Andrei Linde and Inflation

In the last post, I talked about the Penrose-Hawking singularity theorem and the conditions under which the theorem can be violated. In this post, I want to talk about a well-known proposal for a model that does just that, that being the paper written by Andrei Linde in 1983, Chaotic inflation. Inflationary models are one of broadly two majorly popular avenues to avoid the implications of early singularity theorems.

Though Linde wasn't the original architect of inflation his paper argues that inflation will occur under a variety of early universe scenarios, rather than being a peculiar phenomenon that's restricted to only a few class of models (like those of earlier inflationary models, involving supergravity). By "inflation" we mean the condition that $\ddot{a}\left ( t \right )> 0$ for a period between $t_{i}$ and $t_{R}$ the "reheating" phase (the only constraint of which is that it must occur before nuclyosenthesis).

Linde constructs a model in which the vacuum potential of a scalar field is $V\left ( \varphi \right ) = \frac{1}{4} \lambda \varphi ^{4}$ where lambda is less than or approximately equal to 0.1. Before the Planck time a universe like this is assumed to exist in a chaotic state. So the paper asks us to consider just a small homogeneous patch of that universe which expands with a scale factor $a\left ( t \right ) = a_{0} \times e^{\left ( Ht \right )}$ where H is Hubble’s constant and the equation of motion for the field is the Klein-Gordon equation.

It takes the same form as an equation for a harmonic oscillator. What it tells us is if you start with a large scalar field then H will be large and the scalar field potential reduces very slowly. Where $H = \frac{m^{2}}{2}\phi ^{2}$ Then the Hubble constant is nearly constant. 

This means that the typical time it takes for the field to decrease considerably will be much greater than the Planck time $\Delta t \sim \left ( 6\pi \right ) ^{\frac{1}{2}}/\sqrt{\lambda }M_{p}$ and it follows that if we have a large scalar field (with energies at least three times the Planck mass), with a function of positive vacuum energy density it will produce a de Sitter space that looks homogeneous and flat on large scales, while decreasing the density of topological defects. 

For the initial value of the inflaton, Linde takes the field value to be $\sim M_{p} /\left ( \lambda \right )^{1/4}$ therefore, you get inflation (and he assumes also the potential for life) even in a chaotic initial state, for regions of the field that have sufficiently high energies. Whereas under Linde's earlier "new inflation" scenario, life can only arise in regions where the vacuum energies are much lower. In this Chaotic inflationary model two aspects of the field work against each other, if the decay time $\Delta t$ is less the time it takes for the scale factor $a\left (t\right)$ to double, then inflation never ends.

Modern versions of the chaotic inflation scenario include a "stochastic noise" term in the equation of motion for the field (see diagram), to account for the role of quantum fluctuations. Which for large values of the field are more important than any classical relaxation term $V’\left ( \phi \right )$ this means that chaotic inflation also leads to eternal inflation.

The rest of the paper consists of an argument that inflation most likely occurred during the GUT era of the Big Bang. But this scenario proposed by Linde already violates the strong energy condition and avoids the Penrose-Hawking theorem, which assumes $\rho c^{2} + 3p > 0$ and which equates to the assumption that gravity is always an attractive force. 

According to inflationary models density is positive and a constant $E\left ( t \right ) = \rho V \left ( t \right )$ so pressure must be negative, to be consistent with the second Friedman equation. The equation of state for inflation is $p = -\rho$ which appears to be enough to avoid the initial singularity.

The major lines of support for inflation were fairly quickly realised. In the original 1979 paper, Guth proposed old inflation as a solution to the horizon and flatness problems of the Big Bang model, though I don’t find these arguments entirely convincing. As presented the problems for the standard Big Bang model were as follows.

The horizon problem arises because everywhere across the cosmic microwave background cosmologists observe the uniform temperature of ~ 2.7K with a relative accuracy of about 0.001 per cent. As we look back into the early universe, there wasn’t enough time under the standard Big Bang model for distant regions to come into causal contact. Since the Penrose diagram for spacetime between regions lies beyond 45 degrees. 

But under inflation, this is no longer a problem, it allows you start the universe off at much smaller size than the standard model, then if there was an exponential expansion different regions which were close enough together in the early universe, to reach thermal equilibrium can now expand far enough away from each other to escape what would've been their past light cones. In effect, it "stretches" out the Penrose diagram and creates enough room for distant regions to exchange heat. 

The flatness problem is also alleged to have been solved by inflation, it arises because observations show that on large scales the universe appears to be perfectly flat and according to the standard model, following its evolution back in time using the Friedman equations the universe starts off very close to flatness, and it should deviate from flatness with time. So the density of matter (which determines local geometry), has to be $\left | \Omega - 1 \right |< 0.01$ this might seem unnatural because a priori the density would depend on the region. But during inflation $\left | \Omega - 1 \right |$ decreases with time, along with our coomoving horizon and this is sufficient to reverse the trend and create a large enough universe that is almost perfectly flat. 

In 1983 it was also shown by Linde that inflation can solve other problems, too. The number of topological defects is much lower than one might expect from a grand unified theory, but the density of objects like magnetic monopoles and cosmic strings in the observable universe are diluted and decrease substantially with the exponential expansion. 

Inflation is a powerful idea but I am sceptical of this particular model of inflation, (as well as inflation itself but that’s for a future post) a lot of the appeal comes from combining it with string theory but supergravity tends to make the potential for inflation very steep, for values of the field that are large. The most natural solution (e.g., something called "hybrid inflation") involves adding more structure like a second scalar field. 

A much more difficult problem for Linde’s model is that (at least the 1983 paper) is in conflict with COBE and other satellite data. Chaotic inflation tends to predict anisotropies orders of magnitude larger than what are observed, the only way to resolve this is to assume lambda starts off at $\lambda = 10^{-12}$ or smaller. There’s no justification for this on the model. 

Moreover a large number of inflationary models are consistent with the data. So the major draw back of Linde 1983 is that it’s difficult to imagine a way in which a model like chaotic inflation could be testable. In fact, it's worse than that. Linde relies on the weak anthropic principle which makes even a statistical calculation of what is most probable in our sector of the multiverse virtually impossible (given that we admittedly live in a very improbable region). 

Still, the proposal shows that at least in principle there is a reasonable way to avoid Penrose-Hawking singularities in classical gravity with semi-classical Coleman-De Luccia tunneling. There’s no need to quantise the gravitational field to restore an eternal universe. In another post however, I’ll discuss whether inflation by itself can really be eternal.