Average Luminous Mass of Early Galaxies at z = 10 - 20 Predicted as ~10⁹ Solar Masses

T. R. Mongan

Sausalito, CA, USA.

DOI: 10.4236/jmp.2023.143015                

Abstract

This paper predicts the average luminous mass of galaxies that will be detected by JWST space telescope at redshift z ≈ 10 - 20. The prediction, derived in the paper, is based on holographic analysis, developed from quantum mechanics, general relativity, thermodynamics, and Shannon information theory. Consistent with early JWST data, ~10⁹ solar masses is the predicted average luminous mass of early galaxies at z ≈ 10 - 20 that will be detected by JWST.

Keywords

Galaxies: High-Redshift

1. Introduction

Luminous galaxy candidates at z > 10 revealed by JWST indicate “future deep JWST observations may identify relatively bright galaxies to much earlier epochs than might have been expected” [1]. As explained below, holographic analysis based on quantum mechanics, general relativity, thermodynamics, and Shannon information theory indicates galaxies with mass ~10⁹ solar masses should be expected at z = 10 - 20. The analysis does not assume or require a theory specifically relating arrangements of bits of information on holographic screens to distribution (as opposed to amount) of mass within such screens.

Behroozi et al. [2] listed some early predictions for what JWST will see, and then used a supercomputer simulation to predict JWST will find galaxies z > 10 with masses considerably below those revealed by early JWST data. In contrast, the prediction from holographic analysis below, ~10⁹ solar masses for average luminous mass of galaxies at z ≈ 10 - 20, is consistent with early JWST results.

2. Holographic Analysis

Our post-inflationary universe, dominated by vacuum energy, has cosmological constant $\Lambda =1.088\times {10}^{-56}{\text{cm}}^{-2}$ [3] and event horizon radius $R_H=\sqrt{\frac{3}{\Lambda }}=1.661\times {10}^{28}\text{cm}$. Holographic analysis [4] finds only a finite number $N=\frac{\pi }{\ln \left(2\right)}{\left(\frac{R_H}{l_P}\right)}^2=4.741\times {10}^{122}$ of bits of information on the event horizon are available to describe our universe, where Planck length $l_P=\sqrt{\frac{\hslash G}{c^3}}=1.616\times {10}^{-33}\text{cm}$.

Friedmann’s general relativistic equation $H_0^2=\frac{8\pi G}{3}{\rho }_{crit}+\frac{\Lambda c^2}{3}$ for a flat Euclidean universe with critical density and cosmological constant requires ${\Omega }_{\Lambda }\equiv \frac{\Lambda c^2}{3H_0^2}$. PDG 2022 [3] lists present day Hubble expansion rate $H_0=67.4\text{km}/\left(\sec \cdot \text{Mpc}\right)$, critical density ${\rho }_{crit}=\frac{3H_0^2}{8\pi G}=8.53\times {10}^{-30}\text{g}/{\text{cm}}^{\text{3}}$, dark energy density parameter ${\Omega }_{\Lambda }=0.685$, Hubble length $c/H_0=1.37\times {10}^{28}\text{cm}$, and matter density ${\rho }_m=\left(1-{\Omega }_{\Lambda }\right){\rho }_{crit}$. PDG parameters result in $\frac{\Lambda c^2}{3{\Omega }_{\Lambda }{H}_0^2}=0.997$, so our universe is indistinguishable from flat Euclidean space to three significant figures. Mass within the event horizon $M_H=\frac{4}{3}\pi \left(1-{\Omega }_{\Lambda }\right){\rho }_{crit}{R}_H^3=\frac{\left(1-{\Omega }_{\Lambda }\right)c^2}{2G}\sqrt{\frac{3}{\Lambda }}=5.14\times {10}^{55}\text{g}$ is constant in time, and $M_H=\left[\frac{\left(1-{\Omega }_{\Lambda }\right)c^2}{2G}{\left(\frac{H_0}{c}\right)}^2\sqrt{\frac{3}{\Lambda }}\right]R_H^2=\left(0.187\text{g}/{\text{cm}}^{\text{2}}\right)R_H^2$. The constant mass per bit of information $m_{bit}=M_H/N=1.08\times {10}^{-67}\text{g}$.

In a fundamental sense, information specifies location of matter in space, and holographic analysis indicates 4.741 × 10¹²² bits of information on the spherical holographic screen (SHS) of the event horizon are associated with matter within our observable universe. Holographic analysis then indicates information and associated mass M within isolated gravitationally bound systems relates to radii R of spherical holographic screens around system centers of mass by $M=\left(0.187\text{g}/{\text{cm}}^{\text{2}}\right)R^2$.

Cosmic microwave background (CMB) radiation density at redshift z is ${\rho }_r\left(z\right)={\left(1+z\right)}^4{\rho }_r\left(0\right)$, where mass equivalent of today’s radiation energy density ${\rho }_r\left(0\right)=4.59\times {10}^{-34}\text{g}/{\text{cm}}^{\text{3}}$. Matter density ρ(z) is much greater than radiation density and Jeans mass [5] $M_J=\frac{\pi }{48{\rho }_m^2}{\left[\frac{2c}{3}\sqrt{\frac{\pi {\rho }_r\left(0\right)}{G}}\right]}^3=2.30\times {10}^{50}\text{g}$, the upper limit on mass of gravitationally bound systems stable against gravitational collapse, is independent of z . Large scale structures at z > 10 are gravitationally bound systems of individual stars with masses between Jeans mass $M_J$ and minimum stellar mass $m_{\ast \min }\left(z\right)$ at redshift z .

3. Minimum Stellar Mass at Redshift z

Minimum stellar mass $m_{\ast \min }\left(z\right)$ is estimated by setting escape velocity of protons at SHS radius $R_{\ast \min }$ for minimum stellar mass equal average velocity of protons in equilibrium with CMB radiation outside the SHS for $m_{\ast \min }\left(z\right)$. Protons in equilibrium with CMB outside the SHS for stellar systems with mass $<m_{\ast \min }\left(z\right)$ can transfer energy to those systems until they reach $m_{\ast \min }\left(z\right)$. Escape velocity v for protons of mass $m_p$ gravitationally bound at radius R from system center of mass M is calculated from $\frac{1}{2}{m}_p{v}^2=\frac{GM{m}_p}{R}$. Escape velocity of protons on the SHS for minimum mass stars with mass M at redshift z is velocity of protons in thermal equilibrium with CMB, so $\frac{3}{2}kT\left(z\right)=\frac{GM{m}_p}{R}$, where CMB temperature $T\left(z\right)=\left(1+z\right)2.725\text{K}$ and Boltzmann constant $k=1.38\times {10}^{-16}\left(\text{g}\cdot {\text{cm}}^{\text{2}}/{\sec }^2\right)/\text{K}$. With radius $R=\sqrt{M/\left(0.187\text{g}/{\text{cm}}^{\text{2}}\right)}$ for structures of mass M , $m_{\ast \min }\left(z\right)=\frac{1}{0.187}{\left(\frac{1.5k\left(1+z\right)2.725}{G{m}_p}\right)}^2\text{g}$, so $m_{\ast \min }\left(10\right)=8.26M_{\odot }$, $m_{\ast \min }\left(12\right)=11.5M_{\odot }$, and $m_{\ast \min }\left(20\right)=30.1M_{\odot }$, with solar mass $M_{\odot }=2\times {10}^{33}\text{g}$. If outgoing protons at the SHS are in thermal equilibrium with outgoing photon flow from minimum mass stars, stars must have mass $>m_{\ast \min }\left(z\right)$ to appear against the CMB background. Maximum star mass $6\times {10}^{35}\text{g}\approx 300M_{\odot }$ [6] coincided with minimum star mass at $z\approx 65$, consistent with indications stars first formed at $z\approx 65$ [7]. At $z=0$, minimum star mass $\approx 1.4\times {10}^{32}\text{g}=0.07M_{\odot }$, consistent with hydrogen burning mass threshold separating brown dwarfs from lowest mass stars [8].

4. Average Galactic Mass at Redshift z

With no good reason to suspect information describing gravitationally bound systems is not uniformly distributed between mass bins, the number of gravitationally bound systems of mass m at redshift z can be taken as $n\left(m\right)=K/m$ with constant K . Then total mass of observable gravitationally bound systems at redshift z is $M_H\approx {\displaystyle {\int }_{m_{\ast \min }\left(z\right)}^{M_J}m\left(\frac{K}{m}\right)\text{d}m}=K\left(M_J-m_{\ast \min }\left(z\right)\right)$ and $K\approx M_H/M_J=2.24\times {10}^5$. The number of observable gravitationally bound structures in Jeans mass $M_J$ at redshift z is $N\left(z\right)={\displaystyle {\int }_{m_{\ast \min }\left(z\right)}^{M_J}\frac{K}{m}\text{d}m}=K\ln \left(2.30\times {10}^{50}\text{g}/m_{\ast \min }\left(z\right)\right)$. Estimated average total mass of observable gravitationally bound structures at redshift z, $M_{avg}\left(z\right)=M_J/N\left(z\right)$, at z = 10 - 20 is $~1.4\times {10}^{10}{M}_{\odot }$, between estimated Milky Way mass $~{10}^{45}\text{g}~{10}^{12}{M}_{\odot }$ and dwarf galaxy masses $~{10}^{40}\text{g}~{10}^7{M}_{\odot }$.

PDG 2022 [3] lists baryon density fraction of the universe ${\Omega }_b=0.0493$, so stellar mass is ${\Omega }_b/\left(1-{\Omega }_{\Lambda }\right)=0.0720$ times total galactic mass. Then stellar mass $~{10}^9{M}_{\odot }$ of galaxy candidates at redshift z ≈ 10 - 12 identified by Naidu et al. in JWST data is consistent with “early appearance of UV-luminous galaxies with stellar masses as high as $\approx {10}^9{M}_{\odot }$ already at few 100 Myr after the Big Bang” [1].

5. Conclusion

Holographic analysis predicts the average galaxy detected by JWST space telescope at redshift z ≈ 10 - 12 will have luminous mass of about 10⁹ solar masses, consistent with early JWST data.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

References

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