The effect of de Sitter like background on increasing the zero point budget of dark energy

During this work, using subtraction renormalization mechanism, zero point quantum fluctuations for bosonic scalar fields in a de-Sitter like background are investigated. By virtue of the observed value for spectral index, $n_s(k)$, for massive scalar field the best value for the first slow roll parameter, $\epsilon$, is achieved. In addition the energy density of vacuum quantum fluctuations for massless scalar field is obtained. The effects of these fluctuations on other components of the Universe are studied. By solving the conservation equation, for some different examples, the energy density for different components of the Universe are obtained. In the case which, all components of the Universe are in an interaction, the different dissipation functions, $\tilde{Q}_{i}$, are considered. The time evolution of ${\rho_{DE}(z)}/{\rho_{cri}(z)}$ shows that $\tilde{Q}=3 \gamma H(t) \rho_{m}$ has best agreement in comparison to observational data including CMB, BAO and SNeIa data set.


I. INTRODUCTION
Determination of the true value of cosmological constant is a fundamental issue in modern cosmology. The subject has been investigated in different approaches. For instance, some scientists deal with it as an integration constant. However, the cosmological constant has got a given magnitude [1]. For more details, we provide readers with some references cited in [2]. On the other hand, from particle physics point of view the origin of it goes back to 1916 [3], when for the first time W. Nerst supposed that the zero point energy of the electromagnetic fields fill the vacuum right as an aether, although his hypothesis has some drawbacks (i.e. the results which attained by Nerst need to be corrected) [4]. For example S. E. Rugh and H. Zinkeraged [5] have shown that using the density of zero point energy achieved in unpublished work of W. Pauli [6] lead to an unacceptable radius for the Einstein static universe [6], [7]. For modern cosmological discussions the problem is even worse, because even for a well tested quantum field theory cutoff, the density of zero point energy exceeds the critical density of the present epoch universe's energy by many orders of magnitude. Therefore we accost with divergence problem risen from quantum field theory. These divergences have ultra violet nature, similar to those which appear in loop quantum expansion in higher orders [8]. People usually solve this problem by some cogently approaches, e.g. heat kernel expansion regularization, cutoff regularization and renormalization [9]. Heat kernel approach usually uses the zeta function regularization as a fundamental block of the method. Regularization in a simple and short definition is an approach to change an infinite quantity into a finite one [10]. Amongst important regularization schemes we can refer to frequency cutoff and point splitting regularization. For instance, people use the cutoff k c to attain the vacuum energy density of normal modes of a massless scalar field.
In Minkowskian space time this cutoff usually causes to this relation , where subscript c is for dependence of cutoff quantities. Following semiclassical approach brought in [11] and [12], the k 4 c quantity is contributed from flat space time (Minkowskian space) and this term diverges and therefore does not gravitate, so it has to be subtracted from same quantity which risen from Friedmann-Limaitre-Robertson-Walker (FLRW) space time, and therefore k 4 c term should be eliminated. Based on [13], we will study the zero point quantum fluctuations of massless scalar field for a quasi de Sitter background in the FLRW Universe. Therefore the slow roll parameter,ǫ , due to Hubble parameter form in quasi de Sitter model can play the important role to determine the physical quantity like energy density of the quantum fluctuations. Due to time dependence of Hubble parameter, it is explicit that the energy density of zero point quantum fluctuations exchanges energy with other components of the Universe. We will show that in quasi de Sitter background, the contribution of zero point quantum fluctuations are increased and therefore it find an important role as a detectable source of dark energy (DE). Furthermore, the equation of matter energy density, ρ m , is different in comparison to the case which ordinary de Sitter background is considered. In fact ρ m has some extra terms besides a −3 term. These new terms risen from interaction between matter and zero point quantum fluctuations. The conservation equation of energy density for different components of the Universe will obtain. To estimate the contribution of different components of the Universe, we shall use the standard Λ cold dark matter (ΛCDM) considering Cosmic Microwave Background (CMB), Baryonic Acoustic Oscillation (BAO) and Supernovae type Ia (SNeIa) data set as a criterion. The scheme of this paper is as follows, in Sec.I some historical aspects of zero point energy and it's properties as an introduction are discussed. Sec.II is devoted to general calculations. In Sec.III the cosmological role for zero point energy density is discussed, and the results of our work are compared with previous works. In Sec.IV the bounds which risen from time evolution energy density of DE will discuss. And at last, results and conclusions of the work bring in Sec.V .

II. FIELD QUANTIZATION AND CALCULATION OF EXPECTATION VALUE OF ENERGY MOMENTUM TENSOR
To study zero point quantum fluctuation, we consider a minimally coupled scalar field action as where g is the determinant of the metric, and Φ is the massless bosnic scalar field. For quantum field theoretical investigation one can quantize this scalar field as where a k and a † k are annihilation and creation operators respectively [14]. Variation Eq.(1) with respect to (w.r.t) scalar field givesχ where "′" ≡ d/dη, H =ȧ/a, and a(t) is the scale factor of the Universe. It should be noted, in Eq. (3), dot denotes derivative w.r.t the ordinary time and the latest term due to dependence of Φ to the space. Using conformal time η = dt/a(t), one can easily rewrite Eq. (3) as where H = a ′ /a =ȧ is the conformal Hubble parameter. One can see, a(t)H(t) = H, that H could be considered as comoving Hubble parameter. In a quasi de Sitter background, scale factor is taken as a(t) = t 1/ǫ , where ǫ = −Ḣ/H 2 is slow-rolling parameter and ǫ ≪ 1 [15]. Solving Eq. (4), one can obtain where ξ = (2 + 3ǫ)/2. In this solution we ignore the O(ǫ 2 ) and upper orders. Now we need to deal with energymomentum tensor. From Eq.(1) and the definition T µν = (−2/ √ −g) × (δS/δg µν ), one can easily achieve, Using above equation, the 00 and ii components of energy-momentum tensor are as Therefore density energy and pressure of zero point quantum fluctuations, are defiened as Where in a regularization scheme, according to quantum gravity concepts the cutoff, k c , should be considered greater than physical momenta k/a(t). Therefore from Eq. (5) and definition of scale factor (i.e. a(η) = −(1 + ǫ)/Hη), |χ| 2 and |χ| 2 terms could be obtained as Therefore using Eqs. (9) and (10), one can receive Using Eq. (11), solving Eq.(8) leads to The first term in Eqs. (12) and (13) is the Minkowskian contribution of the density of energy and pressure respectively. Therefore using subtraction mechanism which is a powerful method in quantum field theory, these infinities could be eliminated. Comparing the second terms (in Eqs. (12,13)), with ones in the ordinary de Sitter background (ǫ = 0), it is clear that cause to the presence of slow-roll parameter (which causes to time dependency of scale factor), some problems which was appeared in ordinary de Sitter model are removed. Some of those problems are as follows • 1. In ordinary de Sitter background, H (i.e. Hubble parameter) is time independent. Therefore such mechanism, could not explain the evolution of the Universe as a general model.
• 2. In de Sitter frame a conceptual problem maybe risen from the relation between P c and ρ c which is appeared as P c = ω c ρ c , where ω c = 2/3 + ω d and d denotes dominant epoch of any corresponding phases. But in this work, according to Eqs. (12) and (13), because of P c = −1/3ρ c so the conceptual problem is removed, for more discussion we refer the reader to [11], [12].
Now for omitting contribution of cut off dependence terms one can use following approach. Using the subtraction mechanism, for regularization, we introduce counter terms for energy density and pressure as where where M is a constant which risen from quantum gravity and for instance is equal to string or Planck mass. Using Eqs. (16) and (17), the equation of state (EoS) parameter of the vacuum fluctuations after some manipulation is obtained as This relation play a fundamental role in this work and it's importance will be clear in the following. Whereas this model is similar to Casimir approach, therefore we should consider both the negative and the positive signs for energy density, albeit it's sign is dependence to boundary conditions. To considering this note, we redefine ρ Z as where σ = ±1. Considering the σ signs, it is clear that attractive ( σ = +1 ) and repulsive (σ = −1) forces could be made. If we consider natural units, and using definition of Planck mass M P l = 1/ √ G, (G is the Newtonian constant) and critical density (ρ cri = 3H 2 (t)/8πG), the energy density of zero point fluctuations could be rewritten as where Ω Z = β(1 + 3ǫ)/M 2 P l and β = σM 2 /16π 2 . As regards, ρ Z (t) is time dependent, so the conservation equation energy of ρ Z is not satisfied. Thus using Eq.(20) andρ Z = −2ǫH(t)ρ Z , one can obtaiṅ where Therefore the energy density of quantum fluctuations capable to exchange energy with other components of the Universe. To investigate this energy transfer we consider some cases as bellow.

A. Energy transfer between zero point energy and matter
For this case we haveρ combining two above equations yieldρ which indicates, the conservation equation is satisfied. Using Eqs. (21) and (24), ρ m could be achieved as Where Ψ = 2β(3ǫ − 1)/9ǫ 2 andΨ is the integration constant. From Eq. (25) it is realized that in our model the matter density equation is modified, where the first term indicates the matters which risen from quantum fluctuations effect and latter term indicates the remain contribution of matter. One can interpret this extra term as the source of dark matter. In fact this source is risen from interaction of matter component of the Universe with quantum fluctuations contribution.

B. Energy transfer between zero point energy and other contribution of DE
Assume there is an internal interaction between different components of DE, namely ρ Λ and ρ Z . In a certain case which ω Λ = −1 one can writeρ usingρ Z = −2β(1 + 3ǫ)/(ǫ 2 a 3 (t)) and definition of scale factor in quasi de Sitter background, one can easily attain where C 0 is integration constant. Now using extend of ln (1 + x), we have ln[ǫH(t)] = (−5/6) + 3ǫH(t) + O(ǫ 2 ).
Combining Eqs.(26)-(28), the energy density of cosmological constant is reordered as where C 1 = 6C 0 (1 − 3ǫ) and C 2 = 3C 0 ǫ + β(1 + 3ǫ). For the coefficient of H 2 (t) will be positive, thus model reduces to modified ghost DE mechanism, and also when σ = −1 the Eq.(30) could be rewriten as M 2 /16π 2 (1 + 3ǫ) > 3C 0 ǫ. In addition as regards the big bang nucleosynthesis (BBN) constraint which has been discussed in [11], because ofρ Z is not proportional toρ Λ one can not give an upper bound on Ω Z = σM 2 (1 + 3ǫ)/6πM 2 P l . Comparing ρ Z with one in de Sitter model, it is clear that the coefficient (1 + 3ǫ) increases the energy density magnitude. Also whereas ρ DE is a combination of ρ Λ and ρ Z and both of them have interaction, we can not easily eliminate ρ Λ to guess upper bound for Ω Z [16].

C.
Energy Transfer between all components of the Universe In this state we consider a more general case which all components of the Universe have an interaction, in which the conservation equations are rewritten as belloẇ For this propose we assume three dissipated function forQ i as follows where κ, γ and θ indicate the strength of the coupling between different components of the Universe [17].

Solving conservation equation forQ1
Using Eq.(32) andQ 1 the conservation equation for DE components of the Universe is aṡ based on Eqs. (20) and (18), Eq.(33) could be rewritten aṡ Solving this differential equation yields whereD = 2(1 − ǫ)β/ǫ 3 andc is integration constant. By substituting Eq.(35) into Eq.(31) one can attain ρ m as follows In above equationB is integration constant. We propose that in a model which quantum effects are considered, in matter equation it is not only ordinary cold dark matter plays role in the evolution of the Universe, but there are matters which risen from matter and zero point quantum fluctuations interaction. Also it should be noted based on definition ofQ, different equations for matter component of the Universe could be attained.

Solving conservation equation forQ2
By rewriting Eq.(31) and (32) forQ 2 = 3γH(t)ρ m , we havė thus, Eq.(37), could be considered asρ m + 3H(t)(1 − γ)ρ m = 0, and after solving, one can attain By substituting Eq.(39) into Eq.(38), we have where M is the constant of integration. In a certain case which γ = 1/3 the above equation is reduced to This equation shows that ρ Λ , has a different behavior in comparison with ρ Z and it confirms a different source for DE.

Solving conservation equation forQ3
Now we want to investigate interaction between different components of the Universe by virtue ofQ 3 . Thence Eqs. (31) and (32), is reformed asρ Using definition of scale factor in quasi de Sitter background we rewrite Eq.(42) aṡ solving this equation for ρ m , yields whereȂ = θβ(3 + 11ǫ)/3ǫ 2 andK is the integration constant. In following, to solving ρ Λ , from Eq.(43) we proceed asρ and at last solving this equation is given

IV. BOUNDS WHICH RISEN FROM TIME EVOLUTION OF DE
In this section we want to compare the results of our model with results risen from standard ΛCDM model. For this goal, we start from the Friedmann equation. Therefore the ratio of DE density and critical energy density in the standard model as a function of red shift parameter, z, is obtained as Where we use Ω Λ = 0.73, Ω M = 0.23 and ω Λ = −0.98, which are obtained from a combinition fo CMB, BAO and SNeIa data set [11] and [18]. We assume, the components of DE density are quantum fluctuations energy density and cosmological constant, Λ, thus the Friedmann equation in our approach is as By substituting Eq.(20) and definition of ρ DE in Eq.(49) the Friedmann equation can be rewritten as Where Ω i (t) = ρ i (t)/ρ cri (0), i refers to m, Λ and Z, also ρ cri (0) denotes the critical energy density in present epoch. It should be noted the energy densities of curvature and radiation are omitted. From Eq.(20) and definition of dimensionless energy density parameters, we have Considering definition of ρ cri (t), Eq. (50) gives By dividing ρ DE and ρ cri (t), we receive whereΩ m = Ψ/ρ cri (0) and Ω m =Ψ/ρ cri (0). In F IG.1, the evolution of Eq.(54) versus z parameter shows a deviation in comparison to the ordinary ρ DE (t)/ρ cri (t), Eq. (48), that illuminates the effects of Ω Z and ǫ in the evolution of this function.
• d-From Eq. (45): where Ω m =K/ρ cri (0) and Ω ♯ m =Ȃ/ρ cri (0). The energy density parameter in this case, behaves as the case related to Eq.(27). This manner is similar to case a i.e. Eq.(54), and therefore, it does not need to plot it's evolution. It should be noted that the behaviour of ρ DE (z)/ρ cri (z) based on different quantities for Ω Z are plotted in F IG.3.

V. CONCLUSION
Vacuum quantum fluctuations in a quasi de Sitter space have considered. The scalar fields have quantized in a FLRW framework and it have shown that the contribution of vacuum fluctuations have increased in quasi de Sitter background in comparison with de Sitter case. It should be emphasized that we use subtraction approach to eliminate the infinities which have appeared in our calculations. Also using the physical energy density of zero point quantum fluctuation, it is realized that this component of the universe have to had an interaction with other components of the Universe. Also when the energy density of matter are achieved, we find out that beside of ordinary dark matter, there is exist component of matter created due to interaction with zero point quantum fluctuations. Also from time dependent zero point energy density, we have investigated how energy exchanges between different ingredients of the Universe. For different cases we have obtained the matter and Λ energy densities and the behavior of such quantities has been discussed. It is considerable that when we have considered the state which all components of the Universe exchange energy between them, time evolution of ρ DE (z)/ρ cri (z) have been shown that Q 2 = 3γHρ m is in best agrement in comparison to observational data set and also the interaction term asQ 1 = 3κH(t)ρ Λ , had not physical results. Also for more details we have discussed the bounds which have risen from time variation of DE density in comparison with standard Λ cosmology. To compare our results with observational dat, we have considered the time evolution of ρ DE (z)/ρ cri (z) which concluded from a combination of CMB, BAO and SNeIa data sets. From FIG.2 and  FIG.3, It have been illustrated the evolution of Eqs.(54) and (56) versus z and they were compared to observational results, in addition the effect of different values for Ω Z , includes of Ω Z = 0.16 , Ω Z = 0.05 and Ω Z = −0.16, has been considered.