Higgs Field in Universe: Long-Term Oscillation and Deceleration/Acceleration Phases

Abstract

It is shown that the Einstein gravity and Higgs scalar field have (a) a long-term oscillation phase; (b) cosmological regular solutions with deceleration/acceleration phases. The first has a preceding contracting and subsequent expanding phases and between them there exists an oscillating phase with arbitrary time duration. The behavior of the second solution near to a flex point is in detail considered.

1. Introduction

The standard cosmological model (for review, see [1]) gives us an accurate description of the evolution of the Universe. In spite of its success, the standard cosmological model has a series of problems such as the initial singularity, the cosmological horizon, the flatness problem, the baryon asymmetry, and the nature of dark matter and dark energy.

Under the dynamical laws of general relativity, the standard FLRW cosmology becomes singular at the origin of Universe. The matter density and geometrical invariants diverge as the volume of the Universe goes to zero. The Big Bang singularity seems to be an unavoidable aspect of the currently established cosmological model [2] which probably only a full quantum theory of gravity could resolve. A bouncing Universe with an initial contraction to a nonvanishing minimal radius; then subsequently an expanding phase provides a possible solution to the singularity problem of the standard Big Bang cosmology.

Bouncing cosmologies, in which the present era of expansion is preceded by a contracting phase, have been studied as potential alternatives to inflation in solving the problems of standard FRW cosmology. The first explicit semianalytic solution for a closed bouncing FRW model filled by a massive scalar field was found by Starobinskii [3]. Later explicit solutions for a bouncing geometry were obtained by Novello and Salim [4] and Melnikov and Orlov [5]. For the review of the cosmological bounce one can see review [6].

Supernova observations [7, 8] were the first to suggest that our Universe is currently accelerating. For this acceleration now it is believed that as much as 2/3 of the total density of the Universe is in a form which has large negative pressure and which is usually referred to as dark energy. A number of various models have been proposed aiming at the description of dark energy universe (for review, see [9–11]). It is evident that to have deceleration (where 𝑎̈<0) and acceleration (where 𝑎̈>0) phases it is necessary to have the moment with 𝑎̈=0.

Here we would like to show that (a) a Universe bounce can be not only a short time event but also it can be a long-term oscillating process; (b) the gravitating Higgs scalar field may have cosmological solutions with such property. Such solution exists only with a single value of cosmological constant.

2. Long-Term Bouncing

2.1. The Statement of the Problem

In this section we will investigate a cosmological solution for a closed Universe filled with a Higgs scalar field. Our goal is to find a regular solution with contracting, expanding, and oscillating phases.

We start with the Lagrangian 𝑅𝐿=−21+𝜘2𝜕_𝜇𝜙𝜕^𝜇𝜙−𝑉𝜙, (1) where 𝑅 is the 4D scalar curvature and 𝜙 is the scalar Higgs field with the potentials 𝑉(𝜙). The corresponding field equations are 𝑅_𝜇𝜈−12𝑔_𝜇𝜈𝑅=𝜘𝑇_𝜇𝜈,1√𝜕−𝑔𝜕𝑥^𝜇√−𝑔𝑔^𝜇𝜈𝜕𝜙𝜕𝑥^𝜈𝜙=−𝑑𝑉.𝑑𝜙 (2) The potential 𝑉(𝜙) is the Mexican has potential 𝑉𝜙=𝜆4𝜙²−𝑚²²−𝑉₀, (3) where 𝑚 and 𝑉₀ are constants and 𝑉₀ can be considered as a cosmological constant. The energy-momentum tensor 𝑇_𝜇𝜈 for the scalar field is 𝑇_𝜇𝜈=∇_𝜇𝜙∇_𝜈𝜙−𝑔_𝜇𝜈𝐿. (4) For the investigation of Universe having an oscillating phase between contraction and expanding phases, we consider the cosmological metric 𝑑𝑠²=𝑑𝑡²−𝑎²(𝑡)𝑑𝜒²+sin²𝜒𝑑𝜃²+sin²𝜃𝑑𝜑². (5) After the substitution of the metric (5) into field equations (2), we have following equations set: 3𝑎̇²𝑎²+3𝑎²𝜙̇=𝜘²2+𝜆4𝜙²−𝑚²²−𝑉₀,2𝑎̈𝑎+𝑎̇²𝑎²+1𝑎²−𝜙̇=𝜘²2+𝜆4𝜙²−𝑚²²−𝑉₀,̈3𝑎̇𝜙̇𝜙+𝑎=𝜆𝜙𝑚²−𝜙². (6)(7)(8) To begin with we would like to define the notion “bouncing-off of the Universe.” We will say that the Universe undergoes bouncing-off if a long enough the stage of contraction is replaced by the stage of expansion. A long time interval Δ𝑡 means that Δ𝑡≫𝑡_Planck or Δ𝑡≈𝑡_Universe, where 𝑡_Universe is the life time of the Universe.

Why we need such definition? Below we will show that bouncing-off can be a complicated process lasting a long enough time which we will call as an oscillating stage.

The statement of the problem: to find solutions having big enough contraction/expansion stages and between them a long enough oscillation stage.

2.2. Solution with Oscillating Stage

We will investigate solutions with an oscillating stage having a point of inflection with the following conditions: 𝑎̈(𝑡)<0,if𝑡<𝑡₀,=0,if𝑡=𝑡₀,>0,if𝑡>𝑡₀. (9) Here 𝑡₀ is the point of inflection. In the consequence of the existence of the inflection point the functions 𝑎(𝑡), 𝜙(𝑡) can be presented in the form 𝑎(𝑡)=𝑎₀+𝑎₁𝑡−𝑡₀+𝑎₃𝑡−𝑡₀³𝜙(𝑡)+⋯,=𝜙₀+𝜙₁𝑡−𝑡₀+𝜙₂𝑡−𝑡₀²+𝜙₃𝑡−𝑡₀³+⋯. (10) Using the conditions (9) and (6)–(8) one can find the following constraints on the initial conditions 𝑎₀=𝑎(𝑡₀), 𝑎₁=𝑎̇₀=𝑎(̇𝑡₀), 𝜙₀=𝜙(0), and 𝜙₁=𝜙̇₀=𝜙̇(0) and the cosmological constant 𝑉₀2𝑎̇²₀𝑎²₀+2𝑎²₀𝜆=𝜘4𝜙²₀−𝑚²²−𝑉₀,2𝑎̇²₀𝑎²₀+2𝑎²₀=𝜘𝜙̇²₀. (11) It means that the cosmological constant 𝑉₀ is defined uniquely: Λ=𝜘𝑉₀=𝜆4𝜘𝜙²₀−𝑚²²𝑎̇−2²⁰𝑎²₀+1𝑎²₀. (12) For the numerical investigation we introduce the dimensionless time 𝑥=𝑡/√𝜘 and dimensionless functions 𝜙√𝜘→𝜙, 𝑎/√𝜘→𝑎. Then (6)–(8) become 3𝑎̇²𝑎²+3𝑎²=𝜙̇²2+𝜆4𝜙²−𝑚²²−𝑉₀,2𝑎̈𝑎+𝑎̇²𝑎²+1𝑎²=−𝜙̇²2+𝜆4𝜙²−𝑚²²−𝑉₀,̈3𝑎̇𝜙̇𝜙+𝑎=𝜆𝜙𝑚²−𝜙² (13) with the following initial conditions: 𝑎(0)=𝑎₀,𝑎̇(0)=𝑎₀,𝜙(0)=𝜙₀,𝜙̇₀=−2𝑎̇²₀+2𝑎₀. (14) The numerical solutions are presented in Figures 1, 2, 3, 4, and 5. In Figure 1 the profiles for different 𝑎₁ are presented. From this figure we see that there exists the bounce with different time durations. Every such solution can be enumerated with the number of minima or maxima. In Figures 1–4 the profiles of 𝜙(𝑡), Hubble parameter 𝐻(𝑡)=𝑎(̇𝑡)/𝑎(𝑡), and the state equation 𝑤(𝑡) for 𝑎₁=0.08050204989312597 are presented. It is useful to present the profile of a state equation (see Figure 4) in the form 𝑝𝑤=𝜀=𝑇¹¹𝑇⁰⁰=𝜙̇²/2−(𝜆/4)𝜙²−𝑚²+𝑉₀𝜙̇²/2+(𝜆/4)𝜙²−𝑚²−𝑉₀, (15) where 𝑝 is the pressure and 𝜀 is the energy density. We see that in the contracting and expanding stages 𝑤≈−1 and additionally during the oscillation stage there are points where 𝑤≈−1.

Figure 1 

The profiles 𝑎(𝑡). The curve 1 for 𝑎1=0.08, the curve 2 for 𝑎1=0.0805, the curve 3 for 𝑎1=0.080502, the curve 4 for 𝑎1=0.0805020498, the curve 5 for 𝑎1=0.080502049893, and the curve 6 for 𝑎1=0.08050204989312597.

Figure 2 

The profile 𝜙(𝑡) for 𝑎1=0.08050204989312597.

Figure 3 

The profile of Hubble parameter 𝐻(𝑡)=𝑎(̇𝑡)/𝑎(𝑡) for 𝑎1=0.08050204989312597.

Figure 4 

The profile of the equation state 𝑤(𝑡) for 𝑎1=0.08050204989312597.

Figure 5 

The profiles of regular and singular 𝑎(𝑡). The regular solutions: curve 1 for 𝑎1=0.080502049893, curve 2 for 𝑎1=0.08050204989312597. The singular solutions: curve 3 for 𝑎1=0.08050204989312599.

How long could such prolonged Universe oscillation be? For the investigation of this question we are addressing this in Figure 5. We see that for 𝑎₁=0.080502049893,0.08050204989312597 the solution has the oscillations, but for 𝑎₁=0.08050204989312599 the solution is singular one. It means that there exists a special solution with 𝑎₁=𝑎^∗₁, where 0.08050204989312597<𝑎^∗₁<0.08050204989312599. From Figures 1 and 5 we see that such solution should have an infinite long oscillation stage with a contracting phase at 𝑡→−∞ and an expansion phase at 𝑡→+∞. The most interesting is that by 𝑎₁ close enough to 𝑎^∗₁ we may have the stage with oscillations with any duration.

During the oscillations the size of Universe is 𝑎≈𝑎_𝑃𝑙. It means that the existence time of Universe in such state can be somehow long with possible expansion at any time.

3. Deceleration/Acceleration Phases from Gravitating Higgs Field

3.1. Numerical Solution with Deceleration/Acceleration Phases

The aim of this section is to show that in the ordinary Einstein gravity interacting with the Higgs scalar field there exist solutions having the deceleration and acceleration phases. The corresponding Einstein and scalar field equations are 𝑅_𝜇𝜈−12𝑔_𝜇𝜈𝑅=𝜘𝑇_𝜇𝜈,1√𝜕−𝑔𝜕𝑥^𝜇√−𝑔𝑔^𝜇𝜈𝜕𝜙𝜕𝑥^𝜈𝜙=−𝑑𝑉,𝑑𝜙 (16) where 𝑉(𝜙) is 𝑉𝜙=𝜆4𝜙²−𝑚²²−𝑉₀. (17)𝜙 is the Higgs field and the Lagrangian for the scalar field is 1𝐿=2∇_𝜇𝜙∇^𝜇𝜙𝜙,𝑇−𝑉_𝜇𝜈=∇_𝜇𝜙∇_𝜈𝜙−𝑔_𝜇𝜈𝐿. (18) The quantity 𝑉₀ in (17) is identical to a cosmological constant. Later we will see that 𝑉₀ is defined uniquely in a flex point 𝑡₀, where 𝑎(̈𝑡₀)=0.

We consider the cosmological metric 𝑑𝑠²=𝑑𝑡²−𝑎²(𝑡)𝑑𝜒²+sin²𝜒𝑑𝜃²+sin²𝜃𝑑𝜑². (19) The equations for 𝑎(𝑡) and 𝜙(𝑡) are 3𝑎̇²𝑎²+3𝑎²𝜙̇=𝜘²2+𝜆4𝜙²−𝑚²²−𝑉₀,2𝑎̈𝑎+𝑎̇²𝑎²+1𝑎²−𝜙̇=𝜘²2+𝜆4𝜙²−𝑚²²−𝑉₀,̈3𝑎̇𝜙̇𝜙+𝑎=𝜆𝜙𝑚²−𝜙². (20)(21)(22) For the deceleration phase we have 𝑎(̈𝑡)<0, for the acceleration phase: 𝑎(̈𝑡)>0. Consequently there is a flex point 𝑡₀ where 𝑎̈𝑡₀=0. (23) Using the conditions (23) and (20), (21) one can find the following constraints on the initial conditions and cosmological constant: 2𝑎̇²₀𝑎²₀+2𝑎²₀𝜆=𝜘4𝜙²₀−𝑚²²−𝑉₀,2𝑎̇²₀𝑎²₀+2𝑎²₀=𝜘𝜙̇²₀, (24) where 𝑎₀=𝑎(𝑡₀), 𝑎̇₀=𝑎(̇𝑡₀), 𝜙₀=𝜙(0), and 𝜙̇₀=𝜙̇(0). It means that the cosmological constant is defined uniquely: Λ=𝜘𝑉₀=𝜆4𝜘𝜙²₀−𝑚²²𝑎̇−2²⁰𝑎²₀+1𝑎²₀. (25) For the numerical solution we introduce the dimensionless quantities 𝑥=𝑡/√𝜘, 𝜙√𝜘→𝜙, and 𝑎/√𝜘→𝑎. Then (20)–(22) are 3𝑎̇²𝑎²+3𝑎²=𝜙̇²2+𝜆4𝜙²−𝑚²²−𝑉₀,2𝑎̈𝑎+𝑎̇²𝑎²+1𝑎²=−𝜙̇²2+𝜆4𝜙²−𝑚²²−𝑉₀,̈3𝑎̇𝜙̇𝜙+𝑎=𝜆𝜙𝑚²−𝜙² (26) with the following initial conditions: 𝑎(0)=𝑎₀,𝑎̇(0)=𝑎₀,𝜙(0)=𝜙₀,𝜙̇₀=−2𝑎̇²₀+2𝑎₀. (27) The numerical solution is presented in Figures 6-7. We see that there are two type of solutions: regular and singular one. The regular solution exists for −∞<𝑡<+∞ and has bouncing-off point and four flex points. The singular solution has one flex point only. The presented solution is symmetrical one relative to the bouncing-off moment, but there exist nonsymmetrical solutions with the initial conditions different from (27).

Figure 6 

The profiles 𝑎(𝑡) for regular (𝑎̇0=0.08) and singular (𝑎̇0=0.2) solutions. The parameters 𝜆=0.41, 𝑎0=1.0, 𝜙0=−0.0028; 𝜙̇0=−2𝑎̇20+2/𝑎0, and 𝑚=0.603 are the same for both solutions.

Figure 7 

The profiles 𝜙(𝑡) for regular and singular solutions.

The asymptotical behavior of the solution is 𝑎(𝑡)≈𝑎_∞𝑒^{𝑡−𝜘𝑉0/3}𝑎(𝑡),𝑡⟶+∞,≈𝑎_∞𝑒^{−𝑡−𝜘𝑉0/3}𝜙(𝑡),𝑡⟶−∞,≈−𝑚+𝜙_∞𝑒^−𝛼𝑡,𝛼_1,2=−3𝜘𝑉₀4±−3𝜘𝑉₀4−2𝜆𝑚²,|𝑡|⟶∞, (28) where 𝑎_∞, 𝜙_∞ are constants. This solution can describe the inflation of Universe with the posterior standard decay of the scalar field.

It is useful to present the profile of a state equation (see Figure 8) in the form 𝑝𝑤=𝜀=𝑇¹¹𝑇⁰⁰=𝜙̇²/2−(𝜆/4)𝜙²−𝑚²+𝑉₀𝜙̇²/2+(𝜆/4)𝜙²−𝑚²−𝑉₀, (29) where 𝑝 is the pressure and 𝜀 is the energy density. Particularly interesting is the behavior of 𝑤 in the region 𝑥>𝑥₀, that is, in the acceleration region. We see there that 0<𝑤<−1.

Figure 8 

The profiles 𝑤(𝑡) for the regular solution.

3.2. The Deceleration → Acceleration Transition

In the preceding section we have shown that the Universe filled with the Higgs field may have the deceleration/acceleration phases and have presented the solution with 𝑎₀≈𝑙_𝑃𝑙∝√𝜘. In this section we would like to investigate more carefully the solution near the flex point where the transition from the deceleration to acceleration phase has happened and with 𝑎₀≫𝑙_𝑃𝑙∝√𝜘. It is not too hard to find the solution of equations set (20)–(22) in the following form: 𝑎(𝑡)=𝑎₀+𝑎̇₀𝑡+4𝑎̇₀𝑎̇²₀+1𝑎²₀+𝜆𝜙₀2𝜘𝑎̇²₀+1𝑚²−𝜙²₀𝑡³6+⋯,𝜙(𝑡)=𝜙₀−1𝑎₀2𝑎̇²₀+1𝜘𝑡+3𝑎̇₀𝑎²₀2𝑎̇²₀+1𝜘+𝜆𝜙₀𝑚²−𝜙²₀𝑡²2−𝑎̇12²⁰𝑎³₀2𝑎̇²₀+1𝜘𝑎̇+3𝜆⁰𝜙₀𝑎₀𝑚²−𝜙²₀+𝜆𝑎₀2𝑎̇²₀+1𝜘𝑚²−3𝜙²₀𝑡³6+⋯. (30)(31) The deceleration parameter is 𝑞(𝑡)𝑎𝑎̈=−𝑎̇²𝑎≈−⁰𝑎₃𝑎̇²₀𝑎𝑡,₃𝑎̇=4⁰𝑎̇²₀+1𝑎²₀+𝜆𝜙₀2𝜘𝑎̇²₀+1𝑚²−𝜙²₀. (32) More convenient in this approach is the modified deceleration parameter 𝑞(𝑡)=−𝑎−𝑎₀𝑎̈𝑎̇²𝑎≈−³𝑎̇₀𝑡². (33) The Habble constant is 𝐻(𝑡)=𝑎̇𝑎≈𝑎̇₀𝑎₀−𝑎̇²₀𝑎²₀𝑎̇𝑡+³⁰𝑎³₀+𝑎₃2𝑡². (34) Let us remind that the time 𝑡 is counted from the flex point moment 𝑡₀. Unfortunately it is not for a while yet unknown: is the solution with 𝑎₀≫𝑙_𝑃𝑙 regular or singular, that is, has the solution bouncing-off from a cosmological singularity or not.

4. Outlook

We have shown the following.(i)The bouncing-off process of Universe is not a simple process. Implicitly it is supposed that bouncing-off happens at a moment. Here we have shown that it can be a long-term process taking place at the Planck region (oscillation). It allows us to consider a quantum birth of the Universe as the following process: the Universe exists in the state with 𝑎₁=𝑎^∗₁ and in the consequence of quantum fluctuations of 𝑎₁ the Universe passes to the state with 𝑎₁≈𝑎^∗₁. After that the Universe goes into an inflation phase.(ii)The Einstein-Higgs gravity has cosmological solutions with the deceleration/acceleration phases. Additionally these solutions may have bouncing-off from a cosmological singularity. The detailed investigation is made near the moment where the transition from the deceleration epoch to the acceleration one happens.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was partially supported by a Grant no. 1626/GF3 in fundamental research in natural sciences by Science Committee of the Ministry of Education and Science of Kazakhstan and by a grant of VolkswagenStiftung.