Espen Gaarder
email: espenhaug@mac.com

Stéphane Wojnow
email: wojnow.stephane@gmail.com

Investigation of the Casimir E↵ect in the Hubble Universe and Black Holes

Keywords: Hubble sphere, Casimir e↵ect, luminosity, radiation pressure, quantum cosmology, vacuum energy

The Casimir e↵ect can either work over very short distances or at very low temperatures. At the cosmic scale, we naturally deal with long distances, but at the same time, we encounter extraordinarily low temperatures, namely the CMB (Cosmic Microwave Background) temperature. Recently, there has been increased interest in the Casimir e↵ect and its implications in cosmology. Here, we briefly demonstrate that the luminosity and radiation pressure of the CMB is mathematical identical to a theoretical Casimir e↵ect within the Hubble sphere, or actually our solution seems to be valid for any Schwarzschild black hole. The Casimir e↵ect for every Schwarzschild black hole is:

N . This also include the Hubble sphere if we treat it as a Schwarzschild black hole that is related to increased interest in black hole cosmology.

The Hubble-Casimir E↵ect and its Relation to the CMB luminosity

The total Casimir pressure e↵ect is described by the formula (as seen in [START_REF] Casimir | On the attraction between two perfectly conducting plates[END_REF] and elaborated upon by Balian and Duplantier [START_REF] Balian | Geometry of the casimir e↵ect[END_REF]):

P ca = 1 A ( X0 + XT ) = ⇡ 2 240 ~c D 4 ⇡ 2 45 1 1 ( ~c) 3 + 1 ⇡ D 3 e ↵ + • • • (1)
Here, = 1 k b T , where k b is the Boltzmann constant, T is the temperature, D is the distance between the two plates, and ↵ = ⇡ ~c D . Typically, when one thinks of the Casimir 1 e↵ect, it is associated with plates placed at extremely short distances. However, as noted by Balian, Duplantier, and others, the Casimir pressure e↵ect is valid at very short distances or, conversely, at very low temperatures. The longer the distance over which the Casimir e↵ect operates, the lower the temperature must be.

The second term in the equation above represents the black body pressure in what is known as an infinite geometry.

P ca = ⇡ 2 45 1 1 ( ~c) 3 (2)
where again = 1 k b T . Investigating the Casimir e↵ect in relation to cosmology is not very common. However, there are multiple recent papers on the Casimir e↵ect in relation to cosmology, as seen in Leonhardt [START_REF] Leonhardt | The case for a Casimir cosmology[END_REF][START_REF] Leonhardt | Casimir cosmology[END_REF]. One reason is that the Casimir e↵ect is typically related to vacuum, and most of the universe is a vacuum with very low energy, the cosmological vacuum energy.

In cosmology, we certainly do not work on short distances; rather, we focus on very long distances and over very large surfaces ares (in the Hubble sphere). However, we work with extraordinarily low temperatures, specifically the Cosmic Microwave Background (CMB) temperature, which is likely as close as one can get to absolute zero in any observation. The CMB temperature has been measured more accurately and is one of the most precise factors in cosmology. It is approximately 2.72K, as seen in [START_REF] Fixsen | The temperature of the cosmic microwave background at 10 GHz[END_REF][START_REF] Fixsen | The temperature of the cosmic microwave bacground[END_REF][START_REF] Noterdaeme | The evolution of the cosmic microwave background temperature[END_REF][START_REF] Dhal | Calculation of cosmic microwave background radiation parameters using cobe/firas dataset[END_REF]. Whether one can really study the Casimir e↵ect at distances equal to the Hubble radius with only the CMB temperature is a question we leave for further study, but we will assume so here.

Let us now consider the Hubble radius, denoted as R H = c H 0 , for the distance between the plates and the CMB temperature T cmb ⇡ 2.72K. This gives:

P ca = dF c dS = 1 A ( X0 + XT ) = ⇡ 2 240 ~c R 4 H ⇡ 2 45 1 1 ( ~c) 3 + 1 ⇡ R 3 H e ↵ ⇡ 1.39⇥10 14 J •m 3 (3)
where

↵ = ⇡ ~c R H and = 1 k b T cmb .
We have used a Hubble parameter given by the recent study of Kelly and et. al [START_REF] Kelly | Constraints on the Hubble constant from supernova Refsdal's reappearance[END_REF]: H 0 = 66.6 +4. 1 3.3 (km/s)/M pc. Further the second term, the black body radiation pressure at infinite geometry is equal to:

P ca = ⇡ 2 45 1 1 ( ~c) 3 ⇡ 1.39 ⇥ 10 14 J • m 3 (4)
we mention this as it is evident the second term is the main contributor when dealing with the Hubble scale. If we multiply P ca with the surface area of the Hubble sphere we get the Casimier force of

F ca,H = P ca A H = 1.39 ⇥ 10 14 ⇥ 4 3 ⇡R 2 H ⇡ 3.34 ⇥ 10 39
N . Next, we will analyze the radiation pressure related to CMB luminosity within the Hubble sphere. Based on the Stefan-Boltzmann [START_REF] Boltzmann | Ableitung des stefanschen gesetzes, betre↵end die abhängigkeit der wärmestrahlung von der temperatur aus der electromagnetischen lichttheori[END_REF][START_REF] Stefan | Über die beziehung zwischen der wärmestrahlung und der temperatur[END_REF] law, we have a theoretical blackbody luminosity derived from the CMB temperature:

L cmb = 4⇡R 2 H T 4 cmb = 4⇡ ✓ c H 0 ◆ 2 ⇥ 2.72 4 ⇡ 7.5 ⇥ 10 47 W (5)
Alternatively, this can be predicted using a new equation recently published by Haug and Wojnow [START_REF] Haug | The blackbody CMB temperature, luminosity, and their relation to black hole cosmology, compared to Bekenstein-Hawking luminosity[END_REF], where they have derived the CMB luminosity from the Stefan-Boltzmann law, resulting in:

L cmb = ~c6 15360⇡G 2 m 2 p ⇡ 7.50338 ⇥ 10 47 W ( 6
)
The formula is valid for any Schwarzschild black hole size [START_REF] Schwarzschild | über das gravitationsfeld einer kugel aus inkompressibler flussigkeit nach der Einsteinschen theorie[END_REF] and is remarkably independent of the mass of the black hole, unlike the Bekenstein-Hawking luminosity [START_REF] Hawking | Black hole explosions[END_REF], which is massdependent. Based on the Stefan-Boltzmann law, the radiation pressure is given by:

P rad = L 4⇡R 2 c (7)
For a Hubble sphere, when we input the CMB luminosity as L cmb , Haug and Wojnow [START_REF] Haug | The blackbody CMB temperature, luminosity, and their relation to black hole cosmology, compared to Bekenstein-Hawking luminosity[END_REF] have demonstrated that this leads to:

P cmb = ~c 46080⇡ 2 R 2 s l 2 p = ~c 46080⇡ 2 R 2 H l 2 p (8)
Here, we have used the Hubble radius as the Schwarzschild radius, which is permissible when treating the Hubble sphere as a Schwarzschild black hole. The idea of the Hubble sphere as a black hole was likely first introduced in 1972 by Pathria [START_REF] Pathria | The universe as a black hole[END_REF], but is an actively discussed alternative view on cosmology to this day, see for example [START_REF] Stuckey | The observable universe inside a black hole[END_REF][START_REF] Pop | The universe in a black hole in Einstein-Cartan gravity[END_REF][START_REF] Tatum | A Heuristic Model of the Evolving Universe Inspired by Hawking and Penrose in New Ideas Concerning Black Holes and the Universe[END_REF][START_REF] Akhavan | The universe creation by electron quantum black holes[END_REF][START_REF] Lineweaver | All objects and some questions[END_REF][START_REF] Haug | New exact solution to Einsteins field equation gives a new cosmological model[END_REF][START_REF] Wojnow | Alternative cosmology: ⇤CDM-like predictions today[END_REF], this could also be highly relevant for the wider class of R h = ct cosmology models, see for example [START_REF] Melia | The r h = ct universe[END_REF][START_REF] Melia | The r h = ct universe without inflation[END_REF][START_REF] Melia | The linear growth of structure in the r h = ct universe[END_REF][START_REF] Tatum | How a realistic linear r h = ct model of cosmology could present the illusion of late cosmic acceleration[END_REF][START_REF] John | r h = ct and the eternal coasting cosmological model[END_REF][START_REF] John | Generalized chen-wu type cosmological model[END_REF] .

This results in a radiation pressure of:

P cmb = ~c 46080⇡ 2 R 2 H l 2 p ⇡ 1.39 ⇥ 10 14 J • m 3 (9)
where R H = c H 0 . Notably, this is identical to the Casimir pressure calculated previously (Equation 3 and 4). When multiplied by the area of the Hubble surface, we obtain:

P cmb A H = ~c 46080⇡ 2 R H l 2 p 4⇡R 2 H = ~c 11520⇡l 2 p = c 4 G 1 11520⇡ ⇡ 3.34 ⇥ 10 39 N. (10)
The Planck gravitational force is given by

F N,p = G m p m p l 2 p = ~c l 2 p ( 11
)
which is also identical to the Coulomb [START_REF] Coulomb | Premier mémoire sur l'électricité et le magnétisme[END_REF] force for two Planck charges, as we have

F C,p = k e q p q p l 2 p = ~c l 2 p ( 12
)
where q p is the Planck charge which again is equal to q p = e p ↵ , where e is the elementary charge and ↵ is the fine structure constant. This means we also can re-write equation 10 as

P cmb A H = ~c 11520⇡l 2 p = F N,p 1 11520⇡ = F C,p 1 11520⇡ . ( 13
)
This value remains constant and is the same for every Schwarzschild black hole. Consequently, we have:

P cmb A H = F ca,H = G m 2 p l 2 p 1 11520⇡ = k e q 2 p l 2 p 1 11520⇡ = ~c 11520⇡l 2 p ⇡ 3.34 ⇥ 10 39 N ( 14
)
where F ca,H stands for the Casimier force acting on the Hubble sphere. The change in the surface area can be seen as from Planck mass micro black hole to the Hubble sphere, this is consistent with growing black hole cosmology models of the R H = ct type, and likely also a steady state black hole that can be derived from the Haug-Spavieri metric, something we have to come back to in a future paper.

This implies that there could indeed be a Casimir e↵ect within the Hubble sphere and for any Schwarzschild black hole, as the value remains constant at approximately 3.34 ⇥ 10 39

N . All of this should also be seen and investigated further in connection with the recent rapid theoretical progress in understanding the CMB temperature. Tatum et al. [START_REF] Tatum | The basics of flat space cosmology[END_REF] already heuristically suggested the formula to predict the CMB temperature as follows:

T cmb = ~c3 k b 8⇡G p Mm p = ~c3 k b 4⇡G p R s 2l p (15)
The formula was first formally mathematically proven by Haug and Wojnow [START_REF] Haug | How to predict the temperature of the CMB directly using the Hubble parameter and the planck scale using the Stefan-Boltzman law[END_REF] by deriving the CMB temperature from the Stefan-Boltzmann law. Haug and Tatum [START_REF] Haug | The hawking Hubble temperature as a minimum temperature, the Planck temperature as a maximum temperature and the CMB temperature as their geometric mean temperature[END_REF] demonstrated that the CMB temperature formula above could be derived from a more general geometric mean approach, and Haug [START_REF] Haug | Planck quantized bending of light leads to the CMB temperature[END_REF] also showed that the CMB temperature can be derived based on the quantized minimum bending of light at the horizon of a black hole. All in all, it seems that more and more pieces of what we can call the Hubble puzzle are falling into place, and all this can be applied to any Schwarzschild black hole, and can likely relatively easily be extended to Reisner-Nordström [START_REF] Reissner | Über die eigengravitation des elektrischen feldes nach der Einsteinschen theorie[END_REF][START_REF] Nordström | On the energy of the gravitation field in Einstein's theory[END_REF], Kerr [START_REF] Kerr | Gravitational field of a spinning mass as an example of algebraically special metrics[END_REF], Kerr-Newman [START_REF] Newman | Note on the kerr spinning-particle metric[END_REF][START_REF] Newman | Metric of a rotating, charged mass[END_REF] and Haug-Spavieri [START_REF] Haug | Mass-charge metric in curved spacetime[END_REF] black holes also.

Possible interpretation

We have found that the Casimir force for the Hubble sphere is given by

F ca ⇡ G m 2 p l 2 p 1 11520⇡ = k e q 2 p l 2 p 1 11520⇡ ⇡ 3.34 ⇥ 10 39
N , which is close to the Planck gravitational force and also the Coulomb force between two Planck charges over the distance of the Planck length. This is an enormous force. However, we think this can be seen as an aggregated Casimir force coming from the entire Hubble surface A = 4⇡R 2 H , so the Casimir pressure is identical to the CMB pressure of only

P ca = P cmb = ~c 46080⇡ 2 R 2 H l 2 p = Fca A H = J •m 3 = N •m 2 ⇡ 1.39⇥10 14 N •m 2
, which is very low. So even if the Casimir force is very strong for the Hubble sphere, it is divided over a large surface area, so the radiation pressure is very low.

As our formulas end up having Planck units in them, this indicates a strong connection between gravity and Planck units. Einstein [START_REF] Einstein | Näherungsweise integration der feldgleichungen der gravitation[END_REF] already in 1916 suggested that the next step in the development of gravity theory would be a quantum gravity theory. Already in 1918, Eddington [START_REF] Eddington | Report On The Relativity Theory Of Gravitation[END_REF] suggested that such a theory had to be linked to the Planck length (Planck units). Quantum gravity is an actively researched area to this day, and perhaps even our findings of the Casimir e↵ect for the Hubble sphere can one day be incorporated into such theories, but that is outside the scope of this paper.

Another interesting question is how the Casimir force can be identical to the gravitational force and the Coulomb force, as we have demonstrated in the derivations above. It is important to be aware that this is naturally only applicable to black holes. One possible interpretation is that in black holes, at least close to the singularity, mass is so densely packed together that everything is possible. Therefore, gravity and electromagnetism, along with all other forces, become united into one force, as predicted by multiple researchers. However, this should naturally be further investigated and, at this point, is at best only a hypothesis. Still, it's a hypothesis that may gain strength with this paper.

Conclusion

We have tried to demonstrate that radiation pressure, derived from CMB luminosity, which is again linked to CMB temperature, exhibits close similarities to the Casimir force. We have named this phenomenon the Casimir-Hubble e↵ect. Alternatively, it could be referred to as the black hole Casimir e↵ect, as it should apply to all black holes. We have shown that for any Schwarzschild black hole we have: N . That is in a black hole the Casimier force is equal to the gravity force for two Planck mass particles or the Coulomb force of two Planck charges, after we multiply these by 1 11520⇡ .

F ca = G

Declarations

Conflict of interest

The authors declares no conflict of interest.