Observational constraints on the fractal cosmology

Eur. Phys. J. C (2022) 82:960 https://doi.org/10.1140/epjc/s10052-022-10927-4 Regular Article - Theoretical Physics Observational constraints on the fractal cosmology Mahnaz Asghari1,2,a , Ahmad Sheykhi1,2,b 1 Department of Physics, College of Sciences, Shiraz University, Shiraz 71454, Iran 2 Biruni Observatory, College of Sciences, Shiraz University, Shiraz 71454, Iran Received: 3 August 2022 / Accepted: 14 October 2022 © The Author(s) 2022 Abstract In this paper, we explore a fractal model of the universe proposed by Calcagni (J High Energy Phys 03:120, 2010) for a power-counting renormalizable field theory living in a fractal spacetime. Considering a timelike fractal profile, we derived field equations in fractal cosmology, in order to explore the structure formation and the expansion history in fractal universe. Numerical investigations based on matter power spectra diagrams report higher structure growth in fractal cosmology, being in contrast to local galaxy surveys. Additionally, according to the evolution of Hubble parameter diagrams, it can be understood that Hubble constant would decrease in fractal cosmology, which is also incompatible with low redshift estimations of H0 . So, concerning primary numerical studies, it seems that fractal cosmology is not capable to alleviate the tensions between local and global observational probes. Then, in pursuance of more accurate results, we constrain the fractal cosmology by observational data, including Planck cosmic microwave background (CMB), weak lensing, supernovae, baryon acoustic oscillations (BAO), and redshift-space distortions (RSD) data. The derived constraints on fractal dimension β indicate that there is no considerable deviation from standard model of cosmology. 1 Introduction Recent observational data from type Ia supernovae (SNeIa) [1,2], the cosmic microwave background (CMB) anisotropies [3–5], large scale structures [6–8], and baryon acoustic oscillations (BAO) [9–11], confirm CDM as the concordance model of cosmology. On the other hand, CDM model which is based on the theory of general relativity (GR), encounters some observational discrepancies, principally σ8 a e-mail: (corresponding author) and H0 tensions. To be more specific, there is a disagreement between low-redshift determinations and Planck CMB measurements of matter perturbation amplitude, σ8 [5,12]. Moreover, direct measurements of Hubble constant are in significant tension with Planck observations [5,13–16]. So, inconsistencies between local and global data, motivate cosmologists to prob beyond CDM model. Accordingly, it is suggested to consider some corrections on GR, describing as modified theory of gravity [17–20]. On the other hand, Calcagni proposed an effective quantum field theory which is power counting renormalizable and Lorentz invariant, living in a fractal universe [21,22]. The fractal nature firstly introduced by Mandelbrot in 1983 [23] suggests conditional cosmological principle in fractal universe, where the universe appears the same from every galaxy. Thereafter, in 1986, Linde [24] propounded a model of an eternally existing chaotic inflationary universe, explaining a fractal cosmology. Accordingly, there are several investigations on the theory of fractal cosmology in literatures. Rassem and Ahmed [25] in 1996 considered a nonhomogeneous cosmological model with a fractal distribution of matter which evolves to a homogeneous universe as time passes. The conditional cosmological principle in fractal cosmology is discussed in [26,27]. Calcagni [28] studies Multi-fractal geometry. Thermodynamics of the apparent horizon in a fractal universe is explored in [29]. In order to find more theoretical studies on fractal cosmology refer to e.g. [30–34]. Furthermore, it is interesting to explore fractal models with cosmological data as discussed in [35–40] (also see [41] for a review). Correspondingly, in the present work we are going to investigate the fractal universe in background and perturbation levels, as well as studying observational constraints on parameters of fractal cosmology. The paper is organized as follows. Section 2 is dedicated to field equations in a fractal universe. In Sect. 3, we study the fractal model numerically, and further we constrain the model b e-mail: 0123456789().: V,-vol 123 960 Page 2 of 6 Eur. Phys. J. C with current observational data in Sect. 4. We summarize our results in Sect. 5. (2022) 82:960 background level take the form     8π G 1 2β 2 a 2β = ρ̄i , (6) H 1 + β − ωH0 β 6 a0 3 i     1 H 2β 2 a 2β 2 2 + H 3 + 2β + β + ωH0 β (β + 2) a 2 a0 2 2 Field equations in a fractal universe The total action in a fractal spacetime is given by [21] S= 1 16π G  d(x) √ = −8π G   −g R − 2 − ω∂μ v∂ μ v + Sm , (1) where ω is the fractal parameter, v is the fractional function, and d(x) is a Lebesgue–Stieltjes measure. It is possible to derive field equations from action (1) similar to scalar–tensor theories. Thus, in a fractal universe we obtain [21] ∇μ ∇ν v v 1 − Rμν − gμν (R − 2) + gμν 2 v v  1 σ + ω gμν ∂σ v∂ v − ∂μ v∂ν v = 8π GTμν . 2 (2) Furthermore, continuity equation in a fractal spacetime takes the form [21]   ∇μ vTνμ − ∂ν vLm = 0. (3) It should be noted that for v = 1, standard equations in GR will be recovered. Here, we focus on a timelike fractal, then v is only time dependent given by β v = H0  a β a0 , (4) in which β = 4(1 − α) is the fractal dimension, and the parameter α ranges as 0 < α ≤ 1. We consider a fractal universe with the following flat Friedmann–Lemaître–Robertson–Walker (FLRW) metric in the synchronous gauge     ds 2 = a 2 (τ ) − dτ 2 + δi j + h i j dx i dx j , (5) in which  h i j (x, τ ) = 3 d ke (7)    1  k̂i k̂ j h(k, τ )+ k̂i k̂ j − δi j 6η(k, τ ) , 3 where a prime indicates a deviation with respect to the conformal time. It can be easily seen that, β = 0 restores field equations in standard cosmology. According to Eq. (6), total density parameter can be find as  a 2β 1 2β , tot = 1 + β − ωH0 β 2 6 a0 (8) where we have considered a universe filled with radiation (R), baryons (B), dark matter (DM) and cosmological constant (). Also field equations to linear order of perturbations can be written as 1  a  1 + β h  − 2k 2 η = 8π Ga 2 a 2   ρ̄i + p̄i θi , k 2 η = 4π Ga 2 δρi , (9) i (10) i  1  1  a   h + 3η + h + 6η 1 + β − k 2 η = 0, 2 a 2  a  2 + β h  + h  − 2k 2 η = −24π Ga 2 δpi . a (11) (12) i In addition, regarding Eq. (3), conservation equations of fractal cosmology for ith component of the universe in background and perturbation levels become    a  ρ̄i + p̄i = 0, ρ̄i + 3 + β a   2  a  3 + β δi csi − wi a  2     θi 2 a + csi 3 + β 1 + wi 2 − cai a k     1 − 1 + wi θi − 1 + wi h  , 2 (13) δi = − (14) ik.x with scalar perturbations h and η, and k = k k̂ [42]. Then, considering the energy content of the universe as a perfect   fluid with T (...truncated)