It is commonly accepted that the Universe started in an ordered state with low entropy (S), which allowed its entropy to further increase - as observed - in line with the second principle of thermodynamics.
What was then the most robust value of departure ?
One could suppose that the entropy at the Big Bang was S=0 as the Universe was concentrated in one single micro-state (one single configuration). But this is a lot speculative as nobody knows what did the physics look like before Planck’s time (1.351 10-43 s).
At this time, in an hot Big Bang scenario, one could use the values proposed by Planck to the Academy of Berlin in 1899 to compute the entropy :
- Planck’s length : 4.051 10-33 m
- Planck’s mass : 5.456 10-8 kg
- Planck’s temperature : 3.551 1032 K
These values are a combination of universal constants exclusively, thus without « human neutral ». One then finds an entropy : S = 1.381 10-23 J.s-1 , i.e. Planck’s constant, corresponding to (using Boltzmann’s equation : S = k.lnW) a number of micro-states : W = 2.718 (= natural number « e »).
One could also use the formula of Bekenstein-Hawking for the entropy of a black hole, as the Universe was like a black hole at start (very hot, very dense). Using Planck’s length as diameter of the horizon circle of the black hole, we get an entropy : S = 1.652 10-23 J.s-1 , corresponding to W = 3.309 micro-states.
We would thus have an entropy corresponding to 2-3 micro-states at Planck’s time.
Another way would be to assess the entropy just after the presumed inflation stage (ending 10-33 s after the Big Bang). The entropy is then the area of the cosmological horizon in Planck's units, and would be about a million Planck’s lengths of radius, so probably a value of the order of ln W = 4.787 1012 which is still very low compared to the order of 10135 (maximum entropy one can squeeze in the modern cosmological horizon).
Other proposals go in the direction of a very high entropy at the beginning of the Universe, which would itself start from a statistical fluctuation of low entropy : which would then be the order of magnitude of this low entropy ?
In summary, could these evaluations be accepted and, whether yes or no, which would be the best value to have an idea of the entropy of the Universe shortly after the Big Bang ?
Please, read "human intervention" instread of "human neutral". With apologies !
In fact, the paragraph should correctly be read :
"These values are a combination of universal constants exclusively, thus without « human intervention ». One then finds an entropy : S = 1.381 10-23 J.s-1 , i.e. Boltzmann’s constant, corresponding to (using Boltzmann’s equation : S = k.lnW) a number of micro-states : W = 2.718 (= natural number « e »)."
On the grounds of theories, we have till now, entropy calculated by Bekenstein-Hawking formula will be okay to go with. As far as the Planck's units are concerned, i guess it will not be fine to use them because time and space(length) are embedded together as a single quantity and we are yet to define space-time on quantum level. I imagine, the day when will have a theory of Quantum Gravity then we will have a uncertain answer for that.
The entropy does not have the unit J s-1 but rather J K-1. Also, the initial mass of the universe exceeds by far the Planck mass. In an open universe, it is infinity, so no entropy can be calculated using the total mass. The question would then probably be what is the entropy inside the horizon of a causally connected part of the universe. In a closed universe (not a very likely scenario nowadays), the entropy would be finite but much larger than k times two or three.
KK: In an open universe, it is infinity, so no entropy can be calculated using the total mass.
That's the key I think, you have to think in terms of "entropy density" instead. It's not a subject I know about but a search for that term found the lecture linked below which might be a starting point in your research.
http://www.helsinki.fi/~hkurkisu/cosmology/Cosmo6.pdf
In the following PQG model (see eqns. 85 & 86) the value of the entropy of the Universe shortly after the Big Bang is S = 196.6 x 10^(-23) J/K which corresponds to W = 7.46383 x 10^(61) and each microstate having planck energy level which corresponds to macrostate energy level of 1.823x10^(81) GeV which is the total energy of the universe at present epoch including CDM and dark energy.
Article Periodic quantum gravity and cosmology
In a generalized sense, entropy is a measure of disorder. How much disorder can you get in a singularity, id est at a single point? And the entropy only increases, as the 2nd law of thermodynamics tells us. So, here is your answer.
Dear all, I'm back from a week abroad !
@Wasif
I agree with you that this is a bit speculative stuff as long as we don't have a sufficiently convincing quantum gravity (QG) theory. However, with the question, I try to limit as far as possible speculation by starting from Planck's time. What happened before is probably a degree more speculative ! And yes, Bekenstein-Hawking entropy could be the nearest to a QG answer.
@K. Kassner
Yes, you are right, entropy is in J.K-1. Thank you for correcting this mistake. I probably was too much in a hurry leaving eight days ago, that I made so many mistakes (3 for the present ...) in elaborating the question :=(
I agree with you that "the initial mass of the universe exceeds by far the Planck mass" as we probably are in an open universe according to the most recent results of Planck's mission. However, the idea is to start from something as general and not connected to theories as possible : e.g. with physical data in Planck's units which only use universal constants. Yes, much larger than 2 - 3 times k, but how much larger, taking account that the universe evolved a factor 1060 from Planck's time with increasing entropy during all this time ?
@George
Thank you for the link : I still have to study it ...
@Vikram
Thank you for this reference. So the entropy would be about 142 times k (?). I shall further analyse the path followed ....
@Natalia
I'm in line with you. However, there are voices in favour of an high entropy at the beginning. The idea is that the universe started with a local quantum fluctuation of the vacuum from a state of thermodynamic equilibrium (maximum entropy). But even then, just after this fluctuation took place (Planck's time), the entropy could have been very low locally. Yes, space and time were mixed then and modelling such a fluctuation would probably be very speculative. But there should have been some kind of starting point where the level of entropy could be quantitatively assessed using our present investigation means and accepted theories ...
Please read : "...but how much larger, taking account that the universe evolved a factor 1060 from Planck's time with increasing entropy during all this time ?"
Dear Guilbert, it's interesting to read "....when space and time were mixed". You probably heard that R. Feynman said that time is clock, and nothing more. I began to agree with him. Especially, we were initially both from Caltech :).
Guibert, remember also that inflation theory alters the picture considerably. What ever conditions occurred around the Planck time were first reduced to a near vacuum by inflation so probably become negligible. Then new matter and radiation were deposited by the oscillating decay of the field that drove inflation so really it is the value of entropy that results from the process known as "reheating" that governs the rest of the evolution of the universe. The attached link is probably badly out of date but again could give you a starting point:
"It is assumed that the Universe initially expands quasi-exponentially in a vacuum-like state with a vanishing entropy and particle number density. At the stage of inflation, all energy was concentrated in a classical slowly moving inflaton field. Soon after the end of inflation, when an observable universe was the size of a dime, the inflaton field began to oscillate near the minimum of its effective potential V(phi). An almost homogeneous inflaton field phi(t) coherently oscillated with a very large amplitude of the order of the Planck mass phi~Mp. The interaction of the inflaton field with other elementary particles led to creation of many ultra-relativistic particles from the classical inflaton oscillations. Gradually, the inflaton eld decayed and transferred all its energy non-adiabatically to the created particles. They interacted with each other and came to a state of thermal equilibrium at some temperature Tr, which was called the reheating temperature. An enormous total entropy of the observable universe (or the total number of particles inside the horizon), S=1088, was thus produced from the reheating after inflation."
https://cds.cern.ch/record/303943/files/9605155.pdf
@George,
Thank you for this interesting document. I indeed checked this issue (post inflation) in my question but found (via Google) a value of S ~1012, which is 75 orders of magnitude lower than the S ~ 1088 you mention !
Inflation is very much a current research topic and ideas are developing rapidly so please don't take my link as current, I really intend it as a starting point, have a look for more modern papers that perhaps cite this, or take keys phrases from it for searches on arXiv.
Note also, the value of S=1088 is only for the observable universe, not the whole.
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