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Inflation Note
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Inflation Note
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PHYS 1200 Report
Brief review on Inflation
Reading Notes
Copyright © 2023 Wang Liang
TOR PROJECT, UNIVERSITY OF NORTH SOUTH WALES
This project was done under the supervision of Professor Yvonne Y.Y. Wong (New South Wales U.)
for 10 weeks, from Sep 11th to Dec 21th of 2023.
Contents
1 A Glimp at Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1 Universe is expanding 5
1.2 Why universe is expanding? 5
1.2.1 What is the special obeserver? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.2 Why the observer prefer to use FLRW metric? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.3 Why is the universe described above expanding? . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 How does the universe expand exactly? 8
1.3.1 The χ −τ Diagram and Horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3.2 Two Problems within the classical universe expansion . . . . . . . . . . . . . . . . . . . . . . 8
1.3.3 Inflation as Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3.4 Are the two solutions mentioned above consistent? . . . . . . . . . . . . . . . . . . . . . 10
1.3.5 Detailed explaination on two kinds of horizon . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4 What is Inflation? 13
1.4.1 What is the physical requirements for Inflation? . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.5 What dynamic does this sclaer field has? 16
1.5.1 Obtaining the equation of motion for φ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.5.2 Constraints on V (φ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.5.3 Inflation condition equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.5.4 What can V (φ) look like? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2 Quantize Inflaton Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1 Quantum Properties of Scalar Field 21
2.1.1 Quantize perturbation term for the Inflation field . . . . . . . . . . . . . . . . . . . . . . . . 22
4
2.2 Can we directly observe this quantity? 24
2.2.1 What happens in the superhorizon regime? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2.2 Power spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3 Gravitational Waves: 28
3 Perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.1 How perturbation evolve with universe expansion? (Newtonian Version) 31
3.2 How perturbation evolve with universe expansion? (Relativistic Version) 33
3.2.1 Perturbed spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2.2 Gauge Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2.3 Gauge Fixing: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2.4 Perturbed Matter: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2.5 Comoving Curvature Perturbation: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3 Theoretically discuss Perturbation’s evolution 40
3.3.1 Gravitational potential evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.3.2 density contrast evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.4 Evolution of Perturbations in different scenerio 41
3.4.1 Gravitational Potential Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.4.2 Evolution of density contrast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4 Observation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.1 Observable Quantities of CMB: 45
4.1.1 Processing the observing data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.1.2 Correlation Function in CMB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.2 Polarization 48
4.2.1 Quantitative Description of Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2.2 E and B Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2.3 Cross-correlation Observable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.3 Large-Scale Structure (LSS) Observables 49
4.4 Coherent Phase 50
4.4.1 Lyth Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.1 Energy-Momentum Tensor 53
5.2 Thoughts about quantization and the relation to classic 54
5.2.1 Canonical Quantization and Creation and Annihilation Operators . . . . . . . . . . 54
5.3 From metric to SVT 57
5.3.1 Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.3.2 SVT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
1. A Glimp at Inflation
1.1 Universe is expanding
I can’t imagine what the universe would be like if it didn’t expand. Olbers’ paradox tells us that
if the universe were static, night time would be as hot as the day time. At the same time, Hubble’s
observations also tell us that the universe is expanding.
The fact that theIt’s difficult to imagine what the universe would be like if it didn’t expand.
Olbers’ paradox tells us that if the universe were static, the night sky would be as bright as the day,
due to an infinite number of stars in every direction. However, observations made by Hubble and
other scientists have confirmed that the universe is indeed expanding.
The expansion of the universe is now an accepted fact, much like the acceptance of the heliocen-
tric model in the 18th century was a natural progression of understanding.
1.2 Why universe is expanding?
To understand why universe is expanding, let’s first consider what we can observe in the
universe on a cosmological scale. The universe appears to be isotropic (the same in all directions)
and homogeneous (uniform) on large scales. From this observation, we can deduce the following
fundamental principles of cosmology:
Theorem 1.2.1 — Fundamental principles of cosmology. The spatial distribution of matter in
the universe is equally distributed and isotropic when viewed on a large enough scale.
However, this property only holds for a specific class of observers, and these observers can be
described using the FLRW metric to represent the structure of spacetime. Through this specific
spacetime structure, we can derive an expanding universe.
1.2.1 What is the special obeserver?
If we consider an observer, the observational properties mentioned above may not hold. One
can imagine that such an observer would see approaching stellars and planets, leading to a loss of
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