Showing papers on "Particle horizon published in 2022"

A Peek Outside Our Universe

Abstract: According to general relativity (GR), a universe with a cosmological constant Λ, like ours, is trapped inside an event horizon, r<3/Λ. What is outside? We show, using Israel (1967) junction conditions, that there could be a different universe outside. Our universe looks like a black hole for an outside observer. Outgoing radial null geodesics cannot escape our universe, but incoming photons can enter and leave an imprint on our CMB sky. We present a picture of such a fossil record from the analysis of CMB maps that agrees with the black hole universe predictions but challenges our understanding of the origin of the primordial universe.

Bianchi Type I model of universe with customized scale factors

Abstract: According to standard cosmology, the universe is homogeneous and isotropic at large scales. However, some anisotropies can be observed at the local scale in the universe through various ways. Here, we have studied the Bianchi Type I model by customizing the scale factors to understand the anisotropic nature of the universe. We have considered two cases with slight modifications of scale factors in different directions in the generalized Bianchi Type I metric equation, and compared the results with the [Formula: see text]CDM model and also with available cosmological observational data. Through this study, we also want to predict the possible degree of anisotropy present in the early universe and its evolution to current time by calculating the value of density parameter for anisotropy [Formula: see text] for both low and high redshift [Formula: see text] along with the possible relative anisotropy that exist among different directions. It is found that there was a relatively higher amount of anisotropy in the early universe and the anisotropic nature of the universe vanishes at the near past and the present epochs. Thus, at near past and present stages of the universe there is no effective distinction between this anisotropic model and the standard [Formula: see text]CDM model.

Living in a non-flat universe: theoretical formalism

Abstract: Recent analysis of the Planck measurements opened a possibility that we live in a non-flat universe. Given the renewed interest in non-zero spatial curvature, here we re-visit the light propagation in a non-flat universe and provide the gauge-invariant expressions for the cosmological probes: the luminosity distance, galaxy clustering, weak gravitational lensing, and cosmic microwave background anisotropies. With the positional dependence of the spatial metric, the light propagation in a non-flat universe is much more complicated than in a flat universe. Accounting for all the relativistic effects and including the vector and tensor contributions, we derive the expressions for the cosmological probes and explicitly verify their gauge invariance. We compare our results to previous work in a non-flat universe, if present, but this work represents the first comprehensive investigation of the cosmological probes in a non-flat universe. Our theoretical formalism in a non-flat universe will play a crucial role in constraining the spatial curvature in the upcoming large-scale surveys.

Stasis in an expanding universe: A recipe for stable mixed-component cosmological eras

Abstract: One signature of an expanding universe is the time-variation of the cosmological abundances of its different components. For example, a radiation-dominated universe inevitably gives way to a matter-dominated universe, and critical moments such as matter-radiation equality are fleeting. In this paper, we point out that this lore is not always correct, and that it is possible to obtain a form of "stasis" in which the relative cosmological abundances $\Omega_i$ of the different components remain unchanged over extended cosmological epochs, even as the universe expands. Moreover, we demonstrate that such situations are not fine-tuned, but are actually global attractors within certain cosmological frameworks, with the universe naturally evolving towards such long-lasting periods of stasis for a wide variety of initial conditions. The existence of this kind of stasis therefore gives rise to a host of new theoretical possibilities across the entire cosmological timeline, ranging from potential implications for primordial density perturbations, dark-matter production, and structure formation all the way to early reheating, early matter-dominated eras, and even the age of the universe.

Emergence of space and expansion of Universe

Abstract: According to the principle of emergence, the expansion of the universe can be explained as the emergence of space with the progress of cosmic time. We have analytically solved the equation of emergence proposed by Padmanabhan by assuming the Komar energy density $\rho+3P$ as a function of the Hubble parameter. The resulting model describes the evolution of the universe, which proceeds towards a final de Sitter state. Model parameters have been extracted using the cosmological observational data. Further, the horizon entropy evolution of the model has been studied. The model predicts a universe having a transition from a prior decelerated epoch to a late accelerated epoch and reasonably predicts the cosmological constant.

Causality and the arrow of time in the branch‐cut cosmology

Abstract: We basis our initial analysis of the arrow of time on a relationship between the time evolution operator of quantum system and the time-independent density operator which describes the equilibrium state of a many-particle system at temperature $T$. We highlight through this analysis the identification of the imaginary temporal component of the branch-cut complex cosmic form factor with the direction in which the time-parameter flows globally, or the arrow of time. As a novelty, in this work we calculate the number of branches in the branch-cut universe to achieve causality involving the global time of evolution of the universe and the local time of travel of the light around each Hubble horizon. The preliminary result obtained is comparable to 60 e-folds of contraction in the FLRW cosmic scale factor $a(t) $ to overcome causality achieved in the bouncing model.

Unified cosmological scale versus Planck scale: As above, so below!

Abstract: We will demonstrate that the mass (equivalent mass) of the observable universe divided by the universe radius is exactly identical to the Planck mass divided by the Planck length. This only holds true in the Haug universe model that takes into account Lorentz’s relativistic mass, while in the Friedmann model of the universe, the critical mass of the universe divided by the Hubble radius is exactly equal to m p /(2l p ). Furthermore, in a recently suggested quantum gravity model, the mass of the universe divided by the radius of the universe is exactly identical to mp /lp ; that is, the Planck mass divided by the Planck length. This is much more than just a speculative approximation, for the findings are consistent with a new unified quantum gravity theory that links the cosmological scale directly to the Planck scale.

A Heuristic Approach to the Far-Future State of a Universe Dominated by Phantom Energy

Abstract: This work is based on a cosmological scenario of a universe dominated by phantom energy with equation of state parameter w < − 1 and the analysis of its asymptotic behaviour in the far-future. The author discusses whether a Big Rip singularity could be reached in the future. Working in the context of general relativity, it is argued that the Big Rip singularity could be avoided due to the gravitational Schwinger pair-production, even if no other particle-creating contribution takes place. In this model, the universe is described in its far-future by a state of a constant but large Hubble rate and energy density, as well as of a constant but low horizon entropy. Similar conditions existed at the beginning of the universe. Therefore, according to this analysis, not only the Big Rip singularity could be avoided in the far-future but also the universe could asymptotically be led to a new inflationary phase, after which more and more universes could be created.

Th Hyperbolic Universe

Abstract: This paper proposes a relativistic model of the Universe in which the geometry describes a 4D version of the 2-sheeted hyperboloid that is isotropic, homogeneous in space at a given time and inhomogeneous in time. The radius of this metric is temporal as opposed to spatial. It predicts both a Universe and Anti-Universe moving in opposite directions of time undergoing an expansion phase, followed by a collapsing phase. Using only the current age of the Universe and transition redshift, it predicts the accelerated expansion and it is shown that its Hubble diagram fits currently available supernova and quasar data as well as predicting a Hubble constant H0 &asymp; 71.6km/s/M pc. The angular term of the metric describes time dilation caused by the relativistic kinematic precession effect known as Thomas Precession which can be interpreted as spin about the time dimension. The model also makes two novel predictions: that the early Universe should have structures older than expected due to an increased amount of proper time relative to coordinate time in that era and that the background Universe should appear brighter than current models predict

Clarifying some common misconceptions about the expansion of the universe

Abstract: Abstract Modern cosmology is based on a 6-parameter model, called Lambda-CDM, which provides an accurate description of the expansion of the Universe on the basis of measured parameters. However, there are attempts to give ‘phenomenological’ descriptions of the dynamics of the Universe that are inconsistent with this model and sometimes lead to misconceptions. They stem from a Newtonian approach that interprets the expansion as due to an ‘initial kick’ that placed all matter and energy in the Universe on ‘inertial trajectories’. These descriptions do not conform to the relativistic approach. Here we show that the Einstein field equations trace the expansion of cosmic space directly to the increase in the energy content of the observable universe caused by the additional space entering it as a consequence of the forward motion of its horizon. We derive here the connection between variations in the scale factor and the corresponding variations in the energy of the observable universe, in the case of the most general cosmological model. We then obtain explicit closed-form expressions of this relationship both in the case of the Universe of any curvature and containing matter and radiation, and in the case of the de Sitter universe, characterized by the cosmological constant only. Some interesting features are pointed out, both in terms of the energy growth rate of the observable universe in the various cosmological models, and in terms of the dynamics of the scale factor. Finally our demonstration is corroborated by an analogy between the expansion of cosmic space and the gravitational stretching of space at the center of a sphere of uniform density whose radius increases with time. This analogy, which allowed us to obtain the novel closed-form expression of the stretching of space at the centre of such a sphere, confirmed that the paradigm that ‘gravity expands space’ is valid in both systems.

Does diffusion mechanism favor the emergent scenario of the universe?

Abstract: In this work, the flat FLRW Universe has been modeled with cosmic matter in the form of diffusive barotropic fluid. The diffusive fluid undergoes dissipation due to diffusion mechanism in the form of cosmological scalar field [Formula: see text]. From the perspective of nonequilibrium thermodynamics, the evolution equations of the universe have been formulated. By a suitable choice of the cosmological scalar field, emergent scenario of the universe has been obtained.

Universe: the dynamics of the minimum and maximum expansion states (Time: the collateral dimension)

Abstract: Abstract Two hypotheses stand out in describing the evolution of the Universe. The predominant one predicts that the present expansion began at a certain instant and will not preserve any variation of energy that performs work; apparent flat Universe (Ω = 1) is advocated by relativistic calculations and observational data, with an end at its maximum expansion (3D Space). The other hypothesis considers that the Universe is cyclical, always alternating phases of expansion and contraction. This proposal aims to demonstrate that both hypotheses can be correct by not being distinct, but complementary. Supported by the immutability of physical laws, analyses of concepts define an exclusive presence of 1D Space in static states of minimum and maximum expansion of the Universe. With our 3D Space Universe created and existing between these extreme states, the Friedmann-Lemaître-Robertson-Walker (FLRW) metric, becomes inadequate for this analysis; therefore, the concept of absolute rest energy (1D Space) is applied, demonstrating that the complete evolution of the Universe is spatially dynamic in a perpetual time dimension, always recreating our Universe, and making possible a relative maintenance of any existence.

Universe: the dynamics of the minimum and maximum expansion states (Time: the collateral dimension)

Abstract: Abstract Two hypotheses stand out in describing the evolution of the Universe. The predominant one predicts that the present expansion began at a certain instant and will not preserve any variation of energy that performs work; apparent flat Universe (Ω = 1) is advocated by relativistic calculations and observational data, with an end at its maximum expansion (3D Space). The other hypothesis considers that the Universe is cyclical, always alternating phases of expansion and contraction. This proposal aims to demonstrate that both hypotheses can be correct by not being distinct, but complementary. Supported by the immutability of physical laws, analyses of concepts define an exclusive presence of 1D Space in static states of minimum and maximum expansion of the Universe. With our Universe (3D Space) created and existing between these extreme states, the Friedmann-Lemaître-Robertson-Walker (FLRW) metric, becomes inadequate for this analysis; therefore, the concept of absolute rest energy (1D Space) is applied, demonstrating that the complete evolution of the Universe is spatially dynamic in a perpetual time dimension, always recreating our Universe, and making possible a relative maintenance of any existence.

Study about Geometric Scales and Force Fields

Abstract: The Universe holds many profound mysteries which to this time are unknown to the world. This is an attempt to view the concept of quantization of Geometry in the Universe in a very different way from the prevailing views on the subject. It is postulated that the quantum levels of geometry form a geometric progression (like a, ax, ax2, ax3, ax4………….axn) where the scale factor “a” stands for the Planck’s length (lP = 1.616199(97) x 10-35m) and the common ratio “x” stands for e(\(\pi \sqrt{2}\))/3 . Based on observational facts it is further attempted to establish that the Geometric Quantum levels could be grouped into different scales, namely, pre-atomic scale, atomic scale, cosmic scale, super-cosmic scale, etc., with the accompanying force fields. It is further postulated that detection of any super cosmic structure with a length or diameter of the order of magnitude of 20 Billion Light Years would mean that a super-cosmic scale is present beyond the observable Universe. This paper just describes a proposed theoretical framework for New Physics, which could ultimately explain all the observable phenomena in the Universe, without venturing into a detailed mathematical study to support the theory. The objective of this study is to present a model of the Universe with a different outlook on quantum levels of geometry and the force fields specific to such levels.

Universe: the dynamics of the minimum and maximum expansion states

Abstract: Abstract Two proposals stand out in describing the evolution of the Universe; the predominant one predicts that the current expansion began at a certain instant and will not preserve a variation of energy that performs work; an apparent flat Universe (Ω = 1) is advocated by relativistic calculations and observational data, with an end at its maximum expansion (3D Space); the other proposal considers that the Universe is cyclical, always alternating phases of expansion and contraction. This article aims to demonstration that these two perspectives can be correct by not being distinct, but complementary. Supported by the immutability of physical laws, analyses of concepts define an exclusive presence of 1D Space in static states of minimum and maximum expansion of the Universe. With our Universe (3D Space) created and existing between these extreme states, each dynamic is highlighted and completes the usual relativity. The concept of absolute rest energy (1D Space) become applicable; thus, the complete evolution of the Universe is described as spatially dynamic in a perpetual time dimension, always recreating our Universe, and making possible a relative maintenance of any existence.

Universe: the dynamics of the minimum and maximum expansion states (Time: the collateral dimension)

Abstract: Abstract Two hypotheses stand out in describing the evolution of the Universe. The predominant one predicts that the current expansion began at a certain instant and will not preserve any variation of energy that performs work; apparent flat Universe (Ω=1) is advocated by relativistic calculations and observational data, with an end or thermal death at its maximum expansion (3D Space) . The other hypothesis considers that the Universe is cyclical (always alternating phases of expansion and contraction). This proposal aims to demonstrate that both hypotheses can be correct by not being distinct, but complementary. Supported by the immutability of physical laws, analyses of concepts define an exclusive presence of 1D Space in static states of minimum and maximum expansion of the Universe. With our 3D Space Universe created and existing between these extreme states, the Friedmann-Lemaître-Robertson-Walker (FLRW) metric, despite its general effectiveness, becomes inadequate for analysis; therefore, the concept of absolute rest energy (1D Space) is applied, demonstrating that the complete evolution of the Universe is spatially dynamic in a perpetual time dimension, always recreating our universe, and making possible a relative maintenance of any existence.

Horizon thermodynamics and cosmological equations: A holographic-like connection between thermostatistical quantities on a cosmological horizon and in the bulk

Abstract: Horizon thermodynamics is expected to be related to the effective energy based on the energy density calculated from the Friedmann equation for a Friedmann--Robertson--Walker (FRW) universe. In the present study, the effective energy and thermostatistical quantities on a cosmological horizon are examined to clarify the holographic-like connection between them, with a focus on a de Sitter universe. To this end, the Helmholtz free energy on the horizon is derived from horizon thermodynamics. The free energy is found to be equivalent to the effective energy calculated from the Friedmann equation. This consistency is interpreted as a kind of holographic-like connection. To examine this connection, Padmanabhan's holographic equipartition law, which is related to the origin of spacetime dynamics, is applied to a de Sitter universe. It is found that the law should lead to a holographic-like connection. The holographic-like connection is considered to be a bridge between thermostatistical quantities on the horizon and in the bulk. For example, cosmological equations for a flat FRW universe can be derived from horizon thermodynamics by accepting the connection as a viable scenario. In addition, a thermal entropy equivalent to the Bekenstein--Hawking entropy is obtained from the Friedmann equation using the concept of a canonical ensemble in statistical physics. The present study provides new insight into the discussion of horizon thermodynamics and cosmological equations.

The Hyperboloidal Universe

Abstract: This paper proposes a relativistic model of the Universe in which the geometry describes a 4D version of the 2-sheeted hyperboloid that is isotropic, homogeneous in space at a given time and inhomogeneous in time. The radius of this metric is temporal as opposed to spatial. It predicts both a Universe and Anti-Universe moving in opposite directions of time undergoing an expansion phase, followed by a collapsing phase. Using only the current age of the Universe and transition redshift, it predicts the accelerated expansion and it is shown that its Hubble diagram fits currently available supernova and quasar data as well as predicting a Hubble constant $H_0\approx71.6km/s/Mpc$. The angular term of the metric describes time dilation caused by the relativistic kinematic precession effect known as Thomas Precession which can be interpreted as spin about the time dimension. The model also makes two novel predictions: that the early Universe should have structures older than expected due to an increased amount of proper time relative to coordinate time in that era and that the background Universe should appear brighter than current models predict.

The Hyperboloidal Universe

Abstract: This paper proposes a relativistic model of the Universe in which the geometry describes a 4D version of the 2-sheeted hyperboloid that is isotropic, homogeneous in space at a given time and inhomogeneous in time. The radius of this metric is temporal as opposed to spatial. It predicts both a Universe and Anti-Universe moving in opposite directions of time undergoing an expansion phase, followed by a collapsing phase. Using only the current age of the Universe and transition redshift, it predicts the accelerated expansion and it is shown that its Hubble diagram fits currently available supernova and quasar data as well as predicting a Hubble constant $H_0\approx71.6km/s/Mpc$. The angular term of the metric describes time dilation caused by the relativistic kinematic precession effect known as Thomas Precession which can be interpreted as spin about the time dimension. The model also makes two novel predictions: that the early Universe should have structures older than expected due to an increased amount of proper time relative to coordinate time in that era and that the background Universe should appear brighter than current models predict.

Calculation of the Moment of the Beginning of the Existence of Light in the Universe

Abstract: The apparent age of the universe is about T » 13.56´109 years. Light is part of our everyday life, but it has not always existed. According to physicists, it began to exist about ~360 000 to 380 000 years after the Big Bang. We calculated the exact moment of the appearance of light based on a new cosmological model shown in 2019. In this model, we make the following hypotheses: 1) The apparent radius of curvature of the universe increases at the velocity of light. 2) Our universe is in rotation on itself. 3) The tangential speed of the universe’s periphery is the same as an electron. 4) Our universe is made of a ″material universe″ imbricated in a ″luminous universe″. With Einstein’s laws of relativity for spinning disks and the addition of speeds, we can establish the moment where the universe became transparent and emitted light. We evaluate that moment to ~361 108 years after the Big Bang. The assumptions used for the calculations reveal some interesting facts about the structure and characteristics of our universe. This article may be a step-stone for other analyses.

Entangled Dual Universe

Abstract: Advances in cosmology and astronomical observations over the last two decades have revealed significant tensions and many ambiguities within the standard model of cosmology of a spatially flat Universe, the lambda cold dark matter model. Moreover, the recent Planck Legacy 2018 (PL18) release has confirmed the presence of an enhanced lensing amplitude in the cosmic microwave background (CMB) power spectra, which prefers a positively curved early Universe with a confidence level higher than 99%. This paper addresses the study of a quantum mechanism that could replace the concept of dark matter and energy by considering a primordial curvature as preferred by the PL18 release while yielding the present-day spatial flatness. The implied primordial curvature is incorporated as the background curvature to extend the field equations in terms of the brane-world modified gravity. The Universe evolution is modelled by utilizing a new wavefunction of the Universe that propagates in the bulk with reference to the scale factor of the early Universe and its radius of curvature upon the emission of the CMB, which revealed both positive and negative solutions. This characteristic implies that a pair of entangled wavefunctions was created and evolved in opposite directions as a manifestation of distinct matter and antimatter sides of the Universe. The wavefunction indicates a nascent hyperbolic expansion away from early energy is followed by a first phase of decelerating expansion during the first 10 Gyr, and then, a second phase of accelerating expansion in reverse directions, whereby both sides free-fall towards each other under gravitational acceleration. The predicted background curvature evolution demonstrates the fast orbital speed of outer stars owing to external fields exerted on galaxies as they travelled through earlier conformally curved spacetime. Finally, the wavefunction predicts an eventual phase of rapid spatial contraction that culminates in a Big Crunch, signaling a cyclic Universe. These findings reveal that early plasma could be separated and evolved into distinct sides of the Universe that collectively and geometrically inducing its evolution, physically explaining the effects attributed to dark matter and energy.

Bianchi Type I Model of Universe With Customized Scale Factors

Abstract: Abstract According to standard cosmology, the universe is homogeneous and isotropic at large scales. However, some anisotropies can be observed at the local scale in the universe through various ways. Here we have studied the Bianchi type I model with customizing the scale factors to understand the anisotropic nature of the universe. We have considered two cases with slight modifications of scale factors in different directions in the generalized Bianchi Type I metric equation, and compared the results with the ΛCDM model and also with available cosmological observational data. Through this study, we also want to predict the possible degree of anisotropy present in the early universe and its evolution to current time by calculating the value of density parameter for anisotropy (Ω σ ) for both low and high redshift (z) along with the possible relative anisotropy that exist among different directions. It is found that there was a significant amount of anisotropy in the early universe and the anisotropic nature of the universe vanishes at the near past and the present epochs. Thus at near past and present stages of the universe there is no effective distinction between this anisotropic model and the standard ΛCDM model.

Holographic Description of the Dissipative Model of Universe with Curvature

Abstract: We investigate the accelerated expansion of the late-time universe in the Friedmann–Robertson–Walker metric with nonzero curvature, applying a holographic principle based on a generalized holographic dark energy model introduced by Nojiri and Odintsov (2005, 2006). We describe the evolution of the universe using a generalized equation of state in the presence of a viscous fluid. Solutions of the gravitational equation of motions are obtained in explicit form for a constant value of the thermodynamic parameter, and for various forms of the bulk viscosity. We calculate analytic expressions for infrared cut-offs in terms of the particle horizon, and derive the energy conservation law in the holographic picture. We show that the inclusion of nonzero curvature in the Friedmann equation leads to the appearance of additional singularities of type Big Rip in the Universe.

Universe: the dynamics of the minimum and maximum expansion states (Time: the collateral dimension)

Abstract: Abstract Two proposals stand out in describing the evolution of the Universe; the predominant one predicts that the present expansion began at a certain instant and will not preserve any variation of energy that performs work; apparent flat Universe (Ω = 1) is advocated by relativistic calculations and observational data, with an end at its maximum expansion (3D Space); the other proposal considers that the Universe is cyclical, always alternating phases of expansion and contraction. This article aims to demonstrate that these two views can be correct by not being distinct, but complementary. Supported by the immutability of physical laws, analyses of concepts define an exclusive presence of 1D Space in static states of minimum and maximum expansion of the Universe. With our Universe (3D Space) created and existing between these extremes states, the Friedmann-Lemaître-Robertson-Walker (FLRW) metric, becomes inadequate for such an extension; therefore, the concept of absolute rest energy (1D Space) is applied, demonstrating that the complete evolution of the Universe is spatially dynamic in a perpetual time dimension, always recreating our Universe, and making possible a relative maintenance of any existence.

Proposal of a Model of a Three-Dimensional Universe

Abstract: Despite the existence of several exact solutions to the general theory of relativity, it is still difficult to explain the entire structure of the universe. In this paper, we propose a novel three-dimensional spherical (S3) universe model. According to this model, the universe had a powerful gravity source at its origin, and a part of the gravitational source formed a bubble of spacetime in which the universe was born. The universe expanded explosively immediately after the Big Bang. The energy obtained from the gravity source at the birth of the universe is constant, that is, it remains constant from the birth of the universe to the end. The expansion and contraction of the universe is determined by the passage of coordinate time. The S3 universe observed from outside can be considered to be a two-dimensional spherical surface by projecting it onto a three-dimensional space. On the other hand, the visible universe viewed from inside is observed as a three-dimensional sphere with an arbitrary observation point. The origin of the S3 universe can be seen to be evenly spread in the outer shell of the visible universe. We define visible longitude as the difference between an observation point and its farthest light source. At an initial phase of the expansion of the universe, few stars can be seen, because the visible longitude is small. However, when the expansion of the universe progresses, the number of visible stars also increases, since visible longitude increases. In our S3 universe model, the redshifts observed and reported so far in literature appear to indicate that the expansion of the universe is accelerating. Our S3 universe model can thus lead to a novel exact solution to general relativity.

Inevitability and Incompleteness of the Second Law of Thermodynamics in the Expanding Universe

Abstract: A simple line of reasoning, based on the most fundamental concepts of thermodynamics, yields some intriguing results for a better understanding of the processes occurring in the observable Universe. Gravitational mass must be continuously generated within an expanding thermodynamic system for this system to remain closed. The Second Law is a direct consequence of this production of mass. Simple expressions for the entropy and temperature of the Universe were obtained and the results agree well with observable values. Furthermore, it is demonstrated that the conservation laws within the Universe are independent of its energy density. Based on the solution for the quantum state of the Universe, it is conjectured that the Second Law is incomplete and must be complemented to a conservation law, which takes into account the growth of the amount of information within the Universe. Once the Second Law is complemented to a conservation law, the importance of mass generation within the Universe becomes well pronounced – not only gravitational effects play the role of an organising force, but also the amount of mass within the Universe defines both the amount of information within the Universe and the level of the Universe’s complexity.