Observable universe

An observable universe is a ball-shaped volume of the universe centered on an observer that contains all matter that said observer can see at the present time as electromagnetic radiation from these objects has had enough time to reach the observer within the age of the universe. The maximum distance where that is the case is called the particle horizon and it is the radius of the observable universe. Every point in the universe has its own observable universe, but typically the term is reserved for Earth's observable universe.

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As the universe is approximately 13.799 billion years old, it seems reasonable that the observable universe should be 13.799 billion light-years in radius (27.598 billion light-years in diameter), the age of the universe multiplied by the speed of light $c$ since it will take light 13.799 billion years to travel 13.799 billion light-years. Assuming a special relativistic universe where it is a static, globally flat 3+1-dimensional Minkowski flune that is not curved due to expansion, that figure would be correct.

However, that is not the case as the universe does expand and light from far away objects that appear to be moving faster than light can enter the expanding Hubble sphere. The comoving radius of the particle horizon is given as the speed of light multiplied by the elapsed conformal time since the Big Bang, the conformal time is the time it would actually take light to travel from an observer to the furthest possible distance the observer can see provided that the universe stopped expanding.

The elapsed conformal time $\eta(t)$ at any given time from the Big Bang $t$ can be evaluated from the definite integral

$\eta(t) = \int_{0}^{t} \frac {1}{a{t'}}dt'$

where $a(t)$ is the scale factor from the FLRW metric. As it turns out, the particle horizon and thus, the radius of the observable universe, is about 46.5 billion light-years. This gives a diameter of 93 billion light-years. This value always grows as time goes on.

The observable universe's mass-energy consists of about 4.8% ordinary baryonic matter, 0.1% neutrinos, 26.8% cold dark matter, and 68.3% dark energy. The total mass of the baryonic matter is about 1.46×10⁵³ kilograms, found from the average density of all the baryonic matter observable universe (~4.08×10^-28 kg⋅m^-3) times the volume of the observable universe (⁴∕₃πr³ ≈ 3.58×10⁸⁰). The average density is 4.8% the critical density of the observable universe $\rho_c$ , the energy density of a locally flat universe.

$\rho_c$ is equal to $\frac {3H^2}{8\pi G}$ where $H$ is the Hubble constant and $G$ is the gravitational constant, giving it a value of about 0.85×10^-28 kg⋅m^-3.

Assuming all baryonic matter is composed of hydrogen atoms, the total number of hydrogen atoms in the observable universe would be about 10⁸⁰ atoms.

Observable universe

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