Particle horizon

Evolution of the particle horizon

In this section we consider the FLRW cosmological model. In that context, the universe can be approximated as composed by non-interacting constituents, each one being a perfect fluid with density ${\displaystyle {\rho }_i}$, partial pressure ${\displaystyle p_i}$ and state equation ${\displaystyle p_i={\omega }_i{\rho }_i}$, such that they add up to the total density ${\displaystyle \rho }$ and total pressure ${\displaystyle p}$. Let us now define the following functions:

• Hubble function ${\displaystyle H=\frac{\dot{a}}{a}}$
• The critical density ${\displaystyle {\rho }_c=\frac{3}{8\pi G}{H}^2}$
• The i-th dimensionless energy density ${\displaystyle {\mathrm{\Omega }}_i=\frac{{\rho }_i}{{\rho }_c}}$
• The dimensionless energy density ${\displaystyle \mathrm{\Omega }=\frac{\rho }{{\rho }_c}=\sum {\mathrm{\Omega }}_i}$
• The redshift ${\displaystyle z}$ given by the formula ${\displaystyle 1+z=\frac{a_0}{a(t)}}$

Any function with a zero subscript denote the function evaluated at the present time ${\displaystyle t_0}$ (or equivalently ${\displaystyle z=0}$). The last term can be taken to be ${\displaystyle 1}$ including the curvature state equation. It can be proved that the Hubble function is given by

$${\displaystyle H(z)=H_0\sqrt{\sum {\mathrm{\Omega }}_{i0}(1+z{)}^{n_i}}}$$

where the dilution exponent ${\displaystyle n_i=3(1+{\omega }_i)}$. Notice that the addition ranges over all possible partial constituents and in particular there can be countably infinitely many. With this notation we have:

$${\displaystyle \text{The particle horizon }{d}_p\text{ exists if and only if }N>2}$$

where ${\displaystyle N}$ is the largest ${\displaystyle n_i}$ (possibly infinite). The evolution of the particle horizon for an expanding universe (${\displaystyle \dot{a}>0}$) is:

$${\displaystyle \frac{d}{dt}{d}_p=d_p(z)H(z)+c}$$

where ${\displaystyle c}$ is the speed of light and can be taken to be ${\displaystyle 1}$ (natural units). Notice that the derivative is made with respect to the FLRW-time ${\displaystyle t}$, while the functions are evaluated at the redshift ${\displaystyle z}$ which are related as stated before. We have an analogous but slightly different result for event horizon.

Horizon problem

The concept of a particle horizon can be used to illustrate the horizon problem, which is an unresolved issue associated with the Big Bang model. Extrapolating back to the time of recombination when the cosmic microwave background (CMB) was emitted, we obtain a particle horizon of about

$${\displaystyle H_p(t_{\text{CMB}})=c{\eta }_{\text{CMB}}=284\text{ Mpc}=8.9\times {10}^{-3}{H}_p(t_0)}$$

which corresponds to a proper size at that time of:

$${\displaystyle a_{\text{CMB}}{H}_p(t_{\text{CMB}})=261\text{ kpc}}$$

Since we observe the CMB to be emitted essentially from our particle horizon (${\displaystyle 284\text{ Mpc}\ll 14.4\text{ Gpc}}$), our expectation is that parts of the cosmic microwave background (CMB) that are separated by about a fraction of a great circle across the sky of

$${\displaystyle f=\frac{H_p(t_{\text{CMB}})}{H_p(t_0)}}$$

(an angular size of ${\displaystyle \theta \sim {1.7}^{\circ }}$) should be out of causal contact with each other. That the entire CMB is in thermal equilibrium and approximates a blackbody so well is therefore not explained by the standard explanations about the way the expansion of the universe proceeds. The most popular resolution to this problem is cosmic inflation.