Appendix A
Cosmology and Cosmography

The simulations in this work take a look deep into space and time. Therefore it is necessary to take the properties of spacetime into account. This chapter gives a short overview about some aspects of general relativity. The main focus is set on distance measures (cosmography), as they are frequently used in the simulation.

A.1 General Relativity

Einsteins field equations describe how energy, being equivalent to mass, affects space and time, making use of the covariance principle according to Lorentz transformations. Einsteins field equations are

(A.1)

where R is the Ricci tensor (the contraction of the curvature tensor) and R is its contraction, the curvature scalar. g is the metric tensor and T is the energy-momentum tensor. Finally, is the cosmological constant, G is Newtons gravitational constant and c is the speed of light:

Assuming that the universe is - looking at large enough scales - homogeneous and isotropic, one gets a simplification of the metric of the universe. It may be described by the line element ds of the Robertson-Walker metric in polar coordinates x = (t,r,,) as

(A.3)

where a(t) is the scale factor which depends on time, as to describe an expanding universe. Since a(t) has the dimension of length, r is dimensionless and k, which is the curvature of space, may be set to k = 0, +1 or -1 by choosing an appropriate scaling of r.

One may describe the matter in the universe as a continuous ideal fluid. It consists of particles (e.g. galaxies) which have a mean matter density (t) and a pressure P(t). Assuming that the trajectories of those particles may be described as xⁱ = const. (which means for galaxies that they are locked in the Hubble flow, i.e., the expansion of space), one obtains the energy-momentum tensor

(A.6)

with

(A.8)

where

(A.9)

is the critical density, with

(A.10)

where h₁₀₀ is ”a dimensionless number parameterizing our ignorance“ Hogg (1999) and may be fitted to the experimental results you believe in. Finally, one may introduce the density parameters

(A.11)

Figure A.1: Constraints on the cosmological parameters, given by measurements of the CMB power spectrum and the light from supernovae. = m + +k. Taken from the MAXIMA collaboration (Balbi et al., 2000).

The values of the cosmological parameters H₀, _m, and k (or _k) were and are subject to numerous measurements. The most precise values are derived from type Ia supernovae brightness measurements (e.g. Perlmutter et al. (1999)) and from the power spectrum of the cosmic microwave background (CMB). This year, the balloon-borne experiments BOOMERANG and MAXIMA, which were designed to measure the CMB power spectrum, brought new constraints on the cosmological parameters (Jaffe et al. (2000); see Fig. A.1). Throughout this work the best values found from the experiment are used (if not otherwise stated):

There is a problem: The baryonic matter in the universe we can see directly (as stars or illuminated by stars) or indirectly (from absorption or by projections) accounts for at most 10% of _m. The dynamic properties of galaxies and galaxy clusters and the effects of gravitational lensing also suggest that the majority of mass cannot be seen. It is still unclear what those remaining 90% consist of, but as this matter seems to only interact gravitationally with baryonic matter, it is referred to as dark matter.

A.2 Distance Measures

A.2.1 Redshift

One of the most common ways to describe large astronomical distances is by the concept of redshift. Observations show, that the more distant an object is, the more its light is shifted to longer wavelengths. Absorption and emission lines of atoms are commonly used to measure a redshift. The redshift effect may be explained in two ways. One may interpret it as the Doppler effect that light affects due to the movement of the source relative to the observer. That motion then arises from the expansion of the universe. The other way to look at it is a general expansion of space which ”stretches“ the light wave (cosmological redshift).

The redshift z is defined as

(A.13)

where _o is the observed wavelength and _e is the wavelength of the photon in the restframe of the source. The redshift is related to the scale factor a(t) of the universe (as defined in eq. (A.3)):

(A.14)

where t_o is the time when the photon is observed, and t_e is the time the photon was emitted.

Redshift is independent of cosmology, but it does not correspond to a distance one could measure with a ruler. To obtain a distance measure in a proper length scale, one needs to take spacetime into account. Unfortunately, there are many ways to define a distance in an expanding or curved universe.

A.2.2 Line-of-Sight Comoving Distance

As light needs time to get from an object to the observer, one can define a distance that may be measured between the observer and the object with a ruler at the time the light was emitted, the proper distance. For the proper distance between observer and object today, one takes the proper distance at the time the light was emitted times the ratio of scale factors now to then. We call the proper distance today the comoving distance. To obtain this quantity, we consider a beam of light emitted along the line of sight between object and observer. It is emitted at the time t_e from the object and is received by the observer today, i.e., at t = t₀. The light runs a distance c ^. dt, which then is stretched as the universe expands. Integrating all those small distances along the way of the light ray gives the line-of-sight comoving distance

(A.15)

To get the comoving distance parameterized in z, we consider the function

(A.16)

From eq. (A.14) one obtains

(A.17)

so that

(A.19)

with the Hubble distance

(A.20)

As the function E(z) is defined as in eq. (A.16), one has to consider the scale factor and its time derivative, which are described by the Friedmann equation (eq. (A.6)). Starting from there, one gets:

A.2.3 Transverse Comoving Distance

Next thing to be defined is the transverse comoving distance, which is a quantity used to get the comoving distance perpendicular to the line of sight. In other words it is used to get the comoving distance D_S that two objects at the same redshift have which are separated on the sky by an angle . In an Euclidean geometry this distance would be D_S = D_C . As this no longer holds true in a curved spacetime, the transverse comoving distance D_M is defined so, that D_S = D_M  holds true in any spacetime. Using Robertson-Walker metric, one gets (Hogg, 1999):

(A.22)

To get a proper distance out of the transverse comoving distance, one simply has to multiply it by the ratio of scale factors then to now. This results in the angular diameter distance

(A.23)

Using this, one can obtain the proper distance D_P of two objects separated by an angle on the sky, situated at the same redshift:

(A.24)

A.2.4 Luminosity Distance

Very often it is necessary to relate a radiation flux S measured on earth to the object’s bolometric luminosity L. In Euclidean geometry one has to distribute the emitted energy on a sphere of radius r:

(A.25)

One may now define the luminosity distance D_L, so that eq. (A.25) holds true with r = D_L for any spacetime. The luminosity distance is given by (Weinberg (1972), pp 420 ff.):

(A.26)

Eq. (A.25) is valid for the bolometric luminosity and flux (i.e., integrated over all frequencies), but if the differential flux S is concerned, the spectrum of the object becomes important. If L is not constant, due to the redshift the observed flux varies. This effect, the k-correction, is described in more detail in section 1.4. Taking those effects of redshift into account, the following relations can be found:

A.2.5 Comoving Volume

The comoving volume is the volume measure in which the number density of objects remains constant with epoch, assuming no evolution and that the objects are locked in the Hubble flow. Thus, the comoving volume is proper volume times the ratio of scale factors now to then to the third power. To obtain the comoving volume, the comoving volume element is defined. The solid angle d is converted to a proper area using eq. (A.23):

(A.29)

The line-of-sight distance element is calculated using the same method as in A.19. This comoving distance element must be converted into a proper distance element dr by multiplying it with the ratio of scale factors then to now:

(A.30)

Now the proper volume element is dV _P = dA(z) ^.dr(z), which has to be converted to the comoving volume element dV _C:

(A.32)

There is an analytic solution given by Carroll et al. (1992) for the total comoving volume V _C, all sky, from now (redshift zero) out to redshift z:

(A.33)

As the universe is assumed to be homogeneous and isotropic, one may find the comoving volume corresponding to a part of the sky up to redshift z by calculating the all sky comoving volume and multiply the result by a factor [(1/4) ^. area of part of sky]. And to find the comoving volume for a redshift interval from z to z + z one may simply use V _C(z)|_z = V _C(z + z) - V _C(z).

A.2.6 Lookback Time

As light travels with finite speed, it takes time for it to cover the distance related to the redshift it encountered. So, a look into space is always a look back in time. How far this ”time machine“ takes us can be calculated as lookback time. How to do so tells eq. (A.18). Together with eq. (A.14) follows:

(A.34)

Defining the Hubble time

(A.35)

the time interval corresponding to a redshift interval z is

(A.36)

To get the lookback time for redshift z the integration must run from 0 (now) to z. To get the age of the universe at redshift z the integration limits are z and . In order to calculate the age of universe today, one has to integrate from 0 to .

Using the values of the cosmological parameters as given in eq. (A.12) to calculate the age of the universe t₀ results in

(A.37)

The oldest objects known so far are globular clusters. From color magnitude diagrams and models for stellar evolution their age can be estimated. Depending on the underlying model the oldest of those objects are believed to have an age of 10-14 billion years, well in agreement with the cosmological model used here.