# Why does the low entropy at the big bang require an explanation? (cosmological arrow of time)

I have read Sean Carrol's book. I have listened to Roger Penrose talk on "Before the Big Bang". Both are offering to explain the mystery of low entropy, highly ordered state, at the Big Bang.

Since the second law of thermodynamics is considered a fundamental law of nature, and since it states that in a closed system entropy either must stay the same or increase, the entropy at the time of the Big Bang must have been much lower than it is now.

Also, the thermodynamic arrow was explained by Boltzmann in 1896 embodied in $S = k\ln W$ which was inscribed on his tombstone. $W$ is the number of distinct microstates of the system. It seems to me that trivially this number will be less in the past than in the future since entropy obeys the 2nd law. This determines the thermodynamic arrow of time.

Why do we need more explanation of a "fundamental law"?

• I see that Lubos has posted this great link on another question: "Is it physically meaningful to talk about the arrow of time in other universes?"--:"Robert Wald arxiv.org/abs/gr-qc/0507094 who argued that the thermodynamic arrow of time - or low entry of the early Universe - can't have a dynamical origin. – Luboš Motl Jan 21 at 18:34" Many thanks, Lubos. I am not going to argue with Robert Wald. Jan 30, 2011 at 7:15
• Thanks, Gordon, for your good contributions. Just a link so that people may simply click: arxiv.org/abs/gr-qc/0507094 Jan 30, 2011 at 7:19
• All these arguments hinge on the assumption that the Universe is a closed system. If you abandon that assumption the whole ball-game changes. So one has to seriously consider whether this assumption is warranted in the first place. After all no closed systems exist in the world that we observe. Not even black holes. Why should the Universe be any different?
– user346
Feb 1, 2011 at 7:22
• – user4552
Aug 14, 2013 at 18:15

This is somewhat controversial issue. But let me present the reasons, as far as I understood, why people like Sir Penrose thinks so.
Their arguments are roughly as follows:

1. The basic microscopic laws of physics are perfectly time symmetric. They are not biased in any time direction past or future.

2. Second law follows from the fact that that given an initial condition of a system which is not in the most probable state will tend to go towards the most probable state by the same microscopic laws. Since number of disordered states are much higher the system will become more and more disordered with time. Accordingly its entropy will increase until a maximum value when the system comes to the thermal equilibrium.

3. Since the microscopic laws are time symmetric the same argument can be made towards the past time direction as well. Given an an initial condition of a system which is not in the most probable state should go towards more disordered (high entropy) states towards past as well. That's what the mathematics of the laws tells us.

4. This is against our experience. Either all the parts of the universe we are observing (including our memories of past) has just undergone a HUGE fluctuation right now to give the impression that there was a more ordered past (which is crazy) OR the system was already even more ordered (low entropy) and more special in the past. But that means even more huge a fluctuation. This reasoning will lead us to conclude that at the moment of big bang the universe was extra ordinarily ordered and most special. It should be so special that it requires explanation.

Critics often point out that prediction and retrodiction is not the same thing forgetting that when one talks about the very "arrow of time" no one can say with justification which is prediction and which is retrodiction. Other than that it is also questionable whether second law can be applied this way to the whole universe or not.

• See the link to the paper by Robert Wald:arxiv.org/find/all/1/… Jan 30, 2011 at 7:31
• OT: (I went to some of the Waldfest in 2006 with my son...it was fun.) Jan 30, 2011 at 7:36
• I have just read the paper and I agree with it. I agree that the specialness of the big bang is required and one can not invoke dynamical theories like chaotic inflation to explain the specialness, which assumes the universe started with a random initial condition.
– user1355
Jan 30, 2011 at 7:45
• @sbi Yes, Wald does argue that, but the point of my posting the link is that he also does not hold that a cosmological explanation is known or perhaps required. He says that cosmological arguments become circular and uses inflation and anthropic examples. Ultimately, he says that the universe came into existence in a special state, period. He has an addendum saying he disagrees with Carrol and Chen. In my view Wald answers the above question I asked as "It doesn't".. He does agree with you about inflation though. Jan 30, 2011 at 16:08
• Well, I will leave it with Wald's paper and we will disagree. Feb 5, 2011 at 17:57

The key issue is: To allow galaxy and star formation, the universe had to start up in an extremely (but not perfectly) homogeneous state. The fact that the initial inhomogeneities were of size 1 part in a gadzillion (as opposed to zero) is what is puzzling.

• Not really. Quantum vacuum fluctuations were inflated by the inflaton field. Jan 30, 2011 at 6:21
• Not sure what you are commenting on. Are you disputing that the initial universe must have had very small but nonzero inhomogeneity? Jan 30, 2011 at 6:58
• Inflation will increase the entropy. So the initial entropy has to be even smaller. Jan 30, 2011 at 7:18
• @Gordon, Wald argued the very opposite in the paper! Inflation can't explain time's arrow. No theories which assumes initial randomness can. That was his point.
– user1355
Jan 30, 2011 at 7:59
• @sbi Yes, Wald does argue that, but the point of my posting the link is that he also does not hold that a cosmological explanation is known or perhaps required. He says that cosmological arguments become circular and uses inflation and anthropic examples. Ultimately, he says that the universe came into existence in a special state, period. He has an addendum saying he disagrees with Carrol and Chen. Jan 30, 2011 at 16:05

Because this violates the assumption that every state ought to be equally likely.

The low entropy of the big bang leads to the second law of thermodynamics, not the other way around.

In fact, as Boltzmann had shown, for almost all states constrained to have a given macroscopic description at a given time, entropy will increase both toward the past as well as the future. For the entropy to have been continually decreasing as we move back to the past for billions of years requires an incredible amount of fine-tuning of the correlations between almost all the particles in the present.

• I still don't follow (agree). I do agree that the initial conditions lead to the 2nd law, but it is a bit of a tautology. Jan 30, 2011 at 6:11
• What leads to the 2nd law and the arrow is the difference in entropy between then and now. I don't see how incredible fine tuning is required. If (say) the universe started as a bubble (say, Linde's eternal inflation model), it must start in a highly ordered state, and inflation can explain the fine tuning. Or not? Jan 30, 2011 at 6:18
• That is, "the correlation between almost all the particles in the present" could be explained by eternal inflation? Jan 30, 2011 at 6:20
• (The -1 wasn't me :) ) Jan 30, 2011 at 7:06
• I think it would require a lot of fine tuning to start with eternal inflation, but once we have eternal inflation, it will continue indefinitely.
– QGR
Feb 1, 2011 at 15:38