Expanding universe with a variable cosmological term

Z. Naturforsch. 2015; aop

*Corresponding author: Carlos Blanco-Pérez, Universidad Pontificia Comillas, Universidad Pontificia Comillas 3-5, Madrid, Madrid 28049, Spain, E-mail: [email protected] Fernández-Guerrero: Universität Zürich, Zürich, Switzerland

Carlos Blanco-Pérez* and Antonio Fernández-Guerrero

Expanding Universe with a Variable Cosmological TermDOI 10.1515/zna-2015-0314Received June 3, 2015; accepted August 6, 2015

Abstract: We propose a model of expansion of the universe in which a minimal, ‘quantised’ rate is dependent upon the value of the cosmological constant Λ in Einstein’s field equations, itself not a constant but a function of the size and the entropy of the universe. From this perspective, we offer an expression which relates Hubble’s constant with the cosmological constant.

Keywords: Cosmological Constant; Hubble Constant; Quantised Expansion.

1 Theoretical FrameworkAs it is well known, the Theory of General Relativity offers the farthest-reaching physical model to explore reality on a large scale. In the original formulation of Einstein’s field equations, we find a magnitude called ‘the cosmological constant’ [1]:

4

8 GG g Tcµν µν µν

πΛ+ =

(1)

The cosmological constant tends to be contemplated as a measure of the energy density of the vacuum [2, 3], but it has originated a vivid debate in regard to its nature, and the discussion is still open [4, 5], especially concern-ing dark energy and the problem of the expansion of the universe [6].

However, it is generally assumed that Λ is a real con-stant and, moreover, a fundamental constant of nature, of the same kind as Planck’s constant or Newton’s constant of gravitation. Here, we want to explore a different possi-bility, because it could provide us with some deep insights about the connection between cosmology and quantum

principles. From this approach, Λ would be a variable, a function of the ‘potential energy’ available in the universe. Some models with a time-dependent Λ have already been suggested in the literature [7–10]; for a comprehensive and insightful review of all of them and their feasibility in light of empirical data, see Basilakos et al. [11], where, from a model of a time-dependent cosmological constant, some cosmological functions such as the scale factor of the uni-verse, the Hubble expansion rate and the energy densities are defined analytically.

For example, Berman [8] presents a model in which Λ varies and the gravitational constant G may vary or remain unaltered, identifying a law of variation encom-passed by the expression Λ ∝ t−2 (where t is the time), different from the equation Λ ∝ R−2 (where R is the scale factor in Robertson–Walker metric) that had been advo-cated by Chen and Wu [12]. But it is still unclear how this picture of an expanding universe could be harmonised with the quantum principle applied to the spatial units of expansion. The feasibility of this idea, taken as a heuristic principle, will lead us into an interesting result regarding the relationship between, on the one hand, Λ and other fundamental magnitudes and, on the other hand, Λ and Hubble’s constant, as we shall show.

In any case, we face a fundamental difficulty before developing any model based upon the assumption of a variable cosmological constant, namely, the mathemati-cal consistency of this idea with the so-called contracted Bianchi identity:

1 02

R g Rµν µν

µ

− =

(2)

Its implication is clear, and its physical meaning, pro-found: the covariance of Einstein’s tensor must be equal to zero, so that both the energy and the momentum may be conserved in space-time. Choosing the appropriate frame of reference, we can preserve the conservation of momen-tum, but still we would be violating the principle of con-servation of energy. As a result, the conservation of the Tμν

tensor, in such a way that 0T

µν∂

∂τ= (where τ is the proper

time) seems to be essentially incompatible with a variable osmological constant, for 0.T µν

ν=

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2 C. Blanco-Pérez and A. Fernández-Guerrero: Expanding Universe with a Variable Cosmological Term

However, even if Λ were a real constant – as it happens to be in the standard exposition of general rela-tivity – energy would not be conserved in the universe. The reason for this fact is the following: the cosmological constant can be regarded as a representation of the con-stant density of energy in the universe. As the universe expands, so does its volume, leading to a situation in which the total energy increases. In a deeper, theoretical sense the principle of conservation of energy emanates as a consequence of Noether’s theorem, according to which the symmetry of the coordinates yields a conservation law. As in the Theory of General Relativity, the coordinates are time-dependent, invariance under time translation is lost; as a result, the principle of conservation of energy cannot be satisfied.1 The conclusion that can be built from these considerations is that the nature of the law of conserva-tion of energy is itself problematic under the domains of general relativity, so that no firm prohibition against a var-iable cosmological constant can be established a priori, at least if it is based upon the necessity of respecting the principle of conservation of energy.

If our hypothesis were right, Λ would vary over time according to the following magnitudes: the potential energy U of the universe (the ‘free energy’ at disposal, without which its expansion would not be viable) and its area A, an idea encapsulated in the expression

d d dd d d

U Aft t tΛ

= −

(3)

Or, using a linear approximation to f,

1 2

d d dd d d

U AA At t tΛ = −

(4)

where A1 and A2 are constants with the proper dimen-sional units.

1 Alternatively, one could argue in terms of a heuristic solution to this apparent paradox (the violation of the principle of conservation of energy). The cosmological constant would be a function of the ‘cosmic time’ T (the time of existence of the universe), a privileged time which is identical for all observers. Therefore, it does not fall under the scrutiny of the principle of relativity: it should be meas-ured equally by all potential observers, and it should differ from the proper time of the specific system described by the field equations of general relativity. Of course, this idea meets the objection that if the proper time changes, the cosmic time, despite enjoying a privi-leged status, must also change. However, if the cosmic time sets an ‘absolute frame of reference’, the violation of the law of conserva-tion of energy does not affect a ‘local system’, but the universe as a whole, whose content of energy (more specifically, the energy of the vacuum) varies because of its expansion. As a consequence, the principle of conservation of energy can be maintained locally for any observer, although it is violated on a cosmic scale.

Then,2

1 1

0 01 2

d d d d( ) d A A dd d d d

t t

t t

U A U At f t tt t t t

Λ

= − = − ∫ ∫

(5)

Of course, there is a profound analogy with the thermodynamic identity dU=TdS + pdV (where U is the internal energy of the system, T its temperature, S its entropy, p its pressure, and V its volume). The correlate of the cosmological constant is the term pdV, itself a func-tion of dU − TdS, and cosmologists know that there is a connection between the entropy and the area of a singu-larity: the so-called Bekenstein–Hawking equation [13, 14] teaches us that black holes have a defined entropy, being proportional to its area A:

3

2AkcShG

π=

(6)

A combined analysis of (5) and (6) allows us to realise that, in accordance with the hypothesis of a time- dependent Λ, the variation of this magnitude would consist of a function expressed in terms of the potential energy of the universe and its entropy. If we consider the initial universe as a singularity whose physical descrip-tion can be made analogous to that of a black hole, it is possible to draw a fruitful cosmological picture. From this angle, the expansion of the initial universe can be regarded as the increasing of its event horizon, and a plau-sible model can be developed in this way: as the universe grew in size, the value of the cosmological constant grad-ually decreased, leading to an increasing of the overall entropy of the system. This conclusion could be expected from Bekenstein and Hawking’s result that the relation between the entropy of a singularity like a black-hole and its area is kept proportional. As the size of the universe increased, so did its entropy (while the initial potential energy decreased); while the universe is expanding, the potential energy implied by Λ decreases, and the remain-ing energy becomes transformed into the kinetic energy of the expansion. This process continues until Λ reaches a value as small as it can be measured nowadays.

From this heuristic perspective, the cosmological con-stant Λ is susceptible to being interpreted as a measure of the potential energy of vacuum pervading the universe. Therefore, it should be proportional to the fundamental quantum of expansion of the universe, as we show in the next section.

2 In order to avoid the initial singularity (when t = 0) and the con-ceptual difficulties associated with it, the integral could run from the Planck time to the present moment in the history of the universe.



C. Blanco-Pérez and A. Fernández-Guerrero: Expanding Universe with a Variable Cosmological Term 3

affect some constant factors, depending on the geometrical form that has been chosen (spherical, cubic, etc.); the essential parameters would suffer no significant modification. The advantage of choosing a spherical geometry lies in the assumption of isotropy: there is no privileged direction. This offers a more elegant and parsimonious model, of great heuristic potential.

2 Mathematical Development

2.1 The Fundamental Rate of Expansion

Let a ‘quantum of vacuum’ with energy 2hν (the funda-

mental energy level of a simple quantum harmonic oscil-lator, where ν is the frequency of oscillation of the system)3 expand at a rate k0. Assuming that the principle of conser-vation of energy is fulfilled, the kinetic energy associated with this rate should be equal to the gravitational poten-tial of the vacuum:

20

2k Gm

r=

(7)

The fundamental energy of this quantum of vacuum must be equal to the energy that is inherent to any mass m:

2

2h mcν =

(8)

Hence, the mass that exists in connection with this quantum of vacuum can be expressed in the following way:

22

hmcν=

(9)

If we substitute this formula in the initial equation, we get

20

22 2k G h

r cν

=

(10)

The fundamental frequency ν can be regarded as equal to the quotient ,c

λ with λ being the fundamental

wave-length of this quantum of vacuum and c the veloc-ity of light in vacuo (the highest velocity at which this quantum could possibly expand).

From these expressions, we reach the following equation:

2

01Ghk

c rλ=

(11)

If we imagine, in its simplest and most parsimonious form, a spherical quantum with radius r,4 its wavelength

3 This assumption is based on heuristic and extremely idealised grounds, but it will allow us to elaborate a dimensional analysis en-dowed with an interesting cosmological explanatory potential.4 Of course, the geometry adopted to describe this fundamental quantum of expansion could be altered, but this change would only

can be made equal to the wavelength of its circumference. Therefore,

20 2

1 1( 2 )

Gh Gkc r r c rπ

= =

(12)

What is the physical meaning of the term 2

1 ?r The

canonical interpretation of the cosmological constant Λ appearing in the field equations of general relativity contemplates this magnitude as a manifestation of the pressure exerted by the vacuum. Its units are m−2, and the energy density and pressure of the vacuum obey the

equation 4

2

8cc pG

Λ

π=− = (where is the density and p is

the pressure). Let us draw a provisional analogy with this

concept, so that the amount 2

1r is associated with the fun-

damental pressure exerted upon the quantum of vacuum.5 Rather than being arbitrary, it makes sense that our ‘ideal’ quantum of vacuum be subject to a unitary pressure per area, caused by the fundamental energy that the system contains. The plausibility of this hypothesis will emerge once we analyse an immediate consequence of its accept-ance, which is the following equation:

0

GkcΛ=

(13)

This formula involves a profound imbrication between some fundamental constants of nature. As progress in theoretical physics has shown, the presupposition of an underlying unity binding all fundamental constants leads into interesting and fruitful results. For example, the com-bination of some of these fundamental constants with the proper dimensional arrangements produces the so-called

Planck mass ,cG

Planck length 3 ,Gc

and Planck

time 5 .Gc

Assuming that the entropy of a single quantum of vacuum can be measured by Bekenstein-Hawking’s

5 Our interpretation establishes a variable Λ; the plausibility of this hypothesis can be assessed from the connection between Λ and 2

1 :r

since the quantum of space is experiencing dilatation, r must vary accordingly.




equation (6), and because the term r2 is the expression of an area, the fundamental rate k0 must be inversely propor-tional to the entropy:

22

0 4kckS

=

(14)

where k is Boltzmann’s constant (=1.381 × 10−23 m2 kg s−2 K−1). Here we find the crystallisation of a deep physical truth, evoked in the first section of this paper: as the uni-verse expands, its entropy increases. This phenomenon is marked by the decay of Λ, in accordance with the follow-ing equation:

3

4kcG S

Λ =

(15)

In connection with the development which we have just presented, it is useful to remark that if we choose to combine four fundamental quantities of the uni-verse, namely G (Newton’s universal constant of gravi-tation = 6.674 × 10−11 m3 kg−1 s−2), ℏ (reduced Planck’s constant = 1.055 × 10−34 m2 kg s−1), c (the speed of light in vacuum = 2.998 × 108 ms−1), and Λ (10−52 m−2; see Barrow and Shaw [15] – which strictly speaking would not be a constant – it is possible to obtain, in a simple and elegant manner, a number whose dimensions are those of a velocity:

0

GkcΛ=

(16)

Unveiled by pure dimensional analysis, this quantity coincides with the fundamental velocity of expansion that has been reached from the physical representation of how this process of dilatation could be visualised.

Measuring this value according to the International System of Units, we arrive at the following number in meters/seconds:

− − −−× × × ×

=+ ××

11 34 5253

8

6.674 10 1.055 10 104.845 10 m/s

2.998 10 (17)

It is an extremely small value, but it is not null. Given the fact that its calculation involves fundamental quantities of the universe, this magnitude, expressed in dimensions of rate of change in space over time, could be interpreted as the threshold rate of dilatation or con-traction of the universe. Strictly speaking, it would not consist of a velocity (in respect to which frame of ref-erence would this whole system be moving?), but of a

fundamental, quantised rate of expansion which sets a threshold limit beyond which no dilatation or con-traction can possibly occur. In other words, it could be regarded as the ‘quantum’ of expansion/contraction for the universe. If h were equal to zero, no expansion could take place. From this perspective, it is reasonable to speak in terms of a fundamental quantum of expan-sion, stemming from the application of the basic princi-ples of quantum mechanics to the study of the universe as a whole.

2.2 The Fundamental Rate and Hubble’s Constant

Now, it is possible to move farther along the path of the idea which we have presented in the aforementioned sec-tions. If we divide the quantity k0 by the so-called Planck length,

353 1.616 10 m ,PGc

−

× ∼

(18)

we obtain a value which is the inverse of time:

2 18 10

3

2.3 10 sP

Gk c c

Gc

Λ

Λ − −= = = ×

(19)

We know the time of existence of the universe with a considerable degree of accuracy, for the so-called Hubble’s constant H0 (2.3 × 10−18 s−1) is inversely propor-tional to Hubble’s time tH:

0

1

H

Ht

=

(20)

The fact that the value of H0 is so narrowly related with the ratio 0

P

k

can be explained from the following

theoretical point of view. In the distant past, when the value of Hubble’s constant was larger and Hubble’s time was smaller, the value of Λ was also greater. The closer we get to the initial explosion, the larger the value of Λ is. Therefore, Λ is not a constant in the cosmological scale: it suffered a great – and perhaps sudden – decay after the Planck time.

Let us suppose a remote galaxy moving away from us at speed ν = H0d. The increasing of the distance between the galaxy and ourselves is because of the scale of the




and Wu’s formula Λ ∝ R−2 [12]7, encompassing them from simple and powerful postulates. Of course, it consists of a heuristic relationship, and it is necessary to add some remarks regarding the phenomenology of the equation. First of all, it is important to notice that the equation as such cannot be valid at any instant in the history of the universe, especially if we want to offer a theoretical explo-ration of its origin and the physical conditions that might have prevailed at that remote moment. The proportional-ity between the cosmological constant and the square of Hubble’s constant cannot be correct at the background of the universe, for it inevitably implies that the universe is already experiencing a certain degree of acceleration. The turning point mediating from deceleration into accelera-tion cannot be contemplated from the simple and inexo-rably limited postulates upon which we have based our heuristic reasoning. Secondly, a similar constraint to the explanatory potential of the equation emerges if we con-sider the level of structure formation at a cosmological scale. The reason lies in the absence of convincing argu-ments about how to integrate our perspective with the idea of a positive acceleration.

These remarks can be – at least partially – addressed through the introduction of an additive constant in (24). Such a term, instead of being equal to 0 as it is implicitly stated in (24), a priori would always be a small, positive quantity which would allow that, even at an infinite time, the cosmological constant would not be become annulled. Therefore, (24) turns into the following expression:

202

HA

cΛ = +

(25)

Phenomenologically, the aforementioned reasoning involves that the energy density of a vacuum never loses its intrinsic value,8 even at an infinite time. Therefore, the advantage of adding a constant A stems from the possi-bility of considering an ‘asymptotic situation’ in which Hubble’s constant diminishes to zero, but the value of the

universe augmenting at a minimum rate 0 .GkcΛ= 6 If

the quantum of space is expanding at a rate k0, and if we are separated n quanta of space from that remote galaxy, its distance is n times the Planck length (nℓP). Its derivative should be equivalent to the velocity of separation of the galaxy from our present position:

d( )d( distance)d d

Pnt t

=

(21)

Because the Planck length is a constant, we have

0

dd Pn nkt

=

(22)

Therefore, the recession velocity of the galaxy is nk0, but at the same time we know, through Hubble’s law, that it is equal to H0d. The consequence is clear: it must be equal to H0nℓP.

Combining the previous expressions we obtain:

00

P

kH=

(23)

Finally,

202

Hc

Λ =

(24)

This result has been obtained from a physical picture based upon fundamental, quantum mechanical consid-erations. Interestingly enough, however, the cosmologi-cal constant has been unified with this ‘microphysical’ representation, and the conclusion is compatible with some recent proposals that can be found in the litera-ture, such as Berman’s equation Λ ∝ t−2 [8] and Chen

6 If we consider a line element ds in the Robertson-Walker metric, we obtain the expression 2 2 2 2 2 2 2 2d ( d ) ( )[ d ( )( d sin d )],ks c t R t Sχ χ θ θ φ= − + +

2 2 2 2 2 2 2 2d ( d ) ( )[ d ( )( d sin d )],ks c t R t Sχ χ θ θ φ= − + + where R is the scale factor, k the curvature factor (whose admitted values are +1,0,–1, although the curvature factor is gener-ally assumed to be 0) contained in the function Sk(χ), and dχ, dθ, and dφ are the co-moving space coordinates. According to this equa-tion, the evolution of the universe is governed by the scale factor, and the physical distance x between two points is Rdl (where l is the co-moving distance, whose square is expressed by terms inside the square brackets in the previous equation). By differentiating x with respect to time t, we can find the proper velocity of a particle mov-ing in this expanding universe. Our model states that this velocity v must be proportional, in its simplest form, to the fundamental rate of

expansion 0 .Gkc

Λ=

7 From (24), Berman’s formula [8] follows immediately, for we know that Hubble’s constant is inversely proportional to so-called Hubble’s time. From (15), namely

3

,4kcG S

Λ=

we can also unveil

Chen and Wu’s formula [12], because if the cosmological constant is inversely proportional to the square of the scale factor, it can be as-sumed to be inversely proportional to the area and therefore to the entropy. This means that the cosmological constant decreases as the entropy increases, as we have suggested in the previous section.8 Although in a highly speculative form, the reader will allow us to claim that this additive contribution might be related with the Higgs field permeating the whole universe. Its discussion would be extremely complex and must be postponed for a further publication.




cosmological constant does not vanish.9 Also, this picture is coherent with the assumption that in the remote future, the universe will be dominated by dark energy.

In any case, and beyond the inherent theoretical limitations which we have just discussed, in particular regarding the necessity of an additive constant, (24) offers a powerful cosmological insight. Interestingly enough, this formula could have been unveiled from the classic Theory of General Relativity, even without the funda-mental, quantum mechanical assumptions that have guided our investigation. Assuming a homogenous and isotropic universe, we can express the basic equations of the standard cosmological model in accordance with the Robertson–Walker metric [16]. Einstein’s equation yields the following form:

2 283 3

a G ca

π Λρ

= +

(26)

If we define aa

as the Hubble rate H(t) (the time dependence is through the scale factor a, which is a func-tion of time), and ρ as the energy density (which includes separate terms for matter and radiation), we can manifest the relationship between Hubble’s constant and the cos-mological constant in a form that will lead us to a result convergent with (24).

Writing the Hubble rate as a function of the cosmolog-

ical redshift z (where 1 1za

= − ), we obtain the expression

22 8( ) ( )

3 3G cH z zπ Λ

ρ= +

(27)

Defining the critical density as 2

038cr

HG

ρπ

= (where H0 = H(z = 0) is the present-day Hubble rate, namely the Hubble constant), it is possible to reformulate (26) in the following way:

2 22 0( ) ( )

3cr

H cH z z Λρ

ρ= +

(28)

( )

cr

zρ

ρ is equivalent to the density ratio Ω, and it contains

a sum of terms for matter (M) and radiation (R). Making explicit the redshift dependence, this formula becomes:

9 Hubble’s constant would only approach zero at an extremely (or perhaps potentially infinite) large time, that is to say, at the asymp-totic limit that defines the end of the universe. At that stage in cosmic evolution, the universe would still preserve a certain energy density of the vacuum, an idea convergent with the hypothesis of a universe dominated by dark matter at late times.

22 2 4 3

0 M( ) ( ( 1 ) ( 1 ) )3RcH z H z z Λ

Ω Ω= + + + +

(29)

Expressing H2(z) in terms of ΩΛ (which is equal to 2

203cH

Λ), we obtain

2 2 4 30( ) ( ( 1 ) ( 1 ) )M RH z H z z

ΛΩ Ω Ω= + + + + (30)

If we intend to analyse the present time, z must be equal to 0, so that (28) is transformed into the following formula:

22 2

0 0 ( )3M RcH H Λ

Ω Ω= + +

(31)

Cosmological data suggest negligible values for ΩR and ΩM of approximately 0.3, and of 0.7 for ΩΛ. If we take ΩR = 0 as a plausible value for the present time, the result is

2 20

13( 1 )M

H cΛΩ

=−

(32)

Equation (24) pointed to a factor of 1 in the relation-

ship between Λ and 2

02 ,

Hc

whereas (32) suggests that

the value of this factor is close to 0.48. This discrepancy implies that our model establishes a constant factor of 1 for the sum composed by ΩR + ΩM + ΩΛ. In geometri-cal terms, meaningful for the debate concerning the shape of the universe, this conclusion would involve the acceptance of a flat cosmos, an outcome which seems compatible with empirical evidence [17], and more recently, the results provided by the Baryon Oscillation Spectroscopic Survey (BOSS). In a certain sense, it nor-malises the density ratios so that the cosmological con-stant and Hubble’s constant keep a strictly proportional correlation.

However, we know that (24) is incomplete and it needs to be improved in at least one ineluctable way, for it needs an additive constant A in order to preserve the energy of the vacuum throughout the whole history of the universe. But in accordance with astronomical fitting, we realise that the coefficient multiplying the square of Hubble’s constant cannot be equal to 1 as suggested by (24). On the contrary, it seems to be below 1 if we want to be fully faith-ful to empirical evidence. Equation (21) would represent a limit (the perfect correspondence between the cosmologi-cal constant and the square of Hubble’s constant, that is to say, the case of a totally flat universe expanding over a clearly defined period of time), and in order to agree with




it pave the path for solving the discrepancy between the observed-value of the cosmological constant and the standard theoretical predictions? Is the expansion of the universe ultimately based upon an unceasing conversion of dark energy and kinetic energy, whose corollary is a variable cosmological constant?

Acknowledgments: We want to express our gratitude to Prof. Joan Solà (University of Barcelona), Dr. Jonathan Ruel, and Mr. Alexandre Pérez Casares for their valuable and insightful comments.

References[1] A. Einstein, Proc. R. Pru. Aca.Scien. 1, 844 (1915).[2] J. Solà, J. Phys. Conf. Series 453, 012015 (2013).[3] A. Gómez-Valent, J. Solà, and S. Basilakos, J. Cosmol.

Astropart. Phys. 1, 004 (2015).[4] P. J. E. Peebles, and B. Ratra, Rev. Mod. Phys. 75, 559 (2003).[5] S. Weinberg, Rev. Mod. Phys. 61, 1 (1989).[6] E. T. Kipreos, PLoS One 9, e115550 (2014).[7] P. J. E. Peebles and B. Ratra, Astrophys. J. 325, L17 (1988).[8] M. S. Berman, Phys. Rev. D 43, 1075 (1991).[9] I. L. Shapiro, J. Solà, C. España-Bonet, and P. Ruiz-Lapuente,

Phys. Lett. B 574, 149 (2003).[10] K. P. Singh, Turk. J. Phys. 34, 163 (2010).[11] S. Basilakos, M. Plionis, and J. Solà, Phys. Rev. D 80, 3511

(2009).[12] W. Chen and Y. S. Wu, Phys. Rev. D 41, 695 (1990).[13] J. D. Bekenstein, Phys. Rev. D 7, 2333 (1973).[14] S. W. Hawking, Commun. Math. Phys. 43, 199 (1975).[15] J. D. Barrow and D. Shaw, J. Int. J. Mod. Phys. D 20, 2875 (2011).[16] S. Dodelson, Modern Cosmology, Academic Press, New York

2003.[17] P. De Bernardis, P. A. R. Ade, J. J. Bock, J. R. Bond, J. Borrill,

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2810 (2015).

experimental data10 and with theoretical considerations at infinite time, it needs to be corrected in the following way:

2

02

HA

cΛ β= + (33)11

where β is a coefficient lower than 1 and A is an additive constant, hypothetically of a small value.

Of course, our model states that Hubble’s constant is not actually a constant but a time-dependent variable which, in a way similar to Λ, evolves over time, so that in the beginning of the universe its value had to be higher than in the present.

3 ConclusionsOur proposal is capable of integrating previous results from a simple and elegant theoretical framework. However, the physical depiction which we have exposed opens many questions that should be addressed in future investigations. Some of the most crucial myster-ies, capable of broadening the scope of our research, are the following: what is the exact relationship between the cosmological constant and the dark energy pervading the universe? How to integrate the so-called Higgs’ field and the electroweak vacuum associated with the standard model in particle physics with this cosmological picture? Could the ‘quantum of vacuum’ used in our model as a heuristic approach be related with a hypothetical funda-mental particle like the ‘graviton’? The expression con-necting the cosmological constant and Hubble’s constant, although plausible in light of current astronomical data, can be regarded as true at any instant in the evolution of the universe? Does it consist of a simplification of the so-called concordance model (ΛCDM model) and does

10 Given that our reasoning has been guided by heuristic postulates and dimensional analysis, it is almost impossible to deduce a priori the necessity of introducing this non-dimensional coefficient; rather, it must be obtained through astronomical fitting.11 Although stemming from a different starting point, Gómez-Valent et al. [3, p. 7] contemplates a similar possibility, namely, a cosmo-logical constant varying in accordance with the square of Hubble’s constant, an additive constant and a coefficient multiplying Hubble’s constant. See also (especially regarding the empirical plausibility of this expression) Gómez-Valent and Solà [18].



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Expanding universe with a variable cosmological term