Distance measures are used in
physical cosmology
Physical cosmology is a branch of cosmology concerned with the study of cosmological models. A cosmological model, or simply cosmology, provides a description of the largest-scale structures and dynamics of the universe and allows study of f ...
to give a natural notion of the
distance
Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
between two objects or events in the
universe
The universe is all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmological description of the development of the universe. Acc ...
. They are often used to tie some ''observable'' quantity (such as the
luminosity
Luminosity is an absolute measure of radiated electromagnetic power (light), the radiant power emitted by a light-emitting object over time. In astronomy, luminosity is the total amount of electromagnetic energy emitted per unit of time by a st ...
of a distant
quasar
A quasar is an extremely Luminosity, luminous active galactic nucleus (AGN). It is pronounced , and sometimes known as a quasi-stellar object, abbreviated QSO. This emission from a galaxy nucleus is powered by a supermassive black hole with a m ...
, the
redshift
In physics, a redshift is an increase in the wavelength, and corresponding decrease in the frequency and photon energy, of electromagnetic radiation (such as light). The opposite change, a decrease in wavelength and simultaneous increase in f ...
of a distant
galaxy
A galaxy is a system of stars, stellar remnants, interstellar gas, dust, dark matter, bound together by gravity. The word is derived from the Greek ' (), literally 'milky', a reference to the Milky Way galaxy that contains the Solar System. ...
, or the angular size of the acoustic peaks in the
cosmic microwave background
In Big Bang cosmology the cosmic microwave background (CMB, CMBR) is electromagnetic radiation that is a remnant from an early stage of the universe, also known as "relic radiation". The CMB is faint cosmic background radiation filling all spac ...
(CMB) power spectrum) to another quantity that is not ''directly'' observable, but is more convenient for calculations (such as the
comoving coordinates of the quasar, galaxy, etc.). The distance measures discussed here all reduce to the common notion of
Euclidean distance at low redshift.
In accord with our present understanding of cosmology, these measures are calculated within the context of
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, where the
Friedmann–Lemaître–Robertson–Walker solution is used to describe the universe.
Overview
There are a few different definitions of "distance" in cosmology which are all
asymptotic
In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...
one to another for small
redshift
In physics, a redshift is an increase in the wavelength, and corresponding decrease in the frequency and photon energy, of electromagnetic radiation (such as light). The opposite change, a decrease in wavelength and simultaneous increase in f ...
s. The expressions for these distances are most practical when written as functions of redshift
, since redshift is always the observable. They can also be written as functions of
scale factor
In affine geometry, uniform scaling (or isotropic scaling) is a linear transformation that enlarges (increases) or shrinks (diminishes) objects by a '' scale factor'' that is the same in all directions. The result of uniform scaling is similar ...
In the remainder of this article, the
peculiar velocity
Peculiar motion or peculiar velocity refers to the velocity of an object relative to a ''rest frame'' — usually a frame in which the average velocity of some objects is zero.
Galactic astronomy
In galactic astronomy, peculiar motion refers to ...
is assumed to be negligible unless specified otherwise.
We first give formulas for several distance measures, and then describe them in more detail further down. Defining the "Hubble distance" as
:
where
is the
speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
,
is the Hubble parameter today, and is the
dimensionless Hubble constant
Hubble's law, also known as the Hubble–Lemaître law, is the observation in physical cosmology that galaxies are moving away from Earth at speeds proportional to their distance. In other words, the farther they are, the faster they are moving ...
, all the distances are asymptotic to
for small .
We also define a dimensionless Hubble ''parameter'':
:
Here,
and
are normalized values of the present radiation energy density, matter density, and "
dark energy
In physical cosmology and astronomy, dark energy is an unknown form of energy that affects the universe on the largest scales. The first observational evidence for its existence came from measurements of supernovas, which showed that the univer ...
density", respectively (the latter representing the
cosmological constant
In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: ), alternatively called Einstein's cosmological constant,
is the constant coefficient of a term that Albert Einstein temporarily added to his field equ ...
), and
determines the curvature. The
Hubble parameter
Hubble's law, also known as the Hubble–Lemaître law, is the observation in physical cosmology that galaxies are moving away from Earth at speeds proportional to their distance. In other words, the farther they are, the faster they are moving a ...
at a given redshift is then
.
The formula for comoving distance, which serves as the basis for most of the other formulas, involves an
integral
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
. Although for some limited choices of parameters (see below) the comoving distance integral has a closed analytic form, in general—and specifically for the
parameters of our universe—we can only find a solution
numerically. Cosmologists commonly use the following measures for distances from the observer to an object at redshift
along the line of sight (LOS):
*Comoving distance:
::
*Transverse comoving distance:
::
*Angular diameter distance:
::
*Luminosity distance:
::
*Light-travel distance:
::
Alternative terminology
Peebles (1993) calls the transverse comoving distance the "angular size distance", which is not to be mistaken for the angular diameter distance.
Occasionally, the symbols
or
are used to denote both the comoving and the angular diameter distance. Sometimes, the light-travel distance is also called the "lookback distance" and/or "lookback time".
Details
Peculiar velocity
In real observations, the movement of the earth with respect to the
Hubble flow
Hubble's law, also known as the Hubble–Lemaître law, is the observation in physical cosmology that galaxies are moving away from Earth at speeds proportional to their distance. In other words, the farther they are, the faster they are moving a ...
has an effect on the observed redshift.
There are actually two notions of redshift. One is the redshift that would be observed if both the earth and the object were not moving with respect to the "comoving" surroundings (the
Hubble flow
Hubble's law, also known as the Hubble–Lemaître law, is the observation in physical cosmology that galaxies are moving away from Earth at speeds proportional to their distance. In other words, the farther they are, the faster they are moving a ...
), let us say defined by the cosmic microwave background. The other is the actual redshift measured, which depends both on the
peculiar velocity
Peculiar motion or peculiar velocity refers to the velocity of an object relative to a ''rest frame'' — usually a frame in which the average velocity of some objects is zero.
Galactic astronomy
In galactic astronomy, peculiar motion refers to ...
of the object observed and on our own peculiar velocity. Since the solar system is moving at around 370 km/s in a direction between
Leo and
Crater
Crater may refer to:
Landforms
*Impact crater, a depression caused by two celestial bodies impacting each other, such as a meteorite hitting a planet
*Explosion crater, a hole formed in the ground produced by an explosion near or below the surfac ...
, this decreases
for distant objects in that direction by a factor of about 1.0012 and increases it by the same factor for distant objects in the opposite direction. (The speed of the motion of the earth around the sun is only 30 km/s.)
Comoving distance
The comoving distance
between fundamental observers, i.e. observers that are both moving with the
Hubble flow
Hubble's law, also known as the Hubble–Lemaître law, is the observation in physical cosmology that galaxies are moving away from Earth at speeds proportional to their distance. In other words, the farther they are, the faster they are moving a ...
, does not change with time, as comoving distance accounts for the expansion of the universe. Comoving distance is obtained by integrating the proper distances of nearby fundamental observers along the line of sight (LOS), whereas the proper distance is what a measurement at constant cosmic time would yield.
In
standard cosmology, comoving distance and proper distance are two closely related distance measures used by cosmologists to measure distances between objects; the comoving distance is the proper distance at the present time.
The comoving distance (with a small correction for our own motion) is the distance that would be obtained from parallax, because the parallax in degrees equals the ratio of an
astronomical unit
The astronomical unit (symbol: au, or or AU) is a unit of length, roughly the distance from Earth to the Sun and approximately equal to or 8.3 light-minutes. The actual distance from Earth to the Sun varies by about 3% as Earth orbits t ...
to the circumference of a circle at the present time going through the sun and centred on the distant object, multiplied by 360°. However, objects beyond a
megaparsec
The parsec (symbol: pc) is a unit of length used to measure the large distances to astronomical objects outside the Solar System, approximately equal to or (au), i.e. . The parsec unit is obtained by the use of parallax and trigonometry, and ...
have parallax too small to be measured (the
Gaia space telescope measures the parallax of the brightest stars with a precision of 7 microarcseconds), so the parallax of galaxies outside our
Local Group
The Local Group is the galaxy group that includes the Milky Way.
It has a total diameter of roughly , and a total mass of the order of .
It consists of two collections of galaxies in a "dumbbell" shape: the Milky Way and its satellites form ...
is too small to be measured.
There is a
closed-form expression
In mathematics, a closed-form expression is a mathematical expression that uses a finite number of standard operations. It may contain constants, variables, certain well-known operations (e.g., + − × ÷), and functions (e.g., ''n''th roo ...
for the integral in the definition of the comoving distance if
or, by substituting the scale factor
for
, if
. Our universe now seems to be closely represented by
In this case, we have:
::