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cosmic topological defect

cosmic rays High energy subatomic particles. Cosmic-ray primaries are mostly protons and hydrogen nuclei, but they also contain heavier nuclei. Their energies range from 1 to 2 billion electron volts to perhaps 1016 eV, though the higher energies are rare. On colliding with atmospheric particles, they produce many different kinds of lower energy secondary cosmic radiation. Many cosmic rays are produced in the sun, but the highest-energy rays are produced outside the solar system, perhaps in shocks associated with supernova explosions.

cosmic spring Cosmic strings formed in the early universe might have the ability to carry currents. These currents were at some stage believed to have the capability to locally remove the string tension: the energy carried by the current indeed tends to balance the string tension, so that the effective tension could be made to vanish, or even become negative, hence turning a string into a spring (i.e., a tensionless string). Detailed numerical investigations revealed that the maximum allowed current was, in the case of spacelike currents, not enough for this mechanism to take place. The possibility that timelike currents could make up springs is still open, although quite unlikely, since reasonable equations of state show the phase space where it could happen to be very small.

Another possibility is that of static electromagnetically supported string loops (some authors adopt this as a definition of a spring). Here it is not the current inertia that balances the string tension, but the long-range electromagnetic field generated by the current that would support the whole configuration. In this particular case, the string loop would be required to be unnaturally large, due to the slow (logarithmic) growth of the electromagnetic support around the string core. Non-static but still stationary (rotating) configurations are now believed to have a much more important role to play in cosmology. See Carter– Peter model, current saturation (cosmic string), magnetic regime (cosmic string), tension (cosmic string), vorton.

cosmic string A type of cosmic topological defect that arises from symmetry breaking schemes when the low temperature minimum of the potential has a phase structure: φ = φ0e,

where the ϕ is an arbitrary real phase, all values of ϕ having the same (zero) energy. Then, at large distances from the string, the phase can continuously increase around the string, forcing a high energy region along the line describing the string.

Both local (which have an associated gauge vector field that compensates much of the string energy), and global (which have no such gauge vector) strings may be formed depending on whether the broken group is a gauge or a rigid symmetry of the system before the transition, respectively. See Abelian string, cosmic topolog- ical defect, deficit angle (cosmic string), global topological defect, homotopy group, local topo- logical defect.

cosmic texture Cosmic structures in which multicomponent fields provide large scale matter sources. Their dynamics can generate local energy concentrations which act to seed subsequent formation of structure (super cluster of galaxies, etc.) in the universe. See cosmic topological defect.

cosmic topological defect Current understanding of the physics of the early universe is based in part on the spontaneous breaking of fundamental symmetries. These symmetry breaking processes take place during phase transitions, and many of these transitions might have occurred at grand unified energy scales. At these scales spacetime gets “oriented” by the presence of a hypothetical field called generically the “Higgs field”, pervading all the space. Different models for the Higgs field lead to the formation of a whole variety of topological defects with very different characteristics and dimensions.

Some of the proposed theories have symmetry breaking patterns leading to the formation of “domain walls” (mirror reflection discrete symmetry): incredibly thin (thickness comparable to a Compton wavelength associated with particle energy 1015 GeV) planar surfaces, trapping enormous concentrations of mass-energy, which separate domains of conflicting field orientations, similar to two-dimensional sheet-like structures found in ferromagnets.

In other theories, cosmological fields are distributed in such a way that the old (symmetric) phase gets confined into a finite region

© 2001 by CRC Press LLC

cosmochemistry

of space surrounded completely by the new (non-symmetric) phase. This situation leads to the generation of defects with linear geometry called “cosmic strings”. Theoretical reasons require that these strings (vortex lines) not have any loose ends in order that the two phases stay separated. This leaves infinite strings and closed loops as the only possible alternatives for these defects to manifest themselves in the early universe. Point-like defects known as (magnetic) “monopoles” do arise in other particular symmetry breaking schemes. These are extremely important configurations, since their formation is predicted in virtually all grand unified theories whose low energy limit matches the standard model of particle interactions. See cosmic phase transition, cosmic string, cosmic texture, domain wall, Kibble mechanism, monopole, spontaneous symmetry breaking, texture.

cosmochemistry The study of the chemical make-up of solar system bodies, how the chemistry of these bodies has evolved (through radioactive decay, cooling temperatures, etc.), and the chemical reactions that occur between different regions of a body (such as surfaceatmosphere interactions). Cosmochemistry describes which elements will condense from the solar nebula at various temperatures and pressures, which explains why the inner terrestrial planets are composed of dense refractory elements while the outer Jovian planets and their moons are composed of more volatile gases and ices. Cosmochemical reactions tend to move to a state of equilibrium, which explains why certain molecules are found on planetary surfaces and others are found in the planetary atmospheres.

cosmogenic nuclides Nuclides produced by the interaction of cosmic rays with the atmosphere. For instance, if thermal neutrons are captured by atmospheric nitrogen 14N, a proton is emitted and the cosmogenic nuclide 14C results. Formally, this reaction can be written as 14N(n,p)14C. The spallation of atmospheric nitrogen or oxygen, due to the capture of fast protons or neutrons, produces nuclides such as 10Be under emission of nucleons or smaller fragments such as 2H or 4He. Cosmogenic nuclides are produced in the lower stratosphere and can

be transported down to the troposphere. Some cosmogenic nuclides, such as 10Be, are washed out by rain, and their traces are stored in the polar ice sheets; others, such as 14C, are assimilated by living matter and stored, for instance in trees. The production rate of cosmogenic nuclides depends on the intensity of the galactic cosmic radiation and thus varies during the solar cycle as the galactic cosmic radiation is modulated. The isotopes mentioned above frequently are used in palaeoclimatology because their records provide an indirect measure for solar activity. On longer time-scales, the intensity of the galactic cosmic radiation in the atmosphere and, therefore, of the cosmogenic nuclides is also modulated by the geomagnetic field. Thus, the longterm trend (some thousand years) in the cosmogenic nuclides can be used as a measure for the strength of the geomagnetic field. See modulation of galactic cosmic rays.

cosmological constant A constant (usually denoted U) that measures the curvature of an empty space devoid of gravitational fields. In the real universe, where gravitational fields exist throughout the whole space, this curvature would provide a tiny background (approx. 1050 cm2) to the total curvature, but its effects on the evolution of the universe could be profound. Depending on the sign of U, a Friedmann–Lemaître model with positive spatial curvature could go on evolving forever, or the one with negative spatial curvature could recollapse. In the first version of his relativity theory, Einstein did not use U (i.e., effectively he assumed that U = 0). Then it turned out that, contrary to everybody’s expectations, the theory implied that the universe cannot be static (i.e., unchanging in time) if it is spatially homogeneous (see homogeneity). Consequently, Einstein modified his theory to allow for U > 0, and in the modified theory a model of a static universe existed (see Einstein universe). Later, E.P. Hubble discovered that the real universe is nonstatic indeed (see expansion of the universe). When Einstein realized how close he was to a prediction of this discovery (14 years in advance), he called the introduction of the cosmological constant “the biggest blunder of my life”. Nevertheless, the constant is routinely taken into account in solving Einstein’s equa-

© 2001 by CRC Press LLC

cosmological principle

tions, and in some modern theories of the early universe it must be nonzero. (See inflation.) A positive cosmological constant implies a universal repulsive force acting on all objects in the universe. The force is very weak at short distances, but can outbalance the gravitational force at large distances, hence the existence of the static Einstein universe. The Einstein equations written with the cosmological constant are

1

Rµν 2 gµνR Ugµν = 8πGTµν

where Rµν is the Ricci tensor, gµν is the metric tensor, R is the curvature scalar, and Tµν is the stress-energy tensor.

cosmological constant problem In quantum

field theory, all the fields contribute to the vacuum energy density, that is, to the cosmological constant and, therefore, the theoretical value for it is quite big. From dimensional analysis one expects the density of the cosmological constant in the universe to be of the order of (mass of heaviest particle)4, and very heavy particles have been detected, for instance the top-quark with mass about 175 GeV. At the same time, the observable value of cosmological constant is zero, or at least it is very small. The astronomical observations do not give any definite lower bounds for the cosmological constant, but instead put the upper bound for this density corresponding to a mass of about 1047 GeV. Thus, there is an explicit discrepancy between the theoretically based matter field’s contributions to the induced cosmological constant and observations. To eliminate this discrepancy one has to introduce the vacuum cosmological constant, which must cancel the induced one with tremendous precision. The exactness of this cancellation is the cosmological constant problem.

Suggestions for the solution of the cosmological constant problem typically reduce the finetuning of the vacuum cosmological constant to the fine-tuning of some other quantities like the potential for a cosmic scalar field. Other suggestions are based on supersymmetry, which can prevent the contributions of matter fields to the induced cosmological constant via the cancellation of the contributions between bosons and fermions. However, supersymmetry, if it exists in nature, is believed to be broken at the energies

of the order MF , the Fermi scale, (250)4 GeV. Thus, there is no acceptable and reliable solution of the cosmological constant problem at the moment. A solution has been postulated in terms of an anthropic many-world hypothesis. According to this hypothesis our universe is just a single one among many others, and we live in it because the small cosmological constant lets us do so. Most of the other universes are strongly compactified because of the large cosmological constant.

One can also mention that some cosmological and astrophysical theories require small but non-zero cosmological constant. The density of the cosmological constant may, in such theories, serve as a dark matter and, in particular, provide a desirable age of the universe. Recent observations of Ia supernovae in distant galaxies suggest the existence of dark matter in the form of a very small, repulsive cosmological constant. This small cosmological constant has apparently little to do with the cosmological constant problem because its possible existence can reduce the necessary exactness of the cancellation between the induced and vacuum cosmological constants by at most one order of magnitude. See cosmological constant, induced gravity, spontaneous symmetry breaking.

cosmological model A solution of Einstein’s equations (see also metric) that can be used to describe the geometry and large-scale evolution of matter in the universe. The term is sometimes misused (for various reasons) to denote such models of spacetime that do not apply to the real universe. See Bianchi classification, ho- mogeneity, inhomogeneous models, perturba- tive solution.

cosmological principle An assumption that says that every observer in the universe would see the same large-scale distribution of matter around him (see homogeneity), and that for a fixed observer, the large-scale distribution of matter would be the same along every direction (see isotropy). Philosophically, the cosmological principle is the extreme opposite of the pre-Copernican view that the Earth was the center of the universe; according to the cosmological principle all positions in the universe are equivalent. The cosmological principle clearly

© 2001 by CRC Press LLC

cosmology

does not apply at small (e.g., galactic) scales, and several explicit general relativistic models of the universe exist in which the cosmological principle does not apply (see inhomogeneous models, microwave background radiation). A strengthened version of the cosmological principle called a “perfect cosmological principle” gave rise to the steady-state models.

cosmology The science that investigates the whole universe as a single physical system. It combines mathematics (heavily used to find cosmological models from Einstein’s equations or other theories of gravity), physics (that guides the theoretical research on observable effects in cosmological models), and astronomy (that provides observational support or negation of various theoretical results).

The universe is usually assumed to be a continuous medium (usually a perfect fluid or dust) to which the laws of hydrodynamics and thermodynamics known from laboratory apply. In the very early universe, seconds after the Big Bang, the continuous medium is a mixture of elementary particles, and then a plasma. The medium resides in a spacetime, whose metric is that of a cosmological model. This spacetime is an arena in which geodesics (i.e., trajectories of bodies moving under the influence of gravitational fields) can be investigated. Among the geodesics are light-rays that lie on light-cones; they are trajectories along which astronomers receive their observational data from distant galaxies and quasars. In particular, the microwave background radiation consists of photons that travel along light-like geodesics; it brings information from an era shortly after the Big Bang. These notions provide the foundation on which theoretical cosmology is built. Observational cosmology deals mainly with the spectrum and temperature of the microwave background radiation, in particular the dependence of its temperature on the direction of observation, with the formation of the light elements in the early universe, with spatial distribution of matter in the large scale, with properties and evolution of galaxies and quasars, also with gravitational lenses. Cosmology seeks to explain, among other things, the creation and evolution of the large-scale matter distribution in the universe,

formation of galaxies, and the sequence of physical processes following the Big Bang.

coude focus An optical system that directs the beam of light by bending the path at an “elbow”; (“coude” = “elbow”) from the primary mirror of a reflecting telescope down the hollow polar axis of the instrument to a remote fixed focal position.

coulomb Standard international unit of electric charge, equal to the charge that passes through any cross-section of a conductor in 1 sec during a constant current flow of 1 A. See ampere.

Coulomb collisions In a plasma, collisions are mediated through long-range electrostatic (Coulomb) forces between electrons and protons. The dynamics of a particle in the plasma are governed by the electrostatic interactions with all other particles in the plasma. The Coulomb interaction serves to slow down the incident particle, which releases energy in the form of a photon of a given wavelength (e.g., via bremsstrahlung). X-ray production is dominated by close encounters of electrons with protons, while long range Coulomb collisions are primarily responsible for radio bremsstrahlung.

Coulter counter® One of a class of instruments that measures particle size distribution from the change in electrical conductivity as particles flow through a small orifice; originally developed by Coulter Electronics.

Courant number A dimensionless number used to assess the numerical stability of a numerical solution scheme. Commonly used in the study of computational fluid mechanics.

covariant derivative A differential operator defined on the tensors of an arbitrary rank; the map from tensor fields of type (j, k) to tensor fields of type (j, k + 1) on a manifold; i.e., it produces a tensor of one higher covariant rank and the same contravariant rank. The covariant derivative satisfies the properties of derivative operators: (i) linearity, (ii) Leibnitz rule for derivatives of products, and (iii) commuting with the operation of contraction. In

© 2001 by CRC Press LLC

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