# Cosmology Lecture 22

### Motivating Inflation

The Big Bang model of the universe has 3 pillars supporting it:

• It predicts the relative abundances of ${\displaystyle He,H,\dots }$.
• It explains the Hubble expansion.
• It predicted the CMB.

However, there are 4 problems with the original Big Bang model:

• The Horizon/Homogeneity Problem * Why is the universe so uniform on large scales? * The CMB temperature is uniform to a part in ${\displaystyle 10^{5}}$, implying that large areas of the universe should have been in causal contact with one another. * However the “acoustic horizon” at photon decoupling subtends only ${\displaystyle \sim 1^{\circ }}$ on the sky. Note that the CMB angular spectrum depends on ${\displaystyle \Omega _{0}}$, but so does the acoustic horizon size. These effects cancel, making the above statement true for all ${\displaystyle \Omega _{0}}$.
• The Flatness Problem * Why is ${\displaystyle \Omega _{0}}$ so close to 1? * In PS#1 we showed that ${\displaystyle {1-\Omega (a) \over \Omega (a)}={1-\Omega _{0} \over \Omega _{0}}\cdot a^{1+3w}}$. Thus at ${\displaystyle a=10^{-3}}$, when recombination occurred, for a matter-dominated universe:
${\displaystyle \Omega ={\begin{cases}0.9973\to \Omega _{0}=0.3\\1.0\to \Omega _{0}=1.0\\1.00067\to \Omega _{0}=3.0\end{cases}}\,\!}$
• Even worse, at Planck time, ${\displaystyle a_{planck}={T_{0} \over T_{planck}}={{1 \over 4000}eV \over 10^{19}GeV}\sim 10^{-31}}$. Scaling backward through matter-domination to equality time, and then on back through radiation-domination, we get that to get ${\displaystyle \Omega _{0,M}=0.3}$, we need ${\displaystyle \Omega (a_{planck}\sim 1.0-5\cdot 10^{-60}}$. * Why so close to 1.0, but not 1.0? This is level of extreme fine tuning is a little hard to swallow.
• The Monopole Problem * This is the problem which originally motivated Guth to invent inflation. Most GUTs (Grand Unified Theories) predict the possible creation of monopoles. * The monopole mass ${\displaystyle \sim {E_{GUT} \over \alpha }\sim 10^{16}GeV}$ where ${\displaystyle E_{GUT}}$ is the symmetry-breaking energy, and ${\displaystyle \alpha }$ is the coupling constant. * The monopole # density, in terms of the correlation length ${\displaystyle \xi }$ is ${\displaystyle {1 \over \xi ^{3}}\sim 10^{-11}n_{\gamma }}$. Thus, the Energy density of monopoles should be ${\displaystyle m_{mono}c^{2}n_{mono}\sim 8\cdot 10^{-17}{g \over cm^{3}}\gg \rho _{crit}}$. This means the universe should have collapsed long ago.
• The Structure Formation Problem: Where is ${\displaystyle \delta \rho \over \rho }$ and ${\displaystyle \Delta T \over T}$ from? What is the origin of these fluctuations?

Inflation solves all of these problems.