Advances in High Energy Physics

Volume 2018, Article ID 3916727, 15 pages

https://doi.org/10.1155/2018/3916727

## A New Inflationary Universe Scenario with Inhomogeneous Quantum Vacuum

^{1}College of Physics, Jilin University, Changchun, Jilin 130021, China^{2}Department of Chemistry and Physics, State University of New York, Stony Brook, NY 11794, USA^{3}State Key Laboratory of Electroanalytical Chemistry, Changchun Institute of Applied Chemistry, Changchun, Jilin 130022, China

Correspondence should be addressed to Yilin Chen; moc.liamg@32806991niliynehc

Received 7 February 2018; Accepted 1 April 2018; Published 8 May 2018

Academic Editor: Marek Szydlowski

Copyright © 2018 Yilin Chen and Jin Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

We investigate the quantum vacuum and find that the fluctuations can lead to the inhomogeneous quantum vacuum. We find that the vacuum fluctuations can significantly influence the cosmological inhomogeneity, which is different from what was previously expected. By introducing the modified Green’s function, we reach a new inflationary scenario which can explain why the Universe is still expanding without slowing down. We also calculate the tunneling amplitude of the Universe based on the inhomogeneous vacuum. We find that the inhomogeneity can lead to the penetration of the Universe over the potential barrier faster than previously thought.

#### 1. Introduction

Gravity governs the evolution of the Universe. A great breakthrough in gravitational research in the last century is the discovery of general relativity (GR). After Einstein laid down the relationship between the space-time geometry through the curvature and the matter through energy momentum, the general relativity theory has been applied to many fields, especially in astrophysics and cosmology. Another great achievement of modern physics is quantum mechanics. Based on that, quantum field theory (QFT) emerged, which has been tested in many experiments. Due to the success of relativity in the macroscopic world and quantum mechanics in the microscopic world, it is natural to ask how we can combine them together. In cosmology, this issue becomes more apparent after the birth of the theory of inflationary Universe [1, 2] and successful in solving the horizon and flatness problems and eventually quantifying the seeds in terms of the density fluctuations for large scale structure formation and inhomogeneity for the microwave background radiation. In this theory, at the very early history, the Universe expanded exponentially, and the expansion was sustained by the vacuum energy. Despite the success, one issue remains on how the Universe quits this stage. Many proposals were suggested to resolve this issue [3, 4]. It turns out rather difficult to complete and have a graceful exit for the old inflationary scenario with multiple bubbles coalescing in a Universe suggested by Guth [1]. Chaotic inflationary scenario has been suggested with essentially one bubble for a Universe but forever evolving to avoid the exiting issue.

The idea of the inflationary scenario is to combine the quantum vacuum energy for describing the matter and the Einstein’s equation for describing the space-time evolution together. Based on the equivalence principle of GR, each form of the energy influences the space-time in the same way. Quantum vacuum brings a new source of energy. Naively, one can study how the quantum vacuum influences the space-time evolution by simply put the energy-momentum tensor for the quantum vacuum on the right hand side and the Einstein space-time curvature tensor on the left hand side of the Einstein equation of GR.

Unfortunately, there is an issue once one naively puts these two theories of general relativity and quantum mechanics together. This is because there is currently no applicable method for quantizing GR or space-time. This indicates that quantum mechanics and GR are at totally different footing and do not match each other. Therefore, many existing theories suggested certain approximate equations relating these two theories. However, these approaches often show great ambiguity. Firstly, let us look at Einstein’s field equation: This equation in the current form would not make sense if represents the quantum (such as vacuum here) rather than classical matter. This is because the energy-momentum tensor is an operator in quantum world. In quantum mechanics, once we attempt to observe something about a system, the expected observed values correspond to the average values of the corresponding operator for the observable. Therefore, it seems natural to modify the Einstein equation by changing the energy-momentum operator for the average value of it: This describes, at the average level, how the quantum matter influences the space-time evolution. For quantum matter, we know that fluctuations are unavoidable. This is even true for the quantum vacuum. One natural question to ask is how the quantum matter fluctuations influence the space-time evolution. Another way to modify the Einstein equation for taking the fluctuations into account is to take the square of both sides of the Einstein equation and then take the average value on the right hand side: In fact, (2) and (3) are equivalent only if the fluctuation of energy-momentum tensor is zero, but this of course is not the case since the energy-momentum tensor even for quantum vacuum is not zero.

This example illustrates that before we “totally” understand quantum gravity, there could be many ways of combining the general relativity and quantum field theory, which are not equivalent at the semiclassical level. Furthermore, these differences in semiclassical treatments are caused by the fluctuations of the quantum field vacuum. The cosmological constant problem [5] is a good example to demonstrate this issue, where the vacuum energy density predicted by quantum field theory is much larger than the cosmological constant from the observations. A natural question one can ask is whether taking the fluctuation into account can be an effective way to improve the quantitative descriptions of the cosmological evolution driven by the quantum vacuum energy [6]. The idea of considering the quantum vacuum fluctuations was suggested [6] to solve the cosmological constant problem at present. However, quantum fluctuations exist not only in the current Universe with approximately flat space-time but also in the very early history of the Universe. Therefore, it is also important to consider the impacts of quantum vacuum fluctuations on the evolution of the Universe, in particular the early Universe.

In this study, by improving the method developed in [6], we suggest going one step further beyond the conventional semiclassical method for combining the GR and the QFT by taking into account the vacuum fluctuations and, based on that, reaching a new inflationary scenario. In this new scenario, the cosmological constant issue, which arises when trying to combine GR and QFT, can be resolved.

This paper is organized as follows: in Section 2, we illustrate that the quantum vacuum is not homogeneous, but inhomogeneous, due to quantum fluctuation. In Section 3, by introducing the modified Green’s function, we build up a model to quantify the fluctuations of the quantum vacuum. We study the influence of the quantum vacuum fluctuations and its physical interpretation. In Section 4, we consider a simple case and solve the corresponding Einstein’s field equation. In Section 5, by introducing finite temperature field theory, we take into account the temperature, which is a key element in cosmological evolution. In Section 6, based on our solutions to the cases in Sections 4 and 5, we propose a new inflationary scenario, which can help to resolve the cosmological constant problem. In Section 7, we analyze the influence of the inhomogeneous vacuum on the tunneling amplitude of the Universe from nothing.

The units and metric signature are set to be and throughout. And in this paper, 4-vectors are denoted by light italic type, and 3-vectors are denoted by boldface type.

#### 2. The Quantum Fluctuation and Inhomogeneous Vacuum

Vacuum energy plays an very important role in the inflationary theory. In this theory, at the very early time, the Universe expanded exponentially. In this period, vacuum energy dominated the expansion of the Universe. Usually, the vacuum energy density is treated as a constant; for example, just as in (2), the average value of is where is the energy density of a free scalar field, and the high energy cutoff is much greater than the mass in the free scalar field.

By recalling the example in the introduction section, the vacuum fluctuations are not zero. This is because the vacuum is not the eigen state of , but the eigenstate of Hamiltonian (for the detailed discussions, see [6]). Therefore, the fluctuations in energy density should be considered. Due to the vacuum fluctuations, the conventional assumption of homogeneous Universe is only approximately correct. When fluctuations are taken into account, the vacuum is not homogeneous. Thus, a more suitable theory should include the effects of the fluctuations in energy density. To achieve this goal, the strategy we adopt is to modify both sides of the field equation, in order to have the fine structures which are compatible with the fluctuations.

##### 2.1. Generalizing the FLRW Metric

To describe a homogeneous, isotropic expanding Universe, we introduce Friedmann–Lemaître–Robertson–Walker (FLRW) metric [7]: where can be −1, 0, or +1, which indicates the 3-dimensional space is elliptical space (closed), Euclidean space (flat), or hyperbolic space (open), respectively. The factor , known as the scale factor, depends only on . Although the FLRW metric can naturally describe an expanding Universe, it cannot describe the inhomogeneous Universe where the inhomogeneity is caused by the vacuum fluctuations. The corresponding resolution is to allow the scale factor to have spatial dependence. To simplify our model, we only assume that the scale factor has only radius dependence, so the rotational symmetry is preserved. For the spatial part, is chosen (the reason will be explained in Section 5). Now, the Ricci tensor for the metric becomes where the dot represents the derivative with respect to , and the prime represents the derivative with the respect to . Then the Ricci tensor can be substituted into the Einstein’s field equation: where is given by the energy-momentum tensor: where is the trace of the energy-momentum tensor .

##### 2.2. Quantification of the Inhomogeneous Vacuum

After transforming the left hand side of (8), now we can focus on the right hand side. Following the description of the introduction section, the which is often taken as the average value (expectation value) of the tensor over the entire space-time should not be taken as a constant due to the vacuum fluctuations. By only taking the expectation values of the energy-momentum tensor, some fine structures which come from the fluctuations are lost. Here, our main task is to find a correct for (8), which describes the inhomogeneity of the vacuum fluctuations.

Before studying how to find the suitable , we introduce the scalar field to describe the matter field in our toy model. For simplicity, -4 theory is adopted: where . Due to the Higgs mechanism, the symmetry is broken spontaneously at low temperatures. The effective potential becomes To be compatible with the modification of the FLRW metric, we reduce a number of degrees of freedom of the field and preserve the rotational symmetries just as what we did before.

According to Noether’s theorem, the energy-momentum tensor for this field is given as Substituting (13) in (9), we reach Now, we are ready to substitute into Einstein’s equations. At first, let us look at one of the Einstein equations:

Next, the main challenge becomes the evaluations of and . However, it is impossible to obtain the correct and in conventional methods. The key to obtain the correct expectation values of the potential is to first find out . In general, this expectation value is given as where is the full Green’s propagator. To gain insights, the free propagator should be derived at first following the perturbation theory. Due to the spontaneous symmetry breaking, the free propagator cannot be obtained directly. To resolve this issue, we make a shift of the variable, , where . Then we have Now it is easy to see that the square of the effective mass of the new field is After shifting the variables, we can derive the solution for and return back to the original variable: where satisfies the equation of motion for the free scalar field. For example, in the flat space-time, , and in this case, the Green’s function is where . In the curved space, the propagator is not as simple as it is in the flat space. Following [8], by introducing Riemann normal coordinates and expanding the metric, the propagator in the curved space can also be obtained in momentum space: where and are certain functions which are related to the curvature tensor. If the curvature is not quite large, the second and third terms in (21) would be much smaller than the first term. In that case, we can establish a perturbative propagator in the curved space-time. For our toy model, we omit and terms and just preserve the leading term for the approximate flat space-time (because when the scale factor is large, the Ricci scalar for FLRW metric is proportional to , so the curvature, especially after inflation, is very small). Meanwhile, the van Vleck determinant should also be one in this case. Thus, the approximate propagator in the curved space-time becomes where the and are geodesic distances. After that, we can calculate the full propagator. For simplicity, the tree level propagator is considered, and higher order corrections are neglected.

#### 3. Modifications of the Green’s Function

In this section, we aim to explore the fluctuations of the vacuum and take this into account in our model by modifying the propagator; then we can establish an effective field theory taking into account of the effects of the fluctuations by introducing the modified Green’s function.

##### 3.1. Another Approach to Obtaining the Propagator

Following the previous section, the free propagator for the scalar field is given as

Usually, in quantum field theory, the free propagator can be derived by calculating the Green’s function in momentum space and then integrating over with a suitable contour. However, the two-point correlation function obtained by this way is not the expectation value, , which we expect for the energy-momentum tensor. This is because certain hidden structures inside the correlation functions which can cause the fluctuations are ignored. To show the fine structures of the expectation values for the energy-momentum tensor, here we introduce another approach to obtaining the Green’s function. We will write down the explicit solution of and then calculate the correlation directly.

Here, we represent the solution of in terms of the creation and annihilation operators and for the scalar field without interactions: Then the multiplication of the two s becomes Conventionally, in (25), the first two terms do not give contributions to the propagator. This is because their vacuum expectation values are zero. When we take the expectation value of (25), , there would be no difference in the result in (23). However, the crucial thing to consider is how to evaluate the first two terms. To see the significance of the first two terms, here by setting and rewriting (25), we have Similarly, the last two terms indicate zero point energy, and the first two terms are zero when they are taken vacuum expectation value. However, the first two terms are the parts in (26) where the fluctuations emerge Obviously, . However, and are not zero. In fact, they are the reason why the vacuum state is not the eigen state of the energy density. In this case, and are the key elements to show the “hidden” structure of . Here, we have

A simplest way to preserve the part which quantifies the fluctuation is given as Here, we simply preserve the latest nonzero order of and , so as to leave all the parts in (26) “survived” after taking the expectation value. Fortunately, is convergent when and go to infinity and is related to . In order to be independent of the high momentum cutoff, we can rewrite this result by subtracting , so Now, the new Green’s function is totally composed of the terms of the fluctuations and and starts at zero. Meanwhile, the large constant which is proportional to is eliminated. This constant is the vacuum energy which is nonobservable, due to the QFT.

##### 3.2. The Interpretation of Modified Green’s Function

To gain the modified Green’s function, (28) should be computed. However, the integration is not simple. Here is a way which can be used to estimate the integrals for and . At first, divide the interval of the integration: For the first part of integrals, because , we have In this case, the integrals become To see the result of integrals in (34) more clearly, we set and keep and as functions only with variable . Here we have

After direct calculations, and have limits where At the origin, we also have

Then, think about the next part of integrals. In the next part of integrals, because , Similarly, we have For simplicity, setting , we can easily evaluate (39) directly:

Combining (36) and (40), the modified Green’s function is obtained. Because , the approximate result is given as In this case, the limit when time goes infinity becomes

To check the validity of the approximate modified Green’s function derived here, we calculate numerically.

In Figure 2, the numerical result is seen to be close to the result in Figure 1 which is plotted with the approximate Green’s function. The amplitude of the oscillation of remains nearly a constant in Figure 1 at initial times; however, when the time becomes very long, the amplitude of the oscillations of our approximate result approaches zero. The maximal values of the two results derived by different methods are slightly different. In fact, this is not hard to explain. In (41), we drop the and terms, which influence the amplitude and the value of peak. Roughly speaking, comparing to the value of the Green’s function’s limit when , the oscillation’s amplitude is much smaller compared to the overall value. Therefore, it can be omitted. The oscillation of the Green’s function is shown in Figure 3 by computing the Green’s function numerically.