The $q$ parameter is at present -ve which implies accelerated expansion , hence we can approximate our universe now to be dominated by dark energy since dark energy provides exponential growth $~e^{\alpha t}$ It is therefore after matter-dark energy equality time inflation starts e.g $$\epsilon(t)_d=\epsilon(t)_m$$,$$\frac{\epsilon_{0m}}{a^3}=\epsilon_{0d}$$,$$a^3=\frac{\epsilon_{0m}}{\epsilon_{0d}}$$,$$a^3=\frac{\Omega_{m0}(.27)}{\Omega_{d0}(.73)}$$,$$a_e=0.71$$,$$z=.3908$$, Can we say that at this redshift the inflation period of the universe starts and is continuing till now when z=0?? Or inflationary period is not so much long??

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    $\begingroup$ Suggestion: Permute the commas and dollar-signs to avoid flying commas. $\endgroup$ – Qmechanic Feb 9 at 14:59

Yes, you are right about the idea. In the calculation process, you can change your $\Omega_m$ and $\Omega_{\Lambda}$ values with respect to the 2018 Planck data

$\Omega_m=0.3111$ and $\Omega_{\Lambda}=0.6889$

Hence we get

$$a(t)=0.76$$ so $$z=0.3157$$

I am not exactly sure about the inflation part. The dark energy is dominant for the last 4 billion years however I think its still early to say that the inflation period is started. I think we can still use the scale factor related to the matter-lambda universe, $$a(t)=(\Omega_m/\Omega_{\Lambda})^{1/3}sinh^{2/3}(t/t_{\Lambda})$$

for $t_{\Lambda}=\frac {2} {3H_0\sqrt {\Omega_{\Lambda}}}$

  • $\begingroup$ If we take our universe to be dominated by radiation very early and then matter ,in both the cases inflation can't be possible since for both $$\frac{da^2}{dt^2}<0$$ ,so it is this $\Lambda$ dominated (dark energy) dominated universe for which inflation is possible, isn't it?! $\endgroup$ – Apashanka Das Feb 9 at 19:11
  • $\begingroup$ @ApashankaDas I am not sure I understand you correctly but I ll give it a try. In the period of radiation and matter dominanted times, ofc the inflation is not possible. Only after the $\Lambda$ becomes dominant part we can talk about the inflation. $\endgroup$ – Reign Feb 9 at 19:31
  • $\begingroup$ @ApashankaDas think about the deceleration parameter, $q=\Omega_r+\Omega_m/2-\Omega_{\Lambda}$ and when $q<0$ means that universe is accelerating ($\ddot{a}(t)>0$). $\endgroup$ – Reign Feb 9 at 19:34
  • $\begingroup$ Yaa that means inflation starts in the $\Lambda$ dominated phase which is very near in the past(e.g $z$=0.3 and $z_{present}=0$).But one question remains there for solving the horizon problem ,inflation is taken into account for very early times??@Reign $\endgroup$ – Apashanka Das Feb 9 at 20:06
  • $\begingroup$ @ApashankaDas There are two inflation eras. First one happened in the early times and ended at $10^{-34}s$ which again the $\Lambda$ was dominated in the universe. This inflation solves the horizon problem. The other one started to happen at 4 billion years ago, which as you described at $z\approx 0.3$ $\endgroup$ – Reign Feb 9 at 20:22

Inflation is just a model, and we don't have any proof that it actually happened. It is nowhere near as solidly tested by observation as other aspects of our cosmological models. Inflationary theories use crude models of the potential of the field that causes the inflation (Mexican hat potentials), and these crude models actually tend to have a hard time producing reasonable descriptions of the exit from inflation. In fact, the inability to get the exit to work is one of the main unsolved problems with inflationary theories at the present time.

So basically we have no idea whether inflation occurred, and if so, when it ended.

However, if inflation did occur, then it has to have ended before big bang nucleosynthesis started, so $z$ would be many orders of magnitude lower than 1.

  • $\begingroup$ How can we define a z with reapect to the inflation time ? There was no photons at the time. $\endgroup$ – Reign Feb 10 at 6:13
  • $\begingroup$ Can we say inflation starts at the time for which the universe become dark energy dominated ($\Lambda$) otherwise for other models inflation is not possible ?? $\endgroup$ – Apashanka Das Feb 10 at 6:36

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