Cosmological implications of the transition from the false vacuum to the true vacuum state

We study the cosmology with the running dark energy. The parametrization of dark energy with the respect to the redshift is derived from the first principles of quantum mechanics. Energy density of dark energy is obtained from the quantum process of transition from the false vacuum state to the true vacuum state. This is the class of the extended interacting $\Lambda$CDM models. We consider the energy density of dark energy parametrization $\rho_\text{de}(t)$, which follows from the Breit-Wigner energy distribution function which is used to model the quantum unstable systems. The idea that properties of the process of the quantum mechanical decay of unstable states can help to understand the properties of the observed universe was formulated by Krauss and Dent and this idea was used in our considerations. In the cosmological model with the mentioned parametrization there is an energy transfer between the dark matter and dark energy. In such a evolutional scenario the universe is starting from the false vacuum state and going to the true vacuum state of the present day universe. We find that the intermediate regime during the passage from false to true vacuum states takes place. The intensity of the analyzed process is measured by a parameter $\alpha$. For the small value of $\alpha$ ($0<\alpha<0.4$) this intermediate (quantum) regime is characterized by an oscillatory behavior of the density of dark energy while the for $\alpha>0.4$ the density of the dark energy simply jumps down. In both cases (independent from the parameter $\alpha$) the today value of density of dark energy is reached at the value of $0.7$. We estimate the cosmological parameters for this model with visible and dark matter. This model becomes in good agreement with the astronomical data and is practically indistinguishable from $\Lambda$CDM model.


I. INTRODUCTION
The standard cosmological model (ΛCDM model), which describes the Universe, is the most favored by astronomical observations such as supernovae of type Ia or measurements of CMB. In the ΛCDM model, the dark matter is treated as dust and dark energy has the form of the cosmological constant Λ bare . We are looking an alternative for the ΛCDM model The nature of both components of the Universe is unknown up to now but we describe these in terms of useful fiction, the cosmological constant and the cold dark matter which is kind of the dust perfect fluid.
In this paper we concentrate on the interpretation of dark energy rather in terms of running cosmological constant than in term of the pure cosmological constant parameter (Λ bare in our approach). It is consequence of some problems with interpretation of of pure the cosmological constant, namely: 1. We cannot explain why the cosmological constant is not large.
2. We do not know why it is not just equal zero. 3. We cannot explain why energy densities of both dark energy and dark matter, expressed in terms of dimensionless density parameters, are comparable in the current epoch (cosmic coincidence problem).
In our proposition of the explanation of these problems with the cosmological constant parameter, we base on the theories of the cosmological constant in which the vacuum energy is fixed by the fundamental theory [1]. Extending the ΛCDM model beyond the classical regime, we apply quantum mechanics as a fundamental theory, which determines cosmological parameters and explain how cosmological parameters vary during the cosmic evolution.
The cosmological constant is the source of two problems in modern cosmology. The first problem is the cosmological constant problem, which is consequence of the interpretation of dark energy as a vacuum energy. The observed present value of the cosmological constant is 120 orders of magnitude smaller than we expect from quantum physics. The second problem is the coincidence problem. If we assume that the dark energy is always constant then the ΛCDM model cannot explain why the cosmological constant has the same order of magnitude as density of matter today. If the model belongs to the class of running dark energy cosmologies then the first problem of cosmological constant can be solved.
This question seems to be crucial in contemporary physics because its solution would certainly mean a very crucial step forward in our attempts to understand physics from the boundary of particle physics and cosmology. The discussion about the cosmological constant problem can be found in papers [1]- [14].
In our model, the influence of running dark energy densities of both visible and invisible matter is very small. Thus we share Weinberg's opinion, according to which looking for a solution of the coincidence problem, we should consider the anthropic principle. According to Weinberg's argument, any observers should not be in the Universe if the cosmological constant was even three orders of magnitude larger than it is now.
Coleman et al. [15]- [17] discussed the instability of a physical system, which is not at an absolute energy minimum, and which is separated from the minimum by an effective potential barrier. They showed that if the early Universe is too cold to activate the energy transition to the minimum energy state then a quantum decay, from the false vacuum to the true vacuum, is still possible through a barrier penetration via the macroscopic quantum tunneling.
The discovery of the Higgs-like resonance at 125-126 GeV [18]- [21] caused the discussion about the instability of the false vacuum. If we assume that the Standard Model well describes the evolution of the Universe up to the Planck epoch then a Higgs mass m h < 126GeV causes that the electroweak vacuum is in a metastable state [19]. In consequence the instability of the Higgs vacuum should be considered in the cosmological models of the early time Universe.
The idea that properties of the quantum mechanical decay process of metastable states can help to understand the properties of the observed universe was formulated in [22][23][24]. It is because the decay of the false vacuum is the quantum decay process [15][16][17]. This means that state vector corresponding to the false vacuum is a quantum unstable (or metastable) state. Therefore all general properties of quantum unstable systems must also occur in the case of such a quantum unstable state as the false vacuum and as a consequence models of quantum unstable systems can be used to analyze properties of the systems which time evolution starts from the false vacuum state.
In this paper, we assume the Breit-Wigner energy distribution function, which is very often used to model unstable quantum systems, as a model of the process of the energy transition from the false vacuum to the true vacuum. In consequence the parametrization of the dark energy is given by formula where α and E R are model parameters describing the variation from the standard cosmological model. The values of α parameter belong to interval 0, 1). Note that if the α parameter or E R is equal zero than the model is equivalent to the ΛCDM model.
(1) can be rewritten in the equivalent form Here the units 8πG = c = 1 are used.
The functions J(t) and I(t) are defined by the following expressions Integrals J(t) and I(t) can be expressed by the exact solutions of these integrals. Formula J(t) is described by the following expression and I(t) is expressed by V 0 t and V 0 is the volume of the Universe in the Planck epoch. In this paper we assume that V 0 = 1. The function E 1 (z) is called the exponential integral and is defined by the formula: x dx (see [25,26]).

II. PRELIMINARIES: UNSTABLE STATES
As it was mentioned in Sec. 1 we will use the parametrization of the dark energy transition from the false vacuum state to the true vacuum state following from the quantum properties of a such process. This process is a quantum decay process, so we need quantities characterizing decay processes of quantum unstable systems. The main information about properties of quantum unstable systems is contained in their decay law, that is in their survival probability. So if one knows that the system is in the initial unstable state |φ ∈ H, (H is the Hilbert space of states of the considered system), which was prepared at the initial instant t 0 = 0, then one can calculate its survival probability (the decay law), P(t), of the unstable state |φ decaying in vacuum, which equals where A(t) is the probability amplitude of finding the system at the time t in the rest frame O 0 in the initial unstable state |φ , and |φ(t) is the solution of the Schrödinger equation for the initial condition |φ(0) = |φ , which has the following form Here |φ , |φ(t) ∈ H, and H denotes the total self-adjoint Hamiltonian for the system considered. The spectrum of H is assumed to be bounded from below: E min > −∞ is the lower bound of the spectrum σ c (H) = [E min , +∞) of H). Using the basis in H build from normalized eigenvectors |E , E ∈ σ c (H) of H and expanding |φ in terms of these eigenvectors one can express the amplitude A(t) as the following Fourier integral where ω(E) > 0 (see: [27,28]).
Note that in fact the amplitude A(t) contains information about the decay law P φ (t) of the state |φ , that is about the decay rate γ 0 φ of this state, as well as the energy E 0 φ of the system in this state. This information can be extracted from A(t). It can be done using the rigorous equation governing the time evolution in the subspace of unstable states, H ∋ |φ ≡ |φ . Such an equation follows from Schrödinger equation (9) for the total state space H.
The use of the Schrödinger equation (9) allows one to find that within the problem This relation leads to the conclusion that the amplitude A(t) satisfies the following equation where and h(t) is the effective Hamiltonian governing the time evolution in the subspace of unstable states H = PH, where P = |φ φ| (see [37] and also [38,39] and references therein). The subspace H ⊖ H = H ⊥ ≡ QH is the subspace of decay products. Here Q = I − P. There is the following equivalent formula for h(t) [37][38][39]: One meets the effective Hamiltonian h(t) when one starts with the Schrödinger equation and are the instantaneous energy (mass) E φ (t) and the instantaneous decay rate, Γ φ (t) [37][38][39].
Here ℜ (z) and ℑ (z) denote the real and imaginary parts of z respectively. The relations (12), (14) and (16) are convenient when the density ω(E) is given and one wants to find the instantaneous energy E φ (t) and decay rate Γ φ (t): Inserting ω(E) into (10) one obtains the amplitude A(t) and then using (14) one finds the h(t) and thus E φ (t) and Γ φ (t). The simplest choice is to take ω(E) having the Breit-Wigner form where N is a normalization constant and Θ(E) = 1 for E ≥ 0 and Θ(E) = 0 for E < 0.
The parameters E 0 and Γ 0 correspond with the energy of the system in the unstable state and its decay rate at the exponential (or canonical) regime of the decay process. E min is the minimal (the lowest) energy of the system. Inserting ω BW (E) into formula (10) for the amplitude A(t) after some algebra one finds that where Here . The integral I β (t) can be expressed in terms of special functions as follows where E 1 (z) denotes the integral-exponential function defined according to [25,26], (z is a complex number).
Next using this A(t) given by relations (18), (19) and the relation (14) defining the effective Hamiltonian h φ (t) one finds that within the Breit-Wigner model considered where It is important to be aware of the following problem: Namely from the definition of J β (τ ) one . This is because within the model defined by the Breit-Wigner distribution of the energy density, ω BW (E), the expectation value of H, that is φ|H|φ is not finite. So all the consideration based on the use of J β (τ ) are valid only for τ > 0.
Note that simply which allows one to find analytical form of J β (τ ) having such a form for I β (τ ).
We need to know the energy of the system in the unstable state |φ considered. The instantaneous energy E φ (t) of the system in the unstable state |φ is given by the relation (16). So within the Breit-Wigner model one finds that or, equivalently (This relation, i.e. κ(t), was studied, for example in [40,41]).
It is relatively simple to find asymptotic expressions I β τ and J β (τ ) for τ → ∞ directly from (19) and (22) using , e.g., method of the integration by parts. We have for τ → ∞: and These two last asymptotic expressions alows one to find for τ → ∞ the asymptotic form of the ratio (21), (24) and (25), which has much simpler form than asymptotic expansions for I β (τ ) and J β (τ ). One finds that for τ → ∞, Starting from this asymptotic expression and formula (24) or making use of the asymptotic expansion of E 1 (z) [26] and (20), where | arg z| < 3 2 π, one finds, eg. that for t → ∞, This last relation is valid for t > T , where T denotes the cross-over time, ie. the time when exponential and late time inverse power law contributions to the survival amplitude begin to be comparable.
Some cosmological scenario predict the possibility of decay of the Standard Model vacuum at an inflationary stage of the evolution of the universe (see eg. [42] and also [43] and reference therein) or earlier. Of course this decaying Standard Model vacuum is described by the quantum state corresponding to a local minimum of the energy density which is not the absolute minimum of the energy density of the system considered (see, eg. Fig. 1).
The scenario in which false vacuum may decay at the inflationary stage of the universe corresponds with the hypothesis analyzed by Krauss and Dent [22,23]. Namely in the mentioned papers the hypothesis that some false vacuum regions do survive well up to the cross-over time T or later was considered where T is the same cross-over time which is is considered within the theory of evolving in time quantum unstable systems. The fact that the decay of the false vacuum is the quantum decay process means that state vector corresponding to the false vacuum is a quantum unstable (or metastable) state. Therefore all the general properties of quantum unstable systems must also occur in the case of such a quantum unstable state as the false vacuum. This applies in particular to such properties as late time deviations from the exponential decay law and properties of the energy E f alse 0 (t) of the system in the quantum false vacuum state at late times t > T : In [44] it was pointed out the energy of those false vacuum regions which survived up to T and much later differs So within the cosmological scenario in which the decay of false vacuum is assumed the unstable state |φ corresponds with the false vacuum state: |φ = |0 false . Then |0 true is the true vacuum state, that is the state corresponding to the true minimal energy. In such a case E 0 → E false 0 is the energy of a state corresponding to the false vacuum measured at the canonical decay time (the exponential decay regime) and E true 0 is the energy of true vacuum (i.e., the true ground state of the system), so E true 0 ≡ E min . The corresponding quantum mechanical process looks as it is shown in Fig 1. If one wants to generalize the above results obtained on the basis of quantum mechanics to quantum field theory one should take into account among others a volume factors so that survival probabilities per unit volume should be considered and similarly the energies and the decay rate: is the volume of the considered system at the initial instant t 0 , when the time evolution starts. The volume V 0 is used in these considerations because the initial unstable state |φ ≡ |0 false at t = t 0 = 0 is expanded into eigenvectors |E of H at this initial instant t 0 , (where E ∈ σ c (H)) and then this expansion is used to find the density of the energy distribution ω(E). It is easy to see that the mentioned changes E → E V 0 and Γ 0 → Γ 0 V 0 do not changes integrals I β (t) and J β (t) and the relation (25). Similarly in such a situation the parameter β = E 0 −E min Γ 0 does not changes. This means that the relations (24), (25), (30) can be replaced by corresponding relations for the densities ρ de or Λ (see, eg., [40,46,47]). Within such an approach E(t) corresponds to the running cosmological constant Λ(t) and E min to the Λ bare . The parametrization used in next Sections is based on relations (24), (25). Integrals (3), (4) introduced in Sec. 1 are obtained from (22) and (19) replacing β by 1−α α . Similarly solutions (5) and (6) correspond to (20) and to the function J β (τ ) obtained from (20) using (23).
The cosmological model with the parametrization of the dark energy (1) belonging to the class of parametrizations proposed in [40] after putting E R = E 0 − Λ bare assumes the following form of ρ de (we use units 8πG = c = 1) It can be introduced as the covariant theory from the following action where R is the Ricci scalar, L m is the Lagrangian for the barotropic fluid and g µν is the metric tensor. We assume the signature of the metric tensor as (+, −, −, −) and, for the simplicity, that the constant curvature is zero (the flat model). The Ricci scalar for the Friedmann-Lemaitre-Robertson-Walker (FLRW) metric is presented by the following formula where a dot means the differentiation with respect to the cosmological time t.
The Lagrangian for the barotropic fluid is expressed by the formula where ρ tot is the total density of fluid and p tot (ρ tot ) is the total pressure of fluid [48]. We assume that this fluid consists of three components: the baryonic matter ρ b , the dark matter ρ dm and the dark energy ρ de . We treat the baryonic matter and the dark matter like dust.
In consequence the equations of state for them are following: p b (ρ b ) = 0 and p dm (ρ dm ) = 0.
The equation of state for the dark energy is assumed in the form p de (ρ de ) = −ρ de .
Of course, the total density is expressed by ρ tot = ρ b + ρ dm + ρ de and the total pressure is expressed by p tot (ρ tot ) = p de (ρ de ) = −ρ de .
We can find the Einstein equations using calculus of variations method by variation action (32) by the metric g µν . Then we get two equations: the Friedmann equation where H =˙a a is the Hubble function, and the acceleration equation From Eqs. (35) and (36) we can get the conservation equatioṅ ρ tot = −3H(ρ tot + p tot (ρ tot )).
The above equation can be rewritten aṡ where ρ m = ρ b + ρ dm .
Let Q is the interaction between the dark matter and the dark energy. Then Eq. (38) is where the interaction Q is defined by Eq. (41). The interaction between the dark matter and the dark energy can be interpreted as the energy transfer in the dark sector. If Q > 0 then the energy flow is from the dark energy to the dark matter. If Q < 0 then the energy flow is from the dark matter to the dark energy.
For the description of the evolution of the universe is necessary to use the Friedmann equation (35) and the conservation equation (38). These formulas can be rewritten in dimensionless parameters. Let Ω m = ρm       energy (31) can approximated by the following expression   The dark energy is significantly lower than the energy density of matter in the early universe, which has a consequence that the transfer to the dark sector is negligible (see  The conservation equation for the dark energy (41) can be rewritten aṡ where p de is an effective pressure of the dark energy. In this case the equation of state for the dark energy is expressed by the following formula where the function w(t) is given by the expression The diagram of coefficient equation of state w(t) is presented in Fig. 9. The function w(t), for the late time, is a constant and equal −1 which means that it describes the cosmlogical constant parameter. Note that the function w(t) is also equal −1 which means that ρ de is constant as a consequance of the conservation condition (transfer between sectors is negligible). Therefore, the energy transfer is an effective process only during intermediate oscillation period (quantum regime).
Let ρ de ≫ ρ m in the Planck epoch. Then our model predicts an inflation in the Planck epoch. The formula for e-foldings N = H init (t fin −t init ) (see [49]) gets the following expression for our model  The data of supernovae of type Ia, which were used in this paper, are taken from the Union 2.1 dataset [50]. In this context we use the following likelihood function where A = (µ obs − µ th )C −1 (µ obs − µ th ), B = C −1 (µ obs − µ th ), C = TrC −1 and C is a covariance matrix for SNIa. The observer distance modulus µ obs is defined by the formula dz ′ H(z) ). We use the following BAO data: Sloan Digital Sky Survey Release 7 (SDSS DR7) dataset at z = 0.275 [51], 6dF Galaxy Redshift Survey measurements at redshift z = 0.1 [52], and WiggleZ measurements at redshift z = 0.44, 0.60, 0.73 [54]. The likelihood function is defined by the expression where r s (z d ) is the sound horizon at the drag epoch [55,70].
Measurements of the Hubble parameter H(z) of galaxies were taken from [66][67][68]. The likelihood function is given by the following formula The likelihood function for the Alcock-Paczynski test [56,57] has the following form where and AP (z i ) obs are observational data [53,[58][59][60][61][62][63][64][65].  In this paper, the likelihood function for the measurements of CMB [69] and lensing by Planck, and low-ℓ polarization from the WMAP (WP) has the following form where C is the covariance matrix with the errors, x is a vector of the acoustic scale l A , the (Ω m,0 = 0.3009). We assumed that E 0 /(3H 2 0 ) is equal 10 120 , but changing of the value of E 0 /(3H 2 0 ) does not influence for results. Note that the values of the likelihood function are not sensitive to changing of α parameter.
shift parameter R and Ω b h 2 where where z * is the redshift of the epoch of the recombination [70].
In this paper, the final formula for likelihood function is given in the following form The statistical analysis was done by our own code CosmoDarkBox. This code uses the Metropolis-Hastings algorithm [71,72].
We estimated four cosmological parameters: H 0 , Ω m,0 , α and E 0 parameter. Our statistical results are completed in Table I  remains. The jump down mechanism is independent from the parameter α value, which leads to to solving the cosmological constant problem.
In the early Universe the energy density of dark energy is significantly lower than the energy density of dark matter, therefore the change of energy density of the dark matter, which is caused by energy transfer in the dark sector, is negligible. From the statistical analysis of the model we found that the model is generic in the sense that independently of the values of the parameters α and E 0 we can obtain the present value of the energy density of the dark energy. Therefore, the ΛCDM model is an attractor which the all models with different values of parameters α and E 0 can reach at. The final interval of evolution for which we have data at dispose is identical for whole class of models, therefore it is impossible to find best-fitted values of model parameters and indicate one model (degeneration problem).

ACKNOWLEDGMENTS
The work has been supported by Polish National Science Centre (NCN), project DEC-