When you read articles about the chronology of the Universe, you read for example that the duration of the inflation era is roughly $10^{-32}$s. We also know Universe expansion causes a redshift which can be expressed in terms of ratio of two frequencies. If we imagine the comparison of ideal clocks between now and then, knowing the redshift, we could evaluate a time dilation between proper time at the time of inflation, and time measured on our ideal clock today. My question is the following : when one speaks of a duration of $10^{-32}$s for the inflation, are we talking about proper time, i.e. the time that an hypothetical ideal clock would measure at that moment, or are we speaking of "Cosmological time", that is time measured today on our own ideal clock? 

Let me clarify the reason for my question. If indeed, inflation ends at about t = $10^{-32}$ s, as indicated in this [plot](https://en.wikipedia.org/wiki/File:History_of_the_Universe.svg) from Wikipedia, and if this time is measured in our present frame where the clocks are, it means that the duration of inflation, as measured by such clock, has an upper bound of $10^{-32}$s. Therefore, an ideal clock located at inflation time, would measure an even shorter time. Assuming as mentionned [here](https://en.wikipedia.org/wiki/Chronology_of_the_universe) that "*Cosmic inflation expands space by a factor of the order of $10^{26}$ over a time of the order of $10^{−33}$ to $10^{−32}$ seconds*", then adding again another expansion factor of $10^{3}$ between decoupling and now, this would imply that the duration of inflation as measured by a "local" clock at the beginning of inflation, would be less than $10^{-32}\*10^{-29}=10^{-61}$ second! (Here I have used that $1+z=\frac{\nu\_E}{\nu\_R}=\frac{dt\_R}{dt\_E}=\frac{a(t\_R)}{a(t\_E)}$. See Hobson & al., p.368)

I would have figured out that the inflation duration of $10^{-32}$s would have been the proper time duration, and hence that its duration as measured with our current clocks would have been much longer. Not the opposite! 

Any explanation?