Single Field Research Articles

Relativistic single-electron wavepacket in quantum electromagnetic fields: quantum coherence, correlations, and the Unruh effect

Conventional formulation of QED since the 50s works very well for stationary states and for scattering problems, but with newly arisen challenges from the 80s on, where real time evolution of particles in a nonequilibrium setting are required, and quantum features such as coherence, dissipation, correlation and entanglement in a system interacting with its quantum field environment are sought after, new ways to formulate QED suitable for these purposes beckon. In this paper we present a linearized effective theory using a Gaussian wavepacket description of a charged relativistic particle coupled to quantum electromagnetic fields to study the interplay between single electrons and quantum fields in free space, at a scale well below the Schwinger limit. The proper values of the regulators in our effective theory are determined from the data of individual experiments, and will be time-dependent in the laboratory frame if the single electrons are accelerated. Using this new theoretical tool, we address the issues of decoherence of flying electrons in free space and the impact of Unruh effect on the electrons. Our result suggests that vacuum fluctuations may be a major source of blurring the interference pattern in electron microscopes. For a single electron accelerated in a uniform electric field, we identify the Unruh effect in the two-point correlators of the deviations from the electron’s classical trajectory. From our calculations we also bring out some subtleties, involving the bosonic versus fermionic spectral functions.

Inversion of Soil Moisture Content in Cotton Fields Using GBR-RF Algorithm Combined with Sentinel-2 Satellite Spectral Data

Soil moisture content plays a vital role in agricultural production, significantly influencing crop growth, development, and yield. Thoroughly understanding the specific soil moisture content in cotton fields is crucial for enhancing agricultural efficiency and driving sustainable agricultural development. This study utilized the gradient-boosting regression–random forest (GBR-RF) algorithm and the GBR and RF algorithms separately, in conjunction with Sentinel-2 satellite images, to estimate cotton soil moisture content, focusing on the B1–B8 bands and in particular the sensitive B6, B7, and B8 bands. The soil data in the jujube orchard of the study area were collected using soil augers at a depth of 30 cm, with soil data collected from a depth of 20 to 30 cm. The findings revealed that the integrated learning algorithm GBR-RF demonstrated high accuracy, with R2, MAE, and MSE results of 0.8838, 1.0121, and 1.6168, respectively. In comparison, the results using just the GBR algorithm yielded R2, MAE, and MSE values of 0.8158, 1.1327, and 1.9645, respectively, while those obtained from the RF algorithm were 0.8415, 1.0680, and 1.8331, respectively. These results indicate that the algorithms exhibited strong generalization, robustness, and accuracy, with GBR-RF outperforming GBR and RF by 8.34% and 5.03%, respectively, in combination with using the B1–B8 bands for inversion. Furthermore, utilizing the full-band data resulted in R2 values that were up to 24.27% higher than those of the individual bands, affirming the efficacy of band combinations for improved accuracy. This study’s demonstration of the positive impact of integrated learning algorithms on estimating cotton soil moisture content underscores the advantages of multi-band data combinations over single-band data, highlighting their ability to enhance accuracy without significantly impacting errors. Importantly, this study’s findings, while not limited to a single experimental field, have broad applicability in cotton precision agriculture, offering valuable insights for research on yield enhancement and agricultural efficiency.

Reactor field reconstruction from sparse and movable sensors using Voronoi tessellation-assisted convolutional neural networks

The aging of operational reactors leads to increased mechanical vibrations in the reactor interior. The vibration of the in-core sensors near their nominal locations is a new problem for neutronic field reconstruction. Current field-reconstruction methods fail to handle spatially moving sensors. In this study, we propose a Voronoi tessellation technique in combination with convolutional neural networks to handle this challenge. Observations from movable in-core sensors were projected onto the same global field structure using Voronoi tessellation, holding the magnitude and location information of the sensors. General convolutional neural networks were used to learn maps from observations to the global field. The proposed method reconstructed multi-physics fields (including fast flux, thermal flux, and power rate) using observations from a single field (such as thermal flux). Numerical tests based on the IAEA benchmark demonstrated the potential of the proposed method in practical engineering applications, particularly within an amplitude of 5 cm around the nominal locations, which led to average relative errors below 5% and 10% in the L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_2$$\\end{document} and L∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_{\\infty }$$\\end{document} norms, respectively.

Image Processing in MATLAB Enhancement, Encryption and Decryption

Photography was invented in 1826 by French inventor Joseph Nicéphore Niépce. Since then, it has revolutionized the modern world. From medical evidence in court & drawing images of industrial machines to personal photos for memory, not a single field is left untouched by photography. Nowadays when there is so many photo editing software, not a single one ensures true privacy and provides open-source code to ensure that no data is being collected. In six months of project making, we have developed an Image processing app that not only provides enhancement tools but also displays the power of steganography, a user-friendly environment to encrypt your enhanced images before sending them forward, and a magnificent display of Fourier series epicycles which creates an image using Fourier series, an intuitive way to learn and understand the beauty of mathematics. Mathematics could be called the language of science. Our tool proves why it is said so. And what could be a better platform to make all this other than MATLAB which not only provides numerous libraries but also is a powerful debugging tool that can display all the things happening inside the code. This app is built using the Guide function in MATLAB and has the potential to influence this generation toward data security as well as mathematics. Keyword: Enhancement, Cryptography, Epicycles, and Steganography

Rethinking Higher Education through Multidisciplinary Studies

Knowledge must not restrict or put boundaries in one's thoughts about the discipline or academic approach one must follow. A multidisciplinary approach to education releases one from such boundaries and allows an individual to use all the available resources and put them into practice to solve problems. The problems AI (Artificial Intelligence) and ML (Machine Learning) aim to solve are not limited to a single academic field. So, a multidisciplinary foundation is essential; it also allows students to explore specialized areas within AI and ML. A multidisciplinary approach allows students to connect their education to real-world applications and industries.

MolProphet: A One-Stop, General Purpose, and AI-Based Platform for the Early Stages of Drug Discovery.

Artificial intelligence (AI) is an effective tool to accelerate drug discovery and cut costs in discovery processes. Many successful AI applications are reported in the early stages of small molecule drug discovery. However, most of those applications require a deep understanding of software and hardware, and focus on a single field that implies data normalization and transfer between those applications is still a challenge for normal users. It usually limits the application of AI in drug discovery. Here, based on a series of robust models, we formed a one-stop, general purpose, and AI-based drug discovery platform, MolProphet, to provide complete functionalities in the early stages of small molecule drug discovery, including AI-based target pocket prediction, hit discovery and lead optimization, and compound targeting, as well as abundant analyzing tools to check the results. MolProphet is an accessible and user-friendly web-based platform that is fully designed according to the practices in the drug discovery industry. The molecule screened, generated, or optimized by the MolProphet is purchasable and synthesizable at low cost but with good drug-likeness. More than 400 users from industry and academia have used MolProphet in their work. We hope this platform can provide a powerful solution to assist each normal researcher in drug design and related research areas. It is available for everyone at https://www.molprophet.com/.

Scalar gravitational waves can be generated even without direct coupling between Dark Energy and ordinary matter

We point out, the scalar sector of gravitational perturbations may be excited by an isolated astrophysical system immersed in a universe whose accelerated expansion is not due to the cosmological constant, but due to extra field degrees of freedom. This is true even if the source of gravitational radiation did not couple directly to these additional fields. We illustrate this by considering a universe driven by a single canonical scalar field. By working within the gauge-invariant formalism, we solve for the electric components of the linearised Weyl tensor to demonstrate that both the gravitational massless spin-2 (transverse-traceless) tensor and the (Bardeen) scalar modes are generated by a generic astrophysical source. For concreteness, the Dark Energy scalar field is either released from rest, or allowed to asymptote toward the minimum in a certain class of potentials; and we compute the traceless tidal forces induced by gravitational radiation from a hypothetical compact binary system residing in such a universe. Though their magnitudes are very small compared to the tensors', spin zero gravitational waves in such a canonical scalar driven universe are directly sensitive to both the Dark Energy equation of state and the eccentricity of the binary's orbit.

Advancing simulation efficiency for MEMS electrostatic actuators

This paper presents an overview of the research progress on Multiphysics simulation of MEMS electrostatic actuators. MEMS electrostatic actuators have shown significant development potential in applications such as ADWSS and VOA. However, the increasing size and complexity of these devices have posed challenges to traditional Multiphysics simulation methods, leading to lengthy computational times that hinder the overall device development process. In order to enhance simulation efficiency and meet the demands of device development, the use of simplified single physics field models have been explored. The results have demonstrated that the simplified single physics filed simulation model yields comparable results to traditional Multiphysics simulations with an error tolerance of within 10%, while reducing simulation time by 80-90%.

Myofascial trigger point (MTrP) size and elasticity properties can be used to differentiate characteristics of MTrPs in lower back skeletal muscle

Myofascial trigger points (MTrPs) are localized contraction knots that develop after muscle overuse or an acute trauma. Significant work has been done to understand, diagnose, and treat MTrPs in order to improve patients suffering from their effects. However, effective non-invasive diagnostic tools are still a missing gap in both understanding and treating MTrPs. Effective treatments for patients suffering from MTrP mediated pain require a means to measure MTrP properties quantitatively and diagnostically both prior to and during intervention. Further, quantitative measurements of MTrPs are often limited by the availability of equipment and training. Here we develop ultrasound (US) based diagnostic metrics that can be used to distinguish the biophysical properties of MTrPs, and show how those metrics can be used by clinicians during patient diagnosis and treatment. We highlight the advantages and limitations of previous US-based approaches that utilize elasticity theory. To overcome these previous limitations, we use a hierarchical approach to distinguish MTrP properties by patients’ reported pain and clinician measured palpation. We show how US-based measurements can characterize MTrPs with this approach. We demonstrate that MTrPs tend to be smaller, stiffer, and deeper in the muscle tissue for patients with pain compared to patients without pain. We provide evidence that more than one MTrP within a single US-image field increases the stiffness of neighboring MTrPs. Finally, we highlight a combination of metrics (depth, thickness, and stiffness) that can be used by clinicians to evaluate individual MTrPs in combination with standard clinical assessments.

Auditory oddball responses in the human subthalamic nucleus and substantia nigra pars reticulata

The auditory oddball is a mainstay in research on attention, novelty, and sensory prediction. How this task engages subcortical structures like the subthalamic nucleus and substantia nigra pars reticulata is unclear. We administered an auditory OB task while recording single unit activity (35 units) and local field potentials (57 recordings) from the subthalamic nucleus and substantia nigra pars reticulata of 30 patients with Parkinson's disease undergoing deep brain stimulation surgery. We found tone modulated and oddball modulated units in both regions. Population activity differentiated oddball from standard trials from 200 ms to 1000 ms after the tone in both regions. In the substantia nigra, beta band activity in the local field potential was decreased following oddball tones. The oddball related activity we observe may underlie attention, sensory prediction, or surprise-induced motor suppression.

Polyakov blocks for the 1D conformal field theory mixed-correlator bootstrap

We introduce manifestly crossing-symmetric expansions for arbitrary systems of 1D CFT correlators. These expansions are given in terms of certain Polyakov blocks which we define and show how to compute efficiently. Equality of operator product expansion and Polyakov block expansions leads to sets of sum rules that any mixed correlator system must satisfy. The sum rules are diagonalized by correlators in tensor product theories of generalized free fields. We show that it is possible to do a change of a basis that diagonalizes instead mixed correlator systems involving elementary and composite operators in a single field theory. As an application, we find the first nontrivial examples of optimal bounds, saturated by the mixed correlator system ϕ,ϕ2 in the theory of a single generalized free field. Published by the American Physical Society 2024

Postinflationary production of particle dark matter: Hilltop and Coleman-Weinberg inflation

We investigate the production of nonthermal dark matter (DM), χ, during the postinflationary reheating era. For inflation, we consider two slow roll single field inflationary scenarios—generalized version of Hilltop (GH) inflation, and Coleman-Weinberg (CW) inflation. Using a set of benchmark values that comply with the current constraints from cosmic microwave background radiation (CMBR) data for each inflationary model, we explored the parameter space involving mass of dark matter particles, mχ, and coupling between inflaton and χ, yχ. For these benchmarks, we find that tensor-to-scalar ratio r can be as small as 2.69×10−6 for GH and 1.91×10−3 for CW inflation, both well inside 1−σ contour on scalar spectral index versus r plane from 2018+BICEP3+2018 dataset, and testable by future cosmic microwave background (CMB) observations, e.g., Simons Observatory. For the production of χ from the inflaton decay satisfying CMB and other cosmological bounds and successfully explaining total cold dark matter density of the present universe, we find that yχ should be within this range O(10−4)≳yχ≳O(10−20) for both inflationary scenarios. We also show that, even for the same inflationary scenario, the allowed parameter space on reheating temperature versus mχ plane alters with inflationary parameters including scalar spectral index, r, and energy scale of inflation. Published by the American Physical Society 2024

A 600 m2 array of 6.5 m telescopes at the lunar pole.

The proposed lunar telescope for optical and infrared astronomy aims at very large aperture, 600 m2, at a fundable cost. It comprises an array of 18 separate telescopes, each of 6.5 m aperture. The 200 m diameter array will be located within 1/2° (15 km) of a lunar pole on approximately level ground, with a perimeter screen deployed to provide shade and cooling to cryogenic temperature. The 500 m diameter screen will allow unobscured access down to 8° elevation. All 18 telescopes will reflect light into a central beam combiner to form a single image covering wavelengths from 0.4 µm to 10 µm. The initial instrument complement will include high-resolution and multi-object spectrographs to exploit the single combined field of view of two arcminute diameter, with the diffraction limited resolution of 6.5 m aperture. Scientific applications include the search for molecular biosignatures in transiting exoplanets, and the study of galaxy evolution using red-shifted spectra to beyond z = 10. The array cost, including delivery to the Moon by SpaceX Starship for installation using lunar base infrastructure, is around $10 billion, similar to that of the 25 m2 JWST. To test the concept, first a single prototype 6.5 m unit would be operated at the lunar south pole. This article is part of a discussion meeting issue 'Astronomy from the Moon: the next decades (part 2)'.

Generalized Gaussian states in group field theory and su(1,1) quantum cosmology

We define generalized Gaussian states for quantum cosmological models based on the su(1,1) algebra, with particular emphasis on its realization in group field theory for a single field mode, and study their semiclassical properties. These states are generalizations of coherent, squeezed, and thermal states considered previously. As two possible characterizations of semiclassicality, we contrast the requirement of small relative fluctuations in volume and energy with the saturation of the Robertson-Schrödinger uncertainty principle. We find that for the most general class of states, the appearance of small relative fluctuations, which we take as the main criterion relevant for the emergence of cosmology, is mostly determined by the amount of displacement used to define the state. We also observe that defining such generalized Gaussian states is less straightforward in the algebraic approach to canonical quantization of group field theory, and discuss special cases. Published by the American Physical Society 2024

Mapping disability and climate change knowledge base in Scopus using bibliometric analysis.

Climate change and disability are rarely addressed by academic scholars within the spectrum of disabilities and as a single field of study. However, the intersectionality of disability exacerbates the vulnerability of people with disabilities to climate change as climate change frameworks in the Global North and South continue excluding them. This study aims to map the research-based knowledge housed in Scopus on disability and climate change. At the same time, it provides insights into innovative (novelty) ways of thinking and proposes a futuristic research agenda. A bibliometric analysis was conducted on Scopus-indexed articles using VOSviewer to map co-occurrences of keywords and co-authorship, and a manual thematic-scoping review augmented the data analysis. The disability and climate change debate as a joint study evolved from concern among health practitioners to human rights and social inclusion. In conclusion, there is a skewness towards mental health and medical sociology lens, while other sub-groups of persons with disabilities are yet to be engaged in co-creating disability-inclusive climate change knowledge. Thematic areas emerged as gaps that future studies embed principles enshrined in the United Nations Convention for the Rights of Persons with Disabilities and the Sustainable Development Goals.

Hemp cultivation opportunities for marginal lands development.

Agricultural diversification and high-quality products deriving from sustainable crops such as hemp can represent a solution to revitalize marginal areas and reverse land abandonment. This study aimed at comparing four different hemp cultivars (Carmagnola Selezionata, "CS"; Futura 75, "FUT"; Felina 32, "FEL"; Secuieni Jubileu, "JUB") to provide information to select the best suited cultivar for cultivation in mountain marginal areas and for specific end-use applications. Hemp cultivars were cultivated in a single experimental field to compare their ecological and agronomic behavior (duration of life cycle phases, plant size and biomass allocation, and plant resource-use strategies). Through metabolomic analysis of both vegetative and reproductive parts of the plants we tested the presence of substances of nutraceutical interest and traced seed nutritional profile. The four cultivars had different ecological and agronomic behavior, and nutritional profile. We found several compounds with potential pharmaceutical and nutraceutical values in all parts of the plant (leaves, inflorescences, and stems). JUB resulted the most suitable for seed production while CS showed the highest content of bioactive compounds in flowers and leaves. FUT, showed the best suitability for multi-purpose cultivation, while FEL seemed to be not appropriate for the cultivation in mountain area. The multi-disciplinary approach we adopted was effective in distinguish across hemp cultivars and provided information to farmers for the selection of the best hemp cultivar to select. Hemp had a high potential for cultivation in marginal lands, demonstrating to be an economic resource due to its multi-purpose use and to the possibility to generate high-added values products. Our results could serve as a stimulus for the reintroduction of this culture in the study area and in other similar environments.

Development of a Modular, Open-Source, Reduced-Pressure, Drift Tube Ion Mobility Spectrometer.

Toward the goal of minimizing construction costs while maintaining high performance, a new, reduced-pressure, drift tube ion mobility system is coupled with an ion trap mass analyzer through a custom ion shuttle. The availability of reduced-pressure ion mobility systems remains limited due to comparatively expensive commercial options and limited shared design features in the open literature. This report details the complete design and benchmarking characteristics of a reduced-pressure ion mobility system. The system is constructed from FR4 PCB electrodes and encased in a PTFE vacuum enclosure with custom torque-tightened couplers to utilize standard KF40 bulkheads. The PTFE enclosure directly minimizes the overall system expenses, and the implementation of threaded brass inserts allows for facile attachments to the vacuum enclosure without damaging the thermoplastic housing. Front and rear ion funnels maximize ion transmission and help mitigate the effects of radial ion diffusion. A custom planar ion shuttle transports ions from the exit of the rear ion funnel into the ion optics of an ion trap mass analyzer. The planar ion shuttle can couple the IM system to any contemporary Thermo Scientific ion trap mass analyzer. Signal stability and ion intensity remain unchanging following the implementation of the planar ion shuttle when compared to the original stacked ring ion guide. The constructed IM system showed resolving powers up to 85 for various small molecules and proteins using the Fourier transform from a ∼1 m drift tube. Recorded mobilities derived from first principles agree with published literature results with an average error of 1.1% and an average error toward literature values using single field calibration of <1.3%.

Assessing field cancerization and actinic keratosis using ultraviolet-induced fluorescence dermatoscopy after the application of 5-aminolevulinic acid – An observational study

BackgroundActinic keratoses (AK) are one of the most frequent reasons for consultations in dermatology. Ultraviolet-induced fluorescence dermatoscopy (UVFD) is a new method that allows the assessment of lesions in a spectrum of light that originates from the fluorochromes emitting UV-excited luminescence. The aim of this study was to assess the UVFD features of AKs before PDT and their intensity in field cancerization and single lesions. MethodsThis retrospective study was conducted from June to November 2023. Lesions were assessed with the Olsen scale clinically and dermatoscopically (DermLite DL5, 10x magnification) and photographed. UVFD fluorescence was categorized as ‘none’, ‘weak’, ‘moderate’, and ‘intense’. A 1-mm thick layer of 10 % 5-ALA gel was applied to single lesions or cancerization field (depending on the patient) and covered with an occlusive dressing for 3 h. Prior the application of 10 % 5-ALA gel, the lesions were degreased with an alcoholic solution. The occlusion was removed, and the field was cleaned with a 0,9 % saline solution. Afterward, each lesion was photographed in polarized light and UVFD mode. ResultsA total of 194 dermatoscopic images were analyzed, 111 corresponded to field cancerization and 81 to single AKs. Overall, weak fluorescence was noticed in 22 of them (11,3 %), moderate in 107 (55,15 %), and intense in 65 (33,5 %). Amongst field cancerization (111 images), weak fluorescence was seen in 11 (9.9 %), moderate in 68 (61,26 %), and intense in 32 (28,82 %). In single lesions (81 images), weak fluorescence was detected in 11 (13,2 %), moderate in 39 (46,99 %), and intense in 33 (28.83 %) of the lesions. Slightly more intense fluorescence was noticed in higher Olsen grade (p = 0.04). ConclusionsUVFD can enhance our efficacy of pre-procedural examination and might arise as a useful device to predict the therapeutic effect of PDT.

Optical reciprocity-nonreciprocity-amplification conversion based on degenerate four-wave mixing

Optical nonreciprocity plays an important role in optical communication and quantum networks. In this research, we propose and demonstrate a conversion scheme of magnetic-free dual-channel optical reciprocal amplification (RA) and nonreciprocal amplification (NRA) based on the multiple degenerate four-wave mixing (FWM) process in hot atoms. In our experiment, the dual-channel NRA works with the action of a single pump field based on the establishment of FWM in the same direction and breaking in the opposite direction. Based on stable ground-state Zeeman coherence, by introducing a counter-propagating pump field again, NRA can be changed to RA in the opposite direction of the two original amplified conjugate signals. Moreover, the frequencies of RA signals are very dependent on those of co-propagating pump fields. The experimental realization of NRA-RA conversion may have applications for multichannel angular momentum spatial multiplexing and quantum gate manipulation.

Single field inflation in the light of Pulsar Timing Array Data: quintessential interpretation of blue tilted tensor spectrum through Non-Bunch Davies initial condition

In this work, we present a quintessential interpretation of having a blue-tilted tensor power spectrum for canonical single-field slow-roll inflation to explain the recently observed Pulsar Timing Array (NANOGrav 15-year and EPTA) signal of Gravitational Waves (GW). We formulate the complete semi-classical description of cosmological perturbation theory in terms of scalar and tensor modes using the Non-Bunch Davies initial condition. We found that the existence of the blue tilt (nt)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(n_t)$$\\end{document} within the favoured range 1.2<nt<2.5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1.2<n_t<2.5$$\\end{document} can be explained in terms of a newly derived consistency relation. Further, we compute a new field excursion formula using the Non-Bunch Davies initial condition, that validates the requirement of Effective Field Theory in the sub-Planckian regime, |Δϕ|≪Mpl\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|\\Delta \\phi |\\ll M_{\ extrm{pl}}$$\\end{document} for the allowed value of the tensor-to-scalar ratio, r<0.06\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r<0.06$$\\end{document} from CMB observations. In our study, we refer to this result as Anti Lyth bound as it violates the well-known Lyth bound originally derived for Bunch Davies initial condition. Further, we study the behaviour of the spectral density of GW and the associated abundance with the frequency, which shows that within the frequency domain 10-9Hz<f<10-7Hz\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$10^{-9}~{\ extrm{Hz}}<f<10^{-7}~{\ extrm{Hz}}$$\\end{document} the outcome obtained from our analysis is completely consistent with the Pulsar Timing Array (NANOGrav 15-year and EPTA) signal. Also, we found that the behaviour of GW spectra satisfies the CMB constraints at the low frequency, f∗∼7.7×10-17Hz\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f_*\\sim 7.7\ imes 10^{-17}~{\ extrm{Hz}}$$\\end{document} corresponding to the pivots scale wave number, k∗∼0.05Mpc-1.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k_*\\sim 0.05~{\ extrm{Mpc}}^{-1}.$$\\end{document} Finally, the sharp falling behaviour of the GW spectra within the frequency domain 10-7Hz<f<1Hz\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$10^{-7}~{\ extrm{Hz}}<f<1~{\ extrm{Hz}}$$\\end{document} validates our theory in the comparatively high-frequency regime as well.