# Effect of Fast Scale Factor Fluctuations on Cosmological Evolution

## Abstract

**:**

## 1. Introduction

^{−52}m

^{−2}[2]. Cosmological solutions exhibiting fast scale factor oscillations also appear quite generically in many other theoretical frameworks, such as semiclassical quantum gravity [3], cosmological models with oscillating dark energy [4], and various modified gravity theories [5]. It should be noted that the role of the universe scale factor a(t) is somewhat similar to the role of the center of mass coordinate, which describes the motion of some complicated macroscopic object. In classical mechanics, this motion is well-defined. However, in the quantum mechanical picture, the center of mass motion obeys the uncertainty principle, and thus is not well-defined anymore. Like any other object, the center of mass of a system experiences quantum fluctuations. Similar to quantum mechanics, in the semi-classical quantum gravity picture, we should expect a(t) not to be well-defined. Instead, the universe-scale factor a(t) must experience quantum fluctuations, especially since there is no reason to expect that the quantum state of the universe within the scope of the quantum Friedmann model will always be a scale factor eigenstate.

## 2. Materials and Methods

^{3}= μ = const, we may obtain the effective universe potential U(a), which is shown in Figure 1. The maximum of the effective potential corresponds to the Einstein static universe (ESU), which appears to be unstable to small perturbations, and can, therefore, be perturbed into an accelerating emergent cosmology, or into a contracting singular universe. The effective universe potential formalism appears to be quite useful, since, for any functional form of the effective potential U(a), there is an equation of state p = p(ρ) that will produce it.

_{p}= (hG/2π c

^{3})

^{1/2}(see, for example, [9]), so that the expected quantum uncertainty of the scale factor is at least Δa > l

_{p}. Somewhat similar ultra-fast quantum fluctuations in the scale factor may also be expected within the scope of the stochastic interpretation of quantum mechanics [10].

_{0}= Acos(ωt), the equation of motion for the angle θ may be obtained from the Euler–Lagrange equation as

_{s}is a slow varying function over one oscillation cycle and θ

_{f}is the rapidly oscillating component of the angle θ. Based on Equation (6), the rapidly oscillating component is given by

## 3. Results

_{f}of these fast fluctuations must depend on a

_{s}. By taking into account the generally assumed modifications of the uncertainty principle due to quantum gravity [9], we may expect the following simplified functional form of a

_{f}(a

_{s}) dependence

_{p}[9]. Substituting Equation (11) into Equations (2) and (3) (and neglecting the higher order terms) we obtain

_{s}and pressure p

_{s}will need to be determined based on the functional forms of ρ(a) and p(a) dependencies, and where we have kept only the lowest order terms in 1/a

_{s}, while eliminating all the fast-oscillating terms. However, if an analysis at early cosmological times is desirable, higher-order terms in 1/a

_{s}in these equations may be recovered from Equations (15) and (16) in a straightforward fashion.

_{s}= 0 and p

_{s}= 0. Such an unusual solution, with near-zero first and second derivatives of a

_{s}(t), may be obtained if k = 0 and Λ = 1.5a

_{s}

^{−2}(assuming that a

_{s}is large). Note that a

_{f}

^{2}ω

^{2}/c

^{2}~1 condition should be expected for quantum oscillations in the universe-scale factor on the Planck scale.

^{2}ω

^{2}/2a

_{s}

^{2}term. Based on Equation (18), the cosmological constant term begins to dominate at a sufficiently large a

_{s}. The observed “effective Λ“ in the current epoch, which may be characterized as the beginning of an accelerated expansion phase, would be of the order of Λ~1/a

^{2}if a

_{f}

^{2}ω

^{2}/c

^{2}~1 is assumed, which is consistent with the currently measured value of Λ = 1.11 × 10

^{−52}m

^{−2}[2]. It is noteworthy that the oscillating scale factor hypothesis is capable of reproducing both the experimentally observed modification of galaxy dynamics (see Figure 3), and the magnitude of the currently observed Λ with the same choice a

_{f}

^{2}ω

^{2}/c

^{2}~1 of the oscillation parameters, which is consistent with the expected quantum fluctuations in a(t) on the Planck scale.

_{eff}(a) on the short time scales, and demonstrate that corrections due to quantum scale factor fluctuations may indeed considerably modify the initial dynamics of the universe.

_{eff}may now exhibit considerable variations compared to the potential shown in Figure 1, which was derived in the absence of quantum scale factor fluctuations. Some examples of these variations are depicted in Figure 4. Similar to the inverted Kapitza pendulum, a stable local potential energy minimum may appear near the initially unstable static universe solution under the influence of quantum scale factor fluctuations—see, for example, the black and green curves in Figure 4. In the scenario represented by the black curve, the universe may originate due to quantum tunneling from a metastable state at a

_{s}~0.4 into an accelerating state at a

_{s}> 2, thus giving rise to an emergent inflationary cosmology.

## 4. Discussion and Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**(

**a**) A rigid pendulum with a vibrating pivot; (

**b**) Effective potential V

_{eff}(in units of mgl) for an inverted (Kapitza) pendulum for the cases of A

^{2}ω

^{2}/gl = 0 and A

^{2}ω

^{2}/gl = 4.

**Figure 3.**Log–log plot of the observed circular velocity V as a function of the maximum radius R for low-redshift disk galaxies re-plotted using data from Figure 4 of Reference [12]. Low scatter around the “universal slope” 1.84 × 10

^{−3}km/s/ly indicated by the red line implies that this relationship is generic for disc galaxies in the low-redshift universe. As a result, the galaxies behave as “cosmic clocks”, rotating roughly once a billion years at the very outskirts of their discs. This “universal slope” corresponds to α~0.01 [7].

**Figure 4.**Variations in the effective potential of the universe U

_{eff}(a

_{s}) under the influence of scale factor fluctuations, assuming different matter and radiation content. The original potential, calculated without scale factor fluctuations (see Figure 1), is shown by the red line.

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Smolyaninov, I.I. Effect of Fast Scale Factor Fluctuations on Cosmological Evolution. *Universe* **2021**, *7*, 164.
https://doi.org/10.3390/universe7060164

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Smolyaninov II. Effect of Fast Scale Factor Fluctuations on Cosmological Evolution. *Universe*. 2021; 7(6):164.
https://doi.org/10.3390/universe7060164

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Smolyaninov, Igor I. 2021. "Effect of Fast Scale Factor Fluctuations on Cosmological Evolution" *Universe* 7, no. 6: 164.
https://doi.org/10.3390/universe7060164