Project II: Distance Measures in Cosmology

In cosmology (or to be more specific, cosmography, the measurement of the Universe) there are many ways to specify the distance between two points, because in the expanding Universe, the distances between comoving objects are constantly changing, and Earth-bound observers look back in time as they look out in distance. The unifying aspect is that all distance measures somehow measure the separation between events on radial null trajectories, ie, trajectories of photons which terminate at the observer.

In this project, you will learn to compute various cosmological distance measures. I treat the concept of "distance measure" very liberally, so, for instance, the lookback time and comoving volume are both considered distance measures. All formulae are provided below. The results should be presented in five plots:

• Angular diameter distance versus redshift.
• Luminosity distance versus redshift.
• Distance modulus versus redshift.
• Comoving volume of the HDF versus redshift.
• Lookback time versus redshift.

Cosmographic parameters:

The Hubble constant H₀ is the constant of proportionality between recession speed v and distance d in the expanding Universe;

(1)

The subscripted "0" refers to the present epoch because in general H changes with time. The dimensions of H₀ are inverse time, but it is usually written

(2)

where h is a dimensionless number. You can assume h=0.7, in agreement with the most recent observations. The inverse of the Hubble constant is the Hubble time t_H

(3)

and the speed of light c times the Hubble time is the Hubble distance D_H

(4)

These quantities set the scale of the Universe.

The mass density of the Universe and the value of the cosmological constant are dynamical properties of the Universe, affecting the time evolution of the metric, but in these notes we will treat them as purely kinematic parameters. They can be made into dimensionless density parameters _M and by

(5)

(6)

(Peebles 1993, pp. 310-313), where the subscripted "0" indicates that the quantities (which in general evolve with time) are to be evaluated at the present epoch. A third density parameter _k measures the "curvature of space" and can be defined by the relation

(7)

These parameters totally determine the geometry of the Universe if it is homogeneous, isotropic, and matter-dominated. By the way, the critical density = 1 corresponds to 7.5 x 10²¹ h^-1 M D_H^-3, where M is the mass of the Sun.

You will consider the following three cases in your calculations:

name	M
Einstein-de Sitter	1	0
low density	0.05	0
high lambda	0.3	0.7

The fundamental observable in cosmography is the redshift z of an object, which is the fractional doppler shift of its emitted light resulting from radial motion.

(8)

For small v / c, or small distance d, in the expanding Universe, the velocity is linearly proportional to the distance (and all the distance measures, eg, angular diameter distance, luminosity distance, etc, converge)

(10)

where D_H is the Hubble distance (see above).

In terms of cosmography, the cosmological redshift is directly related to the scale factor a (t), or the ``size'' of the Universe. For an object at redshift z

(11)

where a (t₀) is the size of the Universe at the time the light from the object is observed, and a (t_e) is the size at the time it was emitted.

Comoving Distance (line-of-sight)

The comoving distance D_C between two nearby objects in the Universe is the distance between them which remains constant with epoch if the two objects are moving with the Hubble flow. The total line-of-sight comoving distance D_C from us to a distant object is computed by integrating the infinitesimal D_C contributions between nearby events along the radial ray from z = 0 to the object.

Following Peebles (1993, pp. 310-321), we define the function

(13)

where z is the redshift and the three density parameters are defined above. The total line-of-sight comoving distance is then given by integrating these contributions, or

(14)

where D_H is the Hubble distance defined above.

In some sense the line-of-sight comoving distance is the fundamental distance measure in cosmography since, as will be seen below, all others are quite simply derived in terms of it.

Comoving Distance (transverse)

The comoving distance between two events at the same redshift or distance but separated on the sky by some angle is D_M and the transverse comoving distance D_M is simply related to the line-of-sight comoving distance D_C:

(15)

where the trigonometric functions "sinh" and "sin" account for what is called "the curvature of space".

Angular Diameter Distance

The angular diameter distance D_A is defined as the ratio of an object's physical transverse size to its angular size (in radians). It is used to convert angular separations in telescope images into proper separations at the source. It is famous for not increasing indefinitely as z -> ; it turns over at z ~ 1 and thereafter more distant objects actually appear larger in angular size. Angular diameter distance is related to the transverse comoving distance by

(17)

(Peebles 1993, pp. 325-327).

Luminosity Distance

The luminosity distance D_L is defined by the relationship between bolometric (ie, integrated over all frequencies) flux S and bolometric luminosity L:

(19)

It turns out that this is related to the transverse comoving distance and angular diameter distance by

(20)

(Weinberg 1972, pp. 420-424).

The distance modulus DM is defined by

(24)

because it is the magnitude difference between an object's observed bolometric flux and what it would be if it were at 10 pc. The absolute magnitude M is the astronomer's measure of luminosity, defined to be the apparent magnitude the object in question would have if it were at 10 pc, so

(25)

where K is the k-correction. This correction applies when we are dealing not with bolometric quantities but rather with differential flux S and luminosity L, as is usually the case in astronomy. In this case a correction, so-called k-correction, must be applied to the flux or luminosity because the redshifted object is emitting flux in a different band than that in which you are observing. For the purposes of this exercise, you will consider bolometric luminosities, i.e., no k-correction is needed.

Comoving Volume

The comoving volume V_C is the volume measure in which number densities of non-evolving objects locked into Hubble flow are constant with redshift. The comoving volume element in solid angle d and redshift interval dz is

(27)

where D_A is the angular diameter distance at redshift z and E (z) is defined above (Weinberg 1972, p. 486; Peebles 1993, pp. 331-333). The integral of the comoving volume element from the present to redshift z gives the total comoving volume, all-sky, out to redshift z

(28)

where D_H³ is sometimes called the Hubble volume. For the purposes of this exercise, you can assume that the HDF projects onto an angular area of 5.78 arcmin² on the sky. The comoving volume element and its integral are both used frequently in predicting number counts or luminosity densities.

Look-back Time

The lookback time t_L to an object is the difference between the age t₀ of the Universe now (at observation) and the age t_e of the Universe at the time the photons were emitted (according to the object). It is used to predict properties of high-redshift objects with evolutionary models, such as passive stellar evolution for galaxies. It can be calculated using the following expression:

(29)

(Peebles 1993, pp. 313-315; gives some analytic solutions to this equation, but he is concerned with the age t (z), so they integrate from z to ).

References

1. Peebles P. J. E., 1993, Principles of Physical Cosmology, Princeton University Press, Princeton
2. Weinberg S., 1972, Gravitation and Cosmolgy: Principles and Applications of the General Theory of Relativity, John Wiley & Sons, New York