Friedmann–Lemaître–Robertson–Walker metric

From Wikipedia, the free encyclopedia

The Friedmann–Lemaître–Robertson–Walker (FLRW; /ˈfriːdmən ləˈmɛtrə ... /) metric is an exact solution of Einstein's field equations of general relativity; it describes a homogeneous, isotropic, expanding (or otherwise, contracting) universe that is path-connected, but not necessarily simply connected. The general form of the metric follows from the geometric properties of homogeneity and isotropy; Einstein's field equations are only needed to derive the scale factor of the universe as a function of time. Depending on geographical or historical preferences, the set of the four scientists – Alexander Friedmann, Georges Lemaître, Howard P. Robertson and Arthur Geoffrey Walker – are customarily grouped as Friedmann or Friedmann–Robertson–Walker (FRW) or Robertson–Walker (RW) or Friedmann–Lemaître (FL). This model is sometimes called the Standard Model of modern cosmology, although such a description is also associated with the further developed Lambda-CDM model. The FLRW model was developed independently by the named authors in the 1920s and 1930s.

General metric

The FLRW metric starts with the assumption of homogeneity and isotropy of space. It also assumes that the spatial component of the metric can be time-dependent. The generic metric which meets these conditions is

${\displaystyle -c^{2}\mathrm {d} \tau ^{2}=-c^{2}\mathrm {d} t^{2}+{a(t)}^{2}\mathrm {d} \mathbf {\Sigma } ^{2}}$

where ${\displaystyle \mathbf {\Sigma } }$ ranges over a 3-dimensional space of uniform curvature, that is, elliptical space, Euclidean space, or hyperbolic space. It is normally written as a function of three spatial coordinates, but there are several conventions for doing so, detailed below. ${\displaystyle \mathrm {d} \mathbf {\Sigma } }$ does not depend on t — all of the time dependence is in the function a(t), known as the "scale factor".

Reduced-circumference polar coordinates

In reduced-circumference polar coordinates the spatial metric has the form

${\displaystyle \mathrm {d} \mathbf {\Sigma } ^{2}={\frac {\mathrm {d} r^{2}}{1-kr^{2}}}+r^{2}\mathrm {d} \mathbf {\Omega } ^{2},\quad {\text{where }}\mathrm {d} \mathbf {\Omega } ^{2}=\mathrm {d} \theta ^{2}+\sin ^{2}\theta \,\mathrm {d} \phi ^{2}.}$

k is a constant representing the curvature of the space. There are two common unit conventions:

• k may be taken to have units of length⁻², in which case r has units of length and a(t) is unitless. k is then the Gaussian curvature of the space at the time when a(t) = 1. r is sometimes called the reduced circumference because it is equal to the measured circumference of a circle (at that value of r), centered at the origin, divided by 2π (like the r of Schwarzschild coordinates). Where appropriate, a(t) is often chosen to equal 1 in the present cosmological era, so that ${\displaystyle \mathrm {d} \mathbf {\Sigma } }$ measures comoving distance.
• Alternatively, k may be taken to belong to the set {−1,0,+1} (for negative, zero, and positive curvature respectively). Then r is unitless and a(t) has units of length. When k = ±1, a(t) is the radius of curvature of the space, and may also be written R(t).

A disadvantage of reduced circumference coordinates is that they cover only half of the 3-sphere in the case of positive curvature—circumferences beyond that point begin to decrease, leading to degeneracy. (This is not a problem if space is elliptical, i.e. a 3-sphere with opposite points identified.)

Hyperspherical coordinates

In hyperspherical or curvature-normalized coordinates the coordinate r is proportional to radial distance; this gives

${\displaystyle \mathrm {d} \mathbf {\Sigma } ^{2}=\mathrm {d} r^{2}+S_{k}(r)^{2}\,\mathrm {d} \mathbf {\Omega } ^{2}}$

where ${\displaystyle \mathrm {d} \mathbf {\Omega } }$ is as before and

${\displaystyle S_{k}(r)={\begin{cases}{\sqrt {k}}^{\,-1}\sin(r{\sqrt {k}}),&k>0\\r,&k=0\\{\sqrt {|k|}}^{\,-1}\sinh(r{\sqrt {|k|}}),&k<0.\end{cases}}}$

As before, there are two common unit conventions:

• k may be taken to have units of length⁻², in which case r has units of length and a(t) is unitless. k is then the Gaussian curvature of the space at the time when a(t) = 1. Where appropriate, a(t) is often chosen to equal 1 in the present cosmological era, so that ${\displaystyle \mathrm {d} \mathbf {\Sigma } }$ measures comoving distance.
• Alternatively, as before, k may be taken to belong to the set {−1,0,+1} (for negative, zero, and positive curvature respectively). Then r is unitless and a(t) has units of length. When k = ±1, a(t) is the radius of curvature of the space, and may also be written R(t). Note that when k = +1, r is essentially a third angle along with θ and φ. The letter χ may be used instead of r.

Though it is usually defined piecewise as above, S is an analytic function of both k and r. It can also be written as a power series

${\displaystyle S_{k}(r)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}k^{n}r^{2n+1}}{(2n+1)!}}=r-{\frac {kr^{3}}{6}}+{\frac {k^{2}r^{5}}{120}}-\cdots }$

or as

${\displaystyle S_{k}(r)=r\;\mathrm {sinc} \,(r{\sqrt {k}}),}$

where sinc is the unnormalized sinc function and ${\displaystyle {\sqrt {k}}}$ is one of the imaginary, zero or real square roots of k. These definitions are valid for all k.

Cartesian coordinates

When k = 0 one may write simply

${\displaystyle \mathrm {d} \mathbf {\Sigma } ^{2}=\mathrm {d} x^{2}+\mathrm {d} y^{2}+\mathrm {d} z^{2}.}$

This can be extended to k ≠ 0 by defining

${\displaystyle x=r\cos \theta \,}$,

${\displaystyle y=r\sin \theta \cos \phi \,}$, and

${\displaystyle z=r\sin \theta \sin \phi \,}$,

where r is one of the radial coordinates defined above, but this is rare.

Curvature

Cartesian coordinates

In flat ${\displaystyle (k=0)}$ FLRW space using Cartesian coordinates, the surviving components of the Ricci tensor are

${\displaystyle R_{tt}=-3{\frac {\ddot {a}}{a}},\quad R_{xx}=R_{yy}=R_{zz}=c^{-2}(a{\ddot {a}}+2{\dot {a}}^{2})}$

and the Ricci scalar is

${\displaystyle R=6c^{-2}\left({\frac {{\ddot {a}}(t)}{a(t)}}+{\frac {{\dot {a}}^{2}(t)}{a^{2}(t)}}\right).}$

Spherical coordinates

In more general FLRW space using spherical coordinates (called "reduced-circumference polar coordinates" above), the surviving components of the Ricci tensor are

${\displaystyle R_{tt}=-3{\frac {\ddot {a}}{a}},}$

${\displaystyle R_{rr}={\frac {c^{-2}(a(t){\ddot {a}}(t)+2{\dot {a}}^{2}(t))+2k}{1-kr^{2}}}}$

${\displaystyle R_{\theta \theta }=r^{2}(c^{-2}(a(t){\ddot {a}}(t)+2{\dot {a}}^{2}(t))+2k)}$

${\displaystyle R_{\phi \phi }=r^{2}(c^{-2}(a(t){\ddot {a}}(t)+2{\dot {a}}^{2}(t))+2k)\sin ^{2}(\theta )}$

and the Ricci scalar is

${\displaystyle R=6\left({\frac {{\ddot {a}}(t)}{c^{2}a(t)}}+{\frac {{\dot {a}}^{2}(t)}{c^{2}a^{2}(t)}}+{\frac {k}{a^{2}(t)}}\right).}$

Solutions

Main article: Friedmann equations

Einstein's field equations are not used in deriving the general form for the metric: it follows from the geometric properties of homogeneity and isotropy. However, determining the time evolution of ${\displaystyle a(t)}$ does require Einstein's field equations together with a way of calculating the density, ${\displaystyle \rho (t),}$ such as a cosmological equation of state.

This metric has an analytic solution to Einstein's field equations ${\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\frac {8\pi G}{c^{4}}}T_{\mu \nu }}$ giving the Friedmann equations when the energy–momentum tensor is similarly assumed to be isotropic and homogeneous. The resulting equations are:

${\displaystyle \left({\frac {\dot {a}}{a}}\right)^{2}+{\frac {kc^{2}}{a^{2}}}-{\frac {\Lambda c^{2}}{3}}={\frac {8\pi G}{3}}\rho }$

${\displaystyle 2{\frac {\ddot {a}}{a}}+\left({\frac {\dot {a}}{a}}\right)^{2}+{\frac {kc^{2}}{a^{2}}}-\Lambda c^{2}=-{\frac {8\pi G}{c^{2}}}p.}$

These equations are the basis of the standard Big Bang cosmological model including the current ΛCDM model. Because the FLRW model assumes homogeneity, some popular accounts mistakenly assert that the Big Bang model cannot account for the observed lumpiness of the universe. In a strictly FLRW model, there are no clusters of galaxies, stars or people, since these are objects much denser than a typical part of the universe. Nonetheless, the FLRW model is used as a first approximation for the evolution of the real, lumpy universe because it is simple to calculate, and models which calculate the lumpiness in the universe are added onto the FLRW models as extensions. Most cosmologists agree that the observable universe is well approximated by an almost FLRW model, i.e., a model which follows the FLRW metric apart from primordial density fluctuations. As of 2003, the theoretical implications of the various extensions to the FLRW model appear to be well understood, and the goal is to make these consistent with observations from COBE and WMAP.

If the spacetime is multiply connected, then each event will be represented by more than one tuple of coordinates.

Interpretation

The pair of equations given above is equivalent to the following pair of equations

${\displaystyle {\dot {\rho }}=-3{\frac {\dot {a}}{a}}\left(\rho +{\frac {p}{c^{2}}}\right)}$

${\displaystyle {\frac {\ddot {a}}{a}}=-{\frac {4\pi G}{3}}\left(\rho +{\frac {3p}{c^{2}}}\right)+{\frac {\Lambda c^{2}}{3}}}$

with ${\displaystyle k}$, the spatial curvature index, serving as a constant of integration for the first equation.

The first equation can be derived also from thermodynamical considerations and is equivalent to the first law of thermodynamics, assuming the expansion of the universe is an adiabatic process (which is implicitly assumed in the derivation of the Friedmann–Lemaître–Robertson–Walker metric).

The second equation states that both the energy density and the pressure cause the expansion rate of the universe ${\displaystyle {\dot {a}}}$ to decrease, i.e., both cause a deceleration in the expansion of the universe. This is a consequence of gravitation, with pressure playing a similar role to that of energy (or mass) density, according to the principles of general relativity. The cosmological constant, on the other hand, causes an acceleration in the expansion of the universe.

Cosmological constant

The cosmological constant term can be omitted if we make the following replacements

${\displaystyle \rho \rightarrow \rho +{\frac {\Lambda c^{2}}{8\pi G}}}$

${\displaystyle p\rightarrow p-{\frac {\Lambda c^{4}}{8\pi G}}.}$

Therefore, the cosmological constant can be interpreted as arising from a form of energy which has negative pressure, equal in magnitude to its (positive) energy density:

${\displaystyle p=-\rho c^{2}.\,}$

Such form of energy—a generalization of the notion of a cosmological constant—is known as dark energy.

In fact, in order to get a term which causes an acceleration of the universe expansion, it is enough to have a scalar field which satisfies

${\displaystyle p<-{\frac {\rho c^{2}}{3}}.\,}$

Such a field is sometimes called quintessence.

Newtonian interpretation

This is due to McCrea and Milne, although sometimes incorrectly ascribed to Friedmann. The Friedmann equations are equivalent to this pair of equations:

${\displaystyle -a^{3}{\dot {\rho }}=3a^{2}{\dot {a}}\rho +{\frac {3a^{2}p{\dot {a}}}{c^{2}}}\,}$

${\displaystyle {\frac {{\dot {a}}^{2}}{2}}-{\frac {G{\frac {4\pi a^{3}}{3}}\rho }{a}}=-{\frac {kc^{2}}{2}}\,.}$

The first equation says that the decrease in the mass contained in a fixed cube (whose side is momentarily a) is the amount which leaves through the sides due to the expansion of the universe plus the mass equivalent of the work done by pressure against the material being expelled. This is the conservation of mass–energy (first law of thermodynamics) contained within a part of the universe.

The second equation says that the kinetic energy (seen from the origin) of a particle of unit mass moving with the expansion plus its (negative) gravitational potential energy (relative to the mass contained in the sphere of matter closer to the origin) is equal to a constant related to the curvature of the universe. In other words, the energy (relative to the origin) of a co-moving particle in free-fall is conserved. General relativity merely adds a connection between the spatial curvature of the universe and the energy of such a particle: positive total energy implies negative curvature and negative total energy implies positive curvature.

The cosmological constant term is assumed to be treated as dark energy and thus merged into the density and pressure terms.

During the Planck epoch, one cannot neglect quantum effects. So they may cause a deviation from the Friedmann equations.

Name and history

The Soviet mathematician Alexander Friedmann first derived the main results of the FLRW model in 1922 and 1924. Although the prestigious physics journal Zeitschrift für Physik published his work, it remained relatively unnoticed by his contemporaries. Friedmann was in direct communication with Albert Einstein, who, on behalf of Zeitschrift für Physik, acted as the scientific referee of Friedmann's work. Eventually Einstein acknowledged the correctness of Friedmann's calculations, but failed to appreciate the physical significance of Friedmann's predictions.

Friedmann died in 1925. In 1927, Georges Lemaître, a Belgian priest, astronomer and periodic professor of physics at the Catholic University of Leuven, arrived independently at results similar to those of Friedmann and published them in the Annales de la Société Scientifique de Bruxelles (Annals of the Scientific Society of Brussels). In the face of the observational evidence for the expansion of the universe obtained by Edwin Hubble in the late 1920s, Lemaître's results were noticed in particular by Arthur Eddington, and in 1930–31 Lemaître's paper was translated into English and published in the Monthly Notices of the Royal Astronomical Society.

Howard P. Robertson from the US and Arthur Geoffrey Walker from the UK explored the problem further during the 1930s. In 1935 Robertson and Walker rigorously proved that the FLRW metric is the only one on a spacetime that is spatially homogeneous and isotropic (as noted above, this is a geometric result and is not tied specifically to the equations of general relativity, which were always assumed by Friedmann and Lemaître).

This solution, often called the Robertson–Walker metric since they proved its generic properties, is different from the dynamical "Friedmann–Lemaître" models, which are specific solutions for a(t) which assume that the only contributions to stress–energy are cold matter ("dust"), radiation, and a cosmological constant.

Einstein's radius of the universe

Einstein's radius of the universe is the radius of curvature of space of Einstein's universe, a long-abandoned static model that was supposed to represent our universe in idealized form. Putting

${\displaystyle {\dot {a}}={\ddot {a}}=0}$

in the Friedmann equation, the radius of curvature of space of this universe (Einstein's radius) is

${\displaystyle R_{E}=c/{\sqrt {4\pi G\rho }}}$,

where ${\displaystyle c}$ is the speed of light, ${\displaystyle G}$ is the Newtonian gravitational constant, and ${\displaystyle \rho }$ is the density of space of this universe. The numerical value of Einstein's radius is of the order of 10¹⁰ light years.

Evidence

By combining the observation data from some experiments such as WMAP and Planck with theoretical results of Ehlers–Geren–Sachs theorem and its generalization, astrophysicists now agree that the universe is almost homogeneous and isotropic (when averaged over a very large scale) and thus nearly a FLRW spacetime. That being said, attempts to confirm the purely kinematic interpretation of the Cosmic Microwave Background (CMB) dipole through studies of radio galaxies  and quasars show disagreement in the magnitude. Taken at face value, these observations are at odds with the Universe being described by the FLRW metric. Moreover, one can argue that there is a maximum value to the Hubble constant within an FLRW cosmology tolerated by current observations, ${\displaystyle H_{0}=71\pm 1}$ km/s/Mpc, and depending on how local determinations converge, this may point to an explanation beyond the FLRW metric. Such speculations aside, it should be stressed that the FLRW metric is a valid first approximation for the Universe.

Accelerating expansion of the universe

From Wikipedia, the free encyclopedia

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

Lambda-CDM, accelerated expansion of the universe. The timeline in this schematic diagram extends from the Big Bang/inflation era 13.7 billion years ago to the present cosmological time.

Observations show that the expansion of the universe is accelerating, such that the velocity at which a distant galaxy recedes from the observer is continuously increasing with time.

The accelerated expansion was discovered during 1998, by two independent projects, the Supernova Cosmology Project and the High-Z Supernova Search Team, which both used distant type Ia supernovae to measure the acceleration. The idea was that as type Ia supernovae have almost the same intrinsic brightness (a standard candle), and since objects that are further away appear dimmer, we can use the observed brightness of these supernovae to measure the distance to them. The distance can then be compared to the supernovae's cosmological redshift, which measures how much the universe has expanded since the supernova occurred. The unexpected result was that objects in the universe are moving away from one another at an accelerated rate. Cosmologists at the time expected that recession velocity would always be decelerating, due to the gravitational attraction of the matter in the universe. Three members of these two groups have subsequently been awarded Nobel Prizes for their discovery. Confirmatory evidence has been found in baryon acoustic oscillations, and in analyses of the clustering of galaxies.

The accelerated expansion of the universe is thought to have begun since the universe entered its dark-energy-dominated era roughly 4 billion years ago. Within the framework of general relativity, an accelerated expansion can be accounted for by a positive value of the cosmological constant Λ, equivalent to the presence of a positive vacuum energy, dubbed "dark energy". While there are alternative possible explanations, the description assuming dark energy (positive Λ) is used in the current standard model of cosmology, which also includes cold dark matter (CDM) and is known as the Lambda-CDM model.

Background

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

Further information: Cosmological constant, Lambda-CDM model, Hubble's law, Friedmann–Lemaître–Robertson–Walker metric, and Friedmann equations

In the decades since the detection of cosmic microwave background (CMB) in 1965, the Big Bang model has become the most accepted model explaining the evolution of our universe. The Friedmann equation defines how the energy in the universe drives its expansion.where κ represents the curvature of the universe, a(t) is the scale factor, ρ is the total energy density of the universe, and H is the Hubble parameter.

We define a critical density

${\displaystyle \rho _{c}={\frac {3H^{2}}{8{\pi }G}}}$

and the density parameter

${\displaystyle \Omega ={\frac {\rho }{\rho _{c}}}}$

We can then rewrite the Hubble parameter as

where the four currently hypothesized contributors to the energy density of the universe are curvature, matter, radiation and dark energy. Each of the components decreases with the expansion of the universe (increasing scale factor), except perhaps the dark energy term. It is the values of these cosmological parameters which physicists use to determine the acceleration of the universe.

The acceleration equation describes the evolution of the scale factor with time

${\displaystyle {\frac {\ddot {a}}{a}}=-{\frac {4{\pi }G}{3}}\left(\rho +{\frac {3P}{c^{2}}}\right)}$

where the pressure P is defined by the cosmological model chosen. (see explanatory models below)

Physicists at one time were so assured of the deceleration of the universe's expansion that they introduced a so-called deceleration parameter q₀. Current observations indicate this deceleration parameter being negative.

Relation to inflation

According to the theory of cosmic inflation, the very early universe underwent a period of very rapid, quasi-exponential expansion. While the time-scale for this period of expansion was far shorter than that of the current expansion, this was a period of accelerated expansion with some similarities to the current epoch.

Technical definition

The definition of "accelerating expansion" is that the second time derivative of the cosmic scale factor, ${\displaystyle {\ddot {a}}}$, is positive, which is equivalent to the deceleration parameter, ${\displaystyle q}$, being negative. However, note this does not imply that the Hubble parameter is increasing with time. Since the Hubble parameter is defined as ${\displaystyle H(t)\equiv {\dot {a}}(t)/a(t)}$, it follows from the definitions that the derivative of the Hubble parameter is given by

${\displaystyle {\frac {dH}{dt}}=-H^{2}(1+q)}$

so the Hubble parameter is decreasing with time unless ${\displaystyle q<-1}$. Observations prefer ${\displaystyle q\approx -0.55}$, which implies that ${\displaystyle {\ddot {a}}}$ is positive but ${\displaystyle dH/dt}$ is negative. Essentially, this implies that the cosmic recession velocity of any one particular galaxy is increasing with time, but its velocity/distance ratio is still decreasing; thus different galaxies expanding across a sphere of fixed radius cross the sphere more slowly at later times.

It is seen from above that the case of "zero acceleration/deceleration" corresponds to ${\displaystyle a(t)}$ is a linear function of ${\displaystyle t}$, ${\displaystyle q=0}$, ${\displaystyle {\dot {a}}=const}$, and ${\displaystyle H(t)=1/t}$.

Evidence for acceleration

To learn about the rate of expansion of the universe we look at the magnitude-redshift relationship of astronomical objects using standard candles, or their distance-redshift relationship using standard rulers. We can also look at the growth of large-scale structure, and find that the observed values of the cosmological parameters are best described by models which include an accelerating expansion.

Supernova observation

Artist's impression of a Type Ia supernova, as revealed by spectro-polarimetry observations

In 1998, the first evidence for acceleration came from the observation of Type Ia supernovae, which are exploding white dwarfs that have exceeded their stability limit. Because they all have similar masses, their intrinsic luminosity is standardizable. Repeated imaging of selected areas of the sky is used to discover the supernovae, then follow-up observations give their peak brightness, which is converted into a quantity known as luminosity distance (see distance measures in cosmology for details). Spectral lines of their light can be used to determine their redshift.

For supernovae at redshift less than around 0.1, or light travel time less than 10 percent of the age of the universe, this gives a nearly linear distance–redshift relation due to Hubble's law. At larger distances, since the expansion rate of the universe has changed over time, the distance-redshift relation deviates from linearity, and this deviation depends on how the expansion rate has changed over time. The full calculation requires computer integration of the Friedmann equation, but a simple derivation can be given as follows: the redshift z directly gives the cosmic scale factor at the time the supernova exploded.

${\displaystyle a(t)={\frac {1}{1+z}}}$

So a supernova with a measured redshift z = 0.5 implies the universe was 1/1 + 0.5 = 2/3 of its present size when the supernova exploded. In the case of accelerated expansion, ${\displaystyle {\ddot {a}}}$ is positive; therefore, ${\displaystyle {\dot {a}}}$ was smaller in the past than today. Thus an accelerating universe took a longer time to expand from 2/3 to 1 times its present size, compared to a non-accelerating universe with constant ${\displaystyle {\dot {a}}}$ and the same present-day value of the Hubble constant. This results in a larger light-travel time, larger distance and fainter supernovae, which corresponds to the actual observations. Adam Riess et al. found that "the distances of the high-redshift SNe Ia were, on average, 10% to 15% farther than expected in a low mass density Ω_M = 0.2 universe without a cosmological constant". This means that the measured high-redshift distances were too large, compared to nearby ones, for a decelerating universe.

Baryon acoustic oscillations

Main article: Baryon acoustic oscillations

In the early universe before recombination and decoupling took place, photons and matter existed in a primordial plasma. Points of higher density in the photon-baryon plasma would contract, being compressed by gravity until the pressure became too large and they expanded again. This contraction and expansion created vibrations in the plasma analogous to sound waves. Since dark matter only interacts gravitationally it stayed at the centre of the sound wave, the origin of the original overdensity. When decoupling occurred, approximately 380,000 years after the Big Bang, photons separated from matter and were able to stream freely through the universe, creating the cosmic microwave background as we know it. This left shells of baryonic matter at a fixed radius from the overdensities of dark matter, a distance known as the sound horizon. As time passed and the universe expanded, it was at these inhomogeneities of matter density where galaxies started to form. So by looking at the distances at which galaxies at different redshifts tend to cluster, it is possible to determine a standard angular diameter distance and use that to compare to the distances predicted by different cosmological models.

Peaks have been found in the correlation function (the probability that two galaxies will be a certain distance apart) at 100 h⁻¹ Mpc, (where h is the dimensionless Hubble constant) indicating that this is the size of the sound horizon today, and by comparing this to the sound horizon at the time of decoupling (using the CMB), we can confirm the accelerated expansion of the universe.

Clusters of galaxies

Measuring the mass functions of galaxy clusters, which describe the number density of the clusters above a threshold mass, also provides evidence for dark energy. By comparing these mass functions at high and low redshifts to those predicted by different cosmological models, values for w and Ω_m are obtained which confirm a low matter density and a non zero amount of dark energy.

Age of the universe

See also: Age of the universe

Given a cosmological model with certain values of the cosmological density parameters, it is possible to integrate the Friedmann equations and derive the age of the universe.

${\displaystyle t_{0}=\int _{0}^{1}{\frac {da}{\dot {a}}}}$

By comparing this to actual measured values of the cosmological parameters, we can confirm the validity of a model which is accelerating now, and had a slower expansion in the past.

Gravitational waves as standard sirens

Recent discoveries of gravitational waves through LIGO and VIRGO not only confirmed Einstein's predictions but also opened a new window into the universe. These gravitational waves can work as sort of standard sirens to measure the expansion rate of the universe. Abbot et al. 2017 measured the Hubble constant value to be approximately 70 kilometres per second per megaparsec. The amplitudes of the strain 'h' is dependent on the masses of the objects causing waves, distances from observation point and gravitational waves detection frequencies. The associated distance measures are dependent on the cosmological parameters like the Hubble Constant for nearby objects and will be dependent on other cosmological parameters like the dark energy density, matter density, etc. for distant sources.

Explanatory models

The expansion of the Universe accelerating. Time flows from bottom to top

Dark energy

Main article: Dark energy

The most important property of dark energy is that it has negative pressure (repulsive action) which is distributed relatively homogeneously in space.

${\displaystyle P=wc^{2}\rho }$

where c is the speed of light and ρ is the energy density. Different theories of dark energy suggest different values of w, with w < −1/3 for cosmic acceleration (this leads to a positive value of ä in the acceleration equation above).

The simplest explanation for dark energy is that it is a cosmological constant or vacuum energy; in this case w = −1. This leads to the Lambda-CDM model, which has generally been known as the Standard Model of Cosmology from 2003 through the present, since it is the simplest model in good agreement with a variety of recent observations. Riess et al. found that their results from supernova observations favoured expanding models with positive cosmological constant (Ω_λ > 0) and a current accelerated expansion (q₀ < 0).

Phantom energy

Main article: Phantom energy

Current observations allow the possibility of a cosmological model containing a dark energy component with equation of state w < −1. This phantom energy density would become infinite in finite time, causing such a huge gravitational repulsion that the universe would lose all structure and end in a Big Rip. For example, for w = −3/2 and H₀ =70 km·s⁻¹·Mpc⁻¹, the time remaining before the universe ends in this Big Rip is 22 billion years.

Alternative theories

See also: Dark energy § Theories of dark energy

There are many alternative explanations for the accelerating universe. Some examples are quintessence, a proposed form of dark energy with a non-constant state equation, whose density decreases with time. A negative mass cosmology does not assume that the mass density of the universe is positive (as is done in supernova observations), and instead finds a negative cosmological constant. Occam's razor also suggests that this is the 'more parsimonious hypothesis'. Dark fluid is an alternative explanation for accelerating expansion which attempts to unite dark matter and dark energy into a single framework. Alternatively, some authors have argued that the accelerated expansion of the universe could be due to a repulsive gravitational interaction of antimatter or a deviation of the gravitational laws from general relativity, such as massive gravity, meaning that gravitons themselves have mass. The measurement of the speed of gravity with the gravitational wave event GW170817 ruled out many modified gravity theories as alternative explanation to dark energy.

Another type of model, the backreaction conjecture, was proposed by cosmologist Syksy Räsänen: the rate of expansion is not homogenous, but we are in a region where expansion is faster than the background. Inhomogeneities in the early universe cause the formation of walls and bubbles, where the inside of a bubble has less matter than on average. According to general relativity, space is less curved than on the walls, and thus appears to have more volume and a higher expansion rate. In the denser regions, the expansion is slowed by a higher gravitational attraction. Therefore, the inward collapse of the denser regions looks the same as an accelerating expansion of the bubbles, leading us to conclude that the universe is undergoing an accelerated expansion. The benefit is that it does not require any new physics such as dark energy. Räsänen does not consider the model likely, but without any falsification, it must remain a possibility. It would require rather large density fluctuations (20%) to work.

A final possibility is that dark energy is an illusion caused by some bias in measurements. For example, if we are located in an emptier-than-average region of space, the observed cosmic expansion rate could be mistaken for a variation in time, or acceleration. A different approach uses a cosmological extension of the equivalence principle to show how space might appear to be expanding more rapidly in the voids surrounding our local cluster. While weak, such effects considered cumulatively over billions of years could become significant, creating the illusion of cosmic acceleration, and making it appear as if we live in a Hubble bubble. Yet other possibilities are that the accelerated expansion of the universe is an illusion caused by the relative motion of us to the rest of the universe, or that the supernova sample size used wasn't large enough.

Theories for the consequences to the universe

As the universe expands, the density of radiation and ordinary dark matter declines more quickly than the density of dark energy (see equation of state) and, eventually, dark energy dominates. Specifically, when the scale of the universe doubles, the density of matter is reduced by a factor of 8, but the density of dark energy is nearly unchanged (it is exactly constant if the dark energy is the cosmological constant).

In models where dark energy is the cosmological constant, the universe will expand exponentially with time in the far future, coming closer and closer to a de Sitter universe. This will eventually lead to all evidence for the Big Bang disappearing, as the cosmic microwave background is redshifted to lower intensities and longer wavelengths. Eventually, its frequency will be low enough that it will be absorbed by the interstellar medium, and so be screened from any observer within the galaxy. This will occur when the universe is less than 50 times its current age, leading to the end of cosmology as we know it as the distant universe turns dark.

A constantly expanding universe with a non-zero cosmological constant has mass density decreasing over time. In such a scenario, the current understanding is that all matter will ionize and disintegrate into isolated stable particles such as electrons and neutrinos, with all complex structures dissipating away. This scenario is known as "heat death of the universe".

Alternatives for the ultimate fate of the universe include the Big Rip mentioned above, a Big Bounce, Big Freeze or Big Crunch.