The Inflationary Universe

INTRODUCTION

Some of the mysteries mentioned last time can be explained by what is known as the Inflation Theory. To get a handle on the inflation recall how the nature of the four forces of nature changes as the Universe evolves:

At earlier and earlier times, the way matter, radiation, and the Universe interact gets simpler. The events when the forces of the Universe get more complicated are referred to as

Symmetry Breaking

The symmetry breakings are analogous to phase transitions (e.g., liquid water ---> solid water (ice), liquid water ---> vapor, etc.). The transitions occur because lower energy states become available.

• time before 10**(-43) sec, T > 10**32 Kelvin
  Planck era and all forces are unified. This era is mysterious (to say the least).
• Planck era ---> ~ 10**(-35) - 10**(-32) seconds (or something, depends on the model)
  GUTs era and when the symmetry breaking for the strong and electro-weak forces occurred. This is the era when inflation occurred.

  Before I offer a possible cause for the inflation, let me describe in general terms the inflationary model and some of the consequences of the inflationary model. To see what happens, recall:

  -k=1/2[dR(t)/dt]**2 - (4 pi G/3) rho(t) R(t)**2 - (CC/3) R(t)**2

  The inflation is driven by the Cosmological constant term. The thing is that because of the nature of the Cosmological constant (vacuum energy) term, the push does not weaken as the Universe expands. This is different than say for a balloon which expands because it contains hot gas. As the balloon expands, the gas cools and the expansion slows down. For inflation, roughly

  dR(t)/dt ~ sqrt[2 CC/3] R(t) ,

  the slowing down due to gravity is negligible and k is small. If we solve this equation, we find

  R(t) ~ exp[(2 CC/3)t]

  This is fast. How fast? We'll answer this later. Schematically, the scale factor and the temperature evolve as:

  

  The inflation causes the Universe to increase in size greatly and to cool rapidly. At the end of the rapid expansion, the Universe re-heats and then continues its normal evolution. The inflation does not affect the long-term evolution of the Universe; it is needed only to cause a rapid blow-up in size of the Universe at an early time. What are some consequences of this blow-up?

CONSEQUENCES OF INFLATION

For typical models of the Cosmological constant term, the Universe will roughly double in size every ~ 10**(-34) seconds [the exact number depends on the precise model one uses to drive the expansion]. This is a phenomenal rate, e.g.,

• t = 0 sec ---> 10**(-24) cm
• t = 10**(-34) sec ---> 2 x 10**(-24) cm
• t = 2 x 10**(-34) sec ---> 4 x 10**(-24) cm
• t = 3 x 10**(-34) sec ---> 9 x 10**(-24) cm
• ...
• t = 100 x 10**(-34) sec ---> 2**100 x 10**(-24) cm ~ 13 km!!

In the example, the thing grows by a factor of > 10**30 every 10**(-32) seconds.

• Horizon Problem

  Suppose that we start with a large universe and consider a small causally connected chunk say of size 10**(-23) cm. Suppose the Universe now inflates, and that the inflation lasts for say 100 e-fold times, i.e., R increases by exp(100). The causally connected part of the Universe increases to size 3 x 10**20 cm and, interestingly, the part of this bigger patch which will eventually encompass our Universe is now only ~ 30 cm in size. That is, our Universe is embedded in this huge region which was causally connected before inflation,

  

  This explains the horizon problem.

• Flatness Problem

  The Earth is spherical, however, for us small (short people) living on the surface of the Earth it appears flat. It appears flat because we are not tall enough to see over the edge of the Earth. Technically speaking, our height is small compared to the radius of curvature of the Earth. In any event, this means that we perceive a spherical object as being flat. Due to inflation, we have that

The random blow-up of the Universe makes it appear flat to us (and to thus have Omega = 1, identically). This resolves the flatness problem.

A MODEL FOR INFLATION

• As the early Universe expands, it cools and it enters the GUTs era. It evolves smoothly until it reaches the symmetry breaking point where the nuclear (strong) and electro-weak forces want to part ways. However, the transition does not occur immediately.

  Just as water can remain in a supercooled phase where it remains in its liquid phase below 0 degrees Celsius, the Universe may remain in its symmetric phase without breaking.

• The Universe thus gets trapped in this high energy state. The excess energy (above the true vacuum) drives the inflation. The inflation continues until the symmetry breaking occurs. When the tansition finally happens, the pent-up energy is released which causes the re-heating of the Universe.