# On the Number of Galaxies at High Redshift

## Abstract

**:**

## 1. Introduction

## 2. Basic Formulae

**Figure 1.**HgI 435.83 nm line shifts versus the electron density, data as extracted by the author from Figure 7 in [16] (empty stars) and linear regime (full line).

**Figure 2.**Redshift of the $H\alpha $ line versus the electron density, data as extracted by the author from Table 1 in [23] (empty stars) and linear fit (full line).

#### 2.1. Magnitude System

**Figure 3.**The blue absolute magnitude ${M}_{B}$ computed with the nonlinear Equation (11) for 263 galaxies belonging to the FORS Deep Field (FDF) catalog versus the well measured redshift. The lower theoretical curve as represented by the nonlinear Equation (11) is the red thick line when ${m}_{L}$ = 30.33, which is the maximum apparent magnitude of the catalog, ${\mathcal{M}}_{\odot}$ = 5.48 and ${H}_{0}=69.6\phantom{\rule{4pt}{0ex}}\text{km}\phantom{\rule{4pt}{0ex}}{\mathrm{s}}^{-1}\phantom{\rule{4pt}{0ex}}{\text{Mpc}}^{-1}$ (green points). The redshift covers the range $[0,4.5]$.

**Figure 4.**The B absolute magnitude M computed with the nonlinear Equation (11) for 9697 galaxies belonging to the zCOSMOS catalog versus the redshift. The lower theoretical curve as represented by the nonlinear Equation (11) is the red thick line when ${m}_{L}$ = 23.2, ${\mathcal{M}}_{\odot}$ = 4.08 and ${H}_{0}=69.6\phantom{\rule{4pt}{0ex}}\text{km}\phantom{\rule{4pt}{0ex}}{\mathrm{s}}^{-1}\phantom{\rule{4pt}{0ex}}{\text{Mpc}}^{-1}$ (green points). The redshift covers the range $[0,1]$.

#### 2.2. Tired Light

#### 2.3. The Luminosity Function

## 3. N–z Relation

#### 3.1. The Linear Case

#### 3.2. The Nonlinear Case

## 4. Astrophysical Applications

- (1)
- A window in apparent magnitude or flux is chosen around m or f.
- (2)
- All the galaxies which fall in the window are selected.
- (3)
- The mean value in redshift of N selected galaxies is $\langle {z}_{obs}\rangle $ and the uncertainty in the mean, ${\sigma}_{\mu}$, is ${\sigma}_{\mu}=s/\sqrt{N}$ where s is the standard deviation, see formula (4.14) in [34].

#### 4.1. The FDF Catalog

**Figure 6.**The galaxies of the FDF catalog are organized in frequencies versus spectroscopic redshift. The redshift covers the range $[0,4]$ and the histogram’s interval is 0.1.

**Figure 7.**The galaxies of the FDF catalog with $22.08\le m\le 26.81$ or $2.33\frac{{L}_{\odot}}{Mp{c}^{2}}\le f\le 181.39\frac{{L}_{\odot}}{Mp{c}^{2}}$ are organized in frequencies versus spectroscopic redshift. The redshift covers the range $[0,1.5]$ and the histogram’s interval is 0.18. The maximum frequency of observed galaxies is at $z=0.33$, ${\chi}^{2}=$ 77.8, and the number of bins is 8. The full line is the theoretical curve generated by $\frac{dN}{d\Omega dzdf}\left(z\right)$ as given by the application of the Schechter luminosity function (LF) which is Equation (48) with ${\Phi}^{*}=0.01\phantom{\rule{0.166667em}{0ex}}/Mp{c}^{3}$, ${M}^{*}=-17.78$ and $\alpha =-1.07$.

**Figure 8.**The theoretical number of galaxies of the FDF catalog as afunction of redshift and apparent magnitude represented as a 3D surface, parameters as in Figure 7.

#### 4.2. The zCOSMOS Catalog

**Figure 10.**The galaxies of the zCOSMOS catalog are organized in frequencies versus spectroscopic redshift. The redshift covers the range $[0,1.2]$ and the histogram’s interval is 0.02.

**Figure 11.**The galaxies of the zCOSMOS catalog with $17.88\le m\le 19.06$ or $803.43\frac{{L}_{\odot}}{Mp{c}^{2}}\le f\le 2392.36\frac{{L}_{\odot}}{Mp{c}^{2}}$ are organized in frequencies versus spectroscopic redshift. The redshift covers the range $[0,1]$ and the interval in the histogram is 0.1. The error bar is given by the square root of the frequency (Poisson distribution) . The maximum frequency of observed galaxies is at $z=0.213$, ${\chi}^{2}=$ 147.3, and the number of bins is 10. The full line is the theoretical curve generated by $\frac{dN}{d\Omega dzdf}\left(z\right)$ as given by the application of the Schechter LF, which is Equation (48) with ${\Phi}^{*}=0.01\phantom{\rule{0.166667em}{0ex}}/Mp{c}^{3}$, ${M}^{*}=-20.88$ and $\alpha =-1.07$.

**Figure 12.**All the galaxies of the zCOSMOS catalog, organized in frequencies versus spectroscopic redshift. The redshift covers the range $[0,1]$ and the interval in the histogram is 0.1. The error bar is given by the square root of the frequency (Poisson distribution) . The maximum frequency of all observed galaxies is at $z=0.35$, ${\chi}^{2}=$ 1864.65, and the number of bins is 10. The full line is the theoretical curve generated by $\frac{dN}{d\Omega dz}\left(z\right)$ as given by the numerical integration of Equation (51) with ${\Phi}^{*}=0.01\phantom{\rule{0.166667em}{0ex}}/Mp{c}^{3}$, ${M}^{*}=-18$ and $\alpha =-1.07$.

## 5. The Relativistic Case

**Figure 14.**The galaxies of the zCOSMOS catalog with the same parameters of Figure 11 are organized in frequencies versus spectroscopic redshift. The full line is the theoretical curve generated by $\frac{dN}{d\Omega dzdf}\left(z\right)$ as given by the application of the Schechter LF in the relativistic case, which is Equation (68) with ${\Phi}^{*}=0.01\phantom{\rule{0.166667em}{0ex}}/Mp{c}^{3}$, ${M}^{*}=-20.7$, $\alpha =-1.07$ and ${\Omega}_{\mathrm{M}}=0.286$; ${\chi}^{2}$ =95.68 when the number of bin is 10.

**Figure 15.**Average observed redshift, $\langle {z}_{obs}\rangle $, as function of the apparent magnitude for the zCOSMOS catalog (empty stars) and theoretical full line, $\langle z\rangle $, as given by a numerical integration. Theoretical parameters as in Figure 14.

## 6. Evolutionary Effects

- (1)
- A value for the redshift is fixed, z, as well as the thickness of the layer, $\delta z$.
- (2)
- All the galaxies comprised between z and $\delta z$ are selected.
- (3)
- The absolute magnitude can be computed from Equation (28) which represents the distance modulus for tired light.
- (4)
- The distribution in magnitude is organized in frequencies versus absolute magnitude.
- (5)
- The frequencies are divided by the volume, which is $V=\Omega \pi {r}^{2}\delta r$, where r is the considered radius, $\delta r$ is the thickness of the radius, and Ω is the solid angle of ZCOSMOS.
- (6)
- The error in the observed LF is obtained as the square root of the frequencies divided by the volume.

**Figure 16.**The luminosity function data of zCOSMOS are represented with error bars. The continuous line fit represents our beta LF (77), the parameters are $z=0.2$, $\delta z$ = 0.05 and NDIV = 8, which means ${\chi}^{2}=5.35$.

**Figure 17.**The luminosity function data of zCOSMOS are represented with error bars. The continuous line fit represents our beta LF (77), the parameters are $z=0.5$, $\delta z$ = 0.05, and NDIV = 8, which means ${\chi}^{2}$ = 16.71.

**Figure 18.**The luminosity function data of zCOSMOS are represented with error bars. The continuous line fit represents our beta LF (77), the parameters are $z=0.7$, $\delta z$ = 0.03, and NDIV = 4, which means ${\chi}^{2}$ = 8.76.

**Figure 19.**The luminosity function data of zCOSMOS are represented with error bars. The continuous line fit represents our beta LF (77) in the relativistic case. The input parameters are ${\Omega}_{\lambda}=0$, ${\Omega}_{\mathrm{M}}=0.286$, $z=0.7$, $\delta z$ = 0.03 and NDIV = 4, which means ${\chi}^{2}$ = 8.78.

## 7. Conclusions

#### 7.1. Results

#### 7.2. Generalizated Tired Light

#### 7.3. Tired Light versus GR

#### 7.4. The Cells

## Acknowledgments

## Conflicts of Interest

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Zaninetti, L.
On the Number of Galaxies at High Redshift. *Galaxies* **2015**, *3*, 129-155.
https://doi.org/10.3390/galaxies3030129

**AMA Style**

Zaninetti L.
On the Number of Galaxies at High Redshift. *Galaxies*. 2015; 3(3):129-155.
https://doi.org/10.3390/galaxies3030129

**Chicago/Turabian Style**

Zaninetti, Lorenzo.
2015. "On the Number of Galaxies at High Redshift" *Galaxies* 3, no. 3: 129-155.
https://doi.org/10.3390/galaxies3030129