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"?
 A: 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:


*

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

*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.

*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.

*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.
A: 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.
A: 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.
