The Inflationary Universe

# The Inflationary Universe

## Abstract #

The initial conditions defined in the Standard Model present two problems:

1. The early universe is defined to be homogeneous despite the massive distances between regions (causal disconnect)
2. The Hubble constant must be set very finely to produce a flat universe (like ours)

These problems could disappear if the universe (in its early stages) cooled to temperatures twenty-eight or more orders of magnitude below the critical temperature for “some phase transition”.

## I #

The Standard Model’s initial conditions have a singularity at $$t = 0$$ as $$t \rightarrow 0$$, the temperature, $$T$$, approaches infinity. Because of this requirement, no initial value problem can be defined here.

But! Very close to that time, when $$T$$ is near the order of the Planck mass:

\begin{equation} (M_{P} \equiv \frac{1}{\sqrt{G}} = 1.22 \times 10^{19} GeV)\tag{1} \end{equation}

or greater, quantum gravitational effects become dominant and the Standard Model’s equations less reliable.

If we set $$T_0$$ below $$10^{19} GeV$$ (e.g. $$10^{17} GeV$$) we can describe the scenario using the ordinary equations of motion. As noted above, quantum effects only come into dominance at $$10^{19} GeV$$ or higher.

What Guth is doing here is setting a window in which to study the universe’s evolution after the $$T \rightarrow \infty$$ constraint, but before quantum effects become dominant; i.e. while $$T_0$$ < $$M_P$$ and $$t > 0$$

In the Standard Model the initial universe is assumed to be isotropic, homogeneous, and filled with a (effectively) massless gas in thermal equilibrium at $$T_0$$. The Hubble constant’s initial value ($$H_0$$) is defined and the universe completely described in its initial conditions.

### The puzzles from the above assumptions #

#### Horizon Problem #

The universe in these conditions (above) is defined to be homogeneous, but there are $$~10^{83}$$ regions which are “causally disconnected”, which is defined as regions which are so far apart that they have not “had enough time” to communicate with each other via light signals. The result of this implication is that one must be forced to conclude that the creation forces which these regions were born from “violate the laws of causality”.

I think “violate the laws of causality” means faster-than-light (FTL) information

#### Flatness Problem #

“It is known” that the universe’s current energy density ($$\rho$$) is approaching the critical density $$\rho_{cr}$$

When $$\Omega$$ (the density parameter) equals unity (i.e. $$\rho$$ and $$\rho_{cr}$$ are equal) the geometry of the universe is flat (Euclidean). When $$\Omega$$ is greater than unity, the universe is closed. If $$\Omega$$ is less than unity, the universe is open. (See: Shape of the Universe)

With this in mind, we see that above Guth is stating the current geometry of the universe is essentially flat.

With the assumption above we can assume:

\begin{equation} .01 < \Omega_{p} < 1.0\tag{1.1} \end{equation}

where

\begin{equation} \Omega \equiv \frac{\rho}{\rho_{cr}} = (\frac{8\pi}{3})\frac{G \rho}{H^{2}}\tag{1.2} \end{equation}

where the subscript $$p$$ denotes the value at the “present time”. By these constraints we see that $$\Omega \approx 1$$ is unstable. (WHY#1)

The only time referenced in the equations for the radiation-dominated universe described by the Standard Model is the Planck time ($$\frac{1}{M_{p}} = 5.4 \times 10^{-44} seconds$$). Any closed universe will have reached its maximum size by this point, and any open universe will will approach a value of $$\rho$$ much less than $$\rho_{cr}$$ (i.e. $$\Omega « 1$$).

The only way (by the Standard Model’s initial conditions) that a universe could survive as long as ours has ($$\sim 10^{10} years$$), is if $$H$$ and $$\rho$$ are incredibly fined tuned.

By “fine tuned” Guth means one part in $$10^{55}$$, this value is assumed with little explanation in the Standard Model.

Concluding this section, Guth states that the above numbers are not simply a result of his $$T_0$$. He goes on to show that a more modest selection of a $$T_0$$ would still result in massive numbers (albeit, smaller than the above $$10^{55}$$ as an example) and assumed/skepticism-inducing fine tuning.

His following sections will touch on the equations of the Standard Model (II), a description of the Inflationary Universe (III), a discussion of this in the context of the grand unified models (IV), and comments on monopole supression in (V).

His claim is that the above two puzzles can be explained away by a scenario for the behavior of the universe at temperatures well below $$M_{P}$$.