Distance measure

They are often used to tie some observable quantity (such as the luminosity of a distant quasar, the redshift of a distant galaxy, or the angular size of the acoustic peaks in the cosmic microwave background (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.).

In accord with our present understanding of cosmology, these measures are calculated within the context of general relativity, where the Friedmann–Lemaître–Robertson–Walker solution is used to describe the universe.

There are a few different definitions of "distance" in cosmology which are all asymptotic one to another for small redshifts.

The expressions for these distances are most practical when written as functions of redshift

In the remainder of this article, the peculiar velocity 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.

According to the Friedmann equations, we also define a dimensionless Hubble parameter:[1]

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

[citation needed] 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), defined by the cosmic microwave background.

Since the Solar System is moving at around 370 km/s in a direction between Leo and Crater, this decreases

between fundamental observers, i.e. observers that are both moving with the Hubble flow, does not change with time, as comoving distance accounts for the expansion of the universe.

[citation needed] 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 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 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 is too small to be measured.

There is a closed-form expression for the integral in the definition of the comoving distance if

The comoving distance should be calculated using the value of z that would pertain if neither the object nor we had a peculiar velocity.

Together with the scale factor it gives the proper distance of the object when the light we see now was emitted by the it, and set off on its journey to us:

Proper distance roughly corresponds to where a distant object would be at a specific moment of cosmological time, which can change over time due to the expansion of the universe.

[citation needed] Two comoving objects at constant redshift

This is commonly used to observe so called standard rulers, for example in the context of baryon acoustic oscillations.

When accounting for the earth's peculiar velocity, the redshift that would pertain in that case should be used but

should be corrected for the motion of the solar system by a factor between 0.99867 and 1.00133, depending on the direction.

of a distant object is known, we can calculate its luminosity distance by measuring the flux

This quantity is important for measurements of standard candles like type Ia supernovae, which were first used to discover the acceleration of the expansion of the universe.

When accounting for the earth's peculiar velocity, the redshift that would pertain in that case should be used for

should use the measured redshift, and another correction should be made for the peculiar velocity of the object by multiplying by

where z is the measured redshift, in accordance with Etherington's reciprocity theorem (see below).

[citation needed] There is a closed-form solution of the light-travel distance if

(or involving inverse trigonometric functions if the cosmological constant has the other sign).