ABSTRACT

We address the ~122 orders of magnitude discrepancy between the cosmological constant (dark

energy) and the vacuum fluctuation density predicted by quantum field theory and deemed the

so-called “vacuum catastrophe”.

Utilizing a quantized solution to gravity we consider the total mass-energy density in the

geometry of a spherical shell universe (as a first order approximation) and find the result to be

closely correlated with the currently observed critical density of the Universe.

We discuss the validity of such an approach and consider its implications to cosmogenesis and

universal evolution.

THE VACUUM CATASTROPHE AND THE HOLOGRAPHIC MASS

N. HARAMEIN & A. K. F. VAL BAKER

DISCUSSION

In summary we have shown how Haramein’s quantized solution to gravity resolves the 122 orders of

magnitude discrepancy between the mass-energy density at the Planck scale and at the cosmological

scale. Thus, not only resolving this long standing problem in physics but also validating this

geometrical approach.

Theories of quintessence attempt to solve the vacuum catastrophe but offer no physical explanation

and although solutions such as Zlatev[35][36] do not depend on any fine tuning of the initial conditions,

fine-tuning is still required to set the energy density of the scalar field to equal the energy density of

matter and radiation at the present time i.e at the cross-over from matter dominated to scalar field (or

vacuum) dominated. This was the weak point of Hoyle’s[10] steady state universe, as although he was

able to show expansion properties with the introduction of the space-time vector C, no physical

explanation was proposed.

These results have huge implications for astrophysics, cosmogenesis and universal evolution:

Steady state cosmology – previously suggested by Hoyle[10] and Einstein,[37] but instead the mass-

energy density decreases with the increasing size of the Universe, so although the number of particles

in the Universe is increasing with continuous matter creation the energy/information is conserved i.e.

particles passing out of the observable universe are compensated by the creation of new particles

where it is only through the creation of matter that an expanding universe can be consistent with the

conservation of mass within the observable universe.

The structure and dynamics of the Universe, from the quantum to the large scale, is governed by a

simple geometrical relationship.

REFERENCES

1. Reiss, A. G., et al. 1998, Astron. J., Vol. 116, pp. 1009-1038. 20. Capasso, F., Munday, J. N. and Parsegian, V. A. 2009, Nature, Vol. 457, pp. 170-173.

2. Schmidt, B. P., et al. 1998, ApJ, Vol. 507, pp. 46-63. 21. Wilson, C. M., et al. 2011, Nature, Vol. 479, pp. 376-379.

3. Perlmutter, G., et al. 1999, ApJ, Vol. 517, pp. 565-586. 22. Weinberg, S. 1, 1989, Rev. Mod. Phys., Vol. 61.

4. Spergel, D. N., et al. 2003, ApJ. Suppl., Vol. 148, pp. 175-194. 23. Witten, E. 1995, Int. J. Mod. Phys., Vol. 10, pp. 1247-1248.

5. Eisenstein, D. J., et al. 2005, ApJ, Vol. 633, pp. 560-574. 24. Peebles, P. J. E. and Ratra, B. 1988, ApJ, Vol. 325, pp. L17-L20.

6. Hinshaw, G., et al. 2, 2013, ApJ. Suppl., Vol. 208, p. 19H. 25. Ratra, B. and Peebles, P. J. E. 1988, Phys. Rev. D, Vol. 37, p. 3406.

7. Zel'dovich, Y. B. 1968, Sov. Phys. Usp, Vol. 11, pp. 381-393. 26. Wetterich, C. 1988, Nucl. Phys. B, Vol. 302, pp. 668-696.

8. Bludman, S. A. and Ruderman, M. A. 1977, Phys. Rev. Lett., Vol. 38, pp. 255-257. 27. Weinberg, S. 1987, Phys. Rev. Lett., Vol. 59, p. 2607.

9. Carroll, S. M. 1, 2001, Living Reviews in Relativity, Vol. 4. 28. Peebles, P. J. E. 1967, ApJ, Vol. 147, p. 859.

10. Hoyle, F. 1948, MNRAS, Vol. 108, p. 372H. 29. Haramein, N. 4, 2013, Phys. Rev. Res. Int., Vol. 3, pp. 270-292.

11. Guth, A. H. 1981, Phys. Rev., Vol. 23, pp. 347-356. 30. 't Hooft, G. Erice : s.n., 2001. Basics and Highlights in Fundamental Physics.

12. Ade, P. A. R., et al. 2014, A&A, Vol. 571, p. 48. 31. Bekenstein, J. 8, 1973, Phys. Rev. D, Vol. 7, pp. 2333-2346.

13. Sparnaay, M. J. 1958, Physica , Vol. 24, pp. 751-764. 32. Hawking, S. 1975, Commun. math. Phys., Vol. 43, pp. 199-220.

14. Wheeler, J. A. Geometrodynamics. ew York and London : Academic Press, 1962. 33. Wilczek, F. 2001, Physics Today, Vol. 54, pp. 12-13.

15. Sabisky, E. S. and Anderson, C. H. 1973, Phys. Rev. A, Vol. 7, pp. 790-806. 34. Ali, F. A. and Das, S. 2015, Phys. Lett. B, Vol. 741, pp. 276-279.

16. Eberlein, C. 1996, Phys. Rev. Lett., Vol. 76, pp. 3842-3845. 35. Zlatev, I., Wang, L. and Steinhardt, P. J. 1999, Phys. Rev. Lett. , Vol. 82, p. 896.

17. Lamoreaux, S. K. 1997, Phys. Rev. Lett., Vol. 78, pp. 5-8. 36. Zlatev, I., Wang, L. and Steinhardt, P. J. 12, 1999, Phys. Rev. D, Vol. 59.

18. Bordag, M., Mohideen, U. and Mostepanenko, V. M. 2001, Phys. Rep., Vol. 353, pp. 1-205. 37. O'Raifeartaigh, C., et al. 3, 2014, EPJ H, Vol. 39, pp. 353-367.

19. Becka, C. and Mackey, M. C. 2007, Physica A, Vol. 379, pp. 101-110.

INTRODUCTION

The current cosmological model states that we live in a flat, Λ dominated, homogeneous and isotropic

Universe, composed of radiation, baryonic matter and non-baryonic dark matter.[1][2][3][4][5][6]

Assuming the Universe is pervaded by a form of energy (dark energy) that is of constant density in space

and time (i.e. vacuum energy)[7][8][9] then the cosmological constant can be interpreted as an energy

density[10][11] and given in terms of the vacuum density, Λ = 8𝜋𝐺𝜌𝑣𝑎𝑐. Note, this result can also be found

by assuming a static Universe (i.e. 𝑎 = 0).

The Friedman equation thus takes the form:

𝒂

𝒂

𝟐

=𝟖𝝅𝑮

𝟑𝝆𝒃 + 𝝆𝒅 + 𝝆𝒗𝒂𝒄 =

𝟖𝝅𝑮

𝟑𝟎. 𝟎𝟒𝟗𝝆𝒄𝒓𝒊𝒕 + 𝟎. 𝟐𝟔𝟖𝝆𝒄𝒓𝒊𝒕 + 𝟎. 𝟔𝟖𝟑𝝆𝒄𝒓𝒊𝒕 =

𝟖𝝅𝑮

𝟑𝟏 𝝆𝒄𝒓𝒊𝒕

where 𝜌𝑏 is the density due to baryonic matter; 𝜌𝑑 is the density due to dark matter; and 𝜌𝑐𝑟𝑖𝑡~3𝐻𝑜

2

8𝜋𝐺[4][6][12]

Using the current value of 𝐻𝑜 = 67.3 ± 1.4𝑘𝑚𝑠−1𝑀𝑝𝑐−1for Hubble’s constant (12), gives the critical

density at the present time as, 𝝆𝒄𝒓𝒊𝒕 (𝒐) = 𝟖. 𝟓𝟏 × 𝟏𝟎−𝟑𝟎𝒈/𝒄𝒎𝟑 and 𝝆𝒗𝒂𝒄 (𝒐) = 𝟎. 𝟔𝟖𝟑𝝆𝒄𝒓𝒊𝒕 (𝒐) = 𝟓. 𝟖𝟏 ×

𝟏𝟎−𝟑𝟎𝒈/𝒄𝒎𝟑

Particle physicists determine the vacuum energy density by summing the energies ℏ𝜔2 over all oscillatory

modes. However quantum fluctuations predict infinite oscillatory modes[13][14] thus yielding an infinite

result unless renormalized at the Planck cutoff, such that:

𝝆𝒗𝒂𝒄 = 𝝆𝓵 =𝒄𝟓

ℏ𝑮𝟐=

𝒎𝓵

𝓵𝟑= 𝟓. 𝟏𝟔 × 𝟏𝟎𝟗𝟑𝒈/𝒄𝒎𝟑 [15][16][17][18][19][20][21]

A discrepancy of 122 orders of magnitude is found between the cosmological vacuum density and the

Planck vacuum density, with many physicists hoping that an unknown mechanism would set it precisely to

zero and others hoping that such a mechanism would suppress it by just the right amount to equal the

observed value.

POSSIBLE SOLUTIONS

Possible attempts to solve this discrepancy, include

1) A scalar field coupled to gravity → 𝜌𝑣𝑎𝑐 cancels when the scalar field reaches equilibrium.[22]

2) A deep symmetry that constrains parameters such that𝜌𝑣𝑎𝑐 is zero or small.[23]

3) Quintessence which states that the cosmological constant is small because the universe is old and thus imagines

a scalar field that rolls down a potential governed by a field equation.[24][25][26]

4) Anthropic considerations which apply an anthropic bound on +ve 𝜌𝑣𝑎𝑐 by setting the requirement that it should

not be so large as to prevent the formation of galaxies.[27][28]

QUANTIZED SOLUTION TO GRAVITY

In previous work [29], a quantized solution to gravity is given in terms of Planck Spherical

Units (PSUs or ‘voxels’), defined as:

𝑷𝑺𝑼 = 𝑽𝓵𝒔 =𝟒

𝟑𝝅𝒓𝓵

𝟑 = 𝟐. 𝟐𝟏 𝒙 𝟏𝟎−𝟗𝟗 𝒄𝒎𝟑 (where 𝑟𝓁 is the Planck radius)

The Planck density 𝜌𝓁 or quantum vacuum density 𝜌𝑣𝑎𝑐 (𝑙) was subsequently more

appropriately calculated in terms of PSUs: 𝝆𝓵 =𝒎𝓵

𝑷𝑺𝑼= 𝟗. 𝟖𝟔 × 𝟏𝟎𝟗𝟑𝒈/ 𝒄𝒎𝟑

The mass-energy density of any spherical body can therefore be considered in terms of its

PSU packing, and can be defined as a ratio, 𝑅, of any spherical volume, 𝑉 to PSU: 𝑹 =𝑽

𝑷𝑺𝑼,

where the corresponding mass-energy, 𝑅𝜌 can be calculated in terms of Planck mass: 𝑹𝝆 =

𝑹𝒎𝓵

In the case of the proton, 𝑹𝝆 = 𝟐. 𝟒𝟔 𝒙 𝟏𝟎𝟓𝟓 𝒈, which is equivalent to the mass of the

observable universe → indicating that the mass-energy of all protons in the observable

universe could be holographically stored in the mass-energy density of any one proton!

Following the holographic principle of t’hooft,[30] based on the Bekenstein-Hawking

formulae for the entropy of a black hole,[31][32] Haramein[29] defines the holographic bit of

information as the equatorial disc of the oscillating PSU. These PSUs, or Planck voxels,

tile along the area of the proton surface horizon, producing a holographic relationship

with the interior information mass-energy density. The ratio of the information bit to the

spherical surface is thus defined as: 𝜼 =𝑨𝒔

𝑨𝓵𝒄

(where 𝐴𝑠 is the surface area of the spherical body and 𝐴𝓁𝑐 is the area of the PSU

equatorial disc).

It is then shown that the ratio of this interior vacuum energy density to the surface horizon

Planck tiling yields an exact quantized derivation of the Schwarzschild solution to

Einstein’s field equations, and thus a quantized approach to gravity is found, where:

𝒎𝒉 =𝑹

𝜼𝒎𝓵 =

𝒓𝒄𝟐

𝟐𝑮= 𝒎𝒔

The solution applied at the nucleon scale, defines:

𝒎𝒑 = 𝟐𝜼

𝑹𝒎𝓵= 𝟐𝝓𝒎𝓵 = 𝟏. 𝟔𝟕𝟑𝟑𝟒𝟐 × 𝟏𝟎−𝟐𝟒𝒈

(where 𝜙 is the fundamental universal ratio defined as 𝜂/𝑅)

The rest mass of the proton is given as a function of the Planck vacuum oscillators

(PSUs) holographic surface to volume geometric relationship of space time, 𝜙. Thus

resolving the hierarchy problem between the Planck mass and the proton rest-mass!

“We see that the question is poses is not, ’Why is gravity so feeble?’ but rather, ‘Why is

the proton’s mass so small?’ For in natural (Planck) units, the strength of gravity simply

is what it is, a primary quantity, while the protons mass is the tiny number …” Frank

Wilczek, 2001[33]

𝒓𝒑 = 𝟒 𝓵𝒎𝓵

𝒎𝒑= 𝟎. 𝟖𝟒𝟏𝟐𝟑𝟓𝟕 × 𝟏𝟎−𝟏𝟑𝒄𝒎

This quantized solution to gravity successfully predicts the radius of the proton to

within 𝟎. 𝟎𝟎𝟎𝟑𝟔 × 𝟏𝟎−𝟏𝟑𝒄𝒎 of the 2013 muonic charge radius of the proton!

RESOLVING THE VACUUM CATASTROPHE

With this geometrical understanding the change in mass-energy density (by a factor of 10122) can

therefore be determined by expanding the radius of the proton to equal the radius of the observable

universe, 𝑟𝑈, such that the mass-energy density at the cosmological scale is approximately the

critical radius:

𝜌𝑜 =𝑅𝑝𝑚𝓁

𝑉𝑈= 2.25 𝑥 10−30 𝑔/𝑐𝑚3~ 𝜌𝑐𝑟𝑖𝑡

where 𝑉𝑈 = 1.09 × 1085𝑐𝑚3 and was found by taking 𝑟𝑈 as the Hubble radius,

𝑟𝐻 =𝑐

𝐻𝑜= 1.37 × 1028𝑐𝑚.

This solution is in line with the ideas of quintessence in which the mass-energy density is governed

by the scale factor 1

𝜂𝜑, such that

𝝆𝝋 =𝝆𝓵

𝜼𝝋, for 𝜼𝝋 > 𝜼𝓵 or 𝝆𝝋 =

𝝆𝓵

𝟒

𝒓𝓵

𝒓𝝋

𝟐

, for 𝒓𝝋 > 𝒓𝓵

The Friedman equation can then be written in the form:

𝑯𝝋𝟐 =

𝟖𝝅𝑮

𝟑𝝆𝝋 =

𝟖𝝅𝑮

𝟑

𝝆𝓵𝜼𝝋

=𝟖𝝅𝑮

𝟑

𝝆𝓵𝟒

𝒓𝓵𝒓𝝋

𝟐

=𝟐𝝅𝑮

𝟑𝝆𝓵

𝒓𝓵𝒓𝝋

𝟐

These findings are in agreement with those of Ali and Das[34] who include the correction term

Λ𝑄 =𝑟𝓁2

𝐿02 (where 𝐿0 is identified as the current linear dimension of our observable Universe).

However, their solution describes a purely quantum mechanical description of the universe, assuming

quantum gravity affects are practically absent, whereas the results described here show how the density

changes with radius where we have a scaler field that is coupled to gravity and thus rolls down a

potential governed by Haramein’s[29] quantized solution to gravity.