Type Ia supernova

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Type_Ia_supernova

A Type Ia supernova (read: "type one-A") is a type of supernova that occurs in binary systems (two stars orbiting one another) in which one of the stars is a white dwarf. The other star can be anything from a giant star to an even smaller white dwarf.

Physically, carbon–oxygen white dwarfs with a low rate of rotation are limited to below 1.44 solar masses (M_☉). Beyond this "critical mass", they reignite and in some cases trigger a supernova explosion; this critical mass is often referred to as the Chandrasekhar mass, but is marginally different from the absolute Chandrasekhar limit, where electron degeneracy pressure is unable to prevent catastrophic collapse. If a white dwarf gradually accretes mass from a binary companion, or merges with a second white dwarf, the general hypothesis is that a white dwarf's core will reach the ignition temperature for carbon fusion as it approaches the Chandrasekhar mass. Within a few seconds of initiation of nuclear fusion, a substantial fraction of the matter in the white dwarf undergoes a runaway reaction, releasing enough energy (1×10⁴⁴ J) to unbind the star in a supernova explosion.

The Type Ia category of supernova produces a fairly consistent peak luminosity because of the fixed critical mass at which a white dwarf will explode. Their consistent peak luminosity allows these explosions to be used as standard candles to measure the distance to their host galaxies: the visual magnitude of a type Ia supernova, as observed from Earth, indicates its distance from Earth.

Consensus model

Spectrum of SN 1998aq, a type Ia supernova, one day after maximum light in the B band

The Type Ia supernova is a subcategory in the Minkowski–Zwicky supernova classification scheme, which was devised by German-American astronomer Rudolph Minkowski and Swiss astronomer Fritz Zwicky. There are several means by which a supernova of this type can form, but they share a common underlying mechanism. Theoretical astronomers long believed the progenitor star for this type of supernova is a white dwarf, and empirical evidence for this was found in 2014 when a Type Ia supernova was observed in the galaxy Messier 82. When a slowly-rotating carbon–oxygen white dwarf accretes matter from a companion, it can exceed the Chandrasekhar limit of about 1.44 M_☉, beyond which it can no longer support its weight with electron degeneracy pressure. In the absence of a countervailing process, the white dwarf would collapse to form a neutron star, in an accretion-induced non-ejective process, as normally occurs in the case of a white dwarf that is primarily composed of magnesium, neon, and oxygen.

The current view among astronomers who model Type Ia supernova explosions, however, is that this limit is never actually attained and collapse is never initiated. Instead, the increase in pressure and density due to the increasing weight raises the temperature of the core, and as the white dwarf approaches about 99% of the limit, a period of convection ensues, lasting approximately 1,000 years. At some point in this simmering phase, a deflagration flame front is born, powered by carbon fusion. The details of the ignition are still unknown, including the location and number of points where the flame begins. Oxygen fusion is initiated shortly thereafter, but this fuel is not consumed as completely as carbon.

G299 Type Ia supernova remnant.

Once fusion begins, the temperature of the white dwarf increases. A main sequence star supported by thermal pressure can expand and cool which automatically regulates the increase in thermal energy. However, degeneracy pressure is independent of temperature; white dwarfs are unable to regulate temperature in the manner of normal stars, so they are vulnerable to runaway fusion reactions. The flare accelerates dramatically, in part due to the Rayleigh–Taylor instability and interactions with turbulence. It is still a matter of considerable debate whether this flare transforms into a supersonic detonation from a subsonic deflagration.

Regardless of the exact details of how the supernova ignites, it is generally accepted that a substantial fraction of the carbon and oxygen in the white dwarf fuses into heavier elements within a period of only a few seconds, with the accompanying release of energy increasing the internal temperature to billions of degrees. The energy released (1–2×10⁴⁴ J) is more than sufficient to unbind the star; that is, the individual particles making up the white dwarf gain enough kinetic energy to fly apart from each other. The star explodes violently and releases a shock wave in which matter is typically ejected at speeds on the order of 5,000–20,000 km/s, roughly 6% of the speed of light. The energy released in the explosion also causes an extreme increase in luminosity. The typical visual absolute magnitude of Type Ia supernovae is M_v = −19.3 (about 5 billion times brighter than the Sun), with little variation. The Type Ia supernova leaves no compact remnant, but the whole mass of the former white dwarf dissipates through space.

The theory of this type of supernova is similar to that of novae, in which a white dwarf accretes matter more slowly and does not approach the Chandrasekhar limit. In the case of a nova, the infalling matter causes a hydrogen fusion surface explosion that does not disrupt the star.

Type Ia supernovae differ from Type II supernovae, which are caused by the cataclysmic explosion of the outer layers of a massive star as its core collapses, powered by release of gravitational potential energy via neutrino emission.

Formation

Formation process

 

An accretion disc forms around a compact body (such as a white dwarf) stripping gas from a companion giant star. NASA image

 

Supercomputer simulation of the explosion phase of the deflagration-to-detonation model of supernova formation.

Single degenerate progenitors

One model for the formation of this category of supernova is a close binary star system. The progenitor binary system consists of main sequence stars, with the primary possessing more mass than the secondary. Being greater in mass, the primary is the first of the pair to evolve onto the asymptotic giant branch, where the star's envelope expands considerably. If the two stars share a common envelope then the system can lose significant amounts of mass, reducing the angular momentum, orbital radius and period. After the primary has degenerated into a white dwarf, the secondary star later evolves into a red giant and the stage is set for mass accretion onto the primary. During this final shared-envelope phase, the two stars spiral in closer together as angular momentum is lost. The resulting orbit can have a period as brief as a few hours. If the accretion continues long enough, the white dwarf may eventually approach the Chandrasekhar limit.

The white dwarf companion could also accrete matter from other types of companions, including a subgiant or (if the orbit is sufficiently close) even a main sequence star. The actual evolutionary process during this accretion stage remains uncertain, as it can depend both on the rate of accretion and the transfer of angular momentum to the white dwarf companion.

It has been estimated that single degenerate progenitors account for no more than 20% of all Type Ia supernovae.

Double degenerate progenitors

A second possible mechanism for triggering a Type Ia supernova is the merger of two white dwarfs whose combined mass exceeds the Chandrasekhar limit. The resulting merger is called a super-Chandrasekhar mass white dwarf. In such a case, the total mass would not be constrained by the Chandrasekhar limit.

Collisions of solitary stars within the Milky Way occur only once every 10⁷ to 10¹³ years; far less frequently than the appearance of novae. Collisions occur with greater frequency in the dense core regions of globular clusters (cf. blue stragglers). A likely scenario is a collision with a binary star system, or between two binary systems containing white dwarfs. This collision can leave behind a close binary system of two white dwarfs. Their orbit decays and they merge through their shared envelope. A study based on SDSS spectra found 15 double systems of the 4,000 white dwarfs tested, implying a double white dwarf merger every 100 years in the Milky Way: this rate matches the number of Type Ia supernovae detected in our neighborhood.

A double degenerate scenario is one of several explanations proposed for the anomalously massive (2 M_☉) progenitor of SN 2003fg. It is the only possible explanation for SNR 0509-67.5, as all possible models with only one white dwarf have been ruled out. It has also been strongly suggested for SN 1006, given that no companion star remnant has been found there.^[22] Observations made with NASA's Swift space telescope ruled out existing supergiant or giant companion stars of every Type Ia supernova studied. The supergiant companion's blown out outer shell should emit X-rays, but this glow was not detected by Swift's XRT (X-ray telescope) in the 53 closest supernova remnants. For 12 Type Ia supernovae observed within 10 days of the explosion, the satellite's UVOT (ultraviolet/optical telescope) showed no ultraviolet radiation originating from the heated companion star's surface hit by the supernova shock wave, meaning there were no red giants or larger stars orbiting those supernova progenitors. In the case of SN 2011fe, the companion star must have been smaller than the Sun, if it existed. The Chandra X-ray Observatory revealed that the X-ray radiation of five elliptical galaxies and the bulge of the Andromeda Galaxy is 30–50 times fainter than expected. X-ray radiation should be emitted by the accretion discs of Type Ia supernova progenitors. The missing radiation indicates that few white dwarfs possess accretion discs, ruling out the common, accretion-based model of Ia supernovae. Inward spiraling white dwarf pairs are strongly-inferred candidate sources of gravitational waves, although they have not been directly observed.

Type Iax

Main article: Type Iax supernova

It has been proposed that a group of sub-luminous supernovae should be classified as Type Iax. This type of supernova may not always completely destroy the white dwarf progenitor, but instead leave behind a zombie star. Known examples of type Iax supernovae include: the historical supernova SN 1181, SN 1991bg, SN 2002cx, and SN 2012Z.

The supernova SN 1181 is believed to be associated with the supernova remnant Pa 30 and its central star IRAS 00500+6713, which is the result of a merger of a CO white dwarf and an ONe white dwarf. This makes Pa 30 and IRAS 00500+6713 the only SN Iax remnant in the Milky Way.

Observation

Supernova remnant N103B taken by the Hubble Space Telescope.

Unlike the other types of supernovae, Type Ia supernovae generally occur in all types of galaxies, including ellipticals. They show no preference for regions of current stellar formation. As white dwarf stars form at the end of a star's main sequence evolutionary period, such a long-lived star system may have wandered far from the region where it originally formed. Thereafter a close binary system may spend another million years in the mass transfer stage (possibly forming persistent nova outbursts) before the conditions are ripe for a Type Ia supernova to occur.

A long-standing problem in astronomy has been the identification of supernova progenitors. Direct observation of a progenitor would provide useful constraints on supernova models. As of 2006, the search for such a progenitor had been ongoing for longer than a century. Observation of the supernova SN 2011fe has provided useful constraints. Previous observations with the Hubble Space Telescope did not show a star at the position of the event, thereby excluding a red giant as the source. The expanding plasma from the explosion was found to contain carbon and oxygen, making it likely the progenitor was a white dwarf primarily composed of these elements. Similarly, observations of the nearby SN PTF 11kx, discovered January 16, 2011 (UT) by the Palomar Transient Factory (PTF), lead to the conclusion that this explosion arises from single-degenerate progenitor, with a red giant companion, thus suggesting there is no single progenitor path to SN Ia. Direct observations of the progenitor of PTF 11kx were reported in the August 24 edition of Science and support this conclusion, and also show that the progenitor star experienced periodic nova eruptions before the supernova – another surprising discovery.  However, later analysis revealed that the circumstellar material is too massive for the single-degenerate scenario, and fits better the core-degenerate scenario.

In May 2015, NASA reported that the Kepler space observatory observed KSN 2011b, a Type Ia supernova in the process of exploding. Details of the pre-nova moments may help scientists better judge the quality of Type Ia supernovae as standard candles, which is an important link in the argument for dark energy.

In July 2019, the Hubble Space Telescope took three images of a Type Ia supernova through a gravitational lens. This supernova appeared at three different times in the evolution of its brightness due to the differing path length of the light in the three images; at −24, 92, and 107 days from peak luminosity. A fourth image will appear in 2037 allowing observation of the entire luminosity cycle of the supernova.

Light curve

This plot of luminosity (relative to the Sun, L₀) versus time shows the characteristic light curve for a Type Ia supernova. The peak is primarily due to the decay of nickel (Ni), while the later stage is powered by cobalt (Co).

Light curve for type Ia, over the course of one year SN 2018gv

Type Ia supernovae have a characteristic light curve, their graph of luminosity as a function of time after the explosion. Near the time of maximal luminosity, the spectrum contains lines of intermediate-mass elements from oxygen to calcium; these are the main constituents of the outer layers of the star. Months after the explosion, when the outer layers have expanded to the point of transparency, the spectrum is dominated by light emitted by material near the core of the star, heavy elements synthesized during the explosion; most prominently isotopes close to the mass of iron (iron-peak elements). The radioactive decay of nickel-56 through cobalt-56 to iron-56 produces high-energy photons, which dominate the energy output of the ejecta at intermediate to late times.

The use of Type Ia supernovae to measure precise distances was pioneered by a collaboration of Chilean and US astronomers, the Calán/Tololo Supernova Survey. In a series of papers in the 1990s the survey showed that while Type Ia supernovae do not all reach the same peak luminosity, a single parameter measured from the light curve can be used to correct unreddened Type Ia supernovae to standard candle values. The original correction to standard candle value is known as the Phillips relationship and was shown by this group to be able to measure relative distances to 7% accuracy. The cause of this uniformity in peak brightness is related to the amount of nickel-56 produced in white dwarfs presumably exploding near the Chandrasekhar limit.

The similarity in the absolute luminosity profiles of nearly all known Type Ia supernovae has led to their use as a secondary standard candle in extragalactic astronomy. Improved calibrations of the Cepheid variable distance scale and direct geometric distance measurements to NGC 4258 from the dynamics of maser emission when combined with the Hubble diagram of the Type Ia supernova distances have led to an improved value of the Hubble constant.

In 1998, observations of distant Type Ia supernovae indicated the unexpected result that the universe seems to undergo an accelerating expansion. Three members from two teams were subsequently awarded Nobel Prizes for this discovery.

Subtypes

Supernova remnant SNR 0454-67.2 is likely the result of a Type Ia supernova explosion.

There is significant diversity within the class of Type Ia supernovae. Reflecting this, a plethora of sub-classes have been identified. Two prominent and well-studied examples include 1991T-likes, an overluminous ${\displaystyle (M_V\lesssim -19.5)}$ subclass that exhibits particularly strong iron absorption lines and abnormally small silicon features, and 1991bg-likes, an exceptionally dim ${\displaystyle (M_V\gtrsim -18)}$ subclass characterized by strong early titanium absorption features and rapid photometric and spectral evolution. Despite their abnormal luminosities, members of both peculiar groups can be standardized by use of the Phillips relation, defined at blue wavelengths, to determine distance.

Inhomogeneous cosmology

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Inhomogeneous_cosmology

An inhomogeneous cosmology is a physical cosmological theory (an astronomical model of the physical universe's origin and evolution) which, unlike the dominant cosmological concordance model, postulates that inhomogeneities in the distribution of matter across the universe affect local gravitational forces (i.e., at the galactic level) enough to skew our view of the Universe. When the universe began, matter was distributed homogeneously, but over billions of years, galaxies, clusters of galaxies, and superclusters coalesced. Einstein's theory of general relativity states that they warp the space-time around them.

While the concordance model acknowledges this fact, it assumes that such inhomogeneities are not sufficient to affect large-scale averages of gravity observations. Two studies claimed in 1998-1999 that high redshift supernovae were further away than the distance predicted by calculations. It was suggested that the expansion of the universe was accelerating, and dark energy, a repulsive energy inherent in space, was proposed as an explanation. Dark energy became widely accepted, but remains unexplained. Inhomogeneous cosmology falls into the class of models that might not require dark energy.

Inhomogeneous cosmologies assume that the backreactions of denser structures and those of empty voids on space-time are significant. When not neglected, they distort understanding of time and observations of distant objects. Burchert's equations in 1997 and 2000 derive from general relativity, but allow for the inclusion of local gravitational variations. Alternative models were proposed under which the acceleration of the universe was a misinterpretation of astronomical observations and in which dark energy is unnecessary. For example, in 2007, David Wiltshire proposed a model (timescape cosmology) in which backreactions caused time to run more slowly or, in voids, more quickly, thus leading supernovae observed in 1998 to be thought to be further away than they were. Timescape cosmology may also imply that the expansion of the universe is in fact slowing.

History

Standard cosmological model

Main article: Lambda-CDM model

The conflict between the two cosmologies derives from the inflexibility of Einstein's theory of general relativity, which shows how gravity is formed by the interaction of matter, space, and time. Physicist John Wheeler famously summed up the theory's essence as "Matter tells space how to curve; space tells matter how to move." However, in order to build a workable cosmological model, all of the terms on both sides of Einstein's equations must be balanced: on one side, matter (i.e., all the things that warp time and space); on the other, the curvature of the universe and the speed at which space-time is expanding. In short, a model requires a particular amount of matter in order to produce particular curvatures and expansion rates.

In terms of matter, all modern cosmologies are founded on the cosmological principle, which states that whichever direction we look from Earth, the universe is basically the same: homogeneous and isotropic (uniform in all dimensions). This principle grew out of Copernicus's assertion that there were no special observers in the universe and nothing special about the Earth's location in the universe (i.e., Earth was not the center of the universe, as previously thought). Since the publication of general relativity in 1915, this homogeneity and isotropy have greatly simplified the process of devising cosmological models.

Possible shapes of the universe

In terms of the curvature of space-time and the shape of the universe, it can theoretically be closed (positive curvature, or space-time folding in itself as though on a four-dimensional sphere's surface), open (negative curvature, with space-time folding outward), or flat (zero curvature, like the surface of a "flat" four-dimensional piece of paper).

The first real difficulty came with regards to expansion, for in 1915, as previously, the universe was assumed to be static, neither expanding nor contracting. All of Einstein's solutions to his equations in general relativity, however, predicted a dynamic universe. Therefore, in order to make his equations consistent with the apparently static universe, he added a cosmological constant, a term representing some unexplained extra energy. But when in the late 1920s Georges Lemaître's and Edwin Hubble's observations proved Alexander Friedmann's notion (derived from general relativity) that the universe was expanding, the cosmological constant became unnecessary, Einstein calling it "my greatest blunder."

With this term gone from the equation, others derived the Friedmann-Lemaître–Robertson–Walker (FLRW) solution to describe such an expanding universe — a solution built on the assumption of a flat, isotropic, homogeneous universe. The FLRW model became the foundation of the standard model of a universe created by the Big Bang, and further observational evidence has helped to refine it. For example, a smooth, mostly homogeneous, and (at least when it was almost 400,000 years old) flat universe seemed to be confirmed by data from the cosmic microwave background (CMB). And after galaxies and clusters of galaxies were found in the 1970s, mainly by Vera Rubin, to be rotating faster than they should without flying apart, the existence of dark matter seemed also proven, confirming its inference by Jacobus Kapteyn, Jan Oort, and Fritz Zwicky in the 1920s and 1930s and demonstrating the flexibility of the standard model. Dark matter is believed to make up roughly 23% of the energy density of the universe.

Dark energy

Main article: Dark energy

Timeline of the universe according to the CMB

Another observation in 1998 seemed to complicate the situation further: two separate studies found distant supernovae to be fainter than expected in a steadily expanding universe; that is, they were not merely moving away from the earth but accelerating. The universe's expansion was calculated to have been accelerating since approximately 5 billion years ago. Given the gravitation braking effect that all the matter of the universe should have had on this expansion, a variation of Einstein's cosmological constant was reintroduced to represent an energy inherent in space, balancing the equations for a flat, accelerating universe. It also gave Einstein's cosmological constant new meaning, for by reintroducing it into the equation to represent dark energy, a flat universe expanding ever faster can be reproduced.

Although the nature of this energy has yet to be adequately explained, it makes up almost 70% of the energy density of the universe in the concordance model. And thus, when including dark matter, almost 95% of the universe's energy density is explained by phenomena that have been inferred but not entirely explained nor directly observed. Most cosmologists still accept the concordance model, although science journalist Anil Ananthaswamy calls this agreement a "wobbly orthodoxy."

Inhomogeneous universe

All-sky mollweide map of the CMB, created from 9 years of WMAP data. Tiny residual variations are visible, but they show a very specific pattern consistent with a hot gas that is mostly uniformly distributed.

While the universe began with homogeneously distributed matter, enormous structures have since coalesced over billions of years: hundreds of billions of stars inside of galaxies, clusters of galaxies, superclusters, and vast filaments of matter. These denser regions and the voids between them must, under general relativity, have some effect, as matter dictates how space-time curves. So the extra mass of galaxies and galaxy clusters (and dark matter, should particles of it ever be directly detected) must cause nearby space-time to curve more positively, and voids should have the opposite effect, causing space-time around them to take on negative curvatures. The question is whether these effects, called backreactions, are negligible or together comprise enough to change the universe's geometry. Most scientists have assumed that they are negligible, but this has partly been because there has been no way to average space-time geometry in Einstein's equations.

In 2000, a set of new equations—now referred to as the set of Buchert equations—based on general relativity was published by cosmologist Thomas Buchert of the École Normale Supérieure in Lyon, France, which allow the effects of a non-uniform distribution of matter to be taken into account but still allow the behavior of the universe to be averaged. Thus, models based on a lumpy, inhomogeneous distribution of matter could now be devised. "There is no dark energy, as far as I'm concerned," Buchert told New Scientist in 2016. "In ten years' time, dark energy is gone." In the same article, cosmologist Syksy Räsänen said, "It’s not been established beyond reasonable doubt that dark energy exists. But I’d never say that it has been established that dark energy does not exist." He also told the magazine that the question of whether backreactions are negligible in cosmology "has not been satisfactorily answered."

Inhomogeneous cosmology

Inhomogeneous cosmology in the most general sense (assuming a totally inhomogeneous universe) is modeling the universe as a whole with the spacetime which does not possess any spacetime symmetries. Typically considered cosmological spacetimes have either the maximal symmetry, which comprises three translational symmetries and three rotational symmetries (homogeneity and isotropy with respect to every point of spacetime), the translational symmetry only (homogeneous models), or the rotational symmetry only (spherically symmetric models). Models with fewer symmetries (e.g. axisymmetric) are also considered as symmetric. However, it is common to call spherically symmetric models or non-homogeneous models as inhomogeneous. In inhomogeneous cosmology, the large-scale structure of the universe is modeled by exact solutions of the Einstein field equations (i.e. non-perturbatively), unlike cosmological perturbation theory, which is study of the universe that takes structure formation (galaxies, galaxy clusters, the cosmic web) into account but in a perturbative way.

Inhomogeneous cosmology usually includes the study of structure in the Universe by means of exact solutions of Einstein's field equations (i.e. metrics) or by spatial or spacetime averaging methods. Such models are not homogeneous, but may allow effects which can be interpreted as dark energy, or can lead to cosmological structures such as voids or galaxy clusters.

Perturbative approach

Perturbation theory, which deals with small perturbations from e.g. a homogeneous metric, only holds as long as the perturbations are not too large, and N-body simulations use Newtonian gravity which is only a good approximation when speeds are low and gravitational fields are weak.

Non-perturbative approach

Work towards a non-perturbative approach includes the Relativistic Zel'dovich Approximation. As of 2016, Thomas Buchert, George Ellis, Edward Kolb, and their colleagues judged that if the universe is described by cosmic variables in a backreaction scheme that includes coarse-graining and averaging, then whether dark energy is an artifact of the traditional way of using the Einstein equation remains an unanswered question.

Exact solutions

The first historical examples of inhomogeneous (though spherically symmetric) solutions are the Lemaître–Tolman metric (or LTB model - Lemaître–Tolman-Bondi). The Stephani metric can be spherically symmetric or totally inhomogeneous. Other examples are the Szekeres metric, Szafron metric, Barnes metric, Kustaanheimo-Qvist metric, and Senovilla metric. The Bianchi metrics as given in the Bianchi classification and Kantowski-Sachs metrics are homogeneous.

Averaging methods

The simplest averaging approach is the scalar averaging approach, leading to the kinematical backreaction and mean 3-Ricci curvature functionals. Buchert's equations are the most commonly used equations of such averaging methods. The simplest averaging kernels include spheres (cylinders, when viewed with a time component), Gaussians, and hard-momentum cutoffs. The former work well for non-relativistic fluids (dust); the later are more convenient for relativistic fluid calculations (photons and pre-recombination universes).

Timescape cosmology

In 2007, David L Wiltshire, a professor of theoretical physics at the University of Canterbury in New Zealand, argued in the New Journal of Physics that quasilocal variations in gravitational energy had in 1998 given the false conclusion that the expansion of the universe is accelerating. Moreover, due to the equivalence principle, which holds that gravitational and inertial energy are equivalent and thus prevents aspects of gravitational energy from being differentiated at a local level, scientists thus misidentified these aspects as dark energy. This misidentification was the result of presuming an essentially homogeneous universe, as the standard cosmological model does, and not accounting for temporal differences between matter-dense areas and voids. Wiltshire and others argued that if the universe is not only assumed not to be homogeneous but also not flat, models could be devised in which the apparent acceleration of the universe's expansion could be explained otherwise.

One more important step being left out of the standard model, Wiltshire claimed, was the fact that as proven by observation, gravity slows time. Thus, from the perspective of the same observer, a clock will move faster in empty space, which possesses low gravitation, than inside a galaxy, which has much more gravity, and he argued that as large as a 38% difference between the time on clocks in the Milky Way galaxy and those floating in a void exists. Thus, unless we can correct for that—timescapes each with different times—our observations of the expansion of space will be, and are, incorrect. Wiltshire claims that the 1998 supernovae observations that led to the conclusion of an expanding universe and dark energy can instead be explained by Buchert's equations if certain strange aspects of general relativity are taken into account. The arguments of Wiltshire have been contested by Ethan Siegel.

Observational evidence

A 2024 study examining the Pantheon+ Type Ia Supernova dataset conducted a significant test of the Timescape cosmology. By employing a model-independent statistical approach, the researchers found that the Timescape model could account for the observed cosmic acceleration without the need for dark energy. This result suggested that inhomogeneous cosmological models may offer viable alternatives to the standard ΛCDM framework and warranted further exploration to assess their ability to explain other key cosmological phenomena.

Invariant (mathematics)

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https://en.wikipedia.org/wiki/Invariant_(mathematics)

A wallpaper is invariant under some transformations. This one is invariant under horizontal and vertical translation, as well as rotation by 180° (but not under reflection).

In mathematics, an invariant is a property of a mathematical object (or a class of mathematical objects) which remains unchanged after operations or transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used. For example, the area of a triangle is an invariant with respect to isometries of the Euclidean plane. The phrases "invariant under" and "invariant to" a transformation are both used. More generally, an invariant with respect to an equivalence relation is a property that is constant on each equivalence class.

Invariants are used in diverse areas of mathematics such as geometry, topology, algebra and discrete mathematics. Some important classes of transformations are defined by an invariant they leave unchanged. For example, conformal maps are defined as transformations of the plane that preserve angles. The discovery of invariants is an important step in the process of classifying mathematical objects.

Examples

A simple example of invariance is expressed in our ability to count. For a finite set of objects of any kind, there is a number to which we always arrive, regardless of the order in which we count the objects in the set. The quantity—a cardinal number—is associated with the set, and is invariant under the process of counting.

An identity is an equation that remains true for all values of its variables. There are also inequalities that remain true when the values of their variables change.

The distance between two points on a number line is not changed by adding the same quantity to both numbers. On the other hand, multiplication does not have this same property, as distance is not invariant under multiplication.

Angles and ratios of distances are invariant under scalings, rotations, translations and reflections. These transformations produce similar shapes, which is the basis of trigonometry. In contrast, angles and ratios are not invariant under non-uniform scaling (such as stretching). The sum of a triangle's interior angles (180°) is invariant under all the above operations. As another example, all circles are similar: they can be transformed into each other and the ratio of the circumference to the diameter is invariant (denoted by the Greek letter π (pi)).

Some more complicated examples:

• The real part and the absolute value of a complex number are invariant under complex conjugation.
• The tricolorability of knots.
• The degree of a polynomial is invariant under a linear change of variables.
• The dimension and homology groups of a topological object are invariant under homeomorphism.
• The number of fixed points of a dynamical system is invariant under many mathematical operations.
• Euclidean distance is invariant under orthogonal transformations.
• Area is invariant under linear maps which have determinant ±1 (see Equiareal map § Linear transformations).
• Some invariants of projective transformations include collinearity of three or more points, concurrency of three or more lines, conic sections, and the cross-ratio.^[6]
• The determinant, trace, eigenvectors, and eigenvalues of a linear endomorphism are invariant under a change of basis. In other words, the spectrum of a matrix is invariant under a change of basis.
• The principal invariants of tensors do not change with rotation of the coordinate system (see Invariants of tensors).
• The singular values of a matrix are invariant under orthogonal transformations.
• Lebesgue measure is invariant under translations.
• The variance of a probability distribution is invariant under translations of the real line. Hence the variance of a random variable is unchanged after the addition of a constant.
• The fixed points of a transformation are the elements in the domain that are invariant under the transformation. They may, depending on the application, be called symmetric with respect to that transformation. For example, objects with translational symmetry are invariant under certain translations.
• The integral ${\int }_{M}Kd\mu$ of the Gaussian curvature ${\displaystyle K}$ of a two-dimensional Riemannian manifold ${\displaystyle (M,g)}$ is invariant under changes of the Riemannian metric ${\displaystyle g}$. This is the Gauss–Bonnet theorem.

MU puzzle

The MU puzzle is a good example of a logical problem where determining an invariant is of use for an impossibility proof. The puzzle asks one to start with the word MI and transform it into the word MU, using in each step one of the following transformation rules:

1. If a string ends with an I, a U may be appended (xI → xIU)
2. The string after the M may be completely duplicated (Mx → Mxx)
3. Any three consecutive I's (III) may be replaced with a single U (xIIIy → xUy)
4. Any two consecutive U's may be removed (xUUy → xy)

An example derivation (with superscripts indicating the applied rules) is

MI →² MII →² MIIII →³ MUI →² MUIUI →¹ MUIUIU →² MUIUIUUIUIU →⁴ MUIUIIUIU → ...

In light of this, one might wonder whether it is possible to convert MI into MU, using only these four transformation rules. One could spend many hours applying these transformation rules to strings. However, it might be quicker to find a property that is invariant to all rules (that is, not changed by any of them), and that demonstrates that getting to MU is impossible. By looking at the puzzle from a logical standpoint, one might realize that the only way to get rid of any I's is to have three consecutive I's in the string. This makes the following invariant interesting to consider:

The number of I's in the string is not a multiple of 3.

This is an invariant to the problem, if for each of the transformation rules the following holds: if the invariant held before applying the rule, it will also hold after applying it. Looking at the net effect of applying the rules on the number of I's and U's, one can see this actually is the case for all rules:

Rule	#I's	#U's	Effect on invariant
1	+0	+1	Number of I's is unchanged. If the invariant held, it still does.
2	×2	×2	If n is not a multiple of 3, then 2×n is not either. The invariant still holds.
3	−3	+1	If n is not a multiple of 3, n−3 is not either. The invariant still holds.
4	+0	−2	Number of I's is unchanged. If the invariant held, it still does.

The table above shows clearly that the invariant holds for each of the possible transformation rules, which means that whichever rule one picks, at whatever state, if the number of I's was not a multiple of three before applying the rule, then it will not be afterwards either.

Given that there is a single I in the starting string MI, and one is not a multiple of three, one can then conclude that it is impossible to go from MI to MU (as the number of I's will never be a multiple of three).

Invariant set

A subset S of the domain U of a mapping T: U → U is an invariant set under the mapping when ${\displaystyle x\in S⟺T(x)\in S.}$ The elements of S are not necessarily fixed, even though the set S is fixed in the power set of U. (Some authors use the terminology setwise invariant, vs. pointwise invariant, to distinguish between these cases.) For example, a circle is an invariant subset of the plane under a rotation about the circle's center. Further, a conical surface is invariant as a set under a homothety of space.

An invariant set of an operation T is also said to be stable under T. For example, the normal subgroups that are so important in group theory are those subgroups that are stable under the inner automorphisms of the ambient group. In linear algebra, if a linear transformation T has an eigenvector v, then the line through 0 and v is an invariant set under T, in which case the eigenvectors span an invariant subspace which is stable under T.

When T is a screw displacement, the screw axis is an invariant line, though if the pitch is non-zero, T has no fixed points.

In probability theory and ergodic theory, invariant sets are usually defined via the stronger property ${\displaystyle x\in S\iff T(x)\in S.}$ When the map ${\displaystyle T}$ is measurable, invariant sets form a sigma-algebra, the invariant sigma-algebra.

Formal statement

The notion of invariance is formalized in three different ways in mathematics: via group actions, presentations, and deformation.

Unchanged under group action

Firstly, if one has a group G acting on a mathematical object (or set of objects) X, then one may ask which points x are unchanged, "invariant" under the group action, or under an element g of the group.

Frequently one will have a group acting on a set X, which leaves one to determine which objects in an associated set F(X) are invariant. For example, rotation in the plane about a point leaves the point about which it rotates invariant, while translation in the plane does not leave any points invariant, but does leave all lines parallel to the direction of translation invariant as lines. Formally, define the set of lines in the plane P as L(P); then a rigid motion of the plane takes lines to lines – the group of rigid motions acts on the set of lines – and one may ask which lines are unchanged by an action.

More importantly, one may define a function on a set, such as "radius of a circle in the plane", and then ask if this function is invariant under a group action, such as rigid motions.

Dual to the notion of invariants are coinvariants, also known as orbits, which formalizes the notion of congruence: objects which can be taken to each other by a group action. For example, under the group of rigid motions of the plane, the perimeter of a triangle is an invariant, while the set of triangles congruent to a given triangle is a coinvariant.

These are connected as follows: invariants are constant on coinvariants (for example, congruent triangles have the same perimeter), while two objects which agree in the value of one invariant may or may not be congruent (for example, two triangles with the same perimeter need not be congruent). In classification problems, one might seek to find a complete set of invariants, such that if two objects have the same values for this set of invariants, then they are congruent.

For example, triangles such that all three sides are equal are congruent under rigid motions, via SSS congruence, and thus the lengths of all three sides form a complete set of invariants for triangles. The three angle measures of a triangle are also invariant under rigid motions, but do not form a complete set as incongruent triangles can share the same angle measures. However, if one allows scaling in addition to rigid motions, then the AAA similarity criterion shows that this is a complete set of invariants.

Independent of presentation

Secondly, a function may be defined in terms of some presentation or decomposition of a mathematical object; for instance, the Euler characteristic of a cell complex is defined as the alternating sum of the number of cells in each dimension. One may forget the cell complex structure and look only at the underlying topological space (the manifold) – as different cell complexes give the same underlying manifold, one may ask if the function is independent of choice of presentation, in which case it is an intrinsically defined invariant. This is the case for the Euler characteristic, and a general method for defining and computing invariants is to define them for a given presentation, and then show that they are independent of the choice of presentation. Note that there is no notion of a group action in this sense.

The most common examples are:

• The presentation of a manifold in terms of coordinate charts – invariants must be unchanged under change of coordinates.
• Various manifold decompositions, as discussed for Euler characteristic.
• Invariants of a presentation of a group.

Unchanged under perturbation

Thirdly, if one is studying an object which varies in a family, as is common in algebraic geometry and differential geometry, one may ask if the property is unchanged under perturbation (for example, if an object is constant on families or invariant under change of metric).

Invariants in computer science

For other uses of the word "invariant" in computer science, see invariant (disambiguation).

In computer science, an invariant is a logical assertion that is always held to be true during a certain phase of execution of a computer program. For example, a loop invariant is a condition that is true at the beginning and the end of every iteration of a loop.

Invariants are especially useful when reasoning about the correctness of a computer program. The theory of optimizing compilers, the methodology of design by contract, and formal methods for determining program correctness, all rely heavily on invariants.

Programmers often use assertions in their code to make invariants explicit. Some object oriented programming languages have a special syntax for specifying class invariants.

Automatic invariant detection in imperative programs

Abstract interpretation tools can compute simple invariants of given imperative computer programs. The kind of properties that can be found depend on the abstract domains used. Typical example properties are single integer variable ranges like 0<=x<1024, relations between several variables like 0<=i-j<2*n-1, and modulus information like y%4==0. Academic research prototypes also consider simple properties of pointer structures.

More sophisticated invariants generally have to be provided manually. In particular, when verifying an imperative program using the Hoare calculus, a loop invariant has to be provided manually for each loop in the program, which is one of the reasons that this approach is generally impractical for most programs.

In the context of the above MU puzzle example, there is currently no general automated tool that can detect that a derivation from MI to MU is impossible using only the rules 1–4. However, once the abstraction from the string to the number of its "I"s has been made by hand, leading, for example, to the following C program, an abstract interpretation tool will be able to detect that ICount%3 cannot be 0, and hence the "while"-loop will never terminate.

void MUPuzzle(void) {
    volatile int RandomRule;
    int ICount = 1, UCount = 0;
    while (ICount % 3 != 0)                         // non-terminating loop
        switch(RandomRule) {
        case 1:                  UCount += 1;   break;
        case 2:   ICount *= 2;   UCount *= 2;   break;
        case 3:   ICount -= 3;   UCount += 1;   break;
        case 4:                  UCount -= 2;   break;
        }                                          // computed invariant: ICount % 3 == 1 || ICount % 3 == 2
}