Supermassive Black Holes, Large Scale Structure and Holography

doi:10.4236/jmp.2013.47A1006

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Journal of Modern Physics, 2013, 4, 50-54

http://dx.doi.org/10.4236/jmp.2013.47A1006 Published Online July 2013 (http://www.scirp.org/journal/jmp)

Supermassive Black Holes, Large Scale Structure

and Holography

T. R. Mongan

84 Marin Avenue, Sausalito, USA

Email: [email protected]

Received March 21, 2013; revised April 29, 2013; accepted June 7, 2013

permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

ABSTRACT

A holographic analysis of large scale structure in the universe provides an upper bound on the mass of supermassive

black holes at the center of large scale structures with matter density varying as 2

r as a function of distance r from

their center. The upper bound is consistent with two important test cases involving observations of the supermassive

black hole with mass 3.6 × 10−6 times the galactic mass in Sagittarius A* near the center of our Milky Way and the 2 ×

109 solar mass black hole in the quasar ULAS J112001.48 + 064124.3 at redshift z = 7.085. It is also consistent with

upper bounds on central black hole masses in globular clusters M15, M19 and M22 developed using the Jansky Very

Large Array in New Mexico.

Keywords: Supermassive Black Holes; Large Scale Structure; Holographic Principle

1. Introduction

How supermassive black holes “... form and evolve in-

side galaxies is one of the most fascinating mysteries in

modern astrophysics” [1]. This analysis addresses that

issue with a holographic model [2] for large scale struc-

ture in the universe, based on the holographic principle

[3] resulting from the theory of gravitation expressed by

general relativity. The internal dynamics of large scale

structures is analyzed using classical Newtonian gravity

to describe the motion of sub-elements within the struc-

tures and general relativity to describe the supermassive

black holes at their centers. Consistency of the results

with test cases across the range of large scale structures

and redshifts makes it difficult to ascribe those results to

numerical coincidences.

2. Internal Dynamics of Large Scale

Structures

The holographic model for large scale structure [2] iden-

tifies three levels of self-similar large scale structures

(corresponding to superclusters, galaxies and star clusters)

between stellar systems and the totality of today’s ob-

servable universe. The extended holographic principle

employed in that model indicates all information de-

scribing physics of a gravitationally-bound astronomical

system of total mass

is encoded on a spherical

holographic screen enclosing the system. In our vac-

uum-dominated universe, the radius of the holographic

screen encoding all information describing a structure of

0.16

R

mass

is , if the Hubble constant

071 kmsecMpcH



. In the holographic model, the

number of sub-elements of mass in a large scale m

structure is

, where

is constant, so the amount of

information in any mass bin (proportional to

m) is

the same in all mass bins. This is consistent with the 1

behavior of the mass spectrum in the Press-Schechter

formalism [4], and implies lowest mass sub-elements are

the most numerous.

The main idea of this analysis is that matter inside a

core radius much smaller than the holographic radius of a

large scale structure is accumulated in a central black

hole, where the core radius is the radius at which lowest

mass sub-elements can exist without being disrupted and

drawn into the central black hole.

The analysis assumes visible large scale structures

develop within isothermal spherical halos of dark matter.

T. R. MONGAN 51

So, the matter density distribution in large scale structures





r

, where is the distance from the center

of the structure and is constant. The mass

within

the holographic radius

R is

 

aR, requiring 4



. Then,

the mass within radius from the center of a large

scale structure is 2

rr M





 and the

tangential speed t of a sub-element of mass m moving

in a circle of radius around the center is found from

Gam





4GMm

R, where G = 6.67 × 10−8

cm3·g−1·sec−2. So, the tangential speed of sub-elements in

circular orbits around the center,

, does not

depend on distance from the center and sub-elements lie

on a flat tangential speed curve.

With an 2

matter density distribution, sub-elements

orbiting the center of a large scale structure at radius

are equivalent to sub-elements orbiting a point mass with

mass

R. In large scale structures with 2

den-

sity distributions, the mass within a core radius c is

drawn into a central black hole with an innermost stable

circular orbit (ISCO) radius [5] much less than the holo-

graphic radius of the lowest mass sub-element. The core

radius c is the holographic radius of the lowest mass

sub-element of the large scale structure because no sub-

element can exist as an isolated system closer to the cen-

tral black hole than the holographic radius of the lowest

mass sub-element without being disrupted and drawn

into the central black hole. In consequence, the upper

bound on the mass concentrated in the central black hole

of the large scale structure is the mass of the 2

density distribution within c. The upper bound is rea-

ched when the central black hole has accumulated all of

the matter within the central volume inside . With

density distribution













, the mass of the

structure within radius from the center of the large

scale structure is c

. So, when the mass within the

core radius c is concentrated in the central black hole

of the large scale structure, the upper bound on the mass

of the central black hole is c

R. The corresponding

upper bound on the fraction of the mass of large scale

structures concentrated in the central black hole is

min

min

M, where is the mass of the lowest

mass sub-elements of the large scale structures.

Note that the innermost stable circular orbit (ISCO)

radius,

SCO , of a black hole of mass r

 and spin an-

gular momentum J depends on its gravitational radius



10 1

3.0010cmsecc

 

, where , and its spin

J. The ISCO radius is

SCOG for a non-rotating

black hole,

rr

SCO G



for maximal prograde rotation of

the black hole and 9

SCO G



for maximal retrograde

rotation of the black hole [5]. So, the necessary condition

for this analysis, 1

ISCO

, is



90.16 1





 

4.08 10cm

. For

the largest of all structures, with Jeans’ mass 2.61 × 1050

g and holographic radius ,



90.16 0.0044.

 



 Thus, central black holes are

point particles compared to the holographic radius of the

smallest sub-elements in any of the self-similar large

scale structures.

3. Central Black Holes at z = 0

This upper bound on the mass of central black holes in

large scale structures is consistent with two important

test cases. The first is the supermassive black hole in

Sagittarius



near the center of our Milky Way. The

mass of the Milky Way is estimated as 2.52 × 1045 g [6].

If the Hubble constant H0 = 71 km/sec Mpc, the holo-

graphic model of self-similar large scale structure [2]

estimates the mass of lowest mass star cluster sub-ele-

ments of galaxies as 1.0 × 1035 g. Then, the upper bound

on the mass of the central black hole in the Milky Way is

1.6 × 1040 g, about twice the observed 9 × 1039 g mass [7]

of the supermassive black hole in Sagittarius



. The

upper bound on the central black hole mass for galaxies

with the average galactic mass [2] 1.6 × 1044 g is 4.0 ×

1039 g, or 6

210



210 gM



, where is the

solar mass.

In the holographic model for large scale structure, the

constant relating the mass of isolated structures to their

holographic radius, as well as the mass of lowest mass

substructures within large scale structures, depends on

the Hubble constant H0. If the Hubble constant H0 = 65

km/sec Mpc, the upper bound on the mass of the central

black hole in the Milky Way is 9.2 × 1039 g, close to the

estimate from Keck telescope observations [7].

Self-similarity of large scale structures in the holo-

graphic model indicates there should be black holes in

the centers of superclusters and star clusters, just as there

are in galaxies. The largest black hole at



should

T. R. MONGAN

be in the center of the largest supercluster, corresponding

to a supercluster with the Jeans’ mass 2.6 × 1050 g. Using

the estimate of 1.4 × 1040 g for the lowest mass galaxies

from the holographic model [2], the upper bound for the

mass of the largest supermassive black hole in the uni-

verse at is 1.9 × 1045 g. This is about fifty times

the 4.2 × 1043 g mass of one of the largest black holes

found to date, that in NGC 4889 in the Coma con-

stellation [8].

0z



210M

1.6 10g

At the lower end of the range of large scale structures,

the holographic model estimates an average star cluster

mass of 1.2 × 1039 g at z = 0. Using a z = 0

minimum stellar mass of 0.08 , 4.4 ×

1035 g is the upper bound on the central

black hole in such a star cluster. The holographic upper

bound on central black hole mass in star clusters is con-

sistent with upper bounds on central black hole masses in

globular clusters M15, M19 and M22 developed using

the Jansky Very Large Array (JVLA) in New Mexico [9].

The mass of M15 is



M



5.6 10



220 M

 [10], the mass of M19

is 6

101.1



[11], and the mass of M22 is

2.9 1

 [10]. The holographic upper bound for the

mass of the central black hole in M15 of 212



980

, con-

sistent with the JVLA upper bound of

. Cor-

respondingly, the upper bound on the central black hole

mass of 297

 for M19 is consistent with the JVLA

upper bound of 730

, and the upper bound on the

central black hole mass of 150

 for M22 is consis-

tent with the JVLA upper bound of 360

.

4. Supermassive Black Holes at z > 0

In the holographic model [2], the range of the mass

spectrum at any structural level decreases with redshift,

because the mass at the lower end of the mass spectrum

at any structural level increases with redshift. Also, the

number of structural levels increases with redshift. Ac-

cordingly, for a given structure mass, the upper bound on

central black hole mass increases with redshift. The

holographic upper bound on central black hole mass for a

structure with mass equal to that of the Milky Way is 6.1

× 1041 g at z = 0.5 and 8.1 × 1042 g at z = 1, compared to

the upper bound of 1.6 × 1040 g at z = 0. The estimated

mass of an average galaxy at z = 0 in the holographic

model is 1.6 × 1044 g. The upper bound on the central

black hole mass

 for a structure with mass 1.6 × 1044

g is 1.5 × 1041 g at z = 0.5 and 2.1 × 1042 g at z = 1,

compared to 4.0 × 1039 g at z = 0. This is consistent

with indications that the average ratio

galaxy

 increases

with redshift [12].

Volonteri [1] says the “golden era” of 1 billion



supermassive black holes “occurred early on.” So, the

analysis below estimates upper bounds on black hole

masses in the early universe at z > 6, less than a billion

years after the end of inflation [13] and before develop-

ment of self-similar large scale structures present in

today’s universe began at z < 6 [2].

If the Hubble constant H0 = 71 km/sec Mpc, the criti-

30 3

crit

39.510g cm









cal density . Assuming

the universe is dominated by vacuum energy resulting

from a cosmological constant , matter accounts for

about 26% of the energy in today’s universe [14]. So, the

matter density





m at redshift z is ρm(z) = (1 + z)3

ρm(0), where today’s matter density is ρm(0) = 0.26 ρcrit =

2.5 × 10−30 g/cm3. Correspondingly, the cosmic micro-

wave background radiation density at redshift z is ρr(z) =

(1 + z)4 ρr(0), where the mass equivalent of today’s

radiation energy density is ρr(0) = 4.4 × 10−34 g/cm3 [15].

When matter dominates, the speed of pressure waves

affecting matter density at redshift z is



 



41 0

cz c









[16], and the Jeans’ length

Lzc z









[16,17]. The first level

of large scale structure within the universe is determined



by the Jeans’ mass



34 m











 



, where

 



101

zzB

Lz zG z





 



, and



22.8910g cm









 

. So the Jeans’ mass

48 0





z

6,z

is independent of [16].

Consider the era at before self-similar large

scale structure developed [2], when each Jeans’ mass was

populated by early stars, with masses in the range 2 ×

1034 g to 2 × 1035 g (10

 to 100

) [18]. Taking

this range as bounds on the lowest mass of early stellar

systems, the holographic radii of the lowest mass early

stellar systems, and thus the central core radii c of the

matter distribution within the Jeans’ masses, was between

3.6 × 1017 cm and 1.1 × 1018 cm. Then, the analysis

above indicates each Jeans’ mass should harbor a central

supermassive black hole with upper bounds on its mass

in the range 2.3 × 1042 g to 7.2 × 1042 g. This estimate is

consistent with observation [19] of the 4 × 1042 g black

hole in the quasar ULAS J112001.48 + 064124.3 at

redshift

7.085z



. If the lowest mass of early stars was

, the upper bound of 4 × 1042 g on the mass of

black holes in the center of each Jeans’ mass equals the

mass observed in ULAS J112001.48 + 064124.3. Later,

T. R. MONGAN 53

at , as self-similar large scale structure develop-

ed, supermassive black holes formed within lower struc-

tural levels, and almost all systems composed of stars or

star aggregations developed central supermassive black

holes.

6z

5. Central Black Hole Develop ment

Development of visible large scale structures within

isothermal spherical halos of dark matter with 2

density distributions resulted in the commonly observed

flat tangential speed distribution of sub-elements. In the

holographic model of large scale structure, black holes

near the center of nascent large scale structures are pro-

genitors of supermassive black holes. Any sub-element

passing within a distance from the central black hole that

is less than the holographic radius of the sub-element is

disrupted and drawn into the central black hole. So, the

mass within a core radius c is drawn into in the cen-

tral black hole. The core radius c is the holographic

radius of the lowest mass sub-element of the large scale

structure because no sub-element can exist as an isolated

system closer to the central black hole than the holo-

graphic radius of the lowest mass sub-element without

being disrupted and drawn into the central black hole. In

consequence, the upper bound on the mass of the central

black hole within in the













density

distribution of the large scale structure is c

. Again,

the upper bound is reached when the central black hole

has accumulated all of the matter within the central

volume inside .

Stars with masses >100

 developed at .

They had very short lives and many of them collapsed to

black holes [20]. It has been claimed that black holes

resulting from collapse of stars in the 100

10z

 range

might not suffice as seeds for supermassive black holes,

so supermassive stars in the 5

 range should be

considered as seeds for supermassive black holes [21].

That scenario is consistent with the holographic model

for large scale structure [2]. When photon decoupling

took place, at , “hydrogen gas was free to col-

lapse under its own self-gravity (and the added gravita-

tional attraction of the dark matter)” [22]. The extended

holographic principle used in the holographic model of

large scale structure [2] indicates the information de-

scribing a structure of mass

1100z

is encoded on a holo-

graphic screen with radius cm

0.16



R, if the Hub-

ble constant 0. Consider the escape

velocity of protons on the holographic screen for a mass

71HkmsecMpc

with radius

R at , and set it equal to the

average velocity of protons in equilibrium with CMB

radiation outside the screen. Then the holographic model

for large scale structure [2] identifies 10

1100z

 as the

mass of systems in thermal equilibrium with the CMB,

since there is no heat transfer between a system with

mass s

M1100z

 and the CMB at . At z =

1100, protons outside the holographic screen with radius

0.16

R

that are in equilibrium with the CMB

cannot transfer heat (and energy) across the holographic

screen surrounding a system with mass s

M

1100z

 at



. The free fall time [23] for systems with mass

M1100z

 with the matter density at



about 2.6 million years, so there is sufficient time for

those systems to ignite as supermassive stars and subse-

quently collapse to seed black holes with masses near

 [24] leading to formation of supermassive black

holes in the 800 million years before emissions as-

sociated with the 4 × 1042 g black hole in the quasar

ULAS J112001.48 + 064124.3 observed at redshift

7.085z



[19]. Anyway, the first stars apparently pro-

duced seed black holes for subsequent development of

large scale structures.

In the earliest phase of development of large scale

structure, at 6z



, there was only one Jeans’ mass struc-

ture level in the holographic model for large scale struc-

ture [2]. These earliest large scale structures, home of the

earliest quasars, then developed around a seed black hole

near their center. Sub-elements of the earliest large scale

structures were early stars with masses in the

 to 10

100

 range, resulting in the estimated mass for super-

massive black holes in the range mentioned above.

As additional self-similar large scale structure levels de-

veloped, remaining seed black holes at the center of each

emerging large scale structure grew by disrupting and

entraining lowest mass sub-elements of the self-similar

large scale structure.

6z

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