Horizon problem

The most commonly accepted hypothesis to explain the horizon problem is cosmic inflation. Alternative solutions propose a cyclic universe or a variable speed of light.

Background

Astronomical distances and particle horizons

The distances of observable objects in the night sky correspond to times in the past. We use the light-year (the distance light can travel in the time of one Earth year) to describe these cosmological distances. A galaxy measured at ten billion light-years away appears to us as it was ten billion years ago, because the light has taken that long to travel to the observer. If one were to look at a galaxy ten billion light-years away in one direction and another in the opposite direction, the total distance between them is twenty billion light-years. This means that the light from the first has not yet reached the second because the universe is only about 13.8 billion years old. In a more general sense, there are portions of the universe that are visible to us, but invisible to each other, outside each other's respective particle horizons.

Causal information propagation

In accepted relativistic physical theories, no information can travel faster than the speed of light. In this context, "information" means "any sort of physical interaction". For instance, heat will naturally flow from a hotter area to a cooler one, and in physics terms, this is one example of information exchange. Given the example above, the two galaxies in question cannot have shared any sort of information; they are not in causal contact. In the absence of common initial conditions, one would expect, then, that their physical properties would be different, and more generally, that the universe as a whole would have varying properties in causally disconnected regions.

Horizon problem

Contrary to this expectation, the observations of the cosmic microwave background (CMB) and galaxy surveys show that the observable universe is nearly isotropic, which, through the Copernican principle, also implies homogeneity. CMB sky surveys show that the temperatures of the CMB are coordinated to a level of ${\displaystyle \mathrm{\Delta }T/T\approx {10}^{-5},}$ where ${\displaystyle \mathrm{\Delta }T}$ is the difference between the observed temperature in a region of the sky and the average temperature of the sky ${\displaystyle T}$. This coordination implies that the entire sky, and thus the entire observable universe, must have been causally connected long enough for the universe to come into thermal equilibrium.

According to the Big Bang model, as the density of the expanding universe dropped, it eventually reached a temperature where photons fell out of thermal equilibrium with matter; they decoupled from the electron-proton plasma and began free-streaming across the universe. This moment in time is referred to as the epoch of Recombination, when electrons and protons became bound to form electrically neutral hydrogen; without free electrons to scatter the photons, the photons began free-streaming. This epoch is observed through the CMB. Since we observe the CMB as a background to objects at a smaller redshift, we describe this epoch as the transition of the universe from opaque to transparent. The CMB physically describes the 'surface of last scattering' as it appears to us as a surface, or a background, as shown in the figure below.

Note we use conformal time in the following diagrams. Conformal time describes the amount of time it would take a photon to travel from the location of the observer to the farthest observable distance (if the universe stopped expanding right now).

The decoupling, or the last scattering, is thought to have occurred about 300,000 years after the Big Bang, or at a redshift of about ${\displaystyle z_{rec}\approx 1100}$. We can determine both the approximate angular diameter of the universe and the physical size of the particle horizon that had existed at this time.

The angular diameter distance, in terms of redshift ${\displaystyle z}$, is described by ${\displaystyle d_A(z)=r(z)/(1+z)}$. If we assume a flat cosmology then,

${\displaystyle r(z)={\int }_{t_{em}}^{t_0}\frac{dt}{a(t)}={\int }_{a_{em}}^1\frac{da}{a^{2}H(a)}={\int }_0^z\frac{dz}{H(z)}.}$

The epoch of recombination occurred during a matter dominated era of the universe, so we can approximate ${\displaystyle H(z)}$ as ${\displaystyle H^2(z)\approx {\mathrm{\Omega }}_m{H}_0^2(1+z{)}^3}$. Putting these together, we see that the angular diameter distance, or the size of the observable universe for a redshift ${\displaystyle z_{rec}\approx 1100}$ is

${\displaystyle r(z)={\int }_0^z\frac{dz}{H(z)}=\frac{1}{\sqrt{{\mathrm{\Omega }}_m}{H}_0}{\int }_0^z\frac{dz}{(1+z{)}^{3/2}}=\frac{2}{\sqrt{{\mathrm{\Omega }}_m}{H}_0}\left(1-\frac{1}{\sqrt{1+z}}\right).}$

Since ${\displaystyle z\gg 1}$, we can approximate the above equation as

${\displaystyle r(z)\approx \frac{2}{\sqrt{{\mathrm{\Omega }}_m}{H}_0}.}$

Substituting this into our definition of angular diameter distance, we obtain

${\displaystyle d_A(z)\approx \frac{2}{\sqrt{{\mathrm{\Omega }}_m}{H}_0}\frac{1}{1+z}.}$

From this formula, we obtain the angular diameter distance of the cosmic microwave background as ${\displaystyle d_A(1100)\approx 14\mathrm{M}\mathrm{p}\mathrm{c}}$.

The particle horizon describes the maximum distance light particles could have traveled to the observer given the age of the universe. We can determine the comoving distance for the age of the universe at the time of recombination using ${\displaystyle r(z)}$ from earlier,

${\displaystyle d_{\text{hor,rec}}(z)={\int }_0^{t(z)}\frac{dt}{a(t)}={\int }_z^{\mathrm{\infty }}\frac{dz}{H(z)}\approx \frac{2}{\sqrt{{\mathrm{\Omega }}_m}{H}_0}{\left[\frac{1}{\sqrt{1+z}}\right]}_z^{\mathrm{\infty }}\approx \frac{2}{\sqrt{{\mathrm{\Omega }}_m}{H}_0}\frac{1}{\sqrt{1+z}}}$

To get the physical size of the particle horizon ${\displaystyle D}$,

${\displaystyle D(z)=a(z)d_{\text{hor,rec}}=\frac{d_{\text{hor,rec}}(z)}{1+z}}$

${\displaystyle D(1100)\approx 0.03\text{radians}\approx 2^{\circ }}$

We would expect any region of the CMB within 2 degrees of angular separation to have been in causal contact, but at any scale larger than 2° there should have been no exchange of information.

CMB regions that are separated by more than 2° lie outside one another's particle horizons and are causally disconnected. The horizon problem describes the fact that we see isotropy in the CMB temperature across the entire sky, despite the entire sky not being in causal contact to establish thermal equilibrium. Refer to the timespace diagram to the right for a visualization of this problem.

If the universe started with even slightly different temperatures in different places, the CMB should not be isotropic unless there is a mechanism that evens out the temperature by the time of decoupling. In reality, the CMB has the same temperature in the entire sky, 2.726 ± 0.001 K.