Theoretical Topics in Particle Cosmology

Teacher: 鲜于中之

采用自然单位制 $c=\hslash =k_B=1$

其中 $1\mathrm{GeV}\approx {10}^{-24}\mathrm{g}\approx ({10}^{-24}\mathrm{s}{)}^{-1}\approx ({10}^{-16}\mathrm{m}{)}^{-1}$

采用度规 $(-1,1,1,1)$

常用 $M_{pl}=\sqrt{8\pi G_N}\approx 2.4\times {10}^{18}\mathrm{GeV}$

Motivation

Understanding Gravity

As a logical consequence of Lorentz Invariance, of Quantum Theory, and of Spin-2 massless particle.

Understanding QFT

As a parametization of the lack of constraining power of “L-inv. + locality + unitarity”.

Comstructing GR (classical tests)

From massless spin-2 particles.

A Heuristic Introduction to Gravity

Newton’s Law

$$\varphi =-\frac{Gm}{r}$$

Coulomb’s Law

$$\varphi =+\frac{eq}{r}$$

Newtonian Gravity is peculiar in 3 ways:

1. Force at a distance (nonlocal)
2. Strengths controlled by the mass
3. $[G]=-2$, means very high energy scale (weak)

Gravity is the only irrelevant fundamental interaction (to our knowledge).

In fact, gravity id so irrelevant that it is the only relevant int. at large distance.

Introduction to Fundamental Interactions

$$\mathrm{\Lambda }\approx 10\mathrm{TeV}$$$$\mathcal{L}={\mathcal{L}}_{free}+{\mathcal{L}}_{int}{\mathcal{L}}_{int}\ll {\mathcal{L}}_{free}$$$$\begin{array}{c}{\mathcal{L}}_{free}={\mathcal{L}}_{gluon}+{\mathcal{L}}_{EW}+{\mathcal{L}}_{gravitation}\\ +{\mathcal{L}}_{quark}+{\mathcal{L}}_{lepton}+{\mathcal{L}}_{Higgs}\end{array}$$$${\mathcal{L}}_{int}={\mathcal{L}}_{strong}+{\mathcal{L}}_{EW}+{\mathcal{L}}_{gravity}+{\mathcal{L}}_{Higgs}+{\mathcal{L}}_{Yukawa}$$

Move $\mathrm{\Lambda }$ downward, at some scale, dramatic things happen.

At $\mathrm{\Lambda }$~ $O(100\mathrm{GeV})$: EWSB, $⟨h⟩\approx 246\mathrm{GeV}$

W and Z becomes massive, while photon still massless

$$m_W\approx 80\mathrm{GeV}{m}_Z\approx 91\mathrm{GeV}$$

Below $m_W$, weak interation -> 4-Fermi Theory

At $\mathrm{\Lambda }$ ~ $O(1\mathrm{GeV})$: QCD confinement

gluons, quark -> hadrons (baryons & mesons)

proton $m_p\approx 938\mathrm{MeV}$

2 peculiar things: baryon (lepton) number conserved, baryon and anti-baryon asymmetry.

Our universe is in a N-baryon state ($N\approx {10}^{80}$), we are not in a QFT vacuum.

At $\mathrm{\Lambda }\le 1\mathrm{GeV}$: proton, neutron, meson, photon, graviton

Chirel Lagrangian (Yukawa, Weinberg)

At $\mathrm{\Lambda }\le \mathrm{MeV}$: Yukawa int. Becomes IR Div.

Bound States: Nuclei (Nucleosynthesis)

Nuclei $e^{\pm },\nu$, photon, graviton

When $E\le \mathrm{MeV},E\le m_{Nuclei}$, so we use Non-Rel. physics for nuclei

At $\mathrm{\Lambda }<0\mathrm{eV}$, QED Becomes IR div.

Bound States: Chemical Element…

Introduction to QFT (2-Step)

1. Force at a distance is absurd:

   consequence of particle propagation

   all interactions are local

2. Local interactions are absurd:

   Whereas we refer to a “local int.”, we talk with a cut-off scale.

3. QFT: cut-off scale $\mathrm{\Lambda }$

   all int. within the scale $L$ are treated as local

We’ll import Lorentz inv. as a constraint in QFT parametization, it is incredibly powerful for massless spin-2:

• GR is almost unique
• we can bypass most of QFT treatment

Plan

Massless Spin-2

Free Theory (L-inv. -> gauge)

+int. (Scattering Theory)

QFT

Doing with classical tests (leader )