JWST found large galaxies in the early universe, inconsistent with traditional galaxy formation models involving mass increase by clustering and merging [1]. Consistent with JWST observations and using PDG 2025 [2] data, this holographic analysis accounts for total mass of galaxy clusters, galaxies and star clusters with central black holes, constituting visible large-scale structures (LSS) in our closed, homogeneous, isotropic and vacuum-dominated Friedmann universe.

Probability K m of LSS mass m, with constant K, in a given LSS category results in greater numbers of small structures than large structures in each LSS category, and uniform distribution of information and matter across the mass range in each LSS category.

At a fundamental level, information specifies distribution of matter within the universe.

Holographic analysis (based on quantum mechanics, general relativity, black hole thermodynamics, and Shannon information theory) finds only the finite number of bits of information on the event horizon will ever be available to describe distribution of matter in the universe [3]. That implies the universe is a closed system described by discrete mathematics.

The event horizon at distance R H = 3 Λ = 1.661 × 10 28   cm from any observer’s location is the farthest distance observers can ever see out into our vacuum dominated universe, with cosmological constant Λ = 1.088 × 10 − 56   cm − 2 and vacuum energy density accelerating increase in radius of the closed universe.

This holographic analysis identifies the bits of information available to describe matter distribution within the universe as encoded on areas of l P 2 on the event horizon, where the Planck length l P = ℏ G c 3 = 1.61625 × 10 − 33 , ℏ = 1.05457 × 10 − 27 g ⋅ cm 2 / sec , G = 6.67430 × 10 − 8   cm 3 ⋅ g − 1 ⋅ sec − 2 , and c = 2.99792 × 10 10 cm / sec . The maximum number of bits of information that will ever be available to describe the distribution of matter within the universe is N = ( π ln ( 2 ) ) ( R H l P ) 2 = 4.741 × 10 122 . Mass of the observable universe inside the event horizon M H = 4 3 π ( 1 − Ω Λ ) ρ c r i t R H 3 = 5.16 × 10 55   g = ( 0.187   g cm 2 ) R H 2 with Hubble constant H 0 = 67.4   km ⋅ sec − 1 ⋅ Mpc − 1 , critical energy density ρ c r i t 3 H 0 2 8 π G g / cm 3 = 8.533 × 10 − 30 g / cm 3 , and vacuum energy fraction Ω Λ = 0.685 . Half of the bits of information on the horizon provide information on distribution of matter within the horizon and the other half of the bits of information on the horizon provide information on distribution of anti-matter within the horizon. So the mass of matter associated with one bit of information is m b i t = M H 2.371 × 10 122 = 2.17 × 10 − 67   g .

This holographic analysis then finds the bits of information describing an isolated system with definite mass m within the universe are available on a spherical surface surrounding the system with radius r = m M H R H and holographic radii of isolated systems with definite mass m are r = m 0.187   g / cm 2 .

Visible large-scale structures (LSS) in our expanding universe, comprised of individual stars, exist within isothermal spheres of cold dark matter. Three levels of LSS, respectively categorized as galaxy clusters, galaxies, or star clusters, consist of widely separated sub-elements in a sea of cosmic microwave background radiation.

Looking out in the universe at distant luminous objects, light traveling at constant speed c shows those objects at the time in the past when the light was emitted. Wavelength of light emitted by sources moving away from us is increased (redshifted), and redshift z relates wavelength observed λ o b s e r v e d to wavelength emitted λ e m i t t e d by λ o b s e r v e d = λ e m i t t e d ( 1 + z ) .

Radius R ( t ) of our observable closed, homogeneous, isotropic, and expanding Friedmann universe at time t years ago was R ( t ) = R H / ( 1 + z ) , with radius R H today at z = 0 . In our expanding universe, all LSS move away from us and redshifted light from those distant structures reveals conditions in the smaller universe in the past. Total matter density in the universe today is ρ m ( 0 ) = 2.304 × 10 − 30 g / cm 3 and radiation density is ρ r ( 0 ) = 4.519 × 10 − 34 g / cm 3 . Total matter density in the universe at redshift z was

ρ m ( z ) = ρ m ( 0 ) ( 1 + z ) 3 and radiation density was

ρ r ( z ) = ρ r ( 0 ) ( 1 + z ) 4 .

Jeans length L J 1 ( z ) = c s 1 ( z ) π G ρ m 1 ( z ) is the scale of the largest structures of matter (galaxy clusters) stable against gravitational collapse at redshift z , where speed of pressure waves in matter in the universe is c s 1 ( z ) = c 4 ( 1 + z ) 4 ρ r ( 0 ) 12 ( 1 + z ) 4 ρ r ( 0 ) + 9 ( 1 + z ) 3 ρ m 1 ( 0 ) and ρ m 1 ( z ) = ρ m ( 0 ) ( 1 + z ) 3 . So, Jeans mass M J 1 ( z ) = 4 3 π ( L J 1 ( z ) 2 ) 3 ρ m 1 ( z ) = π 5 / 2 6 c s 1 3 ( z ) 1 G 3 / 2 ρ m 1 ( z ) , is maximum galaxy cluster mass at redshift z . M J 1 ( 0 ) = 1.26 × 10 18 M ⊙ , with solar mass M ⊙ = 1.99 × 10 33   g , is consistent with the 2.1 × 10 17 M ⊙ mass of Quipu, the largest cosmic structure found to date [4].

Average density within Jeans mass holographic radius is ρ m 2 ( z ) = M J 1 ( z ) / 4 3 π ( M J 1 ( z ) 0.187 ) 3 / 2 = 3 4 π 0.187 3 / 2 M J 1 ( z ) and second level Jeans mass, the maximum mass of galaxies within galaxy clusters at redshift z , is M J 2 ( z ) = π 5 / 2 6 c s 2 3 ( z ) 1 G 3 / 2 ρ m 2 ( z ) with c s 2 ( z ) = c 4 ( 1 + z ) 4 ρ r ( 0 ) 12 ( 1 + z ) 4 ρ r ( 0 ) + 9 ρ m 2 ( z ) . M J 2 ( 0 ) = 4.48 × 10 13 M ⊙ is consistent with estimated mass about 10 13 M ⊙ for IC 1101, one of the most massive galaxies found to date.

Average density within second level Jeans mass holographic radius is ρ m 3 ( z ) = M J 2 ( z ) / 4 3 π ( M J 2 ( z ) 0.187 ) 3 / 2 = 3 4 π 0.187 3 / 2 M J 2 ( z ) and third level Jeans mass, the maximum mass of star clusters within galaxies at redshift z , is M J 3 ( z ) = π 5 / 2 6 c s 3 3 ( z ) 1 G 3 / 2 ρ m 3 ( z ) with c s 3 ( z ) = c 4 ( 1 + z ) 4 ρ r ( 0 ) 12 ( 1 + z ) 4 ρ r ( 0 ) + 9 ρ m 3 ( z ) . M J 3 ( 0 ) = 1.60 × 10 9 M ⊙ is consistent with an estimated upper limit on total star cluster mass [5].

Average density within third level Jeans mass holographic radius is ρ m 4 ( z ) = M J 3 ( z ) / 4 3 π ( M J 3 ( z ) 0.187 ) 3 / 2 = 3 4 π 0.187 3 / 2 M J 3 ( z ) and fourth level Jeans mass, an upper bound on stellar masses within star clusters at redshift z , is M J 4 ( z ) = π 5 / 2 6 c s 4 3 ( z ) 1 G 3 / 2 ρ m 4 ( z ) with c s 4 ( z ) = c 4 ( 1 + z ) 4 ρ r ( 0 ) 12 ( 1 + z ) 4 ρ r ( 0 ) + 9 ρ m 4 ( z ) and M J 4 ( 0 ) = 56900 M ⊙ .

In what follows, M max ( z ) is maximum mass (Jeans mass) for LSS at a given structural level and M min ( z ) , minimum mass for LSS at that structural level is Jeans mass for the next lower structural level (or, in the case of star clusters, maximum stellar mass).

Star formation results from thermonuclear reactions between strongly interacting protons in the baryon fraction of matter density in the universe. Mass of the smallest gravitationally bound systems (stars) at redshift z is estimated by setting escape velocity of protons at holographic radius R min * ( z ) = M min * ( z ) 0.187   g / cm 2 of stars with minimum mass M min * ( z ) equal to average velocity of protons in thermal equilibrium with CMB radiation at redshift z outside R min * ( z ) .

Escape velocity v E p for protons with mass m p gravitationally bound at radius R from the centroid of a structure with mass M is determined by 1 2 m p v E p 2 = G M m p R 2 . If proton escape velocity v E p at holographic radius R min * ( z ) of minimum mass stars at redshift z is proton velocity in thermal equilibrium with CMB radiation at redshift z , 3 2 k ( 1 + z ) 2.7255   K = G M min * ( z ) m p ( R min * ( z ) ) 2 . With CMB temperature 2.7255 ° K at z = 0 and Boltzmann constant k = 1.381 × 10 − 16 ( g ⋅ cm 2 / sec 2 ) / K , M min * ( z ) = 1 0.187 ( 1.5 k ( 1 + z ) 2.7255 G m p ) 2 g .

If outgoing protons at R min * ( z ) are in thermal equilibrium with outgoing photon flow from minimum mass stars, stars must have mass > M min * ( z ) to appear against the CMB. If population III stars first appeared [6] when M max * ( z ) = M min * ( z ) at z = 65 , they may have had mass M min * ( 65 ) ≈ 300 M ⊙ . Minimum star mass today [7] M m i n * ( 0 ) ≈ 0.07 M ⊙ is consistent with hydrogen burning mass threshold separating brown dwarfs from lowest mass stars.

In isothermal spheres of cold dark matter inhabited by LSS, core radius R c ( z ) of an LSS containing concentrated mass in the central black hole is determined by the holographic radius of sub-elements orbiting the center just outside the core without being disrupted and drawn into the central black hole. Sub-elements of an LSS at a given structural level are LSS in the structural level in the next lower LSS mass range. Central black hole mass is M C B H ( z ) = M L S S ( z ) M s e ( z ) , where M L S S ( z ) is total large-scale structure mass and M s e ( z ) is mass of LSS sub-elements that can occupy a circular orbit just outside the core without being disrupted and drawn into the central black hole.

The most massive black holes allowed by holographic analysis are at the center of the most massive galactic clusters with Jeans mass M J 1 ( 0 ) = 1.26 × 10 18 M ⊙ after all but lowest mass galaxies (with mass M J 3 ( 0 ) = 1.60 × 10 9 M ⊙ ) are engulfed in the central black hole. The 4.49 × 10 13 M ⊙ holographic upper limit on black hole mass is considerably greater than 10 11 M ⊙ mass of the most massive black hole found to date.

Holographic analysis determines total mass M g t o t ( z ) of galaxies at redshift z as M g t o t ( z ) = ∫ M g m i n ( z ) M g m a x ( z ) K m m d m = K ( M g m a x ( z ) − M g m i n ( z ) ) and total number of galaxies at redshift z as N g t o t ( z ) = ∫ M g m i n ( z ) M g m a x ( z ) K m d m = K ln ( M g m a x ( z ) M g m i n ( z ) ) . Average total galaxy mass at redshift z is M g a v g ( z ) = ( M g m a x ( z ) − M g m i n ( z ) ) / ln ( M g m a x ( z ) M g m i n ( z ) ) and M g a v g ( 14.4 ) = 5.191 × 10 10 M ⊙ . Average visible mass 8.143 × 10 9 M ⊙ for galaxies at z = 14.4 identified by holographic analysis is consistent with JWST data finding early galaxies with stellar masses ~ 10 9 M ⊙ when the universe was only a few hundred million years old. Such large galaxies in the early universe were not expected in traditional galaxy formation models involving galaxies increasing in mass by clustering and merging [1].

JWST found abundant compact red sources, called Little Red Dots (LRDs), within large-scale structures. LRDs were largely missed by previous telescopes.

LRDs, with masses ranging from about 10 9 M ⊙ to 10 12 M ⊙ , are within the holographic radii of large-scale structures. Possible explanations for LRDs include:

Primordial galaxies [8]. This holographic analysis finds galaxies with masses ranging from about 10 9 M ⊙ to 10 12 M ⊙ existed in the very early universe at redshift z = 14.4 .Quasi-stars (black hole stars) [9]. This holographic analysis shows central black holes with masses ranging from about 10 9 M ⊙ to 10 11 M ⊙ existed in early galaxies at redshift z = 14.4 . If some of them were quasi-stars, they could appear as LRDs.Star clusters containing supermassive stars [9]. At redshift z = 14.4 , this holographic analysis shows star clusters with masses ranging from 5.66 × 10 4 M ⊙ to 1.59 × 10 9 M ⊙ existed in early galaxies at redshift z = 14.4 , and supermassive stars with masses up to 5.66 × 10 4 M ⊙ could be present.