This idea has been making its appearance in my posts for a while, but its time to say it outright, entropy has nothing to do with the arrow of time. The Second Law of Thermodynamics states that in a closed system, entropy will never decrease. The two main examples are 1) a gas will disperse randomly and never gather again in a corner of a container and 2) heat will never flow from a cold object to a hot one. First the gas. If you have a concentrated area of a gas in a container, the gas will flow from areas of high concentration to areas of low concentration. This is a “downhill” process of movement from high concentration (or pressure) to areas of lower concentration (or pressure). By itself (without the addition of energy), the gas wont flow “uphill” into an area of high concentration or pressure. This attribute of matter happens to be irreversible in time, again, unless you add energy. So, perhaps it is fair to say that within this process there is an arrow of time that can be identified. It is just that what we call (for convenience) disorder is energetically downhill of order. What is it about this attribute of matter that causes physicists to apply entropy to the arrow of time for the whole universe? As I mentioned before, cold can not flow from a cold object to a hot one. Heat is movement, the lack of heat is stillness, and stillness can not flow. Again, this is an attribute of matter that happens to be irreversible in time because of the form of the materials.
Broken cups and scrambled eggs. Physicists love to give the example of the broken cup to describe entropy. The cup is formed by energy and broken by energy. The materials of the cup can not be reformed into the cup once broken, because those materials have been changed (fired in the kiln). If the cup is made out of unfired clay when you drop it, you CAN pick it up and reform the cup. The fact that it takes energy to increase entropy (make the cup) and energy to decrease it (break the cup) makes me wonder why one direction is considered so different from the other. The same with the egg, energy to make it, energy to unmake it. So what if it can’t spontaneously reform, it can’t spontaneously form either.
The issue is that physicists found that physics equations are time symmetric, they work going forward and backward in time. These equations are abstractions, but perhaps the processes they describe could go forward and backward in time (billiard balls for example) but they don’t, do they? For entropy, there are many more disordered states possible than ordered ones, so probabilities send you in that direction. As to why the early universe had such low entropy, well it was a singularity wasn’t it? Wouldn’t that have low (to no) entropy? Is time the direction that is energetically downhill for matter? Then all this organization in the universe must be either going backwards in time, or defying the arrow of time.