Tag Archives: #scalarfield

How Thermal Radiation Affects Gravitational Waves and Space-time Singularities? (Quantum / Cosmology)

The recent observations of gravitational waves and supermassive black holes can be considered as the main probes of General Relativity (GR) in its fundamental aspects which are: 1) the propagation of space-time perturbations, 2) the existence of singularities. Despite of these undeniable successes, several shortcomings affect GR because the whole phenomenology cannot be addressed in the framework of the Einstein picture. The theory is missing at ultraviolet scales because of the lack of a self-consistent theory of Quantum Gravity, and at infrared scales because it is not capable of encompassing clustering phenomena related to large-scale structure and the observed accelerated expansion of the cosmic fluid. These are generically dubbed as dark matter and dark energy but, up to now, no particle counterpart has been discovered to address them at fundamental level.

In this perspective, extensions and modifications of GR are considered as a reliable way out of the above problems assuming that gravitational field has not been completely explored.

These extensions come from effective theories on curved spacetimes or as alternative formulations like teleparallel gravity and its related models.

A main role to test theories is played by cosmology because phenomena connected to the so called dark side can substantially affect structure formation and cosmic dynamics. Their equivalent geometric explanations could be a major step towards a comprehensive theory of gravity at all scales.

In general, dynamical characteristics of gravitational waves can be the features probing a given theory of gravity. Specifically, speed, damping, dispersion, and oscillations of gravitational waves could be used to fix and reconstruct interactions into gravitational Lagrangian and then be a sort of roadmap inside the wide forest of competing theories of gravity. For example, further gravitational polarization modes, besides the two standard ones of GR, emerge when further degrees of freedom are considered into the theory. In general, as soon as modifications or extensions of GR are taken into account, scalar modes are present into dynamics.

Motivated by these considerations, it is possible to investigate the propagation of gravitational waves in various gravitational models. For example, in F(T) extended teleparallel gravity, in domain wall models, in scalar tensor and F(R) gravity theories, in Chern-Simons Axion Einstein gravity and in several media as in strong magnetic fields or in viscous fluids.

Furthermore, the behavior of gravitational waves can be used to test past and future singularities and then contributes in their classification. Another important issue is connected to thermal effects emerging during the cosmic evolution.

“We point out that the contribution of thermal radiation can heavily affect the dynamics of gravitational waves giving enhancement or dissipation effects both at quantum and classical level. These effects are considered both in General Relativity and in modified theories like F(R) gravity.”

— told Odintsov, lead author of the study

Now, Odintsov and colleagues investigated how thermal effects on various cosmological backgrounds affect the propagation of gravitational waves. In particular, they want to take into account such effects in GR, in modified theories of gravity and in presence of future singularities. Lets have a closer look on their findings:

(A) Thermal effects in Cosmology

At first, they considered thermal effects in cosmology and showed that, when hubble parameter (H) is large, the temperature of the universe becomes large and they may expect the generation of thermal radiation as in the case of the Hawking radiation. The Hawking temperature T is proportional to the inverse of the radius rH of the apparent horizon and the radius rH is proportional to the inverse of the Hubble rate H. Therefore, the temperature T is proportional to the Hubble rate H. In simple terms, at cosmological scales, thermal effects emerge with respect to the Hubble radius and then they can strongly affect the cosmic evolution, in particular, at early epochs or nearby singularities. In other words, thermal effects can dynamically affect the cosmological background and then the evolution of phenomena on it.

(B) Thermal effects in future cosmological singularities

It is well known that in the cosmic future several kinds of space-time singularity can happen. Such singularities have been classified as follows:

  • Type I, which is also called as “Big rip”, in this type of singularity, scale factor a(t), the total effective pressure peff and the total effective energy density ρeff diverges strongly.
  • Type II, which is also called as “sudden/pressure singularity” and it is milder than the Big Rip scenario. Here, only the total effective pressure diverges, and the total effective energy density & the scale factor remain finite.
  • Type III, in which both the total effective pressure and the total effective energy density diverges, but the scale factor remains finite. So, this type of singularity is milder than Type I (Big Rip) but stronger than Type II (sudden).
  • Type IV, which is the mildest from a phenomenological point of view.

According to authors, the thermal radiation usually makes the singularities less singular, that is, the Big Rip (Type I) singularity or the Type III singularity transit to the Type II singularity.

(C) Thermal effects in Scalar-field cosmologies

In this sub-section, they showed that the thermal radiation assumes a key role in determining the evolution of the scalar field and then of the universe.

Gravitational waves in a dynamical background

In this section, they considered the propagation of gravitational waves in a dynamical cosmological background where thermal contributions are present. They showed mathematically, how these terms affect the evolution of gravitational waves. First, they reviewed the propagation of gravitational waves in a general medium. Gravitational waves are derived as perturbations of Einstein field equations. In the Einstein equations, not only the curvature but also the energy-momentum tensor depends on the metric and therefore the variation of the energy-momentum tensor gives a non-trivial contribution to the propagation of gravitational waves.

“We showed thermal radiation contribution play a key role in the evolution of the Gravitational waves”

(D) Thermal corrections in quantum matter

In this sub-section, they considered a real scalar field φ as the source of matter. They deal with the scalar field as a quantum field at finite temperature. In the case of high temperature or in the massless case, the scalar field plays the role of radiation. On the other hand, in the limit where the temperature is vanishing but the density is finite, they obtained the dust, which can be considered as cold dark matter.

Enhancement and dissipation of gravitational waves with thermal effects

In this section, they investigated the propagation of gravitational massless spin-two modes, and showed that thermal contribution affects the evolution of the gravitational wave amplitude.

(A) The behavior of gravitational waves near the singularities

In this sub-section, they studied the behavior of gravitational waves near the Type II singularity and the Big Rip (the Type I) singularity. They showed that the enhancement of the gravitational wave occurs near the Type II singularity but it could not occur near the Big Rip (Type I) singularity.

(B) Gravitational waves in the early universe

In this subsection, they showed that if there is no thermal effect, that is α = 0, there is no enhancement or dissipation of the gravitational wave, but if we include the thermal effect, enhancement or dissipation occur.

The propagation of scalar modes with thermal effects

In this section, they developed similar discussion in a generalized context where scalar modes are included. In case of the Type II singularity, just before the singularity, the scalar field oscillates very rapidly. In the case of Big Rip (the Type I) singularity, the amplitude of scalar field increases or decreases very rapidly. While, in the case of bouncing universe, they found that, scalar field (ω(η)) vanishes at t = ±t0 and therefore mass, m² diverges. If m² >, which may depend on the parameters near t = ±t0, the scalar field oscillates very rapidly and if m² < 0, the amplitude of the scalar field increases or decreases very rapidly.

They also mentioned that expanding universe can be realized by the perfect fluid. The perfect fluid also generates scalar waves, whose propagating velocity is known as the sound speed Cs, which is given by:

Therefore if we find the equation of state (EoS), we can find the speed. Thus, in case of the Type II singularity, they found,

Therefore C²s diverges to negative infinity. Because c²s is negative, the amplitude of the perfect fluid wave rapidly decreases or increases without oscillation. But, this behavior is much different from that in the scalar field, where the scalar field oscillates very rapidly.

In case of the Big Rip (the Type I) singularity, they found

which is finite but because C²s is negative, the amplitude of the perfect fluid wave decreases or increases exponentially without oscillation. Even in case of the scalar field, the amplitude of the scalar field increases or decreases very rapidly but m² diverges in the case of scalar field, the increase or decrease is much more rapid.

In case of the bouncing universe, the sound speed is given by,

Which diverges at t=t0 as the m² of the scalar field. Then, the propagation of the perfect fluid wave might be similar to that in the scalar field.

In case of the inflation, H ∼ H0, they found

which is finite but could be negative as long as ηH is small enough. Therefore the perfect scalar wave does not propagate although the scalar wave can propagate.

All the results given above, tell us that the propagation of scalar modes depends on the mechanism which generates the expansion of the universe and, since thermal effects affect the effective mass, they have to be considered in the evolution. And last,

(A) Scalar waves in modified gravity vs compressional waves of cosmic fluid

In this sub-section they showed how we can distinguish scalar modes with respect to the compressional waves of a perfect fluid? They showed that a perfect fluid, where the EoS parameter is w < 0, does not generate a compressional wave. Therefore if we find massive scalar waves, it could be an evidence for modified gravity. However, even in modified gravity, there are various models. In the case of F(R) gravity, however, the coupling of massive scalar with matter is universal, that is, it does not depend on the kind of matter because the coupling appears by the rescaling of metric. This structure is rather characteristic of F(R) gravity and it may give some clue to observationally discriminate F(R) gravity with respect to other modified gravity models.

“In the case of F(R) gravity, the effects of scalar modes or compressional fluids strictly depend on the “representation” of the theory in the Einstein or the Jordan frames. This could constitute an important feature in order to distinguish the true physical frame by the observations”

— wrote authors of the study

Finally, they concluded that dynamics related to the above discussion could be observationally tested by interferometers. In fact, at suitable sensitivities, it seems realistic to disentangle GR contributions with respect to other contributions in the stochastic background of gravitational waves. In a future study, they will develop this topic in detail.


Reference: Salvatore Capozziello, Shin’ichi Nojiri, Sergei D. Odintsov, “Thermal effects and scalar modes in cosmological gravitational waves”, pp. 1-20, Arxiv, 2021. https://arxiv.org/abs/2104.10936


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Q–Physicists Studied Particle Production from Oscillating Scalar Field & Consistency of Boltzmann Equation (Quantum / Maths)

Boltzmann equation plays important roles in particle cosmology in studying the evolution of distribution functions (also called as occupation numbers) of various particles. The success of the standard cosmology or cold dark matter (ΛCDM) essentially relies on the use of the Boltzmann equation in an expanding Universe. In cosmology, however, the stimulated emission or Pauli blocking effect is usually omitted.

Scalar fields may form condensation; examples of such scalar fields include inflaton, curvaton, axion, Affleck-Dine field for the baryogenesis, and so on. (Hereafter, such a scalar condensation is called φ.) If φ can decay into a pair of bosonic particles (called χ) as φ → χχ, the effects of the stimulated emission can be important since the daughter particles may be enormously populated. In particular, today itself, I wrote an article on a mechanism of producing bosonic dark matter from the inflaton decay, in which a stimulated emission of the bosonic dark matter plays an important role.

Such systems have been studied by employing the Boltzmann equation or in the context of the parametric resonance. In particular, particle production from an oscillating scalar field has been intensively investigated, particularly using the Mathieu equation. In the previous studies, the evolution equation of the expectation value of the field operator (which we call wave function) for the final-state particle is converted to the Mathieu equation, based on which the occupation number of the final state particle has been estimated. Then, it has been shown that there exist resonance bands and that the occupation numbers in the resonance bands grow exponentially. In the broad resonance region, it has been known that the particle production is non-perturbative and that the process cannot be described by the Boltzmann equation. On the contrary, sometimes it has been argued that the Boltzmann equation can provide a proper description in the narrow resonance regime.

Now, Moroi and Yin, in this paper, have studied the particle production from an oscillating scalar field (φ) in the narrow resonance regime (or assuming that the final state particle (χ) is very weakly interacting). They have paid particular attention to the consistency of the results from the Boltzmann equation and those from the QFT calculation. They have concentrated on the case that the production of χ is via the process φ → χχ.

“We study the particle production including the possible enhancement due to a large occupation number of the final state particle, known as the stimulated emission or the parametric resonance.

— told Moroi, first author of the study.

First, they have considered particle production in the flat spacetime. In such a case, they have discussed the evolution of the occupation number of each mode (i.e., mode with a fixed momentum k) separately in the narrow resonance regime. They have derived the evolution equations for the occupation number of each mode based on the QFT. A resonance band shows up at ω_k close to 1/2 ×m_φ, which corresponds to the lowest resonance band in the context of the parametric resonance. The modes within the resonance band can be effectively produced. For the timescale much longer than (qmφ)¯1, the occupation numbers of the modes in the resonance band exponentially grow; the growth rate obtained by them in analysis is consistent with that given by the study of the parametric resonance using the Mathieu equation. Then, comparing the occupation number obtained from the QFT calculation with that from the Boltzmann equation, they have found that they do not agree well when the occupation number is larger than ∼ 1. On the contrary, when f_k << 1, they have found a good agreement of two results. They have also argued how their evolution equation based on the QFT could be related to the ordinary Boltzmann equation. When the occupation number is larger than ∼ 1, some of the approximation and assumption necessary for such an argument cannot be justified, which, they expect, causes the disagreement.

Then, they have studied particle production taking into account the effects of cosmic expansion. With the cosmic expansion, the physical momentum redshifts. The momentum of each mode stays in the resonance band for a finite amount of time and then exits the resonance band. The exponential growth of the occupation number occurs only in the resonance band. The growth factor has been studied numerically and analytically, adopting the evolution equations based on the QFT. The agreement between numerical and analytical results is excellent. They have also analyzed the system by using the conventional Boltzmann equation and found that the growth rate obtained by solving the Boltzmann equation is a factor of 2 larger than that based on the QFT. Thus, the occupation number from the Boltzmann equation may become exponentially larger than that from the QFT, and a naive use of the conventional Boltzmann equation may result in a significant overestimation of the number density of χ.

In this paper, we have considered the production of a bosonic particle, concentrating on the lowest resonance band of the parametric resonance. Consideration of the production of fermionic particles and the study of the higher resonance bands, as well as the use of the evolution equations based on the QFT to other phenomena, are left as future works.

— concluded authors of the study.

Reference: Takeo Moroi, Wen Yin, “Particle Production from Oscillating Scalar Field and Consistency of Boltzmann Equation”, ArXiv, pp. 1-23, 2021. https://arxiv.org/abs/2011.12285


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Can Dark Datter Be Produced From Inflaton Decay? (Quantum / Cosmology)

Moroi and Yin in their recent paper, studied a new possibility of light DM production. The light dark matter (DM) is produced from the decay of an inflaton which reheats the Universe. They showed that, if the reheating temperature (T_R) is higher than the inflaton mass (mφ) , the DM particles from the inflaton decays can become cold enough due to the red shift.

A long-standing puzzle of particle physics and cosmology is the origin of dark matter (DM). The DM is known to be (very) weakly coupled to the standard model (SM) particles, stable, cold, and abundant in the present Universe. However the mass, the interactions, and the production mechanism are still not clear.

The DM mass may be small, in which case the stability (or longevity) is easily explained due to the kinematics (or suppressed decay width). From the point of view of DM direct detections, if the mass is small enough, the recoil energy of a nucleon via the DM-nucleon scattering is highly suppressed, which is consistent with the null result of the direct detection experiments for WIMP. Nevertheless, a light DM can be searched for from different approaches in the near future like axion searches or from direct detection with electron recoils.

A difficulty of the light DM is the production in the early Universe. If it were produced thermally, like the WIMP, the number density cannot be larger than that of the Standard Model (SM) photons, and may be too small unless the mass is larger than eV. Moreover, it may be too hot to be consistent with structure formation. For instance, a thermal relic of sterile neutrino has to be heavier than 2 − 5 keV from Lyman-α forest data. If it is produced from freeze-in the bound is even severer.

The production of light DM has been discussed widely. An axion/axion-like particle (ALP) can be produced via the misalignment mechanism. A light hidden photon DM can be produced gravitationally (Gravitational effect during inflation is also important for the axion/ALP DM), the hidden photon DM can be produced from a parametric/tachyonic resonance by coupling to the QCD axion. In addition, the axion (or ALP) production from the parametric resonance of another scalar field was discussed. A light DM may be produced by the decays of heavy particles at an early epoch.

Now, Moroi and Yin studied a new possibility of light DM production. The light DM is produced from the decay of an inflaton (or more generically, an oscillating scalar field), which reheats the Universe.¹ They showed that, if the inflaton coupling to the SM sector has a sizable strength, the reheating temperature (T_R) due to the inflaton decay can be comparable to or higher than the inflaton mass (mφ) . If T_R ≳ mφ, then the DM produced by the inflaton has a momentum smaller than those of particles in the thermal bath (consisting of the SM particles), assuming that the interaction of the DM with the SM particles or itself is negligibly weak. It can help the DM produced by the decay to be cold enough to be consistent with the Lyman-α bound on warm DMs even if the DM mass is smaller than O(1) keV.

Fig. 1: The contour of constant B to realize the present DM density on mχ vs. r plane, taking Ωχh² = 0.12. The maximal possible reheating temperature as a function of r is shown on the right axis. In the gray region, the DM may be too hot to be consistent with Lyman-α forest data by assuming that the DM momenta at t = tR is typically mφ/2. The purple region may be tested in the future. For the case of the Fermionic DM, only the region below the blue-dotted line is relevant because of the Pauli-blocking. The typical parameter range of QCD axion DM and right-handed neutrino DM are shown in light blue and orange © Moroi and Yin

They also studied the bosonic and fermionic DM production from the inflaton decay, taking into account the effect of the stimulated emission and Pauli blocking, respectively. They have shown that, if the DM is bosonic, the production of the DM from the inflaton decay can be enhanced due to the effect of the stimulated emission, like the LASER. The mechanism, called DASER (i.e, the DM amplification by stimulated emission of radiation), can significantly enhance the DM abundance and can make light bosonic DM scenarios viable.

The DASER mechanism is generic for bosonic particles which is weakly coupled to inflaton and SM particles; the candidates include hidden photon and axion-like particle DM. For instance, one can also use the DASER mechanism to produce the hidden photon or axion DM to explain the XENON1T excess. Conversely, such a DASER production process may overproduce stable (or long-lived) bosons which may result in cosmological problems with dark matters or dark radiations.

— told Moroi, first author of the study.

In addition, if the DM is fermionic, the DM produced by the decay may form Dirac sea. The DASER and Dirac sea DMs produced by the inflaton decay may be searched for by future observations of the 21cm lines. The DMs produced by these mechanisms have null isocurvature perturbations.

Finally, they comment that the DMs produced in the present scenarios may have very special momentum distribution, and the information about the production mechanism discussed by authors in their paper, may be embedded in the momentum distribution of the DM. For example, the momentum of the bosonic DM produced by the DASER mechanism would have sharp peak at the IR mode at the momentum ∼ mφ/a[ti]. From a quantum field theory approach, the peak width can be approximated as ∼ √(mφH(ti))/a(ti). By combining the two, they can get mφ/H(ti). In addition, momentum distribution of the modes produced during the preheating depends on the thermal history during the preheating. One may probe the reheating phase or preheating phase if the information about the momentum distribution of the DM becomes available. For this purpose, they concluded that further study of the structure formation with non-standard momentum distribution of the DM is needed.

Note: #1 The parent scalar field does not have to be the inflaton; if it once dominates the universe, any scalar field may play the role. For the minimality of the scenario, in the this analysis, we assume that the parent scalar field is the inflaton.


Reference: Takeo Moroi and Wen Yin, “Light Dark Matter from Inflaton Decay”, ArXiv, pp. 1-33, 30 Mar, 2021. https://arxiv.org/abs/2011.09475


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What Are The Effects Of “Extra Matter” and Scalar Field On A Wormhole? (Cosmology)

Summary:

Ewha Womans University researcher Sung-Won Kim described the effects of extra matter and scalar field on the stability and geometry of wormhole in his 1999 paper.

There are two effects of extra matter fields on the Lorentzian traversable wormhole. The “primary effect” says that the extra matter can afford to be a part of source or whole source of the wormhole when the wormhole is being formed. Thus the matter does not affect the stability of wormhole and the wormhole is still safe.

The “auxiliary effect” is that the extra matter plays the role of the additional matter to the stably-existed wormhole by the other exotic matter. This additional matter will change the geometry of wormhole enough to prevent from forming the wormhole by backreaction.


One of the most important issues in making a practically usable Lorentzian wormhole is just the traversability. If it is traversable, there is a good usability, such as short-cut space travels, the time machine, and inspection of the interior of a black hole.

To make a Lorentzian wormhole traversable, one has conventionally used an exotic matter which violates the well-known energy conditions. For instance, a wormhole in an inflating cosmological model still requires exotic matter to be traversable and to maintain its shape. It is known that the vacuum energy of the inflating wormhole does not change the sign of the exoticity function. A traversable wormhole in the Friedmann-Robertson-Walker(FRW) cosmological model, however, does not necessarily require exotic matter at very early times. The result means that there was an exotic period in the early Universe.

The problem about maintaining a wormhole by other fields relating with the exotic property has also been of interest to us. There are two ways to generalize or modify the Lorentzian traversable wormhole space-time. (From now on the “wormhole” will be simply used in the sense “Lorentzian traversable wormhole”, unless there is a confusion.) One way is generalization of the wormhole in an alternative theory, for example, Brans-Dicke theory, Einstein-Cartan theory, etc. The other is the generalization by adding the extra matter.

In the case of the latter generalization, the added matter will play two kinds of roles in affecting the wormhole spacetime. The first role of such matter (e.g.,scalar fields, charge, spin, etc.) is the “primary effect”, which means that the added matter is a partial or total source of the wormhole. The matter gets involved in the constructing stage of the wormhole.

The wormhole is then safe under the addition of matter, which is a part of the sources for wormhole constructing. This means that if this added extra matter field has exotic properties, it shares exoticity with other matter, or, if the latter is not exotic, the added matter will monopolize the exotic property. In this case the wormhole cannot exist without this extra matter. Its example is the case of the wormhole solution with a scalar fields.

The second role is the “auxiliary effect”. In this effect, the added matter is not a source, but causes an extra effect on an existing wormhole. Therefore, it is not involved on the wormhole construction stage but affects it afterwards. Since this matter makes extra geometry, the wormhole is not safe when this effect dominates the exotic constructing matter. This is back reaction to the wormhole from an additional field.

What Kim found is that a self-consistent solution for a worm-hole with a classical, minimally-coupled, massless scalar field. He also found the back reaction of the scalar field on the wormhole spacetime, to see the stabilities of the wormhole. The result is that the scalar field effect can break the wormhole structure when the field and the variation of the field is large.

He obtained similar consequences in charged wormhole case, in which there is the interaction term in geometry, even though no interaction term in matter. It is natural that the addition of the nonexotic matter will break wormhole if the “auxiliary effect” is large.

In this paper, he neglected the interaction between the extra field and the original matter. If the interaction exists and it is large, it can change the whole geometry drastically. But, if it is very small, it does not change the main structure of the wormhole.


Reference: Sung-Won Kim, “Backreaction to wormhole by classical scalar field: Will classical scalar field destroy wormhole?”, Astronomical Journal, pp. 1-9, 1999. https://arxiv.org/abs/gr-qc/9911099


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How Could We Produce Exotic Matter Required To Hold Wormhole Open: PART 2 (Quantum / Cosmology)

Previously on “How could we produce exotic matter required to hold a wormhole open, classically?”, we saw that in 1997, Dan Vollick, a physicist at University of Victoria, published a paper, in which he showed that it is possible to produce the exotic matter required to hold a wormhole open. This was accomplished by coupling a scalar field to matter that satisfies the weak energy condition. But then, an year later, Dan Vollick came up with a wormhole which was constructed by cutting and joining two spacetimes satisfying the low energy string equations with a dilaton field. But why?

Well friends, to keep a wormhole open it is necessary to thread its throat with matter that violates the averaged null energy condition (also called the averaged weak energy condition). That is, there must exist null geodesics with tangent vectors k^µ = dx^µ/ds that satisfy:

Most discussions of such exotic matter involve quantum field theory effects, such as the Casimir effect. But, after Dan Vollick came across the researches done by Ford et al. and Taylor et al. it changed his view. In 1995, Ford and Roman have showed that the quantum inequalities satisfied by the negative energy densities in scalar and vector quantum field theories tightly constrain the geometry of wormholes. Their analysis showed that the wormhole can either be on the order of ≃ Planck size or that the negative energy density is concentrated in a “thin-shell” at the throat of the wormhole. In another paper Taylor, Hiscock, and Anderson have examined the energy-momentum tensor of a “test” quantized scalar field in a fixed background wormhole spacetime. They found that for five different wormhole geometries the energy-momentum tensor of a Minimally or conformally coupled scalar field does not even come close to having the properties required to support the wormhole. Thus, Vollick realized that the prospect of maintaining a Wormhole through quantum field theory effects is not very promising.

So, an year later, he came up with a wormhole which was constructed by cutting and joining two spacetimes satisfying the low energy string equations with a dilaton field. In spacetimes described by the “string metric” the dilaton energy-momentum tensor need not satisfy the weak or dominant energy conditions (as in case of scalar field+ matter, in last paper). In the cases considered in paper, he showed that the dilaton field violates these energy conditions and is the source of the exotic matter required to maintain the wormhole.

There is also a surface stress-energy, that must be produced by additional matter, where the spacetimes are joined. He showed that wormholes can be constructed for which this additional matter satisfies the weak and dominant energy conditions, so that it could be a form of “normal” matter.

He also briefly discussed creating charged dilaton wormholes in which the coupling of the dilaton to the electromagnetic field is more general than in string theory. I summarized the method below:

Dan Vollick showed that wormholes cannot be constructed by using Field Equations & Spherically Symmetric Solutions method. Since it is necessary that a > 1 for a wormhole throat to exist. So, he suggested one alternative method, where he considered taking A = 0 (where A is an independent parameter) in field equations (5)-(12) given in paper. This implies that Q = 0 and Fµν = 0. The action for gravity coupled to a dilaton and Maxwell field equation becomes independent of ‘a’ and the static spherically symmetric solutions is still the static, spherically symmetric solution with a now being a free parameter. It is well-known that the static, spherically symmetric solution to the usual Einstein-scalar field equations contains an additional free parameter related to the scalar charge. Thus, in string theory a wormhole throat can be created by taking A = 0 and a > 1.

He showed that charged dilaton wormholes with a > 1 can also be produced by similar methods. In this case, the charge associated with the two asymptotically flat regions will be different & should be chosen so as to make the electromagnetic field continuous across the jump. This implies that A and B will generally have different values on M+ and M–.

Note: Where, M+ and M– are two different spacetimes, while, A and B and independent parameters.


Reference: Dan Vollick, “Wormholes in string theory”, Classical and Quantum Gravity, Volume 16, Number 5, 1998. https://iopscience.iop.org/article/10.1088/0264-9381/16/5/309


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How Could We Produce Exotic Matter Required To Hold A Wormhole Open, Classically? (Quantum / Cosmology)

Summary:

○ Dan Vollick in his paper showed that it is possible to produce the exotic matter required to hold a wormhole open classically. This is accomplished by coupling a scalar field to matter that satisfies the weak energy condition.

○ The violation of the weak energy condition (for the total energy-momentum tensor) is produced by the interaction energy-momentum tensor.

○ In addition to satisfying the weak energy condition he showed that the matter and scalar field also satisfy the dominant energy condition. Thus a wormhole can be maintained classically by coupling a scalar field to matter that satisfies the weak and dominant energy conditions.

○ Finally, he showed that it is not possible for the matter energy-momentum tensor to satisfy the weak energy condition if the gravitational field is weak.


A wormhole is a handle which connects two different space-times or two distant regions in the same space-time. To keep a wormhole open it is necessary to thread its throat with matter that violates the averaged weak energy condition. In other words, there exist null geodesics passing through the wormhole, with tangent vectors k^µ = dx^µ/dσ, which satisfy

Such matter obviously violates the weak energy condition which states that T^µν ×Uµ×Uν ≥ 0 for all non-spacelike vectors U^µ. The weak energy condition ensures that all observers will see a positive energy density. Matter that violates the weak energy condition is called exotic. Thus it takes exotic matter to hold a wormhole open.

Most discussions of exotic matter involve quantum field theory effects, such as the Casimir effect. But, Dan Vollick in his paper showed that it is possible to generate the exotic matter required to maintain a wormhole classically. This is accomplished by coupling a scalar field to matter which satisfies the weak energy condition.

I derived the energy-momentum tensor for a scalar field coupled to an ideal fluid. In addition to the energy-momentum tensor for the matter and the scalar field there exists an interaction energy-momentum tensor. The interaction energy-momentum tensor can violate the weak and dominant energy conditions even if the matter & scalar field energy-momentum tensors do not. It is the interaction energy-momentum tensor that allows the wormhole to be maintained.

— said Dan Vollick, author of the study.

To create a wormhole he took two static, spherically symmetric, scalar-vac solutions of the Einstein field equations and join them together. A surface energy-momentum tensor exists on the surface where the two spacetimes are joined. This surface energy-momentum tensor violates the weak energy condition.

However, if the source of the energy-momentum tensor is taken to be a scalar field coupled to matter he showed that the energy-momentum tensor of the matter and scalar field can satisfy the weak and dominant energy conditions. The violation of the weak energy condition (for the total energy-momentum tensor) is produced by the interaction energy-momentum tensor.

In addition to satisfying the weak energy condition, he showed that the matter and scalar field also satisfy the dominant energy condition. The dominant energy condition ensures that the four velocity associated with the local flow of energy and momentum is non-spacelike. Thus a wormhole can be maintained classically by coupling a scalar field to matter that satisfies the weak and dominant energy conditions.

Finally, he showed that it is not possible for the matter energy-momentum tensor to satisfy the weak energy condition if the gravitational field is weak.


Reference: Dan N. Vollick, “Maintaining a wormhole with a scalar field”, Phys. Rev. D 56, 4724 – Published 15 October 1997. https://journals.aps.org/prd/abstract/10.1103/PhysRevD.56.4724 DOI: https://doi.org/10.1103/PhysRevD.56.4724


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Does Scalar Field and Gravitational Waves Helps in Expansion Of the Universe? (Cosmology / Astronomy / Maths)

Summary:

–> By obtaining de Sitter solutions for isotropic homogeneous on average scalar field, gravitational waves (GW) and gravitons for the empty Universe, Marochnik and colleagues showed that the scalar field and quantum gravitational waves generate the de Sitter expansion of the empty (with no matter) space-time i.e. at the start and by the end of its cosmological evolution the Universe is empty.

–> To get the de Sitter accelerated expansion of the empty space-time under influence of scalar fields, gravitons and classical and quantum gravitational waves, one needs to make a mandatory Wick rotation, i.e. one needs to make a transition to the Euclidean space of imaginary time [means from where the Universe started to the end (which will be imaginary)].

–> In short, in order to get the exponentially fast expansion of the empty FLRW space-time one must use time as a complex variable.

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Inflation and dark energy are two unsolved puzzles of modern cosmology. The idea of the necessity of inflation (exponentially fast expansion of the very early Universe) does not yet have direct reliable observational confirmation but it seems very attractive due to its ability to solve three known major cosmological paradoxes (flatness, horizon and monopoles). On the other hand, the existence of dark energy (exponentially fast expansion of the modern very late Universe) is an established observational fact. In case of inflation, it is almost generally accepted that the acceleration of expansion is due to a hypothetical scalar field. Choosing a different form of the potential of this field, the authors attempt to reconcile the theory with a number of e-foldings needed to ensure the flatness of the modern Universe to solve the horizon and monopole problems and get an agreement with the CMB observations. In the case of dark energy, for the lack of a better choice, the cosmological constant and the quintessence ( evolving scalar field) are considered as the popular candidates to provide an acceleration, although both ideas meet insuperable difficulties like one of these is a so-called “old cosmological constant problem”: Why is the Λ -value measured from observations is of the order of 10−¹²² vacuum energy density? The second one is a “coincidence problem”: Why is the acceleration happening during the contemporary epoch of matter domination?

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A common feature of both effects is exponentially rapid expansion of the Universe, and this is an accepted fact. Another common feature is the fact that both effects occur in an empty (or nearly empty) Universe. If an empty space should expand with acceleration then such a mechanism of acceleration should be common for both very early and very late Universe. The crucial importance of this fact was mentioned by Marochnik in his last paper. The modern Universe is about 70% empty, so that we are in the emptying expanding Universe, which asymptotically must become completely empty, i.e. contain no matter. As a result of such emptiness, we observe the dark energy effect. If before the birth of matter the Universe was empty, this empty space-time was to expand in accordance with the de Sitter law. It could explain the origin of inflation. The space-time without matter is not really empty as it always has the natural quantum and Classical fluctuations of its geometry, i.e. gravitons and classical gravitational waves. The question arises: Could not gravitons and/or classical gravitational waves, filling the empty (or nearly empty) Universe, lead to its accelerated expansion?

Now, University of Maryland’s researchers Leonid Marochnik and colleagues have shown that the answer to this question is yes. In their paper, they showed that in the FLRW metric on the average, homogeneous and isotropic scalar field and on the average homogeneous and isotropic ensembles of classical and quantum gravitational waves generate the de Sitter expansion of the empty (with no matter) space-time. i.e. friends, at the start and by the end of its cosmological evolution the Universe is empty.

“The de Sitter solutions for isotropic homogeneous on average scalar field, gravitational waves (GW) and gravitons are obtained for the empty Universe, i.e. they are applicable to the beginning of the Universe evolution (before matter was born) and to the end of it (after the matter disappears).”, said Marochnik.

According to authors, to get the de Sitter accelerated expansion of the empty space-time under influence of scalar fields, gravitons and classical and quantum gravitational waves, one needs to make a mandatory Wick rotation, i.e. one needs to make a transition to the Euclidean space of imaginary time. In other words, to get the exponentially fast expansion of the empty FLRW space-time one must use time as a complex variable.

“As is shown in this paper, the de Sitter accelerated expansion of the empty FLRW space under backreaction of quantum metric fluctuations, classical gravitational waves and/or scalar field require a mandatory transition to the Euclidean space of imaginary time and then return to the Lorentzian space of real time.”, wrote Marochnik in his paper.

On the other hand, they confirmed the de Sitter accelerated expansion of the empty space at the beginning and end of the evolution of the Universe by observational data (dark energy and inflation). One can assume that the very existence of these two effects is the observable evidence to the fact that time by its nature could be a complex value in the empty spacetime of the Universe. If time is a complex variable, what is the physical meaning of its imaginary part?

“I am indebted to Daniel Usikov profound remark that the observable could be only the real part of time, and the imaginary part, for whatever reasons, is unobservable (by analogy with quantum mechanics).”, said Marochnik.

Reference: Marochnik, L. Cosmological acceleration from a scalar field and classical and quantum gravitational waves (Inflation and dark energy). Gravit. Cosmol. 23, 201–207 (2017). https://doi.org/10.1134/S0202289317030082 https://link.springer.com/article/10.1134/S0202289317030082

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