Quantum tunnelling

From Wikipedia, the free encyclopedia

 

Quantum tunnelling or tunneling (see spelling differences) is the quantum mechanical phenomenon where a particle tunnels through a barrier that it classically cannot surmount. This plays an essential role in several physical phenomena, such as the nuclear fusion that occurs in main sequence stars like the Sun.^[1] It has important applications to modern devices such as the tunnel diode,^[2] quantum computing, and the scanning tunnelling microscope. The effect was predicted in the early 20th century and its acceptance as a general physical phenomenon came mid-century.^[3]

Fundamental quantum mechanical concepts are central to this phenomenon, which makes quantum tunnelling one of the novel implications of quantum mechanics. Quantum tunneling is projected to create physical limits to how small transistors can get, due to electrons being able to tunnel past them if they are too small.^[4]

Tunnelling is often explained in terms of Heisenberg uncertainty principle and the wave–particle duality of matter.

History

Quantum tunnelling was developed from the study of radioactivity,^[3] which was discovered in 1896 by Henri Becquerel.^[5] Radioactivity was examined further by Marie Curie and Pierre Curie, for which they earned the Nobel Prize in Physics in 1903.^[5] Ernest Rutherford and Egon Schweidler studied its nature, which was later verified empirically by Friedrich Kohlrausch. The idea of the half-life and the possibility of predicting decay was created from their work.^[3]

In 1901, Robert Francis Earhart, while investigating the conduction of gases between closely spaced electrodes using the Michelson interferometer to measure the spacing, discovered an unexpected conduction regime. J. J. Thomson commented that the finding warranted further investigation. In 1911 and then 1914, then-graduate student Franz Rother, employing Earhart's method for controlling and measuring the electrode separation but with a sensitive platform galvanometer, directly measured steady field emission currents. In 1926, Rother, using a still newer platform galvanometer of sensitivity 26 pA, measured the field emission currents in a "hard" vacuum between closely spaced electrodes.^[6]

Quantum tunneling was first noticed in 1927 by Friedrich Hund when he was calculating the ground state of the double-well potential^[5] and independently in the same year by Leonid Mandelstam and Mikhail Leontovich in their analysis of the implications of the then new Schrödinger wave equation for the motion of a particle in a confining potential of a limited spatial extent.^[7] Its first application was a mathematical explanation for alpha decay, which was done in 1928 by George Gamow (who was aware of the findings of Mandelstam and Leontovich^[8]) and independently by Ronald Gurney and Edward Condon.^{[9][10][11][12]} The two researchers simultaneously solved the Schrödinger equation for a model nuclear potential and derived a relationship between the half-life of the particle and the energy of emission that depended directly on the mathematical probability of tunnelling.

After attending a seminar by Gamow, Max Born recognised the generality of tunnelling. He realised that it was not restricted to nuclear physics, but was a general result of quantum mechanics that applies to many different systems.^[3] Shortly thereafter, both groups considered the case of particles tunnelling into the nucleus. The study of semiconductors and the development of transistors and diodes led to the acceptance of electron tunnelling in solids by 1957. The work of Leo Esaki, Ivar Giaever and Brian Josephson predicted the tunnelling of superconducting Cooper pairs, for which they received the Nobel Prize in Physics in 1973.^[3] In 2016, the quantum tunneling of water was discovered.^[13]

Introduction to the concept

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Animation showing the tunnel effect and its application to an STM

 

Quantum tunnelling through a barrier. The energy of the tunnelled particle is the same but the probability amplitude is decreased.

 

A simulation of a wave packet incident on a potential barrier. In relative units, the barrier energy is 20, greater than the mean wave packet energy of 14. A portion of the wave packet passes through the barrier.

 

Quantum tunnelling through a barrier. At the origin (x=0), there is a very high, but narrow potential barrier. A significant tunnelling effect can be seen.

Quantum tunnelling falls under the domain of quantum mechanics: the study of what happens at the quantum scale. This process cannot be directly perceived, but much of its understanding is shaped by the microscopic world, which classical mechanics cannot adequately explain. To understand the phenomenon, particles attempting to travel between potential barriers can be compared to a ball trying to roll over a hill; quantum mechanics and classical mechanics differ in their treatment of this scenario. Classical mechanics predicts that particles that do not have enough energy to classically surmount a barrier will not be able to reach the other side. Thus, a ball without sufficient energy to surmount the hill would roll back down. Or, lacking the energy to penetrate a wall, it would bounce back (reflection) or in the extreme case, bury itself inside the wall (absorption). In quantum mechanics, these particles can, with a very small probability, tunnel to the other side, thus crossing the barrier. Here, the "ball" could, in a sense, borrow energy from its surroundings to tunnel through the wall or "roll over the hill", paying it back by making the reflected electrons more energetic than they otherwise would have been.^[14]

The reason for this difference comes from the treatment of matter in quantum mechanics as having properties of waves and particles. One interpretation of this duality involves the Heisenberg uncertainty principle, which defines a limit on how precisely the position and the momentum of a particle can be known at the same time.^[5] This implies that there are no solutions with a probability of exactly zero (or one), though a solution may approach infinity if, for example, the calculation for its position was taken as a probability of 1, the other, i.e. its speed, would have to be infinity. Hence, the probability of a given particle's existence on the opposite side of an intervening barrier is non-zero, and such particles will appear on the 'other' (a semantically difficult word in this instance) side with a relative frequency proportional to this probability.

An electron wavepacket directed at a potential barrier. Note the dim spot on the right that represents tunnelling electrons.

 

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Quantum tunnelling in the phase space formulation of quantum mechanics. Wigner function for tunnelling through the potential barrier ${\displaystyle U(x)=8e^{-0.25x^{2}}}$ in atomic units (a.u.). The solid lines represent the level set of the Hamiltonian ${\displaystyle H(x,p)=p^{2}/2+U(x)}$.

The tunnelling problem

The wave function of a particle summarises everything that can be known about a physical system.^[15] Therefore, problems in quantum mechanics center on the analysis of the wave function for a system. Using mathematical formulations of quantum mechanics, such as the Schrödinger equation, the wave function can be solved. This is directly related to the probability density of the particle's position, which describes the probability that the particle is at any given place. In the limit of large barriers, the probability of tunnelling decreases for taller and wider barriers.

For simple tunnelling-barrier models, such as the rectangular barrier, an analytic solution exists. Problems in real life often do not have one, so "semiclassical" or "quasiclassical" methods have been developed to give approximate solutions to these problems, like the WKB approximation. Probabilities may be derived with arbitrary precision, constrained by computational resources, via Feynman's path integral method; such precision is seldom required in engineering practice.^{[citation needed]}

Related phenomena

There are several phenomena that have the same behaviour as quantum tunnelling, and thus can be accurately described by tunnelling. Examples include the tunnelling of a classical wave-particle association,^[16] evanescent wave coupling (the application of Maxwell's wave-equation to light) and the application of the non-dispersive wave-equation from acoustics applied to "waves on strings". Evanescent wave coupling, until recently, was only called "tunnelling" in quantum mechanics; now it is used in other contexts.

These effects are modelled similarly to the rectangular potential barrier. In these cases, there is one transmission medium through which the wave propagates that is the same or nearly the same throughout, and a second medium through which the wave travels differently. This can be described as a thin region of medium B between two regions of medium A. The analysis of a rectangular barrier by means of the Schrödinger equation can be adapted to these other effects provided that the wave equation has travelling wave solutions in medium A but real exponential solutions in medium B.

In optics, medium A is a vacuum while medium B is glass. In acoustics, medium A may be a liquid or gas and medium B a solid. For both cases, medium A is a region of space where the particle's total energy is greater than its potential energy and medium B is the potential barrier. These have an incoming wave and resultant waves in both directions. There can be more mediums and barriers, and the barriers need not be discrete; approximations are useful in this case.

Applications

Tunnelling occurs with barriers of thickness around 1-3 nm and smaller,^[17] but is the cause of some important macroscopic physical phenomena. For instance, tunnelling is a source of current leakage in very-large-scale integration (VLSI) electronics and results in the substantial power drain and heating effects that plague high-speed and mobile technology; it is considered the lower limit on how small computer chips can be made.^[18] Tunnelling is a fundamental technique used to program the floating gates of flash memory, which is one of the most significant inventions that have shaped consumer electronics in the last two decades.

Nuclear fusion in stars

Quantum tunnelling is essential for nuclear fusion in stars. Temperature and pressure in the core of stars are insufficient for nuclei to overcome the Coulomb barrier in order to achieve a thermonuclear fusion. However, there is some probability to penetrate the barrier due to quantum tunnelling. Though the probability is very low, the extreme number of nuclei in a star generates a steady fusion reaction over millions or even billions of years - a precondition for the evolution of life in insolation habitable zones.^[19]

Radioactive decay

Radioactive decay is the process of emission of particles and energy from the unstable nucleus of an atom to form a stable product. This is done via the tunnelling of a particle out of the nucleus (an electron tunnelling into the nucleus is electron capture). This was the first application of quantum tunnelling and led to the first approximations. Radioactive decay is also a relevant issue for astrobiology as this consequence of quantum tunnelling is creating a constant source of energy over a large period of time for environments outside the circumstellar habitable zone where insolation would not be possible (subsurface oceans) or effective.^[19]

Astrochemistry in interstellar clouds

By including quantum tunnelling the astrochemical syntheses of various molecules in interstellar clouds can be explained such as the synthesis of molecular hydrogen, water (ice) and the prebiotic important formaldehyde.^[19]

Quantum biology

Quantum tunnelling is among the central non trivial quantum effects in quantum biology. Here it is important both as electron tunnelling and proton tunnelling. Electron tunnelling is a key factor in many biochemical redox reactions (photosynthesis, cellular respiration) as well as enzymatic catalysis while proton tunnelling is a key factor in spontaneous mutation of DNA.^[19]

Spontaneous mutation of DNA occurs when normal DNA replication takes place after a particularly significant proton has defied the odds in quantum tunnelling in what is called "proton tunnelling"^[20] (quantum biology). A hydrogen bond joins normal base pairs of DNA. There exists a double well potential along a hydrogen bond separated by a potential energy barrier. It is believed that the double well potential is asymmetric with one well deeper than the other so the proton normally rests in the deeper well. For a mutation to occur, the proton must have tunnelled into the shallower of the two potential wells. The movement of the proton from its regular position is called a tautomeric transition. If DNA replication takes place in this state, the base pairing rule for DNA may be jeopardised causing a mutation.^[21] Per-Olov Lowdin was the first to develop this theory of spontaneous mutation within the double helix (quantum bio). Other instances of quantum tunnelling-induced mutations in biology are believed to be a cause of ageing and cancer.^[22]

Cold emission

Cold emission of electrons is relevant to semiconductors and superconductor physics. It is similar to thermionic emission, where electrons randomly jump from the surface of a metal to follow a voltage bias because they statistically end up with more energy than the barrier, through random collisions with other particles. When the electric field is very large, the barrier becomes thin enough for electrons to tunnel out of the atomic state, leading to a current that varies approximately exponentially with the electric field.^[23] These materials are important for flash memory, vacuum tubes, as well as some electron microscopes.

Tunnel junction

A simple barrier can be created by separating two conductors with a very thin insulator. These are tunnel junctions, the study of which requires quantum tunnelling.^[24] Josephson junctions take advantage of quantum tunnelling and the superconductivity of some semiconductors to create the Josephson effect. This has applications in precision measurements of voltages and magnetic fields,^[23] as well as the multijunction solar cell.

Quantum-dot cellular automata

QCA is a molecular binary logic synthesis technology that operates by the inter-island electron tunneling system. This is a very low power and fastest device that can operate at maximum 15 PHz of frequency^[25].

A working mechanism of a resonant tunnelling diode device, based on the phenomenon of quantum tunnelling through the potential barriers.

Tunnel diode

Diodes are electrical semiconductor devices that allow electric current flow in one direction more than the other. The device depends on a depletion layer between N-type and P-type semiconductors to serve its purpose; when these are very heavily doped the depletion layer can be thin enough for tunnelling. Then, when a small forward bias is applied the current due to tunnelling is significant. This has a maximum at the point where the voltage bias is such that the energy level of the p and n conduction bands are the same. As the voltage bias is increased, the two conduction bands no longer line up and the diode acts typically.^[26]

Because the tunnelling current drops off rapidly, tunnel diodes can be created that have a range of voltages for which current decreases as voltage is increased. This peculiar property is used in some applications, like high speed devices where the characteristic tunnelling probability changes as rapidly as the bias voltage.^[26]

The resonant tunnelling diode makes use of quantum tunnelling in a very different manner to achieve a similar result. This diode has a resonant voltage for which there is a lot of current that favors a particular voltage, achieved by placing two very thin layers with a high energy conductance band very near each other. This creates a quantum potential well that has a discrete lowest energy level. When this energy level is higher than that of the electrons, no tunnelling will occur, and the diode is in reverse bias. Once the two voltage energies align, the electrons flow like an open wire. As the voltage is increased further tunnelling becomes improbable and the diode acts like a normal diode again before a second energy level becomes noticeable.^[27]

Tunnel field-effect transistors

A European research project has demonstrated field effect transistors in which the gate (channel) is controlled via quantum tunnelling rather than by thermal injection, reducing gate voltage from ~1 volt to 0.2 volts and reducing power consumption by up to 100×. If these transistors can be scaled up into VLSI chips, they will significantly improve the performance per power of integrated circuits.^[28]

Quantum conductivity

While the Drude model of electrical conductivity makes excellent predictions about the nature of electrons conducting in metals, it can be furthered by using quantum tunnelling to explain the nature of the electron's collisions.^[23] When a free electron wave packet encounters a long array of uniformly spaced barriers the reflected part of the wave packet interferes uniformly with the transmitted one between all barriers so that there are cases of 100% transmission. The theory predicts that if positively charged nuclei form a perfectly rectangular array, electrons will tunnel through the metal as free electrons, leading to an extremely high conductance, and that impurities in the metal will disrupt it significantly.^[23]

Scanning tunnelling microscope

The scanning tunnelling microscope (STM), invented by Gerd Binnig and Heinrich Rohrer, may allow imaging of individual atoms on the surface of a material.^[23] It operates by taking advantage of the relationship between quantum tunnelling with distance. When the tip of the STM's needle is brought very close to a conduction surface that has a voltage bias, by measuring the current of electrons that are tunnelling between the needle and the surface, the distance between the needle and the surface can be measured. By using piezoelectric rods that change in size when voltage is applied over them the height of the tip can be adjusted to keep the tunnelling current constant. The time-varying voltages that are applied to these rods can be recorded and used to image the surface of the conductor.^[23] STMs are accurate to 0.001 nm, or about 1% of atomic diameter.^[27]

Faster than light

Some physicists have claimed that it is possible for spin-zero particles to travel faster than the speed of light when tunnelling.^[3] This apparently violates the principle of causality, since there will be a frame of reference in which it arrives before it has left. In 1998, Francis E. Low reviewed briefly the phenomenon of zero-time tunnelling.^[29] More recently experimental tunnelling time data of phonons, photons, and electrons have been published by Günter Nimtz.^[30]
Other physicists, such as Herbert Winful ^[31], have disputed these claims. Winful argues that the wavepacket of a tunnelling particle propagates locally, so a particle can't tunnel through the barrier non-locally. Winful also argues that the experiments that are purported to show non-local propagation have been misinterpreted. In particular, the group velocity of a wavepacket does not measure its speed, but is related to the amount of time the wavepacket is stored in the barrier.

Mathematical discussions of quantum tunnelling

The following subsections discuss the mathematical formulations of quantum tunnelling.

The Schrödinger equation

The time-independent Schrödinger equation for one particle in one dimension can be written as

${\displaystyle -{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}\Psi (x)+V(x)\Psi (x)=E\Psi (x)}$ or

${\displaystyle {\frac {d^{2}}{dx^{2}}}\Psi (x)={\frac {2m}{\hbar ^{2}}}\left(V(x)-E\right)\Psi (x)\equiv {\frac {2m}{\hbar ^{2}}}M(x)\Psi (x),}$

where ${\displaystyle \hbar }$ is the reduced Planck's constant, m is the particle mass, x represents distance measured in the direction of motion of the particle, Ψ is the Schrödinger wave function, V is the potential energy of the particle (measured relative to any convenient reference level), E is the energy of the particle that is associated with motion in the x-axis (measured relative to V), and M(x) is a quantity defined by V(x) – E which has no accepted name in physics.

The solutions of the Schrödinger equation take different forms for different values of x, depending on whether M(x) is positive or negative. When M(x) is constant and negative, then the Schrödinger equation can be written in the form

${\displaystyle {\frac {d^{2}}{dx^{2}}}\Psi (x)={\frac {2m}{\hbar ^{2}}}M(x)\Psi (x)=-k^{2}\Psi (x),\;\;\;\;\;\;\mathrm {where} \;\;\;k^{2}=-{\frac {2m}{\hbar ^{2}}}M.}$

The solutions of this equation represent travelling waves, with phase-constant +k or -k. Alternatively, if M(x) is constant and positive, then the Schrödinger equation can be written in the form

${\displaystyle {\frac {d^{2}}{dx^{2}}}\Psi (x)={\frac {2m}{\hbar ^{2}}}M(x)\Psi (x)={\kappa }^{2}\Psi (x),\;\;\;\;\;\;\mathrm {where} \;\;\;{\kappa }^{2}={\frac {2m}{\hbar ^{2}}}M.}$

The solutions of this equation are rising and falling exponentials in the form of evanescent waves. When M(x) varies with position, the same difference in behaviour occurs, depending on whether M(x) is negative or positive. It follows that the sign of M(x) determines the nature of the medium, with negative M(x) corresponding to medium A as described above and positive M(x) corresponding to medium B. It thus follows that evanescent wave coupling can occur if a region of positive M(x) is sandwiched between two regions of negative M(x), hence creating a potential barrier.

The mathematics of dealing with the situation where M(x) varies with x is difficult, except in special cases that usually do not correspond to physical reality. A discussion of the semi-classical approximate method, as found in physics textbooks, is given in the next section. A full and complicated mathematical treatment appears in the 1965 monograph by Fröman and Fröman noted below. Their ideas have not been incorporated into physics textbooks, but their corrections have little quantitative effect.

The WKB approximation

The wave function is expressed as the exponential of a function:

${\displaystyle \Psi (x)=e^{\Phi (x)}}$, where ${\displaystyle \Phi ''(x)+\Phi '(x)^{2}={\frac {2m}{\hbar ^{2}}}\left(V(x)-E\right).}$

${\displaystyle \Phi '(x)}$ is then separated into real and imaginary parts:

${\displaystyle \Phi '(x)=A(x)+iB(x)}$, where A(x) and B(x) are real-valued functions.

Substituting the second equation into the first and using the fact that the imaginary part needs to be 0 results in:

${\displaystyle A'(x)+A(x)^{2}-B(x)^{2}={\frac {2m}{\hbar ^{2}}}\left(V(x)-E\right)}$.

To solve this equation using the semiclassical approximation, each function must be expanded as a power series in ${\displaystyle \hbar }$. From the equations, the power series must start with at least an order of ${\displaystyle \hbar ^{-1}}$ to satisfy the real part of the equation; for a good classical limit starting with the highest power of Planck's constant possible is preferable, which leads to

${\displaystyle A(x)={\frac {1}{\hbar }}\sum _{k=0}^{\infty }\hbar ^{k}A_{k}(x)}$

and

${\displaystyle B(x)={\frac {1}{\hbar }}\sum _{k=0}^{\infty }\hbar ^{k}B_{k}(x)}$,

with the following constraints on the lowest order terms,

${\displaystyle A_{0}(x)^{2}-B_{0}(x)^{2}=2m\left(V(x)-E\right)}$

and

${\displaystyle A_{0}(x)B_{0}(x)=0}$.

At this point two extreme cases can be considered.

Case 1 If the amplitude varies slowly as compared to the phase ${\displaystyle A_{0}(x)=0}$ and

${\displaystyle B_{0}(x)=\pm {\sqrt {2m\left(E-V(x)\right)}}}$

which corresponds to classical motion. Resolving the next order of expansion yields

${\displaystyle \Psi (x)\approx C{\frac {e^{i\int dx{\sqrt {{\frac {2m}{\hbar ^{2}}}\left(E-V(x)\right)}}+\theta }}{\sqrt[{4}]{{\frac {2m}{\hbar ^{2}}}\left(E-V(x)\right)}}}}$

Case 2

If the phase varies slowly as compared to the amplitude, ${\displaystyle B_{0}(x)=0}$ and

${\displaystyle A_{0}(x)=\pm {\sqrt {2m\left(V(x)-E\right)}}}$

which corresponds to tunnelling. Resolving the next order of the expansion yields

${\displaystyle \Psi (x)\approx {\frac {C_{+}e^{+\int dx{\sqrt {{\frac {2m}{\hbar ^{2}}}\left(V(x)-E\right)}}}+C_{-}e^{-\int dx{\sqrt {{\frac {2m}{\hbar ^{2}}}\left(V(x)-E\right)}}}}{\sqrt[{4}]{{\frac {2m}{\hbar ^{2}}}\left(V(x)-E\right)}}}}$

In both cases it is apparent from the denominator that both these approximate solutions are bad near the classical turning points ${\displaystyle E=V(x)}$. Away from the potential hill, the particle acts similar to a free and oscillating wave; beneath the potential hill, the particle undergoes exponential changes in amplitude. By considering the behaviour at these limits and classical turning points a global solution can be made.

To start, choose a classical turning point, ${\displaystyle x_{1}}$ and expand ${\displaystyle {\frac {2m}{\hbar ^{2}}}\left(V(x)-E\right)}$ in a power series about ${\displaystyle x_{1}}$:

${\displaystyle {\frac {2m}{\hbar ^{2}}}\left(V(x)-E\right)=v_{1}(x-x_{1})+v_{2}(x-x_{1})^{2}+\cdots }$

Keeping only the first order term ensures linearity:

${\displaystyle {\frac {2m}{\hbar ^{2}}}\left(V(x)-E\right)=v_{1}(x-x_{1})}$.

Using this approximation, the equation near ${\displaystyle x_{1}}$ becomes a differential equation:

${\displaystyle {\frac {d^{2}}{dx^{2}}}\Psi (x)=v_{1}(x-x_{1})\Psi (x)}$.

This can be solved using Airy functions as solutions.

${\displaystyle \Psi (x)=C_{A}Ai\left({\sqrt[{3}]{v_{1}}}(x-x_{1})\right)+C_{B}Bi\left({\sqrt[{3}]{v_{1}}}(x-x_{1})\right)}$

Taking these solutions for all classical turning points, a global solution can be formed that links the limiting solutions. Given the two coefficients on one side of a classical turning point, the two coefficients on the other side of a classical turning point can be determined by using this local solution to connect them.

Hence, the Airy function solutions will asymptote into sine, cosine and exponential functions in the proper limits. The relationships between ${\displaystyle C,\theta }$ and ${\displaystyle C_{+},C_{-}}$ are

${\displaystyle C_{+}={\frac {1}{2}}C\cos {\left(\theta -{\frac {\pi }{4}}\right)}}$

and

${\displaystyle C_{-}=-C\sin {\left(\theta -{\frac {\pi }{4}}\right)}}$

With the coefficients found, the global solution can be found. Therefore, the transmission coefficient for a particle tunnelling through a single potential barrier is

${\displaystyle T(E)=e^{-2\int _{x_{1}}^{x_{2}}\mathrm {d} x{\sqrt {{\frac {2m}{\hbar ^{2}}}\left[V(x)-E\right]}}}}$,

where ${\displaystyle x_{1},x_{2}}$ are the two classical turning points for the potential barrier.

For a rectangular barrier, this expression is simplified to:

${\displaystyle T(E)=e^{-2{\sqrt {{\frac {2m}{\hbar ^{2}}}(V_{0}-E)}}(x_{2}-x_{1})}={\tilde {V}}_{0}^{-(x_{2}-x_{1})}}$.

Primordial black hole

From Wikipedia, the free encyclopedia

Primordial black holes are a hypothetical type of black hole that formed soon after the Big Bang. In the early universe, high densities and inhomogeneous conditions could lead sufficiently dense regions to undergo gravitational collapse, forming black holes. Yakov Borisovich Zel'dovich and Igor Dmitriyevich Novikov in 1966^[1] first proposed the existence of such black holes, but the theory behind their origins was first studied in depth by Stephen Hawking in 1971.^[2] Since primordial black holes did not form from stellar gravitational collapse, their masses can be far below stellar mass (c. 2×10³⁰ kg). Hawking calculated that primordial black holes could weigh as little as 10⁻⁸ kg, about the weight of a human ovum.

Theoretical history

Depending on the model, primordial black holes could have initial masses ranging from 10⁻⁸ kg (the so-called Planck relics) to more than thousands of solar masses. However, primordial black holes with a mass lower than 10¹¹ kg would have evaporated due to Hawking radiation in a time much shorter than the age of the Universe, so they cannot have survived until the present Universe. A noticeable exception is the case of Planck relics that could eventually be stable.^{[clarification needed]} The abundance of primordial black holes could be as important as the one of dark matter, to which they are a plausible candidate. Primordial black holes are also good candidates for being the seeds of the supermassive black holes at the center of massive galaxies, as well as of intermediate-mass black holes.^[3]

Primordial black holes belong to the class of massive compact halo objects (MACHOs). They are naturally a good dark matter candidate: they are (nearly) collision-less and stable (if sufficiently massive), they have non-relativistic velocities, and they form very early in the history of the Universe (typically less than one second after the Big Bang). Nevertheless, tight limits on their abundances have been set up from various astrophysical and cosmological observations, so that it is now excluded that they contribute importantly to the dark matter over most of the plausible mass range.

In March 2016, one month after the announcement of the detection by Advanced LIGO/VIRGO of gravitational waves emitted by the merging of two 30 solar mass black holes (about 6×10³¹ kg), three groups of researchers proposed independently that the detected black holes had a primordial origin.^[4][5][6][7] Two of them found that the merging rates inferred by LIGO are consistent with a scenario in which all the dark matter is made of primordial black holes, if a non-negligible fraction of them are somehow clustered within halos such as faint dwarf galaxies or globular clusters, as expected by the standard theory of cosmic structure formation. The third group claimed that these merging rates are incompatible with an all-dark-matter scenario and that primordial black holes could only contribute to less than one percent of the total dark matter. The unexpected large mass of the black holes detected by LIGO has strongly revived the interest for primordial black holes with masses in the range of 1 to 100 solar masses. It is however still unclear and debated whether this range is excluded or not by other observations, such as the absence of micro-lensing of stars, the cosmic microwave background anisotropies, the size of faint dwarf galaxies, and the absence of correlation between X-ray and radio sources towards the galactic center.

In May 2016, Alexander Kashlinsky suggested that the observed spatial correlations in the unresolved gamma-ray and X-ray background radiations could be due to primordial black holes with similar masses, if their abundance is comparable to the one of dark matter.^[8]

Formation

Primordial black holes could have formed in the very early Universe (less than one second after the Big-Bang), during the so-called radiation dominated era. The essential ingredient for a primordial black hole to form is a fluctuation in the density of the Universe, inducing its gravitational collapse. One typically requires density contrasts ${\displaystyle \delta \rho /\rho \sim 0.1}$ (where ${\displaystyle \rho }$ is the density of the Universe) to form a black hole.^[9] There are several mechanisms able to produce such inhomogeneities in the context of cosmic inflation (in hybrid inflation models, for example axion inflation), reheating, or cosmological phase transitions.

Observational limits and detection strategies

A variety of observations have been interpreted to place limits on the abundance and mass of primordial black holes:

• Lifetime, Hawking radiation and gamma-rays: One way to detect primordial black holes, or to constrain their mass and abundance, is by their Hawking radiation. Stephen Hawking theorized in 1974 that large numbers of such smaller primordial black holes might exist in the Milky Way in our galaxy's halo region. All black holes are theorized to emit Hawking radiation at a rate inversely proportional to their mass. Since this emission further decreases their mass, black holes with very small mass would experience runaway evaporation, creating a massive burst of radiation at the final phase, equivalent to a hydrogen bomb yielding millions of megatons of explosive force.^[10] A regular black hole (of about 3 solar masses) cannot lose all of its mass within the current age of the universe (they would take about 10⁶⁹ years to do so, even without any matter falling in). However, since primordial black holes are not formed by stellar core collapse, they may be of any size. A black hole with a mass of about 10¹¹ kg would have a lifetime about equal to the age of the universe. If such low-mass black holes were created in sufficient number in the Big Bang, we should be able to observe some of those that are relatively nearby in our own Milky Way galaxy exploding today. NASA's Fermi Gamma-ray Space Telescope satellite, launched in June 2008, was designed in part to search for such evaporating primordial black holes. Fermi data set up the limit that less than one percent of dark matter could be made of primordial black holes with masses up to 10¹³ kg. Evaporating primordial black holes would have also an impact on the Big Bang nucleosynthesis and change the abundances of light elements in the Universe. However, if theoretical Hawking radiation does not actually exist, such primordial black holes would be extremely difficult, if not impossible, to detect in space due to their small size and lack of large gravitational influence.
• Lensing of gamma-ray bursts: Compact objects can induce a change in the luminosity of gamma-ray bursts when passing close to their line-of-sight, through the gravitational lensing effect. The Fermi Gamma-Ray Burst Monitor experiment found that primordial black holes cannot contribute importantly to the dark matter within the mass range 5 x 10¹⁴ – 10¹⁷ kg.^[11]
• Capture of primordial black holes by neutron stars: If primordial black holes with masses between 10¹⁵ kg and 10²² kg had abundances comparable to the one of dark matter, neutron stars in globular clusters should have captured some of them, which leads to the rapid destruction of the star.^[12] The observation of neutron stars in globular clusters can thus be used to set a limit on primordial black holes abundances.
• Micro-lensing of stars: If a primordial black hole passes between us and a distant stars, it induces a magnification of these stars due to the gravitational lensing effect. By monitoring the magnitude of stars in the Magellanic Clouds, the EROS and MACHO surveys have put a limit on the abundance of primordial black holes in the range 10²³ – 10³¹ kg. According to these surveys, primordial black holes within this range cannot constitute an important fraction of the dark matter.^[13][14] However, these limits are model dependent. It has been also argued that if primordial black holes are regrouped in dense halos, the micro-lensing constraints are then naturally evaded.^[15]
• Temperature anisotropies in the cosmic microwave background: Accretion of matter onto primordial black holes in the early Universe should lead to energy injection in the medium that affects the recombination history of the Universe. This effect induces signatures in the statistical distribution of the cosmic microwave background (CMB) anisotropies. The Planck observations of the CMB exclude that primordial black holes with masses in the range 100 – 10⁴ solar masses contribute importantly to the dark matter,^[16] at least in the simplest conservative model. It is still debated whether the constraints are stronger or weaker in more realistic or complex scenarios.

At the time of the detection by LIGO of the gravitational waves emitted during the final coalescence of two 30 solar mass black holes, the mass range between 10 and 100 solar masses were still only poorly constrained. Since then, new observations have been claimed to close this window, at least for models in which the primordial black holes have all the same mass:

• from the absence of X-ray and optical correlations in point sources observed in the direction of the galactic center.^[17]
• from the dynamical heating of dwarf galaxies^[18]
• from the observation of a central star cluster in the Eridanus II dwarf galaxy (but these constraints can be relaxed if Eridanus II owns a central intermediate mass black hole, which is suggested by some observations).^[19] If primordial black holes exhibit a broad mass distributions, those constraints could nevertheless still be evaded.
• from the gravitational micro-lensing of distant quasars by closer galaxies, allowing only 20% of the galactic matter to be in the form of compact objects with stellar masses, a value consistent with the expected stellar population.^[20]
• from micro-lensing of distant stars by galaxy clusters, suggesting that the fraction of dark matter in the form of Primordial Black Holes with masses comparable to those found by LIGO must be less than 10%.^[21]

In the future, new limits will be set up by various observations:

• The Square Kilometre Array (SKA) radiotelescope will probe the effects of primordial black holes on the reionization history of the Universe, due to energy injection into the intergalactic medium, induced by matter accretion onto primordial black holes.^[22]
• LIGO, VIRGO and future gravitational waves detectors will detect new black hole merging events, from which one could reconstruct the mass distribution of primordial black holes.^[23] They could allow to distinguish unambiguously between primordial or stellar origins if a merging event involving black holes with a mass lower than 1.4 solar mass are detected. Another way would be to measure the large orbital eccentricity of primordial black hole binaries.^[24]
• Gravitational wave detectors, such as the Laser Interferometer Space Antenna (LISA) and pulsar timing arrays will also probe the stochastic background of gravitational waves emitted by primordial black hole binaries, when they are still orbiting relatively far from each other.^[25]
• New detections of faint dwarf galaxies, and the observations of their central star cluster, could be used to test the hypothesis that these dark-matter dominated structures contain primordial black holes in abundance.
• Monitoring star positions and velocities within the Milky Way could be used to detect the influence of a nearby primordial black hole.
• It has been suggested^[26][27] that a small black hole passing through the Earth would produce a detectable acoustic signal. Because of its tiny diameter, large mass compared to a nucleon, and relatively high speed, such primordial black holes would simply transit Earth virtually unimpeded with only a few impacts on nucleons, exiting the planet with no ill effects.
• Another way to detect primordial black holes could be by watching for ripples on the surfaces of stars. If the black hole passed through a star, its density would cause observable vibrations.^[28][29]

Implications

The evaporation of primordial black holes has been suggested as one possible explanation for gamma-ray bursts. This explanation is, however, considered unlikely.^{[clarification needed][citation needed]} Other problems for which primordial black holes have been suggested as a solution include the dark matter problem, the cosmological domain wall problem^[30] and the cosmological monopole problem.^[31] Since a primordial black hole does not necessarily have to be small (they can have any size), primordial black holes may also have contributed to the later formation of galaxies.

Even if they do not solve these problems, the low number of primordial black holes (as of 2010, only two intermediate mass black holes were confirmed) aids cosmologists by putting constraints on the spectrum of density fluctuations in the early universe.

String theory

General relativity predicts the smallest primordial black holes would have evaporated by now, but if there were a fourth spatial dimension – as predicted by string theory – it would affect how gravity acts on small scales and "slow down the evaporation quite substantially".^[32] This could mean there are several thousand black holes in our galaxy. To test this theory, scientists will use the Fermi Gamma-ray Space Telescope which was put in orbit by NASA on June 11, 2008. If they observe specific small interference patterns within gamma-ray bursts, it could be the first indirect evidence for primordial black holes and string theory.