## Publications

180 results found

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, 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.

Magueijo JCR, 2018, Solving the flatness problem with an anisotropic instanton in Hořava-Lifshitz gravity, *Physical Review D - Particles, Fields, Gravitation and Cosmology*, Vol: 97, ISSN: 1550-2368

In Hořava-Lifshitz gravity a scaling isotropic in space but anisotropic in spacetime, often called “anisotropic scaling,” with the dynamical critical exponent z=3, lies at the base of its renormalizability. This scaling also leads to a novel mechanism of generating scale-invariant cosmological perturbations, solving the horizon problem without inflation. In this paper we propose a possible solution to the flatness problem, in which we assume that the initial condition of the Universe is set by a small instanton respecting the same scaling. We argue that the mechanism may be more general than the concrete model presented here. We rely simply on the deformed dispersion relations of the theory, and on equipartition of the various forms of energy at the starting point.

Magueijo J, 2017, The phenomenology of squeezing and its status in non-inflationary theories, *Journal of Cosmology and Astroparticle Physics*, Vol: 2017, ISSN: 1475-7516

In this paper we skim the true phenomenological requirements behind the concept of inflationary squeezing. We argue that all that is required is that at horizon re-entry the fluctuations form standing waves with the correct temporal phase (specifically, sine waves). We quantify this requirement and relate it to the initial conditions fed into the radiation dominated epoch by whatever phase of the Universe produced the fluctuations. The only relevant quantity turns out to be the degree of suppression of the momentum, p, of the fluctuations, y, which we measure by σ~ ω2 |y|2/|p|2. Even though σ equals the squeezing parameter, s, in the case of inflation and bimetric varying speed of light scenarios, this is not true in general, specifically in some bouncing Universe models. It is also not necessary to produce a large σ at the end of the primordial phase: it is enough that σ be not too small. This is the case with scenarios based on modified dispersion relations (MDR) emulating the dispersion relations of Horava-Lifshitz theory, which produce σ~ 1, enough to comply with the observational requirements. Scenarios based on MDR leading to a slightly red spectrum are also examined, and shown to satisfy the observational constraints.

Brighenti F, Gubitosi G, Magueijo J, 2017, Primordial perturbations in a rainbow universe with running Newton constant, *Physical Review D*, Vol: 95, ISSN: 2470-0010

e compute the spectral index of primordial perturbations in a rainbow universe. We allow the Newton constant G to run at (super-) Planckian energies and we consider both vacuum and thermal perturbations. If the rainbow metric is the one associated to a generalized Horava-Lifshitz dispersion relation, we find that only when G tends asymptotically to 0 can one match the observed value of the spectral index and solve the horizon problem, both for vacuum and thermal perturbations. For vacuum fluctuations the observational constraints imply that the primordial universe expansion can be both accelerating or decelerating, while in the case of thermal perturbations only decelerating expansion is allowed.

Gubitosi G, Magueijo J, 2017, Correlation between opposite-helicity gravitons: Imprints on gravity-wave and microwave backgrounds, *PHYSICAL REVIEW D*, Vol: 95, ISSN: 2470-0010

We examine some of the roots of parity violation for gravitons and uncover a closely related new effect: correlations between right- and left-handed gravitons. Such correlators have spin 4 if they involve gravitons moving along the same direction and spin zero for gravitons moving with opposite directions. In the first case, the most immediate implication would be a degree of linear polarization for the tensor vacuum fluctuations, which could be seen by gravity-wave detectors sensitive enough to probe the primordial background, its degree of polarization and anisotropies. Looking at the anisotropy of the gravity waves linear polarization, we identify the parity respecting and violating components of the effect. The imprint on the cosmic microwave background temperature and polarization would be more elusive, since it averages to zero in the two-point functions, appearing only in their cosmic variance or in fourth-order correlators. In contrast, spin-zero correlations would have an effect on the two-point function of the cosmic microwave background temperature and polarization, enhancing the BB component if they were anticorrelations. Such correlations represent an amplitude for the production of standing waves, as first envisaged by Grishchuk, and could also leave an interesting signature for gravity-wave detectors.

Afshordi N, Magueijo J, 2016, Critical geometry of a thermal big bang, *Physical Review D - Particles, Fields, Gravitation and Cosmology*, Vol: 94, ISSN: 1550-2368

We explore the space of scalar-tensor theories containing two nonconformal metrics, and find a discontinuity pointing to a “critical” cosmological solution. Due to the different maximal speeds of propagation for matter and gravity, the cosmological fluctuations start off inside the horizon even without inflation, and will more naturally have a thermal origin (since there is never vacuum domination). The critical model makes an unambiguous, nontuned prediction for the spectral index of the scalar fluctuations: nS=0.96478(64). Considering also that no gravitational waves are produced, we have unveiled the most predictive model on offer. The model has a simple geometrical interpretation as a probe 3-brane embedded in an EAdS2×E3 geometry.

Alexander S, Jyoti D, Magueijo J, 2016, Inflation and the quantum measurement problem, *Physical Review D*, Vol: 94, ISSN: 1550-7998

We propose a solution to the quantum measurement problem in inflation. Our model treats Fourier modes of cosmological perturbations as analogous to particles in a weakly interacting Bose gas. We generalize the idea of a macroscopic wave function to cosmological fields, and construct a self-interaction Hamiltonian that focuses that wave function. By appropriately setting the coupling between modes, we obtain the standard adiabatic, scale-invariant power spectrum. Because of central limit theorem, we recover a Gaussian random field, consistent with observations.

Barrow J, Alexander S, Magueijo JCR, 2016, Turning on gravity with the Higgs mechanism, *Classical and Quantum Gravity*, Vol: 33, ISSN: 1361-6382

We investigate how a Higgs mechanism could be responsible for theemergence of gravity in extensions of Einstein theory. In this scenario, at highenergies, symmetry restoration could “turn off” gravity, with dramatic implicationsfor cosmology and quantum gravity. The sense in which gravity is muted dependson the details of the implementation. In the most extreme case gravity’s dynamicaldegrees of freedom would only be unleashed after the Higgs field acquires a non-trivialvacuum expectation value, with gravity reduced to a topological field theory in thesymmetric phase. We might also identify the Higgs and the Brans-Dicke fields in sucha way that in the unbroken phase Newton’s constant vanishes, decoupling matter andgravity. We discuss the broad implications of these scenarios.

Gubitosi G, Lagos M, Magueijo J,
et al., 2016, Bayesian evidence and predictivity of the inflationary paradigm, *Journal of Cosmology and Astroparticle Physics*, Vol: 2016, ISSN: 1475-7516

In this paper we consider the issue of paradigm evaluation by applying Bayes' theorem along the following nested hierarchy of progressively more complex structures: i) parameter estimation (within a model), ii) model selection and comparison (within a paradigm), iii) paradigm evaluation. In such a hierarchy the Bayesian evidence works both as the posterior's normalization at a given level and as the likelihood function at the next level up. Whilst raising no objections to the standard application of the procedure at the two lowest levels, we argue that it should receive a considerable modification when evaluating paradigms, when testability and fitting data are equally important. By considering toy models we illustrate how models and paradigms that are difficult to falsify are always favoured by the Bayes factor. We argue that the evidence for a paradigm should not only be high for a given dataset, but exceptional with respect to what it would have been, had the data been different. With this motivation we propose a measure which we term predictivity, as well as a prior to be incorporated into the Bayesian framework, penalising unpredictivity as much as not fitting data. We apply this measure to inflation seen as a whole, and to a scenario where a specific inflationary model is hypothetically deemed as the only one viable as a result of information alien to cosmology (e.g. Solar System gravity experiments, or particle physics input). We conclude that cosmic inflation is currently hard to falsify, but that this could change were external/additional information to cosmology to select one of its many models. We also compare this state of affairs to bimetric varying speed of light cosmology.

Gubitosi G, Lagos M, Magueijo J,
et al., 2016, Bayesian evidence and predictivity of the inflationary paradigm, *Journal of Cosmology and Astroparticle Physics*, Vol: 2016, Pages: 002-002, ISSN: 1475-7516

In this paper we consider the issue of paradigm evaluation by applying Bayes' theorem along the following nested hierarchy of progressively more complex structures: i) parameter estimation (within a model), ii) model selection and comparison (within a paradigm), iii) paradigm evaluation. In such a hierarchy the Bayesian evidence works both as the posterior's normalization at a given level and as the likelihood function at the next level up. Whilst raising no objections to the standard application of the procedure at the two lowest levels, we argue that it should receive a considerable modification when evaluating paradigms, when testability and fitting data are equally important. By considering toy models we illustrate how models and paradigms that are difficult to falsify are always favoured by the Bayes factor. We argue that the evidence for a paradigm should not only be high for a given dataset, but exceptional with respect to what it would have been, had the data been different. With this motivation we propose a measure which we term predictivity, as well as a prior to be incorporated into the Bayesian framework, penalising unpredictivity as much as not fitting data. We apply this measure to inflation seen as a whole, and to a scenario where a specific inflationary model is hypothetically deemed as the only one viable as a result of information alien to cosmology (e.g. Solar System gravity experiments, or particle physics input). We conclude that cosmic inflation is currently hard to falsify, but that this could change were external/additional information to cosmology to select one of its many models. We also compare this state of affairs to bimetric varying speed of light cosmology.

Gubitosi G, Magueijo J, 2016, Reappraisal of a model for deformed special relativity, *Classical and Quantum Gravity*, Vol: 33, ISSN: 1361-6382

We revisit one of the earliest proposals for deformed dispersion relations in the light of recent results on dynamical dimensional reduction and production of cosmological fluctuations. Depending on the specification of the measure of integration and the addition rule in momentum space the model may be completed so as to merely deform Lorentz invariance, or so as to introduce a preferred frame. Models which violate Lorentz invariance have a negative UV asymptotic dimension and a very red spectrum of quantum vacuum fluctuations. Instead, models which preserve frame independence can exhibit running to a UV dimension of two, and a scale-invariant spectrum of fluctuations. The bispectrum of the fluctuations is another point of divergence between the two casings proposed here for the original model.

Gubitosi G, Arzano M, Magueijo J, 2016, Quantization of fluctuations in deformed special relativity: the two-point function and beyond, *Physical Review D*, Vol: 93, ISSN: 1550-7998

We show that the two-point function of a quantum field theory with de Sitter momentum space (herein called DSR) can be expressed as the product of a standard delta function and an energy-dependent factor. The proportionality to a standard delta function is a nontrivial result valid in any theory without a preferred frame and relies crucially on the on-shellness condition. Different theories are distinguished by the specific form of the energy-dependent proportionality factor. Applied to models exhibiting running of the dimensionality of space, this result is essential in proving that vacuum fluctuations are generally scale invariant at high energies whenever there is running to two dimensions. This is equally true for theories with and without a preferred frame, with differences arising only as we consider higher order correlators. Specifically, the three-point function of DSR has a unique structure of “open triangles,” as shown here.

Barrow JD, Magueijo J, 2015, Local varying-alpha theories, *MODERN PHYSICS LETTERS A*, Vol: 30, ISSN: 0217-7323

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Arzano M, Gubitosi G, Magueijo J,
et al., 2015, Anti-de Sitter momentum space, *Physical Review D*, Vol: 92, ISSN: 1550-7998

We investigate the anti-de Sitter (AdS) counterpart to the well-studied de Sitter (dS) model for energy-momentum space, viz “κ-momentum space” space (with a structure based on the properties of the κ-Poincaré Hopf algebra). On the basis of previous preliminary results one might expect the two models to be complementary: dS exhibiting an invariant maximal spatial momentum but unbounded energy, AdS a maximal energy but unbounded momentum. If that were the case AdS momentum space could be used to implement a principle of maximal Planck-scale energy, just as several studies use dS momentum space to postulate of maximal Planck-scale spatial momentum. However, several unexpected features are uncovered in this paper, which limit the scope of the expected complementarity, and interestingly they take different forms in different coordinatizations of AdS momentum space. “Cosmological” AdS coordinates mimic the dS construction used for κ-momentum space, and produce a Carrol limit in the ultraviolet. However, unlike the κ-momentum space, the boundary of the covered patch breaks Lorentz invariance, thereby introducing a preferred frame. In “horospherical” coordinates we achieve full consistency with frame independence as far as boost transformations are concerned, but find that rotational symmetry is broken, leading to an anisotropic model for the speed of light. Finally, in “static” coordinates we find a way of deforming relativistic transformations that successfully enforces frame invariance and isotropy, and produces a Carrol limit in the ultraviolet. Our results are also relevant for a long-standing debate on whether or not coordinate redefinitions in momentum space lead to physically equivalent theories: our three proposals are evidently physically inequivalent, leading to alternative models of Planck-scale effects. As a corollary we study the UV running of the Hausdorff dimension of momentum spa

Arzano M, Gubitosi G, Magueijo J,
et al., 2015, Vacuum fluctuations in theories with deformed dispersion relations, *Physical Review D*, Vol: 91, ISSN: 1550-7998

We examine vacuum fluctuations in theories with modified dispersion relations which represent dimensional reduction at high energies. By changing units of energy and momentum we can obtain a description rendering the dispersion relations undeformed and transferring all the nontrivial effects to the integration measure in momentum space. Using this description we propose a general quantization procedure, which should be applicable whether or not the theory explicitly introduces a preferred frame. Based on this scheme we evaluate the power spectrum of quantum vacuum fluctuations. We find that in all theories which run to two dimensions in the ultraviolet the vacuum fluctuations, in the ultraviolet regime, are scale invariant. This is true in flat space but also for “inside the horizon” modes in an expanding universe. We spell out the conditions upon the gravity theory for this scale invariance to be preserved as the modes are frozen-in outside the horizon. We also digress on the meaning of dimensionality (in momentum and position space) and suggest that the spectral index could itself provide an operational definition of dimensionality.

Amelino-Camelia G, Arzano M, Gubitosi G,
et al., 2015, Gravity as the breakdown of conformal invariance, *International Journal of Modern Physics D*, Vol: 24, ISSN: 0218-2718

In this paper, we propose that at the beginning of the universe gravity existed in a limbo either because it was switched off or because it was only conformally coupled to all particles. This picture can be reverse-engineered from the requirement that the cosmological perturbations be (nearly) scale-invariant without the need for inflation. It also finds support in recent results in quantum gravity suggesting that spacetime becomes two-dimensional at super-Planckian energies. We advocate a novel top-down approach to cosmology based on the idea that gravity and the Big Bang Universe are relics from the mechanism responsible for breaking the fundamental conformal invariance. Such a mechanism should leave clear signatures in departures from scale-invariance in the primordial power spectrum and the level of gravity waves generated.

Barrow JD, Magueijo J, 2014, Redshifting of cosmological black bodies in Bekenstein-Sandvik-Barrow-Magueijo varying-alpha theories, *PHYSICAL REVIEW D*, Vol: 90, ISSN: 1550-7998

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Amelino-Camelia G, Arzano M, Gubitosi G,
et al., 2014, Planck-scale dimensional reduction without a preferred frame, *Physics Letters B: Nuclear Physics and Particle Physics*, Vol: 736, Pages: 317-320, ISSN: 0370-2693

Several approaches to quantum gravity suggest that the standard description of spacetime as probed at low-energy, with four dimensions, is replaced in the Planckian regime by a spacetime with a spectral dimension of two. The implications for relativistic symmetries can be momentous, and indeed the most tangible picture for “running” of the spectral dimension, found within Horava–Lifshitz gravity, requires the breakdown of relativity of inertial frames. In this Letter we incorporate running spectral dimensions in a scenario that does not require the emergence of a preferred frame. We consider the best studied mechanism for deforming relativistic symmetries whilst preserving the relativity of inertial frames, based on a momentum space with curvature at the Planck scale. We show explicitly how running of the spectral dimension can be derived from these models.

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