## Publications

197 results found

Isichei R, Magueijo J, 2023, Minisuperspace quantum cosmology from the Einstein-Cartan path integral, *Physical Review D*, Vol: 107, Pages: 1-8, ISSN: 2470-0010

We derive the fixed-Λ and unimodular propagators using the path integral formalism as applied to the Einstein-Cartan action. The simplicity of the action (which is linear in the lapse function) allows for an exact integration starting from the lapse function and the enforcement of the Hamiltonian constraint, leading to a product of Chern-Simons states if the connection is fixed at the endpoints. No saddle point approximation is needed. Should the metric be fixed at the endpoints, then, depending on the contour chosen for the connection, Hartle-Hawking or Vilenkin propagators are obtained. Thus, in this approach one trades a choice of contour in the lapse function for one in the connection, where appropriate. The unimodular propagators are also trivial to obtain via the path integral, and the previously derived expressions are recovered.

Gielen S, Magueijo J, 2023, Quantum analysis of the recent cosmological bounce in the comoving Hubble length, *Physical Review D*, Vol: 107, Pages: 1-16, ISSN: 2470-0010

We formulate the transition from decelerated to accelerated expansion as a bounce in connection space and study its quantum cosmology, knowing that reflections are notorious for bringing quantum effects to the fore. We use a formalism for obtaining a time variable via the demotion of the constants of nature to integration constants, and focus on a toy universe containing only radiation and a cosmological constant Λ for simplicity. We find that, beside the usual factor-ordering ambiguities, there is an ambiguity in the order of the quantum equation, leading to two distinct theories: one second order, and one first order. In both cases two time variables may be defined, conjugate to Λ and the radiation constant of motion. We make little headway with the second-order theory, but are able to produce solutions to the first-order theory. They exhibit the well-known “ringing” whereby incident and reflected waves interfere, leading to oscillations in the probability distribution even for well-peaked wave packets. We also examine in detail the probability measure within the semiclassical approximation. Close to the bounce, the probability distribution becomes double peaked, with one peak following a trajectory close to the classical limit but with a Hubble parameter slightly shifted downwards, and the other with a value of b stuck at its minimum. An examination of the effects still closer to the bounce, and within a more realistic model involving matter and Λ, is left to future work.

Afshordi N, Magueijo J, 2022, Lower bound on the cosmological constant from the classicality of the early Universe, *Physical Review D*, Vol: 106, Pages: 1-10, ISSN: 2470-0010

We use the quantum unimodular theory of gravity to relate the value of the cosmological constant, Λ, and the energy scale for the emergence of cosmological classicality. The fact that Λ and unimodular time are complementary quantum variables implies a perennially quantum Universe should Λ be zero (or, indeed, fixed at any value). Likewise, the smallness of Λ puts an upper bound on its uncertainty, and thus a lower bound on the unimodular clock’s uncertainty or the cosmic time for the emergence of classicality. Far from being the Planck scale, classicality arises at around 7×1011 GeV for the observed Λ, and taking the region of classicality to be our Hubble volume. We confirm this argument with a direct evaluation of the wave function of the Universe in the connection representation for unimodular theory. Our argument is robust, with the only leeway being in the comoving volume of our cosmological classical patch, which should be bigger than that of the observed last scattering surface. Should it be taken to be the whole of a closed Universe, then the constraint depends weakly on Ωk: for −Ωk<10−3, classicality is reached at >4×1012 GeV. If it is infinite, then this energy scale is infinite, and the Universe is always classical within the minisuperspace approximation. It is a remarkable coincidence that the only way to render the Universe classical just below the Planck scale is to define the size of the classical patch as the scale of nonlinearity for a red spectrum with the observed spectral index ns=0.967(4) (about 1011 times the size of the current Hubble volume). In the context of holographic cosmology, we may interpret this size as the scale of confinement in the dual 3D quantum field theory, which may be probed (directly or indirectly) with future cosmological surveys.

Albertini E, Alexander S, Herczeg G,
et al., 2022, Torsion and the probability of inflation, *Journal of Cosmology and Astroparticle Physics*, Vol: 2022, Pages: 1-14, ISSN: 1475-7516

We revisit the problem of the "probability of inflation" from the point of view of the Einstein-Cartan theory, where torsion can be present off-shell even in the absence of spinorial currents. An informal estimate suggests that the barrier for tunneling from "nothing" into a classical universe becomes thinner and lower, should torsion be present, even if only off-shell. We perform a detailed calculation that supports this informal estimate for the case of torsion eigenstates. Finally, we impose a quantum mechanical analog of the zero-torsion condition by restricting to states for which the expectation value of the torsion vanishes. An explicit family of such states is obtained by building wave-packets from linear superpositions of torsion eigenstates with Gaussian weights centered around zero torsion. The tunneling probability for these wave packets is maximized when the variance of the torsion goes to zero. Hence, by considering these wave-packets as the physical states, we recover a sensible model of quantum cosmology that incorporates quantum fluctuations in the torsion, despite the apparently unacceptable conclusions one draws from naïvely considering the tunneling probabilities for the torsion eigenstates.

Magueijo J, 2022, Connection between cosmological time and the constants of nature, *PHYSICAL REVIEW D*, Vol: 106, Pages: 1-16, ISSN: 2470-0010

We examine in greater detail the proposal that time is the conjugate of the constants of nature. Fundamentally distinct times are associated with different constants, a situation often found in “relational time” settings. We show how in regions dominated by a single constant the Hamiltonian constraint can be reframed as a Schrödinger equation in the corresponding time, solved in the connection representation by outgoing-only monochromatic plane waves moving in a “space” that generalizes the Chern-Simons functional (valid for the equation of state w=−1) for other w. We pay special attention to the issues of unitarity and the measure employed for the inner product. Normalizable superpositions can be built, including solitons, “light-rays” and coherent/squeezed states saturating a Heisenberg uncertainty relation between constants and their times. A healthy classical limit is obtained for factorizable coherent states, both in monofluid and multifluid situations. For the latter, we show how to deal with transition regions, where one is passing on the baton from one time to another, and investigate the fate of the subdominant clock. For this purpose minisuperspace is best seen as a dispersive medium, with packets moving with a group speed distinct from the phase speed. We show that the motion of the packets’ peaks reproduces the classical limit even during the transition periods, and for subdominant clocks once the transition is over. Deviations from the coherent/semiclassical limit are expected in these cases, however. The fact that we have recently transitioned from a decelerating to an accelerating Universe renders this proposal potentially testable, as explored elsewhere.

Alexandre B, Magueijo J, 2022, Possible quantum effects at the transition from cosmological deceleration to acceleration, *Physical Review D*, Vol: 106, Pages: 1-13, ISSN: 2470-0010

The recent transition from decelerated to accelerated expansion can be seen as a reflection (or “bounce”) in the connection variable, defined by the inverse comoving Hubble length (b=˙a, on shell). We study the quantum cosmology of this process. We use a formalism for obtaining relational time variables either through the demotion of the constants of nature to integration constants, or by identifying fluid constants of motion. We extend its previous application to a toy model (radiation and Λ) to the realistic setting of a transition from dust matter to Λ domination. In the dust and Λ model two time variables may be defined, conjugate to Λ and to the dust constant of motion, and we work out the monochromatic solutions to the Schrödinger equation representing the Hamiltonian constraint. As for their radiation and Λ counterparts, these solutions exhibit “ringing,” whereby the incident and reflected waves interfere, leading to oscillations in the amplitude. In the semiclassical approximation we find that, close to the bounce, the probability distribution becomes double peaked, one peak following a trajectory close to the classical limit but with a Hubble parameter slightly shifted downwards, the other with a value of b stuck at its minimum b=b⋆. Still closer to the transition, the distribution is better approximated by an exponential distribution, with a single peak at b=b⋆, and a (more representative) average b biased towards a value higher than the classical trajectory. Thus, we obtain a distinctive prediction for the average Hubble parameter with redshift: slightly lower than its classical value when z≈0, but potentially much higher than the classical prediction around z∼0.64, where the bounce most likely occurred. The implications for the “Hubble tension” have not escaped us.

Mylova M, Moschou M, Afshordi N,
et al., 2022, Non-gaussian signatures of a thermal big bang, *Journal of Cosmology and Astroparticle Physics*, Vol: 2022, Pages: 1-25, ISSN: 1475-7516

What if Big Bang was hot from its very inception? This is possible in a bimetric theory where the source of fluctuations is thermal, requiring the model to live on a critical boundary in the space of parameters and can be realized when an anti-DBI brane moves within an EAdS2 ×E3 geometry. This setup renders the model unique, with sharp predictions for the scalar spectral index and its running. We investigate the non-Gaussian signatures of this thermal bimetric model, or "bi-thermal" for short. We adapt the standard calculation of non-Gaussianities for P(X,ϕ) models to the thermal nature of the model, emphasising how the bi-thermal peculiarities affect the calculation and alter results. This leads to precise predictions for the shape and amplitude of the three-point function of the bi-thermal model (at tree-level): flocalNL = -3/2 and fequilNL = -2 + 4 √(3)π/9 ≃ 0.4. We also discover a new shape of flattened non-gaussianity ∝ (k1 + k2 - k3)-3/2 + permutations, which is expected due to the excited thermal initial conditions. These results, along with our earlier predictions for the scalar power spectrum, provide sharp targets for the future generation of cosmological surveys.

Alexandre B, Magueijo J, 2021, Semiclassical limit problems with concurrent use of several clocks in quantum cosmology, *Physical Review D: Particles, Fields, Gravitation and Cosmology*, Vol: 104, ISSN: 1550-2368

We revisit a recent proposal for a definition of time in quantum cosmology, to investigate the effects of having more than one possible type of clock “at the same time.” We use as a test tube an extension of Einstein gravity with a massless scalar field in which the gravitational coupling GN is only a constant on-shell, mimicking the procedure for Λ in unimodular gravity. Hence we have two “simultaneous” clocks in the theory: a scalar field clock, and the conjugate of GN. We find that attempts to use two coherent clocks concurrently are disastrous for recovering the classical limit. The Heisenberg relations, instead of being saturated, are always realized abundantly above their bound, with strong quantum effects expected at least in parts of the trajectory. Semiclassical states always result from situations where we effectively impose a single clock, either by making the other clock a failed clock (i.e., by choosing a state where its conjugate constant is infinitely sharp) or by choosing a basis of constants where all clocks but one are redundant; i.e., motion or change in phase space does not occur with the passing of their “times.” We show how this conclusion generalizes to fluids with any equation of state. It also applies to systems where “subclocks” of the same type could be used, for example, in mixtures of different fluids with the same equation of state.https://journals.aps.org/prd/abstract/10.1103/PhysRevD.104.124069

Page J, Magueijo J, 2021, Linking the Baum-Hawking-Coleman mechanism with unimodular gravity and Vilenkin's probability flux, *Journal of Cosmology and Astroparticle Physics*, Vol: 2021, Pages: 1-14, ISSN: 1475-7516

We revisit a mechanism proposed by Hawking to resolve the cosmological constant problem (and the controversy it generated) to identify possibly more palatable alternatives and explore new connections and interpretations. In particular, through the introduction of a new action coupling the four-form field strength F = dA to the cosmological constant via a dynamical field λ(x), a novel Baum-Hawking-Coleman type mechanism is presented. This mechanism can be seen as a generalisation of Unimodular Gravity. A theory with a similar coupling to "F2" is also presented, with promising results. We show how in such theories the 3-form is closely related to the Chern-Simons density, and its associated definition of time. On the interpretational front, we propose a method avoiding the standard Euclidean action prescription, which makes use of Vilenkin's probability flux.

Magueijo J, 2021, Cosmological time and the constants of nature, *Physics Letters B: Nuclear Physics and Particle Physics*, Vol: 820, Pages: 1-4, ISSN: 0370-2693

We propose that cosmological time is effectively the conjugate of the constants of nature. Different definitions of time arise, with the most relevant related to the constant controlling the dynamics in each epoch. The Hamiltonian constraint then becomes a Schrodinger equation. In the connection representation, it is solved by monochromatic plane waves moving in a space generalizing the Chern-Simons functional. Normalizable superpositions exist and for factorizable coherent states we recover the classical limit and a seamless handover between potentially disparate times. There is also a rich structure of alternative states, including entangled constants, opening up the doors to new phenomenology.

Magueijo J, 2021, Real Chern-Simons wave function, *Physical Review D*, Vol: 104, Pages: 1-10, ISSN: 2470-0010

We examine the status of the Chern-Simons (or Kodama) state from the point of view of a formulation of gravity that uses only real connection and metric variables and a real action. We may package the real connection variables into the complex self-dual Ashtekar connection (and will do so to make contact with previous work), but that operation is essentially cosmetic and can be undone at any step or even bypassed altogether. The action will remain the (real) Einstein-Cartan action, forgoing the addition of the usual Holst (or Nieh-Yan) term with an imaginary coefficient. It is then found that the constraints are solved by a modification of the Chern-Simons state which is a pure phase (in the Lorentzian theory, we stress), the phase containing only the fully gauge-invariant imaginary part of the Chern-Simons functional. Thus, the state for the “real theory” is nonpathological with regards to the most egregious criticisms facing its “nonreal” cousin, solving the complex theory. A straightforward modification of the real Chern-Simons state is also a solution in quasitopological theories based on the Euler invariant, for which the cosmological constant, Λ, is dynamical. In that case it is enough to shift the usual factor of Λ in the wave function to the inside of the spatial Chern-Simons integral. The trick only works for the quasi-Euler theory with a critical coupling previously identified in the literature. It does not apply to the quasi-Pontryagin theory.

Alexander S, Herczeg G, Magueijo J, 2021, A generalized Hartle-Hawking wave function, *Classical and Quantum Gravity*, Vol: 38, Pages: 1-15, ISSN: 0264-9381

The Hartle–Hawking wave function is known to be the Fourier dual of the Chern–Simons or Kodama state reduced to mini-superspace, using an integration contour covering the whole real line. But since the Chern–Simons state is a solution of the Hamiltonian constraint (with a given ordering), its Fourier dual should provide a solution (i.e. beyond mini-superspace) of the Wheeler DeWitt equation representing the Hamiltonian constraint in the metric representation. We write down a formal expression for such a wave function, to be seen as the generalization beyond mini-superspace of the Hartle–Hawking wave function. Its explicit evaluation (or simplification) depends only on the symmetries of the problem, and we illustrate the procedure with anisotropic Bianchi models and with the Kantowski–Sachs model. A significant difference of this approach is that we may leave the torsion inside the wave functions when we set up the ansatz for the connection, rather than setting it to zero before quantization. This allows for quantum fluctuations in the torsion, with far reaching consequences.

Magueijo J, Zlosnik T, 2021, Quantum torsion and a Hartle-Hawking beam, *Physical Review D: Particles, Fields, Gravitation and Cosmology*, Vol: 103, Pages: 1-11, ISSN: 1550-2368

In the Einstein-Cartan framework the torsion-free conditions arise within the Hamiltonian treatment as second-class constraints. The standard strategy is to solve these constraints, eliminating the torsion from the classical theory, before quantization. Here we advocate leaving the torsion inside the other constraints before quantization, leading at first to wave functions that can be called “kinematical” with regards to the torsion, but not the other constraints. The torsion-free condition can then be imposed as a condition upon the physical wave packets one constructs, satisfying the usual uncertainty relations, and so with room for quantum fluctuations in the torsion. This alternative strategy has the surprising effect of clarifying the sense in which the wave functions solving an explicitly real theory are “delta-function normalizable.” Such solutions with zero (or any fixed) torsion, should be interpreted as plane waves in torsion space. Properly constructed wave packets are therefore normalizable in the standard sense. Given that they are canonical duals, this statement applies equally well to the Chern-Simons state (connection representation) and the Hartle-Hawking wave function (metric representation). We show how, when torsion is taken into account, the Hartle-Hawking wave function is replaced by a Gauss-Airy function, with finite norm, which we call the Hartle-Hawking beam. The Chern-Simons state, instead, becomes a packet with a Gaussian probability distribution in connection space. We conclude the paper with two sections explaining how to generalize these results beyond minisuperspace.

Barrow JD, Magueijo J, 2021, A contextual Planck parameter and the classical limit in quantum cosmology, *Foundations of Physics: an international journal devoted to the conceptual and fundamental theories of modern physics, biophysics, and cosmology*, Vol: 51, ISSN: 0015-9018

We propose that whatever quantity controls the Heisenberg uncertainty relations (for a given complementary pair of observables) it should be identified with an effective Planck parameter. With this definition it is not difficult to find examples where the Planck parameter depends on the region under study, varies in time, and even depends on which pair of observables one focuses on. In quantum cosmology the effective Planck parameter depends on the size of the comoving region under study, and so depends on that chosen region and on time. With this criterion, the classical limit is expected, not for regions larger than the Planck length, lP, but for those larger than lQ=(l2PH−1)1/3, where H is the Hubble parameter. In theories where the cosmological constant is dynamical, it is possible for the latter to remain quantum even in contexts where everything else is deemed classical. These results are derived from standard quantization methods, but we also include more speculative cases where ad hoc Planck parameters scale differently with the length scale under observation. Even more speculatively, we examine the possibility that similar complementary concepts affect thermodynamical variables, such as the temperature and the entropy of a black hole.

Magueijo J, Zlosnik T, Speziale S, 2020, Quantum cosmology of a dynamical Lambda, *Physical Review D: Particles, Fields, Gravitation and Cosmology*, Vol: 102, Pages: 064006 – 1-064006 – 14, ISSN: 1550-2368

By allowing torsion into the gravitational dynamics one can promote the cosmological constant Λ to a dynamical variable in a class of quasitopological theories. In this paper we perform a minisuperspace quantization of these theories in the connection representation. If Λ is kept fixed, the solution is a delta-normalizable version of the Chern-Simons (CS) state, which is the dual of the Hartle and Hawking and Vilenkin wave functions. We find that the CS state solves the Wheeler–De Witt equation also if Λ is rendered dynamical by an Euler quasitopological invariant, in the parity-even branch of the theory. In the absence of an infrared (IR) cutoff, the CS state suggests the marginal probability P(Λ)=δ(Λ). Should there be an IR cutoff (for whatever reason), the probability is sharply peaked at the cut off. In the parity-odd branch, however, we can still find the CS state as a particular (but not most general) solution, but further work is needed to sharpen the predictions. For the theory based on the Pontryagin invariant (which only has a parity-odd branch) the CS wave function no longer is a solution to the constraints. We find the most general solution in this case, which again leaves room for a range of predictions for Λ.

Alexander S, Jenks L, Jirousek P,
et al., 2020, Gravity waves in parity-violating Copernican universes, *Physical Review D: Particles, Fields, Gravitation and Cosmology*, Vol: 102, Pages: 044039 – 1-044039 – 13, ISSN: 1550-2368

In recent work minimal theories allowing the variation of the cosmological constant, Λ, by means of a balancing torsion, have been proposed. It was found that such theories contain parity violating homogeneous and isotropic solutions, due to a torsion structure called the Cartan spiral staircase. Their dynamics are controlled by Euler and Pontryagin quasitopological terms in the action. Here we show that such theories predict a dramatically different picture for gravitational wave fluctuations in the parity violating branch. If the dynamics are ruled solely by the Euler-type term, then linear tensor mode perturbations are entirely undetermined, hinting at a new type of gauge invariance. The Pontryagin term not only permits for phenomenologically sounder background solutions (as found in previous literature), but for realistic propagation of gravitational wave modes. These have the general property that the right and left handed gravitational waves propagate with different speeds. More generally they imply modified dispersion relations for the graviton, with both parity violating and non-violating deformations, including an effective mass for both gravitational wave polarizations. We discuss the observational constraints and predictions of these theories.

Magueijo J, 2020, Equivalence of the chern-simons state and the Hartle-Hawking and Vilenkin wave functions, *PHYSICAL REVIEW D*, Vol: 102, Pages: 044034 – 1-044034 – 7, ISSN: 1550-7998

We show that the Chern-Simons (CS) state when reduced to minisuperspace is the Fourier dual of the Hartle-Hawking (HH) and Vilenkin (V) wave functions of the Universe. This is to be expected, given that the former and latter solve the same constraint equation, written in terms of conjugate variables (loosely, the expansion factor and the Hubble parameter). A number of subtleties in the mapping, related to the contour of integration of the connection, shed light on the issue of boundary conditions in quantum cosmology. If we insist on a real Hubble parameter, then only the HH wave function can be represented by the CS state, with the Hubble parameter covering the whole real line. For the V (or tunneling) wave function, the Hubble parameter is restricted to the positive real line (which makes sense, since the state only admits outgoing waves), but the contour also covers the whole negative imaginary axis. Hence, the state is not admissible if reality conditions are imposed upon the connection. Modifications of the V state, requiring the addition of source terms to the Hamiltonian constraint, are examined and found to be more palatable. In the dual picture, the HH state predicts a uniform distribution for the Hubble parameter over the whole real line; the modified V state a uniform distribution over the positive real line.

Gubitosi G, Magueijo J, 2019, Life of cosmological perturbations in modified dispersion relation models and the prospect of traveling primordial gravitational waves, *Physical Review D: Particles, Fields, Gravitation and Cosmology*, Vol: 100, Pages: 123501-1-123501-9, ISSN: 1550-2368

We follow the life of a generic primordial perturbation mode (scalar or tensor) subject to modified dispersion relations (MDR), as its proper wavelength is stretched by expansion. A necessary condition ensuring that traveling waves can be converted into standing waves is that the mode starts its life deep inside the horizon and in the trans-Planckian regime, then leaves the horizon as the speed of light corresponding to its growing wavelength drops, to eventually become cis-Planckian whilst still outside the horizon, and finally reenter the horizon at late times. We find that scalar modes in the observable range satisfy this condition, thus ensuring the viability of MDR models in this respect. For tensor modes we find a regime in which this does not occur, but in practice it can only be realized for wavelengths in the range probed by future gravity wave experiments if the quantum gravity scale experienced by gravity waves goes down to the PeV range. In this case traveling—rather than standing—primordial gravity waves could be the telltale signature of MDR scenarios.

Magueijo J, Zlosnik T, 2019, Parity violating Friedmann universes, *Physical Review D: Particles, Fields, Gravitation and Cosmology*, Vol: 100, Pages: 1-17, ISSN: 1550-2368

We revisit extensions of the Einstein-Cartan theory where the cosmological constant Λ is promoted to a variable, at the cost of allowing for torsion even in the absence of spinors. We remark that some standard notions about Friedmann-Robertson-Walker (FRW) universes collapse in these theories, most notably that spatial homogeneity and isotropy may now coexist with violations of parity invariance. The parity-violating solutions have nonvanishing Weyl curvature even within FRW models. The presence of parity-violating torsion opens up the space of possible such theories with relevant FRW modifications; in particular the Pontryagin term can play an important role even in the absence of spinorial matter. We present a number of parity-violating solutions with and without matter. The former are the non-self-dual vacuum solutions long suspected to exist. The latter lead to tracking and nontracking solutions with a number of observational problems, unless we invoke the Pontryagin term. An examination of the Hamiltonian structure of the theory reveals that the parity-even and the parity-violating solutions belong to two distinct branches of the theory, with different gauge symmetries (constraints) and different numbers of degrees of freedom (d.o.f.). The parity-even branch is nothing but standard relativity with a cosmological constant which has become pure gauge under conformal invariance if matter is absent, or a slave of matter (and so not an independent d.o.f.) if nonconformally invariant matter is present. In contrast, the parity-violating branch contains a genuinely new d.o.f.

Alexander S, Cortês M, Liddle AR,
et al., 2019, Cosmology of minimal varying Lambda theories, *Physical Review D*, Vol: 100, ISSN: 2470-0010

Inserting a varying Lambda in Einstein’s field equations can be made consistent with the Bianchi identities by allowing for torsion, without the need to add scalar field degrees of freedom. In the minimal such theory, Lambda is totally free and undetermined by the field equations in the absence of matter. Inclusion of matter ties Lambda algebraically to it, at least when homogeneity and isotropy are assumed, i.e., when there is no Weyl curvature. We show that Lambda is proportional to the matter density, with a proportionality constant depending on the equation of state. Unfortunately, the proportionality constant becomes infinite for pure radiation, ruling out the minimal theory prima facie despite of its novel internal consistency. It is possible to generalize the theory still without the addition of kinetic terms, leading to a new algebraically enforced proportionality between Lambda and the matter density. Lambda and radiation may now coexist in a form consistent with big bang nucleosynthesis, though this places strict constraints on the single free parameter of the theory, θ. In the matter epoch, Lambda behaves just like a dark matter component. Its density is proportional to the baryonic and/or dark matter, and its presence and gravitational effects would need to be included in accounting for the necessary dark matter in our Universe. This is a companion paper to Alexander et al. [Phys. Rev. D 100, 083506 (2019)] where the underlying gravitational theory is developed in detail.

Alexander S, Cortês M, Liddle AR,
et al., 2019, Zero-parameter extension of general relativity with a varying cosmological constant, *Physical Review D*, Vol: 100, ISSN: 2470-0010

We provide a new extension of general relativity (GR) which has the remarkable property of being more constrained than GR plus a cosmological constant, having one less free parameter. This is implemented by allowing the cosmological constant to have a consistent space-time variation, through coding its dynamics in the torsion tensor. We demonstrate this mechanism by adding a “quasitopological” term to the Einstein action, which naturally realizes a dynamical torsion with an automatic satisfaction of the Bianchi identities. Moreover, variation of the action with respect to this dynamical Λ fixes it in terms of other variables, thus providing a scenario with less freedom than general relativity with a cosmological constant. Once matter is introduced, at least in the homogeneous and isotropic reduction, Λ is uniquely determined by the field content of the model. We make an explicit construction using the Palatini formulation of GR and describe the striking properties of this new theory. We also highlight some possible extensions to the theory. A companion paper [, , , , , and , following paper, Cosmology of minimal varying Lambda theories, Phys. Rev. D 100, 083507 (2019)] explores the Friedmann-Robertson-Walker reduction for cosmology, and future work will study Solar System tests of the theory.

Alexander S, Magueijo J, Smolin L, 2019, The quantum cosmological constant, *Symmetry*, Vol: 11, ISSN: 2073-8994

We present an extension of general relativity in which the cosmological constant becomes dynamical and turns out to be conjugate to the Chern–Simons invariant of the Ashtekar connection on a spatial slicing. The latter has been proposed Soo and Smolin as a time variable for quantum gravity: the Chern–Simons time. In the quantum theory, the inverse cosmological constant and Chern–Simons time will then become conjugate operators. The “Kodama state” gets a new interpretation as a family of transition functions. These results imply an uncertainty relation between Λ and Chern–Simons time; the consequences of which will be discussed elsewhere.

Magueijo J, Smolin L, 2019, A Universe that Does Not Know the Time, *UNIVERSE*, Vol: 5, ISSN: 2218-1997

Barrow JD, Magueijo J, 2019, Do we live in an eigenstate of the fundamental constants operators?, *Physical Review D*, Vol: 99, ISSN: 2470-0010

We propose that the constants of Nature we observe (which appear as parameters in the classical action) are quantum observables in a kinematical Hilbert space. When all of these observables commute, our proposal differs little from the treatment given to classical parameters in quantum information theory, at least if we were to inhabit a constants’ eigenstate. Noncommutativity introduces novelties, due to its associated uncertainty and complementarity principles, and it may even preclude Hamiltonian evolution. The system typically evolves as a quantum superposition of hamiltonian evolutions resulting from a diagonalization process, and these are usually quite distinct from the original one (defined in terms of the noncommuting constants). We present several examples targeting G, c, and Λ, and the dynamics of homogeneous and isotropic Universes. If we base our construction on the Heisenberg algebra and the quantum harmonic oscillator, the alternative dynamics tends to silence matter (effectively setting G to zero), and make curvature and the cosmological constant act as if their signs are reversed. Thus, the early Universe expands as a quantum superposition of different Milne or de Sitter expansions for all material equations of state, even though matter nominally dominates the density, ρ, because of the negligible influence of Gρ on the dynamics. A superposition of Einstein static universes can also be obtained. We also investigate the results of basing our construction on the algebra of SU(2), into which we insert information about the sign of a constant of Nature, or whether its action is switched on or off. In this case we find examples displaying quantum superpositions of bounces at the initial state for the Universe.

Magueijo J, 2018, Do we live in an eigenstate of the fundamental constants operators?, *Physical Review D - Particles, Fields, Gravitation and Cosmology*, ISSN: 1550-2368

Contaldi CR, Magueijo J, 2018, Unsqueezing of standing waves due to inflationary domain structure, *Physical Review D*, Vol: 98, ISSN: 2470-0010

The so-called trans-Planckian problem of inflation may be evaded by positing that modes come into existence only when they became “cis-Planckian” by virtue of expansion. However, this would imply that for any mode a new random realization would have to be drawn every N wavelengths, with N typically of order 1000 (but it could be larger or smaller). Such a redrawing of realizations leads to a heteroskodastic distribution if the region under observation contains several such independent domains. This has no effect on the sampled power spectrum for a scale-invariant raw spectrum, but at very small scales, it leads to a spectral index bias toward scale invariance and smooths oscillations in the spectrum. The domain structure would also “unsqueeze” some of the propagating waves, i.e., dismantle their standing wave character. By describing standing waves as traveling waves of the same amplitude moving in opposite directions, we determine the observational effects of unsqueezing. We find that it would erase the Doppler peaks in the cosmic microwave background, but only on very small angular scales, in which the primordial signal may not be readily accessible. The standing waves in a primordial gravitational wave background would also be turned into traveling waves. This unsqueezing of the gravitational wave background may constitute a detectable phenomenon.

Arzano M, Gubitosi G, Magueijo J, 2018, Parity at the Planck scale, *Physics Letters B*, Vol: 781, Pages: 510-516, ISSN: 0370-2693

We explore the possibility that well known properties of the parity operator, such as its idempotency and unitarity, might break down at the Planck scale. Parity might then do more than just swap right and left polarized states and reverse the sign of spatial momentum k: it might generate superpositions of right and left handed states, as well as mix momenta of different magnitudes. We lay down the general formalism, but also consider the concrete case of the Planck scale kinematics governed by κ-Poincaré symmetries, where some of the general features highlighted appear explicitly. We explore some of the observational implications for cosmological fluctuations. Different power spectra for right handed and left handed tensor modes might actually be a manifestation of deformed parity symmetry at the Planck scale. Moreover, scale-invariance and parity symmetry appear deeply interconnected.

Gubitosi G, Magueijo J, 2018, Primordial standing waves, *PHYSICAL REVIEW D*, Vol: 97, ISSN: 2470-0010

We consider the possibility that the primordial fluctuations (scalar and tensor) might have been standing waves at their moment of creation, whether or not they had a quantum origin. We lay down the general conditions for spatial translational invariance, and isolate the pieces of the most general such theory that comply with, or break translational symmetry. We find that, in order to characterize statistically translationally invariant standing waves, it is essential to consider the correlator ⟨c0(k)c0(k′)⟩ in addition to the better known ⟨c0(k)c†0(k′)⟩ [where c0(k) are the complex amplitudes of traveling waves]. We then examine how the standard process of “squeezing” (responsible for converting traveling waves into standing waves while the fluctuations are outside the horizon) reacts to being fed primordial standing waves. For translationally invariant systems only one type of standing wave, with the correct temporal phase (the “sine wave”), survives squeezing. Primordial standing waves might therefore be invisible at late times—or not—depending on their phase. Theories with modified dispersion relations behave differently in this respect, since only standing waves with the opposite temporal phase survive at late times.

Gubitosi G, Magueijo J, 2018, Squeezing of scalar and tensor primordial perturbations generated by modified dispersion relations, *PHYSICAL REVIEW D*, Vol: 97, ISSN: 2470-0010

In recent work we analyzed the evolution of primordial perturbations satisfying Planck-scale-modified dispersion relations and showed that there is no cosmological “squeezing” in the critical model that produces perturbations with a scale invariant spectrum. Nevertheless, the perturbations reenter the horizon as standing waves with the correct temporal phase because of the late-time decay of the momentum mode. Here we shed light on the absence of primordial squeezing by reexamining the problem in the dual rainbow frame, where c is set to 1, shifting the varying c effects elsewhere. In this frame gravity switches off at sub-Planckian wavelengths, so that the fluctuations behave as if they were in Minkowski spacetime. This is ultimately why they are not squeezed. However, away from the critical model squeezing does occur if the fluctuations spectrum is red, as is the case for scalar perturbations. Should the primordial gravity waves have a blue spectrum, we predict that they might not reenter the horizon as standing waves, because the momentum mode would be enhanced in the primordial phase.

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