Characteristic length of dynamical reduction models and decay of cosmological vacuum

Characteristic length of mass density resolution in dynamical reduction models is calculated utilizing energy conservation law and viable cosmological model with decreasing energy density of vacuum (dark energy density). The value found, $ \sim 10^{-5}$ cm, numerically coincides with phenomenological spatial short-length cutoff parameter introduced in the Ghirardi-Rimini-Weber model. It seems that our results support the gravity induced mechanism of dynamical reduction.


Introduction
The Schrödinger equation when applied for system that is not isolated and interacts with complicated "environment" (it is typical for macro-systems) results in appearance of the entanglement states involved many degrees of freedom.The superposition of such states for all practical purposes results in the same outcome of measurements as "mixed" state described by diagonal density matrix.However it does not mean that superposition disappears, it still exists globally but is unobservable at either system alone.The apparent decoherence (suppression of interference) for macro-systems can be explained then as result of the entanglement of system with its environment (for a review see, e.g., [1]).The density matrix of the corresponding system in this approach is just means for calculating expectation values or probabilities for outcomes of measurements.
Another approach to decoherence of macro-systems is grounded on idea of stochastic dynamical reduction (for a review see, e.g., [2,3]) and treats the decoherence as real process of state vector reduction that takes place because corresponding system is influenced by stochastic forces, in other words, interacts with some stochastic environment.The nature of the stochastic field is usually not specified there, except for the dynamical reduction models involving gravity, so called Newtonian Quantum Gravity approach [4,5,6,7,8], where stochastic field is associated with non-relativistic gravity potential field.Reasons why gravity can be treated as stochastic field at small space-time scales are known for a long time (see Ref. [9]): inasmuch as refined length measurement requires large momentum, a probe itself disturbs the gravity field curving space-time and distorting the interval one seeks to measure leading to limit of measurability of gravity field (metric tensor).In fact, it is consequence of the equivalence principle.
Noteworthy that spatial reduction of wave function by stochastic forces leads to momentum increase because of the uncertainty principle, and, therefore, to increase of particle energy.It results in apparent violation of the conservation laws, and the rate of increase diverges for a point-like object [7].To alleviate the problem the spatial coarse graining of the mass density (cutoff on spatial mass density resolution) is used in the dynamical reduction models, this cutoff parameter fixes also the distance scale (localization width) beyond which the wave function collapse becomes effective.The value of the short-length cutoff, 10 −5 cm, that was proposed in Ref. [10] (see also [7]), keeps energy non conservation rate below experimental limits.Certainly, to explicitly satisfy the conservation laws one needs account for not only particle contribution, but also contribution of the stochastic field to the conserved quantities.
Because, as we discussed above, the most natural candidate for stochastic field of the dynamical reduction models is gravity, and a value of the scale factor governs the rate of energy transfer from stochastic field to the matter particles, it might be expected that a value of the scale factor is related to the rate of decay of cosmological (gravitational) vacuum and, consequently, related to decrease of the vacuum energy density with time.Note that cosmological scenarios of the universe evolution where Λ(t) decreases slowly with time (Λ represents the energy density of the vacuum) are allowed by the observational cosmology (see, e.g., [11,12]).
In this paper we utilize the correlation function of fluctuations of gravity acceleration field [4] (see also [5]) and viable model of the cosmological vacuum decay [13] to estimate value of the spatial short-length cutoff parameter of the dynamical reduction models. 1 In Section 2 the cosmological model [13] with decaying vacuum energy density is briefly reviewed.The rate of the cosmological vacuum decay found in the model is utilized in Section 3 to calculate the characteristic length of the dynamical reduction models involving gravity.In Section 4 conclusions are given and some possible consequences for biogenesis are briefly discussed.models (see, e.g., [15,11,12]) because it was explicitly demonstrated in Refs.[13,12] that this model is compatible with current cosmological data.
The Einstein field equations are (throughout the paper we use the "natural units" where = c = 1) where T µν is the energy-momentum tensor of "ordinary" (nearly 3% of total energy density) and "dark" (nearly 27%) matter, while Λ is the cosmological constant that is responsible for famous "dark" energy density (nearly 70%) contribution to total energy density, and also for acceleration of the universe expansion at the present epoch.The cosmological constant can be treated as the energy density of the cosmological vacuum, ǫ vac .Since a vacuum has equation of state ǫ vac = −p vac , p is pressure, then (for recent review see, e.g., [16]) The nature of "dark" matter is still unclear, most probably (see, e.g., [17]) some weakly interacting particles are responsible for the "dark" matter energy density contribution to the energy-momentum tensor of matter fields.
According to Bianchi identities therefore if Λ depends on time (i.e.vacuum decays in the course of the expansion) then T µν can not be separately conserved and there is a coupling between T µν and Λ that follows from Eq. ( 1): here u µ (x) is local 4-velocity of cosmological expansion.The assumption of isotropy and homogeneity implies that the large scale geometry can be described by a metric of the form here a(t) is scale (expansion) factor of the universe and flat spatial sections are also assumed.Taking then perfect fluid form for T µν , one can get where the overdot denotes covariant derivative along the world lines (time comoving derivative, for instance, ǫ = u α ∂ α ǫ).For homogeneous and isotropic Friedmann-Robertson-Walker geometry where is the Hubble parameter, it measures the rate of expansion of the universe.Then, if vacuum transfers energy to matter, the question arises where the matter stores the energy received from the vacuum decay process.The traditional approach for the vacuum decay process is vacuum decay into matter particles.Following to Ref. [13], one can assume that vacuum decay results (mainly) in creation of weakly interacting "dark" matter particles and, so, neglect a contribution of the "ordinary" matter to the left hand side of Eq. (7).It allows to relate the time dependence of the energy density of the vacuum with temporal evolution of the "dark" matter.Then, assuming that "dark" matter is pressureless, p d = 0 (cold "dark" matter model), one can get the following equation: Because vacuum decays into "dark" matter the latter will dilute more slowly compared to its standard evolution, ǫ d ∼ a −3 , when the vacuum energy density does not change in the course of the expansion, ǫvac = 0. Then making a specific ansatz for the "dark" matter energy density where δ > 0 is a constant that characterizes the deviation from the standard evolution, ǫ d0 = ǫ d (t 0 ) and a 0 = a(t 0 ) are the current values of the "dark" matter energy density and of the scale factor of the universe respectively, we have where ǫ vac does not depend on time.
It was found from analysis of cosmological data that δ = 0.06±0.10[12].Such a relatively slow decrease of Λ(t) (vacuum energy density) indicates, perhaps, that viable cosmological variable-Λ models with cold "dark" matter (CDM) and standard physics should not differ too drastically from concordance ΛCDM model to be compatible with observational cosmology.
In the next Section we study whether the found in the Refs.[13,12] rate of cosmological vacuum decay is compatible with the rate of particle energy increase presupposed in the dynamical reduction models, and, so, whether the corresponding characteristic length can be defined by Λ(t)CDM cosmology.

Characteristic length for given rate of the cosmological vacuum decay
To perform the corresponding analysis let us assume that, while main energy gain from cosmological vacuum decay is adopted by the "dark" matter, the cosmological vacuum couples not only to the "dark" matter but to the "ordinary" matter as well, and assume that vacuum decay does not lead to creation of "ordinary" matter particles but increases the mean kinetic energy of the ones.Then, if corresponding kinetic energy gain attributed to all the particles in the universe is much smaller than the total loss of vacuum energy in the universe, it can not essentially influence on the results of Ref. [13] that are briefly reviewed in the previous Section.The next step then is to relate the energy gain of a particle, whose mass is locally smeared within the corresponding characteristic volume, with the rate of energy loss of vacuum within the same volume.It gives us, in fact, upper limit to the particle energy increase allowing by the loss of vacuum energy.
For the sake of simplicity, let us consider a energy gain of a single nucleon.Let us assume that decay of the cosmological vacuum induces stochastic gravity field that results in stochastic acceleration of a particle.Then the mean non-relativistic kinetic energy induced by the vacuum decay is where g and v are stochastic acceleration and velocity field respectively, ... means the averaging over the corresponding characteristic volume V c , and we assume that initially v is equal to zero, v(t i ) = 0. To proceed we need in correlation function of fluctuations of acceleration field, g.The analysis of measurability of the Newtonian acceleration field that was done in Ref. [4] (see also [5]) results in the following expression for correlation function (assuming that g = 0): Hereafter we will take Eq. ( 14) as equality.Hence we obtain from Eqs. ( 13) and ( 14) that and then the rate of the particle energy gain, dEpart dt , is Taking into account the energy conservation, we get finally the equation and, therefore, One can see from Eq. ( 12) that Then one can conclude from the above expressions that V c (t) increases (in cosmological sense, i.e. very slowly) with time.Now let us estimate the present (t = t 0 ) value of the characteristic volume, V c (t 0 ).Taking into account that where H 0 is current value of the Hubble parameter, H 0 = H(t 0 ), we get for the characteristic volume and, therefore, for the characteristic length The total energy density of the universe is close to the critical energy density (see, e.g.[16]), ǫ crit (t 0 ) = 3H 2 0 8πG .Then, because "dark" matter energy density is approximately equal to 27% of the total energy density, we get and, finally, Here δ = 0.06, H 0 = 1 1.3•10 28 cm −1 = 0.769•10 −42 GeV, G = 1/M 2 P l , M P l = 1.22•10 19 GeV, and m is assumed to be proton mass, m = 0.938 GeV.Also taking into account that 1 Gev −1 = 1.973 • 10 −14 cm we get finally R c (t 0 ) = 1.06 • 10 −5 cm.Then the characteristic volume multiplied by the number of particles making up "dark" and "ordinary" matter is much less than the volume of the universe, justifying thereby neglect of influence of the vacuum energy transfer to the kinetic energy of particles on temporal evolution of the vacuum energy density.
Interestingly enough that practically the same value, 10 −5 cm, was proposed in Ref. [10] for phenomenological spatial cutoff parameter of Quantum Mechanics with Spontaneous Localizations, and is accepted now [2,18] as low limit for the characteristic length in the dynamical reduction models.Then our finding, perhaps, supports the gravity induced mechanism of dynamical reduction.
Note that m-dependence of R c is rather weak, R c ∼ m 1/6 , and therefore R c does not change much even for electron mass.Also this m-dependence does not spoil the fast decoherence of the distant, r ≫ R c , superposition of macro-objects, because then decoherence time, t dec , is t dec ∼ R c /Gm2 [6] and, consequently, t dec ∼ m −11/6 resulting, as well as for m-independent R c , in extremely small value that is desired property for Schrödinger cat states.

Concluding remarks
Summarizing this work, we point out that apparent violation of the energy conservation is not actually shortcoming of the dynamical reduction models.Moreover, energy conservation law gives deep insight into the physics of the spatial cutoff parameter.Namely, we found that the value, ∼ 10 −5 cm, of the characteristic length is, perhaps, conditioned by the present rate of energy transfer from decaying vacuum to a particle.Because a value of the rate of the vacuum decay need be compatible with the observational cosmology, one can say that, in a certain sense, cosmology dictates the characteristics of dynamical reduction.
One can speculate that increase of V c with cosmological time (see Eq. ( 18)) is related to the problem of appearance of life and consciousness.First of all, note that biogenesis on Earth has occurred very rapidly, during 0.1 +0.5 −0.1 Gyr and, therefore, appearance of life could be highly probable process [19].However, while age of Earth, ≈ 4.566 Gyr, is much lower than age of the universe, ≈ 13.7 Gyr, we do not observe signals of a vital activity of extraterrestrial civilizations ("silence of the universe" problem, see, e.g., [20]).Explanation of the apparent paradox can be the following.If quantum mechanics indeed play a key role in the origin of biological organisms 2 , then spatial cutoff parameter value (that regulates the "borderline" between quantum and classical worlds) is extremely important for origin and operation of biological organisms.Therefore an appropriate value of the cutoff parameter R c fixes the cosmological time when appearance of living organisms becomes possible.Noteworthy that formation of Earth fell roughly at the same time interval when expansion of the universe became accelerated [22].So, one can speculate that necessary conditions for origin of a life appear only recently, during the epoch of accelerated expansion when the characteristic length also increases with acceleration.Then the observational lack of extraterrestrial intelligent life can be consequence of the "time cutoff" for the most early emergence of a life in the universe.