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We address the ~122 orders of magnitude discrepancy between the vacuum energy density at the cosmological scale and the vacuum density predicted by quantum field theory. This disagreement is known as the cosmological constant problem or the “vacuum catastrophe”. Utilizing a generalized holographic model, we consider the total mass-energy density in the geometry of a spherical shell universe (as a first order approximation) and find an exact solution for 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 energy density predicted by quantum field theory disagrees with cosmological observation by approximately 122 orders of magnitude. It is one of the biggest disagreements between theory, experiment and observation and is known as the vacuum catastrophe [

The Einstein field equations of general relativity include a constant Λ known as the cosmological constant. Originally included to allow for static homogenous solutions to Einstein’s equations, it was subsequently removed when the expansion of the universe was discovered [

With the inclusion of the cosmological constant, Einstein’s field equations are:

R μ ν − 1 2 R g μ ν + Λ g μ ν = 8 π G c 4 T μ ν (1)

where R μ ν is the Ricci curvature tensor, g μ ν is the metric tensor, r is the scalar curvature and T μ ν is the stress-energy tensor, which is modeled as a perfect fluid such that:

T μ ν = ( ρ + P / c 2 ) U μ U ν + P g μ ν (2)

The Robertson-Walker solution, which states that the rest frame of the fluid must be the same as the co-moving observer, reduces the Einstein equations to two Friedman equations:

( a ˙ a ) 2 = 8 π G ρ 3 + Λ 3 − c 2 k a 2 R o 2 (3)

a ¨ a = − 4 π G 3 ( ρ + 3 P / c 2 ) (4)

where a is the scaling factor, k is the curvature constant and R o is the radius of the observable universe (i.e. R t = a t r , where r is the co-moving radius).

Based on astronomical observations 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 [

The Friedman equation for a flat universe (i.e. k = 0 ) is thus given in the form:

H 2 = ( a ˙ a ) 2 = 8 π G ρ 3 + Λ 3 (5)

If we then take the assumption that the universe is pervaded by a form of energy (i.e. dark energy), which is the current consensus in both cosmology and particle physics [

In either case the Friedman equation thus takes the form:

H 2 = ( a ˙ a ) 2 = 8 π G 3 ( ρ + ρ Λ ) (6)

Friedman’s solutions suggest that there is a critical density at which the universe must be flat, where the ratio of the total mass-energy density to the critical density is known as the density parameter Ω = ρ ρ c r i t and is currently measured as Ω ∼ 1 [

The contributions to this density parameter come from: the vacuum density (dark energy), Ω Λ = 0.683 ; the dark matter, Ω d = 0.268 ; and the baryonic matter, Ω b = 0.049 , totaling to Ω T = 1 [

The Friedman equation thus takes the form of an Einstein-de Sitter model in which the cosmological constant is coupled to the density:

( a ˙ a ) 2 = 8 π G 3 ( ρ b + ρ d + ρ Λ ) = 8 π G 3 ( 0.049 ρ c r i t + 0.268 ρ c r i t + 0.683 ρ c r i t ) = 8 π G 3 ρ c r i t (7)

where ρ b is the density due to baryonic matter; ρ d is the density due to dark matter; ρ Λ is the density due to dark energy; and ρ c r i t = 3 H o 2 8 π G .

Using the current value of H o = 67.4 ± 0.5 km ⋅ s − 1 ⋅ Mpc − 1 for Hubble’s constant [

However, quantum field theory determines the vacuum energy density by summing the energies ℏ ω / 2 over all oscillatory modes. See reference [

ρ v a c = c 5 ℏ G 2 = m l l 3 = 5.16 × 10 93 g / cm 3 (8)

where m l = 2.18 × 10 − 5 g is the Planck mass and l = 1.616 × 10 − 33 cm is the Planck length. This value is well supported by both theory and experimental results [

The cosmological vacuum energy density determined from observations, ρ v a c = 5.83 × 10 − 30 g / cm 3 , is therefore in disagreement with the vacuum energy density at the Planck cutoff, predicted by quantum field theory, ρ v a c = 5.16 × 10 93 g / cm 3 . This discrepancy is a significant 122 orders of magnitude and is thus known as the “vacuum catastrophe”.

Possible attempts to solve this discrepancy, as reviewed by Weinberg [

Finally, anthropic considerations apply an anthropic bound on +ve ρ v a c by setting the requirement that it should not be so large as to prevent the formation of galaxies [

In previous work [

Following the holographic principle of ‘t Hooft [

P S U = 4 3 π r l 3 (9)

where r l = l 2 .

These PSUs, or Planck “voxels”, tile along the area of a spherical surface horizon, producing a holographic relationship with the interior information mass-energy density (see

In this generalized holographic approach, it is therefore suggested that the information/entropy of a spherical surface horizon should be calculated in spherical bits and thus defines the surface information/entropy in terms of PSUs, such that,

η = A π r l 2 (10)

where the Planck area, taken as one unit of information/entropy, is the equatorial disk of a Planck spherical unit, π r l 2 and A is the surface area of a spherical system. We note that in this definition, the entropy is slightly greater (~ 5 times) than that set by the Bekenstein bound, and the proportionality constant is taken to be unity (instead of 1/4 as in the Bekenstein-hawking entropy). It has been previously suggested that the quantum entropy of a black hole may not exactly equal A/4 [

As first proposed by ‘t Hooft the holographic principle states that the description of a Volume of space can be encoded on its surface boundary, with one discrete degree of freedom per Planck area, which can be described as Boolean variables evolving with time [

Following the definition for surface information η , the information/entropy within a volume of space is similarly defined in terms of PSU as,

R = V 4 3 π r l 3 = r 3 r l 3 (11)

where V is the volume of the spherical entity and r is its radius.

In previous work [

m S = R η m l (12)

where η is the number of PSU on the spherical surface horizon and R is the number of PSU within the spherical volume. Hence, a holographic gravitational mass equivalence to the Schwarzschild solution is obtained in terms of a discrete granular structure of spacetime at the Planck scale, giving a quantized solution to gravity in terms of Planck spherical units (PSUs). It should be noted that this view of the interior structure of the black hole in terms of PSUs, is supported by the concept of black hole molecules and their relevant number densities as proposed by Miao and Xu [

Of course, these considerations lead to the exploration of the clustering of the structure of spacetime at the nucleonic scale, where it was found that a precise value for the mass m p and charge radius r p of a proton can be given as,

m p = 2 η R m l = 2 ϕ m l (13)

r p = 4 l m l m p = 0.841236 ( 28 ) × 10 − 13 cm (14)

where ϕ = η R is defined as a fundamental holographic ratio. Significantly, this value is within an 1 σ agreement with the latest muonic measurements of the charge radius of the proton [

To resolve the vacuum catastrophe, we must first understand where the value for the vacuum energy density at the Planck scale is coming from. As was previously defined [

ρ l = m l P S U = 9.86 × 10 93 g / cm 3 .

The vacuum energy density at the quantum scale is thus ρ l = 9.86 × 10 93 g / cm 3 instead of the value ρ v a c = 5.16 × 10 93 g / cm 3 given in Equation (8).

The generalized holographic model describes how any spherical body can be considered in terms of its PSU packing, or volume entropy, R. The mass-energy M R , in terms of PSU, can therefore be given as M R = R m l and the mass-energy density is given as, ρ R = M R V .

In the case of the proton, the mass-energy in terms of Planck mass was calculated as M R = R m l = 2.45 × 10 55 g , which is equivalent to the mass of the observable universe (i.e. M u = 136 × 2 256 × m p = N E d d m p = 2.63 × 10 55 g in terms of the Eddington number; and M u ≈ 3.63 × 10 55 g from density measurements). Since these values for the mass of the observable universe are just approximations, we will take the mass of the observable universe to be the mass-energy of the proton, as calculated above. The mass-energy density of the universe can thus be defined in terms of the mass-energy density of the proton. Thus, at the cosmological scale the mass-energy density, or vacuum energy density, is calculated to be,

ρ u = ρ R = M R V U = R m l V U = 2.26 × 10 − 30 g / cm 3 = 0.265 ρ c r i t (15)

where V U = 1.08 × 10 85 cm 3 and was found by taking r U as the Hubble radius r H = c / H o = 1.37 × 10 28 cm . Thus, when the vacuum energy density of the Universe is considered in terms of the proton density and the protons PSU packing (i.e. its volume entropy, R) we find the density scales by a factor of 10^{122}. As well, it should be noted that this value for the mass-energy density is found to be equivalent to the dark matter density, ρ d = 0.268 ρ c r i t .

Similarly, the vacuum energy density can be considered in terms of the PSU surface tiling (i.e. its surface entropy, η ), as the radius expands from the Planck scale ρ l to the cosmological scale. The vacuum density at the cosmological scale is thus given as,

ρ u = ρ l η = 8.53 × 10 − 30 g / cm 3 ( = ρ c r i t ) (16)

where η is found by assuming a spherical shell Universe of radius r U = r H . The resulting change in density, from the vacuum density at the Planck scale to that at the cosmological scale yields an exact equivalent to the currently observed critical density of the universe, ρ c r i t . Thus, when we consider the generalized holographic approach, which describes how any spherical body can be considered in terms of its PSU packing, we show the scale relationship between the PSUs and a spherical shell universe and resolve the 122 orders of magnitude discrepancy between the vacuum energy density at the Planck scale and the vacuum energy density at the cosmological scale.

The solution presented here is in line with the ideas of quintessence in which the mass-energy density is governed by the scale factor η φ − 1 , such that ρ φ = ρ l η φ for η φ > η l . Following this approach, the Friedman equation can then be written in the form:

H φ 2 = 8 π G 3 ρ φ = 8 π G 3 ρ l η φ (17)

which can also be given in terms of the varying radius, such that ρ φ = ρ l 4 ( r l r φ ) 2 for r φ > r l and the Friedman equation becomes:

H φ 2 = 8 π G 3 ρ φ = 8 π G 3 ρ l 4 ( r l r φ ) 2 = 2 π G 3 ρ l ( r l r φ ) 2 (18)

These findings are in agreement with those of Ali and Das [

Essentially, they are adding the correction term Λ Q = r l 2 / L 0 2 whereas we include the scale factor r l 2 / r φ 2 . 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, as the density changes with radius we have a scaler field that is coupled to gravity and thus rolls down a potential governed by a generalized holographic quantized solution to gravity [

Similar scale-invariant models have also been proposed by Maeder [

It should as well be noted that the equivalence found between the critical density and that found from the surface entropy (Equation (16)) yields a critical mass that obeys the Schwarzschild solution for a universe with a radius of the Hubble radius,

M c r i t = ρ l η V u = m l ϕ = 9.24 × 10 55 g ( ≡ r s c 2 2 G ) (19)

The idea that the observable universe is the interior of a black hole was originally put forward by Pathria [

Previous attempts to resolve the vacuum catastrophe include large quantum corrections (e.g. [

The solution described in this paper utilizes the generalized holographic approach [

Similarly, Huang [

The nature of the fundamental constants and the large dimensionless numbers resulting from their relationships has been a long-standing puzzle (e.g. [

The standard model of the universe (i.e. concordance ΛCDM) explains the accelerated expansion of the universe in terms of a negative pressure generated by the so-called dark energy. However, although in good agreement with CMB, large scale structure and SNeIa data, it is not yet able to explain the coincidence (fine-tuning) or the cosmological problem. As noted by Corda (2009) [

In summary we have shown how the generalized holographic model resolves the 122 orders of magnitude discrepancy between the vacuum energy density at the Planck scale and the vacuum energy density at the cosmological scale. Thus, not only resolving this long-standing problem in physics but also validating this geometrical approach. The details in terms of matter creation and the expansion rate are beyond the scope of this paper and will be addressed in a forth coming paper. The results presented here have profound implications for astrophysics, cosmogenesis, universal evolution and quantum cosmology giving incentive to further exploration and developments.

The authors would like to thank Dr. Elizabeth Rauscher, Dr. Michael Hyson, Professor Bernard Carr and Dr. Ines Urdaneta for their helpful notes and discussions, Marshall Lefferts and Andy Day for the use of their diagram (

The authors declare no conflicts of interest regarding the publication of this paper.

Haramein, N. and Val Baker, A. (2019) Resolving the Vacuum Catastrophe: A Generalized Holographic Approach. Journal of High Energy Physics, Gravitation and Cosmology, 5, 412-424. https://doi.org/10.4236/jhepgc.2019.52023