that we should observe the largest possible value of Λ consistent with structure formation.

Question 89

Definition of surface brightness

Answer 89

Flux per solid angle