PHYSICAL REVIEW D 102, 103529 (2020)

Dark energy in multifractional

† 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- 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 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 and of the astrophysical objects within, from logical model of dark energy, here we take a class of early [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 - 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 , 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 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 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 [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 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 (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

103529-4 DARK ENERGY IN MULTIFRACTIONAL SPACETIMES PHYS. REV. D 102, 103529 (2020) over a log oscillation [34]. Restricting to a homogeneous the two frames and is not a coordinate transformation in the background and denoting as sense of general relativity. Z In the theory with weighted derivatives, the integer frame 2π 1 ν ϕ → ϕ˜ ðϕ Þ hfðtÞi ≔ dνfðeωtÞð18Þ is defined by field transformations μν μν ;v 2π 0 involving v. In the absence of gravity, the electroweak- strong model of quantum interactions in the fractional the average of a log oscillating function, one has frame is mapped into the standard Standard Model on X 2 þ 2 Minkowski spacetime. In the presence of gravity, the h i¼ h 2 i¼ 2 þ An Bn ð Þ dynamics in the fractional frame is very similar to a Fω A0; Fω A0 2 : 19 n>0 scalar-tensor theory where v plays the role of the scalar, but only in a superficial visual analogy at the level of the The case A0 ¼ 0, never explored in the cosmological equations of motion: v is not a field, since it is given by the phenomenology of these theories, corresponds to a zero nondynamical fixed profile (12) and its spatial generaliza- ≫ average: starting from mesoscopic temporal scales t tPl, tion. In the unphysical integer frame, the theory is mapped both the log-oscillatory pattern and the power-law contri- into general relativity plus a purely geometric, v-dependent bution begin to disappear. In the case A0 ¼ 1, the only one contribution. considered in the cosmology literature so far [17,18], the We insist that in these theories the integer and fractional power-law contribution survives longer, up to and beyond frame are not physically equivalent and that we live in the scales t ≳ t. latter, while the integer frame is just a convenient trick to simplify calculations. The multifractional theory with frac- D. Physical frame tional derivatives does not admit a frame mapping, which is The form of the kinetic operators in the action is the reason why it has been studied much less than the other determined by the choice of symmetries and the require- two. In general, having a multiscale geometry selects a ment that, just like the Hausdorff dimension, in quantum preferred frame, at least in multifractional theories. Other multiscale scenarios such as string theory, asymptotically gravity also the spectral dimension dS (the scaling of dispersion relations) changes with the probed scale. safe quantum gravity, nonlocal quantum gravity and group Thus, the system is defined by a nontrivial dimensional field theory (which includes spin foams and loop quantum flow (varying dH and/or dS) and certain action symmetries. gravity) have a higher degree of symmetry (Lorentz and In general, the metric is nonminimally coupled with matter diffeomorphism invariance are preserved or deformed in a fields due to the anomalous (i.e., nonconstant) geometric- controlled way) and do not require a frame choice, once harmonic structure, which leads to complicated equations a low-energy or semi-classical notion of spacetime is of motion where Lorentz invariance is explicitly broken by recovered. the time scale t (and the spatial scale l, which we do not consider here) appearing in the measure weight v. E. Infrared extension of the measure However, the problem is greatly simplified in two of the Equations (13) and (14) are the profiles typically admissible cases, the theory with q-derivatives and the considered in the literature. The constant unit term corre- theory with weighted derivatives. We briefly recall their sponds to the measure at the scales on which general main points [7]. In the theory with q-derivatives, the original system is relativity holds, hence its normalization to 1. The physical picture usually advocated is that multiscale spacetimes and recast into a simpler one which is identical to the Standard ≲ Model of particle physics where all coordinates are the the dynamics therein are anomalous at short scales t t geometric profiles qμðxμÞ. When gravity is turned on, the and reduce to ordinary spacetimes and general relativity at large scales t ≫ t. Therefore, the general-relativistic original system becomes equivalent to general relativity in ¼ 1 q coordinates, plus matter fields. To perform calculations, regime and its measure weight v constitute the terminal one can temporarily forget that the qμ are composite regime of dimensional flow at large scales. coordinates and work in this integer frame until the However, this is only one of the possible applications of construction of physical observables, at which point one the dimensional-flow theorem [13] giving rise to the most must revert to the physical (or fractional) frame where the general profile (12). In principle, ultra-IR terms dominating geometry is multiscale. In this theory, the fractional frame the measures at scales larger than those of the general- is such that clocks and rods do not adapt with the relativistic regime are possible. For example, ignoring log observation scale, while they do in the integer frame. oscillations for the sake of the argument, the measure Consequently, in the fractional frame we can observe weight (13) can be replaced by a three-term profile dimensional flow through its imprint on certain physical α −1 α −1 t t c observables, while in the artificial integer frame the vðtÞ ≃ þ 1 þ ; ð20Þ μ μ μ geometry is constant. The mapping x → q ðx Þ connects t tc

103529-5 GIANLUCA CALCAGNI and ANTONIO DE FELICE PHYS. REV. D 102, 103529 (2020)

0 α 1 ¼ 2 where t is close to Planck time tPl and < < , while 3a. Many harmonics with N . Different theoretical ≫ tc t and choices for the n-dependence of An and Bn indicate that higher modes are rapidly suppressed and that it is α 1 ð Þ ≲ 10 c > : 21 sufficient to consider nmax [19]. We take ¼ 10 nmax . Also, to maximize the effect we take all The subscript “c” stands for cosmological. According to amplitudes constant and equal to one another: A1 ¼ this extended measure, general relativity is only a transient ¼ … ¼ ¼ ¼ … ¼ A2 Anmax and B1 B2 Bnmax . This regime between a deep-UV limit, where microscopic configuration can be regarded as a multiharmonic quantum-gravity effects are important, and an ultra-IR measure where only the first Oð10Þ harmonics regime which deviates from standard Einstein theory at dominate. cosmological scales. In other words, our clocks (and rulers, 3b. Many harmonics with N ¼ e10. in an inhomogeneous setting) not only changed in the past, but they will also change in the future. We will see that III. THEORY WITH q-DERIVATIVES dark energy can be interpreted as a manifestation of this deviation. A. Action and FLRW dynamics Therefore, here we will explore a model where the In the theory with q-derivatives, the dynamical action is correction to general relativity is not a UV term but a “post- the same as in general relativity, except that all the ” 0 α 1 ∼ μ IR one. The UV correction, with < < and t tPl, coordinatesR x are replaced by the multifractional profile μ μ μ μ is totally negligible at late times and can be ignored. The q ðx Þ¼ dx vμðx Þ, where the index μ is not contracted. final expression for the measure is therefore identical to As we commented in Sec. II D, this is not a coordinate (14) but with a new set of parameters with prior (21): transformation because measurement units differ in the

α −1 fractional and integer frame. Physical clocks and rulers, t c vðtÞ ≃ 1 þ FωðtÞ; ð22aÞ used for actual measurements, are defined on the manifold t μ c spanned by the coordinatesQx . In this fractional frame, the D ð Þ¼ D−1 μð μÞ Xþ∞ action measure d q x μ¼0 dq x reflects the basic ˜ FωðtÞ¼A0 þ FnðtÞ; ð22bÞ postulate of the theory, namely, that spacetime geometry n¼1 changes with the scale. On the other hand, in the integer     frame the qμ are regarded as noncomposite coordinates and,

˜ ð Þ ≃ ω t þ ω t ð Þ therefore, the spacetime measure is the standard Lebesgue Fn t An cos n ln Bn sin n ln ; 22c D tc tc measure d q, with no dimensional flow. In the fractional frame, the gravitational action in the 2πα ω ¼ c ;N¼ 2; 3; 4; …: ð22dÞ absence of a cosmological constant is [17] ln N Z 1 pffiffiffiffiffi S ¼ dDxv jgjqR þ S ; ð Þ In this paper, we will consider the following cases in q 2κ2 matter 25 increasing order of difficulty: q μν q (1) Binomial measure without log oscillations: where the Ricci scalar R ≔ g Rμν, the Ricci tensor q ≔ q ρ ˜ Rμν Rμρν and the Riemann tensor A0 ¼ 1; FnðtÞ¼0: ð23Þ 1 1 q ρ q ρ q ρ (2) Binomial measure with log oscillations up to a finite R μσν ≔ ∂σ Γμν − ∂ν Γμσ vσ vν number of harmonics: q τ q ρ q τ q ρ þ Γμν Γστ − Γμσ Γντ;   Xnmax ˜ ρ 1 ρσ 1 1 1 FωðtÞ¼A0 þ F ðtÞ: ð24Þ qΓ ≔ ∂ þ ∂ − ∂ n μν 2 g μgνσ νgμσ σgμν n¼1 vμ vν vσ

are all made with q-derivatives (no summation over μ) ¼ 1 ¼ 2 2a. One harmonic (nmax ) with N [smallest N in (22d)] and A0, A1 and B1 free parameters. In turn, this ∂ 1 ∂ μ μ ¼ μ μ : ð26Þ and the following can be divided into two subcases, ∂q ðx Þ vμðx Þ ∂x A0 ¼ 0 (no zero mode) and A0 ¼ 1. 2b. One harmonic with very large N. We choose the The matter action is constructed according to the same arbitrary value N ¼ e10 to get a suppression factor criterion. 1=10 in ω (small frequency). The Friedmann equations in D ¼ 4 topological dimen- 1 (3) Binomial measure with many harmonics (nmax > ). sions in the presence of a perfect fluid are [17]

103529-6 DARK ENERGY IN MULTIFRACTIONAL SPACETIMES PHYS. REV. D 102, 103529 (2020)

κ2 weighted conformal time dtvðtÞ=a.2 Thus, the luminosity 2 ¼ 2ρ ð Þ H 3 v ; 27 distance is Z Z 2 t0 ð Þ z ä κ 2 v_ dtv t dzv ¼ − v ðρ þ 3PÞþH : ð28Þ dL ¼ð1 þ zÞ ¼ð1 þ zÞ : ð33Þ a 6 v tðzÞ a 0 H

The continuity equation for a perfect fluid is compatible On the other hand, the Friedmann equation with only dust with the Friedmann equations: and no dark-energy component reads qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρ_ þ 3Hðρ þ PÞ¼0: ð29Þ ¼ Ω ð1 þ Þ3 ð Þ H vH0 m;0 z ; 34 Multiscale geometry effects may be implicit in the energy density and pressure. For instance, for a homogeneous so that scalar field Z 1 þ z ¼ z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidz ð Þ 1 1 dL ; 35 _ 2 _ 2 H0 0 Ω ð1 þ Þ3 ρϕ ¼ ϕ þ WðϕÞ;Pϕ ¼ ϕ − WðϕÞ; ð30Þ m;0 z 2v2 2v2 and its continuity equation is modified both in the friction exactly as in the cold dark matter (CDM) model of Einstein and in the potential term: gravity without cosmological constant. The conclusion is   that this expression cannot possibly fit supernovæ data and _ ̈ v _ 2 that the theory with q-derivatives cannot make the late-time ϕ þ 3H − ϕ þ v W ϕ ¼ 0: ð31Þ v ; universe accelerate just from pure geometry.

IV. THEORY WITH WEIGHTED DERIVATIVES B. No dark energy from geometry For the cosmologist, the theory with weighted deriva- In the multifractional theory with q-derivatives, the tives may result more intuitive than the previous one geometry of spacetime is such that it eases the slow-roll because here the line element is the usual of general condition in inflation [17,18] and, similarly, it could relativity and the only change in the expression for the enhance the (usually insufficient) acceleration triggered luminosity distance (7) is in the profile HðzÞ. by a quintessence field, i.e., it could relax the fine tuning in the initial conditions. However, we do not wish to straighten a general-relativistic model with some extra A. Action and FLRW dynamics exotic ingredient. A genuine alternative explanation of Weighted derivatives are ordinary derivatives with mea- dark energy should work without relying on a quintessen- sure-weight factors inserted to the left and to the right: tial component. Therefore, we would like to get late-time acceleration from pure geometry. 1 2 D ≔ ∂ðvβ·Þ; β ¼ : ð36Þ Unfortunately, this is not possible in the theory with vβ D − 2 q-derivatives because geometry effects cancel out in the luminosity distance. The latter receives two corrections, The gravitational action in the fractional (physical) frame 3 one from the definition of conformal time and one from the is [17] dynamics. Distances, areas and volumes all must be Z 1 pffiffiffiffiffi calculated with the nontrivial multiscale measure weight ¼ D j j½R − γð ÞD Dμ Sv 2 d xv g v μv v vðxÞ and, in particular, integration along the time direction 2κ includes a factor vðtÞ. In particular, the luminosity distance − 2ð − 1Þ ð Þ þ ð Þ D U v Smatter; 37 dL is still proportional to the comoving distance (times the same redshift factor as in general relativity), but now the γ R where and U are functions of the weight v, Smatter is the 0 ðxÞ μν latter is −qðrÞ dq . The FLRW line element minimally coupled matter action, R ≔ g Rμν, Rμν ≔ ρ R μρν and 2 μ μ ν ν 2 2 2 ds ¼ gμνdq ðx Þdq ðx Þ¼−dq ðtÞþa dq ðxÞ ¼ −v2ðtÞdt2 þ a2v2ðxÞdx2; ð32Þ 2Note that this time redefinition does not make the line element conformally flat, unless vðxÞ¼1. However, we keep the mis- “ ” ðxÞ ≔ ð 1Þ ð 2Þ ð 3Þ nomer conformal for the sake of an easier comparison with where v v1 x v2 x v3 x is the spatial measure general relativity. factorized in the coordinates, vanishes for light rays, so that 3Here U has been rescaled by a factor of 2ðD − 1Þ with respect the comoving distance is given by the integration of the to [7,17] in order to make the Friedmann equations simpler.

103529-7 GIANLUCA CALCAGNI and ANTONIO DE FELICE PHYS. REV. D 102, 103529 (2020)

ρ β ρ β ρ β τ β ρ β τ β ρ ¯ R μσν ≔ ∂σ Γμν − ∂ν Γμσ þ Γμν Γστ − Γμσ Γντ; Thus, given the matter content one obtains a¯ðtÞ and, from 1 (38) or (39), UðvÞ. βΓρ ≔ ρσðD þ D − D Þ μν 2 g μgνσ νgμσ σgμν : These equations are in the integer (Einstein) frame, which is unphysical in this theory. The dynamical equations To avoid confusion with ordinary curvature tensors R…,we in the physical (fractional, Jordan) frame, which we write used the curly symbol R… to denote tensors written in terms here for the first time, are obtained by recalling that the of the derivative (36). The metric is covariantly conserved Jordan and Einstein metric are related to each other by − with respect to the weighted covariant derivative ∇σ gμν ≔ pffiffiffi β τ β τ g¯μν ¼ vgμν; a¯ ¼ va; ð42Þ ∂σgμν − Γσμgτν − Γσνgμτ ¼ 0, implying that these space- times are Weyl integrable and that the metric is not conserved where v ¼ vðtÞ is the temporal part of the measure weight. in the ordinary sense, ∇σgμν ¼ðβ∂σ ln vÞgμν. Also, having chosen the FLRW metric in the integer frame As we said in Sec. II D, in the integer frame the theory implies that the same metric holds in the fractional frame resembles a scalar-tensor model in the Einstein frame. − ¯2 ¼ − 2 Denoting with a bar coordinates and metric quantities provided dt vdt , i.e., evaluated in this frame, the Friedmann equations in D ¼ 4 d 1 d ¼ pffiffiffi ; ð43Þ dimensions are [17] dt¯ v dt κ2 Ω ð∂ Þ2 ð Þ 2 t¯v U v hence H¯ ¼ ρ¯ þ þ ; ð38Þ 3 2 v2 v   ¯ 1 1 v_ 2 2 H ¼ pffiffiffi H þ ; ∂¯ a¯ κ UðvÞ v 2 v t ¼ − ðρ¯ þ 3P¯ Þþ ; ð39Þ    a¯ 6 v 1 2 ¯ 1 ̈ 1 ̈ _ _ 2 d a ¼ a þ v þ v − v ð Þ ¯ ¯2 2 H 2 ; 44 where derivatives are with respect to time t¯ and Ω ¼ a dt v a v v v −3 2 þ 9 2γð Þ 4 = v v = is a function of the measure weight. where dots are derivatives with respect to t. To these One can get the second Friedmann equation (39) either expressions, one must add those for the energy density and directly from the equations of motion or by using the pressure of the perfect fluid: continuity equation ρ ρ¯ ¼ ¯ ¼ P ð Þ ∂ ρ¯ þ 3 ¯ ðρ¯ þ ¯ Þ¼0 ð Þ 2 ; P 2 ; 45 t¯ H P : 40 v v 1 2 While, in general, U is required to be nonzero for where the factor =v stems from thep conformalffiffiffi rescaling 3 2 consistency of cosmological solutions, the function γ (42), ðenergyÞ=ðvolumeÞ¼½ðenergyÞ= v=½v = ðvolumeÞ. was originally introduced to make an analogy with sca- Therefore, the Friedmann equations and the master lar-tensor models, with the difference that v is not a equation in the physical frame read dynamical field. However, this term is neither dictated   2 2 by symmetries nor by consistency of the solutions and it 2 v_ κ 1 1 v_ H þ H ¼ ρ þ Ω − þ U; ð46Þ can be dropped without loss of generality or creating any v 3v 2 2 v2 theoretical or conceptual problem. Since our goal is to test   the most minimalistic model with weighted derivatives, i.e., ä 1 v̈ v_ v_ 2 κ2 þ þ H − ¼ − ðρ þ 3PÞþU; ð47Þ with log-oscillating measure and no kinetic-like term, we a 2 v v v2 6v can set γ to vanish, so that Ω ¼ −3=2 (this was done only in    the numerical code; the equations written here are valid in κ2 1 3 _ 2 ̈ _ _ ¼ − ðρ þ Þþ − Ω v − v þ v ð Þ general). H P 2 H : 48 2v 2 2 v v v Two major points of departure with respect to scalar- tensor theories are, first, that v is not a scalar field but a time It is in these equations, not in (38)–(41), where vðtÞ is given profile fixed a priori [by (12) in the fractional frame; see “ ” ð Þ by (11) and (22). Notice that the contributions of the below]. And, second, that the potential term U v is measure weight mimic both a running effective back- ad hoc neither chosen nor reconstructed from observations, ground-evaluated Newton’s constant but it is determined from the dynamics itself. In fact, combining (38) and (39) to eliminate the last term in the ¼ G ð Þ right-hand side, one gets the master equation Geff

103529-8 DARK ENERGY IN MULTIFRACTIONAL SPACETIMES PHYS. REV. D 102, 103529 (2020) acceleration, without the theoretical inconvenience of _ ϵ ≔−H phantom fields (v is not a dynamical degree of freedom 2 H   and is not associated with classical instabilities or negative- 3 ̈− _ 1 3 _ ¼ ð1 − Ω Þþv Hv − − Ω v ð Þ norm quantum states). 2 DE 2 2 2 2 2 2 55 Finally, the continuity equation (40) becomes v H v H v 1 v_ is the first slow-roll parameter. Another quantity of interest 0 ¼ ρ_ þ 3Hðρ þ PÞ − ðρ − 3PÞ: ð50Þ 2 v is the deceleration parameter q (not to be confused with the composite coordinates q) Radiation (w ¼ 1=3), which is the only conformal invariant ä perfect fluid, is conserved also in this frame, while any q ≔− ¼ −1 þ ϵ; ð56Þ other fluid experiences an effective dissipation. aH2 ¼ð−1 þ 2 Þ 3 related to weff by weff q = . B. Effective dark-energy component The running of the Hausdorff dimension dH in this C. Solving the dynamics theory can drive one or more phases of acceleration. It is convenient to recast the contribution of the multiscale The dynamics can be solved numerically from a minimal geometry and dynamics as a dark-energy component in set of differential equations. We choose the number of e- standard general relativity. Thus, the first Friedmann foldings as the main time variable, together with other equation (46) is written as dimensionless varying or constant parameters:

κ2 N ≔ a0 ≔ t h ≔ H H2 ¼ ðρ þ ρ Þ; ð51aÞ ln ;y; ; 3 m DE a t0 H0       t t H0 2 ≔ ≔ c ¼ c ð Þ 3 1 1 v_ v_ 1 b t0H0;yc : 57 ρ ≔ Ω − þ U − H − 1 − ρ ; t0 b DE κ2 2 2 v2 v v m ð51bÞ Since v is a given function of time, it is also a given function of y. Therefore, one could consider y as the or, equivalently, as independent variable of integration. However, we find it useful to have an autonomous system of differential Ω ¼ 1 − Ω ð Þ m DE; 52 equations. In this case, we consider N to be the indepen- dent variable. On doing so, the system of equations can be where the dimensionless energy densities were defined in written as (3). This equation replaces (4). Since dust matter does not Ω ≠ Ω ð1 þ Þ3 2 2 2 2 obey the standard continuity equation, m m;0 z . 2bhvv − 6b h vΩ − 2v Ω − 2vv þ 3v h0 ¼ − ;y m ;y ;yy ;y ; The effective dark-energy pressure PDE is obtained by 4b2v2h reformulating the master equation (48) as its Einstein- ð Þ gravity counterpart: 58 κ2 Ω _ ¼ − ðρ þ ρ þ Þ Ω0 ¼ m ð6b2h2v2 − 6b2h2vΩ þ bhvv H 2 m DE PDE ; m 2b2h2v2 m ;y     1 1 3 v_ 2 v_ v̈ −2v2 Ω − 2vv þ 3v2 Þ; ð59Þ P ≔− þ Ω − 2H − þ 3U ; ð53Þ ;y ;yy ;y DE κ2 2 2 v2 v v 1 y0 ¼ − ; ð60Þ which allows us to find the dark-energy barotropic index bh N ≔ PDE ð Þ where a prime denotes the derivative with respect to .We wDE ρ ; 54 DE also remind the reader that here we will consider Ω ¼ −3=2. In the most general case (12), several dimen- a complicated function of the Hubble parameter, the sionless parameters appear, while in the trinomial case (20) α −1 ð Þ¼1 þ c þ measure weight v and its time derivatives. For v y bcy , where the ellipsis is the UV Ω ¼ −3=2, the first term in the pressure (53) vanishes. correction, negligible at cosmic scales. This is the profile Note that (54) differs from the effective barotropic index vðyÞ considered here, y being a function of N . Among the coming from the contribution of all (matter and dark three dynamical equations (58)–(60), the first corresponds ≔−1 þ 2ϵ 3 energy) components, defined as weff = , where to the second (modified) Einstein equation, whereas the

103529-9 GIANLUCA CALCAGNI and ANTONIO DE FELICE PHYS. REV. D 102, 103529 (2020) second one is determined by using the continuity equation. 2.1 set) [35], today’s value of H0 (given in [5]) and, finally, Finally, the third equation is a direct consequence of the the constraint on the age of the universe t0 coming from definition of the e-folds number. It is also simple to show studying the globular cluster NGC 6752 [36]. This choice that of data sets is given by the fact that we only focus on the late-time evolution of the models we are considering. High- h0 redshift information, both from the background (necessary, q ¼ −1 þ ; ð61Þ h e.g., for baryon acoustic oscillations) and perturbation sides (necessary, e.g., for PLANCK data), require an investigation 1 − 2 ¼ q ð Þ of the multifractional theory which goes beyond the scope wDE 3ðΩ − 1Þ : 62 m of this study. In particular, we want to see if the model introduced here could be compatible with data which only We can integrate the equations of motion from N ¼ 0 up to require knowledge of the dynamics at low redshift. As we N ¼ 6, giving the following initial conditions: will show, although the above data sets are limited, they will be enough to constrain considerably the parameter ð0Þ¼1 hð0Þ¼1 Ω ð0Þ¼Ω ð Þ y ; ; m m0: 63 space of our model. The reason is that the acceleration of the universe is realized by means of the measure vðyÞ,but In any case, we find that the system goes to matter the latter does not represent itself a dark-energy component, domination at N ¼ 6, i.e., Ω → 1 and q → 1=2. Further i.e., a cosmological constant. Since the function vðyÞ is equations are for z and for the luminosity distance: given a priori by the dimensional-flow theorem [13],itis not surprising that some of its approximated forms may not ð1 þ Þ2 succeed to make the universe accelerate. ˜ 0 ¼ ˜ þ z ˜ ≔ 0 ¼ 1 þ d d h ; d H0dL;z z: Regarding the time scales appearing in vðyÞ, the UV scale t received strong bounds from several observations ð Þ 64 and experiments [7,8], ranging from Standard Model forces time scales to the Planck scale: Let us give some time scales for reference. t0 ≈ 14 × 109 ≈ 1017 yr s is the age of the universe today and −61 t −47 ¼ ð Þ 10 ≤ y ¼ ≤ 10 : ð65Þ H0 H t0 . The onset of big-bang nucleosynthesis is at t0 t ≈ 200 s, corresponding to y ≈ 4.5 × 10−17 and N ≈ 21, 4 while matter-radiation equality happens at t ≈ 7 × 10 yr, The lower bound in (65) corresponds to the Planck scale ≈ 5 10−6 N ≈ 8 ¼ y × , . Our numerical integration starts from t tPl, while the upper bound was obtained under the today and ends at N ¼ 6, at the peak of matter domination. assumption that 0 < α < 1 [8]. These constraints apply This choice is due to having focused our attention only to only to the UV part of the trinomial measure weight (20),in late-time data and, in particular, to data which do not need a a regime (particle-physics scales) where the ultra-IR description in terms of perturbation dynamics. In particular, correction in (20), the most important correction in the we will use the constraints from type Ia supernovæ (Union cosmological model we will consider here, is negligible.

Ω 2σ α TABLE I. Preferred values of the parameters h, b and m0 with (95% confidence level) errors. For c we min min indicate the preferred value and the 2σ lower bound αc , while for yc we only show the lower bound yc . The values of A1 and B1 are not shown due to strong degeneracy. The shaded row corresponds to the only case reproducing the observed late-time Hubble parameter.

α αmin min Ω Case nmax NA0 c ( c ) yc hb m0 1 — 0 70þ0.01 1 04þ0.21 0 278þ0.200 0 1 5.1 (0.6) 1.7 . −0.01 . −0.17 . −0.400 0 7.0 (3.4) 3.2 0 70þ0.01 1 05þ0.20 0 270þ0.040 2a . −0.01 . −0.17 . −0.040 12 0 70þ0.01 1 04þ0.22 0 275þ0.200 1 6.7 (0.3) 2.0 . −0.01 . −0.16 . −0.040 0 5.7 (0.4) 1.7 0.70þ0.01 1.05þ0.20 0.270þ0.050 2b 1 10 −0.01 −0.17 −0.080 e 0 70þ0.01 1 04þ0.21 0 276þ0.130 1 5.6 (1.0) 1.9 . −0.01 . −0.17 . −0.050 0 3.8 (3.7) 3.9 0.73þ0.01 1.05þ0.21 0.276þ0.036 3a 10 2 −0.03 −0.17 −0.040 0 73þ0.02 1 05þ0.22 0 276þ0.040 1 3.8 (3.6) 4.1 . −0.04 . −0.17 . −0.044 0 7.2 (3.7) 4.0 0.70þ0.01 1.05þ0.22 0.273þ0.04 3b 10 10 −0.01 −0.17 −0.04 e 0 70þ0.01 1 05þ0.21 0 273þ0.046 1 7.1 (3.4) 4.0 . −0.01 . −0.17 . −0.038

103529-10 DARK ENERGY IN MULTIFRACTIONAL SPACETIMES PHYS. REV. D 102, 103529 (2020)

2 Conversely, the UV correction at near-Planckian scales ∼t minimization of the χ (via different methods such as is completely negligible at late times and we will ignore the Newton one) and verified that results were compatible it here. with the MCMC sampling. The general procedure we followed to obtain the numerical constraints below was to sample the back- D. Results ground-evolution parameter space for the chosen data sets. We performed an MCMC sampling on the parameters (i) Case 1: no oscillations. In the absence of ¼ 1 of the theory, letting the sampler (EMCEE [37]) finding log oscillations, A0 and the model has only α ≔ the minima of the χ2 distribution together with the five free parameters: c, yc, b, h H0= ð100 −1 −1Þ Ω 2σ constraints for each parameter. In order to double check km s Mpc and m0. As priors, we used −3 ≤ α ≤ 10 10−5 ≤ ≤ 20 0 1 ≤ ≤ 3 the MCMC results, we also performed a numerical c , yc , . b ,

FIG. 1. Marginalized constraints on the parameters Ωm0, αc, b, yc, h, A1 and B1 for the theory with weighted derivatives with multiharmonic modes, N ¼ 2 and A0 ¼ 0. Although there is degeneracy in some of the parameters, this model is able to reach higher values of h.

103529-11 GIANLUCA CALCAGNI and ANTONIO DE FELICE PHYS. REV. D 102, 103529 (2020)

10000 500 400 1000 300 200 100 100 0

q 0 H/H 10 -100 -200 1 -300 -400 0.1 1 10 100 -500 1 1.2 1.4 1.6 1.8 2 1+z 1+z

FIG. 2. Behavior of H=H0 for the multiharmonic best fit with FIG. 4. Behavior of the deceleration parameter q for the N ¼ 2 and A0 ¼ 0. Oscillations start at low redshift values. The multiharmonic best fit with N ¼ 2 and A0 ¼ 0. We can see plot for A0 ¼ 1 is the same. the “heartbeat” of dark energy playing a nontrivial role at late times. The plot for A0 ¼ 1 is the same. 0 6 ≤ ≤ 0 8 0 1 ≤ Ω 0 5 . h . , . m 0 < . . The marginalized likelihood 2σ contours in the parameter space hand, the value of h is strongly constrained and it is (2σ preferred values of the parameters) are shown still in tension with the local observations of H0. in Table I. We will discuss the physical implications This situation seems not to depend on the value of N. of the results of this and the other cases in Sec. IV E. When A0 ¼ 1, the upper bound on Ω 0 slightly For the time being, note that the value of h is in m increases. tension with the local observations of H0. 10 (iii) Case 2b: one harmonic, N ¼ e . Having chosen the (ii) Case 2a: one harmonic, N ¼ 2. The above results same priors as before, we find the results of Table I. do not change in the presence of only one harmonic Again, the value of h is in tension with today’s value mode. As priors, we used the same as above plus 0

10 10

DE 0 w

-10

-20 0 1 -30 H/H

-40 1 1.01 1.02 1+z

FIG. 5. Behavior of wDE for the multiharmonic best fit with ¼ 2 ¼ 0 ¼ 1 0.1 N and A0 . The plot for A0 is the same. Thanks to the 1 1.2 1.4 1.6 1.8 2 last oscillation, the universe is able to increase H to values close 1+z to the measured value H0 today. Here a dark-energy epoch is reached due to oscillations which make the universe accelerate FIG. 3. Behavior of H=H0 for the multiharmonic best fit with and decelerate alternatively. Acceleration becomes then a tran- N ¼ 2 and A0 ¼ 0 at low redshifts. The plot for A0 ¼ 1 sient but today the universe does undergo a period of accelerated is the same. expansion.

103529-12 DARK ENERGY IN MULTIFRACTIONAL SPACETIMES PHYS. REV. D 102, 103529 (2020)

Ω α FIG. 6. Marginalized constraints on the parameters m0, c, b, yc, h, A1 and B1 for the theory with weighted derivatives with multiharmonic modes, N ¼ 2 and A0 ¼ 1.

Ω of the Hubble parameter and the upper bound of m0 for all the previous cases (values approximately increases when A0 ¼ 1. equal to 565). The behavior of the Hubble factor (iv) Case 3a: many harmonics, N ¼ 2. This is the most is shown in Fig. 2, while a more detailed dynamics promising case. The likelihood contours for A0 ¼ 0 for H=H0 at low redshifts is depicted in Fig. 3.At are shown in Fig. 1. Notice in Table I the higher the same time, it is interesting to show also the 2σ α value of h and the lower bound at for c. In order deceleration parameter q (Fig. 4) and wDE (Fig. 5). to explore this better fit more in detail, we tried to Thanks to the oscillations, we are able to get a higher enlarge the prior for A1 and B1 to 0

103529-13 GIANLUCA CALCAGNI and ANTONIO DE FELICE PHYS. REV. D 102, 103529 (2020)

factor, H=H0 at low redshifts, q and wDE is virtually one. However, unlike other IR modifications of standard indistinguishable from that shown in Figs. 2–5. cosmology, this theory does not rely on the typical UV/IR (v) Case 3b: many harmonics, N ¼ e10. Here one can divide through a characteristic scale. When the multifrac- appreciate a difference between the N ¼ 2 and the tional geometry is of this type (no log oscillations), it fails 10 N ¼ e results, the latter getting considerably to fit data, since the scale tc is larger than the age of the worse. Once more, the χ2 for the minimum shifts universe. Thus, we do not end up with the typical model to values close to 565. where general relativity is an adequate description of the cosmos during most of the history of the universe and the E. Consequence: Extending multiscale spacetimes recent acceleration is triggered around a certain IR scale. Rather, multifractional effects embodied in logarithmic All of the above cases share some common features. The oscillations are endemic to the cosmological picture of the value of the parameter b is close to 1, which means that theory and mimic, in a subtle way, a standard cosmology the inverse Hubble parameter today is approximately equal to with a late-time dominating dark-energy component. Here the age of the universe, just like in the standard ΛCDM model: the characteristic scale is very large (actually, beyond the t0H0 ≈ 1. Similarly, Ω 0 is close to the ΛCDM value. m Hubble horizon) but, nevertheless, it modifies the evolution The value of h is the most important discriminator of the universe via the harmonic structure of the geometry. selecting viable models. Only those with many harmonics The last piece of information gathered about the geom- and N ¼ 2 (case 3a) are able to recover the estimated value etry of the cosmos reserves a surprise and is about the from late-time observations.4 Concentrating only on case Hausdorff dimension of spacetime. Just as in the case of yc, 3a, the lower bound on yc is α the distribution of values of c is flat past the peak (preferred value) and one cannot establish an upper bound. ¼ tc 3 9 ð Þ yc > . ; 66 However, while the peak of y is rather mild, the one for α t0 c c is very pronounced and allows us to clearly establish a preferred value. In case 3a, the latter is which means that the characteristic time tc is always much larger than the age of the universe t0. On one hand, this is α ≈ 3.8; ð67Þ compatible with the finding that the value of A0 has little c impact on the numerical analysis, i.e., that the zero mode is α with a very small error bar. The preferred value of c is strongly suppressed and hence it does not affect cosmology. larger in the other cases. This means that the theory with On the other hand, the bound (66) is not in contradiction weighted derivatives cannot accommodate at the same time with the fact that multifractional effects can explain particle-physics constraints and explain the late-time accel- ð Þ observations at times O t0 , since the modulation of eration of the universe if we consider only the binomial superposing log oscillations of the highest frequency ω ¼ measure (13) [for which t tc and (66) would be in gross takes place at much shorter scales. Note the specifications contradiction with (65)], while the case with many har- “ ” “ ” of superposing and highest frequency : in the absence monics of maximal frequency can if we allow for an ultra- of log oscillations (case 1), in the presence of only one IR term as in (20). ¼ 1 harmonic (nmax , cases 2a and 2b), or when the The result (67) has a clear geometric interpretation based harmonics frequency is too low (large N, cases 2b and on the calculation of the Hausdorff dimension in a perfectly 3b), the theory is unable to recover the observed late-time homogeneous FLRW background. On this spacetime, the value of the Hubble parameter. Note the importance of measure weight is (12) but, by definition (10), the time having a theoretical upper limit on ω. Without it, it would Hausdorff dimension dtime is computed after averaging over have been more difficult to find a modulation comparable H log oscillations [7,33], i.e., we must take (20) instead of its with the observable patch of the universe and compatible version with log oscillations. When A0 ¼ 0, the averaged with late-time data. measure is v ¼ 1 and dtime ¼ 1 at all times. However, when Concerning the value of the oscillation amplitudes A1 H A0 ¼ 1 the Hausdorff dimension of time is no longer and B1, as one can see in Figs. 1 and 6 their distribution is constant. With a FLRW metric and A0 ¼ 1, the effective uniform inside the prior 0

4Since we have not used early-universe data, we are not in a 5These are always excluded when computing the Hausdorff or position to say anything about the H0 tension. spectral dimension.

103529-14 DARK ENERGY IN MULTIFRACTIONAL SPACETIMES PHYS. REV. D 102, 103529 (2020) Z α α t 1 1 c Vð Þ¼ 0 ð 0Þ¼ t þ þ t ð Þ TABLE II. Time and spacetime Hausdorff dimension in differ- t dt v t α t α ; 69 ent regimes, after averaging out log oscillations. t c tc Regime vðtÞ dtime spacetime H dH so that P j j1−α α α þ 3 α Quantum gravity t=t i¼1 i 1 þ 3 ¼ 4 α α General relativity 11 j j þ þj j c time t=t t t=tc (general background) d ¼ α α : ð70Þ H ð1 α Þj j þ þð1 α Þj j c = t=t t = c t=tc General relativity 11 1 (FLRW) α −1 j j c α ¼ 4 α ¼ 4 0 α 1 α 1 ≪ Late times (FLRW) t=tc c c Assuming < < and c > , when t t one has time ≃ α ≪ ≲ dH . At intermediate scales t t tc, the Hausdorff dimension of time coincides with the topo- anisotropic. These effects quickly die away before or time ≃ 1 logical dimension, dH . This is the regime character- during inflation: the residual IR correction in (20) survives ≪ izing most of the history of the universe. In a late-time but is negligible at times t

103529-15 GIANLUCA CALCAGNI and ANTONIO DE FELICE PHYS. REV. D 102, 103529 (2020)

α ð1Þ conserved during the evolution of the universe (there would For both c and yc we only have an O lower bound for be no dimensional flow in that case), it is restored all the models, that is, a large degeneracy on those two asymptotically if the universe stays exactly homogeneous. variables. A fine tuning in this context would correspond α ≫ 1 Note that the regimes listed in Table II are phenomeno- either to lower bounds c;yc or to an allowed region logically mismatched with respect to the observability with negligible area in the two-dimensional parameter ≫ window of multifractional effects. That is, since tc t0, space. In contrast, to explain dark energy with a cosmo- we cannot observe time dimensional flow now because we logical constant in general relativity a fine tuning of tens of are in the general-relativity plateau, far away both from the digits on the value of Λ is required. ð Þ UV and the ultra-IR regimes of the profile dH t . However, If the oscillation amplitudes are sufficiently large, the the inception of cosmic acceleration happens well before effective dark energy density log periodically dominates the establishment of the ultra-IR regime. We are not aware over the matter energy density, while the effective baro- of similar mechanisms in any other models of dark energy. tropic index wDE log periodically oscillates between pos- itive and negative values infinitely many times during the V. CONCLUSIONS history of the universe. The last oscillation causes a late- time acceleration period (Fig. 5), but not as an isolated Multifractional spacetimes have been previously event: it is the last but the most pronounced of an infinite explored in a range of settings and regimes, from particle sequence of acceleration epochs (Fig. 4). physics to cosmology. Here we applied them to the late- In turn, data suggest that the preferred value of the time universe in order to see what they can say about the Hausdorff dimension of spacetime in the ultra-IR regime is dark-energy problem and, conversely, how supernovæ tantalizingly close to 4, which stimulates further theoretical data can constrain such models. While the theory with research on the spacetime geometry of this theory along the q-derivatives cannot explain late-time acceleration, the lines discussed in Sec. IV E. It will be worth investigating theory with weighted derivatives has a chance. To do so, further the geometric interpretation presented therein, due we considered a term in the spacetime measure that to its important phenomenological consequences as an describes a sort of ultra-IR regime beyond Einstein gravity. alternative explanation of late-time acceleration. The This term was not added ad hoc in the measure expansion: absence of any fine tuning in the parameters of the theory it was already there, admitted by the dimensional-flow could make it an appealing alternative to other geometry- theorem [13], but in the literature it had been overlooked driven or matter-field-driven explanations of dark energy. because this theory was studied mainly in a microscopic Future experiments in addition to the search of other particle physics setting (e.g., [8]) where such term is constraints for these model will shed further light into completely negligible. In this sense, we are not proposing the restoration law proposed here. a “new” class of models but simply continuing the study of old ones in previously untapped regimes. ACKNOWLEDGMENTS The result is that none of the simplified versions of the multifractional theory with weighted derivatives explored A. D. F. was supported by JSPS KAKENHI Grant more frequently in the literature, the one with monotonic No. 20K03969. G. C. is supported by the I þ D grant measure (no log oscillations) and the one with only one FIS2017-86497-C2-2-P of the Spanish Ministry of Science, harmonic, can explain late-time data. Only when enough Innovation and Universities and acknowledges the contri- undamped harmonics are present can the theory explain the bution of the COST Action CA18108. He also thanks the late-time acceleration of the universe without fine tuning Yukawa Institute for kind hospitality during part of the and without invoking extra dynamical degrees of freedom. development of this project.

[1] Y. Akrami et al. (Planck Collaboration), Planck 2018 [4] G. Calcagni, Classical and Quantum Cosmology (Springer, results. I. Overview and the cosmological legacy of Planck, Switzerland, 2017). Astron. Astrophys. 641, A1 (2020). [5] A. G. Riess et al., A 2.4% determination of the local value of [2] N. Aghanim et al. (Planck Collaboration), Planck 2018 the Hubble constant, Astrophys. J. 826, 56 (2016). results. VI. Cosmological parameters, Astron. Astrophys. [6] https://www.euclid-ec.org. 641, A6 (2020). [7] G. Calcagni, Multifractional theories: An unconventional [3] B. P. Abbott et al. (LIGO Scientific and Virgo Collabora- review, J. High Energy Phys. 03 (2017) 138. tions), Tests of General Relativity with GW150914, Phys. [8] A. Addazi, G. Calcagni, and A. Marcianò, New standard model constraints on the scales and dimension of spacetime, Rev. Lett. 116, 221101 (2016). J. High Energy Phys. 12 (2018) 130.

103529-16 DARK ENERGY IN MULTIFRACTIONAL SPACETIMES PHYS. REV. D 102, 103529 (2020)

[9] G. ’t Hooft, Dimensional reduction in quantum gravity, in [25] S. Maity and U. Debnath, Co-existence of modified Chap- Salamfestschrift, edited by A. Ali, J. Ellis, and S. Randjbar- lygin gas and other dark energies in the framework of fractal Daemi (World Scientific, Singapore, 1993). universe, Int. J. Theor. Phys. 55, 2668 (2016). [10] S. Carlip, Spontaneous dimensional reduction in short- [26] A. Jawad, S. Rani, I. G. Salako, and F. Gulshan, Pilgrim distance quantum gravity?, AIP Conf. Proc. 1196, 72 (2009). dark energy models in fractal universe, Int. J. Mod. Phys. D [11] G. Calcagni, Fractal Universe and Quantum Gravity, Phys. 26, 1750049 (2017). Rev. Lett. 104, 251301 (2010). [27] E. Sadri, M. Khurshudyan, and S. Chattopadhyay, An [12] S. Carlip, Dimension and dimensional reduction in quantum interacting new holographic dark energy in the framework gravity, Classical Quantum Gravity 34, 193001 (2017). of fractal cosmology, Astrophys. Space Sci. 363, 230 [13] G. Calcagni, Multiscale spacetimes from first principles, (2018). Phys. Rev. D 95, 064057 (2017). [28] D. Das, S. Dutta, A. Al Mamon, and S. Chakraborty, Does [14] G. Calcagni, Towards multifractional calculus, Front. Phys. fractal universe describe a complete cosmic scenario?, Eur. 6, 58 (2018). Phys. J. C 78, 849 (2018). [15] G. Calcagni and G. Nardelli, Quantum field theory with [29] U. Debnath and K. Bamba, Parametrizations of dark energy varying couplings, Int. J. Mod. Phys. A 29, 1450012 (2014). models in the background of general non-canonical scalar [16] G. Calcagni, Quantum field theory, gravity and cosmology field in D-dimensional fractal universe, Eur. Phys. J. C 79, in a fractal universe, J. High Energy Phys. 03 (2010) 120. 722 (2019). [17] G. Calcagni, Multi-scale gravity and cosmology, J. Cosmol. [30] A. Sheykhi, Z. Teimoori, and B. Wang, Thermodynamics of Astropart. Phys. 12 (2013) 041. fractal universe, Phys. Lett. B 718, 1203 (2013). [18] G. Calcagni, S. Kuroyanagi, and S. Tsujikawa, Cosmic [31] S. Haldar, J. Dutta, and S. Chakraborty, A comparative microwave background and inflation in multi-fractional study of different entropies in fractal universe, arXiv:1601 spacetimes, J. Cosmol. Astropart. Phys. 08 (2016) 039. .01055. [19] G. Calcagni, Complex dimensions and their observability, [32] P. Coles and F. Lucchin, Cosmology, 2nd ed. (Wiley, Phys. Rev. D 96, 046001 (2017). Chichester, United Kingdom, 2002). [20] G. Calcagni, Lorentz violations in multifractal spacetimes, [33] G. Calcagni, Geometry and field theory in multi-fractional Eur. Phys. J. C 77, 291 (2017). spacetime, J. High Energy Phys. 01 (2012) 065. [21] G. Calcagni, S. Kuroyanagi, S. Marsat, M. Sakellariadou, N. [34] R. R. Nigmatullin and A. Le M´ehaut´e, Is there geometrical/ Tamanini, and G. Tasinato, Quantum gravity and gravita- physical meaning of the fractional integral with complex tional-wave astronomy, J. Cosmol. Astropart. Phys. 10 (2019) exponent?, J. Non-Cryst. Solids 351, 2888 (2005). 012. [35] N. Suzuki et al. (Supernova Cosmology Project Collabo- [22] K. Karami, M. Jamil, S. Ghaffari, K. Fahimi, and R. ration), The Hubble Space Telescope cluster supernova Myrzakulov, Holographic, new agegraphic and ghost dark survey: V. Improving the dark energy constraints above energy models in fractal cosmology, Can. J. Phys. 91, 770 z>1 and building an early-type-hosted supernova sample, (2013). Astrophys. J. 746, 85 (2012). [23] O. A. Lemets and D. A. Yerokhin, Interacting dark energy [36] A. Renzini et al., The white dwarf distance to the globular models in fractal cosmology, arXiv:1202.3457. cluster NGC 6752 (and its age) with the Hubble Space [24] S. Chattopadhyay, A. Pasqua, and S. Roy, A study on Telescope, Astrophys. J. Lett. 465, L23 (1996). some special forms of holographic Ricci dark energy in [37] D. Foreman-Mackey, D. W. Hogg, D. Lang, and J. fractal universe, ISRN High Energy Phys. 2013 (2013) Goodman, emcee: The MCMC hammer, Publ. Astron. 251498. Soc. Pac. 125, 306 (2013).

103529-17