# How long was the hyper inflation phase?

How long was the hyper inflation phase? Haw can we know that it lasted for a period of time and not more or les time?

• What is the difference between inflation and hyperinflation (which I personally never heard about) ? Feb 7 '15 at 8:56

Googling "hyperinflation" mostly returns articles about Zimbabwe and the Weimar Republic, so I'm going to assume that you are referring to what is usually just called (cosmological) inflation.

We are still pretty uncertain about when and for how long inflation occured, but we do however have some sort of constraints. In general, inflation is thought to be caused by some sort of scalar field, usually written as $\phi(\mathbf{r},t)$, and its associated potential energy $V(\phi)$. Various forms of how this field changes as a function of time exist, but a popular one is, or has been at least, the slow-roll field, which makes it behave like a cosmological constant, enabling it to drive an exponential inflation.

Inflation can be initiated by a symmetry break when the unified forces of electromagnetism and the weak and strong forces are separated in the strong and the electroweak force (the "GUT" era, for Grand Unified Theory), at temperatures of $T\sim10^{28}$ K. It will continue until the field has "rolled" from its metastable false vacuum state to a true vacuum. During this time, the Universe expands something like $e^{100}$ times, and its temperature drops by the same factor, i.e. to a freezing temperature of $T\sim10^{-15}$ K. This doesn't last long, though, because the energy of the field is converted to photons and other relativistic particles. This is called reheating.

The factor by which the Universe expands during inflation is essentially set by the minimum factor needed to explain in particular three cosmological "problems", viz. the flatness of the Universe, the causality of the horizon, and the missing monopoles, and the maximum factor allowed in order not to blow stuff too far apart that no structure would have had the time to form before gravitational attraction was too small, so that we wouldn't be here to ask the question. This requires at least 60 e-foldings (i.e. expanding by a factor of $e^{60}$), but could be 100 or maybe more. The factor is discussed e.g. in Galaxy Formation And Evolution by Mo, Bosch & White, Sec. 3.6.2.

Since the size evolution of the Universe during inflation is exponential, the scale factor can be written as $$a(t) \propto e^{Ht},$$ where $H$ is the Hubble constant during inflation, so for an expansion of, say, $e^{100}$ we have $$e^{100} = \frac{a(t_{\mathrm{end}})}{a(t_{\mathrm{begin}})} = \frac{e^{H t_{\mathrm{end}}}}{e^{H t_{\mathrm{begin}}}} = e^{H(t_{\mathrm{end}} - t_{\mathrm{begin}})},$$ or $$\Delta t \equiv t_{\mathrm{end}} - t_{\mathrm{begin}} = 100/H,$$ where $t_{\mathrm{begin}}$ and $t_{\mathrm{end}}$ marks the time of beginning and end of inflation, respectively.

During inflation, the Hubble constant should not evolve, but have a constant value of $H=(\Lambda/3)^{1/2}$, where $\Lambda$ is a cosmological constant. If inflation started when the GUT era ended, which was around $t_{\mathrm{GUT}} \sim 10^{-36}$ s, the Hubble constant was of the order $H \sim t_{\mathrm{GUT}}^{-1} \sim 10^{36}$ s. Then a hundred e-foldings will take you to $t_{\mathrm{end}} \sim 10^{-34}$ s. Wikipedia states that inflation should last somewhat longer, something like $10^{-33}$ to $10^{-32}$ s. This might be if you do a more careful analysis, or maybe a different model. For instance, H is not exactly constant during inflation, but does evolve a little.

Various observations can constrain the physical parameters that enter the equations, e.g. cosmic microwave background and baryon acoustic oscillations. The latest results from the Planck satellite can be found here. Wikipedia also has a summary of the observational status of inflation.

So, I guess the answer to your question is, "We don't, really, but we have several theories that together with several observational constraints tell us that the duration of inflation should be of this order of magnitude. If it were much shorter, it wouldn't solve the cosmological problems that gave rise to the idea of inflation, and if it were much longer, the Universe would be blown apart".

• It would be nice to see some calculations from where those numbers come out, if you can. I tried thoroughly through the web, but I did not find anything easily understandable. Feb 11 '15 at 9:28
• Okay, thanks for urging me to do some math, @Py-ser. I included a derivation of the duration based on the factor by which the Universe expanded. The value of this factor itself takes a little more math, but will be found in many textbooks on the topic (also included a ref for this).
– pela
Feb 11 '15 at 14:17