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Due to the non-smoothness of optimization problems in Machine Learning, generalized smoothness assumptions have been gaining a lot of attention in recent years. One of the most popular assumptions of this type is $(L_0,L_1)$-smoothness (Zhang et al., 2020). In this paper, we focus on the class of (strongly) convex $(L_0,L_1)$-smooth functions and derive new convergence guarantees for several existing methods. In particular, we derive improved convergence rates for Gradient Descent with (Smoothed) Gradient Clipping and for Gradient Descent with Polyak Stepsizes. In contrast to the existing results, our rates do not rely on the standard smoothness assumption and do not suffer from the exponential dependency from the initial distance to the solution. We also extend these results to the stochastic case under the over-parameterization assumption, propose a new accelerated method for convex $(L_0,L_1)$-smooth optimization, and derive new convergence rates for Adaptive Gradient Descent (Malitsky and Mishchenko, 2020)., Comment: 51 pages, 1 figure

Non-Gaussianity in the primordial curvature perturbation is a crucial element of the early universe due to its significant impact on the primordial black hole (PBH) production. In this work, we focus on the effects of negative non-Gaussianity on PBH abundance through the lens of the compaction function criterion for PBH formation. Our setup utilizes an effective field theory of non-singular bounce, including the standard slow-roll inflation with an ultra-slow roll phase for amplifying the curvature perturbations to form PBHs. We investigate with two separate values of the non-Gaussianity parameter, $f_{\rm NL}=(-39.95,-35/8)$, found within the ekpyrotic contraction and the matter bounce scenarios, respectively, and show that a negatively large amount of $f_{\rm NL}$ can provide sizeable abundance, $10^{-3}\leq f_{\rm PBH}\leq 1$, and completely mitigates the PBH overproduction issue. We also highlight that the case with the effective sound speed $c_{s}=1$, coupled with $f_{\rm NL}=-39.95$, provides an agreement under $1\sigma$ for the scalar-induced gravitational wave explanation of the latest PTA (NANOGrav15 and EPTA) signal. Lastly, we extract an upper bound on the most negative value of, $f_{\rm NL}\sim -60$, below which we show breaching of the underlying perturbativity constraints on the power spectrum amplitude., Comment: 51 pages, 6 figures, 2 tables, Comments are welcome

In this paper, we review in detail different mechanisms of generation of large primordial fluctuations and their implications for the production of primordial black holes (PBHs) and scalar-induced secondary gravity waves (SIGW), with the ultimate aim of understanding the impact of loop correction on quantum correlations and the power spectrum. To accomplish the goal, we provide a concise, comprehensive, but in depth review of conceptual and technical details of the standard model of the universe, namely, causal structure and inflation, quantization of primordial perturbations and field theoretic techniques such as "in-in" formalism needed for the estimation of loop correction to the power spectrum. We discuss at length the severe constraints (no-go) on PBH production in single-field inflation imposed by appropriately renormalized quantum loop corrections, computed while maintaining the validity of the perturbation framework and assuming sufficient inflation to address the causality problem. Thereafter, we discuss in detail the efforts to circumvent the no-go result in Galileon inflation, multiple sharp transition (MST)-induced inflation, and stochastic single field inflation using an effective field theoretic (EFT) framework applicable to a variety of models. We provide a thorough analysis of the Dynamical Renormalization Group (DRG) resummation approach, adiabatic and late-time renormalization schemes, and their use in producing solar and sub-solar mass PBHs. Additionally, we give a summary of how scalar-induced gravitational waves (SIGWs) are produced in MST setups and Galileon inflation.Finally, the PBH overproduction issue is thoroughly discussed., Comment: 322 pages, 80 figures, 7 tables, Invited Physics Reports review, Dedicated to the memory of Alexei A. Starobinsky, Criticism and comments are most welcome

In order to unravel the present situation of the PBH overproduction problem, our study emphasizes the critical role played by the equation of state (EoS) parameter $w$ within the framework of effective field theory (EFT) of non-singular bounce. Our analysis focuses on a wide range of EoS parameter values that are still optimal for explaining the latest data from the pulsar timing array (PTA). As a result of our study, the most advantageous window, $0.31 \leq w \leq 1/3$, is identified as the location of a substantial PBH abundance, $f_{\rm PBH} \in (10^{-3},1)$ with large mass PBHs, $M_{\rm PBH}\sim {\cal O}(10^{-7}-10^{-3})M_{\odot}$, in the SIGW interpretation of the PTA signal. When confronted with PTA, we find that the overproduction avoiding circumstances are between $1\sigma-2\sigma$, while the EoS parameter lies inside the narrow window, $0.31

We present a complete and consistent exposition of the regularization, renormalization, and resummation procedures in the setup of having a contraction and then non-singular bounce followed by inflation with a sharp transition from slow-roll (SR) to ultra-slow roll (USR) phase for generating primordial black holes (PBHs). We consider following an effective field theory (EFT) approach and study the quantum loop corrections to the power spectrum from each phase. We demonstrate the complete removal of quadratic UV divergences after renormalization and softened logarithmic IR divergences after resummation and illustrate the scheme-independent nature of our renormalization approach. We further show that the addition of a contracting and bouncing phase allows us to successfully generate PBHs of solar-mass order, $M_{\rm PBH}\sim {\cal O}(M_{\odot})$, by achieving the minimum e-folds during inflation to be $\Delta N_{\rm Total}\sim {\cal O}(60)$ and in this process successfully evading the strict no-go theorem. We notice that varying the effective sound speed between $0.88\leq c_{s}\leq 1$, allows the peak spectrum amplitude to lie within $10^{-3}\leq A \leq 10^{-2}$, indicating that causality and unitarity remain protected in the theory. We analyse PBHs in the extremely small, $M_{\rm PBH}\sim {\cal O}(10^{-33}-10^{-27})M_{\odot}$, and the large, $M_{\rm PBH}\sim {\cal O}(10^{-6}-10^{-1})M_{\odot}$, mass limits and confront the PBH abundance results with the latest microlensing constraints. We also study the cosmological beta functions across all phases and find their interpretation consistent in the context of bouncing and inflationary scenarios while satisfying the pivot scale normalization requirement. Further, we estimate the spectral distortion effects and shed light on controlling PBH overproduction., Comment: 103 pages, 17 figures, Reference updated and typos corrected, Comments are welcome

We present a study of the Effective Field Theory (EFT) generalization of stochastic inflation in a model-independent single-field framework and its impact on primordial black hole (PBH) formation. We show how the Langevin equations for the "soft" modes in quasi de Sitter background is described by the Infra-Red (IR) contributions of scalar perturbations, and the subsequent Fokker-Planck equation driving the probability distribution for the stochastic duration ${\cal N}$, significantly modify in the present EFT picture. An explicit perturbative analysis of the distribution function by implementing the stochastic-$\delta N$ formalism is performed up to the next-to-next-to-next-to-leading order (NNNLO) for both the classical-drift and quantum-diffusion dominated regimes. In the drift-dominated limit, we perturbatively analyse the local non-Gaussianity parameters $(f_{\rm NL}, g_{\rm NL}, \tau_{\rm NL})$ with the EFT-induced modifications. In the diffusion-dominated limit, we numerically compute the probability distribution featuring exponential tails at each order of perturbative treatment., Comment: 83 pages, 19 figures, 1 table, Comments are welcome, Reference list updated, New clarifications and discussions added, Updated version

Renormalization of quantum loop effects generated from large fluctuations is a hugely debatable topic of research these days which rules out the Primordial Black Hole (PBH) formation within the framework of single-field inflation. In this article, we briefly discuss that the correct implementation of regularization, renormalization, and resummation techniques in a setup described by an ultra-slow-roll phase sandwiched between two slow-roll phases in the presence of smooth or sharp transitions can lead to a stringent constraint on the PBH mass (i.e. ${\cal O}(10^{2}{\rm gm}$)), which we advertise as a new No-go theorem. Finally, we will give some of the possible way-outs using which one can evade this proposed No-go theorem and produce solar/sub-solar mass PBHs., Comment: 15 pages (10 pages material + 5 pages references), Essay written and received honorable mention for the Gravity Research Foundation 2024 Awards for Essays on Gravitation, Comments are welcome, Accepted for publication in Int. J. Mod. Phys. D (for the special issue on Gravity Research Foundation 2024 Awards for Essays on Gravitation)

Adaptive methods are extremely popular in machine learning as they make learning rate tuning less expensive. This paper introduces a novel optimization algorithm named KATE, which presents a scale-invariant adaptation of the well-known AdaGrad algorithm. We prove the scale-invariance of KATE for the case of Generalized Linear Models. Moreover, for general smooth non-convex problems, we establish a convergence rate of $O \left(\frac{\log T}{\sqrt{T}} \right)$ for KATE, matching the best-known ones for AdaGrad and Adam. We also compare KATE to other state-of-the-art adaptive algorithms Adam and AdaGrad in numerical experiments with different problems, including complex machine learning tasks like image classification and text classification on real data. The results indicate that KATE consistently outperforms AdaGrad and matches/surpasses the performance of Adam in all considered scenarios., Comment: 27 pages, 12 figures

We demonstrate the explicit realisation of the ultra-slow roll phase in the framework of the effective field theory of single-field Galileon inflation. The pulsar timing array (PTA) collaboration hints at the scalar-induced gravity waves (SIGW) from the early universe as an explanation for the origin of the observed signal, which, however, leads to an enhancement in the amplitude of the scalar power spectrum giving rise to the overproduction of primordial black holes (PBHs). In the setup under consideration, we examine the generation of SIGW consistent with PTA (NANOGrav15 and EPTA) data, in addition to which we also consider the impact from QCD crossover at the nHz frequencies and address the PBH overproduction issue assuming linear approximations for the over-density without incorporating non-Gaussian effects from the comoving curvature perturbation. The framework is shown to give rise to SIGWs well consistent with the PTA signal with comfortable PBH abundance, $10^{-3} \lesssim f_{\rm PBH} < 1$, of near solar-mass black holes., Comment: 62 pages, 23 figures, Comments are welcome, Revised version accepted for publication in JCAP

Our work highlights the crucial role played by the equation of state (EoS) parameter $w$ within the context of single field inflation with Multiple Sharp Transitions (MSTs) to untangle the current state of the PBH overproduction issue. We examine the situation for a broad interval of EoS parameter that remains most favourable to explain the recent data released by the pulsar timing array (PTA) collaboration. Our analysis yields the interval, $0.2 \leq w \leq 1/3$, to be the most acceptable window from the SIGW interpretation of the PTA signal and where sizeable PBHs abundance, $f_{\rm PBH} \in (10^{-3},1)$, is observed. We also obtain $w=1/3$, radiation-dominated era, to be the best scenario to explain the early stages of the Universe and address the overproduction problem. Within the range of $1 \leq c_{s} \leq 1.17$, we construct a regularized-renormalized-resummed scalar power spectrum whose amplitude obeys the perturbativity criterion while being substantial enough to generate EoS dependent scalar induced gravitational waves ($w$-SIGWs) consistent with NANOGrav-15 data. Working for both $c_{s} = 1\;{\rm and}\;1.17$, we find the $c_{s}=1.17$ case more favourable for generating large mass PBHs, $M_{\rm PBH}\sim {\cal O}(10^{-6}-10^{-3})M_{\odot}$, as potential dark matter candidates with substantial abundance after constraints coming from microlensing experiments., Comment: 75 pages, 13 figures, 1 table, Reference list updated, Comments are welcome, Substantially revised version, Added the details of renormalization and its implications

We investigate the explicit role of negative local non-Gaussianity, $f_{\rm NL}$, in suppressing the abundance of primordial black holes (PBHs) in the single-field model of Galileon inflation. PBH formation requires significantly enhancing the scalar power spectrum, which greatly affects their abundance. The associated frequencies in the nHz regime are also sensitive to the generation of scalar-induced gravitational waves (SIGWs) which may explain the current data from the pulsar timing arrays (PTAs). Our analysis using the threshold statistics on the compaction function demonstrates that Galileon theory not only avoids PBH overproduction using the curvature perturbation enhancements that give $f_{\rm NL} \sim {\cal O}(-6)$, but also generates SIGWs that conform well with the PTA data., Comment: 21 pages (9 pages material+7 pages refs.+5 pages Appendices), 5 figures, Comments are welcome, Accepted for publication in Physics Letters B

Deploying \textit{multiple sharp transitions} (MSTs) under a unified framework, we investigate the formation of Primordial Black Holes (PBHs) and the production of Scalar Induced Gravitational Waves (SIGWs) by incorporating one-loop corrected renormalized-resummed scalar power spectrum. With effective sound speed parameter, $1 \leq c_s \leq 1.17$, the direct consequence is the generation of PBH masses spanning $M_{\rm PBH}\sim{\cal O}(10^{-31}M_{\odot}- 10^{4}M_{\odot})$, thus evading well known \textit{No-go theorem} on PBH mass. Our results align coherently with the extensive NANOGrav 15-year data and the sensitivities outlined by other terrestrial and space-based experiments (e.g.: LISA, HLVK, BBO, HLV(O3), etc.)., Comment: 15 pages, 3 figures, Reference updated, Comments are welcome, Accepted for publication in Physics of the Dark Universe

We consider the production of secondary gravity waves in Galileon inflation with an ultra-slow roll (USR) phase and show that the spectrum of scalar-induced gravitational waves (SIGWs) in this case is consistent with the recent NANOGrav 15-year data and with sensitivities of other ground and space-based missions, LISA, BBO, DECIGO, CE, ET, HLVK (consists of aLIGO, aVirgo, and KAGRA), and HLV(03). Thanks to the non-renormalization property of Galileon theory, the amplitude of the large fluctuation is controllable at the sharp transitions between SR and USR regions. We show that the behaviour of the GW spectrum, when one-loop effects are included in the scalar power spectrum, is preserved under a shift of the sharp transition scale with peak amplitude $\Omega_{\rm GW}h^2\sim {\cal O}(10^{-6})$, and hence it can cover a wide range of frequencies within ${\cal O}(10^{-9}{\rm Hz} - 10^{7}{\rm Hz})$. An analysis of the allowed mass range for primordial black holes (PBHs) is also performed, where we find that mass values ranging from ${\cal O}(1M_{\odot} - 10^{-18}M_{\odot})$ can be generated over the corresponding allowed range of low and high frequencies., Comment: 29 pages, 5 figures, Comments are welcome, Accepted for publication in Nuclear Physics B

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 ($n_t$) within the favoured range $1.2

We present a detailed study of the generation of large primordial non-Gaussianities during the slow-roll (SR) to ultra-slow roll (USR) transitions in the framework of Galileon inflation. We found out that due to having sharp transitions in the USR phase, which persist with a duration of $\Delta {\it N}_{\rm USR} \sim 2$ e-folds, we are able to generate the non-Gaussianity amplitude of the order: $|f_{\rm NL}| \sim {\it O}(10^{-2})$ in the SRI, $-5 < f_{\rm NL} < 5$ in the USR, and $-2 < f_{\rm NL} < 2$ in the SRII phases. As a result, we are able to achieve a cumulative average value of $|f_{\rm NL}| \sim {\it O}(1)$. This implies that our results strictly satisfy Maldacena's no-go theorem in the squeezed limit only for SRI, while they strictly violate the same condition in both the USR and SRII phases. The non-renormalization theorem in the Galileon theory helps to support our results regarding the generation of large mass primordial black holes along with large non-Gaussianities, which we show to be dependent on the specific positions of the transition wave numbers fixed at low scales., Comment: 51 pages, 11 figures, 1 table, Typos corrected, and new references added, Comments are welcome, Revised version accepted for publication in JCAP

Distributed and federated learning algorithms and techniques associated primarily with minimization problems. However, with the increase of minimax optimization and variational inequality problems in machine learning, the necessity of designing efficient distributed/federated learning approaches for these problems is becoming more apparent. In this paper, we provide a unified convergence analysis of communication-efficient local training methods for distributed variational inequality problems (VIPs). Our approach is based on a general key assumption on the stochastic estimates that allows us to propose and analyze several novel local training algorithms under a single framework for solving a class of structured non-monotone VIPs. We present the first local gradient descent-accent algorithms with provable improved communication complexity for solving distributed variational inequalities on heterogeneous data. The general algorithmic framework recovers state-of-the-art algorithms and their sharp convergence guarantees when the setting is specialized to minimization or minimax optimization problems. Finally, we demonstrate the strong performance of the proposed algorithms compared to state-of-the-art methods when solving federated minimax optimization problems., Comment: ICLR 2024

We consider Galileon inflation in the Effective Field Theory (EFT) framework and examine the possibility for PBH formation during slow roll (SR) to ultra slow roll (USR) transitions. We show that loop corrections to the power spectrum, in this case, do not impose additional constraints on the masses of PBHs produced. We indicate that the remarkable non-renormalization property of Galileans due to generalized shift symmetry dubbed as Galilean symmetry is responsible for protecting PBH formation from quantum loop corrections., Comment: 45 pages, 2 figures, Comments are welcome, Revised version accepted for publication in JCAP

We present a detailed exposition on the prospects of the formation of Primordial Black Holes (PBHs) during Slow Roll (SR) to Ultra Slow Roll (USR) sharp transitions in the framework of single-field inflation. We use an effective field theory (EFT) approach in order to keep the analysis model-independent and applicable to both the canonical and non-canonical cases. We show in detail how renormalizing the power spectrum to one loop order in $P(X,\phi)$ theories severely limits the prospects for PBH formation in a single-field inflationary framework. We demonstrate that for the allowed range of effective sound speed, $1

Single-call stochastic extragradient methods, like stochastic past extragradient (SPEG) and stochastic optimistic gradient (SOG), have gained a lot of interest in recent years and are one of the most efficient algorithms for solving large-scale min-max optimization and variational inequalities problems (VIP) appearing in various machine learning tasks. However, despite their undoubted popularity, current convergence analyses of SPEG and SOG require a bounded variance assumption. In addition, several important questions regarding the convergence properties of these methods are still open, including mini-batching, efficient step-size selection, and convergence guarantees under different sampling strategies. In this work, we address these questions and provide convergence guarantees for two large classes of structured non-monotone VIPs: (i) quasi-strongly monotone problems (a generalization of strongly monotone problems) and (ii) weak Minty variational inequalities (a generalization of monotone and Minty VIPs). We introduce the expected residual condition, explain its benefits, and show how it can be used to obtain a strictly weaker bound than previously used growth conditions, expected co-coercivity, or bounded variance assumptions. Equipped with this condition, we provide theoretical guarantees for the convergence of single-call extragradient methods for different step-size selections, including constant, decreasing, and step-size-switching rules. Furthermore, our convergence analysis holds under the arbitrary sampling paradigm, which includes importance sampling and various mini-batching strategies as special cases., Comment: 37th Conference on Neural Information Processing Systems (NeurIPS 2023)

Using the Effective Field Theory (EFT) framework of single field inflation, we investigate the possibility of the formation of Primordial Black Holes (PBHs) in the Slow Roll (SR) to Ultra Slow Roll (USR) sharp transition. We demonstrate that, due to one-loop correction to the power spectrum, causality is violated ($c_s>1$) for the mass range of PBHs, $M_{\rm PBH}>10^{2}{\rm gm}$ created during the said transition. We find that non-canonical features with $c_s<1$ worsen the predictions of the canonical framework of single-field inflation., Comment: 13 pages, 5 figures, Updated version, Title changed and substantially elaborated version, Conclusion unchanged, Comments are welcome, Accepted for publication in Physics Letters B

We examine the possibility of Primordial Black Holes (PBHs) formation in single-field models of inflation. Using the adiabatic or wave function renormalization scheme in the short-range modes, we show that one-loop correction to the power spectrum is free from quadratic UV divergence. We consider a framework in which PBHs are produced during the transition from Slow Roll (SR) to Ultra Slow Roll (USR) followed by the end of inflation. We demonstrate that the renormalized power spectrum softens the contribution of the logarithmic IR divergence and severely restricts the possible mass range of produced PBHs in the said transition, namely, $M_{\rm PBH}\sim 10^{2}{\rm gm}$ ala a no-go theorem. In particular, we find that the produced PBHs are short-lived ($t^{\rm evap}_{\rm PBH}\sim 10^{-20}{\rm sec}$) and the corresponding number of e-folds in the USR region is restricted to $\Delta N_{\rm USR}\approx 2$., Comment: 46 pages, 4 figures, Substantially revised version, Reference added and all the typos corrected, Comments are welcome, Revised version accepted for publication in European Physical Journal C

In this work, we study the signatures of quantum entanglement by computing entanglement negativity between two causally unrelated regions in $3+1$ dimensional global de Sitter space. We investigate a bipartite quantum field theoretic setup for this purpose, driven by an axionic Bell pair resulting from Type IIB string compactification on a Calabi-Yau three fold. We take into account a spherical surface that divides the spatial slice of the global de Sitter space into exterior and interior causally unrelated sub regions. For the computational purpose we use the simplest possible initial choice of quantum vacuum, which is Bunch-Davies state. The quantitative quantum information theoretic measure for entanglement negativity turns out be consistent with the results obtained for entanglement entropy, even we have to say it is better than that from quantum information theoretic point of view. We design the problem in a hyperbolic open chart where one of the causally unrelated observers remains constrained and the scale dependence enters to the corresponding quantum information theoretic entanglement measure for axionic Bell pair.We find from our analysis that in the large scales initially maximally entangled Bunch-Davies state turns out to be strongly entangled or weakly entangled depending on the axionic decay constant and the supersymmetry breaking scale. We also find that at the small scales the initial entanglement can be perfectly recovered.We also discuss the possibility of having a biverse picture, which is a mini version of the multiverse in the present theoretical set up. Last but not the least, we provide the necessary criteria for generating non vanishing quantum entanglement measures within the framework of quantum field theory of global de Sitter space as well as well as in primordial cosmology due to the axion derived from string theory., Comment: 96 pages, 17 figures, Updated version, Typos corrected, Accepted for publication in Fortschritte der Physik - Progress of Physics

In this work, we study the time-dependent behaviour of quantum correlations of a system of an inverted oscillator governed by out-of-equilibrium dynamics using the well-known Schwinger-Keldysh formalism in presence of quantum mechanical quench. Considering a generalized structure of a time-dependent Hamiltonian for an inverted oscillator system, we use the invariant operator method to obtain its eigenstates and continuous energy eigenvalues. Using the expression for the eigenstates, we further derive the most general expression for the generating function as well as the out-of-time-ordered correlators (OTOC) for the given system using this formalism. Further, considering the time-dependent coupling and frequency of the quantum inverted oscillator characterized by quench parameters, we comment on the dynamical behaviour, specifically the early, intermediate and late time-dependent features of the OTOC for the quenched quantum inverted oscillator. Next, we study a specific case, where the system of inverted oscillator exhibits chaotic behaviour by computing the quantum Lyapunov exponent from the time-dependent behaviour of OTOC in presence of the given quench profile., Comment: 25 pages, 8 figures, Accepted for publication in Symmetry (Feature Papers 2024: https://www.mdpi.com/journal/symmetry/special_issues/7PP2510774)

In this work, we explore the effects of a quantum quench on the circuit complexity for a quenched quantum field theory having weakly coupled quartic interaction. We use the invariant operator method, under a perturbative framework, for computing the ground state of this system}. We give the analytical expressions for specific reference and target states using the ground state of the system. Using a particular cost functional, we show the analytical computation of circuit complexity for the quenched and interacting field theory. Further, we give a numerical estimate of circuit complexity with respect to the quench rate, $\delta t$ for two coupled oscillators. The parametric variation of the unambiguous contribution of the circuit complexity for an arbitrary number of oscillators has been studied with respect to the dimensionless parameter $(t/\delta t$). We comment on the variation of circuit complexity for different values of coupling strength, different number of oscillators, and even in different dimensions., Comment: 17 pages, 4 figures, Revised version, Accepted for publication in Symmetry (section: Physics and Symmetry/Asymmetry, Special issue: "Symmetry and Asymmetry in Quantum Mechanics")

In this work, we explore the effects of a quantum quench on the entanglement measures of a two-body coupled oscillator system having quartic interaction. We use the invariant operator method, under a perturbative framework, for computing the ground state of this system. We give the analytical expressions for the total and reduced density matrix of the system having non-Gaussian, quartic interaction terms. Using this reduced density matrix, we show the analytical calculation of two entanglement measures viz., Von Neumann entanglement entropy using replica trick and Renyi entanglement entropy. Further, we give a numerical estimate of these entanglement measures with respect to the dimensionless parameter $(t/\delta t$) and show its behaviour in the three regimes, i.e; late time behaviour, around the quench point and the early time behaviour. We comment on the variation of these entanglement measures for different orders of coupling strength. The variation of Renyi entropy of different orders has also been discussed., Comment: 20 pages, 4 figures, Title changed, Revised version, Accepted for publication in Physical Review D

In this paper, we study the Krylov complexity in quantum field theory and make a connection with the holographic "Complexity equals Volume" conjecture. When Krylov basis matches with Fock basis, for several interesting settings, we observe that the Krylov complexity equals the average particle number showing that complexity scales with volume. Using similar formalism, we compute the Krylov complexity for free scalar field theory and find surprising similarities with holography. We also extend this framework for field theory where an inverted oscillator appears naturally and explore its chaotic behavior., Comment: 11 pages, 2 figures, Updated version, Accepted for publication in Nuclear Physics B

In this paper we study the application of four-mode squeezed states in the cosmological context, studying two weakly coupled scalar fields in the planar patch of the de Sitter space. We construct the four-mode squeezed state formalism and connect this concept with the Hamiltonian of the two coupled inverted harmonic oscillators having a time-dependent effective frequency in the planar patch of the de Sitter space. Further, the corresponding evolution operator for the quantum Euclidean vacuum state has been constructed, which captures its dynamics. Using the Heisenberg picture coupled differential equations describing the time evolution for all squeezing parameters (amplitude, phase and angle) have been obtained, for the weakly coupled two scalar field model. With the help of these evolutions for the coupled system, we simulate the dynamics of the squeezing parameters in terms of conformal time. From our analysis, we observe interesting dynamics, which helps us to explore various underlying physical implications of the weakly coupled two scalar field system in the planar patch of the de Sitter cosmological background., Comment: 22 pages, 6 figures, Accepted for publication in Fortschritte der Physik - Progress of Physics

In this paper, we study the Krylov complexity ($K$) from the planar/inflationary patch of the de Sitter space using the two mode squeezed state formalism in the presence of an effective field having sound speed $c_s$. From our analysis, we obtain the explicit behavior of Krylov complexity ($K$) and lancoz coefficients ($b_n$) with respect to the conformal time scale and scale factor in the presence of effective sound speed $c_s$. Since lancoz coefficients ($b_n$) grow linearly with integer $n$, this suggests that universe acts like a chaotic system during this period. We also obtain the corresponding Lyapunov exponent $\lambda$ in presence of effective sound speed $c_s$. We show that the Krylov complexity ($K$) for this system is equal to average particle numbers suggesting it's relation to the volume. Finally, we give a comparison of Krylov complexity ($K$) with entanglement entropy (Von-Neumann) where we found that there is a large difference between Krylov complexity ($K$) and entanglement entropy for large values of squeezing amplitude. This suggests that Krylov complexity ($K$) can be a significant probe for studying the dynamics of the cosmological system even after the saturation of entanglement entropy., Comment: 12 pages, 3 figures, Reference updated, Accepted for publication in Fortschritte der Physik - Progress of Physics

In this article, we investigate the physical implications of the causality constraint via effective sound speed $c_s(\leq 1)$ on Quantum Circuit Complexity(QCC) in the framework of Cosmological Effective Field Theory (COSMOEFT) using the two-mode squeezed quantum states. This COSMOEFT setup is constructed using the St$\ddot{\text{u}}$ckelberg trick with the help of the lowest dimensional operators, which are broken under time diffeomorphism. In this setup, we consider only the contribution from two derivative terms in the background quasi de Sitter metric. Next, we compute the relevant measures of circuit complexity and their cosmological evolution for different $c_s$ by following two different approaches, Nielsen's and Covariance matrix method. Using this setup, we also compute the Von-Neumann and R\'enyi entropy, which finally establishes an underlying connecting relationship between the entanglement entropy and circuit complexity. Essentially, we study the behaviour of the circuit complexity measures and entanglement entropy with respect to the scale factor and $c_s$ and find various interesting unexplored features within the window, $0.024\leq c_s\leq 1$, which is supported by both causality and cosmological observation. Finally, we also comment on the connection between the circuit complexity, entanglement entropy and equilibrium temperature for different $c_s$ lying within the mentioned window., Comment: 35 pages, 8 figures, New references updated, Accepted for publication in Fortschritte der Physik - Progress of Physics

Motivated by recent studies of circuit complexity in weakly interacting scalar field theory, we explore the computation of circuit complexity in $\mathcal{Z}_2$ Even Effective Field Theories ($\mathcal{Z}_2$ EEFTs). We consider a massive free field theory with higher-order Wilsonian operators such as $\phi^{4}$, $\phi^{6}$ and $\phi^8.$ To facilitate our computation we regularize the theory by putting it on a lattice. First, we consider a simple case of two oscillators and later generalize the results to $N$ oscillators. The study has been carried out for nearly Gaussian states. In our computation, the reference state is an approximately Gaussian unentangled state, and the corresponding target state, calculated from our theory, is an approximately Gaussian entangled state. We compute the complexity using the geometric approach developed by Nielsen, parameterizing the path ordered unitary transformation and minimizing the geodesic in the space of unitaries. The contribution of higher-order operators, to the circuit complexity, in our theory has been discussed. We also explore the dependency of complexity with other parameters in our theory for various cases., Comment: 35 pages, 5 figures, 3 tables, reference list updated and version modified, Accepted for publication in Symmetry (section: Physics and Symmetry/Asymmetry, Special issue: Symmetry and Geometry in Physics II)

In this article, we investigate various physical implications of quantum circuit complexity using squeezed state formalism of Primordial Gravitational Waves (PGW). Recently quantum information theoretic concepts, such as entanglement entropy, and complexity are playing a pivotal role to understand the dynamics of quantum system even in the diverse fields such as, high energy physics and cosmology. This paper is devoted in studying quantum circuit complexity of PGW for various cosmological models, such as de Sitter, inflation, radiation, reheating, matter, bouncing, cyclic and black hole gas model etc. We compute complexity measure using both Covariance and Nielsen's wave function method for three different choices of quantum initial vacua: Motta-Allen, $\alpha$ and Bunch-Davies. Besides computing circuit complexity, we have also computed Von-Neumann entanglement entropy. By making the comparison of complexity with entanglement entropy, we are able to probe various features regarding the dynamics of evolution for different cosmological models. Because entanglement entropy is independent of the squeezing angle, we are able to understand more details of the system using Nielsen's measure of complexity which is dependent on both squeezing parameter and angle. This implies that quantum complexity could indeed be a useful probe to study quantum features in cosmological scale. Quantum complexity is also becoming a powerful technique to understand the chaotic behaviour and random fluctuations of quantum fields. Using the growth of complexity, we are able to compute quantum Lyapunov exponent for various cosmological models and comment on it's chaotic nature., Comment: 113 pages, 18 figures, 1 appendix, Revised version, Accepted for publication in Symmetry (section: Physics and Symmetry/Asymmetry, Special issue: "Role of Black Holes in Testing Modified Theories of Gravity")

The SYK model has a wormhole-like solution after averaging over the fermionic couplings in the nearly $AdS_2$ space. Even when the couplings are fixed the contribution of these wormholes continues to exist and new saddle points appear which are interpreted as "half-wormholes". In this paper, we will study the fate of these wormholes in a model without quenched disorder namely a tensor model with $O(N)^{q-1}$ gauge symmetry whose correlation function and thermodynamics in the large $N$ limit are the same as that of the SYK model. We will restate the factorization problem linked with the wormhole threaded Wilson operator, in terms of global charges or non-trivial cobordism classes associated with disconnected wormholes. Therefore for the partition function to factorize especially at short distances, there must exist certain topological defects which break the global symmetry associated with wormholes and make the theory devoid of global symmetries. We will interpret these wormholes with added topological defects as our "half-wormholes". We will also comment on the late time behavior of the spectral form factor, particularly its leading and sub-leading order contributions coming from higher genus wormholes in the gravitational sector. Finally we will show how, the other non-trivial saddles from "half-wormhole" dominate and give rise to unusual thermodynamics in the bulk sector due to non-perturbative effects., Comment: 26 pages, 14 figures, Revised version, Accepted for publication in Physical Review D

In this paper, using the concept of Lewis Riesenfeld invariant quantum operator method for finding continuous eigenvalues of quantum mechanical wave functions we derive the analytical expressions for the cosmological geometric phase, which is commonly identified to be the Pancharatnam Berry phase from primordial cosmological perturbation scenario. We compute this cosmological geometric phase from two possible physical situations,(1) In the absence of Bell's inequality violation and (2) In the presence of Bell's inequality violation having the contributions in the sub Hubble region ($-k\tau\gg 1$), super Hubble region ($-k\tau\ll 1$) and at the horizon crossing point ($-k\tau= 1$) for massless field ($m/{\cal H}\ll 1$), partially massless field ($m/{\cal H}\sim 1$) and massive/heavy field ($m/{\cal H}\gg 1$), in the background of quantum field theory of spatially flat quasi De Sitter geometry. The prime motivation for this work is to investigate the various unknown quantum mechanical features of primordial universe. To give the realistic interpretation of the derived theoretical results we express everything initially in terms of slowly varying conformal time dependent parameters, and then to connect with cosmological observation we further express the results in terms of cosmological observables, which are spectral index/tilt of scalar mode power spectrum ($n_{\zeta}$) and tensor-to-scalar ratio ($r$). Finally, this identification helps us to provide the stringent numerical constraints on the Pancharatnam Berry phase, which confronts well with recent cosmological observation., Comment: 60 pages, 20 figures, 1 table, Revised version accepted for publication in Fortschritte der Physik - Progress of Physics

In this article, we investigate the quantum circuit complexity and entanglement entropy in the recently studied black hole gas framework using the two-mode squeezed states formalism written in arbitrary dimensional spatially flat cosmological Friedmann-Lema$\hat{i}$tre-Robertson-Walker (FLRW) background space-time. We compute the various complexity measures and study the evolution of these complexities by following two different prescriptions viz. Covariant matrix method and Nielsen's method. Independently, using the two-mode squeezed states formalism we also compute the R\'enyi and Von-Neumann entanglement entropy, which show an inherent connection between the entanglement entropy and quantum circuit complexity. We study the behaviour of the complexity measures and entanglement entropy separately for three different spatial dimensions and observe various significant different features in three spatial dimensions on the evolution of these quantities with respect to the scale factor. Furthermore, we also study the underlying behaviour of the equilibrium temperature with two of the most essential quantities i.e. rate of change of complexity with scale factor and the entanglement entropy. We observe that irrespective of the spatial dimension, the equilibrium temperature depends quartically on entanglement entropy., Comment: 27 pages, 28 figures, 1 table, This project is part of the non-profit virtual international research consortium "Quantum Aspects of Space-Time and Matter (QASTM)", Accepted for publication in Physical Review D without any change in the draft

In this article, we study the quantum field theoretic generalization of the Caldeira-Leggett model in general curved space-time considering interactions between two scalar fields in a classical gravitational background. The thermalization phenomena is then studied from the obtained de Sitter solution using quantum quench from one scalar field model obtained from path integrated effective action. We consider an instantaneous quench in the time-dependent mass protocol of the field of our interest. We find that the dynamics of the field post-quench can be described in terms of the state of the generalized Calabrese-Cardy (gCC) form and computed the different types of two-point correlation functions in this context. We explicitly found the conserved charges of $W_{\infty}$ algebra that represents the gCC state after a quench in de Sitter space and found it to be significantly different from the flat space-time results. We extend our study for the different two-point correlation functions not only considering the pre-quench state as the ground state, but also a squeezed state. We found that irrespective of the pre-quench state, the post quench state can be written in terms of the gCC state showing that the subsystem of our interest thermalizes in de Sitter space. Furthermore, we provide a general expression for the two-point correlators and explicitly show the thermalization process by considering a thermal Generalized Gibbs ensemble (GGE). Finally, from the equal time momentum dependent counterpart of the obtained results for the two-point correlators, we have studied the hidden features of the power spectra and studied its consequences for different choices of the quantum initial conditions., Comment: 129 pages, 25 figures, Detailed explanations, and derivations added, Accepted for publication in Nuclear Physics B

The underlying physical concept of computing out-of-time-ordered correlation (OTOC) is a significant new tool within the framework of quantum field theory, which now-a-days is treated as a measure of random fluctuations. In this paper, by following the canonical quantization technique, we demonstrate a computational method to quantify the two different types of cosmological auto-correlated OTO functions during the epoch when the non-equilibrium features dominates in primordial cosmology. In this formulation, two distinct dynamical time scales are involved to define the quantum mechanical operators arising from the cosmological perturbation scenario. We have provided detailed explanation regarding the necessity of this new formalism to quantify any random events generated from quantum fluctuations in primordial cosmology. We have performed an elaborative computation for the two types of two-point and four-point auto-correlated OTO functions in terms of the cosmological perturbation field variables and its canonically conjugate momenta to quantify random auto-correlations in the non-equilibrium regime. For both of the cases, we found significantly distinguishable non-chaotic, but random, behaviour in the OTO auto-correlations, which was not pointed out before in this type of study. Finally, we have also demonstrated the classical limiting behaviour of the mentioned two types of auto-correlated OTOC functions from the thermally weighted phase-space averaged Poisson brackets, which we found to exactly match the large time limiting behaviour of the auto-correlations in the super-horizon regime of the cosmological scalar mode fluctuation., Comment: 152 pages, 38 figures, This project is the part of "Quantum Structures of the Space-Time and Matter (QASTM)". Published in Symmetry (Special Issue "Physics and Symmetry Section: Feature Papers 2021"). This work is the sequel of arXiv:2005.11750 [hep-th]

Computation of circuit complexity has gained much attention in the Theoretical Physics community in recent times to gain insights into the chaotic features and random fluctuations of fields in the quantum regime. Recent studies of circuit complexity take inspiration from Nielsen's geometric approach, which is based on the idea of optimal quantum control in which a cost function is introduced for the various possible path to determine the optimum circuit. In this paper, we study the relationship between the circuit complexity and Morse theory within the framework of algebraic topology, which will then help us study circuit complexity in supersymmetric quantum field theory describing both simple and inverted harmonic oscillators up to higher orders of quantum corrections. We will restrict ourselves to $\mathcal{N} = 1$ supersymmetry with one fermionic generator $Q_{\alpha}$. The expression of circuit complexity in quantum regime would then be given by the Hessian of the Morse function in supersymmetric quantum field theory. We also provide technical proof of the well known universal connecting relation between quantum chaos and circuit complexity of the supersymmetric quantum field theories, using the general description of Morse theory., Comment: 41 pages, 10 figures, 4 tables, This project is the part of the non-profit virtual international research consortium "Quantum Aspects of Space-Time and Matter (QASTM)", Revised version, References and some of the explanations elaborated and updated, Accepted for publication in Symmetry

Recently in various theoretical works, path-breaking progress has been made in recovering the well-known Page Curve of an evaporating black hole with Quantum Extremal Islands, proposed to solve the long-standing black hole information loss problem related to the unitarity issue. Motivated by this concept, in this paper, we study cosmological circuit complexity in the presence (or absence) of Quantum Extremal Islands in the negative (or positive) Cosmological Constant with radiation in the background of Friedmann-Lema$\hat{i}$tre-Robertson-Walker (FLRW) space-time i.e the presence and absence of islands in anti-de Sitter and the de Sitter spacetime having SO(2, 3) and SO(1, 4) isometries respectively. Without using any explicit details of any gravity model, we study the behaviour of the circuit complexity function with respect to the dynamical cosmological solution for the scale factors for the above-mentioned two situations in FLRW space-time using squeezed state formalism. By studying the cosmological circuit complexity, Out-of-Time Ordered Correlators, and entanglement entropy of the modes of the squeezed state, in different parameter spaces, we conclude the non-universality of these measures. Their remarkably different features in the different parameter spaces suggest their dependence on the parameters of the model under consideration., Comment: 75 pages, 29 figures, 4 tables, Dr. Sayantan Choudhury would like to dedicate this work to his lovable father and prime inspiration Professor Manoranjan Choudhury who recently have passed away due to COVID 19. Updated and revised version, Accepted for publication in Symmetry (section: Physics and Symmetry/Asymmetry, Special issue: Manifest and Hidden Symmetries in Field and String Theories)

In this work, our prime objective is to study the phenomena of quantum chaos and complexity in the machine learning dynamics of Quantum Neural Network (QNN). A Parameterized Quantum Circuits (PQCs) in the hybrid quantum-classical framework is introduced as a universal function approximator to perform optimization with Stochastic Gradient Descent (SGD). We employ a statistical and differential geometric approach to study the learning theory of QNN. The evolution of parametrized unitary operators is correlated with the trajectory of parameters in the Diffusion metric. We establish the parametrized version of Quantum Complexity and Quantum Chaos in terms of physically relevant quantities, which are not only essential in determining the stability, but also essential in providing a very significant lower bound to the generalization capability of QNN. We explicitly prove that when the system executes limit cycles or oscillations in the phase space, the generalization capability of QNN is maximized. Finally, we have determined the generalization capability bound on the variance of parameters of the QNN in a steady state condition using Cauchy Schwartz Inequality., Comment: 42 pages, 9 figures, 1 table, , This project is the part of the non-profit virtual international research consortium "Quantum Aspects of Space-Time and Matter (QASTM)". Revised version accepted for publication in Journal of High Energy Physics (JHEP)

$Circuit~ Complexity$, a well known computational technique has recently become the backbone of the physics community to probe the chaotic behaviour and random quantum fluctuations of quantum fields. This paper is devoted to the study of out-of-equilibrium aspects and quantum chaos appearing in the universe from the paradigm of two well known bouncing cosmological solutions viz. $Cosine~ hyperbolic$ and $Exponential$ models of scale factors. Besides $circuit~ complexity$, we use the $Out-of-Time~ Ordered~ correlation~ (OTOC)$ functions for probing the random behaviour of the universe both at early and the late times. In particular, we use the techniques of well known two-mode squeezed state formalism in cosmological perturbation theory as a key ingredient for the purpose of our computation. To give an appropriate theoretical interpretation that is consistent with the observational perspective we use the scale factor and the number of e-foldings as a dynamical variable instead of conformal time for this computation. From this study, we found that the period of post bounce is the most interesting one. Though it may not be immediately visible, but an exponential rise can be seen in the $complexity$ once the post bounce feature is extrapolated to the present time scales. We also find within the very small acceptable error range a universal connecting relation between Complexity computed from two different kinds of cost functionals-$linearly~ weighted$ and $geodesic~ weighted$ with the OTOC. Furthermore, from the $complexity$ computation obtained from both the cosmological models and also using the well known MSS bound on quantum Lyapunov exponent, $\lambda\leq 2\pi/\beta$ for the saturation of chaos, we estimate the lower bound on the equilibrium temperature of our universe at late time scale. Finally, we provide a rough estimation of the scrambling time in terms of the conformal time., Comment: 104 pages, 41 figures, 9 tables, This project is the part of the non-profit virtual international research consortium "Quantum Aspects of Space-Time and Matter (QASTM)", Accepted for publication in SciPost Physics Core

The concept of out-of-time-ordered correlation (OTOC) function is treated as a very strong theoretical probe of quantum randomness, using which one can study both chaotic and non-chaotic phenomena in the context of quantum statistical mechanics. In this paper, we define a general class of OTOC, which can perfectly capture quantum randomness phenomena in a better way. Further we demonstrate an equivalent formalism of computation using a general time independent Hamiltonian having well defined eigenstate representation for integrable supersymmetric quantum systems. We found that one needs to consider two new correlators apart from the usual one to have a complete quantum description. To visualize the impact of the given formalism we consider the two well known models viz. Harmonic Oscillator and one dimensional potential well within the framework of supersymmetry. For the Harmonic Oscillator case we obtain similar periodic time dependence but dissimilar parameter dependences compared to the results obtained from both micro-canonical and canonical ensembles in quantum mechanics without supersymmetry. On the other hand, for one dimensional potential well problem we found significantly different time scale and the other parameter dependence compared to the results obtained from non-supersymmetric quantum mechanics. Finally, to establish the consistency of the prescribed formalism in the classical limit, we demonstrate the phase space averaged version of the classical version of OTOCs from a model independent Hamiltonian along with the previously mentioned these well cited models., Comment: 125 pages, 40 figures, This project is the part of the non-profit virtual international research consortium "Quantum Aspects of Space-Time and Matter (QASTM)", We would like to dedicate this work for the people those who are helping us to fight against COVID-19 pandemic across the globe, Final published version

In this paper, our prime objective is to apply the techniques of parameter estimation theory and the concept of Quantum Metrology in the form of Fisher Information to investigate the role of certain physical quantities in the open quantum dynamics of a two entangled qubit system under the Markovian approximation. There exist various physical parameters which characterize such system, but can not be treated as any quantum mechanical observable. It becomes imperative to do a detailed parameter estimation analysis to determine the physically consistent parameter space of such quantities. We apply both Classical Fisher Information (CFI) and Quantum Fisher Information (QFI) to correctly estimate these parameters, which play significant role to describe the out-of-equilibrium and the long range quantum entanglement phenomena of open quantum system. Quantum Metrology, compared to classical parameter estimation theory, plays a two-fold superior role, improving the precision and accuracy of parameter estimation. Additionally, in this paper we present a new avenue in terms of Quantum Metrology, which beats the classical parameter estimation. We also present an interesting result of revival of out-of-equilibrium feature at the late time scales, arising due to the long range quantum entanglement at early time scale and provide a physical interpretation for the same in terms of Bell's Inequality Violation in early time scale giving rise to non-locality., Comment: 14 pages, 9 figures, This project is the part of the non-profit virtual international research consortium "Quantum Structures of the Space-Time and Matter (QASTM)", Revised version, Accepted for publication in SciPost Physics Core

The out-of-time-ordered correlation (OTOC) function is an important new probe in quantum field theory which is treated as a significant measure of random quantum correlations. In this paper, with the slogan "Cosmology meets Condensed Matter Physics" we demonstrate a formalism using which for the first time we compute the Cosmological OTOC during the stochastic particle production during inflation and reheating following canonical quantization technique. In this computation, two dynamical time scales are involved, out of them at one time scale the cosmological perturbation variable and for the other the canonically conjugate momentum is defined, which is the strict requirement to define time scale separated quantum operators for OTOC and perfectly consistent with the general definition of OTOC. Most importantly, using the present formalism not only one can study the quantum correlation during stochastic inflation and reheating, but also study quantum correlation for any random events in Cosmology. Next, using the late time exponential decay of cosmological OTOC with respect to the dynamical time scale of our universe which is associated with the canonically conjugate momentum operator in this formalism we study the phenomena of quantum chaos by computing the expression for {\it Lyapunov spectrum}. Further, using the well known Maldacena Shenker Stanford (MSS) bound, on Lyapunov exponent, $\lambda\leq 2\pi/\beta$, we propose a lower bound on the equilibrium temperature, $T=1/\beta$, at the very late time scale of the universe. On the other hand, with respect to the other time scale with which the perturbation variable is associated, we find decreasing but not exponentially decaying behaviour, which quantifies the random correlation at out-of-equilibrium. Finally, we have studied the classical limit of the OTOC to check the consistency with the large time limiting behaviour., Comment: 211 pages, 21 figures, This work is written in the memory of the great physicist Professor Freeman J. Dyson with whom I had the chance to meet during my visit at IAS, Princeton on the first week of December, 2019. This project is the part of "Quantum Structures of the Space-Time and Matter (QASTM)". Accepted for publication in Symmetry (Special Issue "New Advances of Cosmology and Astrophysics")

We study the indirect detection of Cosmological Constant from an open quantum system of interacting spins, weakly interacting with a thermal bath, a massless scalar field minimally coupled with the static de Sitter background, by computing the spectroscopic shifts. By assuming pairwise interaction between spins, we construct states using a generalisation of the superposition principle. The corresponding spectroscopic shifts, caused by the effective Hamiltonian of the system due to Casimir Polder interaction, are seen to play a crucial role in predicting a very tiny value of the Cosmological Constant, in the static patch of de Sitter space, which is consistent with the observed value from the Planck measurements of the cosmic microwave background (CMB) anisotropies., Comment: 28 pages, 7 figures, This project is the part of the non-profit virtual international research consortium "Quantum Structures of the Space-Time & Matter (QASTM)", Accepted for publication in Annals of Physics