I'm trying to understand the Horizon Problem. Inflation describes how non-causally connected parts of the universe (that we see today) could once have been in causal contact. The reasoning goes that those regions must have been in causal contact to achieve the uniform temperature that we observe and, thus, we need Inflation to explain the uniform temperature of the sky.

If the universe started out with the same energy and volume everywhere, and the volume at every point in space grew at exactly the same rate, then how could one part of space be a different temperature than another?

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    – rob
    Jan 30 '19 at 20:34

Thermodynamics is a classical theory that axiomatically should hold wherever it can be applied. As it is also emergent from statistical mechanics the frameworks where its laws apply have been extended to the universe as observed and modeled with the original Big Bang theory, which fitted a lot of the observations. Because General Relativity is involved, mathematically there are regions in space that thermodynamic equilibrium, which is an axiomatic concept, cannot be reached at the early ages of the universe.

The original Big Bang model predicts that there should be random discrepancies in the cosmic microwave background, regions that could not interact with radiations and "scatterings" should not display the uniformity observed, of order of $10^{-5}$ in the data. This forced a modification to the original Big Bang model

The high degree of uniformity throughout the observable universe and its faint but measured anisotropy lend strong support for the Big Bang model in general and the ΛCDM ("Lambda Cold Dark Matter") model in particular. Moreover, the fluctuations are coherent on angular scales that are larger than the apparent cosmological horizon at recombination. Either such coherence is acausally fine-tuned, or cosmic inflation occurred.

You ask:

If the universe started out with the same energy and volume everywhere, and the volume at every point in space grew at exactly the same rate, then how could one part of space be a different temperature than another?

Because thermodynamic equilibrium cannot be reached when there are no interaction paths. Even if in the original Big Bang there was uniformity at $(0,0,0,0)$ plus an infinitessimal dt, the fact that there are regions with no causal connections and the fact that thermodynamics emerges from statistical mechanics in a probabilistic formulation makes the uniformity highly improbable, and thus the conclusion above:

Either such coherence is acausally fine-tuned, or cosmic inflation occurred.

As fine tuning is not something that is popular for physics models , cosmic inflation was invoked to solve the discrepancy.

edit after comments:

Here it is stated that

Before a time classified as a Planck time, $10^{-43}$ seconds, all of the four fundamental forces are presumed to have been unified into one force. All matter, energy, space and time are presumed to have exploded outward from the original singularity. Nothing is known of this period.

italics mine

In this link it is stated that there are about $10^4$ acausal regions in the CMB map we have of the observable universe. Nevertheless the black body curve from the CMB is one that fits very well the theoretical expression, at a level of $10-5}$ anisotropy.

Thermodynamics is based on statistical mechanics so there is a probability for the initial energy even though the same in each present acausal region to develop more or less particles if considering only a GUTS type model with gravitons included.

In addition there is no guarantee that the same number of zero mass particles ( before symmetry breakings) will be generated from the primordial energy in each acausal region of the $10^4$ . May be more protons are produced in one slice and only neutrinos in another by statistical fluctuation: at symmetry breaking this could make a large difference between energy bound in masses and kinetic energy. It is kinetic energy that defines temperature.

Thus either one has to assume acausal fine tuning in the theory, or develop a quantum mechanical model that could homogenize the universe, before the symmetry breaking time of the GUTS particles. The inflation model seems to be successful in this.

  • $\begingroup$ $$PV=nRT$$ Where does room for statistics enter this relation? Is the volume probabilistic? Is the number of particles in a given volume probabilistic? Which one of these assumptions has room for random variations (other than the quantum fluctuations which are not removed by Inflation). $\endgroup$
    – user32023
    Jan 29 '19 at 13:15
  • $\begingroup$ Before the inflation model introduced inflatons, it was supposed that something like a GUTS model with gravitons included: unbroken symmetries with the zero mass particles filling up the space time, with causally unconnected regions. The formula you write works for connected regions, if there are no interactions between regions there is no reason to arrive at the same temperature as the fluctuations in the generation of the number of particles/radiation need not be the same, unless assumed which is the fine tuning option $\endgroup$
    – anna v
    Jan 29 '19 at 14:08
  • $\begingroup$ If you have the same amount of matter/energy in causally disconnected areas of equal volume, and you increase those volumes at the same rate, at the end of some period, t, why would one volume have a different temperature than another? Do you have an argument that at $t=5.39\times 10^{−44}\space s$ that matter/energy was not evenly distributed through the universe? $\endgroup$
    – user32023
    Jan 29 '19 at 14:54
  • $\begingroup$ You are correct, and I quote a link above that we do not know anything before Planck1. but uniformity is detected at the time of the photon decoupling, and there is the whole window after Planck one to quark confinement formation and further where the acausal regions have a probability of developing differently, that is all I am saying. see the timeline : hyperphysics.phy-astr.gsu.edu/hbase/Astro/timlin.html#c1%22 $\endgroup$
    – anna v
    Jan 29 '19 at 18:01
  • $\begingroup$ @annav - So if the universe is uniform at $t=5.39\times 10^{−44}\space s$ and is uniform at $t=372,000\space yr$, what mechanism is there for one unit volume of space that is non-causally connected to another unit volume of space to have a different temperature? From the physics I know, the only variable here is volume. There's a lot of talk about how Inflation solves the Horizon Problem, but I've yet to see how the problem evolves from uniformity. $\endgroup$
    – user32023
    Jan 29 '19 at 18:39

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