# How did the Universe behave just at the end of inflation?

Edit When the inflaton field $$\phi$$ rolls down the potential $$V(\phi)$$ to the value $$\phi=\phi_f$$ where the slow-roll parameter $$\epsilon(\phi_f)\to 1$$, inflation ends. By calculating the behaviour of the scale factor at this time, how the Universe was radiation-dominated? How does the exponential scale factor $$a(t)\sim e^{Ht}$$ goes to the power law $$a(t)\sim t^{1/2}$$?

• Why would it be matter dominated ? Commented Nov 25, 2020 at 20:34
• I meant to ask, how can we show the universe will be radiation dominated Commented Nov 25, 2020 at 20:39
• The inflation happened around $t\approx 10^{-36}$. So during this period the universe was radiation dominated..You can find the matter radiation equality in terms of redshift (which is aboit 3400) And we know that inflation occured much earlier. So the universe was radiation dominated Commented Nov 25, 2020 at 22:31
• Please see the edited question. By radiation domination, I would like to understand how an exponential scale factor goes over to a power law. Commented Nov 26, 2020 at 3:46
• As you have said, at some point the inflation ends. So the contribution becomes unimportant and the universe becomes again radiation dominated. If it was not ended the universe would continue to grow exponentially. Commented Nov 26, 2020 at 6:58

Let me denote the inflation field energy density w.r.t to the critical density as $$\Omega_{\rm f}$$.

We can write the Hubble function as

$$H(z) = H_0\sqrt{\Omega_{\rm m,0}(1+z)^3 + \Omega_{\rm r,0} (1+z)^4 + \Omega_{\Lambda} + \Omega_{f}(z)}$$

Before the inflation, the universe was radiation dominated. This implies that

$$H(z) \approx H_0\sqrt{\Omega_{r,0}(1+z)^{4}}$$

During the inflation, what happens is that the $$\Omega_{f}(z)$$ starts to become important and starts to dominate the energy density of the universe so during the inflation period we obtain

$$H(z) \approx H_0\sqrt{\Omega_{f}(z)}$$

When the inflation fields start to oscillate the $$\Omega_{\rm f}(z)$$ starts to decrease and after some period it vanishes.

So we are back to the phase where

$$H(z) \approx H_0\sqrt{\Omega_{r,0}(1+z)^{4}}$$

The inflation field does not affect the universe in any other way. It is just a field that increases the energy density of the field and accelerates the expansion of the universe. There's no need for fancy equations to understand the process.

Without the inflation field, the universe was in already radiation dominated phase. The inflation field just accelerates the expansion of the universe briefly and vanishes. The inflation field is like a cosmological constant but just happens momentarily in the early universe.