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... One of the long-standing fundamental challenges of theoretical cosmology is the problem of an initial singularity. This problem first appeared within the Friedmann cosmology for the simplest cosmological models [1,2]. Many years later, Penrose and Hawking had formulated the cornerstone singularity theorems of General Relativity [3][4][5] (for some historical details see e.g. ...
... In 2011 Horndeski gravity has been rediscovered in the context of generalized Galileon theories [16], and since the interest in this model has only growing. 1 The important subclass of Horndeski gravity is represented by models with a non-minimal derivative coupling of a scalar field φ with the Einstein tensor with the action ...
... In this paper we have explored bounce scenarios in the framework of homogeneous and isotropic cosmological models with arbitrary spatial curvature in the theory of gravity with non-minimal derivative coupling given by the action (1). In general, the model depends on five independent dimensionless parameters: the coupling parameter ζ, and density parameters Ω 0 , Ω 2 , Ω 3 , Ω 4 (see Eqs. (9), (10)), and a cosmological evolution is described by the modified Friedmann equation (13) together with the scalar field equation (12) and constraint (14). ...
We explore bounce scenarios in the framework of homogeneous and isotropic cosmological models with arbitrary spatial curvature in the theory of gravity with non-minimal derivative coupling. As expected, we find that there are no turning points and/or bounces in cosmological models with negative or zero spatial curvature. At the same time, both a turning point and a bounce can exist in the model with positive spatial curvature. In particular, the bounce is happened at when , where is a dimensionless cosmic time. It is important fact that the value depends {\em only} on and , and does {\em not} depend on , and . We find that near the bounce and , where . Thus, the scale factor , the Hubble parameter , and all corresponding geometrical invariants have a regular behavior near the bounce. As well the values characterizing matter energy densities, such as and , are regular near the bounce. Nevertheless, though the spacetime geometry and energy densities remain to be regular near the bounce, the scalar field has a singular behavior there. Namely, as . As a result, we conclude that the complete dynamical system describing the cosmological evolution in theory of gravity with non-minimal derivative coupling is singular near the bounce. On our knowledge, such the scenario, when the spacetime geometry and matter energy densities remain to be regular at approaching the universe evolution to the moment of bounce, while the behavior of scalar field becomes singular, was unknown before. For this reason, we term this scenario as a {\em `singular' bounce}.
... Years later Einstein's paper, Alexander Friedmann proposed a novel cosmological solution in which the universe expands (Friedman, 1922). However, due to a positive curvature (closed universe), it ultimately achieves a maximum radius before collapsing in the future. ...
... The case with negative curvature was also explored by Friedmann in subsequent years (Friedmann, 1924). Shortly thereafter, Einstein published a note referring to Friedman (1922) as an error. However, he was later convinced to recognize the validity of Friedmann's solution, which motivated him to publish a retraction. ...
... Transcription of the Einstein's notes in reference to Friedman (1922) First Note: ...
The discovery that we live in an accelerating universe changed drastically the paradigm of physics and introduced the concept of \textit{dark energy}. In this work, we present a brief historical description of the main events related to the discovery of cosmic acceleration and the basic elements of theoretical and observational aspects of dark energy. Regarding the historical perspective, we outline some of the key milestones for tracing the journey from Einstein's proposal of the cosmological constant to the type Ia supernovae results. Conversely, on the theoretical/observational side, we begin by analyzing cosmic acceleration within the context of the standard cosmological model, i.e., in terms of the cosmological constant. In this case, we show how a positive cosmological constant drives accelerated expansion and discuss the main observational aspects, such as updated results and current cosmological tensions. We also explore alternative descriptions of dark energy, encompassing dynamic and interacting dark energy models.
... Data collected from missions such as COBE [2], WMAP [3], and Planck [4] suggests that the universe evolved according to the standard Big Bang model. In this framework, the time evolution of the geometry of local sections of the universe can be described by the Friedmann-Lemaître-Robertson-Walker (FLRW) metric [5,6,7]. This implies that the universe can be modeled as a 4-manifold, U = M × R, where M is the 3-manifold corresponding to the spatial component, and R represents the temporal dimension. ...
... We defined an overarching algorithm which uses the already mentioned algorithms above, to compute in parallel across the cluster. Compute n j _values for each point in final_points; 3 Select indices using select_points_for_c(n j _values, M desired ); 4 Extract selected points and their transformed versions; 5 Construct points_images as tuples of points and their images; 6 return selected_points, selected_transformed_points, points_images; ...
... Increasing the amount of images we can collect for each point refines the amount of detail 5 Assuming k ∈ (0, 10) with a resolution of 400. Small deviations depending on the manifold are possible. ...
The topology of the universe remains a fundamental question in cosmology, with compact hyperbolic 3-manifolds offering a compelling framework to explain anomalies in the cosmic microwave background (CMB) anisotropies. This thesis investigates the vibrational eigenmodes of such manifolds by solving the Laplace-Beltrami eigenvalue problem, . Using the `method of ghosts,' we attempt to numerically construct eigenfunctions on the universal cover , ensuring invariance under the isometry group , which acts freely and discontinuously on to yield the compact space . Despite a meticulous implementation of the methodology outlined in \cite{eigenvalueprob}, we were unable to replicate the reported numerical results for eigenvalues and their multiplicities. Our findings suggest the need for further refinement and validation of these methods to establish a robust computational framework for investigating compact hyperbolic manifolds.
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I deeply appreciate the patience and understanding of both of my supervisors Marcello and Daan as I tried my best to make my way through this thesis. Although our initial idea didn't end up coming to much fruition in the end, I learned what it felt like to have to figure much of the content that was a part of the thesis on my own and rely on what I've learned to build a substantially large codebase. Not to mention that I now will look at any paper from the arXiv with much more scrutiny, since this one we examined was just a complete nightmare of typos, mistakes, and lack of explanations. I also thank Robbert Scholtens very much for being available to join our meetings and having answered many stupid questions Ive had along the way. I also deeply appreciate my family for giving me the opportunity to be able to complete two bachelors as a non-EU student, and always being there emotionally to console me through the times when I really felt like just not working those extra hours. Lastly I also thank my best friends, Miquel, Yuri, Nikola and Leo who supported me throughout the whole process and were there for me when I needed them.
... An exact definition of a homogeneous and isotropic universe is given in Section 3. Theorem 1.1 is fundamental in relativistic cosmology and is therefore very important. Metric (1) was originally considered in [1][2][3][4][5][6][7][8][9][10][11]. We make a few comments on those parts of the original papers that are related to the form of the metric. ...
... An exact definition of a homogeneous and isotropic universe is given in Section 3. Theorem 1.1 is fundamental in relativistic cosmology and is therefore very important. Metric (1) was originally considered in [1][2][3][4][5][6][7][8][9][10][11]. We make a few comments on those parts of the original papers that are related to the form of the metric. ...
... Friedmann pioneered the use of metric (1) in cosmological models of general relativity [1,2]. He did not write about homogeneous and isotropic universe, instead, he simply required that all spatial cross sections of constant time t = const are to be constantcurvature spaces and assumed that the metric has form (1). ...
Some basic theorems on Killing vector fields are reviewed. In particular, the topic of a constant-curvature space is examined. A detailed proof is given for a theorem describing the most general form of the metric of a homogeneous isotropic space-time. Although this theorem can be considered to be commonly known, its complete proof is difficult to find in the literature. An example metric is presented such that all its spatial cross sections correspond to constant-curvature spaces, but it is not homogeneous and isotropic as a whole. An equivalent definition of a homogeneous and isotropic space-time in terms of embedded manifolds is also given.
... The present standard cosmology starts from the assumption of a complete homogeneity of the cosmic energy distribution in the frame of the so-called "cosmological principle" [6,7]. This homogeneity and, connected with it, the imputed curvature isotropy then allow with the help of the Robertson-Walker metric which then adequately fits the problem to convert Einstein's general relativistic tensorial field equations to only two non-trivial differential equations which describe by the functions Ṙ and R̈ the velocity and the acceleration of the scale R of the universe as function of cosmic time t [8,9]. In these Friedman-Lemaître equations it also has been assumed, that the massive cosmic particles in the sense of Einstein [10] lead to a homogeneous, but world-time and scale-variable mass density ρ = ρ(R) which due to particle number conservation and particle mass conservation is reversely proportional to the world volume, i.e. given by the following expression: ...
... Hereby the physical nature of that energy has not been understood, Einstein [10] only created this term for the sake of creating a stable, static universe which he thought he had to look for at his times. Einstein's Λ-term thus is not connected with real particles, but represents a "volume-specific"-energy term, the physical nature of which has not been understood up to the present days, but its mathematical nature becomes easily evident in the Friedman-Lemaître equations [8,9]: It namely has an accelerative nature, in contrast to the decelerative nature of all the other particle-specific terms, and thus is the only term supporting the expansion of the universe. Concerning the cosmologic action of this term this leads to a game of the competing parameters: as consequence of the relative strengths of the matter density and the vacuum energy density in relation to the so-called critical density ρ c = 3H 0 2 /8πG (H 0 = Hubble constant; G = gravitational constant) one gets different forms for the cosmic scale R(t) as function of the cosmic time t (illustrated in Figure 1) which evidently would predict different forms of the past and future of the universe given for different values of the critical density ρ c . ...
For many people of our present, thinking mankind it appears already as a deep intellectual sin to aim at a physics-based natural, purely scientific explanation of what happens all round in the huge universe: it is in fact seen as the so-called "cosmologic sin"! This is because such an arrogant attempt is directly ranked by most people confronted with it as a clear disdivination of the holy creation of the total world, identifyable with a revocation of any independence and internal beauty of the cosmic creation. It is, as if already the simple attempt to look for a physical explanation of the cosmic evolution would convert the world into a manmade universe and would represent a full dechantment of its gloriosity - rather degrading the latter to a trivial mechanical clockwork. But is it not just too marvelous, seen in a little bit alternative view, that the human attempt of the rational interpretation of the manifest cosmic world just represents a wonderful indication, that this universe - as a completely transcendental being for the human consciousness - talks to this human brain and thereby even installs in fact a tight, valid and mutually supported interaction?
... This is one of Friedmann's equations 99,100 . Note that this simple consequence of thermodynamics follows from the conservation of the energy-momentum tensor 101 . ...
... This is the other of Friedmann's equations 99,100 . The integration constant k quantifies the spatial curvature of the universe 98 . ...
In spatially uniform, but time-dependent dielectric media with equal electric and magnetic response, classical electromagnetic waves propagate exactly like in empty, flat space with transformed time, called conformal time, and so do quantum fluctuations. In empty, flat space the renormalized vacuum energy is exactly zero, but not in time-dependent media, as we show in this paper. This is because renormalization is local and causal, and so cannot compensate fully for the transformation to conformal time. The expanding universe appears as such a medium to the electromagnetic field. We show that the vacuum energy during cosmic expansion effectively reduces the weights of radiation and matter by characteristic factors. This quantum buoyancy naturally resolves the Hubble tension, the discrepancy between the measured and the inferred Hubble constant, and it might resolve other cosmological tensions as well.
... by the important Hubble's discovery of the expanding Universe [8,9], and theoretical models proposed by Friedmann and Lamaitre [10] -finally leading to the establishment of the standard model of cosmology in the following decades [11,12]. The standard model of cosmology, based on GR and assumptions of large-scale homogeneity and isotropicity, was tested by a plethora of observations -such as supernova and microwave background measurements, consistency of the age of the Universe with the age of astronomical objects, abundances of chemical elements, growth of cosmological perturbations etc. [13]. ...
... This means that the contribution of the effective energy density contained in the modified terms needs to increase with time, in order to become significantly large to destroy all bounded systems in the Universe. Starting from (29), (30) and (31) modified Einstein's equation can be written as in (10). Using the conservation of the energy-momentum tensor for matter, and ∇ µ G µν = 0, it also follows that ∇ µ T ef f µν = 0, leading to the equation for the evolution of the effective energy density componenṫ ...
In this work we propose a new general model of eternal cyclic Universe. We start from the assumption that quantum gravity corrections can be effectively accounted by the addition of higher order curvature terms in the Lagrangian density for gravity. It is also taken into account that coefficients associated with these curvature corrections will in general be dependent on a curvature regime. We therewith assume no new ingredients, such as extra dimensions, new scalar fields, phantom energy or special space-time geometries. Evolution of the Universe in this framework is studied and general properties of each phase of the cycle - cosmological bounce, low curvature (CDM) phase, destruction of bounded systems and contracting phase - are analysed in detail. Focusing on some simple special cases, we obtain analytical and numerical solutions for the each phase confirming our analysis.
... 13 Originally, Einstein (and most of the astronomical community) believed the universe to be static, but he could only find static solutions to the universe by introducing a cosmological constant. Shortly after hearing of Einstein's work, both Friedman 14 and LeMaitre 15 proposed unsteady solutions that suggested the universe might be expanding. Subsequent astronomical observation by Hubble and coworkers 16 confirmed that indeed it was. ...
A model of an infinite universe is postulated in which both space and time expand together and are scaled by a time-dependent length scale, δ(t). The Ricci tensor and Ricci scalar both vanish identically, so the Einstein field equations reduce to a balance between the time-dependent spatially averaged stress energy tensor Tμν and its scalar invariant T/2 times the metric tensor. The divergence of Tμν−Tgμν/2 is zero, so the conserved quantity is G*=ρc2/Gδ2(t), where c is the speed of light, ρ is the rest mass density, and G* is Quantum field theory prediction—the so-called “worst prediction in the history of physics.” The implications of this single time-dependent length scale hypothesis for our physical space are explored using the rules of tensor analysis. These imply that the length scale grows linearly with t which itself varies exponentially with atomic clock time. The Hubble parameter is just H(t)=1/t, where t is the age of the universe, so the universe expansion rate is slowing down. The Hubble parameter can be expressed in terms of the red-shift parameter z as H(z(t))=Ho[1+z], where Ho is its current value. Ho=63.6 km/s/Mpc is in excellent agreement with a large number of observations and implies that the universe began 15.4 × 10⁹ years ago. Excellent agreement is demonstrated by recent astronomical measurements with neither dark matter nor dark energy.
... In order to prevent a collapsing universe, Albert Einstein introduced in 1917 the cosmological constant Λ in his field equations 1 . However, during the twenties the development of the Friedmann-Lemaître-Robertson-Walker (FLRW) metric [2][3][4][5] shows that the cosmological constant was not needed to avoid this collapse, even not needed to explain the universe expansion discovered by Edwin Hubble 6 . ...
We introduce a periodic cartesian diagonal metric. The derivation of the Einstein fields equations without including the cosmological constant shows that this metric describes a periodic galactic pressure-free expanding universe. Interestingly, the derived expanding rate equation is that of the flat isotropic pressure-free homogeneous universe FLRW model including a non-null cosmological constant. This geometrically derived cosmological constant is proportional to the square of the ratio of the inter galaxies distance versus their bulge diameter. When the universe becomes homogeneous, the expanding rate equation reduces to that of the cosmological constant-free FLRW model. The model avoids the need of the dark energy thesis.
... That they are generic possibilities is the object of the second group of formal discoveries. Friedmann (1922) not only derived the equations governing cosmological dynamics from GR, but also explicitly showed that cosmological models typically describe expanding universes. But given the delayed attention to his work, which-when it finally came-was mentioned in one breath with Lemaître's independent re-discovery, this discovery is, on the change-driver model, most plausibly attributed to both Lemaître and Friedmann, as independent co-discoverers. ...
What constitutes a scientific discovery? What role do discoveries play in science, its dynamics and social practices? The paper explores these questions by first critically examining extant philosophical explications of scientific discovery—the models of scientific discovery, propounded by Kuhn, McArthur, Hudson, and Schindler. As an alternative, we proffer the “change-driver model”. In a nutshell, it conceives of discoveries as problems or solutions to problems that have epistemically advanced science. Here we take a problem to be generated by a datum that we want to account for and make sense of—by putting it in contact with our wider web of scientific knowledge and understanding. The model overcomes the shortcomings of its precursors, whilst preserving their insights. We demonstrate its intensional and extensional superiority, especially with respect to the link between scientific discoveries and the dynamics of science, as well as with respect to its reward system. Both as an illustration, and as an application to a recent scientific and political controversy, we apply the considered models of discovery to one of the most momentous discoveries of science: the expansion of the universe. We oppose the 2018 proposal of the International Astronomical Union as too simplistic vis-à-vis the historical complexity of the episode, and as problematically reticent about the underlying—and in fact crucial—philosophical-conceptual presuppositions regarding the notion of a discovery. The change-driver model yields a more nuanced and circumspect verdict: (i) The redshift-distance relation shouldn’t be named the “Hubble-Lemaître Law”, but “Slipher-Hubble-Humason Law”; (ii) Its interpretation in terms of an expanding universe, however, Lemaître ought to be given credit for; (iii) The Big Bang Model, establishing the expansion of the universe as an evidentially fully warranted result in the 1950s or 1960s (and a communal achievement, rather than an individually attributable one), doesn’t qualify as a discovery itself, but was inaugurated by, and in turn itself led to, several discoveries.
... But since the extremely hot cosmic matter has relativistic temperatures, this also leads to relativistically enhanced mass sources, and thus to even stronger centripetal gravitational fields connected with them. That may at first glance appear contra-visionary, but as can clearly be shown by the two cosmological Friedmann equations describing the cosmic scale R as function of the cosmic time t, it becomes evident, perhaps as a surprise, that the relativistically hot, enhanced cosmic matter increases the centripetal gravity field such that no explosive cosmic motion, but just the opposite -an implosion -would be caused [4,5]. The hotter the matter is in the mass singularity, the more the situation resembles that of a singular "black hole". ...
In preceding papers we have shown that an initial Big-Bang explosion of the universe can not have happened as simply caused by a singularity of extremely hot, highly condensed cosmic matter due to the enhanced centripetal gravity field, enhanced by relativistic cosmic masses [1-3]. Instead, as we argue here, the initial "Bang" must have started from a pressurized cosmic vacuum. We analyse how to adequately describe this cosmic vacuum pressure and how to formulate the initial scale expansion of the universe as a reaction to it. We find that for a needed positive vacuum pressure the thermodynamic polytrope relation between vacuum energy density and vacuum pressure only allows for a range of the vacuum polytrope indices ξ of 3 ˂ ξvac ˂ 5. Furthermore we find that for the preferred value ξvac = 4 one can derive a complete description of the cosmic vacuum energy as function of the cosmic scale and the cosmic time with inclusion of a process of cosmic matter generation by a specific vacuum condensation process producing quantized matter. As result one obtains a matter universe well acquainted to all present day astronomers, however, without the need for an initial, material Big-Bang of a mass singularity. As a surprise, however, the Hubble expansion of the post-recombination universe under the action of cosmic vacuum pressure drives the baryonic distribution function into a more and more non-equilibrium shape with over-Maxwellian-ized populations of the high velocity wings demonstrating surprisingly enough that the cosmic matter temperatures in this expansion phase are in fact increasing, opposite to classical expectations which properly speaking would clearly predict adiabatic temperature decreases.
... Despite being much less known than the ⇤-CDM model, such cosmological models continue to be actively discussed to this day [2][3][4][5][6][7][8][9][10][11]. For example, the Schwarzschild [12,13] radius, given by r s = 2GM c 2 , is well known to be identical to the Hubble radius when applying the Friedmann [14] critical mass to the universe. The critical Friedmann mass is: ...
We calculate the viscous force of the Hubble sphere in a black hole RH = ct universe. We then prove that for the viscous force to be equal to the gravitational force of the Hubble sphere, the speed of the fluid must be the speed of light c. This supports the idea that the Hubble sphere vacuum energy is a kind of superfluid that is indeed a form of energy, which, in turn, can be interpreted as dark energy. Our analysis demonstrates that the superfluid, which is likely dark energy, must move at a velocity of c to counteract the gravitational force of the entire cosmos and prevent a black hole universe from collapsing into a central singularity. For the universe to undergo accelerating expansion, this velocity must be greater than c, which brings us closer to the Λ-CDM model, where we must account for the expansion of space itself.
... We here assume that the mass of the Hubble sphere is the critical Friedmann [37] mass, ...
In the Hubble sphere, we assume that the wavelength of pure energy spreads out in all directions. The maximum wavelength in the Hubble sphere is then the circumference of the Hubble sphere. We assume the minimum wavelength occurs in a Planck mass black hole, which is given by 4pi R_{s,p}=8pi*l_p. Here, we build further on the geometric mean CMB approach by Haug and Tatum and conclude that the CMB temperature is simply given as: T_cmb=Sqrt(T_min*T_max), which is the geometric mean of the minimum and maximum physically possible temperatures in the Hubble sphere. This is again means the CMB temperature simply is the geometric mean of the Hawking temperature of the Hubble sphere (in black hole cosmology) and the Hawking temperature of the Planck mass black hole, se we have also T_cmb=Sqrt(T_{Haw,H}*T_{Haw,p)).
... The metric in 2-dimensional Friedmann-Lemaítre-Robertson-Walker spacetime [53][54][55][56][57][58] is, ...
Quantum simulation is a rapidly evolving tool with great potential for research at the frontiers of physics, and is particularly suited to be used in computationally intensive lattice simulations, such as problems with non-equilibrium. In this work, a basic scenario, namely free fermions in an expanding universe, is considered and quantum simulations are used to perform the evolution and study the phenomena involved. Using digital quantum simulations with the Jordan-Wigner transformation and Trotter expansion, the evolutions of fermion number density, correlation functions, polarization, and chiral condensation are analyzed. A spread out phenomenon can be observed in the simulation, which is a consequence of momentum redshift. This work also demonstrates the simplicity and convenience of using quantum simulations when studying time-evolution problems.
... Friedmann's pioneering work [1,2], which described the solutions for both expanding and contracting universes, went largely unnoticed for years [3]. Even Einstein, at first, rejected Friedmann's findings, arguing that they were inconsistent with GR. ...
Based on Newtonian mechanics, in this article, we present a heuristic derivation of the Friedmann equations, providing an intuitive foundation for these fundamental relations in cosmology. Additionally, using the first law of thermodynamics and Euler’s equation, we derive a set of equations that, at linear order, coincide with those obtained from the conservation of the stress-energy tensor in general relativity. This approach not only highlights the consistency between Newtonian and relativistic frameworks in certain limits, but also serves as a pedagogical bridge, offering insights into the physical principles underlying the dynamics of the universe.
... Para Friedmann, a dinâmica do universo estava associada a um impulso dado por uma expansão inicial, que poderia combater a força da gravidade. No mesmo trabalho, ele propôs soluções para as equações de campo que descrevem modelos isotrópicos e homogêneos [23]. ...
O presente trabalho, caracterizado como um texto paradidático que pode ser utilizado nas aulas de física, revisita a história do desenvolvimento dos modelos cosmológicos, desde o surgimento da cosmologia relativística de Einstein até as teorias mais aceitas sobre a expansão do universo. Com o objetivo de divulgar um texto paradidático acessível a professores e alunos do ensino básico, o trabalho revisita a história do desenvolvimento do modelo cosmológico padrão (MCP), ou teoria do Big Bang, e seus inúmeros personagens, bem como as divergências e controvérsias ao longo do desenvolvimento da cosmologia moderna.
... , where m p is the Planck mass and M c is the critical Friedmann [21] mass. Even though the Stefan-Boltzmann derivation and the geometric mean lead to the same CMB temperature, the geometric mean is likely the most intuitive. ...
The geometric mean plays a central role in several parts of thermodynamics. Here, we demonstrate that the geometric mean of the maximum and minimum values of the properties in the Hubble sphere leads to equations that can predict the CMB temperature now, as well as be used to link different properties of cosmology into a consistent mathematical framework. Our theory does not seem compatible with a series of assumptions in the Λ-CDM model, but it seems fully consistent with RH = ct black hole cosmology. This means that in RH = ct cosmology, we can easily predict the CMB temperature now—something the Λ-CDM model cannot do.
... where~is the reduced Planck constant, k b the Boltzmann constant, M c the critical Friedmann mass [20], m p the Planck mass, and l p the Planck length; see [21,22]. Haug and Wojnow [23,24] in 2023 proved this formula to hold based on the Stefan-Boltzmann law, providing a solid theoretical basis for our model. ...
We propose to apply the ideal gas law to the Hubble sphere. This attempt is based on the recently demonstrated relationship between the temperature of the cosmic microwave background and the Hubble constant. This approach fits into the conceptual framework of cosmological models in deterministic quantum thermodynamics of type: R_H = ct.
... 2GH 0 , in the Friedmann [27] universe. Further: ...
We will claim the velocity of the particles making up the CMB must be given by v_{cmb}=sqrt{k_bT_cmb/m_g}=c, where is the Boltzmann constant and T_cmb is the CMB temperature. This we will see leads to several interesting results such as the Hubble energy law: E_c=Nm_gv_cmb^2=Nm_gc^2=Nk_bT_cmb=T_p^3/T_0^2k_b/64 \pi^2. The findings here are fully consistent with the recent geometric mean approach of finding the CMB temperature by Haug and Tatum and also other related work we will refer to. }
... The infinitesimal line element ds, whose coefficients determine the metric of the spacetime, in a Friedman-Lemaitre-Robertson-Walker universe is given by Refs. [23][24][25][26][27][28][29] : ...
We use digital quantum computing to simulate the creation of particles in a dynamic spacetime. We consider a system consisting of a minimally coupled massive quantum scalar field in a spacetime undergoing homogeneous and isotropic expansion, transitioning from one stationary state to another through a brief inflationary period. We simulate two vibration modes, positive and negative for a given field momentum, by devising a quantum circuit that implements the time evolution. With this circuit, we study the number of particles created after the universe expands at a given rate, both by simulating the circuit and by actual experimental implementation on IBM quantum computers, consisting of hundreds of quantum gates. We find that state-of-the-art error mitigation techniques are useful to improve the estimation of the number of particles and the fidelity of the state.
... ∇ µ T µν = 0 , T µν = (ρ + p)u µ u ν + pg µν , with a linear, barotropic equation of state p = Kρ where c s = √ K is the speed of sound of the fluid. Equations (1.1) consistute one of the main matter models used in cosmology since the origin of its rigorous study over a century ago [9]. Depending on the value of K, the equations describes radiation (K = 1/3), dust (K = 0) or a fluid with speed of sound interpolating between both values. ...
We prove the nonlinear stability of homogeneous barotropic perfect fluid solutions in fixed cosmological spacetimes undergoing decelerated expansion. The results hold provided a specific inequality between the speed of sound of the fluid and the expansion rate of spacetime is valid. Numerical studies in our earlier complementary paper provide strong evidence that the aforementioned condition is sharp, i.e. that instabilities occur when the inequality is violated. In this regard, our present result covers the regime of slowest possible expansion which allows for fluids to stabilize, depending on their speed of sound. Our proof relies on an energy functional which is universal in the sense that it also applies to the case of linear expansion and enables a significantly simplified proof of bounds for fluids on linearly expanding spacetimes. Finally, we consider the special cases of dust and radiation fluids in the decelerated regime and prove shock formation for arbitrarily small perturbations of homogeneous solutions.
... Before the work of Einstein on the general theory of relativity in 1915 (Einstein 1915), cosmology, the science of the origin and development of the universe, was not even considered science, and the scientific community considered the universe as an entity not subject to change, without spacial limits and without temporal limits (hypothesis H 0 ). In 1922, Friedmann, building on Einstein's works, published a paper Friedman (1922) proposing the first theory of an expanding universe. The work of Friedmann can be seen (continued) as first-order evidence (D 1 ) that attacks H 0 . ...
Models developed by the knowledge representation and reasoning community permit us to study defeasible inference based on argumentation and data. Scientific reasoning progresses by evaluating scientific hypotheses based on data and meta-evidence. Meta-evidence can be understood as arguments for discounting or even ignoring data or other meta-evidence. Non-monotonic reasoning is underpinned by non-monotonic logic. We here develop a method for modelling scientific inferences within formal argumentation models. We show how these models capture hypothesis testing, meta-analysis, strong inference and non-monotonic consequence relations.
... A Riemann integration procedure, which sums the contributions of these poles across the complexified spacetime manifold, provides a new scale factor, denoted as ln −1 [ ( )], and results in a topological foliated spacetime quantum structure. The line element in the BCQG quantum gravity, resulting from the complexification of the FLRW metric (Friedman 1922;Lemaître 1927;Robertson 1935;Walker 1937), is given as (Zen Vasconcellos et al. 2019, 2021a, 2021b ...
We explore the implications of Branch‐Cut Quantum Gravity (BCQG), a novel framework leveraging non‐commutative geometry within a symplectic phase‐space, on the accelerated expansion of the universe. Non‐commutativity, introduced through a deformation of the Poisson algebra and enhanced by a symplectic metric, provides a robust mechanism for addressing key challenges in cosmology, such as the youngness paradox and the fine‐tuning of initial conditions in standard inflationary models. By embedding quantum dual‐field dynamics within a Riemannian‐foliated spacetime, BCQG naturally integrates short‐ and long‐range spacetime effects into a unified formalism. This approach offers an alternative to standard inflationary models by predicting cosmic acceleration through geometric restructuring rather than finely‐tuned initial states. In contrast to models like ΛCDM or String Theory, BCQG introduces unique corrections to cosmic scale factors and predicts a novel transition between contraction and expansion phases via topological branch‐cuts, circumventing the singularity problem. Moreover, BCQG's non‐commutative formulation provides testable predictions, such as modifications in cosmic microwave background (CMB) anisotropies and large‐scale structure evolution. We discuss the mathematical foundation, observational implications, and future avenues for validating BCQG through astrophysical data, positioning it as a promising theoretical alternative for understanding the universe's accelerated growth.
... Haug and Tatum [2] have recently shown that the Friedmann [3] equation can be expressed in thermodynamic form, indicating that the critical density must be given by: ...
We will demonstrate that the photon energy density parameter Ωγ of the universe can be derived exactly in Rh = ct cosmology. We find that it must be Ωγ=1/5760π ≈5.526 × 10−5 which lies well within the 95% confidence interval for the photon radiation density reported by the Particle Data Group (PDG). Our exact result indicates that there is no uncertainty in the radiation density within at least some sub-classes of Rh = ct cosmology and that such model fit observations.
... A deeper understanding of the universe depends on uncovering matter's fundamental forces and behavior under extreme conditions. According to Friedmann's solution [1] of Einstein's gravitational equation, the universe experienced an expansion from a singularity point at time zero which has been confirmed by the formulation of Hubbel's law for the redshift of distant galaxies [2]. At the dawn of the universe, just microseconds after the Big Bang, a state of matter unlike anything observed today is thought to have existed: quark-gluon plasma (QGP). ...
In this study, we investigate the thermodynamic properties of an ideal Quark-Gluon Plasma (QGP) at a vanishing chemical potential, under the influence of quantum gravitational effects, specifically incorporating the Linear-Quadratic Generalized Uncertainty Principle (LQGUP). We analyze the impact of LQGUP on key thermodynamic quantities, including the grand canonical potential, pressure, energy density, entropy, speed of sound, and the bulk viscosity's response to changes in the speed of sound. Furthermore, we extend our analysis to examine the time evolution of the universe's temperature in the presence of LQGUP effects.
... Friedmann's groundbreaking work [1,2] initially struggled to gain recognition. For years, his solutions, which revealed the possibility of both expanding and contracting universes, were largely overlooked by the scientific community [3]. ...
In this essay, we offer a heuristic derivation of the Friedmann equations rooted in Newtonian mechanics, providing an intuitive perspective on these key cosmological relations. This method emphasizes the agreement between Newtonian and relativistic descriptions within specific limits and serves as a pedagogical tool to illuminate the fundamental physical principles governing the universe's dynamics.
... About a hundred years ago, Friedmann (1922;1924) wrote two papers applying Einstein's theory of General Relativity (GR) to homogeneous and isotropic cosmological models. Such models are known as Friedmann or Friedmann-Robertson-Walker (FRW) models. 1 Friedmann investigated those containing dust (the cosmologists' term for pressureless matter), a cosmological constant (positive, negative, or zero), both, or neither, which also implies all possible spatial curvatures (positive, negative, or zero), which I refer to as restricted Friedmann models. ...
... In the critical Friedmann [15] universe, it is well known that the critical mass (equivalent) is given by: ...
We demonstrate that black holes likely have an energy or mass gap, E_g/c^2=m_g, that is of the order m_g\approx m_p^2/M_{BH}. Interestingly, the mass of the black hole divided by the mass gap seems closely related to the Bekenstein-Hawking entropy and thereby potentially leads to a quantization of black holes. Even if mathematically trivial, this could be a potentially important step toward better understanding the potential to quantize black holes. Our focus is mainly on Schwarzschild black holes, but we also briefly discuss Reissner-Nordström black holes. It is also important that this results leads to minimal gravitational acceleration, creating a gravitational gap that could potentially eliminate dark matter.
... In the critical Friedmann [15] universe, it is well known that the critical mass (equivalent) is given by: ...
We demonstrate that black holes likely have an energy or mass gap, E_g/c^2=m_g, that is of the order m_g\approx m_p^2/M_{BH}. Interestingly, the mass of the black hole divided by the mass gap seems closely related to the Bekenstein-Hawking entropy and thereby potentially leads to a quantization of black holes. Even if mathematically trivial, this could be a potentially important step toward better understanding the potential to quantize black holes. Our focus is mainly on Schwarzschild black holes, but we also briefly discuss Reissner-Nordström black holes.
... Friedmann's pioneering work [1,2], which described the solutions for both expanding and contracting universes, went largely unnoticed for years [3]. Even Einstein, at first, rejected Friedmann's findings, arguing that they were inconsistent with GR. ...
Based on the Newtonian mechanics, in this article, we present a heuristic derivation of the Friedmann equations, providing an intuitive foundation for these fundamental relations in cosmology. Additionally, using the first law of thermodynamics and Euler's equation, we derive a set of equations that, at linear order, coincide with those obtained from the conservation of the stress-energy tensor in General Relativity. This approach not only highlights the consistency between Newtonian and relativistic frameworks in certain limits but also serves as a pedagogical bridge, offering insights into the physical principles underlying the dynamics of the universe.
... We now derive the modified cosmological equations that incorporate quantum informational effects into standard cosmological evolution. Following the seminal work of Friedmann [87] and building upon modern cosmological frameworks [4], we begin with the extended field equations that integrate quantum information measures into gravity [32]: ...
Recent observations from the Dark Energy Spectroscopic Instrument (DESI) suggest that dark energy's influence may be systematically weakening over cosmic time. This paper presents a theoretical framework that explains these observations by incorporating quantum information measures-specifically entanglement entropy and quantum complexity-into Einstein's field equations. We demonstrate that when spacetime geometry is coupled to quantum information dynamics, the resulting equations predict a gradually weakening effective cosmological constant that precisely matches DESI measurements. Our framework generates specific, testable predictions for large-scale structure formation, gravitational wave propagation, and cosmic expansion history that can be verified by upcoming cosmological surveys. The theory maintains mathematical and physical consistency with both quantum mechanics and general relativity while offering a potential resolution to the long-standing cosmological constant problem. Multiple independent analyses of current observational data show statistical significance ranging from 2.5-3.9σ for dark energy evolution, providing preliminary support for this quantum information perspective on cosmic acceleration.
... The application of general relativity to cosmology happened quite quickly. Friedmann's metric appeared as early as 1922 [16], but Einstein was already hard at work trying to describe the universe with his new theory right from the get-go, publishing a seminal paper on the topic in 1917 [17]. Einstein's solution was crafted in a clever but ad hoc way. ...
Notes prepared for the introductory general relativity course PHYSICS 748 at The University of Auckland. They are designed to introduce general relativity to upper-year undergraduate students directly using the modern language of differential geometry but in a physically motivated way, and throughout keeping a logical flow from section to section and chapter to chapter. In doing so, they necessarily cover a number of topics either not normally treated in an introductory course, or from a novel perspective. These include for example: affine spaces, comparing and contrasting rank-2 tensors with matrices, integration on manifolds, the Rindler metric, including a proper near-source boundary condition for the Schwarzschild metric, approaching Kruskal-Szekeres coordinates from a Rindler perspective, and more.
... We are entirely out of the dilemma of whether the Universe is static or expanding. Different research wings like Supernovae type Ia [1][2][3] strongly support the feature of the expanding universe that was first coined and established mathematically from GR theory by Friedmann in the year 1922 [4]. Later, it was confirmed by Hubble [5] through his observational concept. ...
In this work, we investigate the cosmological dynamics of the Universe in scenarios where the tachyonic field dominates, exhibiting both its normal characteristics and phantom behavior, over the energy density of matter and radiation. When the time derivative of the tachyon field, ϕ̇ , is very small, we observe that the nature of the scalar field and the potential play pivotal roles in the evolution of the Universe. We further analyze the dynamic features using various parameters, including the deceleration parameter q, the equation of state (EoS) parameter, the squared sound speed, and the Om diagnostic. Additionally, we discuss the Universe’s evolution in the parametric planes of {r, s} and ω- ω′ .
We present the Λ plasma model in an 11-dimensional (11D) framework, aligning with M-theory, where plasma oscillations drive a dynamic cosmological constant, eliminating dark matter and energy, and unifying all fundamental forces via ultra-low frequency (ULF) resonance. This mechanism generates gravitons and fermion masses, revealing a granular spacetime at the Planck scale. The Dynamic Lambda Propulsion System (DLPS) enables faster-than-light (FTL) travel. Mathematical derivations and experimental predictions establish this as a complete Theory of Everything (ToE), rooted in observable physics.
We present the Λ plasma model in an 11-dimensional (11D) framework, aligning with M-theory, where plasma oscillations drive a dynamic cosmological constant, eliminating dark matter and energy, and unifying all fundamental forces via ultra-low frequency (ULF) resonance. This mechanism generates gravitons and fermion masses, revealing a granular spacetime at the Planck scale. The Dynamic Lambda Propulsion System (DLPS) enables faster-than-light (FTL) travel. Mathematical derivations and experimental predictions establish this as a complete Theory of Everything (ToE), rooted in observable physics.
This work introduces the Core–Ring Photon Model (CRPM), a novel framework that reinterprets the photon as a composite system consisting of a mass-bearing core and an orbiting massless charge. By integrating classical electromagnetism, quantum mechanics, and general relativity, the CRPM naturally derives fundamental relations such as
E = h f
and
p = h/λ,
while addressing experimental anomalies in polarization, scattering, and gravitational interactions. Extensions of the model incorporate thermodynamic and advanced algebraic concepts, laying a robust foundation for a unified theory that spans multiple disciplines.
This chapter is devoted to the presentation of the basics of quantum mechanics—one of the main branches of physical science, which studies the laws of the micro and nano world. These laws find their effective application in medical physics for the creation of modern precision methods of medical diagnostics. First of all, this concerns such widely used tomographic methods as X-ray computed tomography (CT) using a layer-by-layer scanning to obtain medical imaging due to the gradual passage of an X-ray beam through a thin layer of tissues of the human body; magnetic resonance imaging (MRI) based on the phenomenon of nuclear magnetic resonance (NMR); positron emission tomography (PET), being a method of medical radioisotope diagnostics based on the phenomenon of electron and positron annihilation. As in all previous chapters, not only a brief general description of the basic ideas underlying the tomographic methods of medical diagnostics (see Sect. 7.2), but also a strict mathematical and physical justification of basic theoretical provisions of quantum mechanics is offered to the attention of the reader. As well as the mentioned tomographic methods of medical diagnostics (see Sect. 7.3). In addition, in connection with the growing interest in the use of PET not only as a diagnostic method, but also as a therapeutic radioisotope method for the treatment of oncological diseases, in the last Sect. 7.3.4 the intriguing problem of the law of conservation of mass in the reaction of annihilation of matter and antimatter is discussed (in particular, electron and positron in the method of positron emission tomography), as well as the general most mysterious problem of natural science about the existence of dark energy and dark matter.
Dirac's Large Number Hypothesis (LNH), proposed in 1937, has captivated the scientific community with its exploration of profound correlations between cosmic and atomic scales. This hypothesis delves into the intricate interplay between the infinitesimally small and the vastly large, offering potential insights into the fundamental nature of the universe. As we continue to uncover the mysteries of the cosmos, the LNH oscillates on the precipice of scientific acceptance—neither fully validated nor entirely dismissed. In this review paper, we embark on a comprehensive journey through Dirac's LNH, shedding light on its theoretical underpinnings, its implications for universe models, and its resonance with the Anthropic Principle. We delve into the possibilities of variable gravitational constants and continuous mass creation, inviting further exploration into the intricacies of our cosmic symphony. By embracing the ongoing quest for understanding, we endeavor to unravel the profound harmonies that connect the infinitesimal with the infinite, contributing to the symphony of knowledge in theoretical physics and cosmology.
Numerous models have been proposed theoretically to explain the observed phenomena of the accelerated expansion of the current cosmology. Among these, f(R) gravity theories emerge as a competitive framework, as they not only account for the observed acceleration but also provide an explanation for the early inflation of the cosmology. To further investigate the validity of these theories, we investigate coupled f(R) gravitational models from the perspective of thermodynamics, under the assumption of non-minimal coupling between matter and geometry. This paper explores the thermodynamic properties of three different f(R) forms of gravitational models on cosmological scales, deriving their corresponding entropy functions and applying the generalized second law of thermodynamics. Finally, by employing the generalized second law of thermodynamics and analyzing relevant astronomical observation data, the model parameters for these three f(R) gravitational models are constrained. The results of these constraints not only identify the parameter ranges that satisfy the thermodynamic requirements but also point to matter candidates that realize the accelerated expansion of the cosmology.
In the year 1924, a paper by Carl Wirtz appeared in ‘ Astronomische Nachrichten ’, entitled ‘De Sitter's cosmology and the radial motion of spiral galaxies’. This paper and its author remained largely unnoticed by the community, but it seems to be the first cosmological interpretation of the redshift of galaxies as a time dilation effect and the expansion of the Universe. Edwin Hubble knew Wirtz' publications quite well. The modern reader would find Wirtz' own understanding diffuse and contradictory in some aspects, but that reflected the early literature on nebulae, to which he himself made important contributions. The 100th anniversary provides a good opportunity to present an English transcription to the community, which can be found in the appendix. This anniversary also provokes to ask for the present status of cosmology which many authors see in a crisis. From an observational viewpoint, it shall be illustrated that until today there is no consistent/convincing understanding of how the Universe evolved.
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