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I have a simple question :

How to prove the following relation :

The comoving distance to an object at redshift $z$ can be computed as

$$r(z)=dfrac{c}{H_{0}} int_{0}^{z} dfrac{mathrm{d} z}{E(z)}$$

from the relation :

$$r(t)=int_{0}^{t} dfrac{cmathrm{d} t}{R(t)}$$

I tried to use with the definition : $1+z= dfrac{R_{0}}{R(t)}$ but I can't conclude.

Any help is welcome.

**UPDATE 1 :** @Tosic's demonstration seems to be correct. But the factor $R_{0}$ is not disappearing. Indeed, If I do :

$$dfrac{ ext{d}(1+z)}{ ext{d}t} = dfrac{ ext{d}z}{ ext{d}t} = -dfrac{H(t)}{R(t)},R_{0}$$

which implies :

$$int_{0}^{z} dfrac{c ext{d}z}{H(z)} = int_{0}^{t}c ext{d}tdfrac{R_{0}}{R(t)} = R_{0} int_{0}^{t}dfrac{c ext{d}t}{R(t)}$$

How to get rid of the factor $R_{0}$ ? Since if I multiply the comoving coordinate $r(t)$ by $R_{0}$, I get the cosmological horizon (the limit of observable universe if I integrate up to $z=1100$), don't I ?

According to this, the Hubble constant for redshift z is $H_0E(z)$. Meaning we need to prove that $$int_{0}^{z_0}frac{cdz}{H(z)} = int_{0}^{t_0}frac{cdt}{R(t)}$$ Take the first derivative of both sides of your second equation to obtain, by the chain rule, and the equation (this is the definition of the Hubble constant) $H = frac{R(t)'}{R(t)}$ the following: $$frac{dz}{dt} = -frac{1}{R(t)^2}*R'(t) = -frac{H(t)}{R(t)}$$ That the redshift is zero for the current time, and some value greater than zero for some time before the current time should explain the minus sign. After multiplying with $c$ and placing the small changes on both sides this becomes $$frac{cdz}{H(z)}=-frac{cdt}{R(t)}$$, which looks like it can be integrated to obtain what we need to prove (the integral from z to 0 is the one from 0 to t).

This proof is not very formal, but it's the best I can do, so I hope it is somewhat correct and that someone will give a more detailed answer if this is not good enough.

## Comoving distance

In standard cosmology, **comoving distance** and **proper distance** are two closely related distance measures used by cosmologists to define distances between objects. *Proper distance* roughly corresponds to where a distant object would be at a specific moment of cosmological time, measured using a long series of rulers stretched out from our position to the object's position at that time, and which can change over time due to the expansion of the universe. *Comoving distance* factors out the expansion of the universe, giving a distance that does not change in time due to the expansion of space. Comoving distance and proper distance are defined to be equal at the present time. That is, the two distances only differ at times other than the time at which they are measured: The universe's expansion results in the proper distance changing, while the comoving distance is unchanged by this expansion. [* clarification needed *]

## Comoving coordinates

Although general relativity allows one to formulate the laws of physics using arbitrary coordinates, some coordinate choices are more natural or easier to work with. Comoving coordinates are an example of such a natural coordinate choice. They assign constant spatial coordinate values to observers who perceive the universe as isotropic. Such observers are called "comoving" observers because they move along with the Hubble flow.

A comoving observer is the only observer who will perceive the universe, including the cosmic microwave background radiation, to be isotropic. Non-comoving observers will see regions of the sky systematically blue-shifted or red-shifted. Thus isotropy, particularly isotropy of the cosmic microwave background radiation, defines a special local frame of reference called the comoving frame. The velocity of an observer relative to the local comoving frame is called the peculiar velocity of the observer.

Most large lumps of matter, such as galaxies, are nearly comoving, so that their peculiar velocities (owing to gravitational attraction) are low.

Comoving coordinates separate the exactly proportional expansion in a Friedmannian universe in spatial comoving coordinates from the scale factor *a(t)*. This example is for the Λ CDM model.

The **comoving time** coordinate is the elapsed time since the Big Bang according to a clock of a comoving observer and is a measure of cosmological time. The comoving spatial coordinates tell where an event occurs while cosmological time tells when an event occurs. Together, they form a complete coordinate system, giving both the location and time of an event.

Space in comoving coordinates is usually referred to as being "static", as most bodies on the scale of galaxies or larger are approximately comoving, and comoving bodies have static, unchanging comoving coordinates. So for a given pair of comoving galaxies, while the proper distance between them would have been smaller in the past and will become larger in the future due to the expansion of space, the comoving distance between them remains *constant* at all times.

The expanding Universe has an increasing scale factor which explains how constant comoving distances are reconciled with proper distances that increase with time.

## Thread: Comoving distance vs Proper distance?

When I'm reading articles on Cosmology there always those 2 concepts thrown around. I think they refer to 2 different kind of distances , is that correct? Can somebody provide a basic explanation of the difference?

For example, in the actual standard Cosmology, the Lambda-CDM model, I understand that the Hubble radius is about 13.9 billion light years in lenght, and that the radius of the observable Universe is about 46 billion light years in lenght. Are those quantities in Comoving distance or Proper distance?

Comoving distance is easy: it's the distance between two objects which isn't changed by the expansion of the universe. So, if galaxy A is (right now) 10 Mpc from galaxy B in comoving coordinates, and then the universe expands by a factor of 2, the distance between the two galaxies is STILL 10 Mpc in comoving coordinates. If you made a movie of the universe using comoving coordinates, then you wouldn't see the galaxies move away from each other on the screen you might see galaxies jiggle around a little bit due to motions within their groups and clusters, but overall, the view would be pretty static.

A movie filmed with proper distance WOULD show the galaxies moving away from each other on the screen. To measure proper distance, you need a long, long line of engineers, each holding his own meterstick. If the line stretches from galaxy A to galaxy B, and each engineer measures the distance from him to the guy on his left and his right, and then you add up all those distances . you get the proper distance. As time passes, you'll need to add engineers to the line to account for the expansion of space.

## An Etymological Dictionary of Astronomy and AstrophysicsEnglish-French-Persian

1) A distance in → *comoving coordinates* between two points in space at a given cosmological time. In other words, the distance between two nearby objects in the Universe which remains constant with epoch if the two objects are moving with the → *Hubble flow*. More specifically, it is the → *proper distance* divided by the ratio of the → *scale factor* of the Universe between then, *a*(*t*) em , and now, *a*(*t*) obs : *D* C = *D* proper . [*a*(*t*) obs /*a*(*t*) em ]. In terms of → *redshift* (*z*), it is the proper distance multiplied by (1 + *z*). At the present epoch, i.e. *a* = *a*(*t* obs ) = 1, *D* C = *D* proper . If the objects have no peculiar velocity their comoving distance at any time is the same as their distance today.

The comoving distance of the → *cosmic horizon* is about 48 × 10 9 → *light-year*s.

2) *Transverse comoving distance*: In a non-flat Universe, the comoving distance between two events at the same → *redshift* but separated on the sky by some angle. It is expressed by trigonometric functions of → *curvature*, → *comoving distance*, and the → *Hubble distance* accounting for the curvature of space. In a flat universe (Ω *k* ) it is the same as the → *comoving distance*.

3) *Line-of-sight comoving distance*: The total line-of-sight comoving distance from us to a distant object computed by integrating the infinitesimal comoving distance contributions between nearby events along the radial ray from the time *t* emit , when the light from the object was emitted, to the time *t* obs , when the object is observed.

A → *reference frame* that is attached to a moving object. The object in this frame is therefore at rest.

## Homogeneity and Isotropy

To say the Universe is homogeneous means that any measurable property of the Universe is the same everywhere. This is only approximately true, but it appears to be an excellent approximation when one averages over large regions. Since the age of the Universe is one of the measurable quantities, the homogeneity of the Universe must be defined on a surface of constant proper time since the Big Bang. Time dilation causes the proper time measured by an observer to depend on the velocity of the observer, so we specify that the time variable t in the Hubble law is the proper time since the Big Bang for comoving observers .

## Many Distances

With the correct interpretation of the variables, the Hubble law ( v = HD ) is true for all values of D, even very large ones which give v > c. But one must be careful in interpreting the distance and velocity. The distance in the Hubble law must be defined so that if A and B are two distant galaxies seen by us in the same direction, and A and B are not too far from each other, then the difference in distances from us, D(A)-D(B), is the distance A would measure to B. But this measurement must be made "now" -- so A must measure the distance to B at the same proper time since the Big Bang as we see now. Thus to determine D now for a distant galaxy Z we would find a chain of galaxies ABC. XYZ along the path to Z, with each element of the chain close to its neighbors, and then have each galaxy in the chain measure the distance to the next galaxy at time t o since the Big Bang. The distance to Z, D(us to Z), is the sum of all these subintervals: And the velocity in the Hubble law is just the change of D now per unit time. It is close to cz for small redshifts but deviates for large ones. The space-time diagram below repeats the example from Part 1 showing how a change in point-of-view from observer A to observer B leaves the linear velocity vs. distance Hubble law unchanged:

but now showing the lightcones. Note how the lightcones must tip over along with the worldlines of the galaxies, showing that in these cosmological variables the speed of light is c with respect to local comoving observers.

The time and distance used in the Hubble law are not the same as the x and t used in special relativity, and this often leads to confusion. In particular, galaxies that are far enough away from us necessarily have velocities greater than the speed of light:

The light cones for distant galaxies in the diagram above are tipped over past the vertical, indicating v > c . The space-time diagram below shows a "zero" (really very low) density cosmological model plotted using the D now and t of the Hubble law.

Worldlines of comoving observers are plotted and decorated with small, schematic lightcones. The red pear-shaped object is our past light cone. Notice that the red curve always has the same slope as the little light cones. In these variables, velocities greater than c are certainly possible, and since the open Universes are spatially infinite, they are actually required. But there is no contradiction with the special relativistic principle that objects do not travel faster than the speed of light, because if we plot exactly the same space-time in the special relativistic x and t coordinates we get:

The grey hyperbolae show the surfaces of constant proper time since the Big Bang. When we flatten these out to make the previous space-time diagram, the worldlines of the galaxies get flatter and giving velocities v = dD now /dt that are greater than c. But in special relativistic coordinates the velocities are less than c. We also see that our past light cone crosses the worldline of the most distant galaxies at a special relativistic distance x = c*t o /2. But the Hubble law distance D now , which is measured now, of these most distant galaxies is infinity (in this model). Furthermore, this galaxy with infinite Hubble law distance and hence infinite Hubble law velocity is visible to us, since in this model the observable Universe is the entire Universe. The relationships between the Hubble law distance and velocity (D now & v) and the redshift z for the zero density model are given below: Note that the redshift-velocity law is not the special relativistic Doppler shift law which only applies to special relativistic coordinates, not to cosmological coordinates.

While the Hubble law distance is in principle measurable, the need for helpers all along the chain of galaxies out to a distant galaxy makes its use quite impractical. Other distances can be defined and measured more easily. One is the angular size distance , defined by where "size" is the transverse extent of an object and "theta" is the angle (in radians) that it subtends on the sky. For the zero density model, the special relativistic x is equal to the angular size distance, x = D A .

The predicted curve relating one distance indicator to another depends on the cosmological model. The plot of redshift vs distance for Type Ia supernovae shown earlier is really a plot of cz vs D L , since fluxes were used to determine the distances of the supernovae. This data clearly rule out models that do not give a linear cz vs D L relation for small cz. Extension of these observations to more distant supernovae have started to allow us to measure the curvature of the cz vs D L relation, and provide more valuable information about the Universe.

The perfect fit of the CMB to a blackbody allows us to determine the D A vs D L relation. Since the CMB is produced at great distance but still looks like a blackbody, a distant blackbody must look like a blackbody (even though the temperature will change due to the redshift). The luminosity of blackbody is where R is the radius, T em is the temperature of the emitting blackbody, and sigma is the Stephan-Boltzmann constant. If seen at a redshift z, the observed temperature will be and the flux will be where the angular radius is related to the physical radius by Combining these equations gives Models that do not predict this relationship between D A and D L , such as the chronometric model or the tired light model, are ruled out by the properties of the CMB.

Here is a Javascript calculator that takes H o , Omega M , the normalized cosmological constant lambda and the redshift z and then computes all of the these distances. Here are the technical formulae for these distances. The graphs below show these distances vs. redshift for three models: the critical density matter dominated Einstein - de Sitter model (EdS), the empty model, and the accelerating Lambda-CDM model (LCDM) that is the current consensus model.

Note that all the distances are very similar for small distances, with D = cz/H o , but the different types of distances deviate substantially at large redshifts. Also note that these deviations depend on what kind of Universe we live in. Precise measurements of deviations of D L from cz/H o are what tell us that the expansion of the Universe is accelerating.

## Scale Factor a(t)

Because the velocity or dD now /dt is strictly proportional to D now , the distance between any pair of comoving objects grows by a factor (1+H*dt) during a time interval dt. This means we can write the distance to any comoving observer as where D G (t o ) is the distance D now to galaxy G now , while a(t) is universal scale factor that applies to all comoving objects. From its definition we see that a(t o ) = 1.

We can compute the dynamics of the Universe by considering an object with distance D(t) = a(t) D o . This distance and the corresponding velocity dD/dt are measured with respect to us at the center of the coordinate system. The gravitational acceleration due to the spherical ball of matter with radius D(t) is g = -G*M/D(t) 2 where the mass is M = 4*pi*D(t) 3 *rho(t)/3. Rho(t) is the density of matter which depends only on the time since the Universe is homogeneous. The mass contained within D(t) is independent of the time since the interior matter has slower expansion velocity while the exterior matter has higher expansion velocity and thus stays outside. The gravitational effect of the external matter vanishes: the gravitational acceleration inside a spherical shell is zero, and all the matter in the Universe with distance from us greater than D(t) can be represented as union of spherical shells. With a constant mass interior to D(t) producing the acceleration of the edge, the problem reduces to the problem of a body moving radially in the gravitational field of a point mass. If the velocity is less than the escape velocity, the expansion will stop and recollapse. If the velocity equals the escape velocity we have the critical case. This gives For rho less than or equal to the critical density rho(crit), the Universe expands forever, while for rho greater than rho(crit), the Universe will eventually stop expanding and recollapse. The value of rho(crit) for H o = 71 km/sec/Mpc is 9E-30 = 9*10 -30 gm/cc or 6 protons per cubic meter or 1.4E11 = 1.4*10 11 solar masses per cubic Megaparsec. The latter can be compared to the observed 1.85E8 = 1.85*10 8 solar luminosities per Mpc 3 , requiring a mass-to-light ratio of 760 in solar units to close the Universe. If the density is anywhere close to critical most of the matter must be too dark to be observed. Current density estimates suggest that the matter density is between 0.2 to 1 times the critical density, and this does require that most of the matter in the Universe is dark.

## Expresion of comoving distance - Astronomy

**3.4 Observations in Cosmology**

The various distances that we have discussed are of course not directly observable all that we know about a distant object is its redshift. Observers therefore place heavy reliance on formulae for expressing distance in terms of redshift. For high-redshift objects such as quasars, this has led to a history of controversy over whether a component of the redshift could be of non-cosmological origin:

For now, we assume that the cosmological contribution to the redshift can always be identified.

DISTANCE-REDSHIFT RELATION The general relation between comoving distance and redshift was given earlier as

For a matter-dominated Friedmann model, this means that the distance of an object from which we receive photons today is

Integrals of this form often arise when manipulating Friedmann models they can usually be tackled by the substitution *u* 2 = *k* ( - 1) / [((1 + *z*)]. This substitution produces **Mattig's formula** (1958), which is one of the single most useful equations in cosmology as far as observers are concerned:

A more direct derivation could use the parametric solution in terms of conformal time, plus *r* = _{now} - _{emit}. The generalisation of the sine rule is required in this method: *S _{k}* (

*a - b*) =

*S*.

_{k}(a) C_{k}(b) - C_{k}(a) S_{k}(b)Although the above is the standard form for Mattig's formula, it is not always the most computationally convenient, being ill defined for small . A better version of the relation for low-density universes is

It is often useful in calculations to be able to convert this formula into the corresponding one for *C _{k} (r)*. The result is

remembering that *R*_{0} = (*c / H*_{0}) [( - 1) / *k*] -1/2 .

It is possible to extend this formula to the case of contributions from pressureless matter (_{m}) and radiation (_{r}):

There is no such compact expression if one wishes to allow for vacuum energy as well (Dabrowski & Stelmach 1987). The comoving distance has to be obtained by numerical integration of the fundamental *dr / dz*, even in the *k* = 0 case. However, for all forms of contribution to the energy content of the universe, the second-order distance-redshift relation is identical, and depends only on the deceleration parameter:

[problem 3.4]. The sizes and flux densities of distant objects therefore determine the geometry of the universe only once an equation of state is assumed, so that *q*_{0} and _{0} can be related.

CHANGE IN REDSHIFT Although we have talked as if the redshift were a fixed parameter of an object, having a status analogous to its comoving distance, this is not correct. Since 1 + *z* is the ratio of scale factors now and at emission, the redshift will change with time. To calculate how, we differentiate the definition of redshift and use the Friedmann equation. For a matter-dominated model, the result is [problem 3.2]

(e.g. Lake 1981 Phillipps 1982). The redshift is thus expected to change by perhaps 1 part in 10 8 over a human lifetime. In principle, this sort of accuracy is not completely out of reach of technology. However, in practice these cosmological changes will be swamped if the object changes its peculiar velocity by more than 3 m s -1 over this period. Since peculiar velocities of up to 1000 km s -1 are built up over the Hubble time, the cosmological and intrinsic redshift changes are clearly of the same order, so that separating them would be very difficult.

AN OBSERVATIONAL TOOLKIT We can now assemble some essential formulae for interpreting cosmological observations. Since we will mainly be considering the post-recombination epoch, these apply for a matter-dominated model only. Our observables are redshift, *z*, and angular difference between two points on the sky, *d* . We write the metric in the form

so that the *comoving* volume element is

The *proper* transverse size of an object seen by us is its comoving size *d* *S _{k} (r)* times the scale factor at the time of emission:

Probably the most important relation for observational cosmology is that between monochromatic flux density and luminosity. Start by assuming isotropic emission, so that the photons emitted by the source pass with a uniform flux density through any sphere surrounding the source. We can now make a shift of origin, and consider the RW metric as being centred on the source however, because of homogeneity, the comoving distance between the source and the observer is the same as we would calculate when we place the origin at our location. The photons from the source are therefore passing through a sphere, on which we sit, of proper surface area 4 [*R*_{0} *S _{k} (r)*] 2 . But redshift still affects the flux density in four further ways: photon energies and arrival rates are redshifted, reducing the flux density by a factor (1 +

*z*) 2 opposing this, the bandwidth

*d*is reduced by a factor 1 +

*z*, so the energy flux per unit bandwidth goes down by one power of 1 +

*z*finally, the observed photons at frequency

_{0}were emitted at frequency

_{0}(1 +

*z*), so the flux density is the luminosity at this frequency, divided by the total area, divided by 1 +

*z*:

A word about units: *L* in this equation would be measured in units of W Hz -1 . Recognizing that emission is often not isotropic, it is common to consider instead the luminosity emitted into unit solid angle - in which case there would be no factor of 4, and the units of *L* would be W Hz -1 sr -1 .

The flux density received by a given observer can be expressed by definition as the product of the **specific intensity** *I* (the flux density received from unit solid angle of the sky) and the solid angle subtended by the source: *S* = *I* *d* . Combining the angular size and flux-density relations thus gives the relativistic version of surface-brightness conservation. This is independent of cosmology (and a more general derivation is given in chapter 4):

where *B* is **surface brightness** (luminosity emitted into unit solid angle per unit area of source). We can integrate over _{0} to obtain the corresponding total or **bolometric** formulae, which are needed e.g. for spectral-line emission:

The form of the above relations lead to the following definitions for particular kinds of distances:

At least the meaning of the terms is unambiguous enough, which is not something that can be said for the term **effective distance**, sometimes used to denote *R*_{0} *S _{k} (r)*. Angular-diameter distance versus redshift is illustrated in figure 3.7.

The last element needed for the analysis of observations is a relation between redshift and age for the object being studied. This brings in our earlier relation between time and comoving radius (consider a null geodesic traversed by a photon that arrives at the present):

So far, all this is completely general to complete the toolkit, we need the crucial input of relativistic dynamics, which is to give the distance-redshift relation. For almost all observational work, it is usual to assume the matter-dominated Friedmann models with differential relation

PREDICTING BACKGROUNDS The above machinery can be given an illustrative application to obtain one of the fundamental formulae used for calculations involving background radiation. Suppose we know the emissivity *j* for some process as a function of frequency and epoch, and want to predict the current background seen at frequency _{0}. The total spectral energy density created during time *dt* is *j* (_{0}[1 + *z*], *z*) *dt* this reaches the present reduced by the volume expansion factor (1 + *z*) -3 . The redshifting of frequency does not matter: scales as 1 + *z*, but so does *d* . The reduction in energy density caused by lowering photon energies is then exactly compensated by a reduced bandwidth, leaving the spectral energy density altered only by the change in proper photon number density. Inserting the redshift-time relation, and multiplying by *c* / 4 to obtain the specific intensity, we get

This may seem a bit of a cheat: we have considered how the energy density evolves at a point in space, even though we know that the photons we see today originated at some large distance. This approach works well enough because of large-scale homogeneity, but it may be clearer to obtain the result for the background directly. Consider a solid angle element *d* : at redshift *z*, this area on the sky for some radial increment *dr* defines a proper volume

This volume produces a luminosity *V* *j* , from which we can calculate the observed flux density *S* = *L* / [4 (*R*_{0} *S _{k}*) 2 (1 +

*z*)]. Since surface brightness is just flux density per unity solid angle, this gives

## What does Comoving mean?

**Comoving distance** is obtained by integrating the proper **distances** of nearby fundamental observers along the line of sight (LOS), where the proper **distance** is what a measurement at constant cosmic time would yield.

One may also ask, what does cosmic time mean? **Cosmic time is** the **time** coordinate commonly used in the Big Bang models of physical **cosmology**. It **is** defined for homogeneous, expanding universes as follows: Choose a **time** coordinate so that the universe has the same density everywhere at each moment in **time**.

Also Know, what is the scale factor of the universe?

**Scale factor** in cosmology is the parameter that describes how the size of the **universe** is changing with respect to its size at the current time. It is the ratio of proper distance between 2 objects at some time to the proper distance between the 2 objects at some reference time .

What does the Hubble constant mean?

The **Hubble Constant** is the unit of measurement used to describe the expansion of the universe. The cosmos has been getting bigger since the Big Bang kick-started the growth about 13.82 billion years ago. The universe, in fact, is getting faster in its acceleration as it gets bigger.

## Expresion of comoving distance - Astronomy

Astro::Cosmology - calculate cosmological distances, volumes, and times

This module provides a set of routines to calculate a number of cosmological quantities based on distance and time. Some are a bit complex - e.g. the volume element at a given redshift - while some, such as the conversion between flux and luminosity, are more mundane.

To calculate results for a given cosmology you create an Astro::Cosmology object with the desired cosmological parameters, and then call the object's methods to perform the actual calculations. If you aren't used to objects, it may sound confusing hopefully the SYNOPSIS section below will help (after all, a bit of code is worth a thousand words). The advantage of using an object-orientated interface is that the object can carry around the cosmological parameters, so you don't need to keep on specifying them whenever you want to calculate anything it also means you can write routines which can just accept an Astro::Cosmology object rather than all the cosmological parameters.

This module requires that the PDL distribution is installed on your machine PDL is available from CPAN or http://pdl.perl.org/

Whilst I believe the results are accurate, I do not guarantee this. Caveat emptor, as the Romans used to say.

If H0 is set to 0, then the units used are the Hubble distance, volume per steradian, or time. If greater than zero, distances are measured in Mpc, volumes in Mpc^3/steradian, and time in years.

- The comoving volume routine gives a slightly smaller answer than Figure 6 of Carroll, Press & Turner for z

100. It could be due to differences in the numerical methods, but I've not yet investigated it thoroughly.

The following calculations were cobbled together from a number of sources, including the following (note that errors in the documentation or code are mine, and are not due to these authors):

In the following all values are in ``natural'' units: Hubble distance, volume, or time.

Symbols used in the following:

For cosmologies with no lambda term, the luminosity distances ( dl ) are calculated by the standard formulae:

For non-zero lambda cosmologies, the luminosity distance is calculated using:

where dc is the comoving distance, calculated by numerical integration of the following from 0 to z :

The comoving distance is always calculated by numerical integration of the above formula. The angular diameter and proper motion distances are defined as dl/(1+z)^2 and dl/(1+z) respectively.

If dm is the proper motion distance, then the comoving volume vc is given by

The differential comoving volume, dvc , is calculated using the proper motion distance, dm , and the differential proper motion distance, ddm , by

The lookback time is calculated by integration of the following formula from 0 to z :

The conversion between absolute and apparent magnitudes is calculated using:

The conversion between flux and luminosity is calculated using

Note that these equations do not include any pass-band or evolutionary corrections.

All integrations are performed using Romberg's method, which is an iterative scheme using progressively higher-degree polynomial approximations. The method stops when the answer converges (ie the absolute difference in the values from the last two iterations is smaller than the ABSTOL parameter, which is described in the new method).

Typically, the romberg integration scheme produces greater accuracy for smooth functions when compared to simpler methods (e.g. Simpson's method) while having little extra overhead for badly-behaved functions.

Currently the following constants are available via use Astro::Cosmology qw( :constants ) :

- LIGHT - the speed of light in m/s.

Please do not use this feature, as it will be removed when an 'Astronomy constants' is created - e.g. see the astroconst package at http://clavelina.as.arizona.edu/astroconst/ .

This document uses the $object->func(. ) syntax throughout. If you prefer the func($object. ) style, then you need to import the functions:

Most functions have two names a short one and a (hopefully) more descriptive one, such as pmot_dist() and proper_motion_distance() .

Most of the routines below include a sig: line in their documentation. This is an attempt to say how they `thread' (in the PDL sense of the word). So, for routines like lum_dist - which have a sig line of dl() = $cosmo->lum_dist( z() ) - the return value has the same format as the input $z value supply a scalar, get a scalar back, send in a piddle and get a piddle of the same dimensions back. For routines like abs_mag - with a sig line of absmag() = $cosmo->abs_mag( appmag(), z() ) - you can thread over either of the two input values, in this case the apparent magnitude and redshift.

Create the object with the required cosmological parameters. Case does not matter and you can use the minimum number of letters which remain unique (the parsing is done by the PDL::Options module).

The options can be specified directly as a list - as shown in the first example above - or in a hash reference - as shown in the second example. You can not mix the two forms within a single call. The options are:

If H0 is set to 0, then answers are returned in units of the Hubble distance, volume, or time, otherwise in Mpc, Mpc^3/steradian, or years.

ABSTOL (absolute tolerance) is used as a convergence criteria when integrating functions as well as whether values are close enough to 0. You should not have to worry about it.

Returns the version number of the Astro::Cosmolgy module as a string. This method is not exported, so it has to be called using either of the two methods shown above.

Returns a string representation of the object. The operator ``'' is overloaded by this function, so that print $cosmo gives a readable answer.

Change the cosmological parameters of the current object. The options are the same as for new.

If supplied with an argument, sets the value of Omega_matter . Returns the current value of the parameter.

If supplied with an argument, sets the value of Omega_lambda . Returns the current value of the parameter.

If supplied with an argument, sets the value of H0 . Returns the current value of the parameter.

returns the luminosity distance, for a given redshift, $z , for the current cosmology.

returns the angular diameter distance, for a given redshift, $z , for the current cosmology.

returns the proper motion distance, for a given redshift, $z , for the current cosmology.

returns the line-of-sight comoving distance, for a given redshift, $z , for the current cosmology.

returns the comoving volume out to a given redshift, $z , for the current cosmology. Does not work if omega_matter and omega_lambda are both 0.0.

returns the differential comoving volume at a given redshift, $z , for the current cosmology. Does not work if omega_matter and omega_lambda are both 0.0.

Returns the lookback time between $zmin and $zmax . If $zmin is not supplied it defaults to 0.0.

Returns the absolute magnitude - excluding K and evolutionary corrections - for the given apparent magnitude.

Returns the apparent magnitude for a given absolute magnitude. As with abs_mag, the K- and evolutionary-corrections are left up to the user.

Returns the luminosity of a source of a given flux. As with abs_mag, the K- and evolutionary-corrections are left up to the user.

The spatial units of the flux must be cm^-2 , so a flux in erg/cm^2/s will be converted into a luminosity in erg/s .

Returns the flux of a source of a given luminosity. As with abs_mag , the K- and evolutionary-corrections are left up to the user.

The spatial units of the flux is cm^-2 , so a luminosity in erg/s will be converted into a flux in erg/cm^2/s .

Add ability to request a particular unit for example have $cosmo->lum_dist() return cm rather than Mpc .

Add the ability to use Pen's approximations (``Analytical Fit to the Luminosity Distance for Flat Cosmologies with a Cosmological Constant'', 1999, ApJS, 120, 49).

There is currently no method to calculate the age of the universe at a given redshift.

Thanks to Brad Holden for trying out early versions of this module and for providing some of the test code.

The cosmology routines make use of code based on routines from

The ``Integration Technique'' section of the documentation is based on that from from the Math::Integral::Romberg module by Eric Boesch (available on CPAN).

Copyright (C) Douglas Burke <djburke @ cpan.org> 1999, 2000, 2001.

All rights reserved. There is no warranty. This program is free software you can redistribute it and/or modify it under the same terms as Perl itself.

## How to find distances of far away galaxies on the NED catalog site

Busy at the moment with doing more then only capturing: busy finding info on the objects I capture. And I have thrown myself into the deep with a kind of Deep Field capture with lots of (very) faint galaxies.

One of them is a tiny blob/point of magnitude 21.7. It has the exotic name of WISEA J112755.96+582235.2.

When I look for info on the NED site, I get of course a lot of info that (for now) is meaningless to me.

But I am looking for distance now. On the NED site there 3 mentions of distance, 2 in lightyears and 1 (CMB) [Mpc]:

- Light Travel-Time = 3.923 Gyr
- Age at Redshift 0.343384 = 9.881 Gyr
- Hubble Distance (CMB) [Mpc] = 1518.35 +/- 106.29

As I understand it, the Hubble Distance has to do with the expansion of the universe. Still, 1 MPC = 3,261,566 light years so that would mean that that distance is almost 5 Gyr . which is rather different then the other two distances.

My knowledge on all this is not a lot (yet). I have some basic understanding on the redshift.