Identificación de los Aspectos Relevantes para la Planificación de Acciones de

A meaningful general definition of distances in the Universe requires the knowledge of the metric for the global geometry. For the isotropic and homogeneous Universe (maximally symmetric space time) the general relativistic solution is the Robertson-Walker-Metric (RWM). The space time line element in spherical coordinates with curvature constant k and scale factorR(t) can be written as

ds2 =c2dt2−R2(t) " dr2 1−kr2 +r 2³_sin2_{θ dφ}2_+dθ2´ # (3.17) with proper timet, radial comoving coordinater, and angular coordinates φand θ. As the global space time metric, the RWM is valid for all distances. For many applications the light propagation along radial coordinates is of prime interest. Since light travels on null geodesics, i.e. ds2_{= 0, Equ. 3.17 reduces to the simple form}

c2_dt2_−R2_(t) dr2

1−kr2 = 0 . (3.18)

For the following discussion of distance measures, the specialization for a flat Universe (k≡0) is assumed, which is valid for concordance cosmology. Small deviations from a flat geometry could still be possible, however, the additional correction functions sin(R0r) for

k= 1 and sinh(R0r) for k=−1 for non-flat space times would inhibit the readability and

clarity for this discussion and are therefore omitted.

Integrating Equ. 3.18 (for k= 0) yields the fundamental comoving distance D, i.e. the distance an object at redshift z has today,

D(z) =R0·r(z) =R0 Z _r 0 dr 0 _=cZ t0 t dt0 a(t0₎ =c Z _z 0 dz0 H(z0₎ = c H0 Z _z 0 dz0 E(z0₎ . (3.19) The second last step took advantage of Equ. 3.8 and its time derivative dt =−dz(1 + z)−1_H(z)−1 _{to relate time tand redshift z, and the final step made use of Equ. 3.16. From}

the observational point of view, two other alternative cosmological distance measures are of prime importance. Firstly, the luminosity distance dlum relates the luminosity L of an

object to its observed fluxf in a way to reproduce the inverse square law of flat Euclidean space f=L/(4πd2

lum). Secondly, the angular diameter distance dang is defined to yield the

Euclidean relation dl =θ·dang for the apparent angular size θ of an object with physical

size dl. Both observationally motivated distance measures can be expressed in terms of the comoving distance D as

dlum =D·(1 +z) (3.20)

dang =

D

The three main cosmological distance measures D, dlum, and dang are illustrated in the

left panel of Fig. 3.16 _{for concordance model parameters. The fact that the angular size}

distancedang has a maximum atz≈1.6 has the important observational consequence that

the apparent angular size of objects is practically constant at z >1, as shown in the right panel of Fig. 3.1. This implies that high-redshift galaxies with typical physical sizes of 10 kpc are observed at angular scales of 1.200_{, 100 kpc cluster cores at 12}00_{, and 1 Mpc scale} clusters of galaxies at 20_.

Combining the relationdlum =dang·(1+z)2 and the fact that the solid angledΩ covered

by an object follows dΩ ∝ d2

ang leads to the cosmological surface brightness dimming

(Tolman’s Law)

Ibol(observed) =

Ibol(emitted)

(1 +z)4 . (3.22)

For monochromatic flux measurements, the surface brightness dimming reduces to Imono(observed) ∝ (1 +z)−3, since the factor (1 +z)−1 for the effective narrowing of the

observed rest-frame bandwidth is recovered.

If the rest-frame spectral energy distribution (SED) has a non-zero slope at the spectral interval of interest, then the observed flux will not only depend on the luminosity distance dlum but also on the actually observed part of the rest-frame spectrum redshifted by the

factor (1 +z). This is taken into account with the so-called K-correction, which can be defined as an additive term KQR for the observed apparent magnitude mR in band-pass R

mR =MQ+DM +KQR , (3.23)

whereMQ is the absolute magnitude of the source in the rest-frame band-pass Qand DM

is the distance modulus

DM =m−M = 25+5 log (dlum[Mpc])−5 logh70 . (3.24)

This last relation follows directly from the definition of absolute magnitudes transformed toh−1

70 Mpc units7. For concordance cosmology, the last term is zero, anddlum can be read

off from Fig. 3.1. TheK-correction term KQR depends on the filter band, the redshift, and

the spectral properties of the object under investigation (see Fig. 11.4). Positive correction terms imply a resulting K-dimming due to a decreasing SED with increasing frequency; negative terms result inK-brightening, i.e.objects actually appear brighter than expected at higher redshift. For efficient observations of the high-redshift Universe, theK-correction is an important consideration for the observing strategy as discussed in Sect. 7.1.

With the distance measures at hand, it is now possible to define comoving volume elements dVcom as

6_{The models for Figs. 3.1, 3.2, and 3.3 were computed with the cosmological calculators athttp://www.}

astro.ucla.edu/%7Ewright/CosmoCalc.html and http://faraday.uwyo.edu/%7Echip/misc/Cosmo2/ cosmo.cgi.

7_{The last term vanishes for concordance cosmology, but reflects the constant offset when changing the}

dVcom=D2dΩdr=

cD2

H(z)dΩdz . (3.25)

Here D is the comoving distance, dΩ the solid angle, and dr the radial thickness8 _{of the}

volume element. In the last step, the radial component was expressed in terms of a redshift intervaldz as in Equ. 3.19. Figure 3.2 (right panel) displays the enclosed comoving volume out to redshift z per square degree for concordance cosmology parameters. The left panel depicts the observed (bolometric) flux for a selection of fiducial object luminosities as a function of redshift and illustrates the luminosity dependent search volumes for flux-limited observations.

When integrating a homogeneously distributed population of sources in Euclidean space over all luminosities (with arbitrary luminosity function), one obtains a universal relation for the cumulative number countsN(> S) above flux S9

N(> S)∝S−3_{2 . (3.26)}

The logN(> S)–logS plot thus shows a constant slope of −3/2 down to the flux limit of the observation, where the counting becomes incomplete (see also Equ. 3.45). The slope is due to the fact, that a lower flux reaches out further in distance (for any luminosity interval) dlum ∝ S−

1

2, and the enclosed volume grows proportional to the third power of

the distance. Since this is only valid for static flat Euclidean space, the relation is expected to intrinsically flatten at a certain flux level when the effects of the expanding Universe and evolving populations become significant (see Sects 3.3 & 4.3).

For a population of objects with a constant comoving number density n0=dN/dVcom,

the redshift distribution dN/dz is obtained by using Equ. 3.25 dN dz =n0 dVcom dz =n0 c D2 H(z)dΩ. (3.27)

The unique link between redshift z and cosmic distance measures can also be generalized for cosmic time t. The time derivative of Equ. 3.8 yields ˙a=−(1+z)−2_{dz/dt. Dividing}

bya(t) and substituting with relation 3.7 results in H(z) =−(1+z)−1_{dz/dt, which can be}

readily integrated by separation of variables to arrive at the lookback time tlookback

tlookback(z) = Z _t t0 dt0 ₌ 1 H0 Z _z 0 dz0 (1 +z0_)·E(z0₎ , (3.28) i.e. the time light from an object at redshift z has been travelling (light travelling time). Integrating to z =∞ returns the age of the Universe t0, and the difference yields the age

of the Universe at a given redshifttage=t0−tlookback(z). Fig. 3.3 illustratestage andtlookback

for a concordance cosmology witht0= 13.46 Gyr.

Table 3.1 summarizes the main cosmological parameters of the concordance model fol- lowing Spergel et al.(2007). These parameters will be used throughout this thesis for any background cosmological model unless otherwise stated.

8_{Strictly speakingd(R}

0r), as the radial coordinateris dimensionless.

Figure 3.1: Cosmological distances and scales for objects at redshift z using concordance model parameters. Left: Comoving distanceD(blue solid line), luminosity distanced_lum(black), and angular size distancedang (red) as a function of redshift. The Hubble radiusdH is indicated by the horizontal dashed line and corresponds to the comoving distance at redshiftz= 1.46. Right: Physical and angular scale evolution with redshift. The black line depicts the physical comoving size of a fixed angular scale of 100 _{in units [kpc/arcsec], the blue line traces the apparent angular size of a fixed physical scale} of 1 Mpc in units [arcmin/Mpc]. Due to the global maximum of dang at z ≈1.6, both functions assume their extrema at this redshift. However, at z >1 the scales are almost redshift independent at approximately 8.4 kpc arcsec−1 _{and correspondingly 2.0 arcmin Mpc}−1_.

Figure 3.2: Bolometric flux evolution and comoving volumes as a function of redshift. Left: Received bolometric flux for three fiducial objects at redshift z with luminosities 1043_{erg s}−1 _(red line), 1044erg s−1 (blue line), and1045erg s−1 (black line). A limited detection bandwidth will shift the curves down with an offset that depends on the flux fraction outside the filter band. At a given flux limit (e.g.10−14_{erg s}−1_cm−2 _{for the dashed horizontal line) objects are detectable out to a maximum} redshift zmax, which is increasing with luminosity. Right: Comoving volume Vcom(< z) per square degree sky coverage from redshift 0 to z. The search volume increases roughly by a factor of 5 from redshift 0.5 to 1.0, and doubles again out to z= 1.5.

Figure 3.3: Cosmic time as a function of redshift for concordance model parameters. Left: The black solid line indicates the age of the Universe tage at redshift z with the total age t0= 13.46Gyr as represented by the horizontal dashed line. The blue line illustrates the lookback-time tlookback from the current epoch to redshift z. The second quarter of the lifetime of the Universe (6.8 Gyr<_∼ tlookback <_∼ 10.1 Gyr) is spanned by the redshift range 0.8<_∼z <_∼1.9; from redshift z= 1 to z= 1.5 the lookback-time increases from 7.7 Gyr to 9.3 Gyr. Right: Redshift zversus lookback time tlookback (black line with left scale). Many physical processes related to structure formation have characteristic timescales of ∼1Gyr, which translate into redshift in a non-linear way. The blue line (right scale) illustrates the changing redshift interval per Giga year lookback timedz/dt. Out toz <_∼0.6the redshift intervals are almost constant at ∼0.1Gyr−1_{,i.e. redshift and time are linear to first order. By z= 1} the redshift interval per Giga year has increased to 0.25Gyr−1_{, at z= 1.5 to 0.4Gyr}−1_{, and at z= 2} it is 0.6Gyr−1_.

Parameters Value Parameters Value

matter density Ωm= 0.3 total energy density Ω0 = 1 power spectrum normalization σ8 = 0.8 primordial PS slope nS = 1 Dark Energy density ΩΛ= 0.7 age of Universe [Gyr] t0 = 13.46

Dark Energy EoS w=−1 recombination redshift zrec ∼1100 Hubble constant [km s−1_Mpc−1 _{] H}

0 = 70 matter-radiation equality zequ ∼3500 baryon density Ωb = 0.045

Table 3.1: Cosmological concordance model parameters as used throughout this thesis. Left: Pa- rameters for which galaxy cluster samples are particularly sensitive to. Right: Additional concordance model parameters that will be assumed for the background cosmology. The parameters σ8 and nS describe the properties of the initial power spectrum of density perturbations, all other parameters are linked to the global geometry of the Universe, which is assumed to be flat (k= 0) in concordance cosmology. The chosen parameter values follow Spergel et al.(2007).