_{1}

The Hubble equation was considered valid enough to calculate the recession velocity of galaxies, until further observations showed that there would be an accelerated recession in the Hubble flow, necessarily tied to an accelerated expansion of the Universe. So, this paper postulates the existence of a Hubble field as a possible cause for such an accelerated expansion, with some conditions: it must be a scalar field whose intensity should be a constant in respect to distance and whose Poisson equation should not be zero nor a function of mass; such field could rather be a property of the space-time. The obvious expression for acceleration should be the derivative of the Hubble equation respect to time, which gives two opposed-signs terms whose substitution by the De-Sitter equation drives to a permanent negative acceleration, similarly to that obtained by the 2
^{nd} Friedmann equation. Otherwise, the inclusion of the ? term in the gravitational Einstein equation has led to a two opposed-signs terms expression, resembled to a non-published Newton equation. The negative term expresses the gravitational attraction and the positive one expresses the accelerated expansion as a ? function, which usually is attributed to dark energy. In this paper it is shown that Λ is proportional to the squared Hubble parameter and that the uncertain dark energy may be substituted by the calculable Hubble field intensity to obtain an equation for the net Universe acceleration. Equations for the Hubble parameter as functions of time and radius are also deduced. A relation is shown between the various assumed masses of the Universe and its critical radius. Additional Universe parameters are estimated such as the deceleration factor and a solution for the Poisson equation in the Hubble field. A brief comment t on high-standard candles is included.

Since the A. Riess et al. [_{L}, as

The dark energy presents two problems. The first one is that such alleged energy has not been detected or measured experimentally. The second one is that its density is usually expressed as an equivalent mass density (kg∙m^{−3}) though a so-crucial relationship such as

has not been proved yet. Its present numerical value has been estimated from the WMAP experiments [

Besides, there is a problematic complement of dark energy theory: it is assumed that it generates a negative pressure or vacuum energy that pulls the Universe to expand itself. Though the equation of state is feasible

where −p is the negative pressure generate d) the implied numerical values would not be big enough to pull the entire Universe back. Reference [

In explaining the Universe acceleration, several alternative theories have been published based on extensions of the relativistic theory of gravity and MOND theory [

The Hubble parameter, H(t) is defined as:

where a is the radial Universe function, related to the distance r by the co-moving equation

x is the constant commoving coordinate and

The time derivative of this equation would represent the accelerated radial expansion:

The problem here is that

Bergstron and Goobar [

the two first terms form the Einstein tensor, G_{m}_{n}; g_{m}_{n} stands for the fundamental covariant tensor; G is the gravitational constant, while T_{m}_{n} represents the energy-momentum tensor. This expression assumes a unitary value for light velocity. Einstein added Lto his equations in 1917 in order to match a static Universe concept. However, A. Friedmann did not take L into account in obtaining two basic solutions, five years later [_{ }

_{b} is the Universe baryonic density; p is th total pressure; k is a curvature parameter of the Universe, and c is the light velocity in a vacuum. Accordingly to R. Tolman [

u^{m} and u^{n} are the 4-velocity vectors that, in a comoving frame, are u = (1, 0, 0, 0). Therefore, T^{oo} = rs

(When the value for L is substituted).

By applying to (1.7) the conservation criteria for matter in the Universe, plus the co-moving equation and the baryonic density concept as well, the present Universe acceleration would be:

where M_{b} is the baryonic (o if preferred, gravitational mass) contained in a sphere of radius r. Equation (1.8) is the same of Newton for gravitational acceleration, so implying that it remains negative as r increases, a conclusion that would result opposed to recent works [^{m}^{n} = 0. In what follows, a L value and its relationship to the Hubble parameter are calculated. Thereafter, an equation for the Hubble field is deduced, and two equations for H as functions of time and distance are also proposed. The criticality and deceleration parameters are calculated as well as a Poisson equation for the scalar Hubble field. A brief comment on high-z standard candles is included at the end of the paper.

By applying the FLRW metric to the Einstein equation, reference [

T is equation has been also deduced by L. Calder and O. Lahav in a landmark paper [

Since L had not been considered a function of time, reference [_{U} curve at the present time. Such a constant unitary value (assumed as vacuum energy) intersecting the W_{U} curve at the present time is also mentioned by reference [_{U} is the ratio between the Universe total density and its own critical density. These and other authors [

Calder and Lahav [

where C is an arbitrary constant. By comparing Equations ((2.2) and (2.3)), they conclude that L must be proportional to the entire mass of the Universe, M_{b} (a constant) as:

However, the constancy of L would require C to be a true constant. It may be calculated at the equilibrium point (

Equating the expressions for C from (2.4) and (2.5) it gives:

So showing that L is a constant being related to the critical radius. Otherwise, the equation presented by reference [_{L}) would require knowing the dark energy density, an uncertain parameter at the present time.

The proposition of M. Carmeli and T. Kuzmenko [

t_{o} is the age of the Universe; if it is t_{o} = 14.0 × 10^{9} (y) [^{−35} (s^{−2}) [

Accordingly to Equation (1.1) the Hubble parameter is a function of time. Its present (constant) value, H_{o} is defined as:

Therefore, from (2.7) and (2.8):

Multiplying Equation (2.9) by r_{o} gives:

If this equation is valid at the present time, it is postulated in this work to be valid also at the critical time, i.e.

and therefore, at any time:

Equation (2.10) shows that the second term of Equation (2.2) does imply a positive acceleration expansion, with a constant value as proposed by reference [

Equation (2.13) represents the net acceleration of the Universe expansion, i.e., the difference between the attractive gravitational field and the expansive Hubble field. It seems clear that, when the Universe radius was small, the gravitational field intensity was dominant, but nowadays, at a bigger Universe radius, the Hubble field intensity should be overbearing, as shown in

In what follows, the variable intensity of the gravitational field is represented by G and the constant intensity of the Hubble field is written as Г_{H}, i.e.

Therefore, the net Universe acceleration may be expressed, from Equation (2.13), as the difference:

Since the potential energy in the gravitational field is always negative (U < 0), the gravitational potential (energy per unit mass) is negative too; it is expressed as:

By definition [

it is therefore negative, as expressed in Equation (3.1).

Similarly, the Hubble field intensity could be defined as the gradient of a positive scalar Hubble field potential:

The substitution of Equation (3.2) in (3.6) gives a definition of the Hubble field intensity

By assuming from Equation (2.12) that G_{H} = constant in Equation (3.7), the radial integration of this equation gives an expression for the Hubble potential:

(alternative units applied in Equation (3.8) point out that there is no mass involved in the Hubble potential).

Assuming that Equation (2.12) is valid, it is possible to estimate the constant Hubble field intensity by using the present values of H_{o} and the Universe radius, r_{o} [

This would be the value of the Hubble field intensity, i.e. the Hubble acceleration of the Universe at any time, if there was not a gravitational field. Since it is not feasible to assign to the Hubblefield any known physical entity, it may be assumed that it corresponds, rather, to a property of the space-time. The present net acceleration results, from Equation (2.13),

The assumption for Г_{H} to be constant allows obtain a general expression for H as a function of distance, from Equations ((2.12) and (3.9)):

The present H_{o} value has been defined as the reciprocal of the Universe age (Equation (2.8)) but there is not a general expression to determine H(t). The same Equations ((2.12) and (3.9)) could allow calculate the Hubble parameter at any time if the distance is expressed as a function of time in a continuously accelerated movement:

Therefore, Equations ((3.10) and (3.11)) are proposed as general functions for H(r) and H(t).

From R. Johnson [^{52} kg; even figure, as modified by the intergalactic and interstellar media, is 1.7 × 10^{53} kg; another one, based in the Hoyle-Carvalho equation [_{h} = 1.84 × 10^{53} (kg) as the value covering the Hubble length (1.37 × 10^{26} m). All these figures have been obtained by research inside the observable Universe, i.e. into the Hubble sphere. So, accordingly to the cosmological principle, trying to determine the total mass contained in the total volume of the Universe is a valid problem. From the above given data, the density of the observable Universe is r_{h} = 1.8 × 10^{−26} (kg∙m^{−3}) a value here assumed for the entire Universe whose radius is estimated to be r_{o} = 4.4 × 10^{26} (m) [_{u} = 6.5 × 10^{54} (kg). Besides, dark mass could eventually be included as a gravitational mass, giving a total of M_{g} = 3.25 × 10^{55} (kg).

The critical point of the Universe may be defined as the time when the Universe becomes flat, i.e. when it may change from positive to a negative curvature. That implies

In Equation (4.1) it has not been specified the kind of mass to apply, which must be selected in any case.

Since the l.h.s. is a constant (G_{H}), it is possible to directly obtain the value of the critical radius from:

Since G and G_{H} are constants, this equation gives

In

The deceleration parameter, q, is defined as:

Its present value results q = −1.3, which confirms the possibility of an accelerated Universe.

The spatial derivative of Equation (3.7) is a solution of the Poisson equation in the Hubble field:

It means that there exists a force-flow in the Hubble field.

As assumed above, the constancy of G_{H} would require some kind of justification. Subsequent studies to that of Riess et al. [^{2}r in Equation (2.12).

Reference [

Mass of the Universe (kg) | Critical radius r_{c} (m) | Critical time t_{c} (Gy) |
---|---|---|

M_{b} = 1.84 × 10^{53} (in the Hubble sphere)^{ } | 7.0 × 10^{25 } | 7.1 |

M_{u} = 6.54 × 10^{54} (in the Universe)^{ } | 4.3 × 10^{26 } | 13.6 |

M_{g} = 3.25 × 10^{55} (including dark matter)^{ } | 1.0 × 10^{27 } | 24.0 |

[G_{G}] > G_{H} epoch (in the total Universe case, curve sphere case, curve G_{h}). Consequently, the G_{H} = [G_{G}] step would have defined the critical time in two possible cases. If dark matter would have been added, the critical point would still have to wait for another t_{o} period. The validity of the model here proposed could only be proven if future SN observations detect additional increases in the expected distances, at several (z > 1) values, and if they match with the net acceleration given by Equation (3.3). Some important experiments at higher z values (8.6 and 11.0) have been performed in two recent projects [

Anyway, there is a limit to distance measurements by present methods: it comes from the estimation of reference [^{3} [_{o} is today about 1 (mm), the emitted l_{e} should have been about 1 (mm) i.e. infrared photons traveling till now, in a co-moving coordinate, since the decoupling time.

1) The value of the cosmological constant, L is proportional to the reciprocal of the squared Universe age (Equation (2.9)). So, it results proportional to the present value of the squared Hubble parameter.

2) Since L cannot be directly associated to any known field, the present work substitutes the L term by a constant Hubble field intensity term (Equation (2.10)), to obtain the Equation (2.13), expressed in

3) The Hubble field intensity is defined as the gradient of a positive Hubble potential (Equation (3.6)). Its constant value is 2.2 × 10^{−9} (m∙s^{−2}). However, the present net acceleration is 1.7 × 10^{−9} (m∙s^{−2}). The expansion velocity at the astronomical radius results today

4) The assumption of the constancy of the Hubble field intensity drives to obtain two general functions of time and distance for the Hubble parameter (Equation (3.10), Equation (3.11)).

5) Equation (2.13) gives the net acceleration of the Universe, as well as its critical radius when_{U} balance. The critical conditions of the Universe are found to depend of the Universe’s mass chosen; here they were considered 3 cases: baryonic into the Hubble sphere, baryonic in the assumed total volume of the Universe and baryonic plus dark matter in the total volume. They are shown in _{H}, G_{G}) as functions of the r/r_{o} ratio and points out both the critical radius and critical time for the 3 cases above mentioned.

6) The deceleration parameter gave a negative value, so showing that the Universe is self-expanding. The Poisson equation for the Hubble field is Ñ^{2}V_{H} = H^{2}, since matter is not a component of the Hubble field.

7) The validity of the model here proposed could only be proven if future SN observations detect additional increases, such as it is being assumed in a recent academic project.

Author thanks to Sc. M. M. A. Zúñiga, C.P. V.M. Torres Tovar, D. Camargo, B. M. Cota and R. Cervantes, by their kind collaboration in the final version of this paper.

Juan Lartigue, (2016) The Hubble Field vs Dark Energy. Journal of Modern Physics,07,1607-1615. doi: 10.4236/jmp.2016.712145