Introduction
At the end of the 20th century, the study of the dynamics of the universe by measuring the luminosity distance of SN1A supernovae led to the introduction of a « repulsion » parameter called dark energy. Since the cosmological constant produces this type of effect, it has been considered as a possible solution.
This point has been dealt with in the article, A new paradigm for cosmology : Planck’s force as a constant of gravity.(12/12/21) among other subjects, here we take up this subject only.
The cosmological constant is a geometric parameter that results from an initial condition
From a geometric point of view, it can be considered that it is a parameter resulting from initial conditions.
Indeed, there are 3 classes of spacetime with maximum symmetry (Minkowski with zero curvature, De Sitter with positive constant curvature and Anti-De Sitter with constant negative curvature). These spacetimes, all of which have 10 symmetries, are considered to be the « fundamental » states of the physical spacetimes that matter and energy distort.
We have « arbitrarily » selected spacetime with zero curvature (Minkowski) in the cosmological model but, from a strictly geometric point of view, nothing imposes it as an initial condition, these spacetimes being « equivalent » in symmetry.
All these choices are equal, so selecting De Sitter’s spacetime does not reduce the generality of the problem! This offers a simple solution: « The initial condition is a De Sitter spacetime ». [1]
If this is the case, can we associate this « geometric » property with a physical phenomenology in the universe that characterizes it and that would allow this hypothesis to be given substance?
In this case, the Planck force, the dimensional link between geometric quantities and physical quantities, which is included in Einstein’s equation, should be implicitly involved (see the link cited).
The energy of a vacuum
One solution considered for this physical justification is what is called « vacuum energy » resulting from the creation/annihilation of particle/antiparticle pairs locally.
Reminder: the standard calculation
Usually, we think of the vacuum as « empty » and massless, and we have determined that its density is in any case less than e-30 g/cm3, or about 1 proton per m3, today, which is a very high vacuum, because the universe has expanded a lot.
But in quantum field theory, the vacuum is not really empty because quantum mechanics teaches us that there is a minimum energy that is not zero. It appears to be filled with a bubbling of virtual particles that result from the creation/annihilation of particle-antiparticle pairs, for a time that is all the shorter the greater the energy of these pairs, this being determined by Heisenberg’s indeterminacy relation.
It is estimated that there is a virtual particle of mass m, by volume equal to the cube of λ = h/mc, deduced from the relation E=hc/λ= mc² which defines the wavelength λ of the photon which has the same energy E as this particle, where h is the Planck constant, cthe speed of light and λ is the Compton length associated with Mr. As this volume is very small, it is a lot.
So the supposed density of the vacuum is ρ = m4c3/h3 which is enormous.
In other words, seen from today, the mass of the entire universe was contained in the size of a proton! But let’s remember what we said about the dimensions of objects in this primordial universe, because even reduced to this size it can have great complexity. We should not be impressed by these gigantic figures, but put them in context.
The observed vacuum energy density (e-30 g/cm3) is at least 121 orders of magnitude lower than the crude estimate of quantum mechanics, so there must be a « compensation » mechanism that is obviously very, but not totally, efficient.
If a small residue of the vacuum energy remains, this translates into a cosmological constant, which is a mechanism proposed to be compatible with the constraints imposed by the model with Ωo = 1, which led to an age of the Universe: to = (2/3)/Ho = 10 Ga.
Since the age of the oldest globular clusters is 16 ± 4 Ga, this value of the age of the universe is too small!
The cosmological constant makes it possible to increase the age of the universe to make it compatible with these data.
But this calculation, which assumes that what is true at the local level is true at the level of the global universe by « adding » the supposedly independent local effects, which would be the case in a Newtonian-type space where space is absolute and independent of time, gives a result, for the equivalent « cosmological constant », different from a factor of 10,122 with respect to the measured value of the cosmological constant that produces the acceleration that we observe.
A fundamental argument: Relativity defines a spacetime.
We argue that the previously mentioned calculation does not apply in relativity, where it is a spacetime that is defined, because each event introducing a local perturbation of spacetime, by an interaction other than gravitation, in this case a « quantum » effect, has a global action at the level of the universe (spacetime). It is a consequence of the theory of general relativity defining a spacetime.
This is accompanied by gravitational waves that propagate the disturbance in the universe, (this is not « instantaneous »). If at the local level quantum fluctuations are assumed to be « independent », each one has its own global action on the universe with the associated gravitational waves.
The result on the universe results from the combination of all these independent perturbations, not additively at the energy level, but dependent on the phase parameter of the gravitational waves generated by these fluctuations.
These waves will combine, interfering, in their propagation in spacetime that they cause to « vibrate », and if we make a spectral analysis of this vibration, we know that, in a body of finite size, the maximum amplitude of these waves corresponds to the one whose period is the size of the universe [2].
We can see that we must then refer to the size of the universe to estimate the magnitude of the phenomenon, which gives a (very) different result.
This point was developed in a chapter of the page on Planck’s force as reminded before and, considering it as a constant of physics because it is this dimensional constant that intervenes in Einstein’s equation to ensure its homogeneity, we have shown, by « dimensional » arguments, that this factor of 10122 results, very precisely, from the difference in scale between the Planck scale and the scale of the universe.
Since the gigantic deviation corresponds exactly to the gigantic deviation in scale, this would rehabilitate the hypothesis of the cosmological constant represented by the energy of the vacuum.
Other phenomenology associated with quantum fluctuations
Another point to be taken into account is inspired by the hypothesis considering the phenomenon of exponential expansion of quantum fluctuations by inflation in the primordial universe where quantum fluctuations at the beginning of inflation will be very dilated and give large structures while those towards the end of inflation will be only slight, which produces an almost scale-invariant power spectrum.
We see that the expansion of the universe, especially during inflation when it is exponential, « dissymmetrizes » the phenomenon of quantum fluctuation.
Indeed, unlike a quantum fluctuation in a space without expansion (in spacetime) which leaves a basic state by violating physical laws of conservation, but for a time that is all the shorter the larger the violation, which induces a symmetric phenomenology, during inflation we do not return to the dissymmetry base state, the particle/antiparticle pair is « dissociated ».
This gives a physical character to the geometric phenomenon of space expansion: this expansion is materialized by particles and antiparticles.
Since at this energy level we have a very hot plasma, where the thermal energy is much higher than the mass energy of the particles and antiparticles, they are in thermal equilibrium and can coexist, at least « statistically ».
The exponential expansion continues to amplify the « fluctuation » throughout inflation. This exponential expansion violates Heisenberg’s indeterminacy principle, since particles/antiparticles do not recombine according to this principle.
Once inflation is over, the current rate of evolution of the universe is not violent enough for this violation to last, but this phenomenon of quantum fluctuations persists locally in the same way, with an effect that is repulsive, because this bubbling creation/annihilation, which is interpreted by a cosmological constant, corresponds to an energy that heats the universe and makes it gas, which expands it.
Notes
[1] This solution, for spacetime describing the universe, proposed by De Sitter in 1918, is often considered as a « universe in exponential expansion », on the grounds that if we consider 2 points they move away from each other in the course of time. In fact, it is a Newtonian type of interpretation where time and space are considered as the fundamental physical entities, whereas in relativity only spacetime is physical. De Sitter’s spacetime is modeled by a manifold with a constant, positive, and certainly not expanding spatiotemporal curvature. This expansion is an « internal » property of the variety that can be observed in phenomenology resulting from a particular layering.
[2] This can be compared to the structure of the density wave decomposition of the RFC (CMB).