Timeline of the universe

From Wikipedia, the free encyclopedia

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

Diagram of Evolution of the universe from the Big Bang (left) to the present

The timeline of the universe begins with the Big Bang, 13.799 ± 0.021 billion years ago,  and follows the formation and subsequent evolution of the Universe up to the present day. Each era or age of the universe begins with an "epoch", a time of significant change. Times on this list are relative to the moment of the Big Bang.

First 20 minutes

Nature timeline
−13 — – −12 — – −11 — – −10 — – −9 — – −8 — – −7 — – −6 — – −5 — – −4 — – −3 — – −2 — – −1 — – 0 —	Dark Ages Reionization Matter-dominated era Water on Earth Life Multicellular life Vertebrates	← Earliest stars ← Earliest galaxy ← Earliest quasar / black hole ← Omega Centauri ← Andromeda Galaxy ← Milky Way spirals ← NGC 188 star cluster ← Alpha Centauri ← Accelerated expansion ← Earth / Solar System ← Earliest known life ← Atmospheric oxygen ← Sexual reproduction ← Earliest fungi ← Earliest land plants ← Cambrian explosion ← Earliest mammals ← Earliest apes / humans
(billion years ago)

Planck epoch

• c. 0 seconds (13.799 ± 0.021 Gya): Planck epoch begins: Big Bang occurs in which ordinary space and time develop out of a primeval state described by a quantum theory of gravity or "theory of everything". All matter and energy of the universe is contained in a hot, dense point (gravitational singularity)

Grand unification epoch

• c. 10⁻⁴³ seconds: Gravity separates and begins operating on the universe—the remaining fundamental forces stabilize into the electronuclear force, also known as the Grand Unified Force or Grand Unified Theory (GUT), mediated by (the hypothetical) X and Y bosons which allow early matter at this stage to fluctuate between baryon and lepton states.

Inflation

• c. 10⁻³⁵ seconds: inflation, expands the universe by a factor of the order of 10²⁶ over a time of the order of 10⁻³³ to 10⁻³² seconds. The universe is supercooled from about 10²⁷ down to 10²² kelvin.
• c. 10⁻³² seconds: Cosmic inflation ends. The familiar elementary particles now form as a soup of hot ionized gas called quark–gluon plasma;

Quark epoch

• c. 10⁻¹² seconds: Electroweak phase transition: The weak nuclear force is now a short-range force as it separates from electromagnetic force, so matter particles can acquire mass and interact with the Higgs Field. The quark–gluon plasma persists (Quark epoch). The universe cools to 10¹⁵ kelvin.

Quark-hadron transition

• c. 10⁻⁶ seconds: As the universe cools to about 10¹⁰ kelvin, a quark-hadron transition takes place in which quarks become confined in more complex particles—hadrons.

Lepton epoch

• c. 1 second: Lepton epoch begins: The universe cools to 10⁹ kelvin. At this temperature, the hadrons and antihadrons annihilate each other, leaving behind leptons and antileptons – possible disappearance of antiquarks. Gravity governs the expansion of the universe: neutrinos decouple from matter creating a cosmic neutrino background.

Photon epoch

• c. 10 seconds: Photon epoch begins: Most leptons and antileptons annihilate each other. As electrons and positrons annihilate, a small number of unmatched electrons are left over – disappearance of the positrons.
• c. 10 seconds: Universe dominated by photons of radiation – ordinary matter particles are coupled to light and radiation.
• c. 3 minutes: Primordial nucleosynthesis: nuclear fusion begins as lithium and heavy hydrogen (deuterium) and helium nuclei form from protons and neutrons.
• c. 20 minutes: Primordial nucleosynthesis ceases

Matter era

Matter and radiation equivalence

• c. 47,000 years (z = 3600): Matter and radiation equivalence
• c. 70,000 years: As the temperature falls, gravity overcomes pressure allowing first aggregates of matter to form.

Cosmic Dark Age

All-sky map of the CMB, created from nine years of WMAP data

• c. 370,000 years (z = 1,100): The "Dark Ages" is the period between decoupling, when the universe first becomes transparent, until the formation of the first stars. Recombination: electrons combine with nuclei to form atoms, mostly hydrogen and helium. Ordinary matter particles decouple from radiation. The photons present during the decoupling are the same photons seen in the cosmic microwave background (CMB) radiation.
• c. 10–17 million years: The "Dark Ages" span a period during which the temperature of cosmic microwave background radiation cooled from some 4,000 K (3,730 °C; 6,740 °F) down to about 60 K (−213.2 °C; −351.7 °F).

Reionization

• c. 100 million years: Gravitational collapse: ordinary matter particles fall into the structures created by dark matter. Reionization begins: smaller (stars) and larger non-linear structures (quasars) begin to take shape – their ultraviolet light ionizes remaining neutral gas.
• 200–300 million years: First stars begin to shine: Because many are Population III stars (some Population II stars are accounted for at this time) they are much bigger and hotter and their life cycle is fairly short. Unlike later generations of stars, these stars are metal free. Reionization begins, with the absorption of certain wavelengths of light by neutral hydrogen creating Gunn–Peterson troughs. The resulting ionized gas (especially free electrons) in the intergalactic medium causes some scattering of light, but with much lower opacity than before recombination due the expansion of the universe and clumping of gas into galaxies.
• 200 million years: The oldest-known star (confirmed) – SMSS J031300.36−670839.3, forms.
• 300 million years: First large-scale astronomical objects, protogalaxies and quasars may have begun forming. As Population III stars continue to burn, stellar nucleosynthesis operates – stars burn mainly by fusing hydrogen to produce more helium in what is referred to as the main sequence. Over time these stars are forced to fuse helium to produce carbon, oxygen, silicon and other heavy elements up to iron on the periodic table. These elements, when seeded into neighbouring gas clouds by supernova, will lead to the formation of more Population II stars (metal poor) and gas giants.
• 320 million years (z = 13.3): HD1, the oldest-known spectroscopically-confirmed galaxy, forms.
• 380 million years: UDFj-39546284 forms, current record holder for unconfirmed oldest-known quasar.
• 600 million years: HE 1523-0901, the oldest star found producing neutron capture elements forms, marking a new point in ability to detect stars with a telescope.
• 630 million years (z = 8.2): GRB 090423, the oldest gamma-ray burst recorded suggests that supernovas may have happened very early on in the evolution of the Universe

Galaxy epoch

• < 1 billion years, (13 Gya): first stars in the central bar portion of the Milky Way are born,
• 2.6 billion years (11 Gya): first stars in the thick disk region of the Milky Way are formed.
• 4 billion years (10 Gya): Gaia Enceladus merges into Milky Way.
• 5 or 6 billion years, (8 or 9 Gya): first stars in the thin disk region of the Milky Way are formed.

Acceleration

• 8.8 billion years (5 Gya, z = 0.5): Acceleration: dark-energy dominated era begins, following the matter-dominated era during which cosmic expansion was slowing down.

Notable cosmological and other events of the natural history depicted in a spiral. In the center left the primal supernova can be seen and continuing the creation of the Sun, the Earth and the Moon (by Theia impact) can be seen

Epochs of the formation of the Solar System

Main article: Formation and evolution of the Solar System

• 9.2 billion years (4.6–4.57 Gya): Primal supernova, possibly triggers the formation of The Solar System.
• 9.2318 billion years (4.5682 Gya): Sun forms – Planetary nebula begins accretion of planets.
• 9.23283 billion years (4.56717–4.55717 Gya): Four Jovian planets (Jupiter, Saturn, Uranus, Neptune) evolve around the Sun.
• 9.257 billion years (4.543–4.5 Gya): Solar System of Eight planets, four terrestrial (Mercury, Venus, Earth, Mars) evolve around the Sun. Because of accretion many smaller planets form orbits around the proto-Sun some with conflicting orbits – early heavy bombardment begins. A large planetoid strikes Mercury, stripping it of outer envelope of original crust and mantle, leaving the planet's core exposed – Mercury's iron content is notably high.
• 9.266 billion years (4.533 Gya): Formation of Earth-Moon system following giant impact by hypothetical planetoid Theia (planet). Moon's gravitational pull helps stabilize Earth's fluctuating axis of rotation.
• 9.271 billion years (4.529 Gya): Major collision with a pluto-sized planetoid establishes the Martian dichotomy on Mars
• 9.3 billion years (4.5 Gya): Sun becomes a main sequence yellow star: formation of the Oort cloud and Kuiper belt
• 9.396 billion years (4.404 Gya): Liquid water may have existed on the surface of the Earth
• 9.7 billion years (4.1 Gya): Resonance in Jupiter and Saturn's orbits moves Neptune out into the Kuiper belt causing a disruption among asteroids and comets there. As a result, Late Heavy Bombardment batters the inner Solar System. Meteorite impact creates the Hellas Planitia on Mars, the largest unambiguous structure on the planet.
• 10.4 billion years (3.5 Gya): Earliest fossil traces of life on Earth (stromatolites)
• 10.6 billion years (3.2 Gya): Martian climate thins to its present density: groundwater stored in upper crust (megaregolith) begins to freeze, forming thick cryosphere overlying deeper zone of liquid water – dry ices composed of frozen carbon dioxide form
• 10.8 billion years (3 Gya): Beethoven Basin forms on Mercury – unlike many basins of similar size on the Moon, Beethoven is not multi ringed and ejecta buries crater rim and is barely visible
• 11.6 billion years (2.2 Gya): Last great tectonic period in Martian geologic history: Valles Marineris, largest canyon complex in the Solar System, forms – although some suggestions of thermokarst activity or even water erosion, it is suggested Valles Marineris is rift fault.

Recent history

11.8 billion years (2 Gya): Olympus Mons, the largest volcano in the Solar System, is formed

12.1 billion years (1.7 Gya): Sagittarius Dwarf Spheroidal Galaxy captured into an orbit around Milky Way Galaxy

12.7 billion years (1.1 Gya): Copernican Period begins on Moon: defined by impact craters that possess bright optically immature ray systems

12.8 billion years (1 Gya): Interactions between Andromeda and its companion galaxies Messier 32 and Messier 110. Galaxy collision with Messier 82 forms its patterned spiral disc: galaxy interactions between NGC 3077 and Messier 81; Saturn's moon Titan begins evolving the recognisable surface features that include rivers, lakes, and deltas

13 billion years (800 Mya): Copernicus (lunar crater) forms from the impact on the Lunar surface in the area of Oceanus Procellarum – has terrace inner wall and 30 km wide, sloping rampart that descends nearly a kilometre to the surrounding mare

13.175 billion years (625 Mya): formation of Hyades star cluster: consists of a roughly spherical group of hundreds of stars sharing the same age, place of origin, chemical content and motion through space

13.2 billion years (600 Mya): Whirlpool Galaxy collides with NGC 5195 forming a present connected galaxy system. HD 189733 b forms around parent star HD 189733: the first planet to reveal the climate, organic constituencies, even colour (blue) of its atmosphere

13.6–13.5 billion years (300-200 Mya): Sirius, the brightest star in the Earth's sky, forms.

13.7 billion years (100 Mya): Formation of Pleiades Star Cluster

13.780 billion years (20 Mya): Possible formation of Orion Nebula

13.792 billion years (7.6 Mya): Betelgeuse forms.

13.8 billion years (Without uncertainties): Present day.

Closed-form expression

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Closed-form_expression

In mathematics, an expression or equation is in closed form if it is formed with constants, variables, and a set of functions considered as basic and connected by arithmetic operations (+, −, ×, /, and integer powers) and function composition. Commonly, the basic functions that are allowed in closed forms are nth root, exponential function, logarithm, and trigonometric functions. However, the set of basic functions depends on the context. For example, if one adds polynomial roots to the basic functions, the functions that have a closed form are called elementary functions.

The closed-form problem arises when new ways are introduced for specifying mathematical objects, such as limits, series, and integrals: given an object specified with such tools, a natural problem is to find, if possible, a closed-form expression of this object; that is, an expression of this object in terms of previous ways of specifying it.

Example: roots of polynomials

The quadratic formula

$${\displaystyle x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}}$$

is a closed form of the solutions to the general quadratic equation ${\displaystyle a{x}^2+bx+c=0.}$

More generally, in the context of polynomial equations, a closed form of a solution is a solution in radicals; that is, a closed-form expression for which the allowed functions are only nth-roots and field operations ${\displaystyle (+,-,\times ,/).}$ In fact, field theory allows showing that if a solution of a polynomial equation has a closed form involving exponentials, logarithms or trigonometric functions, then it has also a closed form that does not involve these functions.

There are expressions in radicals for all solutions of cubic equations (degree 3) and quartic equations (degree 4). The size of these expressions increases significantly with the degree, limiting their usefulness.

In higher degrees, the Abel–Ruffini theorem states that there are equations whose solutions cannot be expressed in radicals, and, thus, have no closed forms. A simple example is the equation ${\displaystyle x^5-x-1=0.}$ Galois theory provides an algorithmic method for deciding whether a particular polynomial equation can be solved in radicals.

Symbolic integration

Symbolic integration consists essentially of the search of closed forms for antiderivatives of functions that are specified by closed-form expressions. In this context, the basic functions used for defining closed forms are commonly logarithms, exponential function and polynomial roots. Functions that have a closed form for these basic functions are called elementary functions and include trigonometric functions, inverse trigonometric functions, hyperbolic functions, and inverse hyperbolic functions.

The fundamental problem of symbolic integration is thus, given an elementary function specified by a closed-form expression, to decide whether its antiderivative is an elementary function, and, if it is, to find a closed-form expression for this antiderivative.

For rational functions; that is, for fractions of two polynomial functions; antiderivatives are not always rational fractions, but are always elementary functions that may involve logarithms and polynomial roots. This is usually proved with partial fraction decomposition. The need for logarithms and polynomial roots is illustrated by the formula

$${\displaystyle \int \frac{f(x)}{g(x)}dx=\sum _{\alpha \in \mathrm{Roots}(g(x))}\frac{f(\alpha )}{g^{\prime }(\alpha )}\ln (x-\alpha ),}$$

which is valid if ${\displaystyle f}$ and ${\displaystyle g}$ are coprime polynomials such that ${\displaystyle g}$ is square free and ${\displaystyle \mathrm{deg}f<\mathrm{deg}g.}$

Alternative definitions

Changing the basic functions to include additional functions can change the set of equations with closed-form solutions. Many cumulative distribution functions cannot be expressed in closed form, unless one considers special functions such as the error function or gamma function to be basic. It is possible to solve the quintic equation if general hypergeometric functions are included, although the solution is far too complicated algebraically to be useful. For many practical computer applications, it is entirely reasonable to assume that the gamma function and other special functions are basic since numerical implementations are widely available.

Analytic expression

This is a term that is sometimes understood as a synonym for closed-form (see "Wolfram Mathworld".) but this usage is contested (see "Math Stackexchange".). It is unclear the extent to which this term is genuinely in use as opposed to the result of uncited earlier versions of this page.

Comparison of different classes of expressions

The closed-form expressions do not include infinite series or continued fractions; neither includes integrals or limits. Indeed, by the Stone–Weierstrass theorem, any continuous function on the unit interval can be expressed as a limit of polynomials, so any class of functions containing the polynomials and closed under limits will necessarily include all continuous functions.

Similarly, an equation or system of equations is said to have a closed-form solution if and only if at least one solution can be expressed as a closed-form expression; and it is said to have an analytic solution if and only if at least one solution can be expressed as an analytic expression. There is a subtle distinction between a "closed-form function" and a "closed-form number" in the discussion of a "closed-form solution", discussed in (Chow 1999) and below. A closed-form or analytic solution is sometimes referred to as an explicit solution.

	Arithmetic expressions	Polynomial expressions	Algebraic expressions	Closed-form expressions	Analytic expressions	Mathematical expressions
Constant	Yes	Yes	Yes	Yes	Yes	Yes
Elementary arithmetic operation	Yes	Addition, subtraction, and multiplication only	Yes	Yes	Yes	Yes
Finite sum	Yes	Yes	Yes	Yes	Yes	Yes
Finite product	Yes	Yes	Yes	Yes	Yes	Yes
Finite continued fraction	Yes	No	Yes	Yes	Yes	Yes
Variable	No	Yes	Yes	Yes	Yes	Yes
Integer exponent	No	Yes	Yes	Yes	Yes	Yes
Integer nth root	No	No	Yes	Yes	Yes	Yes
Rational exponent	No	No	Yes	Yes	Yes	Yes
Integer factorial	No	No	Yes	Yes	Yes	Yes
Irrational exponent	No	No	No	Yes	Yes	Yes
Exponential function	No	No	No	Yes	Yes	Yes
Logarithm	No	No	No	Yes	Yes	Yes
Trigonometric function	No	No	No	Yes	Yes	Yes
Inverse trigonometric function	No	No	No	Yes	Yes	Yes
Hyperbolic function	No	No	No	Yes	Yes	Yes
Inverse hyperbolic function	No	No	No	Yes	Yes	Yes
Root of a polynomial that is not an algebraic solution	No	No	No	No	Yes	Yes
Gamma function and factorial of a non-integer	No	No	No	No	Yes	Yes
Bessel function	No	No	No	No	Yes	Yes
Special function	No	No	No	No	Yes	Yes
Infinite sum (series) (including power series)	No	No	No	No	Convergent only	Yes
Infinite product	No	No	No	No	Convergent only	Yes
Infinite continued fraction	No	No	No	No	Convergent only	Yes
Limit	No	No	No	No	No	Yes
Derivative	No	No	No	No	No	Yes
Integral	No	No	No	No	No	Yes

Dealing with non-closed-form expressions

Transformation into closed-form expressions

The expression: $${\displaystyle f(x)=\sum _{n=0}^{\mathrm{\infty }}\frac{x}{2^n}}$$ is not in closed form because the summation entails an infinite number of elementary operations. However, by summing a geometric series this expression can be expressed in the closed form: $${\displaystyle f(x)=2x.}$$

Differential Galois theory

Main article: Differential Galois theory

See also: Nonelementary integral

The integral of a closed-form expression may or may not itself be expressible as a closed-form expression. This study is referred to as differential Galois theory, by analogy with algebraic Galois theory.

The basic theorem of differential Galois theory is due to Joseph Liouville in the 1830s and 1840s and hence referred to as Liouville's theorem.

A standard example of an elementary function whose antiderivative does not have a closed-form expression is: $${\displaystyle e^{-x^2},}$$ whose one antiderivative is (up to a multiplicative constant) the error function: $${\displaystyle \mathrm{erf}(x)=\frac{2}{\sqrt{\pi }}{\int }_0^x{e}^{-t^2}dt.}$$

Mathematical modelling and computer simulation

Equations or systems too complex for closed-form or analytic solutions can often be analysed by mathematical modelling and computer simulation (for an example in physics, see).

Closed-form number

See also: Transcendental number theory

Three subfields of the complex numbers C have been suggested as encoding the notion of a "closed-form number"; in increasing order of generality, these are the Liouvillian numbers (not to be confused with Liouville numbers in the sense of rational approximation), EL numbers and elementary numbers. The Liouvillian numbers, denoted L, form the smallest algebraically closed subfield of C closed under exponentiation and logarithm (formally, intersection of all such subfields)—that is, numbers which involve explicit exponentiation and logarithms, but allow explicit and implicit polynomials (roots of polynomials); this is defined in (Ritt 1948, p. 60). L was originally referred to as elementary numbers, but this term is now used more broadly to refer to numbers defined explicitly or implicitly in terms of algebraic operations, exponentials, and logarithms. A narrower definition proposed in (Chow 1999, pp. 441–442), denoted E, and referred to as EL numbers, is the smallest subfield of C closed under exponentiation and logarithm—this need not be algebraically closed, and corresponds to explicit algebraic, exponential, and logarithmic operations. "EL" stands both for "exponential–logarithmic" and as an abbreviation for "elementary".

Whether a number is a closed-form number is related to whether a number is transcendental. Formally, Liouvillian numbers and elementary numbers contain the algebraic numbers, and they include some but not all transcendental numbers. In contrast, EL numbers do not contain all algebraic numbers, but do include some transcendental numbers. Closed-form numbers can be studied via transcendental number theory, in which a major result is the Gelfond–Schneider theorem, and a major open question is Schanuel's conjecture.

Numerical computations

For purposes of numeric computations, being in closed form is not in general necessary, as many limits and integrals can be efficiently computed. Some equations have no closed form solution, such as those that represent the Three-body problem or the Hodgkin–Huxley model. Therefore, the future states of these systems must be computed numerically.

Conversion from numerical forms

There is software that attempts to find closed-form expressions for numerical values, including RIES, identify in Maple and SymPy, Plouffe's Inverter, and the Inverse Symbolic Calculator.