Recombination (cosmology)

Recombination (cosmology)

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In cosmology
Cosmology is the discipline that deals with the nature of the Universe as a whole. Cosmologists seek to understand the origin, evolution, structure, and ultimate fate of the Universe at large, as well as the natural laws that keep it in order...

, recombination refers to the epoch
Epoch (astronomy)
In astronomy, an epoch is a moment in time used as a reference point for some time-varying astronomical quantity, such as celestial coordinates, or elliptical orbital elements of a celestial body, where these are subject to perturbations and vary with time...

 at which charged electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

s and proton
The proton is a subatomic particle with the symbol or and a positive electric charge of 1 elementary charge. One or more protons are present in the nucleus of each atom, along with neutrons. The number of protons in each atom is its atomic number....

s first became bound
Bound state
In physics, a bound state describes a system where a particle is subject to a potential such that the particle has a tendency to remain localised in one or more regions of space...

 to form electrically neutral
Electric charge
Electric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...

Hydrogen is the chemical element with atomic number 1. It is represented by the symbol H. With an average atomic weight of , hydrogen is the lightest and most abundant chemical element, constituting roughly 75% of the Universe's chemical elemental mass. Stars in the main sequence are mainly...

The atom is a basic unit of matter that consists of a dense central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...

s.Note that the term recombination is a misnomer, considering that it represents the first time that electrically neutral hydrogen formed. After the Big Bang
Big Bang
The Big Bang theory is the prevailing cosmological model that explains the early development of the Universe. According to the Big Bang theory, the Universe was once in an extremely hot and dense state which expanded rapidly. This rapid expansion caused the young Universe to cool and resulted in...

, the universe was a hot, dense plasma
Plasma (physics)
In physics and chemistry, plasma is a state of matter similar to gas in which a certain portion of the particles are ionized. Heating a gas may ionize its molecules or atoms , thus turning it into a plasma, which contains charged particles: positive ions and negative electrons or ions...

 of photon
In physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...

s, electrons, and protons. This plasma was effectively opaque to electromagnetic radiation, as the distance each photon could travel before encountering a charged particle was very short. As the universe expanded
Metric expansion of space
The metric expansion of space is the increase of distance between distant parts of the universe with time. It is an intrinsic expansion—that is, it is defined by the relative separation of parts of the universe and not by motion "outward" into preexisting space...

, it also cooled. Eventually, the universe cooled to the point that the formation of neutral hydrogen was energetically favored, and the fraction of free electrons and protons as compared to neutral hydrogen decreased to about 1 part in 10,000.

Shortly after, photons decoupled from matter
Matter is a general term for the substance of which all physical objects consist. Typically, matter includes atoms and other particles which have mass. A common way of defining matter is as anything that has mass and occupies volume...

 in the universe, which leads to recombination sometimes being called photon decoupling, although recombination and photon decoupling are distinct events. Once photons decoupled from matter, they traveled freely through the universe without interacting with matter, and constitute what we observe today as cosmic microwave background radiation
Cosmic microwave background radiation
In cosmology, cosmic microwave background radiation is thermal radiation filling the observable universe almost uniformly....

. Recombination occurred when the universe was roughly 360,000 years old, or at a redshift
In physics , redshift happens when light seen coming from an object is proportionally increased in wavelength, or shifted to the red end of the spectrum...

 of z = .

Derivation of recombination epoch

It is possible to find a rough estimate of the redshift of the recombination epoch, starting by considering that during the era preceding recombination, the photons were primarily coupled to matter through the reaction

This reaction requires that the photon (γ) have an energy of at least 13.6 electron volts. As long as photons are coupled to matter, this reaction will be in statistical equilibrium, and so the Saha equation can be applied to determine the equilibrium values of the constituents. This results in the equation

where n represents the number density of the subscripted particle, me is the mass of the electron
Electron rest mass
The electron rest mass is the mass of a stationary electron. It is one of the fundamental constants of physics, and is also very important in chemistry because of its relation to the Avogadro constant...

, kB is Boltzmann's constant, T is the temperature, ħ is the reduced Planck's constant, and Q is the binding energy
Binding energy
Binding energy is the mechanical energy required to disassemble a whole into separate parts. A bound system typically has a lower potential energy than its constituent parts; this is what keeps the system together—often this means that energy is released upon the creation of a bound state...

 of hydrogen. Noting that charge neutrality requires ne = np, and then defining the fractional ionization as

the Saha equation can be rewritten as

The final step is to put this equation in terms of the number density of photons, which is related to the number density of baryons through the baryon-to-photon ratio η. The reason for this is that the baryon-to-photon ratio can be measured, and the number density of photons is given by

where c is the speed of light
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...

. Then

Solving this equation for a 50 percent ionization yields a recombination temperature of roughly . This, in turn, gives the redshift as approximately z = , as the temperature of the radiation in the universe is given by . In units of electronvolts, this temperature is roughly . This is nearly two orders of magnitude lower than the binding energy
Binding energy
Binding energy is the mechanical energy required to disassemble a whole into separate parts. A bound system typically has a lower potential energy than its constituent parts; this is what keeps the system together—often this means that energy is released upon the creation of a bound state...

 of hydrogen
Hydrogen atom
A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively-charged proton and a single negatively-charged electron bound to the nucleus by the Coulomb force...

, which may seem strange, as equilibrium often occurs when the energy scales of two phenomena are roughly equal. The reason for the difference is because photons greatly outnumber baryons; the baryon-to-photon ratio is approximately 10−9. If there are roughly the same number of photons with an energy greater than the binding energy of hydrogen as there are hydrogen atoms, then the gas will remain ionized. There will be some photons in the Wien region of the black body spectrum with an energy greater than kT, and the number of photons with an energy greater than 13.6 eV does not drop below the number of hydrogen atoms until the temperature is roughly 4000K or 0.3 eV. A different statement of this is that recombination was delayed due to the high entropy
Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...

 of the universe.

This derivation relied on the assumptions of thermodynamic equilibrium
Thermodynamic equilibrium
In thermodynamics, a thermodynamic system is said to be in thermodynamic equilibrium when it is in thermal equilibrium, mechanical equilibrium, radiative equilibrium, and chemical equilibrium. The word equilibrium means a state of balance...

 and recombination directly to the ground state
Ground state
The ground state of a quantum mechanical system is its lowest-energy state; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state...

 of hydrogen, each of which simplifies the calculation but also modifies the result. Recombination to an excited state of hydrogen means that recombination proceeds more slowly than that predicted with the Saha equation. A more careful treatment of the physics of recombination yields a value closer to z = .


Prior to recombination, photons were not able to freely travel through the universe, as they constantly scattered
Thomson scattering
Thomson scattering is the elastic scattering of electromagnetic radiation by a free charged particle, as described by classical electromagnetism. It is just the low-energy limit of Compton scattering: the particle kinetic energy and photon frequency are the same before and after the scattering...

 off the free electrons and protons. This scattering causes a loss of information, and "there is therefore a photon barrier at a redshift" near that of recombination that prevents us from using photons to directly learn about the universe at larger redshifts. Once recombination had occurred, however, the mean free path of photons greatly increased due to the lower number of free electrons. Shortly after recombination, the photon mean free path became larger than the Hubble length, and photons traveled freely without interacting with matter. For this reason, recombination is closely associated with the last scattering surface, which is the name for the last time at which the photons in the cosmic microwave background interacted with matter. However, these two events are distinct, and that in a universe with different values for the baryon-to-photon ratio and matter density, recombination and photon decoupling need not have occurred at the same epoch.