PHYSICAL REVIEW D 102, 103529 (2020)
Dark energy in multifractional spacetimes
† Gianluca Calcagni 1,* and Antonio De Felice2, 1Instituto de Estructura de la Materia, CSIC, Serrano 121, Madrid, 28006 Spain 2Center for Gravitational Physics, Yukawa Institute for Theoretical Physics, Kyoto University, 606-8502, Kyoto, Japan
(Received 7 August 2020; accepted 20 October 2020; published 23 November 2020)
We study the possibility to obtain cosmological late-time acceleration from a geometry changing with the scale, in particular, in the so-called multifractional theories with q-derivatives and with weighted derivatives. In the theory with q-derivatives, the luminosity distance is the same as in general relativity and, therefore, geometry cannot act as dark energy. In the theory with weighted derivatives, geometry alone is able to sustain a late-time acceleration phase without fine tuning, while being compatible with structure- formation and big-bang nucleosynthesis bounds. This suggests to extend the theory, in a natural way, from just small-scale to also large-scale modifications of gravity. Surprisingly, the Hausdorff dimension of spacetime is constrained to be close to the topological dimension 4. After arguing that this finding might not be a numerical coincidence, we conclude that present-day acceleration could be regarded as the effect of a new restoration law for spacetime geometry.
DOI: 10.1103/PhysRevD.102.103529
I. INTRODUCTION an alternative explanation to dark energy. Performing this trick once again would only contribute to pile up phenom- To date, all cosmological observations agree on the fact enological models while we wait for more precise mea- that general relativity, incarnated in the so-called ΛCDM surements of the late-time expansion, for instance from model, gives a valid and accurate description of the history EUCLID [6]. However, instead of proposing a phenomeno- of the universe and of the astrophysical objects within, from logical model of dark energy, here we take a class of early times [1,2] until today [3]. Despite the success reaped gravitational theories, named multifractional spacetimes by Einstein theory, however, there is still room for theoretical speculation. In particular, late-time acceleration [7,8], originally presented as deviations from general and the nature of dark energy are unsolved enigmas that do relativity at the fundamental level and check what they not quite fit in the picture. On one hand, even if cosmic have to say about late-time acceleration. This class of top- acceleration is compatible with a plain cosmological down approaches may be more appealing than ad hoc constant Λ, its origin and small value do not find a models because the requirement they have to satisfy to convincing explanation in general relativity and ordinary recover a four-dimensional geometry at post-microscopic particle physics, for several reasons (see, e.g., [4], chapter 7 scales usually make their theoretical predictions at all scales for an overview on the cosmological constant problem). On rigid and, therefore, more easily falsifiable. This also happens for the macroscopic constraints on the models the other hand, the value of the Hubble parameter today H0 computed from the cosmic microwave background in the considered here, that will select one among several pos- PLANCK Legacy release [2] is incompatible with the one sibilities. The ensuing cosmological scenario will turn out found in local (Cepheids-based) observations [5] by more to be worth investigating because it illuminates an unsus- than three standard deviations, a fact that might be imputed pected relation between late-time acceleration and space- to the underlying assumption of the ΛCDM model. time geometry. Our aim in this work is to add another contender to the In all known theories of quantum gravity, the dimension enormous amount of exotic dark-energy models already of the effective spacetimes originated from quantum proposed in the literature, but with a new twist. It is well geometry changes with the probed scale, a phenomenon known that deviations from general relativity with a called dimensional flow related to the renormalizability of – canonical scalar field can easily generate accelerating quantum gravity at short scales [7,9 12]. Multifractional phases which, with the appropriate motivation, can provide spacetimes implement this feature kinematically in the spacetime measure, which admits a universal parametriza- tion [7,13], and in the dynamics via a model-dependent *[email protected] kinetic operator. Taken as independent theories, there are † [email protected] only four types of multifractional scenarios: with ordinary,
2470-0010=2020=102(10)=103529(17) 103529-1 © 2020 American Physical Society GIANLUCA CALCAGNI and ANTONIO DE FELICE PHYS. REV. D 102, 103529 (2020) weighted, q- or fractional derivatives in the dynamical point of view is that a sensible theory beyond Einstein action. The first three have received much attention [7], should be able to describe cosmic acceleration without while the fourth, which we will ignore from now on, is still introducing dark-energy fields by hand. In the case of under construction [7,14]. Unfortunately, the renormaliz- multifractional spacetimes, this acceleration should come ability of perturbative quantum gravity was shown not to only from geometry [17]. The theory with ordinary improve in the theories with q- and weighted derivatives derivatives is less attractive than the others due to some [7,15], a result which indicated that these theories do not technical issues [7]; for this reason, we will turn to the provide a fundamental microscopic description of Nature. theories with q-derivatives and with weighted derivatives, However, the results of this paper will point out that one of which admit a more rigorous structure. these scenarios (the theory with weighted derivatives) can In particular, we will show that geometry alone in the be a very interesting infrared (IR), rather than (or on top of) theory with q-derivatives is unable to sustain a dark-energy ultraviolet (UV), modification of gravity. It is important to phase, contrary to the more interesting case with weighted stress that this extension of multifractional theories is derivatives. We will place constraints on the parameter admitted by a general theorem governing the form of the space of this theory and find its compatibility with spacetime measure [13] and, therefore, it is not introduced observations only when the Hausdorff dimension of space- ¼ 4 by hand. In fact, this holds for any theory of quantum time is kept at the constant value dH , i.e., the number of gravity admitting dimensional flow in the Hausdorff topological dimensions. We interpret this result as the dimension and a continuum approximation [7,13]. The manifestation of something we might call a restoration main implication of this result is that, even if the theories law, not apparent in previous works. It is as if late-time we will study here are not fundamental in the sense acceleration was the expression of the tendency of the specified above, the feature responsible for late-time Hausdorff spacetime dimension to stay close, or at least get acceleration is expected to be present in other theories back, to 4. The value of 4 is not conserved exactly but it is of quantum gravity with a scale-dependent Hausdorff restored asymptotically. We give an argument why this dimension.1 Therefore, our findings may be interesting might not be just a numerical coincidence, although it is too not only because they complete the study of multifractional weak to establish a full-fledged physical principle yet. – spacetimes and enrich their phenomenology to unsuspected In Secs. II A II D, we briefly introduce some well- directions, but also because they could be kept in mind known cosmological quantities and review multifractional when exploring the cosmology of quantum gravities with theories, quoting their main features on a homogeneous dimensional flow. The very idea to link dark energy to background. The reader interested in an in-depth discussion quantum gravity goes against the grain of effective field can consult [7] and references therein. In Sec. II E,we theory, since it is usually believed that quantum gravity can extend the spacetime measure of the theories to include only have Planck-size corrections, which cannot possibly beyond-general-relativistic correction terms, which will have an impact on the late-time universe. However, dimen- play a crucial role in the interpretation of our numerical sional flow is essentially a nonperturbative phenomenon results. The cosmological equations of the theory with where other elements apart form the UV/IR divide enter the q-derivatives are reviewed in Sec. III A, while in Sec. III B game, such as the discrete scale invariance responsible for we show that geometry alone cannot accelerate the universe the cosmic log-oscillations we will describe in due course. in this theory. This result is exact. The FLRW equations of The starting point for the study of the cosmology of these the theory with weighted derivatives are presented in theories are the Friedmann–Lemaître–Robertson–Walker Sec. IVA, while in Sec. IV B we encode all the effects (FLRW) equations obtained from the full background- of multiscale geometry into an effective dark-energy independent equations of motion [16,17]. While there are component obeying standard Friedmann equations. The numerical analysis and its discussion are presented in some studies about inflation [18,19] and gravitational-wave – propagation [20,21] that constrain or propose possible Secs. IV C IV E, while Sec. V is devoted to conclusions. effects in the cosmic microwave background and in gravitational-wave signals in multifractional spacetimes, II. GENERAL SETUP the problem of dark energy has received less attention, A. Cosmology except in the case with ordinary derivatives. The absence of We work in four topological dimensions (D ¼ 4) and cosmological vacuum solutions in the theory with ordinary with signature ð−; þ; þ; þÞ. The Hubble parameter is derivatives [16] led some to explore scenarios with more or – less exotic dark-energy components [22 29] (see also a_ [30,31] for a thermodynamical approach). However, our H ≔ ; ð1Þ a
1An example is the class of theories based on pre-geometric where a dot denotes a derivative with respect to cosmic time structures with discrete labels, such as group field theory and loop t. The evolution of the universe can be parametrized by the quantum gravity. redshift 1 þ z ¼ a0=a, where a0 ¼ að0Þ¼1 is the scale
103529-2 DARK ENERGY IN MULTIFRACTIONAL SPACETIMES PHYS. REV. D 102, 103529 (2020) factor today. We use a subscript 0 for any quantity a0 dL ¼ r ¼ð1 þ zÞ½τ0 − τðzÞ ð6Þ evaluated today: the Hubble constant H0, the age of the a universe t0, and so on. Z Z t0 1 ¼ð1 þ Þ dt ¼ð1 þ Þ da The universe is assumed to be filled by perfect fluids z z 2 ð Þ a ð Þ Ha with energy-momentum tensor Zt z a z z dz ¼ðρ þ Þ þ ð Þ ¼ð1 þ zÞ : ð7Þ Tμν P uμuν gμνP 2 0 H and equation of state P ¼ wρ. In all models, the content of Observations of standard candles such as type I supernovæ ρ ¼ ρ the universe will be nonrelativistic matter (dust, m, allow one to determine the luminosity distance dL, the local ¼ 0 ¼ 1 3 w ) plus a radiation component (w = ). Both Hubble parameter H0 and, from (4), a constraint on the curvature and the cosmological constant are set to zero, dark-energy barotropic index wDE. K ¼ 0 ¼ Λ, the first from observations [1,2] and the second In multifractional theories, the redshift law for time because we want to get acceleration purely from geometry. intervals and frequencies is the same and (6) does not The latter will give an effective contribution we will dub as change, but the relation between conformal time and the ρ DE, where DE stands for dark energy. dynamics is modified and (7) receives several corrections, For each matter component, we will use the dimension- which we will discuss in due course. less energy-density parameter
ρ ρ 3 2 B. Metric and geometric-harmonic structures Ω ≔ m Ω ≔ DE ρ ≔ H ð Þ m ρ ; DE ρ ; crit 2 ; 3 Let D ¼ 4. Multifractional spacetimes are endowed with crit crit κ two mutually independent structures, a metric one and a where κ2 ¼ 8πG is Newton’s constant. The late-time geometric-harmonic one. The metric structure is given by standard Friedmann equation in general relativity in the the metric gμν, in this work the flat FLRW metric with absence of curvature is components qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 3ð1þ Þ ¼ −1 ¼ ð Þ ¼ 1 2 3 ð Þ ≃ Ω ð1 þ Þ þ Ω ð1 þ Þ wDE ð Þ g00 ;gii a t ;i; ; : 8 H H0 m;0 z DE;0 z ; 4 The geometric-harmonic structure is the integro- for a dark-energy component with constant barotropic differential structure of the theory and is determined by index w . This equation is modified in multifractional DE the choice of spacetime measure spacetimes, as we will see below. To compare the theory with supernovæ data, we will use 4 4 d qðxÞ¼d xvðxÞð9Þ the luminosity distance dL, defined by imposing that the flux F of light reaching an observer is the power L emitted in the action and by the type of derivatives in kinetic terms. by a distant source per unit of area, measured on a sphere of In standard general relativity, v ¼ 1 and the action measure radius d : pffiffiffiffiffi L is d4x (times the jgj volume weight, which is determined by the metric structure). The action measure weight vðxÞ is F ≕ L ð Þ 4π 2 : 5 the easiest to understand. The Hausdorff dimension of dL spacetime is defined as theR scaling of the Euclidean- VðlÞ¼ l 4 ð Þ This is a definition and it is completely model independent. signature volume ball;cube d xv x of a 4-ball or The flux F is measured from Earth, while L is determined 4-cube with linear size l, by the astrophysical properties of the source. Standard hVðlÞi candles such as supernovæ are sources for which L is spacetime ≔ d ln ð Þ dH ; 10 known. From this, one finds dL. In turn, the luminosity d ln l distance can be expressed in terms of the dynamics and, in particular, of the dark-energy equation of state. It is this last where h·i means average over the harmonic structure (see 4 part that depends on the specific cosmological model. below). In ordinary spacetime, the volume scales as l ,so spacetime ¼ 4 In standard general relativity, the luminosity distance a that dH . However, when the Hausdorff dimen- photon traveled from some source at redshift z to Earth sion of spacetime varies with the probed scale and this (z ¼ 0)isa0r, as measured by the observed at t0. Since variation is slow and smooth at large scales or late times, its 2 2 dτ ¼ dr on the light cone, dL is proportional to con- form is given by a unique parametrization [7,13], that we formal time τ0 − τðzÞ. Taking into account the redshift of present here only in the time direction because we will be power L ¼ðenergyÞ=ðtimeÞ ∝ a=ð1=aÞ¼a2 of photons interested in homogeneous solutions on a homogeneous reaching the observer at different times, one gets [32] metric background. The full homogeneous measure weight
103529-3 GIANLUCA CALCAGNI and ANTONIO DE FELICE PHYS. REV. D 102, 103529 (2020)
α −1 vðtÞ¼q_ðtÞð11Þ t vðtÞ ≃ 1 þ FωðtÞ; ð14aÞ t is, in the most general case, an infiniteP superposition of α þ ω qðtÞ¼ γ jt=t j l i l Xþ∞ complex powers of time, l l l , where ˜ FωðtÞ¼A0 þ F ðtÞ; ð14bÞ αl; ωl ∈ R and γl ∈ C are dimensionless constants and n n¼1 tl ∈ R are fundamental time scales of the geometry. However, reality of the measure constrains the parameters γ ω ˜ t t l and l of this sum to combine into a real-valued FnðtÞ¼An cos nωln þBn sin nωln ; ð14cÞ expression, a generalized polynomial deformed by loga- t t rithmic oscillations: in their presence. The positive parameter α is related to the
Xþ∞ α −1 Hausdorff dimension of spacetime in the UV, t ;l vðtÞ¼ FlðtÞ; ð12aÞ 3 l¼0 tl X spacetimeUV≃ α þ α ð Þ dH i; 15 Xþ∞ i¼1 ˜ FlðtÞ¼A0;l þ Fn;lðtÞ; ð12bÞ α n¼1 where i are the fractional exponents inQ the spatial dDxvðx0Þ D−1 v ðxiÞ directions. The full measure is i¼1 i , ˜ t t where each profile v ðxiÞ along the spatial directions has Fn;lðtÞ¼An;l cos nωl ln þ Bn;l sin nωl ln ; i tl tl the same parametric form as Eq. (14) with t replaced by xi α → α → l ð12cÞ and different parameters i, t i, and so on. For a fully isotropic measure, α ¼ αi. In virtually all the where l runs over the number of fundamental scales of literature on the subject, the range of values is chosen as 0 α 1 geometry and 0