Category Archives: Quantum Mechanics

Why Strong Interaction at Large Distances And Weaker At Short? (Particle Physics / Quantum)

Recently, Subhash Kak proposed that space is e-dimensional, rather than the commonly supposed 3-dimensions, where, e is Eulers number that is about 2.71828, and he used this to provide a resolution to the problem of Hubble tension, which is the disagreement between values of the rate at which the universe is expanding obtained using two different methods. Now, Kak has provided further support to this theory by a model that may explain why quarks, constituents of matter, don’t interact with each other when they are up close, but are bound strongly when pulled apart.

“Noninteger dimensionality leads to the anomalous situation of strong interaction at large distances and much weaker interaction at short distances, which is a characteristic of asymptotic freedom,” writes Subhash Kak, Professor in the School of Electrical and Computer Engineering at Oklahoma State University.

His new paper shows that, as the dimensionality falls below the value of critical dimension (dcrit), which lies between 2 and 3, there arises strange behavior where increasing energy reduces the strength of interaction between the particles. 

And when the value is 2 or below, the potential becomes constant (independent of separation) and force between objects or particles completely disappears. This new version of asymptotic freedom, which arises from the squeezing the dimensionality of space, could be of use in studying the anomalous mechanical properties of metamaterials.

“Future investigation will reveal whether this phenomenon based on dimensionality of space has any connection with other models of asymptotic freedom,” he concludes.

Featured image: Potential with respect to dimensionality; blue line p3,B(d), and red line p3,C(d) © S. Kak

Reference: Kak, S. Asymptotic freedom and noninteger dimensionality. Sci Rep 11, 3406 (2021).

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Reviewed by S. Kak

New Viable Ways Of Storing Information For Quantum Technologies? (Quantum)

Quantum information could be the source of the next technological revolution. By analogy with the bit in classical computing, the qubit is the basic element of quantum computing. But demonstrating the existence of this information storage unit and using it is still complex and therefore limited. In a study published on August 3, 2021 in Physical Review X , an international research team, composed by Fabio Pistolesi, CNRS researcher and two foreign researchers, succeeded by theoretical calculations to show that it is possible to realize a new type of qubit where information is stored in the oscillation amplitude of a carbon nanotube. 

Indeed, these nanotubes are able to perform a large number of oscillations without fading, which shows their weak interaction with the environment and makes them excellent potential qubits. This property would allow greater reliability in quantum computation. However, a problem persisted in reading and writing the information stored in the first two energy levels of these oscillators. 

Scientists have succeeded in proving that it is possible to read this information by exploiting the coupling between electrons, a negatively charged particle, and the bending mode of these nanotubes. This makes it possible to sufficiently change the spacing between the first energy levels and thus make them accessible independently of the other levels to read the information they contain. It now remains to experimentally verify these promising theoretical predictions.

To find out more: A computer to be reinvented for quantum computing

Featured image: Representation of the bending mode of a nanotube shown here in turquoise blue, and the locations of electrons in red and brown in the tube. © Fabio Pistolesi


Proposal for a Nanomechanical Qubit . F. Pistolesi, AN Cleland and A. Bachtold. Physical Review X , August 3, 2021.

Provided by CNRS

Researchers Developed Quantum Sensing System To Detect Pipeline Leaks More Quickly (Quantum)

To minimize potential damage from underground oil and gas leaks, Oak Ridge National Laboratory is co-developing a quantum sensing system to detect pipeline leaks more quickly.

Currently, fiber-optic sensing cables running through or around pipes detect fluid flow and leaks with signals from classical light sources. The new system by the University of Oklahoma, Louisiana State University and ORNL will replace classical light with quantum light originating from entanglement. Quantum-entangled light sources create less background noise than classical light and are sensitive to smaller signals.

“We’ve shown in many other systems that there are signals that are so small that they’re hidden by the classical noise, but you can see them with the quantum noise,” ORNL’s Raphael Pooser said.

OU researchers are building the machine that produces entangled particles. The team will evaluate the system at ORNL and then perform a 5,000-foot deep well test at LSU. – Alexandra DeMarco

Featured image: ORNL’s particle entanglement machine is a precursor to the device that University of Oklahoma researchers are building, which will produce entangled quantum particles for quantum sensing to detect underground pipeline leaks. Credit: ORNL, U.S. Dept. of Energy

This science news has been confirmed by us from ORNL

Provided by ORNL

What Leads To Factorization Problem? How Half Wormholes Can Fix It? (Maths / Cosmology / Quantum)

Wormholes not only play a key role in understanding the nonperturbative physics of quantum black holes, for instance: the eternal traversable wormhole; the long-time behavior of the spectral form factor and correlation functions, the Page curve etc. but also, it leads to puzzles, in particular the factorization problem. Imagine two decoupled boundary systems in the AdS/CFT context, labelled L and R. From the boundary perspective, if one evaluates a partition function in the combined system the result is just the product of the results for the two component systems:

It factorizes. But, if the bulk calculation of ZLR includes a wormhole linking L and R then superficially at least ZLR ≠ ZLZR. It fails to factorize. Some of the phenomena recently explained by wormholes, in particular the spectral form factor and squared matrix elements, are described by decoupled boundary systems and so the wormhole explanation give rise to a factorization puzzle.

But, you can remove this factorization puzzle by averaging the L and R systems over the same ensemble, denoted by (·), with the help of the SYK model. The factorization puzzle solves because (ZLZR) need not to be same as (ZL) (ZR). And infact this link between wormholes and ensembles is not a new one, it dates back to the 1980s. However, it has been recently applied in AdS/CFT context.

We can create a new form of factorization puzzle in such ensembles by asking what happens to the wormholes connecting decoupled systems when we focus on just 1 element of the ensemble. Now, Phil Saad and colleagues addressed this question in the SYK model where instead of averaging the L and R systems they picked a fixed set of couplings between the fermions.

These pictures represent saddle points of the SYK path integral, associated to the sketched bulk topology by the pattern of correlation. As the wormhole contribution is self-averaging, they have depicted it with a small red “x” to indicate the small amount of randomness. The half-wormhole contributions are not self-averaging, so they have depicted them as “half” of a wormhole with a jagged red boundary to indicate the large amount of randomness. They have included a red line linking the pair of half-wormholes on the LHS, to remind them that the LR collective fields are present, but set to zero, distinguishing this contribution from the unlinked pair of half wormholes on the RHS. © Phil Saad et al.

After averaging over fermion couplings, SYK model has a collective fields called G and Σ, that sometimes has “wormhole” solutions. Phil Saad and colleagues studied the fate of these wormholes when the couplings are fixed.

Working mainly in a simple model, they found that the wormhole saddles persist, and the dependence on the couplings is weak. The wormhole is “self-averaging”. But, that new saddles also appear elsewhere in the integration space, which they interpret as “half-wormholes.” The half-wormhole contributions depends sensitively on the particular choice of couplings.

Finally, they showed that, the half-wormholes are crucial for factorization (or restore factorization) of decoupled systems with fixed couplings, but they vanish after averaging, leaving the non-factorizing wormhole behind.

Reference: Phil Saad, Stephen H. Shenker, Douglas Stanford, and Shunyu Yao, “Wormholes without averaging”, Arxiv, pp. 1-34, 2021.

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Can A Traversable Wormhole Exist Only At The Planck Scale? (Quantum / Cosmology)

Hideki Maeda presented a simple traversable wormhole solution which violate energy conditions only at the planck scale

Wormhole is a configuration of spacetimes containing distinct non-timelike infinities. In particular, a wormhole that contains a casual curve connecting such infinities are referred to as a traversable wormhole. Several examples of traversable wormholes have been already discussed by us in our articles. Now, Hideki Maeda presented a simple static spacetime which describes a spherically symmetric traversable worm-hole characterized by a length parameter, l and reduces to Minkowski in the limit l → 0. His findings recently appeared in Arxiv.

He showed that, this wormhole connects two distinct asymptotically flat regions with vanishing ADM mass and the areal radius of its throat is exactly l. Additionally, all the standard energy conditions i.e. null-energy condition, weak energy condition, dominant energy condition and standard energy condition are violated outside the proper radial distance approximately 1.60 l from the wormhole throat.

Finally, he computed the total amount of negative energy required to support this wormhole and found that, if l is identified as the Planck length lp (≃ 1.616 × 10¯35 m), the total amount of the negative energy supporting this wormhole is only E ≃ −2.65 mpc², which is the rest mass energy of about – 5.77 × 10¯5 g. For a “humanly traversable” wormhole with l = 1m, he obtained mass of −3.57 × 1027 kg, which is about –1.88 times as large as Jupiter’s mass.

“Ofcourse, an important task is to identify a theory of gravity which admits the static spacetime as a solution. Once such theory is identified, a subsequent task is to study the stability. Since the region of the NEC violation is tiny, our wormhole could be a dynamically stable configuration in the semiclassical regime. These important tasks are left for future investigations.”

— he concluded.

Reference: Hideki Maeda, “A simple traversable wormhole violating energy conditions only at the Planck scale”, Arxiv, pp. 1-4, 2021.

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How Would Be Thick Branes In Mimetic Gravity? (Quantum)

Qun-Ying Xie and colleagues investigated thick branes generated by a scalar field in mimetic gravity theory, which is inspired by considering the conformal symmetry under the conformal transformation of an auxiliary metric. They obtained a series of analytical thick brane solutions and found that, perturbations of the brane system are stable and the effective potentials for the tensor and scalar perturbations are dual to each other. Findings of this study recently appeared in the Journal Symmetry.

Mimetic gravity is a Weyl-symmetric extension of General Relativity, related to the latter by a singular disformal transformation, wherein the appearance of a dust-like perfect fluid can mimic cold dark matter at a cosmological level. Within this framework, it is possible to provide an unified geometrical explanation for dark matter, the late-time acceleration, and inflation, making it a very attractive theory. This theory was also extended to Horava-like theory and applied to galactic rotation curves. It was also applied to other gravity theories such as f(R) gravity, Horndeski gravity and Gauss–Bonnet gravity.

On the other hand, Lisa Randall and Raman Sundrum proposed that our four-dimensional world could be a brane embedded in five-dimensional space-time, in order to solve gauge hierarchy problem and the cosmological constant problem. With the warped extra dimension, it was further found that the size of extra dimension can be infinitely large without conflicting with Newtonian gravitational law. This charming idea has attracted substantial researches in particle physics, cosmology, gravity theory, and other related fields. In the RS model, the brane is geometrically thin, therefore the space-time is singular at the brane. Although in the thin brane approximation many interesting results have been obtained, in some situations the effects of the brane thickness cannot be neglected.

“In five-dimensional problems, the thin brane approximation is valid as long as the brane thickness cannot be resolved, in other words, if the energy scale of the brane thickness is much higher than those in the bulk and on the brane. In contrast, when thickness becomes as large as the scale of interest, its effect is no longer negligible.”

Now, Qun-Ying Xie and colleagues investigated the super-potential method with which the second-order equations can be reduced to the first-order ones for thick brane models in modified gravity with Lagrange multiplier. The main step of this method is to introduce a pair of auxiliary super-potentials, i.e., W(φ) and Q(φ). With these two super-potentials, the field equations are rewritten as Equations (1)–(5).

Then, they used this method to find a series of analytical thick brane solutions via some polynomial super-potentials, period super-potentials, and mixed super-potentials.

Finally, they analyzed the tensor and scalar perturbations of the brane system. It was shown that both equations of motion of the perturbations can be transformed into Schrodinger-like equations. They added that, both perturbations are stable and the effective potentials for the tensor and scalar perturbations are dual to each other. Moreover, the tensor zero mode can be localized on the brane while the scalar zero mode cannot. Thus, the four-dimensional Newtonian potential can be recovered on the brane and there is no additional fifth force contradicting with the experiments.

Featured image: The effective potential VS(z(y)) for solution I. The parameter is set as n = 1 (red solid thick lines), n = 3 (blue dashed lines), and n = 5 (black solid thin lines). © Xie et al.

Reference: Xie, Q.-Y.; Fu, Q.-M.; Sui, T.-T.; Zhao, L.; Zhong, Y. First-Order Formalism and Thick Branes in Mimetic Gravity. Symmetry 2021, 13,

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How Quantum Fields Could Be Used To Break Low-temperature Records? (Quantum / Physics)

At first glance, heat and cold do not have much to do with quantum physics. A single atom is neither hot nor cold. Temperature can traditionally only be defined for objects that consist of many particles. But at TU Wien, in collaboration with FU Berlin, Nanyang Technological University in Singapore and the University of Lisbon, it has now been possible to show what possibilities arise when thermodynamics and quantum physics are combined: One can specifically use quantum effects to cool a cloud of ultracold atoms even further.

No matter what sophisticated cooling methods have been used before—with this technique, which has now been presented in the scientific journal Physical Review X-Quantum, it is possible to come a little closer to absolute zero. A lot of work is still needed before this new cooling concept can be turned into an actual quantum refrigerator, but initial experiments already show that the necessary steps are possible in principle.

A new field of research: quantum thermodynamics

“For a long time, thermodynamics has played an important role for classical mechanical machines—think of steam engines or combustion engines, for example. Today, quantum machines are being developed on a tiny scale. And there, thermodynamics has hardly played a role there so far” says Prof. Eisert from the Free University of Berlin.

“If you want to build a quantum heat machine, you have to fulfill two requirements that are fundamentally contradictory,” says Prof. Marcus Huber from TU Wien. “It has to be a system that consists of many particles and in which you cannot control every detail exactly. Otherwise you cannot speak of heat. And at the same time, the system must be simple enough and sufficiently precisely controllable not to destroy quantum effects. Otherwise, you can’t talk about a quantum machine.”

“Back in 2018, we came up with the idea of transferring the basic principles of thermal machines to quantum systems by using quantum field descriptions of many-body quantum systems,” says Prof. Jörg Schmiedmayer (TU Wien). Now the research team from TU Wien and FU Berlin examined in detail how such quantum heat machines can be designed. They were guided by the operating principle of an ordinary refrigerator: initially, everything has the same temperature—the interior of the refrigerator, the environment and the coolant. But when you evaporate the coolant inside the refrigerator, heat is extracted there. The heat is then released outside when the coolant is liquefied again. So by raising and lowering the pressure it is possible to cool the inside and transfer the heat to the environment.

The question was whether there could also be a quantum version of such a process. “Our idea was to use a Bose-Einstein condensate for this, an extremely cold state of matter,” says Prof. Jörg Schmiedmayer. “In recent years, we have gained a lot of experience in controlling and manipulating such condensates very precisely with the help of electromagnetic fields and laser beams, investigating some of the fundamental phenomena at the borderline between quantum physics and thermodynamics. The logical next step was the quantum heat machine.”

How quantum fields could be used to break low-temperature records
Credit: Vienna University of Technology

Energy redistribution at the atomic level

A Bose-Einstein condensate is divided into three parts, which initially have the same temperature. “If you couple these subsystems in exactly the right way and separate them from each other again, you can achieve that the part in the middle acts as a piston, so to speak, and allows heat energy to be transferred from one side to the other,” explains Marcus Huber. “As a result, one of the three subsystems is cooled down.”

Even at the beginning, the Bose-Einstein condensate is in a state of very low energy—but not quite in the lowest possible energy state. Some quanta of energy are still present and can change from one subsystem to another—these are known as “excitations of the quantum field.”

“These excitations take on the role of the coolant in our case,” says Marcus Huber. “However, there are fundamental differences between our system and a classical refrigerator: In a classical refrigerator, heat flow can only occur in one direction—from warm to cold. In a quantum system, it is more complicated; the energy can also change from one subsystem to another and then return again. So you have to control very precisely when which subsystems should be connected and when they should be decoupled.”

So far, this quantum refrigerator is only a theoretical concept—but experiments have already shown that the necessary steps are feasible. “Now that we know that the idea basically works, we will try to implement it in the lab,” says Joao Sabino (TU Wien). “We hope to succeed in the near future.” That would be a spectacular step forward in cryogenic physics—because no matter what other methods you use to reach extremely low temperatures, you could always add the novel ‘quantum refrigerator’ at the end as a final additional cooling stage to make one part of the ultracold system even colder. “If it works with cold atoms, then our ideas can be implemented in many other quantum systems and lead to new quantum technology applications,” says Jörg Schmiedmayer.

Featured image: João Sabino in the lab. Credit: Vienna University of Technology

Reference: Marek Gluza et al, Quantum Field Thermal Machines, PRX Quantum (2021). DOI: 10.1103/PRXQuantum.2.030310

Provided by Vienna University of Technology

Exploring Quantum Systems That Don’t Find Equilibrium (Quantum)

Some physical systems, especially in the quantum world, do not reach a stable equilibrium even after a long time. An ETH researcher has now found an elegant explanation for this phenomenon.

If you put a bottle of beer in a big bathtub full of ice-cold water, it won’t be long before you can enjoy a cold beer. Physicists discovered how this works more than a hundred years ago. Heat exchange takes place through the glass bottle until equilibrium is reached.

However, there are other systems, especially quantum systems, that don’t find equilibrium. They resemble a hypothetical beer bottle in a bath of ice-​cold water that doesn’t always and inevitably cool to the temperature of the bath water, but rather reaches different states depending on its own initial temperature. Until now, such systems have puzzled physicists. But Nicolò Defenu, a postdoc at the ETH Zurich Institute for Theoretical Physics, has now found a way to elegantly explain this behaviour.

A more distant influence

Specifically, we are talking about systems in which the individual building blocks influence not only their immediate neighbours, but also objects further away. One example would be a galaxy: the gravitational force of their individual stars and planetary systems acts not only on the neighbouring celestial bodies, but far beyond that – albeit ever more weakly – on the other components of the galaxy.

Defenu’s approach begins by simplifying the problem to a world with a single dimension. In it, there is a single quantum particle that can reside only in very specific locations along a line. This world resembles a board game like Ludo, where a little token hops from square to square. Suppose there is a game die whose sides are all marked “one” or “minus one”, and suppose the player whose token it is now rolls the die over and over again in succession. The token will hop to a neighbouring square, and from there it will either hop back or else on to the next square. And so on.

The question is, What happens if the player rolls the die an infinite number of times? If there are only a few squares in the game, the token will return to its starting point every now and then. However, it is impossible to predict exactly where it will be at any given time because the throws of the die are unknown.

Back to square one

It’s a similar situation with particles that are subject to the laws of quantum mechanics: there’s no way to know exactly where they are at any given time. However, it is possible to establish their whereabouts using probability distributions. Each distribution results from a different superposition of the probabilities for the individual locations and corresponds to a particular energy state of the particle. It turns out that the number of stable energy states coincides with the number of degrees of freedom of the system and thus corresponds exactly to the number of allowed locations. The important point is that all the stable probability distributions are non-​zero at the starting point. So at some point, the token returns to its starting square.

The more squares there are, the less often the token will return to its starting point; eventually, with an infinite number of possible squares, it will never return. For the quantum particle, this means there are an infinite number of ways in which the probabilities of the individual locations can be combined to form distributions. Thus, it can no longer occupy only certain discrete energy states, but all possible ones in a continuous spectrum.

None of this is new knowledge. There are, however, variants of the game or physical systems where the die can also contain numbers larger than one and smaller than minus one, i.e. the steps allowed per move can be larger – to be precise, even infinitely large. This fundamentally changes the situation, as Defenu has now been able to show: in these systems, the energy spectrum always remains discrete, even when there are infinite squares. This means that from time to time, the particle will return to its starting point.

Peculiar phenomena

This new theory explains what scientists have already observed many times in experiments: systems in which long-​range interactions occur do not reach a stable equilibrium, but rather a meta-​stable state in which they always return to their initial position. In the case of galaxies, this is one reason they develop spiral arms rather than being uniform clouds. The density of stars is higher inside these arms than outside.

An example of quantum systems that can be described with Defenu’s theory are ions, which are charged atoms trapped in electric fields. Using such ion traps to build quantum computers is currently one of the largest research projects worldwide. However, for these computers to really deliver a step change in terms of computational power, they will need a very large number of simultaneously trapped ions – and that is exactly the point at which the new theory becomes interesting. “In systems with a hundred or more ions, you would see peculiar effects that we can now explain,” says Defenu, who is a member of ETH Professor Gian Michele Graf’s group. His colleagues in experimental physics are getting closer every day to the goal of being able to realise such formations. And once they’ve got there, it might be worth their while to have a cold beer with Defenu.

Featured image: Not only quantum systems, but also large objects such as the spiral galaxy NGC 1300 can adopt a meta-stable state that leads to surprising effects. (Picture: Hubble Heritage Team, ESA, NASA)


Defenu N: Metastability and discrete spectrum of long-range systems. Proceedings of the National Academy of Sciences, July 26 2021: doi: 10.1073/pnas.2101785118

Provided by ETH Zurich

How Production Of Primordial Black Holes Take Place In NonMinimal Derivative Coupling Inflation? (Cosmology / Quantum)

Heydari and Karami investigated the generation of Primordial Black Holes (PBHs) with the aid of gravitationally increased friction mechanism originated from the NonMinimal field Derivative Coupling (NMDC) to gravity framework, with the quartic potential. By assigning a coupling parameter as a two-parted function of inflation field and fine-tuning of 4 parameter cases of the model they showed that they could acquire an epoch of ultra slow-roll inflation on scales smaller than CMB scale making the inflaton slow down, sufficient to generate PBHs. Their study recently appeared in Arxiv.

You may have heard the concept of primordial black holes (PBHs) generation from the over-dense regions of the early universe. For the generation of primordial black holes (PBHs) during Radiation Dominated (RD) era, production of a large enough amplitude of primordial curvature perturbations (R) during inflationary epoch is necessary. Overdense regions can be formed when superhorizon scales associated with the large amplitude of R become subhorizon during RD era, and gravitationally collapse of these overdensities generate PBHs.

Meaning, PBHs generation requires an enhancement in the power spectrum of R to order 10¯2 at scales smaller than CMB scales. Several techniques have been employed for multiplying the amplitude of the power spectrum of R at small scale by seven orders of magnitude in comparison with CMB scales. One of the proper ways to achieve a rise in the scalar power spectrum is a brief period of Ultra Slow-Roll (USR) inflation due to declining the speed of inflaton field via gravitationally enhanced friction. The framework of NonMinimal Derivative Coupling to gravity (NMDC) beside the fine-tuning of the parameters of the model can give rise to increase friction gravitationally.

The nonminimal derivative coupling model is a subclass of a generic scalar-tensor theory with second-order equations of motion namely Horndeski theory, which prevents the model from negative energy and pertinent instabilities. A characteristic of the nonminimal field derivative coupling to gravity is that the gravitationally increased friction mechanism can be applied for generic steep potentials such as quartic potential.

Now, Heydari and Karami investigated the generation of Primordial Black Holes (PBHs) with the aid of gravitationally increased friction mechanism originated from the NonMinimal field Derivative Coupling (NMDC) to gravity framework, with the quartic potential.

© Heydari and Karami

By assigning the coupling parameter as a two-parted function of inflaton field, and fine-tuning of the four parameter cases (A, B, C, and D) of the model (see Table I), they showed that, we could acquire an epoch of ultra slow-roll inflation on scales smaller than CMB scale making the inflaton slow down due to high friction. This enables them to achieve enough enhancement in the amplitude of curvature perturbations power spectra to generate PBHs with masses of order 10 M for Case A (stellar mass), 10¯6 M for Case B (earth mass), 10¯13 M for Case C, and 10¯15 M for Case D (asteroid mass).

Their results indicated that PBHs of case A is suitable to describe GWs and LIGO events, Case B can be useful to expound microlensing events in OGLE data, and PBHs of cases C and D can be interesting candidates for composing around 98.32% and 99.11% of dark matter (DM) content of the universe (see Table II and Fig. 1)

FIG. 1. The PBHs abundance (fPBH) in terms of PBHs mass (M) for Case A (purple line), Case B (green line), Case C (red line), and Case D (blue line). The red spots indicate the upper limit on the PBH abundance owing to the upper limit on the LIGO event merger rate. The brown shadowy zone signifies the permitted zone of PBH abundance from the ultrashort-timescale microlensing events in the OGLE data. The other shadowy areas demonstrate the current observational restrictions on the fractional abundance of PBHs comprising extragalactic gamma rays from PBH evaporation (EGγ), galactic center 511 keV γ-ray line (INTEGRAL), white dwarf explosion (WD), microlensing events with Subaru HSC (Subaru HSC), with the Kepler satellite (Kepler), with EROS/MACHO (EROS/MACHO), and accretion constraints from CMB © Heydari and Karami

Additionally, they inquired generation of the induced GWs subsequent to PBHs formation for all cases of their model. Their calculation of current density parameter spectra (ΩGW0) indicated that, all cases have climaxes at contrasting frequencies with nearly identical heights of order 10¯8. The climaxes of ΩGW0 for cases A and B have placed at frequencies 10¯10Hz and 10¯7 Hz, respectively, and both cases can be traced via the SKA detector. Moreover, the spectra of ΩGW0 for Cases A and B have climaxes localized at mHz and cHz bands which are tracked down by LISA, TaiJi, and TianQin (see Fig. 2). Hence, validity of their model can be assessed in view of GWs via the extricated data of these detectors.

Fig 2. The present induced GWs energy density parameter (ΩGW0) with respect to frequency. The solid purple, green, red, and blue lines associate with Cases A, B, C, and D of Table I, respectively. The power-law form of ΩGW0 is illustrated by black dashed line for Case D. © Heydari and Karami

Finally, they demonstrated that in the vicinity of climaxes, the spectra of density parameter behave as a power-law function with respect to frequency (ΩGW0 (f) ∼ (f/fc)^n). Also, in the infrared regime f<<fc, the power index satifies the relation n = 3 – 2/ ln(fc/f).

Reference: Soma Heydari, Kayoomars Karami, “Primordial black holes in nonminimal derivative coupling inflation driven by quartic potential”, Arxiv, pp. 1-28, 2021.

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