A variable metric universe model

*basic field* \(\Xi\) of the universe is introduced. It is called *Space-Time Field* because it builds both the Space-Time and the particles in it, using an original formulation of quantum oscillator (IQuO). This field is fed with energy flowing from a background (Θ) having the structure of no-field because it is composed by no-coupled IQuO. Note that the mass creation has a double consequence: gravity increase (curvature) and space increase (expansion). This allows to formulate a universe model with a variable metric (open in the past, flat in the present and closed in the future) and to explain some fundamental aspects of the universe: the Hubble’s law by creating the mass-space, the acceleration of galaxies as effect of a pressure of increasing Space-Mass \(\Lambda\), the age of the universe as time needed to reach the flat metric phase.

In modern cosmology the expansion of space-time and its zero curvature are not derived from any fundamental principle of physics, but they are deﬁned as empirical properties of space and time in the universe.

In this study we will try to prove that the two physical realities are attributable to a single source in which particles and space-time (ST) are closely related. Then we formulate the hypothesis of the existence of a fundamental field on which the universe, deﬁned like a space-time with fields-particles, is built. In order to support this idea we begin (\ref{spacetimefield}) highlighting the strong connection between the fields-particles and *empirical concepts* of space and time, defined in any reference system, which induces us to use it as a *field-object*.

Some physical aspects as:

the invariance of the light speed \(c\), which value is set as a constant of structure for the spacetime;

the relevant role of massive objects building a reference systems and defining their curvature;

the universe expansion like stretching of a object-space;

lead us to conjecture the existence of a *ST-Field*.

As a first step for better the hypothesis of ST field we must introduce (\ref{iquo}) a new idea of quantum oscillator: the Intrinsic Quantum Oscillator (IQuO), composed by two sub-oscillators each endowed of two half-quanta.

One builds thus a scalar field \(\Xi\) made by lattice of 1-dim. chains of IQuO reciprocally couplings.

Besides, the structure of an IQuO in vacuum eigenstate (one sub-oscillator with vacuum energy), leads us to admit the existence of a quantum vacuum that it is different than the standard vacuum: a set of IQuO elastically uncoupled without the structure of a field. We define this system ad *quantum vacuum of no-field*, and called it \(\Theta\).

In (\ref{massive}), following the general relativity, where the mass takes the role of agent characteristic (gravitational charge, \(\Gamma\)), and considering the hypothesis of a connatural structure between agent (massive particles) and gravitational field, we can derive that the mass of a particle must be expressed by fundamental property of the ST field-object \(\Xi\).

Then we conjecture that the mass is given by a particular *transversal additional coupling* (called massive coupling) between the oscillator chains of the field \(\Xi^0\).

This additional coupling builds a field with lattice structure \(\Xi\) where is given a *time step* and a *space step*, and which gives empirical sense to reference frame associated with a massive particle with Compton wave length \(\lambda_c\).

In this way, the mass becomes an essential property to deﬁne the spacetime in the universe, which we will deﬁne (\ref{cosmos}) as *Universe Space Time Field*, where each particle could be identiﬁed by a distinct and articulated structure of the IQuOs of the field.

We denote by \(\Xi_U\) the USTF having as components the base fields (\(\Xi^0_i, \Xi_i\)).

Moreover it becomes possible to assign to a set of identical massive particles, with mass \(m_i\), a ﬁeld with a lattice structure which we will call *lattice-Universe*, denoted by \(U_i\). The universe \(U\) will thus be expressed as a set of lattices \(U_i\) associated to the respective basic massive particles \(m_i\).

In (\ref{cosmos}) we underline the equivalence between the relativities and cosmological principles. Combining these hypothesis we obtain a universe in which the *apparent* relative motion of each reference system must be *radial* compared to any observator (isotropy) and this is compatible with an expansion or a future contraction of the universe. The fact that we are in a phase of expansion and not of contraction could not be derived from the relativities principles, but from a particular condition of the physical state of the universe that derives from a property of the spacetime field.

Because the space implies the existence of a space step, \(\lambda_C\), it follows that the expansion is amenable to the mass, and in the same way the flow of the cosmic time is amenable to the time step.

Then, we note that when a massive particle is created with its spacetime lattice, we are forced to *add* space and time in universe \(\Xi_U\) because every massive particle is itself space and time.

Also, we will show that the origin of the expansion of the universe is to be found in a sort of *creation of space* following the appearance of massive particles in the field \(\Xi_U\).

We can say the same thing about the cosmic time.

We note that if the universe expansion is detected to be a mutual spacing between galaxies, then saying that this is caused by an increasing *quantity* of space interposed between them, could be equally acceptable to consider the idea of the expansion as a *stretching* of the space.

Rather we show that the expansion of the universe cannot be a consequence of any *stretching* of a certain spatial length \(L\), which can be described by the parameter \(a(t)\) (Friedmann 1922, Friedmann 1924).

In (\ref{hubble_law}) it is shown that the Hubble law can be derived precisely from the *creation of space* resulting by the birth of massive particles in \(\Xi\).

The mechanism of IQuO creation and duplication, with resulting the creation of space and mass, is subject to the presence of a *background* energy composed by uncoupled sub-oscillators (like atoms of a gas). This background conicides with \(\Theta\), the no-field. We show that only a particular way of space growth, determines the Hubble law caused by energy coming from the background \(\Theta\).

See the universe expansion as increasing space (\ref{h_function}) it is equivalent to reaffirm that the expansion is the expression of a particular property of the space (indicated with \(P_{exp}\)). We remind that gravity is generated by the spacetime curvature, property denoted by us with \(P_{grv}\). This distinction denies that gravity can influence the expansion and vice versa.

Thus it can be argued that the velocity of a galaxy, experimentally revealed as \(V_{gal}\), consists of a *speed of gravitational falling* \(v_{fall}\) composed with a recession velocity \(v_{exp}\) caused by the expansion^{1}.

So, the *new* Hubble law will be annotated with \(v = H_U d\), where \(H_U\) is the Hubble’s function in presence of gravitation, which it is different from the standard \(H_H\), function of the linear Hubble law.

Distinguishing between different forms of the Hubble function (\(H_U\), \(H_0\), \(H_H\)), we expect (using a correct distance indicator, for example the (Ia) supernovae) different observational results concerning the galactic red shift (Perlmutter 2003).

Let us note that the three cases (\(H_{U^-}\), \(H_{U^0}\), \(H_{U^+}\)) correspond to the three solutions of Friedmann’s equation, that always include the gravity. However, the experimental data seem to admit a \(v^*_{rev}\) greater than the recession velocity for an open universe. In particular, it seems to be a further velocity \(v_{ext}\) obtained by an acceleration opposite to the gravity (Perlmutter 2003).

In standard cosmology the \(v_\text{ext}\) is consequence of n repulsive action generated by a dark energy or by the energy of quantum vacuum.

We instead consider the expansion as an effect of mass-space increase and that \(v_\text{exp} + v_\text{ext}\) is the expression of a variable metric which in the past has been *open* (curvature negative) but with an *accelerated* increase rate due to an additional pressure that can only result from increase of mass-space.

Thus determining a universe built on a increasing field \(\Xi_U\) with a speciﬁc *law of increase* (\ref{accretion_law}) in space and mass, which leads us to expansion law coincident with that Hubble.

In (\ref{space_mass_balance}) it is shown that a universe with flat geometry exists only if there is a balance between gravity (originating from the mass) and space (also originated from the mass).

This balancing (\ref{universe_formation}) is achieved by each lattice \(U_i\) only at a certain moment of its evolution, during which the lattice \(U_i\) grows in mass (increasing the number \(N\) of particles) and in space. In this way \(U_i\) is a universe with a *variable metric*: its curvature switch from a negative value (open universe) to a positive value (closed universe) going through an intermediate critic phase with a flat metric.

It is proved that the time \(\tau_c\) to reach the critical phase is connected to the gravitational coupling gravitational \(\alpha_i\) of each \(U_i\).

In (\ref{universe_intersection}) we deﬁne the universe-lattice \(U^\ast\) like the *intersection* between the pair of universes-lattice \(U_n\), \(U_e\), because it is assumed (by the standard model), that our universe \(U\) is built on a particular set of basic universe-lattice \(U_i\), such as that of the electron \(U_e\) for leptons and that of the nucleon \(U_n\) for the baryons (quarks).

Then, we consider that the hydrogen atom builds (through the intermediation of the photons, the quanta of the electromagnetic field) the intersection between the two lattices, \(U^\ast_{(p,e)} = U^\ast_H\). This allows us to say that our universe is an *Hydrogen Universe*.

Se we can determine the factor \(k_s\) that allows to *overlap* the Compton wave lengths of the two universe-lattice \(U_p\), \(U_e\).

We obtain (\ref{our_universe}) by simple calculations that the \(\tau_c\) of \(U^\ast_{n,e}\) is coincident (within the errors range) with the present age of our universe, defined only by astronomical observations (Collaboration 2014).

This leads us to believe that our universe is in flat metric phase because it is near to the next critical phase of a universe with a variable metric.

The value of \(H_c\), corresponding to the critical time \(\tau_c\), is calculated. We observe that this value approaches tothe one calculated observing the nearest galaxies: from this result we deduce that our universe is in a phase of flat metric.

We also evaluate the critical density \(\rho_c\) of \(U^\ast_{(n, e)}\): the value is greater than the critical density \(\rho^{(n)}_c\) calculated using the nucleons mass highlighting the need for the existence of a not-visible mass (the missing mass).

Instead, in the past, the universe was found in a phase with open metric, nevertheless from the data it is noticed that the distant galaxies are accelerated more than expected: we explain this fact (\ref{increasing_pressure}) applying an additional pressure generated by the creation of space. We also formulate the relation that defines such pressure and we calculate its value as a function of the parameter \(\Lambda\), the Einstein’s cosmological constant.

A good analogy could be a man walking on a treadmill↩

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