World Science Scholars
1.2 Inflation Review
summary
The inflationary theory is one of the biggest paradigm shifts since the Big Bang.

• Until the end of the 1970s, the standard Big Bang model was thought to be sufficient and many scientists did not see a need to look beyond it.
• Then, some problems emerged in the connections between cosmology and particle physics.
• For instance, early models of the events after the Big Bang predicted an enormous density of magnetic monopoles—a hypothetical particle whose presence was expected by many Grand Unified Theories. Yet, there is no conclusive experimental evidence to say that magnetic monopoles exist at all.
• This forced physicists to look at questions that were previously thought to be almost metaphysical:
• What was before the Big Bang?
• Why is our universe so homogeneous?
• Why is it not exactly homogeneous?
• Why is it isotropic?
• Why did all of its parts begin expanding simultaneously?
• Why is it flat? (Ω = 1)
• Why is it so large?

The only one of these questions that hasn’t been answered by the inflationary model is the first. However, inflation lets you look at this question in a totally different way and suggests some tentative answers.

Consider the energy required in the standard Big Bang theory.

• According to the Big Bang theory, the total mass of matter shortly after the Big Bang was greater than 1085 grams.
• To create our universe, we would require an enormous amount of “high-tech explosives” compressed to a size of less than 1 cm, and then exploding essentially simultaneously, within an accuracy of 10−43 seconds.
• Before the Big Bang, there was apparently nothing, and then suddenly a huge mass appeared. Where did it come from and how did this mechanism happen?
• The inflationary theory solves many problems in the old Big Bang theory, and can explain how the whole universe could be created from less than a milligram of matter while preserving conservation of energy.

Inflationary models have evolved to fix known problems and answer new questions.

• In 1980, Alexei Starobinsky suggested a theory that led to an exponential expansion of the universe, by using a rather complicated version of Einstein’s gravity with quantum corrections. It assumed that the universe started in de Sitter space, with fixed curvature and no singularity, so the history of the universe could be traced back in time infinitely—but the solution was unstable, giving the universe a finite lifetime, meaning the universe could not start at time $t=-\infty$.
• Alan Guth developed a theory of inflation independently, wherein a scalar field would be sitting in a false vacuum and could tunnel to a lower energy state through a process known as bubble nucleation, where bubbles of “true” vacuum could be produced, rapidly expanding at the speed of light.
• Guth was able to use his inflationary theory as a solution to many of these previously unsolved questions. However, Guth’s model was still somewhat problematic—for instance, the homogeneous and isotropic universe we see today could not be preserved in the tunneling process.
• This led Andrei Linde (and, independently, Andreas Albrecht and Paul Steinhardt) to develop the theory of “new” inflation, in which the scalar field instead rolls down a gentle slope rather than tunneling out of the false vacuum phase.
• In 1983, Linde developed his theory further, where it became known as chaotic inflation.

Chaotic inflation results in an eternal inflationary epoch.
• We can look at the potential energy of a scalar field in a very simple form:$$V\left(\varphi\right)=\frac{m^2}{2}\varphi^2$$where $V$ represents the potential energy and $\varphi$ represents the scalar field. The energy depends quadratically on the scalar field’s distance from equilibrium.
• When the scalar field is small, the energy oscillates like a normal pendulum. But when the scalar field is large, the equations show a large “frictional force” is created, damping the field’s motion toward its minimum. The potential energy falls very, very slowly—practically remaining constant. Thus, the inflationary phase lasts forever, at least in some regions of the universe. Chaotic inflation is sometimes referred to as eternal inflation.
• To examine the mathematics a bit further, we can look at simplified versions of the Einstein equation and the Klein-Gordon equation:$$H^2=\left(\frac{\dot{a}}{a}\right)^2=\frac{m^2}{6}\varphi^2\tag{Einstein equation}$$ $$\ddot{\varphi}+3H\dot{\varphi}=-m^2\varphi\tag{Klein-Gordon equation}$$where $H$ is the Hubble constant. If we compare the latter to the equation for a damped harmonic oscillator,$$\ddot{x}+\alpha\dot{x}=-kx$$we can see that the two equations are quite similar.
• In the Klein-Gordon equation, the $3H\dot{\varphi}$ term is akin to the friction term in the harmonic oscillator equation. The Einstein equation tells us that the Hubble constant, $H$, is large when the scalar field is large.
• Thus, the logic of chaotic inflation is as follows:
• Large $\varphi$ $\longrightarrow$ Large $H$ $\longrightarrow$ Large “friction”.
• Field $\varphi$ moves very slowly, so that its potential energy remains nearly constant.$$H=\frac{\dot{a}}{a}=\frac{m\varphi}{\sqrt{6}}\approx \text{constant}$$
• And so the scale factor of the universe has proportionality$$a\sim e^{Ht}$$This is the stage of inflation.

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