Minimum length uncertainty relations in the presence of dark energy

We introduce a dark energy-modified minimum length uncertainty relation (DE-MLUR) or dark energy uncertainty principle (DE-UP) for short. The new relation is structurally similar to the MLUR introduced by K{\' a}rolyh{\' a}zy (1968), and reproduced by Ng and van Dam (1994) using alternative arguments, but with a number of important differences. These include a dependence on the de Sitter horizon, which may be expressed in terms of the cosmological constant as $l_{\rm dS} \sim 1/\sqrt{\Lambda}$. Applying the DE-UP to both charged and neutral particles, we obtain estimates of two limiting mass scales, expressed in terms of the fundamental constants $\left\{G,c,\hbar,\Lambda, e\right\}$. Evaluated numerically, the charged particle limit corresponds to the order of magnitude value of the electron mass ($m_e$), while the neutral particle limit is consistent with current experimental bounds on the mass of the electron neutrino ($m_{\nu_e}$). Possible cosmological consequences of the DE-UP are considered and we note that these lead naturally to a holographic relation between the bulk and the boundary of the Universe. Low and high energy regimes in which dark energy effects may dominate canonical quantum behaviour are identified and the possibility of testing the model using near-future experiments is briefly discussed.

The concept of superposition is the very essence of quantum theory. As the mathematical embodiment of wave-particle duality, it determines the state space structure of canonical non-relativistic quantum mechanics (QM) and its relativistic extension, quantum field theory (QFT). However, despite the unparalleled success of both QM and QFT in describing the micro-world, such duality does not manifest itself in our every day experience: the macro-world does not admit superpositions of states. This gives rise to the so-called measurement problem, recognised since the early days of quantum theory, whereby a classical 'observer' (an experimenter or apparatus not subject to the quantum formalism) is required to reduce the quantum superposition via the act of 'measurement'.
This glaring ontological disparity, yet otherwise arbitrary distinction between observer and observed, has led many physicists to argue that canonical quantum theory is incomplete. Though proposals for the resolution of the measurement problem are varied (see [1][2][3] for reviews of contemporary approaches, plus [4] for a discussion of foundational issues), many involve modifications of the quantum dynamics that lead to spontaneous reduction of the state vector in some mesoscopic regime, which interpolates between the microscopic (quantum) and macroscopic (classical) worlds [5][6][7][8][9][10]. In modern terminology, this spontaneous reduction is known as decoherence, and is believed to be caused by the interaction of the system with its environment [11]. Thus, prior to the act of measurement, micro-systems are weakly coupled to their environment, whereas meso-or macro-systems are strongly coupled. The former behave quantum mechanically, whereas the latter behave classically.
With the measurement problem in mind, it is natural to consider the weakness of gravity, as compared to the three other known fundamental forces -electromagnetic, weak nuclear and strong. Indeed, classical gravitational interactions may typically be ignored in the micro-world and only become relevant on macroscopic, even astrophysical or cosmological, scales [12]. Nonetheless, the exact nature of quantum gravitational interactions is unknown and their description remains the holy grail of theoretical physics research [13,14]. It is therefore natural to suppose that what is missing from canonical quantum theory is not an adequate description of the observer, visa-vis the observed, but gravity. Since the gravitational interaction is universal, affecting all forms of matter and energy, it may be hoped that gravity, or space-time itself, may play a fundamental role in the 'spontaneous' decoherence of quantum systems.
In fact, the idea that quantum gravitational effects may play an important role in the resolution of the measurement problem encountered in canonical nongravitational QM has a long and distinguished history [15][16][17][18][19][20][21][22][23][24][25][26][27]. Originally published in 1966, Károlyházy's model [15,16] was one of the first to consider the possibility of gravitationally-induced wave function collapse. The fundamental idea proposed in [15] is that quantum fluctuations of the metric give rise to an intrinsic and irremovable 'haziness' in the space-time background, corresponding to a superposition of classical geometries. As a result, an initially pure state vector develops, over time, into a mixed state. Coherence is maintained only over a small region, known as a 'coherence cell', whose size depends on the space-time curvature induced by the body and, hence, on its mass. For micro-objects, the effect of curvature is small, giving rise to canonical quantum behaviour but, for macro-objects, the maximum size of a coherence cell lies within the classical radius of the body itself. Thus, the quantum nature of the macro-body remains hidden, as the wave function associated with its centre of mass (CoM) spontaneously decoheres on extremely small scales: the larger the body, the smaller the size of the cell.
From a theoretical perspective, a major advantage of the Károlyházy model is that it contains no free parameters. It is therefore able to make clear predictions regarding gravitational modifications of the canonical quantum dynamics, utilising only the known constants G, c and . Specifically, the existence of a minimum length uncer-tainty relation (MLUR), representing a modification of the canonical Heisenberg uncertainty principle (HUP), necessarily follows from the intrinsic haziness of spacetime assumed in the K-model. The resulting uncertainty, inherent in the measurement of a space-time interval s, is where l Pl = G/c 3 is the Planck length [15,16]. For space-like intervals, this represents the minimum possible uncertainty in the position of a quantum mechanical particle, used to 'probe' the distance s. When ∆s is identified with the Compton wavelength, λ C = /(mc), s may be identified with Károlyházy's estimate of the width of a coherence cell for a fundamental particle, Though motivated by an attempt to resolve the measurement problem, the MLUR (1) represents an important theoretical prediction in its own right. Since its inception, the literature on quantum gravity phenomenology has expanded significantly and many modifications of the HUP, known as generalised uncertainty principles (GUPs), have been proposed [28][29][30]. These share the common feature of giving rise to a minimum resolvable resolvable length in nature, which is usually assumed to be of the order of the Planck length [31,32]. Hence, the existence of some form of MLUR is now regarded as a generic feature of candidate quantum gravity models [33,34].
In the present paper, we will not concern ourselves with the measurement problem per se, though the possible implications of our model for this important open question are briefly discussed in Sec. V. Instead, we will focus on the second major prediction stemming from the introduction of a 'hazy' space-time, i.e., that of a fundamental MLUR in nature. In particular, we will focus on a major advance in fundamental physics, which should have radical implications for any model of gravitationally-induced wave function collapse, as well as for quantum gravity phenomenology in general, including MLURs [35][36][37], namely, the discovery of dark energy [38,39].
Though the precise microphysical origin of dark energy remains unknown, and is an active area of research within the cosmology/astrophysics community, the current best-fit to all available cosmological data favours a 'cosmological concordance' or ΛCDM model [40], in which dark energy takes the form of a positive cosmological constant, Λ > 0. This accounts for approximately 69% of the total energy density of the Universe, whereas cold dark matter (CDM) accounts for around 26% and ordinary (visible) matter for around 5% [41,42].
For our purposes, it is important to note that, although dynamical dark energy models cannot be excluded on the basis of presently available data, any viable dark energy model must give rise to an effective cosmological constant at late times, comparable to the present epoch.
(See [43][44][45][46] for reviews of current dark energy research.) Furthermore, though Λ may, ultimately, turn out to have a particle physics origin (i.e., the dark energy field may correspond to a form of 'matter' in the usual sense, albeit of an exotic kind), its precise origin is unimportant for the derivation of dark energy-modified MLURs. What is important are its gravitational effects. Specifically, regarding the influence of dark energy on physical bodies, it makes no difference whether we write the Λ-dependent term on the left-hand side or the right-hand side of Einstein's field equations. On the right, it may be interpreted as a form of matter, on the left, as a geometrical effect.
As a geometrical effect, Λ may be interpreted as a minimum space-time curvature, or minimum gravitational field strength. This clearly has implications for any model of gravitationally-induced wave function collapse, including the K-model, as well as for any MLUR purporting to include quantum gravity effects, irrespective of the measurement problem. Nonetheless, even if the true origin of dark energy is of a particle nature, the exotic form of matter to which it corresponds necessarily sources a minimum positive curvature, Λ > 0, in otherwise 'empty' space. As we will see, this has profound implications for Károlyházy's model, which originally assumed quantum fluctuations of asymptotically flat (i.e. Minkowski) space [15,16].
By contrast, we embed a K-type model in a realistic background geometry, incorporating the effects of dark energy. A key consequence of the existence of a positive cosmological constant is the existence of a fundamental horizon for all observers (including quantum mechanical 'particles'), the de Sitter horizon, l dS ∼ 1/ √ Λ. We argue that this necessarily implies a modification of the MLUR (1), including minimum curvature/finite-horizon effects.
As with the original model presented in [15,16], our model has the theoretical advantage of involving no free parameters. The main difference is that the MLUR obtained by considering a hazy space-time,à la Károlyházy, in the presence of dark energy, necessarily involves G, c, and Λ. The structure of this paper is as follows. In Sec. II A, we consider classical perturbations of the cosmological Friedmann-Lemâitre-Robertson-Walker (FLRW) metric, induced by the presence of point particles. Although the FLRW metric is not valid on local scales, we note that its perturbed form, at the present epoch, is similar to the Schwarzschild-de Sitter metric. Thus, it predicts approximately the same gravitational potential (up to numerical factors of order unity) in the vicinity of a local compact object. This allows us to view the local fieldfor example, around a microscopic particle located close to the surface of the Earth -as a perturbation away from the cosmological background geometry. Throughout our analysis, Λ is treated as a fundamental constant of nature which gives rise to a constant dark energy density, and minimum curvature, at all points in space. In Sec. II B, we show how the formula for the perturbed line element relates to Károlyházy's scheme for measuring the minimum positional uncertainty of a gravitating, quantum mechanical, 'point' particle. Sections II C-II D review the original derivation of the MLUR given in [15,16] and the alternative derivation given in [47,48], respectively, while Sec. II E outlines motivations for dark energy-induced modifications of the standard result. The physical basis of the dark energy uncertainty principle (DE-UP) is laid out in Secs. III A-III B and its basic properties, including applications to both neutral and electrically charged particles (Secs. III C-III D), as well as its implications for the holographic conjecture [49,50] (Sec. III E), are explored. Possible cosmological consequences of the DE-UP are considered in Sec. IV and Sec. V contains a summary of our main conclusions together with a brief discussion of prospects for future work. Potential conceptual issues regarding the limits of applicability of the model, which arise at various points throughout the text, are discussed at greater length in the Appendix.

II. KÁROLYHÁZY'S MLUR -NEW PERSPECTIVES
In [15,16], Károlyházy et al consider 'resolving' a space-time interval s, traversed by a quantum mechanical particle of mass m, by projecting it into the lab frame using light signals emitted by the particle over the course of its path. They claim that classically, the observed interval s is related to the original ('true') interval s via where r S (m) = 2Gm/c 2 is the Schwarzschild radius associated with the mass m. By explicitly taking into account the quantum nature of the particle traversing s, they then obtain an estimate of the minimum uncertainty in the measurement of s, denoted ∆s. The derivation of the MLUR given in [15,16] is considered in detail in Sec. II C and Károlyházy's measurement procedure is illustrated in Fig. 1. In Sec. II A, we show that a formally similar result, in which the quantities s and s in Eq. (3) have different physical meanings, may be obtained using gravitational perturbation theory. In this formulation, the quantities s and s do not a priori represent 'true' (CoM frame) and 'measured' (lab frame) values of the length of a space-time interval but, instead, the lengths of an interval in an unperturbed background space and in the perturbed space induced by the presence of the particle, respectively. Nonetheless, the new formulation may be reconciled with Károlyházy's picture, since we are free to consider receiving light signals in a lab frame far away from the particle's CoM, in which the gravitational perturbation induced by it is small. The formal equivalence of the two pictures is shown explicitly in Sec II B.
A. Classical intervals in perturbed and unperturbed backgrounds: s and s We now consider the classical perturbation induced by the presence of a point particle in a realistic space-time background, requiring the perturbed metric to satisfy the linearised Einstein equations. Bt 'particle' we mean a spherically symmetric compact object that is point-like with respect to large -in principle, up to cosmologicallength-scales.
In the presence of dark energy, represented by a positive cosmological constant Λ > 0, the gravitational action is and the field equations take the form where g µν denotes the space-time metric, G µν = R µν − (1/2)Rg µν is the Einstein tensor, R µν is the Ricci tensor, R = g µν R µν is the scalar curvature and T µν is the matter energy-momentum tensor. For a perfect fluid, T µν may be represented covariantly as where ρ denotes the rest-mass density, p is the isotropic pressure and u µ is the 4-velocity of an infitesimal fluid element. The Friedmann-Lemaître-Roberston-Walker (FLRW) metric, describing a homogenous, isotropic, expanding Universe, may be written as where τ is the cosmic time and a(τ ) is the cosmological scale factor which is normalized to one at the present epoch, a(τ 0 ) = a 0 = 1. In spherical polar coordinates, dΣ 2 takes the form where dΩ 2 = r 2 (dθ 2 + sin 2 θdφ 2 ) is the line-element for the unit 2-sphere and k is the Gaussian curvature, with dimensions [L] −2 . In appropriate units, k ∈ {−1, 0, +1} for negative, zero, and positive curvature, respectively. Substituting Eqs. (6), (7) and (8) where a dot represents differentiation with respect to τ [51]. For future reference, we note that the Hubble parameter as is defined as and that its present day value is H 0 = 67.74 ± 0.46 kms −1 Mpc −1 , or H 0 = 2.198 × 10 −18 s −1 (ignoring error bars) in cgs units [42]. The critical density is defined as giving ρ crit = 8.639 × 10 −30 gcm −3 . This is the value of ρ required to give zero curvature (k = 0) in the absence of a cosmological constant (Λ = 0). Dividing Eq. (9) by H 2 0 = 8πGρ crit /3, it may be rewritten in terms of the density parameters Ω r , Ω M , Ω k and Ω Λ . These denote the present day contributions, as fractions of the critical density, to the total energy density of the Universe for radiation, matter, curvature and dark energy, respectively. To three significant figures, the values obtained from current observations are Ω r = 0.00, Ω M = 0.31, Ω k = 0.00 and Ω Λ = 0.69, where the matter sector is composed of both non-relativistic baryons Ω b = 0.05 and non-relativistic (cold) dark matter Ω DM = 0.26 [42]. Thus, where Ω(τ ) = ρ total (τ )/ρ crit . In other words, the present day density is very close to the critical density (Ω 0 = 1.00) and the Universe is approximately flat on large scales, with the exception of the minimal curvature induced by Λ.
In an arbitrary spatial coordinate system, Eq. (7) may be written in the general form where γ ij is the spatial part of the metric, and an arbitrary metric perturbation may be written as The gauge invariant tensor perturbations ('gravitons') satisfy the transverse-traceless conditions, Let us now switch back to spherical polar coordinates and consider a spherically symmetric perturbation, induced by the 'birth' of a particle of mass m, at some time τ < τ 0 . Our ansatz for the perturbative part of the energy-momentum tensor T µ ν then takes the form where Θ is the Heaviside step function and all other components are zero. Strictly, Eq. (16) models the birth of a particle, at τ = τ , which remains at rest with respect to a comoving coordinate system at all later times. It also holds approximately for particles that are not subjected to extreme accelerations. In this case, dynamical tensor perturbations, which would otherwise lead to gravitational wave emission, may be neglected. In addition, we may set B i = h 0i = 0 since, at linear order, vector perturbations are associated with vorticity in the cosmic fluid and do not arise in this scenario [52,53]. The full evolution of the scalar and tensor-type perturbations for the birth of a point-like mass may be determined by following a procedure analogous to that used in [54], though such a detailed treatment is unnecessary for our current purposes. Instead, we note that the covariant metric (14) contains four extraneous degrees of freedom associated with coordinate invariance. In the Newtonian gauge, which holds approximately for situations in which h 0i 0 and where wave-like tensor perturbations can be neglected, this 'gauge' freedom may be used to diagonalise the perturbed metric, giving where Ψ and Φ are Newtonian potentials obeying Poisson's equation [52,53]. In our scenario, this is consistent with the fact that, since the source term is timedependent only instantaneously, the time-dependence of the perturbations must be small on scales r r lc (τ −τ ), where r lc (τ − τ ) is the maximum extent of the particle's light cone. In other words, we assume that the metric perturbation induced by the particle's creation propagates radially outwards at the speed of light, but remains approximately 'static', with respect to comoving coordinates, within its horizon. Any additional timedependence is confined to a thin spherical shell at r r lc (τ − τ ).
In the absence of anisotropic stresses, Φ = Ψ [52,53], and Poisson's equation for a mass distribution ρ m immersed in a dark energy background in an expanding Universe is∇ where is the dark energy density and∇ 2 is the Laplacian, defined with respect to comoving coordinates. For spherically symmetric systems, this reduces The current experimental value of Λ, inferred from observations of high-redshift type 1A supernovae (SN1A), Large Scale Structure (LSS) data from the Sloan Digital Sky Survey (SDSS) and Cosmic Microwave Background (CMB) data from the Planck satellite, is Λ = 1.114 × 10 −56 cm −2 [41,42]. This is equivalent to the vacuum energy density ρ Λ = 5.971 × 10 −30 gcm −3 . Now let us consider the case in which ρ m is given by a δ-function density profile corresponding to a classical point-like mass m, ρ m (τ, r) ∝ mδ(r)/a 2 (τ )r 2 . In this scenario, Eq. (18) is simply Poisson's equation with two source terms, a regular point-mass (m > 0) and an 'irregular' constant negative density, −ρ Λ . (Recall that, when written on the right-hand side of the field equations, Λ may be interpreted as a negative energy density belonging to the matter sector.) This is satisfied by the modified Newtonian potential which gives rise to the gravitational field strength [55] g Thus, the cosmological constant corresponds to an effective gravitational repulsion whose strength increases linearly with the comoving distance ar and we note that, for r ≤ (≥)r grav , where and l dS = 3/Λ = 1.641×10 28 cm, the force between two particles is attractive (repulsive). l dS is the asymptotic de Sitter horizon and is of the same order of magnitude as the present day radius of the Universe r U 1.306 × 10 28 cm (13.8 billion light years). In [57,58], it was also referred to as the first Wesson length, after the pioneering work [59], and denoted l W .
In the Newtonian picture, r grav marks the separation distance beyond which the effective gravitational force between two spherically symmetric bodies becomes repulsive (i.e. beyond which the repulsive effect of dark energy overcomes the canonical gravitational attraction). Up to numerical factors of order unity, the same result may be obtained in general relativity by evaluating the Kretschmann invariant, K = R αβγδ R αβγδ , for the Schwarzschild-de Sitter metric, and noting the value of r at which it changes sign.
Including contributions to ρ m from the background baryonic and dark matter densities -that is, embedding the perturbation in a full FLRW background -similar arguments yield where H is a solution to Eqs. (9)-(10), so that This is known as the gravitational turn-around radius, and may also be derived rigorously in a fully general relativistic context [56]. For τ → ∞, the matter density is diluted to such an extent that H 2 → c 2 /l 2 dS and the space-time becomes asymptotically de Sitter, yielding Eq. (23). The implications of de Sitter-type cosmological evolution are discussed further in Sec. III B.
In more complex local environments, we may expect H(τ ) appearing in Eq. (24) to be replaced by a local Hubble parameter, which is not a solution of Eqs. (9)-(10). However, if Λ is a genuine constant of nature, giving rise to a constant dark energy density at all points in space (as assumed in this analysis), there exists a correction term to the canonical Newtonian potential, Φ ∼ Gm/R, determined by the local Hubble parameter, H/c ≥ 1/l dS .
Hence, in general, the infitesimal line-elements of the perturbed and unperturbed metrics, ds and ds, are related via where minimum value of the Hubble term is set by the dark energy scale. Here, ds denotes the lineelement for the flat, unperturbed, space-time. Next, we rewrite the unperturbed line-element Eq. (14) as Restricting ourselves to time-like intervals within the present day horizon then gives ds cdτ .
Note that, since explicit r-dependence drops out of the expression for the unperturbed line-element, the coordinate distance r need not be equal to s(τ ). Formally, Eq. (28) gives the difference between the perturbed and unperturbed line-elements, traversed by a light-like signal (e.g. a photon) over time τ , as seen by an observer at r. The flight time of the photon(s) and the position of the observer relative to the mass m are independent. Hence, so are r and s(τ ).
This corresponds to the following experimental procedure. Suppose we place a 'detector' at a coordinate distance r from a specified origin. (We assume throughout that our detector represents an idealized observer whose gravitational field may be considered negligible, even compared to that of the perturbing particle: though unrealistic, this is a valid assumption in our idealized gedanken experiment.) If the massive particle is absent, a photon travelling for a time τ traverses a space-like interval s(τ ) = cτ . In flat space, this may simply be identified with the coordinate distance, so that s(τ ) = cτ = r.
However, if, instead, we assume the photon is emitted by a massive particle located at r = 0, and absorbed by a detector at r > 0 after the same time τ , the traversed interval 'seen' by an observer at r is s (r, τ ) (28). The simple relationship between the coordinate distance and space-like interval is destroyed by the gravitational field of the particle and, in general, the light signal will not reach the same value of r at the same time τ (i.e., r = cτ ).
Furthermore, τ need not correspond to the flight time of single photon. Instead, we may consider spitting the measurement of the interval s (r, τ ) into two (or more) parts. For simplicity, however, we consider only a two part measurement process. In the first part, a photon travels from the perturbing particle at r = 0 to the detector at r > 0. In the second, an additional photon travels from a (generally different) point, to r. If the total flight time of both photons is τ , the space-like interval that would have been traversed if the particle had not been present is s(τ ) = cτ , but the interval traversed in the perturbed space in s (r, τ ).
Since r can label any point in space, regardless of the value of τ , which we here identify with flight times of the photons used to perform the measurement, it follows that the measured interval s (r, τ ) depends on where we place our detector in relation to the perturbing particle. This fact also enables us to reinterpret Eq. (28) in terms of an experimental procedure to resolve time-like intervals traversed by massive, self-gravitating particles,à la Károlyházy. During the photon flight time τ , the CoM of a classical non-relativistic particle also traverses a timelike interval approximately equal to s(τ ) = cτ . Hence, s (r, τ ) and s(τ ) may be interpreted as the 'observed' (lab frame) and 'true' (CoM frame) values of the spacetime interval traversed by a massive particle, as claimed in [16]. This procedure is discussed in greater detail in Sec. II B and is illustrated in Fig. 2.
For r r grav (τ ), the Hubble expansion term in Eq. (28) dominates, so that r r grav (τ ) marks the limit of the validity of the perturbative Newtonian gauge picture. Physically, r r grav (τ ) corresponds to a region in which the effect of the perturbation is negligible and the standard Hubble expansion takes over. Thus, the Hubble expansion gives rise to small, additive correction term to Károlyházy's formula (3), plus a modification of the original canonical gravitational term, corresponding to the substitution r → ar. Since the additive term is subdominant within the region of physical interest, r r grav (τ ), the latter modification is the most important.
Equation (28) holds both at the current epoch τ 0 4.352 × 10 17 s (13.8 billion years) and at all earlier times for which the FLRW metric is valid, including epochs where the average curvature was far higher than today. In addition, it holds for regions of the present day Universe in which space-time curvature is well above the FLRW background level (k 0). This may be seen by taking the static, spherically symmetric, weak-field limit of the full Einstein equations (5), which also reduce to Eq. (24) with a(τ 0 ) = a 0 1 and H(τ 0 )/c = H 0 /c l dS , for r r grav (τ 0 ), regardless of the profile of the gravitational field on scales r grav (τ 0 ) r l dS . This limit applies to all experiments carried out on (or near) the surface of the Earth at the present epoch.
Taking both these factors into account, it is reasonable to suppose that Eq. (28) holds (at least approximately), far more generally, remaining valid at any epoch under non-extreme conditions. We may expect it to break down close to the inflationary era [60][61][62], or for spacetime intervals close to the event horizon of a black hole. However, we note that, using H 2 0 = 4.830 × 10 −36 s −1 c 2 /l 2 dS = 3.338 × 10 −36 s −1 and substituting the Newtonian potential (21) into Eq. (17), we obtain a time-like metric component directly proportional to that of the Schwarzschild-de Sitter solution, which describes a black hole in the presence of a cosmological constant Λ > 0. It therefore seems probable that Eq. (28) is valid in all physically interesting scenarios. Based on the arguments presented in Sec. II A, we see that, in addition to describing the difference between the 'true' and 'measured' values of a time-like interval traversed by a self-gravitating particle, Eq. (3) also describes the difference between the perturbed space-time interval, s , induced by the presence of the particle, and the unperturbed space-time interval, s, that would have existed if the particle had not been present (assuming Λ = 0 and a a 0 = 1). Physically, this makes sense, since we may consider projecting the particle's worldline onto a detector in the lab frame, a distance r from the CoM, at which the induced gravitational potential is Φ(τ 0 , r) −Gm/r. For relatively small r, we project this interval onto a region of locally curved space (i.e., a region in which the curvature is above the background level of the FLRW metric), induced by the particle's self gravity.
Similar arguments apply even when the background curvature is well above the FLRW average, for example, due to the presence of macroscopic lab equipment, or the lab's proximity to the surface of the Earth. Practically, we may restrict our attention to projections within a very small region in the vicinity of the CoM, over which the particle's (extremely small) self-gravity may be considered non-negligible compared to the background level, whatever this may be. Classically, such a region is well defined for any perturbed metric and traces out a 'world-tube' of width r grav (c.f. Eq. (25)) surrounding the CoM world-line [15,16]. Projecting the world-line onto a 'detector' within this tube gives rise to significant deviations in the measured value of the interval, as compared to its 'true' value, due to the space-time curvature induced by the particle.
Clearly, once the 'fuzziness' of the CoM due to canonical quantum mechanics is taken into account things become even more complicated, as a second radius -the Compton radius -may be associated with the particle. Nonetheless, in our model, we will find that the counterintuitive results implied by the considerations above remain the same: once the particle's self-gravity is taken into account, physical measurements of space-time intervals -for example, the space-like position of a particle, relative to a predefined origin -yield more accurate results if the measurements are made from further away. Below a certain optimum length-scale, attempting to probe the position of the particle's CoM with greater accuracy becomes self-defeating. The resulting 'gravitational uncertainty' caused by the fuzziness of the spacetime close to the particle's CoM outweighs the gain in localising the canonical quantum wave packet. By contrast, far away from the CoM, metric fluctuations reduce to the background level (assumed to be of the order of the Planck length) and canonical quantum behaviour is recovered. The measurement scheme considered above is shown, for particles with both classical gravitational (turn-around) and quantum mechanical (Compton) radii, in Fig. 1.
The explicit connection between this procedure and the perturbed space-time induced by the presence of the particle is illustrated in Fig. 2. For simplicity, let us begin by assuming that the gravitational effect of the particle mass can be neglected, so that s s in our notation. This scenario is represented by the flat blue line. Now let us consider measuring a space-like distance by means of a photon, emitted from the particle at r = 0 and absorbed by a detector in the lab frame at some distance r = ct, where t is the proper time measured by particle's CoM. Note that, in general, this need not be identified with the cosmic time τ , so that we are free to consider t τ . If the particle's recoil velocity is non-relativistic, it may be considered negligible at the classical level, so that dt dτ . Thus, if t is small compared to the cosmic time (t τ ), we may set a a 0 = 1. In this case, it is clear that the time-like interval traversed by the particle in time t is identical to the spacelike interval measured by the experimental apparatus (i.e the particle-photon-detector system). In Károlyházy's notation, we have s (t) = s(t) = ct ≡ s (r) = s(r) = r, where s and s denote the world-lines traversed by the particle and measured in the lab frame, respectively. Now let us consider the more general case, in which the space-time curvature induced by the presence of the particle cannot be ignored. This scenario is represented by the curved red line in Fig. 2. In this case, if the photon travels from the particle at r = 0 to the detector FIG. 1: Measurement of the time-like interval traversed by a massive particle located at r 0, by projecting light-like signals emitted over the course of its path onto a 'detector' at r > 0. The outer tube surrounding the centre of mass (CoM) represents the region r < rgrav, in which the particle's gravitational field may be considered non-negligible compared to the background curvature. Placing the detector within rgrav leads to significant differences between the measured (lab frame) and 'true' (CoM frame) values, even in the classical regime (28). The inner tube represents the fuzziness of the particle's CoM due to the nonzero width of the canonical quantum wave packet. Generally, the tubes defined by the gravitational and quantum mechanical radii have different thicknesses, but coincide for the minimum-mass particle predicted by the DE-UP. (See Sec. III C.) at r > 0 in time t, this corresponds to the measurement of a space-like interval s (t, r) (1 − r S /2r)ct. As shown in Sec. II A, once the gravitational field of the particle is taken into account, the simple relation between the coordinate distance r, traversed by the photon, and the space-like interval this corresponds to breaks down (s (r) = r). Likewise, the simple relationship between the coordinate distance and the time elapsed no longer holds (r(t) = ct).
The time-like interval traversed by the particle is still s(t) = ct, so that s (t, r) (1 − r S /2r)s(t), as in Eq. (3). However, s(t) = ct also represents the space-like interval that would have been measured, had the particle's mass not perturbed the background. Hence, Károlyházy's interpretation of the symbols s and s , as representing the 'true' (CoM) frame and measured (lab frame) values of the space-time interval traversed by the particle, is equivalent to ours, in which they represent intervals in the non-perturbed and perturbed backgrounds, respectively.
As stated in Sec. II A, we now show explicitly that Eq. (3) holds even more generally. Suppose that, rather than measuring the space-like interval between the particle and the detector -which corresponds to the coordi- If the gravitational field of the particle is considered negligible, space-time is approximately flat. In this case, a photon emitted from the particle at r = 0 travels to the point r(t) = ct in time t. This completes a measurement of the space-like interval s(t) = ct. During this time (ignoring recoil), the particle traverses a time-like interval s (t) = ct, so that s (t) = s(t) = r(t) = ct. Taking the particle's gravity into account, if the photon travels from r = 0 to r > 0 in time t, this corresponds to a measurement of the spacelike interval s (t, r) (1 − rS/2r)ct, (r = ct). The time-like interval traversed by the particle is still s(t) = ct, so that s (t, r) (1 − rS/2r)s(t). This formula relates the perturbed line element s (t, r) to the unperturbed line element s(t) or, equivalently, the space-like interval measured at r to the 'true' time-like interval traversed by the particle. Hence, the relation between s and s obtained from Károlyházy's measurement procedure is equivalent to the perturbative result.
nate distance r, even if the two are not equivalent -we instead choose to measure a much larger interval. For example, let us imagine that the particle is surrounded by a horizon, at a (classically) fixed distance s from its CoM. Furthermore, let us imagine that, if the gravitational field of the particle were absent, the horizon would be located at a fixed distance s = l * rather than s .
Our experimental procedure is then as follows. A photon is emitted from the particle at r = 0 and absorbed by the detector in the lab frame (as before) after a time t 1 . This completes a measurement of the space-like interval Simultaneously, or near simulataneously, a photon emitted from a point on the horizon at t 2 = t 1 − t * , where t * = l * /c, also arrives at r and is absorbed by the detector. This completes a measurement of the space-like interval s 2 (1−r S /2r)s 2 , where s 2 = −ct 2 > 0. This result follows directly from the independence of the space-time coordinates r and t where, in our experimental procedure, t is identified with the flight time of a photon and r is identified with the position of the detector. Together, these interactions complete the measurement of a space-like interval given by The time-like interval traversed by the particle during the flight time of both photons is s = ct * = l * , so that this procedure is equivalent to projecting the entire world-line of the particle, traced out over t * , onto the detector at r.
Modifying this argument to include the effects of universal expansion, dark energy (Λ > 0) and the background matter density on the Newtonian potential induced by the perturbation, gives which is simply Eq. (28) with s(τ ) → l * (τ ). In this case, we may identify t τ , and the relevant horizon is the particle horizon, l * (τ ) = r H (τ ), given by where η(τ ) is the conformal time [55] Hence, the measured value of the space-like distance between the particle and the horizon depends on where we place our detector in relation to each. This is a simple consequence of the fact that the perturbation breaks the global symmetry (i.e. homogeneity or, equivalently, isotropy about every point) of the FLRW background. If r is very small, the detector sits within a (relatively) deep potential well, in which the difference between the curvature of the perturbed and the unperturbed backgrounds is large. From Károlyházy's viewpoint, the time-like interval traversed by the particle, over the time taken for a photon to reach the horizon, is projected onto a detector in the lab frame at r. If r r grav (τ ) r * , where r * is the coordinate distance corresponding to the position of the horizon, the distortion induced by the gravitational field of the particle renders the measured value significantly different from the true (CoM frame) value.
Implicitly, this argument assumes that the particle formed in the very early Universe (τ 0). However, even if this is not the case, r H (τ ) still marks the furthest point in causal contact with the particle at the cosmic epoch τ . As such, it still represents the largest distance that can be measured by means of the particle-photondetector system, at time τ . Strictly, for 0 τ τ 0 , Károlyházy's interpretation is not applicable to Eq. (30), since the world-line of the particle is much shorter than l * (τ ) = r H (τ ). Nonetheless, this formula remains physically meaningful in relation to the gedanken experiment described above, in which the detector at r receives signals from both the particle at r = 0 and its horizon at r H (τ ).
The above argument demonstrates the classical equivalence of Károlyházy's measurement scheme and the perturbative result, Eq. (28). In canonical QM, the picture of the classical point-particle is replaced by the wave function ψ, representing a superposition of position or, equivalently, momentum states of the particle's CoM. Thus, it is not difficult to imagine that, in the quantum regime, the classical region over which the particle's selfgravity cannot be neglected gives rise to an irreducible haziness of the underlying space-time metric, induced by the presence of the wave function. This is equivalent to an irreducible 'smearing' out of the particle mass or, equivalently, of the CoM associated with ψ.
This observation, which formed the basis of Károlyházy's predictions [15,16], will also form the basis of our own analysis, though we will depart from his original prescription in a number of crucially important ways. In particular, we will attempt to incorporate the effects of a space-filling dark energy, which exists in the form of a cosmological constant Λ > 0, with effective energy density and pressure given by Eq. (19).
We note that in this model, as in Károlyházy's original [15,16], Dirac δ-function position states do not exist. Even if the position of a quantum particle is ideally localised, from the perspective of the gravitationallymodified quantum theory, its CoM remains 'smeared' over some minimum length-scale, which is a function of the size, mass and possibly charge of the body, and of fundamental physical constants. This point is discussed in detail in Sec. III, in which the dark energy-modified MLUR is derived.
C. Derivation of the MLUR (Károlyházy, 1968) To highlight both the similarities and the differences between the arguments presented in [15,16] and those presented in the present work, we briefly review the original derivation of Károlyházy's MLUR. Special emphasis is placed on the physical assumptions that underly the model and on the chain of reasoning that gives rise to the final result. For clarity, where new or supplementary assumptions are introduced for the first time, they are explicitly stated.
Beginning with Eq. (3), Károlyházy effectively defines the uncertainty in s in terms of an assumed uncertainty in m, via where β is a positive numerical constant of order unity. In fact, following Eq. (3), β is set exactly equal to one in Károlyházy's original derivation [15,16]. We explicitly include it, from here on, for the sake of comparison with the results of Ng and van Dam [47,48], presented in Sec. II C, and their modification in the presence of dark energy, given in Sec. III. While this idea is reasonable from a gravitational perspective -where one may expect statistical fluctuations in space-time configurations to be equivalent to fluctuations in the mass that 'sources' the gravitational field (or at least correlated with them) -it is problematic from the quantum point of view, since 'uncertainty' refers to the statistical spread of measurement outcomes, where the physical quantity in question is represented by a Hermitian operator. However, in both canonical QM and QFT, mass is a parameter, not an operator.
In [15,16], Károlyházy obtains the expression for ∆m from the 'canonical' uncertainty relation ∆E∆t , though this too is potentially problematic, as time t is not an operator in the canonical non-relativistic theory. Defining the uncertainty in the rest-energy of the particle as and using s ct to infer ∆s c∆t, yields ∆m /(c∆s) .
By substituting (35) into (33), then assuming that the self-gravity associated with the particle's wave function is non-negligible only over the interval 0 ≤ r ∆s [i.e., replacing r → ∆r ∆s in (33)] and noting that the minimal value of ∆s is (∆s ) min ∆s, then yields where we define the Planck length l Pl and mass m Pl , for later convenience, as In his original papers [15,16], Károlyházy's MLUR was related to the concept of a coherence cell via a special gravitationally-modified dispersion relation, which enabled estimates of the cell width, a c , and Eq. (36) to be satisfied simultaneously. However, in the present paper, we will not consider the implications of dark energy for models of gravitationally-induced wave function collapse. Their detailed examination is left to future work [63].
D. An alternative derivation (Ng and van Dam, 1994) An alternative derivation of Eq. (36) is based on a gravitational extension of the MLUR obtained in canonical QM, and was originally proposed by Ng and van Dam [47,48]. That an MLUR exists, even in the canonical non-gravitational theory, can be seen by considering the dependence of the positional uncertainty ∆x on the time interval t over which measurements are made. (Note that we again distinguish between this and the cosmic time τ .) The approximate dependence of ∆x on the time interval t may be determined from the non-relativistic quantum dispersion relations, ω = ( /2m)k 2 , which give rise to the group velocity The uncertainties in v group and k at any time t are related via ∆v group (t) ( /2m)∆k(t), or, equivalently Using the fact that ∆x(0) ∆v group (t)t for t > 0 then gives Next, we define the uncertainty over all measurements, made at both t = 0 and t > 0, as the geometric mean of the canonical uncertainties at both times, i.e.
This yields where λ C = /(mc) is the Compton wavelength and where we have defined the distance r = ct, assuming that the wave function is spherically symmetric and spreads radially outwards. Interestingly, Eq. (42) may also be derived using the 'canonical' energy-time uncertainty relation, More rigorously, it may be obtained as a direct solution to the Schrödinger equation in the Heisenberg picture [64,65]. In the absence of an external potential (V = 0), the time evolution of the position operatorx(t) is given by which may be solved directly, yieldinĝ The spectra of any two Hermitian operators,Â andB, obey the general uncertainty relation [66,67] ∆A∆B and Using the definition of ∆x canon. (t), Eq. (41), together with t = r/c, we recover Eq. (42).
Historically, this result was first obtained by Salecker and Wigner using a gedanken experiment in which a quantum 'particle' is used to measure a distance r by means of the emission and reabsorption of a photon [68]. In this description ∆x canon. (r), given by Eq. (42), represents the minimum possible canonical quantum uncertainty in the measurement of r.
The argument presented in [68] proceeds as follows. Suppose we attempt to measure r using a 'clock' consisting of a classical mirror and a quantum mechanical device (e.g. a charged particle such as an electron), initially located at r = 0, that both emits and absorbs photons. A photon is emitted at t = 0 and reflected by the mirror, which is placed at some unknown distance r > 0. The photon is then reabsorbed by the particle after a time t = 2r/c (not t = r/c).
Assuming that the velocity of the particle remains well below the speed of light, it may be modelled nonrelativistically. By the standard Heisenberg uncertainty principle (HUP), the uncertainty in its velocity at any time t ≥ 0 obeys the inequality where ∆x(t) is the positional uncertainty obtained by evolving the initial wave function ψ(x, 0) via the Schrödinger equation (i.e. neglecting recoil). However, if the initial positional uncertainty is ∆x(0) then, in the time required for the photon to travel to the mirror and back, t, the particle acquires an additional positional uncertainty The total canonical positional uncertainty is now defined as and obeys the inequality Minimizing this expression with respect to ∆x(t), or equivalently ∆v(t), and using the fact that ∆v max /(2m∆x min ), gives where we have again used r = 2ct.
We note that similar arguments apply if we consider a modified experimental set up, in which a photon is emitted by the particle at r = 0 and absorbed by a device in the lab frame at r = ct, or vice versa. (In other words, we note that reflection by the mirror is not an essential part of the experimental procedure and, in addition, that it does not affect the order of magnitude estimates of the minimum quantum uncertainty inherent in the measurement.) We also note that requiring r > r S = 2Gm/c 2 (i.e., that photons cannot be emitted from within the Schwarzschild radius of our 'probe' particle), we obtain (∆x canon. ) min = l Pl . Alternatively, requiring r > λ C , the measurement process devised by Salecker and Wigner gives rise to a MLUR which is consistent with the standard Compton bound of the non-relativistic theory.
For fundamental particles, it is therefore interesting to ask, what happens if a photon is emitted from the particle and reabsorbed within the interval r ∈ (r S , λ C ]? Strictly, the answer is that, for r < λ C , the non-relativistic theory breaks down and we must switch to a field theoretic picture. In this, the 'measurement' of r corresponds to a self-interaction, described by a one-loop process in the relevant Feynman diagram expansion, in which the photon remains virtual. However, it is important to remember that interactions corresponding to 'measurements' of r < ∆x canon. (r, m) < λ C (m) in the non-relativistic theory are physical. It is therefore reasonable to apply the non-relativistic formulae, such as Eq. (52) and its gravitational 'extensions', in this regime, on the understanding that 'measuring' distances r < λ C via photon emission/reabsorption corresponds to virtual photon exchange via a one-loop process.
A related point concerns the existence of superluminal velocities for r λ C , as implied by Eq. (52). However, though virtual particles can travel faster than the speed of light, this does not imply a violation of causality, as information is not transmitted outside the light cone of a given space-time point [69]. In fact, a similar effect occurs with respect to the standard Heisenberg term: for ∆x λ C , the HUP implies ∆p mc, or equivalently ∆v c. Hence, superluminal velocities and sub-Compton probe distances in the non-relativistic theory are associated with the regime in which field theoretic effects become important. Nonetheless, we may continue to apply the non-relativistic formulae in this region, subject to the caveats stated above. These issues are discussed in detail in the Appendix.
It is straightforward to extend the arguments presented in [64,65] and [68] to include an estimate of the uncertainty in the position of the particle due to gravitational effects, ∆x grav . By assuming that this is proportional to the Schwarzschild radius r S , Ng and van Dam defined the the total uncertainty due to canonical quantum effects, plus gravity, as where β > 0, which is also assumed to be of order unity [47,48]. (For β = 2, we recover ∆x grav = r S exactly.) Minimizing Eq. (53) with respect to m yields and, substituting this back into Eq. (53), we obtain Neglecting numerical factors of order unity, and relabelling ∆s → ∆x total in Eq. (36), in accordance with standard QM notation for distance measurements, we see that Eq. (55) is equivalent to Károlyházy's result with r = ct ≡ s(t).
Equivalently, (∆x total ) min may be written as a function of m, using Eq. (54). By performing the minimization procedure with respect to m, we have effectively asked the question "what mass must the probe particle have, in order to measure the distance r with minimum quantum uncertainty?". Physically, this is equivalent to asking, "if our particle has mass m, what distance r can be measured with minimum uncertainty?". However, although Eq. (54) fixes the relation between m and r for an uncertainty-minimizing measurement, we note that there is no minimum of the function ∆x total (r, m), given by Eq. (53), in the r-direction of the (r, m) plot. Intuitively, we may expect to be able to minimize ∆x total (r, m) with respect to either m or r, and to obtain the same result in either case, since this gives rise to a procedure which is self-consistent in the limit r → λ ± C (i.e. when the 'probe' distance r tends to the Compton wavelength of the particle, either from above or below). This point is discussed further in Sec. II E.
Finally, before concluding the present subsection, we note that similar results hold, even for electrically neutral particles, whose interactions are mediated by massive, short-range bosons. For electrically charged particles, real photons may be emitted or absorbed, or virtual photon exchange may take place via a one-loop self-interaction. For uncharged particles, photons (either real or virtual) are replaced by the appropriate forcemediating boson(s). For example, in the case of the weak nuclear force, the W ± and Z 0 bosons are massive, and hence short-range, giving rise to short-range probe distances r ≤ λ C r H (τ ). To realize the measurement scheme outlined in Sec. II B, in which a neutral particle communicates with -and effectively 'measures' the distance to -its own horizon, we must instead imagine a higher order self-interaction processes involving W ± exchange, taking place on some scale r ≤ λ C r H (τ ), coupled with the exchange of virtual photons between the intermediate W ± bosons and a charged particle located at ∼ r H (τ ).

E. Motivations for the DE-UP
As shown in Sec. III C, in [15,16] Eq. (36) was obtained by considering a gedanken experiment to measure the length of a space-time interval with minimum quantum uncertainty. This derivation relies on the fact that the mass of the measuring device (probe particle) m distorts the background space-time. Equating the uncertainty in the particle's rest energy with uncertainty in its mass then implies an irremovable uncertainty or 'haziness' in the space-time in the vicinity of the particle itself. This results in an absolute minimum uncertainty in the precision with which a gravitating system can be used to measure the length of any given world-line, s. By contrast, the arguments presented in [47,48] circumvent the need to assume quantum fluctuations in the rest mass, and hence the need to define a rest-energy Hamiltonian, H rest =mc 2 .
Nonetheless, Károlyházy's arguments [15,16] are similar to those of Ng and van Dam [47,48], in that β ∼ O(1) arises as a direct result of the assumption that the Schwarzschild radius of a body, r S (m) = 2Gm/c 2 , represents the minimum 'gravitational uncertainty' in its position. In fact, for MLURs of the form (36)/(55), it is usually assumed that β ∼ O(1) in most of the existing quantum gravity literature [33,34]. For all the scenarios leading to Eq. (55) considered above, this is directly equivalent to assuming a minimum gravitational uncertainty of order r S (m).
An important physical consequence is that, since Eq. (55) holds if and only if Eq. (54) also holds, it is straightforward to verify Substituting the minimization condition for ∆x total (r), Eq. (54), into Eq. (56) then gives For β ∼ O(1), we require the '>' inequality in Eq. (57), since many arguments imply that l Pl represents the minimum resolvable length-scale due to quantum gravitational effects. (See, for example [70][71][72], plus [33,34] for reviews of minimum length scenarios in phenomenological quantum gravity.) This implies that the '>' inequality also holds in Eq. (56) and, hence, that the minimum quantum gravitational uncertainty predicted by Károlyházy/Ng and van Dam is always greater than the Compton wavelength of the particle that minimizes it.
However, from a physical perspective, the assumption ∆x grav (m) r S (m) may be questioned on at least two grounds. First, we see that, for fundamental particles with masses m m Pl , ∆x grav (m) r S (m) l Pl . Although the total uncertainty may remain super-Planckian, the assumption of simple additivity, ∆x total (r, m) = ∆x canon. (r, m) + ∆x grav (m), on which Eq. (55) is ultimately based, implies that canonical quantum uncertainty and the gravitational uncertainty arise independently, without influencing one other (i.e., that the gravitational uncertainty remains fixed, regardless of how dispersed the quantum wave packet becomes). It is therefore not clear whether a gravitational uncertainty given by ∆x grav (m) r S (m) < l Pl is physically meaningful. Second, gravity is a long range force. Intuitively, we may expect that, however it is defined, the gravitational uncertainty induced by the presence of a point-like or near point-like particle at r = 0 should fall with the gravtational field strength. Naïvely, we may assume that the gravitational uncertainty varies in proportion to the classical Newtonian potential, ∆x grav (r, m) ∝ |Φ(r, m)| ∝ β(r)r S (m) as r → ∞.
If this is indeed the case, we see that, rather than being a simple constant, β(r) must take the form of a ratio, β(r) = β l * /r, where β ∼ O(1) and l * l Pl is a phenomenologically significant length-scale which is well motivated by fundamental physical considerations. In the context of a dark energy Universe, it is clear that the de Sitter horizon, l dS = 3/Λ, fulfils this criterion. As we shall see, one consequence of this is that states for which r > l Pl and (∆x total ) min < λ C become possible, in contrast to the predictions obtained from Eqs. (53)- (55). We also note that replacing β = const. → β(r) = β l * /r in Eq. (53) allows us to minimize (∆x total ) min (r, m) with respect to either m or r. It is straightforward to demonstrate that this minimum is unique and is independent of both r and m. As a result, the minimization procedure remains self-consistent in the limit r → λ ± C . In Sec. III, we derive MLURs in which the minimum uncertainty in a physical quantity Q is given by the cube root of of three (possibly distinct) scales, Q 1 , Q 2 , Q 3 , but which differ from relations derived from Eqs. (36)/ (55) in two important ways. First, the new relations attempt to incorporate the effects of dark energy, in the form of a cosmological constant, on the 'smearing' of space-time and, thus, on the minimum quantum gravitational uncertainty inherent in a measurement of position and related physical observables. Second, they lead to substantially different but physically reasonable predictions in a number of scenarios. Specifically, they may be combined with other results obtained in general relativity and canonical quantum theory to give estimates of both the electron (e − ) and electron neutrino (ν e ) masses, in terms of fundamental constants. These estimates yield the correct order of magnitude values obtained from experiment.
In deriving the new relations we follow a procedure analogous to that used by Ng and van Dam [47,48] (outlined in Sec. III B) but assume the existence of an asymp-totically de Sitter/FLRW, rather than Minkowski, spacetime. The results are obtained in two different ways. In the first, it is unnecessary to assume fluctuations in basic parameters, such as the mass m. This avoids the need to promote parameters to observables, represented by Hermitian operators in the non-relativistic quantumgravitational regime. (From a technical point of view, it removes the need to define the operatorm or, equivalently, the rest HamiltonianĤ rest =m/c 2 .) In this case, it is, however, necessary to make certain assumptions about the properties of space-time superpositions in the Newtonian limit. In particular, we assume the existence of an upper bound on ∆x grav , given by the difference between line-elements in two classical space-times: one in which the particle is present and one in which it is absent. This is equivalent to assuming that the 'spread' of quantum states cannot exceed the difference between the two classical extremes.
In the second, we promote the classical Newtonian potential to an operator, Φ = −Gm/r →Φ = −Gm/r,à la Károlyházy, and estimate the associated uncertainty, ∆Φ, by considering a superposition of CoM position states. We then relate ∆Φ to ∆x grav by considering the associated uncertainty induced in the measurement of space-time line-elements. From here on, we refer to all minimum quantity uncertainty relations of the form (∆Q) min (Q 1 Q 2 Q 3 ) 1/3 as 'cubic', due to the value of the exponent on the right-hand side. Like Károlyházy, for τ τ 0 , we take Eq. (3) as our starting point for the quantum mechanical definition of a 'hazy' space-time. In this case, the Hubble flow correction term in Eq. (28) is subdominant within the turnaround radius, r ≤ r grav . However, rather than following the steps expressed in Eqs. (34)- (35), leading to Eq. (36), we instead make the following physical assumption.
We assume that the quantum mechanical uncertainty in the space-like interval between a particle of mass m (located at r = 0) and the coordinate distance r, is of the order of the difference between the classical values s (r, m) and s(r), where s(r) = s (r, m)| m=0 .
Classically, the presence of the particle induces a perturbation in the background space-time, whose magnitude at r is given by so that our assumption is equivalent to setting ∆s(r, m) ∆s pert (r, m) = |s (r, m) − s(r)| , where s(r) and s (r, m) represent the two (classical) extremes.
In the classical picture, the underlying space-time may be in one of two distinct states. In the first, in which the particle is absent, the underlying metric corresponds to the unperturbed line element s(r). In the second, in which the particle is present, the metric corresponds, instead, to the perturbed line element s (r, m). It is reasonable to suppose that, whatever the final theory of quantum gravity may be, a wave function of the form describing a superposition of space-time background states, is possible in at least some limiting cases. Here, we use the notation Ψ(t, r) to distinguish between wave functions representing space-time superpositions and ψ(t, r), which represents a canonical quantum wave function that exists on a definite classical space-time background. Though the mathematical formalism of a theory that contains both is not developed in the present work, we have in mind a composite wave function, that reduces to ψ(t, r) when Ψ(t, r) corresponds to a particular geometry. More realistically, we may assume that the space-time background on which the canonical quantum wave function |ψ propagates is, in fact, in a superposition of an infinite number of states, each corresponding to a unique classical line element s, i.e.
An expansion of this form will yield Eq. (59) if either the limits of integration are such that s i = s, the unperturbed line element, and s f = s , the perturbed line element, or, more generally, if s i and s f take arbitrary values but the wave packet |Ψ st maintains a standard deviation of order |s − s|. This holds true even for s i → 0, s f → ∞, as s i and s f reach the extremal classical limits. Though a complete theory still eludes us, we may imagine a path integral over some kind of phase space, in which space-times corresponding to all other possible line-elements contribute negligible amplitudes to the total state vector expansion. These would include states corresponding to flat or negative curvature in the presence of m, as well as states giving rise to extreme positive curvature, which could only be sourced classically by much larger masses. This scenario is illustrated graphically in Fig. 3.
Incorporating the effects of universal expansion, we have ∆s(r, m) → ∆s(t, r, m) , where t represents the time taken to complete the measurement (r = ct). Note that ∆s becomes a function of time even if we choose to neglect the subdominant Hubble flow term in the perturbed Newtonian potential, since we must still shift to comoving coordinates r → a(t)r. For the measurement procedure considered explicitly in Figs represents the space-like interval, at time t, between the particle (at a(t)r = 0) and the 'detector' (at a(t)r > 0), in the unperturbed space-time. This gives ∆s(t, r, m) Gm c 2 a(t)r s(a(t)r) .
However, as described in Sec. II B, we may use a more general measurement procedure to measure much larger space-like intervals, up to and including the particle horizon r H (τ ). In this case, t τ and the space-time uncertainty takes the form where r H (τ ) is given by Eq. (31). In Eq. (64), direct r-dependence drops out of the expression for s, since this now represents the distance to the horizon, which may be expressed purely in terms of the cosmic time.
With Eq. (64) as our new starting point, we may now ask the question: how is this scenario affected by the presence of dark energy, in the form of a cosmological constant Λ? Clearly, the main physical consequence at the present epoch (τ τ 0 , a a 0 = 1) is the existence of a cosmological horizon at a fixed distance from any observer for all τ τ 0 . This is the de Sitter horizon, which corresponds to the (unperturbed) space-like interval s l dS = 3/Λ, and its formation is discussed in Sec. III B.
Thus, in applying Eq. (64) to particles at the present epoch, we have in mind a particle interacting simultaneously with an object at r, close to its CoM, and with the furthest reaches of its environment, represented by r H (τ 0 ) l dS . For r > λ C , this object may be a detector in the lab frame, which simultaneously receives signals (e.g. photons) from the particle and from distant objects close to l dS . However, for r < λ C , the local object with which the probe particle interacts is simply itself and the local interaction involves the exchange of virtual particles. In principle, the long-range interaction between r < λ C and r H (τ 0 ) l dS may also involve the exchange of virtual particles -if necessary, via an appropriate higher-loop process.
For our purposes, the fact that the interaction between the particle and its horizon may involve the exchange of virtual rather than real particles is extremely important. In effect, such interactions constantly 'measure' the distance from the particle's CoM -or, more specifically, from a point r < λ C close to the CoM -to its horizon. Hence, any irremovable uncertainty present in the result of this measurement is directly equivalent to an irremovable uncertainty in the position of the particle. Classically, both the position of the CoM and the position of the horizon are well-defined, so that any quantum uncertainty in the distance between them is equivalent to an uncertainty in the position of either (or both).
We may obtain the same result using an operational procedure in canonical quantum theory, as follows. In the classical picture, a point-particle of mass m, located at r (i.e., represented by the function δ( r − r )), generates a well-defined gravitational potential at a general point r, given by In the quantum picture, the classical potential is promoted to an operator, Φ →Φ, such that In other words, acting on the canonical position eigenstate δ( r − r ),Φ recovers the classical potential Φ. For superpositions of position states, ψ(t, r ), the gravitational potential at r will also be given by a superposition of states. We then have Φ = ψ|Φ|ψ , yielding and higher-order moments Φ n may be defined in like manner.
In the limit ψ(t, r ) → δ( r − r ), the above definition can easily be modified to ensure that the classical limit is recovered, i.e. that Φ (r, m) → Φ(r, m). The tricky part is dealing with the fact that position eigenstates cannot be normalized, though, in principle, this causes no more fundamental problems here than it does in canonical QM. However, as we shall see (and as mentioned previously in Sec. II B), the presence of irremovable gravitational uncertainty makes the physical realisation of canonical quantum δ-function states impossible. Such a modification is therefore not required: Eq. (66) remains formally valid but the minimum positional uncertainty is greater than zero, for any physically realizable state.
As an example, we consider spherically symmetric Gaussian states, for which where we have chosen our coordinate system so that the wave packet CoM is located at r = 0 and ∆x denotes the canonical quantum uncertainty. For Gaussian states, this is given by where σ = ∆x(0) is the initial spread at t = 0 and r ≡ ct.
For σ (∆x canon. ) min √ λ C r (r σ 2 /λ C ), the spread of the wave function is given approximately by ∆x(r) σ, whereas, for σ √ λ C r (r σ 2 /λ C ), the late-time spread is given by ∆x(r) λ C r/σ. Hence, any 'measurements' (including one-loop self-interactions) occurring on time-scales t σ 2 /(cλ C ) are effectively 'instantaneous' and do not significantly disturb the initial (t = 0) quantum state. However, as t = 0 is an idealization, which is likely not physically realizable, we restrict our attention henceforth to time-scales t σ 2 /(cλ C ). Equation (68) then gives The two expressions above coincide for σ λ C -which is a reasonable assumption in the canonical theory -yielding The next step is to determine the relationship between ∆Φ and ∆s, the uncertainty in the measured space-time interval. This can be done by setting τ τ 0 in Eq. (28) and ignoring the sub-dominant Hubble flow term for r ≤ r grav , giving where s(τ 0 ) l dS = const.. In the quantum picture, we then haveŝ giving ∆s (r, m) as claimed. ∆s (r, m) can then be identified with ∆x grav , as before. In fact, even if we consider alternative timeintervals, t τ 0 , and identify s(t) with the total flight time of photon(s) in the generalized measurement procedure outlined in Sec. II B, an analogous argument still holds. Since we may set s(t) = ct, and because t is a parameter, not an operator, in canonical QM,ŝ(t) may still be regarded as a 'constant' from an operator perspective.
Note that here, as in the derivations of the canonical quantum uncertainty given in [64,65] and [68] (outlined in Sec. II D), we continue to identify r = ct in deriving the expression for ∆Φ. In this sense, the geometric nature of the gravitational field is not explicitly accounted for in this step and (65) is treated like any other potential existing on a flat space background. This is an unavoidable limitation of working within the framework of canonical QM up to this point.
However, combining this with the classical relation given by Eqs. (28)/(72), and 'quantizing' the latter by promoting the classical potential to an operator Φ →Φ, allows us to obtain an expression for the standard deviation of the space-time line-element operatorŝ which takes us beyond canonical QM. Thereafter, the geometric nature ofΦ is made explicit -via its relation toŝ -and the r appearing in Eq. (73) cannot be identified with the flat-space interval corresponding to the unperturbed line-element, i.e. r = s(t) = ct. Nonetheless, it is interesting to note that such a procedure yields results analogous to Eq. (59), which is (explicitly) based on the physical picture illustrated in Fig. 3. At the very least, we may say that this picture does not contradict the results of canonical QM, but allows us to reinterpret them in terms of a Károlyházy-type 'hazy' space-time.

B. Physical basis of the DE-UP
For later convenience, we now define the first and second de Sitter length-scales as together with the associated mass-scales where unprimed quantities are referred to as 'first' and primed ones as 'second', respectively. (In [57,58], these were referred to as the Wesson length/mass-scales).
As discussed in Sec. II A, l dS is the distance to the asymptotic de Sitter horizon, which is of the order of the present day radius of the Universe, l dS r U 10 28 cm. This, in turn, is a manifestation of the so-called 'coincidence problem', which refers to the fact that the current epoch marks the transition between deccelerating and accelerating phases of universal expansion [55]. For τ τ 0 (excluding any short-lived inflationary phase in the very early Universe), Ω r (τ ) + Ω M (τ ) Ω Λ (τ ) and the gravitational attraction of matter and radiation dominated over the repulsive effect of dark energy. At the present epoch, τ τ 0 , we have Ω M = 0.31 Ω Λ = 0.69, and a phase of late-time accelerated expansion has begun. From the Friedmann equations (9)-(10), we see that, for τ → ∞, the dilution of ρ m relative to ρ Λ implies H =ȧ/a → c/l dS , yielding a(τ ) a 0 exp (cτ /l dS ) , (a 0 = 1) .
Hence, any Universe in which dark energy exists in the form of a cosmological constant undergoes exponential expansion as τ → ∞, since Ω Λ → 1. Furthermore, the transition point between deccelerating and accelerating phases of expansion occurs when Ω M Ω Λ ⇐⇒ τ τ 0 l dS /c, r U l dS .
In our Universe, we live at precisely this point of transition, beyond which the FLRW metric becomes approximately equal to the de Sitter metric. In standard spherical polar coordinates, this takes the form where dΣ 2 is given by Eq. (8) with Gaussian curvature k = 0. However, introducing static coordinates, defined by the transformations this may be rewritten as where dΩ 2 = dθ 2 + r 2 sin 2 θ dφ 2 . In this coordinate system, the existence of a Universal horizon at r = l dS , for all time, is made explicit. (The interested reader is referred to [73] for further discussion of this point.) In the presence of a perturbation induced by a point mass m, the local late-time metric tends, instead, to the Schwarzschild-de Sitter solution, which is obtained by inserting an additional −2Gm/c 2 r term into the non-trivial metric components in Eq. (80). Identifying Φ/c 2 = −Gm/c 2 − r 2 /2l 2 dS , and using the fact that (1 − 2Φ/c 2 ) −1 1 + 2Φ/c 2 for small m, we recover the late-time limit implied by Eqs. (17) and (21) for spherically symmetric perturbations. Though, technically, the perturbation induced by m shifts the outer (cosmic) horizon slightly, relative to its position in unperturbed (pure) de Sitter space, the effect is negligible for all practical purposes.
Let us now consider the mass-scales (76). E dS = m dS c 2 represents the intrinsic mass-energy of a 'particle' whose wavelength is of the order of the de Sitter horizon. However, as shown in [58], a particle with rest mass m dS and Compton wavelength l dS would be unstable, having insufficient self-gravity to overcome the effects of dark energy repulsion. Hence, E dS = m dS c 2 = hc/l dS represents the energy of a minimum-energy photon, whose wavelength is equal to its maximum possible value, l dS . By contrast, E dS = m dS c 2 represents the total mass-energy contained in the dark energy field, within the de Sitter horizon (l dS ) of pure de Sitter space (Ω Λ = 1). Since, at the present epoch, Ω Λ = 0. 69 1 and r U l dS , this is of the order of the present day mass-energy of the Universe [58]. Returning to the length-scales (75), we note that l dS is sub-Planckian, so that its physical meaning is unclear, though we include it in our definitions for the sake of formal completeness. Finally, we note that the primed and unprimed scales are related via Hence, since the particle's communication with the outside world is effectively confined within the de Sitter radius -that is, within the region r ∈ [0, l dS ) -the minimum value of the gravitational uncertainty, induced at a given point r from its CoM, is ∆x grav (r, m) Gm c 2 r l dS .
To this we must add the canonical uncertainty due to the gradual diffusion of the wave function, predicted by the canonical (non-gravitational) theory. We here assume that the respective uncertainties are additive, which is consistent with the perturbative approach to the gravitational sector, considered in Sec. II. We then have ∆x total (∆v, r, m) = ∆x canon. (∆v, r, m) + ∆x grav (r, m) ≥ ∆x(∆v) + ∆x recoil (∆v, r, m) + ∆x grav (r, m) ≥ (∆x canon. ) min (r, m) + ∆x grav (r, m) .
Instead of using the order of magnitude estimates for ∆x recoil and ∆x grav , obtained in Secs. II D and III A-III B, together with an order of magnitude inequality ' ', we assume that Eq. (83) holds exactly when these quantities are defined precisely, up to appropriate numerical factors. Hence, we introduce two new parameters, α , β > 0, which are assumed to be not hierarchically larger than unity. Equation (83) may then be rewritten as where α , β ∼ O(1). However, note that, for this reason, they do not count as free parameters of the model. If the value of either constant were permitted to be hierarchically larger (or smaller) than unity, this would, in effect, alter the existing mass/length-scales present in the theory, indicating new physics. The physical basis of Eq.
and hence From here on, we refer to Eq.
It is straightforward to show that minimizing the DE-UP-1 with respect r, followed by m or ∆v, or with respect to m, followed by ∆v or r, yields the same final result (90). Viewed as a function of all three variables, ∆x total (∆v, r, m) has a unique minimum.
After completely minimizing ∆x total (∆v, r, m) to obtain the DE-UP-3 (90), an interesting critical mass-scale is obtained by setting the recoil velocity of the particle equal to the speed of light, The unique properties of this mass, including its relevance for holography in an asymptotically de Sitter Universe, were considered in [74]. In addition, we note that all expressions, Eqs. (84)- (90), are invariant under simultaneous re-scalings of the form where α Q > 0 is a positive real parameter, which does not depend on any of the three variables ∆v, m, or r.
As we shall see in Sec. III D, α Q may depend at most on the charge Q of the probe particle. Equation (91) then becomes In summary, the first two terms in Eq. (84), ∆x and ∆x recoil , give the canonical quantum uncertainty inherent in the measurement of a distance r. This distance is measured by means of a force-mediating boson emitted from a 'probe' particle of mass m, whose CoM is initially located at r = 0, and its subsequent absorption by a 'detector' at r > 0. For r ∈ (0, λ C ], the detector is simply the particle itself, and the boson remains virtual. This uncertainty is equivalent to the canonical uncertainty in the position of the particle, as viewed by an observer situated at r. For charged particles, the relevant boson is a photon but, for neutral particles, it may correspond instead to one of the weak force mediators (the W ± or Z 0 bosons), or perhaps even a graviton in the case of dark matter particles. We note that, in the canonical (nongravitational) theory, the distance r and the time taken to perform the measurement t are related via r = ct.
The third term ∆x grav represents the gravitational uncertainty at r due to the 'haziness' of the underlying space-time metric, induced by the presence of the particle. This is equivalent to the irremovable uncertainty inherent in a measurement of the horizon distance r H (τ ). The measurement is completed by means of real or virtual boson exchange between the probe particle and a 'detector' at r > 0, and between r H (τ ) and r. As with the canonical uncertainty, for r ∈ (0, λ C ], the detector is simply the particle itself. In the non-relativistic picture, the space-time haziness is related to the haziness of the Newtonian potential, which exists in a superposition of states (71). We note that, once gravitational effects are taken into account, the simple relationship between the time taken to perform the measurement t and the coordinate distance r no longer holds, r = ct.
Together, all three terms give the total uncertainty, incorporating both the uncertainty in the space-time metric -including the effects of dark energy in the form of a cosmological constant Λ -and the canonical uncertainty in the position of the particle's CoM.
Finally, we also note that, using ∆p = m∆v, the canonical recoil term can be rewritten as ∆x recoil = α ∆pr/(mc). Hence, in the limit r → r S (m), Salecker and Wigner's MLUR reduces to the string-inspired GUP [75][76][77][78]. In this, the term proportional to ∆p may be interpreted as the uncertainty induced by the gravitational interaction between the probe particle and the mediating boson (typically a photon), as shown by Adler et al [31,32]. Though we may choose to include an additional term of this form in Eq. (84), we note that, for fundamental particles, r S < l Pl < r, so that it is automatically subdominant to ∆x recoil . We therefore choose to neglect it when applying the DE-UP to fundamental particles. Nonetheless, the existence of an Alder-type term may be relevant if we wish to apply the DE-UP to black holes. This possibility is discussed in the Conclusions, Sec. V, though its explicit application is left to a future work.

C. Basic properties of the DE-UP
We now investigate the basic properties of the DE-UP. Since l Pl is expected to form a fundamental lower bound on the resolvability of all physically measurable lengthscales [34,[70][71][72], we start by imposing the conditions (∆x canon. ) min , ∆x grav , r ≥ l Pl . As we shall see, imposing all three constraints gives rise to a fundamental lower bound on the mass of a system obeying Eq. (86). Furthermore, this bound may be derived independently by combining minimum-density requirements, obtained from the generalized Buchdahl inequalities for a spherically symmetric system in the presence of dark energy (Λ > 0) [79][80][81][82], with the simple requirement of the existence of a Compton wavelength [58].
Hence, beginning with the independently derived result, we see that both the canonical and gravitational terms in the DE-UP-2 (86), together with the probe distance r, remain super-Planckian under physically reasonable conditions. More generally, imposing (∆x canon. ) min , ∆x grav ≥ l Pl places constraints on the ratio r/m, or, equivalently, on the range of validity of r as a function of m and, thus, on the range of validity of Eq. (86).
Since we require ∆x grav ≥ l Pl , let us paramaterize it such that giving Likewise, setting (∆x canon. ) min ≥ l Pl so that gives For later convenience, we now define the ratio and, comparing Eqs. (95) and (97), we have To within numerical factors of order unity, N equals the number of Planck-sized bits on the present day boundary of the observable Universe or, equivalently, the number of cells with volume ∼ (∆x total ) 3 min in the present day bulk [57]. This point is discussed in detail in Sec. III E.
Let us also require r ≥ l Pl , setting Combining this with Eqs. (95) and (97) yields and respectively, which themselves combine to give We now define a new mass-scale, It is straightforward to demonstrate that m Λ is the minimum mass of a stable, spherically symmetric, gravitating, charge-neutral and quantum mechanical object. This result was first obtained in [58], though we briefly review its derivation for the sake of clarity.
In [81], it was shown that the density of a stable, spherically symmetric, gravitating, charge-neutral and classical compact object must satisfy the inequality where ρ Λ is the dark energy density given by Eq. (19). Though the proof of this statement, which follows directly from the generalised Buchdahl inequalities [79][80][81][82], is rather complicated, its physical meaning is intuitively obvious: compact objects with energy densities significantly lower than the vacuum density have insufficient self-gravity to overcome the repulsive effect of dark energy. For bodies of fixed mass m, classical radius R and initial density ρ < ρ min , the spatial expansion caused by Λ > 0, which acts as repulsive force, causes R to expand indefinitely and the object is unstable. For a quantum mechanical object, whose mass m is localized within a sphere of radius R = λ C , we then have Since c 1 , c 2 , c 3 ≥ 1 by construction, we may then identify For later convenience, we define which is simply the reduced Compton wavelength associated with the minimum mass m Λ . Thus, requiring both the individual components of the DE-UP-2 (86) and the probe length r to be super-Planckian ensures its consistency with both general relativistic and quantum mechanical constraints. These, in turn, allow us to fix the relation between the parameters α and β on purely theoretical grounds. However, we must remember that, in reality, a length-scale of order l Pl (up to numerical factors of order unity) may be the true fundamental cut-off for resolvable length-scales in nature, so that this relation must be taken as tentative and some ambiguity still remains.
Equivalently, we see that, beginning with the result m ≥ m Λ and reversing our previous logic, the existence of a minimum stable mass for self-gravitating quantum mechanical objects ensures that all three length-scales (∆x canon. ) min , ∆x grav and r, appearing in the DE-UP-2 (86), remain super-Planckian under appropriate conditions. We now investigate these conditions for masses in the range m Λ ≤ m ≤ 2 −1/2 m Pl , which corresponds to the fundamental particle regime.
Rearranging Eq. (99) and imposing c 2 ≥ 1 yields while imposing c 1 ≥ 1 gives Substituting (110) into (96) then gives l Pl ≤ (∆x canon. ) min (r, m) ≤ 2α β l Pl l dS , (111) and where the upper bound is equivalent to the condition ∆x grav (r, m) ≥ l Pl . Next, we impose the following condition, stemming from Eq. (95) with c 1 ≥ 1: Hence, setting allows us to recover the standard constraint which defines the fundamental particle regime. To treat black hole states, we must instead impose together with Eq. (114), giving However, the possibility of applying the DE-UP to black hole physics are considered in discussed in Sec. V and we here confine our attention to fundamental particles.
Having fixed the values of the parameters α and β via purely theoretical considerations, Eqs. (84)-(90) can be rewritten as (∆x canon. ) min = 2 −1/4 λ C r , (∆x canon. ) min = 2 1/3 (l 2 Pl l dS ) 1/3 , and (∆x total ) min = 27 4 We note that Eqs. (119), and hence the first term in Eq. (120), deviate slightly from the canonical result obtained by Salecker and Wigner (52), and by Ng and van Dam using more rigorous methods. Equations (91) and (93) then become and Hence, for fundamental particles, the ranges of m and r are restricted such that For the limiting mass scales m Λ and m Pl / √ 2, and the critical mass scale m crit , we have These mass-scales also have interesting gravitational properties. To within numerical factors of order unity, the smallest possible mass m Λ is the the unique mass scale satisfying the equation In other words, it is the unique mass-scale whose quantum mechanical (Compton) radius is equal to its classical gravitational (turn-around) radius in the presence of dark energy. This gives an alternative interpretation of the stability condition m ≥ m Λ -for smaller masses, the gravitational turn-around radius lies within the Compton wavelength of the particle. Considering the ranges of r for which the canonical quantum uncertainty is greater or less than the gravitational uncertainty in the DE-UP-2 (86), we have Setting r eq equal to the present day turn-around radius of an object of mass m (and again neglecting numerical factors of order unity), yields For the critical mass m crit , we have r eq r min λ C (∆x canon. ) min ∆x grav (∆x total ) min (l 2 where we recall that r min is the probe distance that minimizes the total uncertainty, yielding Eq. (90), and r grav (τ 0 ) (l 4 Pl l 5 dS ) 1/9 .
In general, we note that, when r is approximately equal to the present day turn-around radius, we have Beyond this range, the classical gravitational influence of the particle is effectively negligible, in comparison to the repulsive effect of dark energy. In terms of spacetime curvature, for r r grav (τ 0 ), the additional contribution to the total curvature due to m is less than the background value ∼ Λ. However, in order for the quantum gravitational influence of the particle to be considered negligible, it must induce metric fluctuations smaller than the background average, which are believed to be of order ∼ l Pl [70][71][72]. We now consider this scenario in detail.
To begin with, we note that, for (∆x canon. ) min to be super-Planckian at r grav (τ 0 ) requires m m dS = m 2 Pl /m dS , which is clearly satisfied for any physically realizible mass, up to and including the present day mass of the Universe. However, for ∆x grav to be super-Planckian at the turn-around radius requires m m Λ √ m Pl m dS .
This result implies that metric fluctuations of order ∼ l Pl are associated with pure (empty) de Sitter space, since m Λ may also be interpreted as the mass of an effective dark energy 'particle' [58]. It therefore follows that, for any mass larger than m Λ , the quantum gravitational influence of the particle at its turn-around radius will be non-negligible, in comparison to the magnitude of the background metric fluctuations, even if its classical gravitational influence may be ignored. This is an important point, and may be relevant to future experimental attempts to distinguish between classical and quantum gravitational phenomenology predicted by the DE-UP model. Finally, it is straightforward to determine the ranges of ∆v (or equivalently ∆p), r and m for which the three terms in the DE-UP-1 (84) satisfy ∆x ≥ ∆x recoil ≥ ∆x grav , or any other ordering. The results are summarized, for general values of α and β , in Table 1.
For r (l Pl l 2 dS ) 1/3 (m/m Pl ), low-momentum states are given by 1, intermediate-momentum states by 2-3 and high-momentum states by 4. As r → 0, the limits in 1 and 4 tend to zero and infinity, respectively. For r (l Pl l 2 dS ) 1/3 (m/m Pl ), low-momentum states are given by 5, intermediate-momentum states by 6-7 and highmomentum states by 8. As r → ∞, the limits in 5 and 8 tend to zero and infinity, respectively.
Hence, for r (l Pl l 2 dS ) 1/3 (m/m Pl ), ∆x grav may dominate ∆x recoil , but not ∆x, in the low-momentum regime, or ∆x, but not ∆x recoil , in the high-momentum regime. However, it may also dominate both in the intermediatemomentum regime. For r (l Pl l 2 dS ) 1/3 (m/m Pl ), the situation is similar in the 'low-' and 'high-' momentum regimes -though these now correspond to different physical ranges of momentum uncertainty -but is reversed in the intermediate regime, where ∆x grav is subdominant to both ∆x and ∆x recoil .
From the point of view of future experiments, the r (l Pl l 2 dS ) 1/3 (m/m Pl ) regime is more accessible, and we are free to choose the ratio of the probe distance to the mass of the probe particle, r/m, to lie in this range. In this case, the very high-and very low-momentum regimes are where we may hope to observe modifications of canonical quantum dynamics. Nonetheless, the observability of these effects depends, ultimately, on the ratio of ∆x grav , to the remaining (non-negligible) canonical uncertainty term.
When the DE-UP-1 (84) is minimized with respect to ∆v, yielding the DE-UP-2 (86), ∆x ∆x recoil (∆x canon. ) min and the value of ∆p is fixed in terms of r by Eq. (85). Under these conditions, Table 1 Hence, for r (l Pl l 2 dS ) 1/3 (m/m Pl ), ∆x grav is always subdominant to (∆x canon. ) min . That said, the two need not, necessarily, be of comparable magnitude in order for ∆x grav to be detectable. The possibility of experimentally testing the DE-UP-1 (84) using current technology will be addressed in a future publication, but is discussed briefly in Sec. V.
Before concluding this subsection, we note that the minimum mass-scale m Λ = 4.832 × 10 −36 g (104) is compatible with the current upper bound on the average neutrino mass obtained from the Planck mission data is m ν ≤ 0.23 eV = 4.100 × 10 −34 g [42]. According to the arguments presented here, m Λ may be interpreted as the mass of the electron neutrino, which corresponds to the mass of the lightest possible neutral particle in a dark energy Universe with Λ 10 −56 cm −2 .
As shown in [58], m Λ may also be interpreted as the effective mass of a dark energy particle. In this picture, the dark energy field is composed of a 'sea' of quantum particles, each occupying a volume ∼ l 3 Λ . Under these conditions, and if dark energy particles are charge-neutral but fermionic, the usual laws of quantum mechanics imply that they will readily pair-produce. However, this is impossible without a concomitant expansion in space itself. (In short, 'empty' space is, in fact, full of dark energy particles.) Borrowing a term from basic chemistry to describe this state of affairs, we may say that the space is saturated. It is straightforward to see that, if the probability of pair-production remains constant, the scale factor of the Universe will grow exponentially since the number of particles produced in any given volume, per unit time, is proportional to the volume itself. This leads naturally to a de Sitter-type expansion, da/dτ ∝ a, in which the macroscopic dark energy energy density remains constant, in spite of spatial expansion. For particles of mass m Λ , the additional (positive) energy of the newly created rest mass is exactly counterbalanced by the additional (negative) energy of its gravitational field, which may be seen by considering the Komar mass [85].
However, if this picture is correct, we may expect 'empty' three-dimensional space to exhibit granularity on scales ∼ l Λ . For this reason, it is particularly intriguing that recent experiments provide tentative hints of fluctuations in the strength of the gravitational field on scales comparable to l Λ , which is of order ∼ 0.1 mm [86,87]. Though many theoretical models may account for this, including those exhibiting spatial variation of the gravitational constant G, the influence of dark energy particles on sub-millimetre gravitational interactions cannot be discounted a priori.

D. DE-UP as MLUR -application to charged particles
In this subsection, we consider the implications of the DE-UP derived in Sec. III B for charged particles. As we saw in Sec. III C, combining the existence of a classical minimum density, which follows from the generalised Buchdahl inequalities for uncharged particles in the presence of dark energy [79][80][81][82], with the standard expression for the Compton wavelength, gives rise to a minimum mass for compact, stable, gravitating, chargeneutral and quantum mechanical objects. Furthermore, this mass-scale is physically interesting as it is comparable to present day bounds on the mass of the lightest known particle, the electron neutrino [58]. Combining the minimum-mass bound for neutral particles with the DE-UP also yields interesting results, since it implies that both the canonical and gravitational uncertainty terms, (∆x canon. ) min and ∆x grav , as well as the probe distance r, always remain super-Planckian.
Similarly, generalised Buchdahl inequalities exist for charged particles, both in the presence and absence of dark energy [57,83,84]. However, in this case, they fix only the minimum value of the radius-to-mass ratio, R/m, of a stable compact object, where R is the classical radius. Alternatively, they fix the minimum classical radius in terms of m, or vice versa. This bound may again be combined with the existence of a minimum quantum mechanical radius, λ C ∝ 1/m, and with the existence of a minimum total uncertainty given by Eq. (90). The latter implies that the mass of the object may be written in terms of the critical mass, m crit (m 2 Pl m dS ) 1/3 , multiplied by an arbitrary constant α Q , as in Eq. (93).
By combining all three mass bounds -that is, by assuming that a charged particle exists in nature whose total uncertainty minimizes the DE-UP, according to Eq. (90), whose classical radius satisfies the appropriate generalised Buchdahl inequalities [57,83,84], and whose Compton radius is given by the canonical formula -we fix the value of the free parameter α Q in terms of the the physical charge (Q) of the system. This, in turn, allows us to obtain an explicit expression for the mass m in terms of Q and the physical constants {G, c, , Λ}. Setting Q = ±e and evaluating this expression numerically, the mass-scale obtained is comparable to the measured value of the lightest charged particle, the electron [57]. According to our procedure, this may be interpreted as the minimum possible mass for a compact, stable, gravitating, charged and quantum mechanical object, which also obeys the DE-UP proposed in Sec. III B.
We proceed as follows. The generalized Buchdahl inequality for a charged compact object in the presence of a positive cosmological constant is [83] 2Gm Hence, for R 2 Λ 1, the effect of dark energy is subdominant to electrostatic repulsion and Eq. (139) reduces to 2Gm This expression can be Taylor expanded to give 2Gm so that, to leading order, we have In this limit (and to within numerical factors of order unity), we recover the standard expression for the classical radius of a 'particle' with mass m and charge Q, i.e. the radius at which the electrostatic potential energy associated with the object is equal to its rest energy, mc 2 .
In special relativity, this is roughly the radius the object would have if its mass were due only to electrostatic potential energy. Nevertheless, Eq. (142), which was originally obtained in [84], is a fully general-relativistic result. The fact the the standard formula for the classical radius of a charged particle is recovered via the Taylor expansion (141) simply reflects the fact that Eqs. (139)-(140) remain valid, even in the weak gravity limit.
Next, we note that a natural way to define the quantum gravitational regime for a fundamental particle is to require its positional uncertainty, due to combined canonical and quantum gravitational effects, to be greater than or equal to its classical radius, ∆x total = (∆x canon. ) min + ∆x grav ≥ R. This is essentially the inverse of the requirement for classicality, that the macroscopic radius of an object be larger than its total positional uncertainty. Thus, the conditions correspond to a regime in which the particle behaves 'quantum-gravitationally', but in which specific quantum gravitational effects are subdominant to the standard Compton uncertainty.
Assuming that the total uncertainty takes its minimum possible value, given by the DE-UP-3 (90), we may then set where γ ≤ 1, in this regime. Likewise, we may set where ξ ≥ 1, if we expect the object to display no classical behaviour. Clearly, with equality holding if and only if γ = ξ = 1. For convenience, we now rewrite the three independent expressions we have obtained for m throughout the preceding sections of this work, namely where q Pl = √ c is the Planck charge. Equations (147a) and (147b) are simply Eqs. (93) and (144) restated. Equation (147c) corresponds to saturating the bound in Eq. (142) by assuming that R = (∆x total ) min /ξ represents the value of the classical radius that minimizes the ratio R/m, for a sphere of mass m and charge Q. (For the sake of generality, we have retained the 3/4 numerical factor in Eq. (142) but kept the numerical constants α and β as unfixed parameters for now.) Thus, m in Eqs. (147a)-(147b) is the mass of the body for which the total uncertainty of the object, given by the DE-UP, is minimized for (∆v) max = α −1 Q c, whereas the m in Eq. (147c) is the mass of a body for which the classical bound (142) is saturated. As shown in [57], this is also the radius at which the classical gravitational energy is minimized. We proceed by assuming the equivalence of the two masses, which is equivalent to assuming that the particle saturates all available bounds simultaneously.
The resulting model has much in common with Dirac's extensive model of the electron [88], which was intended to remove singularities from the electric and gravitational fields of charged particles, except that, here, the classical electron is considered as a three-dimensional fluid sphere, rather than a two-dimensional shell. Nonetheless, the relevant Buchdahl bounds can be re-formulated in terms of two-dimensional (surface) quantities [57].
By equating the three expressions for m in Eqs. (147a)-(147c), we may fix the relations between the three unknowns γ, ξ and α Q , explicitly. For our purposes, the key point is that, for ξ ∼ O(1) (i.e. when (∆x total ) min R, its minimum possible value), we have γ α Q Q 2 /q 2 Pl . Equations (147b) and (147c) immediately imply or, equivalently, This gives an interesting (and self-consistent) interpretation of the Planck charge q Pl as the maximum possible charge of a stable, gravitating, quantum mechanical object, obeying the DE-UP. The bound (149) may also be obtained in a more direct way by combining the general relativistic result (142) with canonical quantum theory. Rewriting this as Q 2 ≤ (4/3)q 2 Pl Rmc 2 and taking the limit R → λ C yields the same result.
For the sake of concreteness, we now set yielding and choose the values α = 1/2 √ 2 and β = √ 2 obtained previously in (114), so that and We then have Though the precise numerical factors chosen here are to some degree arbitrary, we see that, for ξ ∼ O(1), the following, general, order of magnitude relations hold, where α Q is given by Eq. (151). Keeping in mind the alternative measurement procedure outlined in Sec. III B, the physical picture we obtain is as follows. A particle of mass m and charge Q 'measures' the distance to its outermost horizon, the de Sitter radius, by means of a two-stage photon exchange. In the first stage, photons (either real or virtual) are exchanged between the particle CoM and a 'detector' at r. The 'detector' simultaneously (or near simultaneously) receives real or virtual photons from the de Sitter horizon. The minimum total uncertainty in the position of the particle is also the minimum uncertainty in the measurement of l dS .
However, as discussed in Sec. III B (and at length in the Appendix), for r < λ C , the 'detector' is simply the particle itself and the first part of the 'measurement' corresponds to a self-interaction. What the relations above show is that the total uncertainty given by the DE-UP-1 (84) obtains its minimum possible value, given by the DE-UP-3 (90), when the charge-squared to mass ratio of the particle Q 2 /m, and the corresponding self-interaction distance r min , are fixed according to Eq. (155). Under these circumstances, the order of magnitude values of R, (∆x total ) min and λ C are also fixed, yielding a strict hierarchy of length-scales associated with m. These are related via the parameter α Q = Q 2 /q 2 Pl according to Eq. (156).
That the minimum uncertainty in the position of the particle is larger than the probe distance r need not concern us, since r min may be associated with the energy scale of the self-interaction via the usual Compton formula, giving E max c/r min (q 4 Pl /Q 2 )/λ C as a natural UV 'cut-off' in the DE-UP model. Though not strictly a cut-off, attempting to probe self-gravitating particles on scales r < r min (E > E max ) is self-defeating, since this only increases ∆x total .
Thus, in this picture, a particle that interacts with its environment (including self-interactions), over the range r min ≤ r ≤ l dS , naturally acquires a charge-squared to mass ratio that satisfies the bound This is obtained simply by rewriting the expression for m Eq. (155) and reinserting the directional inequality originally present in Eq. (142). Thus, it is straightforward to see that, to within numerical factors of order unity, saturating the bound (157) is equivalent to setting Q 2 = e 2 , which yields the correct order of magnitude value of the electron mass, i.e. m = α e (m 2 Pl m dS ) 1/3 = 7.332 × 10 −28 g m e = 9.109 × 10 −28 g , where α e = e 2 /q 2 Pl is the usual fine structure constant. Alternatively, Eq. (157) which close to the best-fit value obtained from current cosmological observations [41,42]. The result (159) was previously obtained by Harko and Boehmer in [82], in which it was expressed in the form Λ l 4 Pl /r 6 e , where r e = e 2 /(m e c 2 ) is the classical electron radius, and justified on the basis of a 'Small Number Hypothesis' (SNH). By analogy with Dirac's Large Number Hypothesis (LNH), which posits that the numerical equality between two very large quantities with a very similar physical meaning cannot be a simple coincidence [89][90][91][92], Harko and Boehmer proposed the same for small numbers, though we note that the reciprocal of a large number is a small number, so that the two hypotheses may, in fact, be considered equivalent. (For contemporary viewpoints on the LNH and current status reports, see [93][94][95]. ) We stress, however, that in this work, the identification (159) is not based on numerical coincidence. Rather, our requirement that the total uncertainty ∆x total , incorporating canonical quantum and gravitational effects according to the DE-UP, be minimized for a stable, compact, charged, gravitating and quantum mechanical object realised in nature, leads inevitably to Eq. (159).
Remarkably, an algebraic formula for Λ, having the same general form as Eq. (159), namely where m is fundamental mass-scale found in atomic physics, was originally proposed by Zel'dovich in 1968 [96]. The origins of this proposal go back to Dirac's formulation of the LNH in 1937, in which he noted the approximate order of magnitude equivalence between several large dimensionless numbers obtained from atomic physics and cosmology [89]. These included the ratio of the present day radius of the Universe r U to the classical electron radius r e and the ratio of the electric and gravitational forces between an electron and a proton, namely r U r e 10 40 , e 2 Gm e m p 10 39 , where m p = 1.673 × 10 −24 g is the proton mass. Assuming that this equivalence was not coincidental, he for-mulated the Large Number Hypothesis (LNH), which required the existence of a time-varying gravitational constant, G(t) ∼ 1/t, under the assumption that Λ = 0 [89,91,92].
In 1968, Zeldovich noted the same (approximate) equivalence between the ratio of r U and the Compton wavelength of the proton, λ p ≡ h/(m p c), and between λ p and the proton's Schwarzschild radius, r S (m p ). In addition, he noted that, if Λ = 0 and r U ∼ 1/ √ Λ (contrary to Dirac's original assumptions), then [96] However, here, it is important to note that the numerical equivalence in Eq. (162) holds only if λ p denotes the true Compton wavelength, defined with respect to Planck's constant h, not the reduced Compton wavelength, defined with respect to . For λ p ≡ /(m p c), we obtain Eq. (160) with m = m p , yielding Λ m 6 p G 2 / 4 10 −53 cm −2 , since (2π) 4 1559. Clearly, the latter estimate is incompatible with current observational bounds on the value of Λ [41,42]. (See [97] for further discussion of this point.) However, Zel'dovich's observation that if a positive cosmological constant (and hence a de Sitter radius) exist in nature, the physics of sub-atomic particles may be profoundly affected, remains valid. In particular, exchanging m p → m e /α e in the formula (160) yields the upper bound given by Eq. (159), which is compatible with the current experimental value of Λ.
Finally, we note that, if the identification (159) results from fundamental physical considerations (as claimed here) and is not simply a numerical coincidence, it is all the more remarkable since it not only implies a connection between cosmological and atomic physics,à la Dirac and Zel'dovich, but, perhaps even more surprisingly, an intimate connection between the very essence of 'dark' and 'light' physics (i.e., Λ and e) [97]. In fact, several models incorporating non-minimal couplings between dark energy and the electromagnetic sector have already been proposed in the literature, as solutions to problems in contemporary cosmology [98][99][100][101][102]. The cosmological implications of the Λ ∝ α −6 e model, based on Eq. (159), and its various motivations [57,[103][104][105], were investigated in [106]. An alternative form of MLUR, also incorporating effects of dark energy/the de Sitter radius (though not based on the arguments presented in Sec. III B), was given in [107]. The possible relation of (generic) cosmological horizons with the GUP were also considered in [108].

E. Holography
It is straightforward to see that, for any particle which minimizes the total uncertainty given by the DE-UP according to Eq. (90), a holographic relation holds between the bulk and the boundary of the Universe. Specifically, so that the number of Planck sized 'bits' on the de Sitter boundary is equal to the number of minimum-volume 'cells', V cell (∆x total ) 3 min , in the bulk [57]. It is interesting to note that (∆x total ) min may also be regarded as the classical radius of a 'particle' with both minimum energy, E dS = m dS c 2 , and minimum energy density, As shown in [58], and discussed in Sec. III B, a massive particle with rest energy E dS would be unstable due to the effects of dark energy. However, E dS may correspond to the energy of a photon with maximum wavelength, λ l dS . Thus, (∆x total ) min may also be interpreted as the classical radius of a localized, minimum-energy photon. A space-filling 'sea' of such photons would have the same energy density as the dark energy field [58].
In addition, we may consider a maximum-mass, maximum-density state, for which ρ ρ Pl and the total energy is E dS m dS c 2 . The classical radius thereby obtained corresponds to the smallest possible volume within which the total mass of the present day horizon may be confined, without exceeding ρ Pl . We then have [57] Thus, the length-scale (∆x total ) min (l 2 Pl l dS ) 1/3 corresponds to at least three physically interesting scenarios in the context of the DE-UP model. It may be interpreted as (i) the maximum classical radius of a minimum-energy, minimum-density 'particle', (ii) the minimum classical radius of a maximum-energy, maximum-density 'particle', and (iii) the classical radius/minimum total uncertainty of the electron, for cosmic epochs greater than or equal to the present day, τ τ 0 . All three interpretations satisfy the general holographic relation, Eq. (163), which also remains valid for earlier epochs under the substitution l dS → r H (τ ).
Furthermore, we note that, if the probability of a single cell of space 'pair-producing' within a time interval ∆τ = t Pl = l Pl /c, due to the production of dark energy particles, is given by where V 0 denotes the initial volume, this leads naturally to a de Sitter-type expansion, modeled by the differential equation or, equivalently [85]. The production of a single dark energy particle then requires the production of n cell = V Λ /V cell l 3 Λ /(l 2 Pl l ds ) = N 1/4 cells of space which, in turn, implies that the probability of a dark energy particle pairproducing within ∆τ = t Pl is given by Since there are n DE l 3 dS /l 3 Λ = N 3/4 dark energy particles within the de Sitter horizon, this implies that one dark energy particle is produced somewhere in the observable Universe during every Planck-time interval. Remarkably, this rate of pair-production is capable of giving rise to the accelerated expansion of the Universe observed at the current epoch.
In this model, the observed vacuum energy is really the energy associated with the dark energy field: its fundamental dynamics remain unknown, but are assumed to be associated with the mass-scale m Λ , and excitations of the vacuum state correspond to the production of chargeneutral particles with this mass. Thus, λ C (m Λ ) = l Λ provides a natural a cut-off for the field modes -with higherenergy excitations yielding pair-production of dark energy particles throughout space -so that The precise dynamics, or 'true' nature of the dark energy field, are essentially unobservable at the current epoch as the field remains 'trapped' in a Hagedorntype phase in which any increase in kinetic energy, even that caused by random collisions between neighbouring dark energy particles due to quantum uncertainty, results in pair-production rather than an increase in temperature/kinetic energy. (The interested reader is referred to [85] for a more in-depth discussion of this point.) The temperature associated with the field is therefore constant, on large scales, and is comparable to the present day temperature of the CMB, Here, the factor of 8π is included by analogy with the expression for the Hawking temperature, Pl /m Λ again denotes the dual mass, which is equal to the total mass-energy contained in the dark energy field within the de Sitter horizon.
Though this may seem like another 'miraculous' coincidence, in the dark energy model implied by the DE-UP it is simply a restatement of the standard coincidence problem of cosmology, whereby the Universe begins a phase of accelerated expansion at the present epoch, when r U l dS and Ω M Ω Λ and, hence, T CMB T Λ . The coincidence remains: why do we live at precisely this epoch? However, no new coincidences are required, in order to explain Eq. (171) in the context of the DE-UP.

IV. COSMOLOGICAL CONSEQUENCES OF THE DE-UP
At epochs prior to the present day, τ τ 0 , the cosmic horizon is smaller than the de Sitter radius and, strictly, we must substitute l dS → r H (τ ) in Eq. (84) and all subsequent formulae derived from it. In this case, the upper bound on the charge-squared to mass ratio for stable charged particles obeying the DE-UP, Eq. (157), is lowered and drops below the charge-squared to mass ratio of present day electrons. Hence, the DE-UP model strongly suggests time-variation of either, or both, e and m e , assuming that {G, c, , Λ} are genuine universal constants. Similar arguments apply to the minimum mass for neutral particles, which is required to ensure that (∆x canon. ) min , ∆x grav and r each remain super-Planckian.
In the case of a running gravitational coupling [109][110][111][112], variable speed of light [113][114][115][116][117][118][119][120], or dynamical dark energy field [43][44][45][46], the situation is even more complicated, and it may be extremely difficult, in practice, to distinguish variation in e and/or m e , or in the minimum neutral particle mass, from other effects. (See [121][122][123][124][125] for current bounds on varying α e theories, including their effects on cosmic string phenomenology [126,127] and [128][129][130][131][132] for more general models involving temporal and/or spatial variations of multiple physical constants.) However, though a thorough analysis of the cosmological implications of the DE-UP model must be left to a later publication, we regard this prediction as a positive aspect of the model since, in principle, future observations and/or analysis of currently available data may be capable of falsifying it. Alternatively, it may be possible that, despite the two not being in causal contact for τ τ 0 , the existence of an asymptotic de Sitter horizon affects sub-atomic particle dynamics through some non-local mechanism, such as (acausal) entanglement [133], so that Eq. (157) remains valid at all epochs. Nonetheless, based on the analysis presented in Sec. III, the DE-UP model strongly favours time-variation of the ratio e 2 /m e , in line with the bound This corresponds to a minimum holographic cell radius which is similar to the MLUR for an expanding Universe recently suggested by Ng [134], but with the cosmological horizon r H (τ ), given by Eq. (31), in place of the Hubble horizon, H(τ )/c. The equivalent time-variation of the neutral particle limit is where m H (τ ) = /(r H (τ )c) is the Compton mass associated with the horizon distance at time τ . We note that, in general, the problem of how (if at all) local physics is affected by the cosmological expansion remains an important open question [135]. Nonetheless, it is interesting to note the similarity of the minimum particle mass (175) with the (running) dark energy mass-scale predicted by 'agegraphic' [136,137] and holographic [138] dark energy models previously proposed in the literature. A further subtlety of the model stems from the fact that, even if we assume Eqs. (173)- (175) to be true, it is not clear whether the resulting time-dependent quantities should be interpreted as bare values or renormalized values of m e , m νe and e. Since the standard model couplings and masses are energy-dependent due to renormalization group flow and, since a reduction in r H (τ ) is equivalent to increasing the IR cut-off for interactions in the DE-UP model, the relationship between these two (energydependent) factors may be non-trivial. What is clear is that, within the limits of the non-relativistic (i.e., non-Lorentz invariant) theory formulated here, such questions may be very difficult to answer. To satisfactorily address them, we need to go beyond the non-relativistic approximation.
It is therefore interesting to note that the relation (159) was originally found by Nottale [103] using a renormalization group approach. He argued that, like other fundamental 'constants', the cosmological constant is in fact a scale-dependent quantity, obeying an (as yet unknown) renormalization group equation. If so, its present day value may be split into a 'bare' gravitational part plus a scale-dependent part, corresponding to the quantum mechanical vacuum energy, i.e. Λ(r) = Λ G + Λ QM (r). Following Zel'dovich [96], who noted that the bare zeropoint energy is unobservable, he then argued that the observable contribution is given by the gravitational energy of virtual particle-antiparticle pairs, continually created and annihilated in the vacuum, so that where m(r) /(cr) is the effective mass of the particles at scale r. This gives rise to a scale-dependent formula for the vacuum energy density, where ρ Pl = (3/4π)m Pl /l 3 Pl is the Planck density. Assuming a renormalization group equation of the form where γ(ρ vac ) is an unknown function, which (he also assumed) could be expanded to first order for ρ vac ρ Pl , giving γ(ρ vac ) γ 0 + γ 1 ρ vac , yields where ρ 0 = −γ 1 /γ 0 and r 0 is an integration constant. Comparing Eqs. (178)-(179) then gives γ 1 = −6, ρ 0 = ρ Pl (γ 0 = 6/ρ Pl ) and r 0 = l Pl . Hence, although Λ is a manifestly scale-dependent quantity, its low-energy asymptotic value, predicted by Eq. (179), is scaleindependent, in agreement with present day observations [41,42]. Next, he argued that e + e − pair-production represents the main contribution to the vacuum energy at late times (τ τ 0 ), so that the transition between the scaledependence and scale-independence of Λ should be identified with the cross-section for this interaction. Finally, he argued that the latter is equal to the Thomson scattering cross-section, which is approximately equal to the square of the classical electron radius, σ T πr 2 e . This is equal to the e + e − annihilation cross-section evaluated at E m e c 2 . In other words, the Thomson scattering length/classical electron radius r e represents the radius of the annihilation cross-section -which is an energy-dependent quantity r(E) -evaluated at the rest-mass scale m e . Hence, by identifying ρ vac ≡ ρ Λ and r ≡ r e in Eq. (179), he obtained the relation which is equivalent to Eq. (159). This is a remarkable achievement. However, we note that the argument above implicitly assumes that the 'gravitational cut-off', i.e., the UV cut-off in the expression for the gravitational self-energy of a particle pairproduced in the vacuum, Eq. (176), is equal to the average inter-particle distance. A priori, there is no reason why this should be the case. In fact, the most natural assumption, for virtual particles pair-produced in the vacuum, is that the average inter-particle distance is comparable to the Compton wavelength, in this case λ C (m e ). In this, more general scenario, Eqs. (176)-(177) are replaced by respectively, where r min denotes the UV cut-off for the gravitational self-energy. Interestingly, if we set r min α 2 e λ C (m e ), the minimum 'probe' distance for a particle of charge ±e predicted by the DE-UP (155), identifying ρ vac ≡ ρ Λ in Eq. (183) also yields Eq. (181). In this sense, the predictions of the DE-UP model may also be considered as compatible with Nottale's analysis.
Finally, we note that, in an expanding Universe, a vacuum energy of the form coupled with a Nottale-type analysis, analogous to that performed above, gives rise to Eqs. (173)- (175). Alternatively, if for τ τ 0 , this implies and where m H (τ ) = /H(τ ) is the mass associated with the Hubble horizon. As mentioned above, Eq. (187) was previously suggested by Ng [134], and naturally implies a holographic relation between the bulk and the boundary of the Universe.

V. CONCLUSIONS
We have proposed a new minimum length uncertainty relation (MLUR), defined by Eqs. (84)- (90), which incorporates both canonical quantum and gravitational effects in the presence of dark energy, given by a positive cosmological constant Λ > 0. In this model Λ is assumed to be a fundamental constant of nature, giving rise to a constant minimum (vacuum) energy density ρ Λ ∝ Λ at all points in space. The new relation, termed the dark energy uncertainty principle, or DE-UP, is structurally similar to the MLUR proposed by Károlyházy, Eq. (36) [15,16], and reproduced by Ng and van Dam using alternative arguments, Eq. (55) [47,48].
However, while both derivations of Eq. (36)/(55) considered gravitational corrections to canonical (nongravitational) quantum theory, each did so under the assumption that the background space-time was both asymptotically flat and static. Though these assumptions are valid in many physically interesting regimes, it is clear that the discovery of dark energy [38,39] gives rise to a new fundamental length-scale in physics, namely, the de Sitter horizon l dS ∼ 1/ √ Λ, as well as to an associated minimum curvature given by Λ. On cosmological time-scales, it is also clear that the effects of universal expansion on local physics must somehow be taken into account [135]. In the DE-UP, the effects of minimum curvature and of a maximum horizon distance for all observers, including quantum mechanical 'particles', are explicitly accounted for, and the effects of universal expansion are incorporated into the MLUR.
At a technical level, our derivation of the DE-UP closely resembles Ng and van Dam's derivation of Eq. (55). The primary difference is that, whilst they assumed the gravitational uncertainty of a fundamental particle is given by its Schwarzschild radius, we assume it is, instead, given by the irremovable quantum uncertainty inherent in a 'measurement' of the particle's horizon distance, r H (τ ), where τ is the cosmic time. The physical basis for this assumption is straightforward. Since, classically, the distance between the particle and its horizon is exact, any quantum uncertainty inherent in the measurement of r H (τ ) is equivalent to an irremovable uncertainty in the position of the particle itself.
Hence, in order to estimate the uncertainty in a measurement of r H (τ ), including the effects of the particle's gravitational field, we assumed a simple relationship between the classical perturbation of the space-time line element, induced by the presence of the particle (∆s pert ), and the quantum mechanical spread in a superposition of background geometries (∆s), i.e. ∆s pert ∆s, Eq. (59). (See also Fig. 3.) This, in turn, allowed us to demonstrate the equivalence of Károlyházy's procedure for 'resolving' space-time intervals, using quantum mechanical particles as 'probes', and the interaction of a particle with its outermost horizon.
Whilst, clearly, this assumption cannot remain valid for macroscopic objects, and must break down at some critical mass and/or length-scale, it leads to a number of interesting and physically viable predictions based on the DE-UP (84)- (90). We note that the scale(s) at which this assumption becomes invalid may be naturally related to Károlyházy's concept of a coherence cell [15,16], though a detailed investigation of the this possibility lies beyond the scope of the present paper.
Applying the DE-UP to neutral particles, and requiring all potentially observable length-scales to remain super-Planckian, implies the existence of minimum massscale in nature, which can be expressed in terms of the fundamental constants {G, c, , Λ}. Furthermore, this mass-scale can be derived independently by combining classical minimum mass bounds for stable compact objects, in the presence of dark energy, with the simple requirement of the existence of a Compton wavelength [57]. The DE-UP is thus naturally consistent with known gravitational and quantum mechanical effects, as well as with the presumed minimum resolution due to quantum gravitational effects at the Planck scale [70][71][72].
Evaluating the minimum mass for neutral particles numerically, it is of order 10 −3 eV, and is consistent with current experimental bounds on the mass of the electron neutrino obtained from Planck satellite data [42]. This mass-scale may also be interpreted as the effective mass of a dark energy 'particle' [57]. Such a model implies that, though the dark energy density is approximately constant on large scales, it may become granular on length-scales of order 0.1 mm, the associated Compton wavelength. With this in mind, it is particularly intriguing that recent submilimetre tests of Newtonian gravity reveal tentative evidence for periodic variation in the gravitational field strength over precisely this lengthscale [86,87].
Applying the DE-UP to electrically charged particles, we defined the quantum gravity regime as the regime in which the minimum total uncertainty, including both canonical quantum and gravitational contributions, was larger than (or equal to) the classical radius, but smaller than (or equal to) the Compton radius. Evaluating this condition for a particle of charge e, at the current cosmological epoch τ 0 , we obtained the minimum mass of a stable, compact, charged, gravitating and quantum mechanical object, obeying the DE-UP, in terms of the constants {G, c, , Λ, e}. Numerically, this is of order 10 −28 g, which is consistent with the current measured value of the electron mass m e [58].
At all epochs, the DE-UP implies the existence of a holographic relation between the bulk and the boundary of the Universe, in which the number of minimumuncertainty 'cells' in the bulk equals the number of Planck sized 'bits' on the boundary, Eq. (163). However, this strongly implies time-variation of the minimum charge-squared to mass ratio of a stable charged object, under the assumption that {G, c, , Λ} remain constant. Hence, for τ < τ 0 , in which r H (τ ) < l dS ∼ 1/ √ Λ, the ratio e 2 /m e becomes a function of the horizon distance in the DE-UP model. The resulting bound, Eq. (174), closely resembles the MLUR for an expanding Universe recently proposed by Ng [134], but with the particle horizon r H (τ ) in place of the Hubble horizon, H(τ )/c. Similar arguments also imply time-variation of the minimummass bound for neutral particles, according to Eq. (175).
Hence, although the DE-UP proposed herein suffers from a number of drawbacks, including an incomplete picture of the communication between a particle and its cosmological horizon, and a reliance on the assumption of an intimate connection between classical perturbations and space-time superpositions, we believe it it yields sufficiently interesting predictions to be worthy of further study. Therefore, with future high-precision quantum experiments in mind, we have identified two regimes, listed in Table 1, in which the gravitational uncertainty term in the DE-UP dominates at least one of the two positional uncertainty terms obtained from canonical quantum theory. These, together with its prediction of precise timevariation of the ratio e 2 /m e and of the minimum neutral particle mass, may render the model falsifiable using table-top measurements and/or cosmological data in the near future.
Specifically, in regard to future lab-based experiments, we note that, since the MLUR proposed herein is structurally similar to that predicted by the K-model [15,16], and since the assumed relation ∆s pert ∆s must break down at some mass/length-scale, which may naturally be identified with a dark energy-modified version of Károlyházy's concept of a 'coherence cell', precision measurements of decoherence may be crucial in this regard. Although the total decoherence of micro-objects may be unobservable over realistic time-scales (as in the original K-model), partial-decoherence [139][140][141][142] may be probed using existing experimental platforms such as mesoscopic suspended atomic clouds [143,144], opto-mechanical experiments involving trapped micro-spheres or micromirrors [145,146], space-based macroscopic quantum resonators [147,148], and neutrino flavour oscillations in existing detection facilities such as IceCube [149][150][151][152]. Such an analysis lies beyond the scope of the present paper and is left to a future work [63].
Finally, we briefly address the question of the implications (if any) of the DE-UP model for black hole physics. As discussed in Sec. III C, it is by no means clear whether Eqs. (84)-(90) apply to objects with masses m m Pl , or whether a different kind of positional uncertainty applies to black holes. (See for [153][154][155][156][157][158] for recent works on this topic.) Realistically, it seems likely that the identification of small classical perturbations with quantum mechanical spreads, postulated as a physical basis for the DE-UP in Eq. (59), breaks down for macroscopic objects. Furthermore, this idea is consistent with Károlyházy's original concept of a mass-limited coherence cell [15,16], as discussed above. Nonetheless, the fact that the DE-UP provides a natural realisation of the holographic conjecture [49,50] is intriguing, and is it is worth exploring its information theoretic implications in the context of the black hole Information Loss Paradox (ILP) [159][160][161][162][163][164][165][166][167]. If valid for m m Pl , Eqs. (84)- (90) should also have non-trivial implications for the potential observability of black holes in collider experiments, such as at the LHC [168,169].
It is therefore certainly worthwhile to attempt to extend the DE-UP into this region, which may be done naïvely by replacing the rest mass m with the 'dual' ADM mass, m → m ADM ≡ m 2 Pl /m ADM m 2 Pl /(m + m 2 Pl /m). This gives rise to a unified Compton-Schwarzschild line connecting the black hole and particle regimes (see [153][154][155][156][157][158] and [170,171]). Since the DE-UP naturally implements holography in the particle sector, it may be hoped that an extended version maintains it for black holes, which may have implications for the ILP [159][160][161]. In this context, it is also interesting to note that Eq. (159) was previously derived using information-theoretic arguments [105] (see also [97] for a critical appraisal of this work).
In addition, we note that the derivations presented in Sec. III can (in principle) be easily generalized to incorporate modified gravity theories, either by substituting modified classical mass bounds in place of Eqs. (139)-(142) (see, for example [85] and [172,173]) and/or by substituting modified line-elements and metric functions in place of Eqs. (17), (18) and (21). Such modifications may also have non-trivial implications for quantum gravity phenomenology on cosmological scales. The impact of generalized uncertainty on the cosmological evolution equations should also be considered for various combinations of classical modified gravity models/MLURs. (See [174] and references therein for an analysis of GUPinduced modifications of the canonical Friedmann equations.) r min α 2 Q λ C α Q (∆x total ) min (α Q ≤ 1), at which relativistic quantum gravitational effects become important, this too may be regarded as physical, even if the full relativistic theory of quantum gravity required to treat it in detail is lacking.
Hence, if ∆x total (r) represents the total (and irremovable) positional uncertainty of a quantum particle, as seen by an observer located at a distance r from its CoM or, equivalently, the irremovable uncertainty in any measurement of l dS , obtained via the two-stage measurement process outlined in Sec. III B, we may ask the question: is it physically meaningful to consider r < ∆x total (r)?
In general, for a particle of a given mass m, we may solve the inequality r ∆x total (r) to find the critical value r crit , below which this condition holds. Intriguingly, and at first sight somewhat bizarrely, the analysis presented in Secs. III-IV suggests that ∆x total (r), given by Eq. (84), is minimized for r min α Q (∆x total ) min α Q (l 2 Pl l dS ) 1/3 , where α Q = Q 2 /q 2 Pl ≤ 1. In other words, when the uncertainty in the measured value of ∆x total (r) is as small as it can be, it is larger than the 'probe' distance r. To interpret this result correctly, we must reconsider the gedanken experiment proposed by Salecker and Wigner and consider in detail the physical conditions that permit the emission (absorption) of a photon from (by) the 'probe' particle in canonical quantum mechanics. We may then consider the modified conditions induced by the DE-UP.
Classically, a particle of finite extension cannot spontaneously emit another without reducing its internal or kinetic energy [175]. In canonical QM, a non-composite particle does not have internal (i.e. binding) energy, but the wave function of its CoM corresponds to a superposition of position or, equivalently, momentum states. Thus, a given positional uncertainty ∆x corresponds to a momentum uncertainty ∆p, and therefore to an uncertainty in the kinetic energy of order ∆E (∆p) 2 /2m. This allows the spontaneous emission of additional particles -for example, the emission of photons from electrons -without violating conservation of energy or momentum. With this is mind, we now reconsider Salecker and Wigner's thought experiment under two different sets of conditions. In the first, the particle 'tries', and succeeds, in emitting a photon with wavelength λ > λ C . In the second, it 'tries', and fails, to emit a photon with λ < λ C .
Prior to the act of measurement, either by an external detector that absorbs it, or via its reabsorption by the particle after reflection at a mirror placed at a distance r, the photon is in a superposition of states corresponding to ∆λ /∆p / √ 2m∆E. The emission of its wave packet takes a time ∆t ∆λ/c /(c √ 2m∆E). Thus, if ∆E mc 2 , then ∆λ λ C : the photon wave packet is larger than the particle's Compton wavelength and may escape to communicate with the outside world. Specifically, it may traverse a distance 2r, where r > ∆λ > λ C , reflect off a mirror and be reabsorbed, yielding a measurement of r.
Clearly, if r < λ C , the 'mirror' cannot lie outside the wave packet of the massive particle and the act of 'measurement' involves a self-interaction, in which the particle emits a photon and reabsorbs it within a time ∆t r/c. This is inevitable if ∆λ c∆t < λ C , since the wave packet of the photon will not have sufficient spatial extension, or have travelled far enough over the time-interval ∆t, to escape to the outside world. Thus, for ∆λ λ C (∆E mc 2 ), the would-be emitted photon wave packet is 'trapped' within the Compton radius of the particle and the associated photon remains virtual.
Strictly, at this point, the conceptual apparatus of canonical quantum mechanics breaks down and we must switch to the Feynman diagram interactions predicted by QFT. In this picture, the particle emits (and reabsorbs) a virtual photon of wavelength λ over a time-scale t λ/c. The photon is never made real as this would require λ λ C or, equivalently, E mc 2 , which is above the threshold for pair-producing particles of mass m. Nonetheless, in the canonical QM picture this result may be obtained from Salecker and Wigner's bound by setting r ∆λ in Eq. (42), giving (∆x canon. ) min (∆λ) λ C ∆λ ∆λ ⇐⇒ ∆λ λ C . (A-1) To obtain the equivalent bound in the non-canonical theory, represented by Eq. (86), we set (∆x total )(∆λ) λ C r + l 2 Pl l dS λ C r ∆λ . which automatically ensures ∆λ λ C for α Q ≤ 1.
To summarize: In canonical quantum mechanics, photon wave packets with ∆λ λ C remain 'trapped' within the massive particle wave packet, whose minimum extent is given by (∆x canon. ) min √ λ C r λ C . In the non-canonical, dark-energy modified theory, the minimum spatial extent of the CoM wave packet and the Compton wavelength of the particle no longer coincide. Instead, (∆x canon. ) min α Q λ C , which is identified with the classical particle radius, R. Photon wave packets with ∆λ λ C still remain trapped, but only those with ∆λ α 2 Q λ C also minimize the positional uncertainty of the CoM.
This suggests that, in the QFT picture, gravitationally-induced modifications of the Feynman diagram structure should yield an expansion in which the main contribution to the particle's self-energy comes the emission and reabsorption of virtual photons with a specific wavelength, λ α 2 Q λ C , In other words, self-interactions with photons of this wavelength should have maximum amplitude, or 'weight' in the path integral approach.
Hence, we argue that it is physically meaningful to consider length-scales r < (∆x total (r)) min < λ C in dark energy-modified quantum mechanics. Though interactions between the particle and its surroundings are possible only for r λ C ∆x total (r), self-interaction is possible within the contiguous regions α 2 Q λ C r α Q λ C and α Q λ C r λ C . Interestingly, the boundary between the two, r (∆x total ) min α Q λ C , marks the length-scale at which renormalization becomes important for charged particles in QED [69,176], and our naïve picture correctly reproduces a phenomenologically significant length-scale from the relativistic (but nongravitational) quantum theory of charged particles. We may therefore conjecture that, in a more complete theory, including relativistic quantum effects from both dark energy and canonical gravity, the length-scale r min α Q (∆x total ) min α 2 Q λ C should naturally emerge as an effective cut-off, which minimizes the self-interaction energy of charged particles due to the irremovable 'haziness' of the space-time in their vicinity.
Finally, we note that, for electrons, the key lengthscale is r min α 2 e λ e 2.054 × 10 −15 cm, which corresponds to an energy E max m e c 2 /α 2 e 9.596 GeV, well below the 13 TeV maximum operating energy of the Large Hadron Collider (LHC). However, the LHC is a proton-proton collider and the relevant energy-scale for protons is E max m u c 2 /(2α e /3) 2 101 GeV where m u 2.4 MeV/c 2 is the mass of the up quark [85]. Though this is also well within the maximum operating energy of 13 TeV, we must remember that only a fraction of the total beam energy is used in any particular quark-quark collision. Nevertheless, the corresponding length-scale is r min (2α e /3) 2 λ u 1.95 × 10 −16 cm, where λ u = /(m u c), which is close to the smallest distances likely to be probed at LHCb. The possibility of directly testing quantum gravity phenomenology predicted by the DE-UP at present day or, more realistically, next generation colliders is therefore tantalizingly close.