Caption: The horizon problem illustrated. The diagram is not-to-scale.

Features:

1. The cosmic background radiation (CBR) (which at cosmic present t_0 (equal to the age of the observable universe = 13.797(23) Gyr (Planck 2018)) is the cosmic microwave background (CMB, T = 2.72548(57) K (Fixsen 2009)) reaches us from starting points that are near the particle horizon: i.e., the edge of the observable universe which is currently at a proper distance ∼ 14.3 Gpc ≅ 46.6 Gly (see Wikipedia: Observable universe).

   These starting points, in fact, form a sphere centered us called the last scattering surface (LSS).

   In the diagram, the LSS is the big sphere.

2. Note

         r_proper = a(t)*r_comoving ,

   where r_proper is proper distance, r_comoving is (cosmological) comoving distance, a(t) is the cosmic scale factor, and t is cosmic time measured from time zero at the big bang singularity of the Friedmann-equation Λ models---which is before by a fraction of second when the Friedmann-equation Λ models can be true.

   Comoving distances are constant with respect to cosmic time.

   For the usual convention that cosmic scale factor a(t=present) = 1, the proper distances are equal to the comoving distances at cosmic present = to the age of the observable universe = 13.797(23) Gyr (Planck 2018).

   So the radius of the cosmic present observable universe is always ∼ 14.3 Gpc ≅ 46.6 Gly in comoving distance.

3. The CBR started toward us at recombination era t = 377,770(3200) Jyr = 1.192*10**13 s (z = 1089.80(21)).

   At that time cosmic time, the cosmic scale factor a(t)/a_0 = 1/(1+z) ≅ 1/1100.

   So the observable universe was much smaller then. The expansion of the universe caused it to grow to its size at cosmic present = to the age of the observable universe = 13.797(23) Gyr (Planck 2018).

   However, its comoving volume was the same as now.

4. The diagram shows to LSS in lookback time with an an out-of-date value which is somewhat too large. The current value for lookback time is nearly age of the observable universe = 13.797(23) Gyr (Planck 2018).

5. The particle horizon at the recombination era was much smaller than at cosmic present in proper distance.

6. The small spheres in the diagram represent particle horizons for the points they are centered on at the recombination era with radii in lookback time for that cosmic time with somewhat out-of-date values.

7. NO signal could have reached from one small sphere to the other since time zero in the Friedmann-equation Λ models. They are NOT causally connected in those models and CANNOT influence each other.

   The fastest a signal could travel relative to its local inertial frame (which would usually be at least approximately comoving frame) is the vacuum light speed c = 2.99792458*10**8 m/s (exact by definition) ≅ 3*10**8 m/s = 3*10**5 km/s ≅ 1 ft/ns.

   Of course, if the Big Bang singularity actually happened, then all points may have been one point at that instant (i.e., the fiducial time zero of the Friedmann-equation Λ models). But we do NOT believe that the Big Bang singularity actually happened even if what came before time zero is still highly speculative.

   In fact, the farthest back in cosmic time, we can even imagine running the Friedmann-equation Λ models) is ∼ Planck time t_plank = sqrt(ħG/c**5) = 5.39125*10**(-44) s when they would have a density ∼ Planck density ρ_Planck = c**5/(ħ*G**2) = 5.15500*10**96 kg/m**3 (see Wikipedia: Planck units).

8. However, the uniformity of the CMB shows that all points on the LSS had the same temperature to about 1 part in 10**5 (see Wikipedia: Cosmic Background Explorer: Intrinsic anisotropy of CMB).

   The LSS was in self thermodynamic equilibrium (very nearly).

   How could this be if those points are NOT causally connected?

   There is NO way that the points could be in near thermodynamic equilibrium (i.e., at nearly the temperature) by influencing each other in the Friedmann-equation Λ models.

   Other evidence shows the observable universe at recombination era t = 377,770(3200) Jyr = 1.192*10**13 s (z = 1089.80(21)) was very homogeneous (and isotropic) in all respects.

9. One explanation for the homogeneity is just to say the observable universe started out as extremely homogeneous at some early point in cosmic time as an initial condition---a "just so story".

   However, cosmologists and physicists in general do NOT like the idea of a beginning of time out of nothing (i.e., ex nihilo)---especially with fine-tuned initial conditions.

   So the extreme homogeneity of the early universe is a problem which is called horizon problem.

   The name is NOT so good since it has to be explained. The horizon of a point is sphere at the limit of causal connection and somehow points beyond the horizon have extreme homogeneity in the observable universe which is has to be explained. A better name than the horizon problem is the homogeneity problem, but that name is NOT much used.

   Note that in science, "problem" often means a perplexing feature of observations or theory that needs to be solved in some manner in order to exorcise perplexity. Of course, "solutions" are often the cause of new "problems". But that is why science progresses.

10. The most considered possible solution to the horizon problem since circa 1979 is the inflation paradigm which posits a false-vacuum universe out of which our observable universe formed by inflation super rapid exponential expansion (much faster than in the Friedmann-equation Λ models) from a tiny piece of the false-vacuum universe that was entirely causally connected and homogeneous and, in particular, was in self thermodynamic equilibrium.

    For an explication of inflation paradigm, see Cosmology file: inflation_paradigm.html.

    Of course, the inflation paradigm creates new problems---but physicists like new problems to solve.

11. By the by, Steven Weinberg (1933--2021) considers the horizon problem the one initial condition problem of the observable universe for which there are NO plausible alternatives to the inflation paradigm (see Weinberg, Steven, 2008, p. 208). Some disagree.