Ali, models are not "wrong" since they are adjusted to fit data. Ptolemy model was not "wrong" either, it fitted data. These kind of models are a necessary step in the process of discovering because they organize data and reveal their global properties. But they are not necessarily "logic", their goal is to obtain equations that fit data and how that is achieved is not really important. You do well in trying to build a logic path but you will find out that sometimes it does not exist in these kind of models - don't give too much importance to it, these are just mathematical models. Nor you should believe that dark matter is a physical entity, it is just an ad hoc parameter; or that really there was a "cosmic inflation" or even a Big Bang - we know nothing of what happened before the CMB, the BB is just a mathematical extrapolation.

The standard model is the best model available, but this does not mean that it is perfect, or "the truth". In Physics it is always like that, a going on discovery process; the cosmological model, the atomic model, QM are just the best models available in each field but they all have difficulties. The concepts of "right" or "wrong" do not apply, only the concept of "best model available". As people tend to classify things in right and wrong, the difficulties of the models are not usually spoken because then people would fall in the simpleton conclusion that they are "wrong". 

Note, however, that there is a kind of model in relation to which the concepts of right and wrong apply, the concept of falsifiabilty applies: a model with no ad hoc parameters, no unknown entities. These models cannot be adjusted to data, either they fit it or not. Like Newton's model (that is why he said "hypotheses non fingo"). To obtain such a model is not an easy task...

I am saying these things because obviously you have a clear mind and you seem able to scope with the complex nature of things... and so it is possible that you will later be able of giving some important contribution to improve the models - eventually a model with no unknown entities.... Who knows?

Ali,

Your question is slightly confused.

Read Mukhanov again! He has explained that inflation is a brief period of acceleration, to solve the horizon and flatness problems,but what happened before it he has not specified. And it must come to an end with the gracefil exit, whereafter a Friedmann (FLRW) deceleration took over - as required to fulfill all the predictions of FLRW..

The rate of change of the scale factor was indeed decreasing until z~4 when the acceleration called Dark Energy started.

Matt,

First, I thank you for your reply. So can we say that the universe first accelerated(inflation), then decelerated(graceful exit) and then it accelerated again(now)? Moreover, according to the attached diagram, the rate of change of the scale factor is approaching 0 as t goes to infinity. How can the condition that \dot{ai}

Note that \dot{a0}=0 because a0 is a constant: the present scale. The relation \dot{ai}

How can a0 be constant at the present scale if we have an expanding universe, at the present scale as well? In addition,  I don't understand why a graceful exit should be associated with a decelerating Friedman universe?

a0 is constant by definition.

The inflation must end, or we would not have the Universe that we have, for instance the precisely understood Big Bang Nucleosynthesis. In my book Introduction to Cormology, Fourth Edition,  all this is explained. I cannot repeat it in Research Gate.

Ali

If you begin questioning the standard model instead of believing in it, you will find a lot of strange things. Because the standard model is just a mathematical model adjusted to fit data by the use of ad hoc parameters. An entity with just one property is never a physical entity, it is just an ad hoc parameter. That is the case of dark energy and dark matter. On the other hand, cosmic inflation must be understood as the way to adapt the concept of Big Bang to the properties of CMB which, truly, are incompatible with a Big Bang. But science is built on solutions, not on doubts, so that was the possible solution for current problems of cosmology. 

If you take a good look, you will find that standard model is just a modern version of Ptolemy model, with dark energy in the role of celestial spheres (dragging stars ways instead of dragging around) and dark matter in the role of epicycles/deferent (explaining local motions). Furthermore, both are only valid at distance "in the skies" and both consider that the distant universe is made of entities unknown locally. This is not a coincidence, this happens because both are mathematical models presuming the invariance of an observer... that is varying! 

if you want to see what is a Physics model of cosmic data, considering no unknown entities, you can take a look to a preliminary document at: http://vixra.org/abs/1107.0016

Alfredo, 

Thank you for your inspirational comment. At the moment, I'm not questioning anything, I'm just trying to understand a property that seems (to me) contradictory. I believe that these models cannot be wrong (since I'm still a student, I'm no allowed to do that at the moment), so there must be something wrong in my understanding, and I need someone's help to fix it. Matts gave some good comments, yet I still didn't understand how the contradiction went away

Ali, models are not "wrong" since they are adjusted to fit data. Ptolemy model was not "wrong" either, it fitted data. These kind of models are a necessary step in the process of discovering because they organize data and reveal their global properties. But they are not necessarily "logic", their goal is to obtain equations that fit data and how that is achieved is not really important. You do well in trying to build a logic path but you will find out that sometimes it does not exist in these kind of models - don't give too much importance to it, these are just mathematical models. Nor you should believe that dark matter is a physical entity, it is just an ad hoc parameter; or that really there was a "cosmic inflation" or even a Big Bang - we know nothing of what happened before the CMB, the BB is just a mathematical extrapolation.

The standard model is the best model available, but this does not mean that it is perfect, or "the truth". In Physics it is always like that, a going on discovery process; the cosmological model, the atomic model, QM are just the best models available in each field but they all have difficulties. The concepts of "right" or "wrong" do not apply, only the concept of "best model available". As people tend to classify things in right and wrong, the difficulties of the models are not usually spoken because then people would fall in the simpleton conclusion that they are "wrong". 

Note, however, that there is a kind of model in relation to which the concepts of right and wrong apply, the concept of falsifiabilty applies: a model with no ad hoc parameters, no unknown entities. These models cannot be adjusted to data, either they fit it or not. Like Newton's model (that is why he said "hypotheses non fingo"). To obtain such a model is not an easy task...

I am saying these things because obviously you have a clear mind and you seem able to scope with the complex nature of things... and so it is possible that you will later be able of giving some important contribution to improve the models - eventually a model with no unknown entities.... Who knows?

When cosmological constant is smaller that a certain critical value then one of two possible solutions is periodic - and necessarily decelerating.

Abir,

First, I would like to thank you for your answer. Second, actually the problem is as follows:

1- according to the diagram attached, \dot{a0} goes to 0 as t goes to infinity. So if someone is living at "infinity" reads this sentence (having this diagram in hand) he will definitely say that that diagram is wrong, iff \dot{ai}0?

3- Why is the graceful exit associated with a DECELERATING universe, and not an accelerating one, but with a smaller rate than the one during inflation?

Thanks 

Dear Ali Rida Khalife,

What we think to know about Universe expansion history is as follows:

- At very early time, it accelerates and the scale factor (a) second derivative with respect to time is positive (a">0).

- Then expansion decelerates (a"

Thank you very much Stephane Fay for finally clearing things for me

I. A. Urusovskii, Multidimensional   Treatment of the Expanding Universe,

 Physical Science International Journal, 4(8): 1110-1144, 2015

Igor,

another model that works is Ijjas and Steinhardt, arXiv:1507.03875 (astro-ph.CO).

Ali,

Maybe you will better understand the idea of Mukhanov by having a look on enclosed slides of one of his recent conferences (Solvay Lectures of 18.10.2015 in Brussels). You find the same figure on slide 20. Only, you see that the beginning of inflation corresponds to an "a dot" lower than what he calls "a0 dot" and lower than the value of "a dot" at present time. This is why he writes that "the rate of change of the scale factor today should be greater than the rate of change at the beginning of inflation". He also made calculations to show that the inflation could start from quantum fluctuations (see reference of article on slide 19).

Afterwards, i.e. after the end of the inflation stage, there is - in line with General Relativity (for a flat Universe) - a Friedman deceleration, first with Hubble parameter H=1/2t (radiation-dominated era) up to end of photons/baryons decoupling, then with H=2/3t (matter-dominated era) until now and further on. You can find an attempt to visualize what happens during these phases in an article I wrote on "A Simple Procedure to Assess the Time Dependence of Cosmological parameters" (1999).

You probably also will find interesting elements in the RG discussion on "What is the expansion rate of the Universe ?" (see enclosed link).

Kind regards  

https://www.researchgate.net/post/what_is_the_Expansion_Rate_of_the_Universe

Here is the correct link to the RG  Discussion (starts with "W" capital).

https://www.researchgate.net/post/What_is_the_Expansion_Rate_of_the_Universe