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EXAMS Summer 2023 > Early Universe > Flashcards

Early Universe Flashcards

1
Q

What are the key properties of the FRW universe?

A

Homogenous and Isotropic (on the scale of ~100MPc)

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2
Q

What parametrisation of the FRW metric do we usually use?
Why is this useful?

A

We parametrise the metric in terms of conformal time and co-moving distance.

This gives a metric of minkowski form: locally flat, ???

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3
Q

What is the relation between time and conformal time?

A

d(eta) = 1/a dt

a is the expansion scalefactor of the universe

**a = 1 in the present epoch

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4
Q

What is the relation between distance and co-moving distance?

A

The full co-ordinate transform is slightly complicated but basically the same as conformal time (i.e: divide normal distance by a).

**a = 1 in the present epoch

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5
Q

What is the equation of state for dark energy?

A

Pressure = -density

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6
Q

What is the effective equation of state for curvature?

A

Pressure = -1/3 density

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7
Q

What is the equation of state for matter?

A

Pressure ~ 0

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8
Q

What is the equation of state for radiation?

A

Pressure = 1/3 density

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9
Q

What is the definition of the hubble parameter?

A

differential of a w.r.t. time / a

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10
Q

How can we express density of a species in terms of an exponent of a?
(Using the continuity equation)

A

Density ~ a^(-3(1 + e.o.s.))

where

Pressure = e.o.s. * density

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11
Q

What is hbar c equal to?

A

197 fm MeV

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12
Q

What observations do we have to guide our understanding of the early universe?

A

Hubble expansion
The CMB
Galactic distribution (specifically galaxy power spectrum).

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13
Q

How do we quantify anisotropies of the CMB?

A

Expand in terms of spherical harmonics.
Extract the coefficients (specifically the average for each value l) [2l+1 values of m for each l]

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14
Q

Why are l=0 and l=1 contributions to the CMB usually subtracted?

A

l=0 contribution will always be 0 if we define temperature deviations relative to the mean temp of the CMB.

l=1 contribution (proportional to cos(theta)) can be attributed to the relativistic doppler effect due to the motion of the earth relative to the cosmic rest frame.

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15
Q

How is the two-point density contrast correlation function defined?

A

The ensemble average of the product of density contrasts at two different points separated by distance r at time t.

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16
Q

What is the ergodic principle?

A

Ensemble average is technically the average over all possible realisations of the universe, but we can assume that this is equivalent to averaging over the choice of origin within one universe.

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17
Q

What is the galaxy power spectrum?

A

Proportional to the fourier transform of the two-point density contrast correlation function.

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18
Q

What is the transfer function?

A

The transfer function describes how an initial spectrum of density perturbations evolve in time.

Density contrast (now) = transfer function * density contrast (initial)

Power spectrum (now) = transfer function^2 * power spectrum (initial)

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19
Q

What equations do we need to describe the universe using first order perturbative Newtonian dynamics?

A

Newton’s 2nd law (forces are pressure and gravitational).

Poisson’s equation (couples mass to gravitational potential)

Continuity equation (relates density and velocity of cosmic fluid)

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20
Q

What is the nature of the final differential equation in the density perturbation we get from a first-order Newtonian treatment?

A

(actually an equation in the FT of the density perturbation)

Basically an equation describing propagating sound waves in an expanding medium.

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21
Q

What does growth of fourier modes in the density perturbation correspond to?

A

Collapse (density perturbations become shorter wavelength).

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22
Q

What is the basic concept behind performing relativistic perturbation theory?

A

Express the metric tensor as the sum of the minkowski metric and a perturbing metric h. (we are using conformal coordinates so there is a factor a^2)

The magnitude of the perturbing metric is «1. Therefore neglect terms higher than first-order in the perturbation.

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23
Q

What is the basic concept behind the synchronous gauge?

A

Basically sets coordinates such that cold dark matter is always stationary with respect to them.

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24
Q

What is the significance of the closed equation we get for vorticity from relativistic perturbation theory?

A

Tells us that vorticity perturbations decay in time.

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25
Q

How many equations does relativistic perturbation theory ultimately give us (not counting the vorticity equation)?

A

2N+1 with N species.

2 because there are density perturbation and (div)velocity perturbation equations for each species.

+1 because there is additionally a metric perturbation equation.

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26
Q

How can we justify thinking of conformal time as equivalent to the co-moving size of the horizon?

A

Consider the path of light (null geodesic) : ds = 0.

Therefore ad(eta) = ad(co-moving size of horizon)
=> eta = co-moving size of horizon

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27
Q

What do we consider “sub-horizon modes”?

A

co-moving wavelengths are small compared to the size of the cosmic horizon:

eta*k > > 1

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28
Q

What do we consider “super-horizon modes”?

A

co-moving wavelength larger than the size of the cosmic horizon:

eta*k < < 1

(we can always use this limit if we go early enough in the universe)

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29
Q

What is the Meszaros effect?

A

Matter perturbations do not grow (much) during the radiation era. (sub-horizon modes)

[super-horizon modes do grow in the synchronous gauge]

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30
Q

Describe the key features of a graph of ln(matter density perturbation) against ln(eta).

A

For modes that enter the horizon during the radiation era:
-eta^2 growth until eta ~ 1/k (enter horizon) then no growth until matter era.
-eta^2 growth in matter era until vacuum era (no growth).

For modes that enter the horizon during the matter era:
-eta^2 growth until vacuum era.

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31
Q

What method do we use to try and solve for initial conditions?

A

Assume regular solutions as eta*k -> 0.
=> make power series solution in eta.
Equate coefficients of eta.

(radiation era in early universe)

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32
Q

What results do we get for the possible initial conditions of the universe?

A

There is 0 radiation density perturbation and radiation velocity perturbation at very early times.

Initial perturbations can be a linear superposition of matter or metric perturbations.

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33
Q

What are adiabatic initial conditions?

A

a.k.a: curvature perturbation, isentropic perturbation.

There is no matter perturbation, just perturbation in the metric (intially).

This leads to:
radiation perturbation = 4/3 * matter perturbation (super-horizon limit).

i.e: resulting perturbations in matter and radiation are consistent.

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34
Q

What are isocurvature initial conditions?

A

a.k.a: entropy perturbation.

There is no metric perturbation, just perturbation in matter density (initially).

This leads to:
radiation perturbation = 4/3 * matter perturbation + constant.

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35
Q

How do we know that hot dark matter cannot describe our universe?

A

*i.e: relativistic dark matter

Leads to a sharp cutoff in the transfer function at high k (due to diffusion). Therefore a universe with HDM would not develop small-scale structures - no galaxy formation.

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36
Q

What are the key aspects of the baryon species?

A

Negligible pressure compared with radiation pressure.

Before the recombination time radiation and baryons are tightly coupled: we can consider them basically as one fluid.

After decoupling baryons behave as CDM but with nonzero velocity perturbation (synchronous gauge allows us to choose only one stationary species).

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37
Q

How many of the photons emitted at the recombination time have not yet interacted with matter?

A

90%

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38
Q

What factors lead to the CMB distribution that we see?

A

Density fluctuations at the recombination time.

Velocity fluctuations at the recombination time (due to doppler effect).

Gravitational redshift effects (both due to the gravitational potential of the surface of last scattering, and that of the space the light passes through)

39
Q

How do we find the effect of density induced anisotropies on the CMB?

A

Use Stefan’s law: radiation density ~ T^4

=> temp anisotropy = 1/4 * radiation density perturbation

(temp anisotropy = temp perturbation / average temp)

40
Q

How do we find the effect of velocity induced anisotropies on the CMB?

A

Use the doppler shift equation, this gives:

check notes

41
Q

How do temperature anisotropies in the CMB due to density and velocity of radiation at the recombination time combine?

A

They both follow sinusoidal distributions but are maximally out of phase.

42
Q

What do we wish to convert temperature anisotropies at the recombination time to in order to compare with the observed CMB?

(“projecting onto the sky”)

A

Basically convert the two-point temp fluctuation correlation function to bessel function form.

We then want to plot these coefficients. This usually results in plots of l(l+1) C_l against log(l) {l~eta*k}.

43
Q

What is Silk damping?
What effect does it have on the structure of the CMB?

A

Due to the fact that matter-radiation decoupling is not instantaneous.
This leads to small perturbations (large k, l) being damped in the power spectrum.

This damping goes like exponential decay.

44
Q

What are the Sachs-Wolfe and Integrated Sachs-Wolfe effects?

Why must we consider the ISW effect?

A

*gravitational redshift
SW effect due to the difference in gravitational potential between the earth and the surface of last scattering.

ISW effect due to the gravitational potentials that CMB photons pass through to get to us. This is only relevant as the universe has been expanding during this time, if the universe was static there would be no ISW effect.

(the ISW effect is smaller anyway)

45
Q

What does the power spectrum {l(l+1) C_l against log(l)} look like?

A

Check notes

46
Q

What is the effect of baryons on the CMB power spectrum?

A

Baryons add inertia but ~ not pressure. ???spring analogy??? - changes the “zero point” of acoustic oscillations.

-Boosts the relative height of odd peaks.
-Also shifts distribution due to change in speed of sound.
-Also increases Silk damping as decoupling takes longer.

47
Q

How do we define (upside-down omega)?

A

Omega * H_0^2

(Omega is density / crit density)

48
Q

What is the sum of all relative densities (omega’s)?

A

1 (in order for the Friedmann equation to make sense)

49
Q

Which spherical harmonics correspond to azimuthally symmetric distributions?

A

Those with m=0

50
Q

What is the effect of the curvature of the universe on the CMB?

Explain the effect of Omega_k on the CMB power spectrum.

A

Will change the apparent horizon size at the time of last scattering.
-A hyperspherical (k=1) universe has smaller apparent angular CMB distributions so apparent horizon size is larger.
-A hyperbolic (k=-1) universe has larger apparent angular CMB distributions so apparent horizon size is smaller.

Omega_k ~ -k so increasing Omega_k (more hyperbolic) shifts peaks to higher l (smaller {actual} features).

51
Q

What is the effect of the cosmological constant on the CMB?

Explain the effect of Omega_Lambda on the CMB power spectrum.

A

The effect is degenerate with that of curvature of the universe. The apparent size of the sonic horizon is changed.

Omega_Lambda has the inverse effect to Omega_k:
-increased Omega_k = increased hubble rate = faster expansion = features appear larger = shift to smaller l.

52
Q

What is the effect of dark matter on the CMB power spectrum?

Explain what happens if Omega_m is larger.

A

Increasing Omega_m (the amount of dark matter) makes the matter-radiation equality time EARLIER.

As decoupling is approximately during the matter era, this makes eta_eq and eta_dec further apart.
This damps peaks in the power spectrum as acoustic oscillations are more damped in the matter era.

i.e: size of peaks decreases as Omega_m increases.

53
Q

How does changing the adiabatic initial conditions change the CMB power spectrum?

Explain changes to both the overall normalisation and the spectral index.

A

Shifting overall normalisation simply shifts peaks up/down.

Altering spectral index changes the “tilt” of the spectrum.

54
Q

What are two motivations for inflation?

A

The horizon and flatness problems.

55
Q

What is the horizon problem?

How does inflation solve this problem?

A

The CMB is extremely uniform in all directions (isotropic), including regions that should not be causally connected.

Inserting a period of accelerated expansion before the decoupling time can result in arbitrarily large areas having been in causal contact previously.

56
Q

In the horizon problem, what is the angular size of spaces that should have been causally connected? (without inflation)

A

decoupling time / time from decoupling to now

~ eta_dec / eta_0
~ only 1.8 degrees

(in conformal time)

57
Q

How much expansion do we need during inflation to solve the horizon problem?
(assuming exponential inflation a ~ e^{Ht})

A

The universe must expand during A period of inflation by a factor at least as much as it has expanded since inflation.

*can express this in terms of a fraction of a’s.

Maybe 60 e-foldings.

58
Q

How is the number of e-foldings, N, defined?

How can we estimate the number of e-foldings since the hot big bang?

A

N = log(a(t_f) / a(t_i))

-for some reason* the fraction of a’s is equivalent to an inverse fraction of temps, so log( 10^15 GeV / 10^-3 eV )

*combination of radiation era density (a-dependence) and stefan’s law

59
Q

What is the flatness problem?

How does inflation solve this?

A

The universe appears spatially flat on large scales.
This is a fine-tuning problem as curvature=0 is unstable if (1+3w)>0 {which it has been since big bang}.

A period of time with (1+3w)<0 {inflation} solves this problem by driving curvature towards zero.

60
Q

What do we need from inflation in order to solve the flatness problem?

A

w<-1/3
However, w<-1 (negative pressure) solves both the horizon and flatness problems.

i,e: a ~ e^Ht during inflation.

61
Q

Why do we neglect the laplacian(phi) term in the EOM of the inflaton field?

A

Spatially homogenous universe.

62
Q

How do we get w=-1 for the inflaton field?

A

Require slow-roll conditions.

63
Q

Why do we need slow-roll conditions for the inflaton field?

A

Slow-roll conditions give negative pressure (w=-1) for the inflaton field which is what we need to solve the horizon and flatness problems.

64
Q

What are the slow-roll equations?

A

H^2 = ( 8piG / 3 ) * V
(freidmann eq. with slow-roll cond.)

3H phi(dot) = -dV/dphi
(K-G equation with slow-roll cond.)

65
Q

What do we require of the slow-roll parameters for slow-roll conditions to be met?

A

both epsilon and eta «1.

66
Q

How can N be calculated for a particular potential of the inflaton field?

A

-check notes

67
Q

How did inflation end?

A

Inflation “automatically” stops when the slow-roll conditions no longer apply (accelerated expansion becomes sufficiently fast).

The inflaton field reaches a minima, and potential energy has been converted to KE of inflaton particles.
These decay (oscillation is damped) to other particles, causing the hot big bang, known as “reheating”.

68
Q

What are the initial conditions that we want inflation to provide us with?

A

Adiabatic (only curvature perturbation)
Nearly scale-invariant (spectral density ~ 1)
Gaussian (perturbation fourier coefficients drawn from gaussian distribution)

Ideally also give us overall normalisation for the power spectrum.

69
Q

What is the idea behind k modes of interest during inflation?

A

We only care about modes of wavlength comparable to the co-moving hubble distance or ~*1000 less.
These are the only perturbations resolvable in our observable universe.

70
Q

What is the dependence of the co-moving horizon size on a during and after inflation?

(basically explain that graph)

A

aH ~ 1/a in radiation era (after inflation).

a ~ e^Ht during inflation, so H ~ constant in a. => aH ~ a

Hubble sphere decreases during accelerated expansion, and increases during deceleration. (it is now decreasing again)

71
Q

What is the significance of the region marked * in the graph of co-moving hubble distance?

A

Showing that modes of interest are super-horizon during the reheating period, and only exit the horizon in the modern universe.

It turns out that fluctuations leading to these modes FREEZE in magnitude when super-horizon, so these perturbations due to quantum fluctuations in the inflaton field are independent of the reheating process.
(The curvature perturbation R is conserved during this time)

72
Q

How is the curvature perturbation defined?

A

Check notes

73
Q

How can the time period marked * be characterised?

A

1/k > > 1/(aH)

i.e: these modes are super-horizon during *

74
Q

How do we find initial conditions from quantum fluctuations of the inflaton field?

A

Using the curvature perturbation, which is conserved during time period *.
Calculate curvature perturbation due to inflaton field quantum fluctuations.
Equate this to the form of the curvature perturbation in the radiation era.

75
Q

How do quantum fluctuations in the inflaton field lead to adiabatic initial conditions?

A

Fluctuations in the scalar field lead to inflation ending at different times in different places.
This leads to space being stretched more in some places than in others.
This is equivalent to adiabatic perturbations.

76
Q

What is the hubble sphere?

A

The comoving size of the observable universe. The sphere beyond which objects moving with the Hubble flow recede superluminally.

77
Q

What can be said about the two-point correlation function of a gaussian random field?

A

The two-point correlation function contains all correlation information about the field.

78
Q

Why do the initial conditions correspond to a gaussian random field in curvature?

A

These perturbations are generated by quantum fluctuations in the inflaton field.
These are gaussian, as they arise from oscillation around the ground state potential.
The expectation value distribution for these oscillations {for SHM potential at the minima} has gaussian amplitude.

79
Q

How many MPc is 1 billion light years?

A

300

80
Q

How do you make a taylor expansion?

A

Check wall note

81
Q

When does recombination happen?

A

Just inside the matter era

82
Q

How do super-horizon matter perturbations behave in the radiation era?

What about sub-horizon modes?

A

They grow with eta^2

Sub-horizon modes do not grow in the radiation era (Mezaros effect).

83
Q

How do sub-horizon matter perturbations behave in the matter era?

What about super-horizon?

A

In the matter era all modes grow as eta^2.

84
Q

How do matter perturbations behave in the Cosmic Constant domination era?

A

Matter perturbations freeze.

85
Q

What is the Synchronous Gauge in early universe physics?

A

We have gauge freedom in defining the metric perturbation h_mu_nu, due to the choice of space-time coordinates.
In synchronous gauge we define h_mu_0 = 0, which gives the simplest form of the christoffel symbols.

In synchronous gauge we can set the matter velocity perturbation to zero (i.e: matter particles fix the coordinate system).

86
Q

Describe the angular power spectrum of the cosmic microwave background.

A

-axis labels.
-indicative l values, what is l?
-4 main features and physics responsible

87
Q

What are some general results from relativistic perturbation theory about the initial conditions for the universe?

A

ZERO rad. and rad. velocity perturbation at very early times.

Initial perturbations can therefore only be some superposition of matter and metric perturbations.

88
Q

What is the relation between the scalefactor and redshift?

A

1/a = 1+z

89
Q

Radiation era:
a with time
a with conformal time

A

a ~ t^1/2

a = sqrt(little_omega_r) * eta

90
Q

Matter era:
a with time
a with conformal time

A

a ~ t^2/3

a = 1/4 * little_omega_m * eta^2

91
Q

Density dependence on a of:
Radiation
Matter
“K”
Lambda

A

Rad: a^-4

Matter: a^-3

“K”: a^-2 (from freidmann)

Lambda: const. (w = -1)

92
Q

How does a evolve during inflation?
Why?

A

To solve the horizon and flatness problems, we need w = -1.

This gives constant H in the freidmannn equation. Considering the dependence of H on a:

a ~ e^Ht

93
Q

What are the slow roll conditions?

A

-check notes

94
Q

What order do the things happen in since the big bang?

A

Inflation ends, EW phase transition, Q-H transition, Nucleosynthesis, Matter-radiation equality, Recombination

EXAMS Summer 2023 (6 decks)

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