Symmetric polynomial

From Wikipedia, the free encyclopedia

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

 

In mathematics, a symmetric polynomial is a polynomial P(X₁, X₂, …, X_n) in n variables, such that if any of the variables are interchanged, one obtains the same polynomial. Formally, P is a symmetric polynomial if for any permutation σ of the subscripts 1, 2, ..., n one has P(X_σ(1), X_σ(2), …, X_σ(n)) = P(X₁, X₂, …, X_n).

Symmetric polynomials arise naturally in the study of the relation between the roots of a polynomial in one variable and its coefficients, since the coefficients can be given by polynomial expressions in the roots, and all roots play a similar role in this setting. From this point of view the elementary symmetric polynomials are the most fundamental symmetric polynomials. A theorem states that any symmetric polynomial can be expressed in terms of elementary symmetric polynomials, which implies that every symmetric polynomial expression in the roots of a monic polynomial can alternatively be given as a polynomial expression in the coefficients of the polynomial.

Symmetric polynomials also form an interesting structure by themselves, independently of any relation to the roots of a polynomial. In this context other collections of specific symmetric polynomials, such as complete homogeneous, power sum, and Schur polynomials play important roles alongside the elementary ones. The resulting structures, and in particular the ring of symmetric functions, are of great importance in combinatorics and in representation theory.

Examples

The following polynomials in two variables X₁ and X₂ are symmetric:

${\displaystyle X_1^3+X_2^3-7}$

${\displaystyle 4X_1^2{X}_2^2+X_1^3{X}_2+X_1{X}_2^3+(X_1+X_2{)}^4}$

as is the following polynomial in three variables X₁, X₂, X₃:

${\displaystyle X_1{X}_2{X}_3-2X_1{X}_2-2X_1{X}_3-2X_2{X}_3}$

There are many ways to make specific symmetric polynomials in any number of variables (see the various types below). An example of a somewhat different flavor is

${\displaystyle \prod _{1\le i<j\le n}(X_i-X_j{)}^2}$

where first a polynomial is constructed that changes sign under every exchange of variables, and taking the square renders it completely symmetric (if the variables represent the roots of a monic polynomial, this polynomial gives its discriminant).

On the other hand, the polynomial in two variables

${\displaystyle X_1-X_2}$

is not symmetric, since if one exchanges ${\displaystyle X_1}$ and ${\displaystyle X_2}$ one gets a different polynomial, ${\displaystyle X_2-X_1}$. Similarly in three variables

${\displaystyle X_1^4{X}_2^2{X}_3+X_1{X}_2^4{X}_3^2+X_1^2{X}_2{X}_3^4}$

has only symmetry under cyclic permutations of the three variables, which is not sufficient to be a symmetric polynomial. However, the following is symmetric:

${\displaystyle X_1^4{X}_2^2{X}_3+X_1{X}_2^4{X}_3^2+X_1^2{X}_2{X}_3^4+X_1^4{X}_2{X}_3^2+X_1{X}_2^2{X}_3^4+X_1^2{X}_2^4{X}_3}$

Applications

Galois theory

Main article: Galois theory

One context in which symmetric polynomial functions occur is in the study of monic univariate polynomials of degree n having n roots in a given field. These n roots determine the polynomial, and when they are considered as independent variables, the coefficients of the polynomial are symmetric polynomial functions of the roots. Moreover the fundamental theorem of symmetric polynomials implies that a polynomial function f of the n roots can be expressed as (another) polynomial function of the coefficients of the polynomial determined by the roots if and only if f is given by a symmetric polynomial.

This yields the approach to solving polynomial equations by inverting this map, "breaking" the symmetry – given the coefficients of the polynomial (the elementary symmetric polynomials in the roots), how can one recover the roots? This leads to studying solutions of polynomials using the permutation group of the roots, originally in the form of Lagrange resolvents, later developed in Galois theory.

Relation with the roots of a monic univariate polynomial

Consider a monic polynomial in t of degree n

${\displaystyle P=t^n+a_{n-1}{t}^{n-1}+\cdots +a_2{t}^2+a_{1}t+a_0}$

with coefficients a_i in some field K. There exist n roots x₁,…,x_n of P in some possibly larger field (for instance if K is the field of real numbers, the roots will exist in the field of complex numbers); some of the roots might be equal, but the fact that one has all roots is expressed by the relation

${\displaystyle P=t^n+a_{n-1}{t}^{n-1}+\cdots +a_2{t}^2+a_{1}t+a_0=(t-x_1)(t-x_2)\cdots (t-x_n).}$

By comparing coefficients one finds that

${\displaystyle \begin{array}{rl}{a}_{n-1}& =-x_1-x_2-\cdots -x_n\\ a_{n-2}& =x_1{x}_2+x_1{x}_3+\cdots +x_2{x}_3+\cdots +x_{n-1}{x}_n=\sum _{1\le i<j\le n}{x}_i{x}_j\\ & ⋮\\ a_1& =(-1{)}^{n-1}(x_2{x}_3\cdots x_n+x_1{x}_3{x}_4\cdots x_n+\cdots +x_1{x}_2\cdots x_{n-2}{x}_n+x_1{x}_2\cdots x_{n-1})=(-1{)}^{n-1}\sum _{i=1}^n\prod _{j\ne i}{x}_j\\ a_0& =(-1{)}^n{x}_1{x}_2\cdots x_n.\end{array}}$

These are in fact just instances of Viète's formulas. They show that all coefficients of the polynomial are given in terms of the roots by a symmetric polynomial expression: although for a given polynomial P there may be qualitative differences between the roots (like lying in the base field K or not, being simple or multiple roots), none of this affects the way the roots occur in these expressions.

Now one may change the point of view, by taking the roots rather than the coefficients as basic parameters for describing P, and considering them as indeterminates rather than as constants in an appropriate field; the coefficients a_i then become just the particular symmetric polynomials given by the above equations. Those polynomials, without the sign ${\displaystyle (-1{)}^{n-i}}$, are known as the elementary symmetric polynomials in x₁, …, x_n. A basic fact, known as the fundamental theorem of symmetric polynomials, states that any symmetric polynomial in n variables can be given by a polynomial expression in terms of these elementary symmetric polynomials. It follows that any symmetric polynomial expression in the roots of a monic polynomial can be expressed as a polynomial in the coefficients of the polynomial, and in particular that its value lies in the base field K that contains those coefficients. Thus, when working only with such symmetric polynomial expressions in the roots, it is unnecessary to know anything particular about those roots, or to compute in any larger field than K in which those roots may lie. In fact the values of the roots themselves become rather irrelevant, and the necessary relations between coefficients and symmetric polynomial expressions can be found by computations in terms of symmetric polynomials only. An example of such relations are Newton's identities, which express the sum of any fixed power of the roots in terms of the elementary symmetric polynomials.

Special kinds of symmetric polynomials

There are a few types of symmetric polynomials in the variables X₁, X₂, …, X_n that are fundamental.

Elementary symmetric polynomials

Main article: Elementary symmetric polynomial

For each nonnegative integer k, the elementary symmetric polynomial e_k(X₁, …, X_n) is the sum of all distinct products of k distinct variables. (Some authors denote it by σ_k instead.) For k = 0 there is only the empty product so e₀(X₁, …, X_n) = 1, while for k > n, no products at all can be formed, so e_k(X₁, X₂, …, X_n) = 0 in these cases. The remaining n elementary symmetric polynomials are building blocks for all symmetric polynomials in these variables: as mentioned above, any symmetric polynomial in the variables considered can be obtained from these elementary symmetric polynomials using multiplications and additions only. In fact one has the following more detailed facts:

• any symmetric polynomial P in X₁, …, X_n can be written as a polynomial expression in the polynomials e_k(X₁, …, X_n) with 1 ≤ k ≤ n;
• this expression is unique up to equivalence of polynomial expressions;
• if P has integral coefficients, then the polynomial expression also has integral coefficients.

For example, for n = 2, the relevant elementary symmetric polynomials are e₁(X₁, X₂) = X₁ + X₂, and e₂(X₁, X₂) = X₁X₂. The first polynomial in the list of examples above can then be written as

${\displaystyle X_1^3+X_2^3-7=e_1(X_1,X_2{)}^3-3e_2(X_1,X_2)e_1(X_1,X_2)-7}$

(for a proof that this is always possible see the fundamental theorem of symmetric polynomials).

Monomial symmetric polynomials

Powers and products of elementary symmetric polynomials work out to rather complicated expressions. If one seeks basic additive building blocks for symmetric polynomials, a more natural choice is to take those symmetric polynomials that contain only one type of monomial, with only those copies required to obtain symmetry. Any monomial in X₁, …, X_n can be written as X₁^α₁…X_n^α_n where the exponents α_i are natural numbers (possibly zero); writing α = (α₁,…,α_n) this can be abbreviated to X^α. The monomial symmetric polynomial m_α(X₁, …, X_n) is defined as the sum of all monomials x^β where β ranges over all distinct permutations of (α₁,…,α_n). For instance one has

${\displaystyle m_{(3,1,1)}(X_1,X_2,X_3)=X_1^3{X}_2{X}_3+X_1{X}_2^3{X}_3+X_1{X}_2{X}_3^3}$,

${\displaystyle m_{(3,2,1)}(X_1,X_2,X_3)=X_1^3{X}_2^2{X}_3+X_1^3{X}_2{X}_3^2+X_1^2{X}_2^3{X}_3+X_1^2{X}_2{X}_3^3+X_1{X}_2^3{X}_3^2+X_1{X}_2^2{X}_3^3.}$

Clearly m_α = m_β when β is a permutation of α, so one usually considers only those m_α for which α₁ ≥ α₂ ≥ … ≥ α_n, in other words for which α is a partition of an integer. These monomial symmetric polynomials form a vector space basis: every symmetric polynomial P can be written as a linear combination of the monomial symmetric polynomials. To do this it suffices to separate the different types of monomial occurring in P. In particular if P has integer coefficients, then so will the linear combination.

The elementary symmetric polynomials are particular cases of monomial symmetric polynomials: for 0 ≤ k ≤ n one has

${\displaystyle e_k(X_1,\dots ,X_n)=m_{\alpha }(X_1,\dots ,X_n)}$ where α is the partition of k into k parts 1 (followed by n − k zeros).

Power-sum symmetric polynomials

Main article: Power sum symmetric polynomial

For each integer k ≥ 1, the monomial symmetric polynomial m_(k,0,…,0)(X₁, …, X_n) is of special interest. It is the power sum symmetric polynomial, defined as

${\displaystyle p_k(X_1,\dots ,X_n)=X_1^k+X_2^k+\cdots +X_n^k.}$

All symmetric polynomials can be obtained from the first n power sum symmetric polynomials by additions and multiplications, possibly involving rational coefficients. More precisely,

Any symmetric polynomial in X₁, …, X_n can be expressed as a polynomial expression with rational coefficients in the power sum symmetric polynomials p₁(X₁, …, X_n), …, p_n(X₁, …, X_n).

In particular, the remaining power sum polynomials p_k(X₁, …, X_n) for k > n can be so expressed in the first n power sum polynomials; for example

${\displaystyle p_3(X_1,X_2)=\frac{3}{2}{p}_2(X_1,X_2)p_1(X_1,X_2)-\frac{1}{2}{p}_1(X_1,X_2{)}^3.}$

In contrast to the situation for the elementary and complete homogeneous polynomials, a symmetric polynomial in n variables with integral coefficients need not be a polynomial function with integral coefficients of the power sum symmetric polynomials. For an example, for n = 2, the symmetric polynomial

${\displaystyle m_{(2,1)}(X_1,X_2)=X_1^2{X}_2+X_1{X}_2^2}$

has the expression

${\displaystyle m_{(2,1)}(X_1,X_2)=\frac{1}{2}{p}_1(X_1,X_2{)}^3-\frac{1}{2}{p}_2(X_1,X_2)p_1(X_1,X_2).}$

Using three variables one gets a different expression

${\displaystyle \begin{array}{rl}{m}_{(2,1)}(X_1,X_2,X_3)& =X_1^2{X}_2+X_1{X}_2^2+X_1^2{X}_3+X_1{X}_3^2+X_2^2{X}_3+X_2{X}_3^2\\ & =p_1(X_1,X_2,X_3)p_2(X_1,X_2,X_3)-p_3(X_1,X_2,X_3).\end{array}}$

The corresponding expression was valid for two variables as well (it suffices to set X₃ to zero), but since it involves p₃, it could not be used to illustrate the statement for n = 2. The example shows that whether or not the expression for a given monomial symmetric polynomial in terms of the first n power sum polynomials involves rational coefficients may depend on n. But rational coefficients are always needed to express elementary symmetric polynomials (except the constant ones, and e₁ which coincides with the first power sum) in terms of power sum polynomials. The Newton identities provide an explicit method to do this; it involves division by integers up to n, which explains the rational coefficients. Because of these divisions, the mentioned statement fails in general when coefficients are taken in a field of finite characteristic; however, it is valid with coefficients in any ring containing the rational numbers.

Complete homogeneous symmetric polynomials

Main article: Complete homogeneous symmetric polynomial

For each nonnegative integer k, the complete homogeneous symmetric polynomial h_k(X₁, …, X_n) is the sum of all distinct monomials of degree k in the variables X₁, …, X_n. For instance

${\displaystyle h_3(X_1,X_2,X_3)=X_1^3+X_1^2{X}_2+X_1^2{X}_3+X_1{X}_2^2+X_1{X}_2{X}_3+X_1{X}_3^2+X_2^3+X_2^2{X}_3+X_2{X}_3^2+X_3^3.}$

The polynomial h_k(X₁, …, X_n) is also the sum of all distinct monomial symmetric polynomials of degree k in X₁, …, X_n, for instance for the given example

All symmetric polynomials in these variables can be built up from complete homogeneous ones: any symmetric polynomial in X₁, …, X_n can be obtained from the complete homogeneous symmetric polynomials h₁(X₁, …, X_n), …, h_n(X₁, …, X_n) via multiplications and additions. More precisely:

Any symmetric polynomial P in X₁, …, X_n can be written as a polynomial expression in the polynomials h_k(X₁, …, X_n) with 1 ≤ k ≤ n.

If P has integral coefficients, then the polynomial expression also has integral coefficients.

For example, for n = 2, the relevant complete homogeneous symmetric polynomials are h₁(X₁, X₂) = X₁ + X₂ and h₂(X₁, X₂) = X₁² + X₁X₂ + X₂². The first polynomial in the list of examples above can then be written as

${\displaystyle X_1^3+X_2^3-7=-2h_1(X_1,X_2{)}^3+3h_1(X_1,X_2)h_2(X_1,X_2)-7.}$

As in the case of power sums, the given statement applies in particular to the complete homogeneous symmetric polynomials beyond h_n(X₁, …, X_n), allowing them to be expressed in terms of the ones up to that point; again the resulting identities become invalid when the number of variables is increased.

An important aspect of complete homogeneous symmetric polynomials is their relation to elementary symmetric polynomials, which can be expressed as the identities

${\displaystyle \sum _{i=0}^k(-1{)}^i{e}_i(X_1,\dots ,X_n)h_{k-i}(X_1,\dots ,X_n)=0}$, for all k > 0, and any number of variables n.

Since e₀(X₁, …, X_n) and h₀(X₁, …, X_n) are both equal to 1, one can isolate either the first or the last term of these summations; the former gives a set of equations that allows one to recursively express the successive complete homogeneous symmetric polynomials in terms of the elementary symmetric polynomials, and the latter gives a set of equations that allows doing the inverse. This implicitly shows that any symmetric polynomial can be expressed in terms of the h_k(X₁, …, X_n) with 1 ≤ k ≤ n: one first expresses the symmetric polynomial in terms of the elementary symmetric polynomials, and then expresses those in terms of the mentioned complete homogeneous ones.

Schur polynomials

Main article: Schur polynomial

Another class of symmetric polynomials is that of the Schur polynomials, which are of fundamental importance in the applications of symmetric polynomials to representation theory. They are however not as easy to describe as the other kinds of special symmetric polynomials; see the main article for details.

Symmetric polynomials in algebra

Symmetric polynomials are important to linear algebra, representation theory, and Galois theory. They are also important in combinatorics, where they are mostly studied through the ring of symmetric functions, which avoids having to carry around a fixed number of variables all the time.

Alternating polynomials

Main article: Alternating polynomials

Analogous to symmetric polynomials are alternating polynomials: polynomials that, rather than being invariant under permutation of the entries, change according to the sign of the permutation.

These are all products of the Vandermonde polynomial and a symmetric polynomial, and form a quadratic extension of the ring of symmetric polynomials: the Vandermonde polynomial is a square root of the discriminant.

Galaxy rotation curve

From Wikipedia, the free encyclopedia

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

 

Rotation curve of spiral galaxy Messier 33 (yellow and blue points with error bars), and a predicted one from distribution of the visible matter (gray line). The discrepancy between the two curves can be accounted for by adding a dark matter halo surrounding the galaxy.

The rotation curve of a disc galaxy (also called a velocity curve) is a plot of the orbital speeds of visible stars or gas in that galaxy versus their radial distance from that galaxy's centre. It is typically rendered graphically as a plot, and the data observed from each side of a spiral galaxy are generally asymmetric, so that data from each side are averaged to create the curve. A significant discrepancy exists between the experimental curves observed, and a curve derived by applying gravity theory to the matter observed in a galaxy. Theories involving dark matter are the main postulated solutions to account for the variance.

The rotational/orbital speeds of galaxies/stars do not follow the rules found in other orbital systems such as stars/planets and planets/moons that have most of their mass at the centre. Stars revolve around their galaxy's centre at equal or increasing speed over a large range of distances. In contrast, the orbital velocities of planets in planetary systems and moons orbiting planets decline with distance according to Kepler’s third law. This reflects the mass distributions within those systems. The mass estimations for galaxies based on the light they emit are far too low to explain the velocity observations.

The galaxy rotation problem is the discrepancy between observed galaxy rotation curves and the theoretical prediction, assuming a centrally dominated mass associated with the observed luminous material. When mass profiles of galaxies are calculated from the distribution of stars in spirals and mass-to-light ratios in the stellar disks, they do not match with the masses derived from the observed rotation curves and the law of gravity. A solution to this conundrum is to hypothesize the existence of dark matter and to assume its distribution from the galaxy's center out to its halo.

Though dark matter is by far the most accepted explanation of the rotation problem, other proposals have been offered with varying degrees of success. Of the possible alternatives, one of the most notable is modified Newtonian dynamics (MOND), which involves modifying the laws of gravity.

History

In 1932, Jan Hendrik Oort became the first to report that measurements of the stars in the solar neighborhood indicated that they moved faster than expected when a mass distribution based upon visible matter was assumed, but these measurements were later determined to be essentially erroneous. In 1939, Horace Babcock reported in his PhD thesis measurements of the rotation curve for Andromeda which suggested that the mass-to-luminosity ratio increases radially. He attributed that to either the absorption of light within the galaxy or to modified dynamics in the outer portions of the spiral and not to any form of missing matter. Babcock's measurements turned out to disagree substantially with those found later, and the first measurement of an extended rotation curve in good agreement with modern data was published in 1957 by Henk van de Hulst and collaborators, who studied M31 with the newly commissioned Dwingeloo 25 meter telescope. A companion paper by Maarten Schmidt showed that this rotation curve could be fit by a flattened mass distribution more extensive than the light. In 1959, Louise Volders used the same telescope to demonstrate that the spiral galaxy M33 also does not spin as expected according to Keplerian dynamics.

Reporting on NGC 3115, Jan Oort wrote that "the distribution of mass in the system appears to bear almost no relation to that of light... one finds the ratio of mass to light in the outer parts of NGC 3115 to be about 250". On page 302–303 of his journal article, he wrote that "The strongly condensed luminous system appears imbedded in a large and more or less homogeneous mass of great density" and although he went on to speculate that this mass may be either extremely faint dwarf stars or interstellar gas and dust, he had clearly detected the dark matter halo of this galaxy.

The Carnegie telescope (Carnegie Double Astrograph) was intended to study this problem of Galactic rotation.

In the late 1960s and early 1970s, Vera Rubin, an astronomer at the Department of Terrestrial Magnetism at the Carnegie Institution of Washington, worked with a new sensitive spectrograph that could measure the velocity curve of edge-on spiral galaxies to a greater degree of accuracy than had ever before been achieved. Together with fellow staff-member Kent Ford, Rubin announced at a 1975 meeting of the American Astronomical Society the discovery that most stars in spiral galaxies orbit at roughly the same speed, and that this implied that galaxy masses grow approximately linearly with radius well beyond the location of most of the stars (the galactic bulge). Rubin presented her results in an influential paper in 1980. These results suggested either that Newtonian gravity does not apply universally or that, conservatively, upwards of 50% of the mass of galaxies was contained in the relatively dark galactic halo. Although initially met with skepticism, Rubin's results have been confirmed over the subsequent decades.

If Newtonian mechanics is assumed to be correct, it would follow that most of the mass of the galaxy had to be in the galactic bulge near the center and that the stars and gas in the disk portion should orbit the center at decreasing velocities with radial distance from the galactic center (the dashed line in Fig. 1).

Observations of the rotation curve of spirals, however, do not bear this out. Rather, the curves do not decrease in the expected inverse square root relationship but are "flat", i.e. outside of the central bulge the speed is nearly a constant (the solid line in Fig. 1). It is also observed that galaxies with a uniform distribution of luminous matter have a rotation curve that rises from the center to the edge, and most low-surface-brightness galaxies (LSB galaxies) have the same anomalous rotation curve.

The rotation curves might be explained by hypothesizing the existence of a substantial amount of matter permeating the galaxy outside of the central bulge that is not emitting light in the mass-to-light ratio of the central bulge. The material responsible for the extra mass was dubbed dark matter, the existence of which was first posited in the 1930s by Jan Oort in his measurements of the Oort constants and Fritz Zwicky in his studies of the masses of galaxy clusters. The existence of non-baryonic cold dark matter (CDM) is today a major feature of the Lambda-CDM model that describes the cosmology of the universe.

Halo density profiles

In order to accommodate a flat rotation curve, a density profile for a galaxy and its environs must be different than one that is centrally concentrated. Newton's version of Kepler's Third Law implies that the spherically symmetric, radial density profile ρ(r) is:

${\displaystyle \rho (r)=\frac{v(r{)}^2}{4\pi G{r}^2}\left(1+2\frac{d\log v(r)}{d\log r}\right)}$

where v(r) is the radial orbital velocity profile and G is the gravitational constant. This profile closely matches the expectations of a singular isothermal sphere profile where if v(r) is approximately constant then the density ρ ∝ r⁻² to some inner "core radius" where the density is then assumed constant. Observations do not comport with such a simple profile, as reported by Navarro, Frenk, and White in a seminal 1996 paper.

The authors then remarked that a "gently changing logarithmic slope" for a density profile function could also accommodate approximately flat rotation curves over large scales. They found the famous Navarro–Frenk–White profile, which is consistent both with N-body simulations and observations given by

${\displaystyle \rho (r)=\frac{{\rho }_0}{\frac{r}{R_s}{\left(1+\frac{r}{R_s}\right)}^2}}$

where the central density, ρ₀, and the scale radius, R_s, are parameters that vary from halo to halo. Because the slope of the density profile diverges at the center, other alternative profiles have been proposed, for example the Einasto profile, which has exhibited better agreement with certain dark matter halo simulations.

Observations of orbit velocities in spiral galaxies suggest a mass structure according to:

${\displaystyle v(r)=(rd\mathrm{\Phi }/dr{)}^{1/2}}$

with Φ the galaxy gravitational potential.

Since observations of galaxy rotation do not match the distribution expected from application of Kepler's laws, they do not match the distribution of luminous matter. This implies that spiral galaxies contain large amounts of dark matter or, alternatively, the existence of exotic physics in action on galactic scales. The additional invisible component becomes progressively more conspicuous in each galaxy at outer radii and among galaxies in the less luminous ones.

A popular interpretation of these observations is that about 26% of the mass of the Universe is composed of dark matter, a hypothetical type of matter which does not emit or interact with electromagnetic radiation. Dark matter is believed to dominate the gravitational potential of galaxies and clusters of galaxies. Under this theory, galaxies are baryonic condensations of stars and gas (namely hydrogen and helium) that lie at the centers of much larger haloes of dark matter, affected by a gravitational instability caused by primordial density fluctuations.

Many cosmologists strive to understand the nature and the history of these ubiquitous dark haloes by investigating the properties of the galaxies they contain (i.e. their luminosities, kinematics, sizes, and morphologies). The measurement of the kinematics (their positions, velocities and accelerations) of the observable stars and gas has become a tool to investigate the nature of dark matter, as to its content and distribution relative to that of the various baryonic components of those galaxies.

Further investigations

Comparison of rotating disc galaxies in the present day (left) and the distant Universe (right).

The rotational dynamics of galaxies are well characterized by their position on the Tully–Fisher relation, which shows that for spiral galaxies the rotational velocity is uniquely related to their total luminosity. A consistent way to predict the rotational velocity of a spiral galaxy is to measure its bolometric luminosity and then read its rotation rate from its location on the Tully–Fisher diagram. Conversely, knowing the rotational velocity of a spiral galaxy gives its luminosity. Thus the magnitude of the galaxy rotation is related to the galaxy's visible mass.

While precise fitting of the bulge, disk, and halo density profiles is a rather complicated process, it is straightforward to model the observables of rotating galaxies through this relationship. So, while state-of-the-art cosmological and galaxy formation simulations of dark matter with normal baryonic matter included can be matched to galaxy observations, there is not yet any straightforward explanation as to why the observed scaling relationship exists. Additionally, detailed investigations of the rotation curves of low-surface-brightness galaxies (LSB galaxies) in the 1990s and of their position on the Tully–Fisher relation showed that LSB galaxies had to have dark matter haloes that are more extended and less dense than those of galaxies with high surface brightness, and thus surface brightness is related to the halo properties. Such dark-matter-dominated dwarf galaxies may hold the key to solving the dwarf galaxy problem of structure formation.

Very importantly, the analysis of the inner parts of low and high surface brightness galaxies showed that the shape of the rotation curves in the centre of dark-matter dominated systems indicates a profile different from the NFW spatial mass distribution profile. This so-called cuspy halo problem is a persistent problem for the standard cold dark matter theory. Simulations involving the feedback of stellar energy into the interstellar medium in order to alter the predicted dark matter distribution in the innermost regions of galaxies are frequently invoked in this context.

Alternatives to dark matter

There have been a number of attempts to solve the problem of galaxy rotation by modifying gravity without invoking dark matter. One of the most discussed is modified Newtonian dynamics (MOND), originally proposed by Mordehai Milgrom in 1983, which modifies the Newtonian force law at low accelerations to enhance the effective gravitational attraction. MOND has had a considerable amount of success in predicting the rotation curves of low-surface-brightness galaxies, matching the baryonic Tully–Fisher relation, and the velocity dispersions of the small satellite galaxies of the Local Group.

Using data from the Spitzer Photometry and Accurate Rotation Curves (SPARC) database, a group has found that the radial acceleration traced by rotation curves could be predicted just from the observed baryon distribution (that is, including stars and gas but not dark matter). The same relation provided a good fit for 2693 samples in 153 rotating galaxies, with diverse shapes, masses, sizes, and gas fractions. Brightness in the near infrared, where the more stable light from red giants dominates, was used to estimate the density contribution due to stars more consistently. The results are consistent with MOND, and place limits on alternative explanations involving dark matter alone. However, cosmological simulations within a Lambda-CDM framework that include baryonic feedback effects reproduce the same relation, without the need to invoke new dynamics (such as MOND). Thus, a contribution due to dark matter itself can be fully predictable from that of the baryons, once the feedback effects due to the dissipative collapse of baryons are taken into account. MOND is not a relativistic theory, although relativistic theories which reduce to MOND have been proposed, such as tensor–vector–scalar gravity (TeVeS), scalar–tensor–vector gravity (STVG), the f(R) theory of Capozziello and De Laurentis, not to mention a version of Superfluid Vacuum theory based on the Logarithmic Schrödinger equation.

A model of galaxy based on a general relativity metric was also proposed, showing that the rotation curves for the Milky Way, NGC 3031, NGC 3198 and NGC 7331 are consistent with the mass density distributions of the visible matter, avoiding the need for a massive halo of exotic dark matter.

According to a 2020 analysis of the data produced by the Gaia spacecraft, it would seem possible to explain at least the Milky Way's rotation curve without requiring any dark matter if instead of a Newtonian approximation the entire set of equations of general relativity is adopted.

In March 2021, Gerson Otto Ludwig published a model based on general relativity that explains galaxy rotation curves with gravitoelectromagnetism.