Black hole information turbulence and the Hubble tension

Abstract

Fractals are complex geometric patterns whose structure look similar at different scales of magnification. Examples of fractals in astrophysics are diverse: the cosmic microwave background (CMB) or the distribution of matter in the universe show patterns consistent with fractals. A major outstanding challenge in cosmology is the discrepancy between the Hubble constant obtained from early and late universe measurements — the Hubble tension. By examining cosmological evolution through the lens of information growth within a black hole, we demonstrate the emergence of two distinct fractal growth processes characterizing the early and late universe epochs. These fractal patterns induce space expansion rates of (62.79 ± 4.56) Km/s/Mpc and (70.07 ± 0.39) Km/s/Mpc, remarkably close to current values of the Hubble constants involved in the tension. Such a result suggest that the Hubble tension arises not from unexpected large-scale structures or multiple unrelated measurement errors, but rather from innate properties underlying the universe dynamics.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-026-44713-z.

Introduction

Fractal geometry appears throughout cosmic structure at multiple scales, from temperature fluctuations in the cosmic microwave background (CMB)¹ to the distribution of galaxies² and the large-scale cosmic web^3,4. Understanding the geometric properties of spacetime at different scales may prove crucial for resolving one of cosmology’s most pressing challenges: the Hubble tension⁵. Leading astrophysical experiments have achieved unprecedented precision in measuring the universe’s expansion rate – the Hubble constant, – through two fundamentally different approaches. Early-universe observations from CMB experiments produce one value, while late-time measurements based on Cepheid variable stars and Type Ia supernovae produce another, statistically incompatible value. This persistent discrepancy suggests either systematic errors (which seems to be non-Gaussian^6,7) in our measurements or new physics in our understanding of cosmic evolution. The remarkable precision of each measurement method imposes strict constraints on potential solutions, which may require fundamentally new physical phenomena or a deeper understanding of how spacetime structure influences cosmic dynamics.

One theoretical avenue explores whether spacetime geometry itself – particularly at quantum scales – might influence cosmological observables in unexpected ways. Developments in quantum gravity suggest that spacetime is not a fundamental entity but rather an emergent property arising from quantum entanglement. Building on the Ryu-Takayanagi formula, which relates entanglement entropy in boundary conformal field theories (CFT) to minimal surfaces in bulk Anti-de Siter (AdS) spacetime⁸, Van Raamsdonk demonstrated that the connectivity of spacetime regions is fundamentally tied to quantum entanglement between their associated degrees of freedom⁹. This perspective suggests that entanglement doesn’t merely correlate with geometry, it may literally construct spacetime itself – with implications that could extend from black hole (BH) horizons to cosmological scales. These insights converge with BH physics through the ER = EPR conjecture, which proposes that Einstein-Rosen bridges (wormholes) connecting entangled BHs are equivalent to Einstein-Podolsky-Rosen pairs (quantum entanglement)¹⁰. BHs, which store the maximum possible information for a given region¹¹, serve as natural laboratories for studying quantum-gravitational effects. Within the AdS/CFT framework, BHs have been modelled as quantum circuits, revealing deep connections between gravitational dynamics and quantum information processing¹². Recent work has shown that the growth of Einstein-Rosen bridge interiors corresponds to the growth of computational complexity in boundary quantum circuits^13,14. This complexity-geometry correspondence suggests that as entangled BHs process information, the wormhole connecting them grows in a manner directly tied to the computational complexity of the quantum circuit that describes their entanglement. These dynamics can be modelled as random quantum circuits that rapidly scramble quantum information — the fast scrambling conjecture^15,16. While the resolution of the BH information paradox — what happens to the information of objects that fall into a BH when it eventually evaporates – remains under active investigation, several theoretical approaches support information-preserving (unitary) dynamics. Though these results primarily concern theoretical BHs in AdS spacetime — which differ from astrophysical BHs in our asymptotically flat universe — they reveal fundamental principles connecting quantum entanglement, spacetime geometry, and information processing. If the entanglement structure of quantum fields influences spacetime geometry at cosmological scales, it could potentially manifest in observable ways, including through fractal-like patterns in cosmic structure or even through effects on cosmic expansion measurements.

Results

In this context, featured in the AdS/CFT framework, the inside of a BH has been described¹⁷ as a quantum circuit where the evolution of qubits, in a space of states of qubits, obeys,

1

where an average number of new infections, , are produced in the circuit next step (using the notation and , with integer). Previous research transformed the iteration time, , into a continuous variable¹⁷, thereby converting the original difference equation into a differential equation. However, maintaining the discrete time step of the original formulation reveals that Eq. (1) is equivalent to the well-known logistic map (see Methods) — a fundamental model in nonlinear dynamics^18,19,

2

with control parameter, . Thus , which indicates a proliferation of qubits when . However, this high-entropy regime lacks a particular dynamical significance, as the logistic map with merely converges to a stationary fixed point , exhibiting no complex behaviour despite the elevated information content.

Black hole information dynamics with limited resources

It is important to recognize that the number of qubits in the BH -space is finite. Consequently, the quantum states available in any given generation must be constrained by those already occupied in previous generations. This parallels resource limitations in population dynamical models. Such a finite-resource framework can be modelled through a regulation mechanism where the next generation, , depends on both the current population, , and a regulation term from the preceding generation, . With these considerations in mind Eq. (2) transforms into the Delayed Regulation Model (DRM)²⁰.

3

This map has a Hopf bifurcation at and as Eq. (2) also leads to chaos^21–23. Notably, the DRM can be deduced from the incompressible form of the Navier-Stokes equation in the frequency domain²⁴. At any generation , the leading terms of the eigenvalues governing the dynamics are second-order polynomials whose coefficients are themselves nested second-order polynomials, with this nesting repeated times -- at the Methods section an example is shown for the particular case . The recursive generation of these coefficients can be described precisely²⁴. This recursion enables the construction of an analytical cascade that exhibits the scaling law characteristic of the power spectrum in isotropic homogeneous turbulence. The cascade’s first steps are depicted in the Fig. 1. This branching structure can be represented as the fractal seen in Fig. 2a, where the value of the three coefficients for the initial condition (second order: , linear: and independent: , terms) are mapped to segments of size on the unit interval, and subsequently subdivided following the coefficients generating rule (see details in²⁴ and in Methods). In this work, the generated fractal will be considered a sufficiently valid coarse grain approximation of a turbulent process underlying the population growth. As for large enough the control parameter is close to the Hopf bifurcation value, , the population of qubits inside the BH describes critical dynamics. After a few steps , the coefficients governing the qubits population describe a fully developed turbulent state characterized by a power law spectra, i.e., a state of information turbulence in a space partitioned in – a fast growing number – states. After a probably large, but finite number of iterations, the population of qubits should exhaust its resources in the BH bounded space. As a result, after such a large – much larger than the exemplified in Fig. 2a – no additional incoming energy will feed the cascade and one may expect turbulence to recede from its fractal limit set. A distant observer of such a receding situation would see the same cascade backwards – as in this space there is nothing able to perturb the backward trajectory –. However, such an observer would be unable to witness the final fully developed cascade but would rather see a later stage with turbulence already receding, i.e., this later state acts as a horizon. Figure 2 illustrates this process: the direct cascade happening inside the BH (Fig. 2a) is mirrored by the inverse cascade (Fig. 2c).

Fig. 1.

Tree representation for the generation of the analytical cascade. First steps in the generation of the analytical fractal cascade shown in the Fig. 2. Note that the process is not symmetric, i.e., it is clearly anisotropic. This figure fixes a typo in a similar one published before²⁴.

Fig. 2.

Fractal turbulent cosmology. a The coefficients describing the evolution of a population of qubits inside the BH form a fractal cascade as the iterations increase (described from right to left). b The fraction of lacunar zero coefficients (squares) increases while the fraction of coloured non-zero coefficients (circles) decreases describing a cantor-like dust in a, while grows exponentially (see Fig. 3). a was calculated with , after iterating generations, giving rise to coefficients. Colours describe the coefficient’s intensities in the nested cascade (see²⁴ and Methods). Non-zero coefficients are colored blue and brown and the lacunar component is violet. c Once the cascade stops, it is inverted and d increases while decreases reaching the initial proportions and . In b and d lines are a best fit of the CDM model with parameters , , and (see²⁴ for the meaning of and ). Note that the inverse cascade does not runs in a time-reversal way. The time arrow is depicted at the top, where it is shown that time runs from right to left. Fig. a and b are similar to one published by the author in²⁴, but here they are represented in a different way.

Measuring the fractal cascade

To quantify the evolution of the cascade components during both, the direct and the recession stages, at a given generation, , we count the number of coloured, , and lacunar, , components in the fractal. With these quantities we can calculate the normalised number of coloured non-zero coefficients, , and of lacunar zero coefficients, . With increasing iterations, and grow exponentially (Fig. 3), while and describe the curves seen at Figs. 2b and d and 3. In the long run, during recession, dominates over . It is striking the similarity of the inverse cascade’s and with the evolution of the fractional energy density of nonrelativistic matter and dark energy components (Note that in this construction the total energy density is always conserved and equals ). To better quantify such a similarity, we fitted²⁵ seven different cosmological models^26–29 in terms of the redshift (Supplementary Information (SI) Eqs. 3–10) and found that the CDM and two Early Dark Energy models produced better results (see Fig. 4) -- the fractions of the fractal components are not the same as the fractional energy densities but using the fact that both are dimensionless quantities behaving in a similar way we extract parameters from fitting. We included the parameter to establish a relationship between and , i.e., . These three mentioned models give similar ’s (SI Table 1). Large redshifts therefore correspond to the origin of the cascade, while moderate redshifts correspond to an era dominated by turbulent states. According to this description the backward turbulence progression roughly describes the universe evolution from the instant when the original turbulence - developed by a population of qubits inside a BH -, started to recede. It must be stressed that in this framework we deal with information turbulence, whose relationship with a barotropic cosmological perfect fluid is, to the best of the author knowledge, currently unknown.

Fig. 3.

Behavior of the cascade components. (Right top:) Number of colored components and (right bottom:) number of lacunar components in the inverted fractal cascade. (Left:) Fraction of colored and lacunar components in the inverted cascade as seen by an observer at the origin. In all the cases the continuous line is the best fit

Fig. 4.

Fitting cosmological models. (Dots) vs. fitted by (top) the CDM model and (bottom) an Early Dark Energy (EDE) model^26–28 using a Markov Chain Monte Carlo analysis. The red line correspond to the fit used in Fig. 2b and d. In the CDM model , while in the EDE model (see SI Sect. 2)

How the fractal space growths

There is further evidence supporting the present description. The cascade is formed by non-zero colored coefficients and lacunar ones. While growing, the fractal structure generates a space where both components are intertwined. There is no structure if both components are not present. The space-filling characteristic of the generated space is measured by its fractal dimension. In particular, we may consider two different assertions of this quantity: a coarse grain dimension (in the sense specified above), (Fig. 5), and a motif dimension, (Fig. 6), both defined in Methods. It must be noted that for the inverse cascade shown in Fig. 2c, it is satisfied that (where applies to or , and refers to the inverse cascade). Now we can estimate how the fractal space grows calculating (Figs. 5c and 6c),

Fig. 5.

Coarse grain dimension. a takes the value at , the first two iterates resulted in a constant value as the fractal is beginning to take shape; b starting from , the consecutive differences, , peaks at and decays to at ; c , calculated using the represented data in a and b, note that at the result diverges as , and is not represented (dropped). Data dots are joined by lines.

Fig. 6.

Motifs dimension. a takes the value at . b starting from , the consecutive differences, , decays to at . c , calculated using the represented data in a and b. Data dots are joined by lines.

4

where is the successive difference of and (Figs. 5b and 6b).

The Hubble constants

Figure 7a and c show the prominent similitude of the rate of growth of the fractal space with the evolution of as calculated with the CDM model³⁰. To be more precise, we fitted to,

Fig. 7.

Determination of the Hubble constant from the relative growth of the fractal space. a and c show the result of fitting to , using as given by the CDM model (Eq. (11) in the SI). b and d show the result of evaluating the function given by Eq. (30) (red line) while the plus signs show an interpolating function whose extrapolation to yield the value remarked with the green dot at b Km/s/Mpc and at d Km/s/Mpc. Further information about the error is given in the SI last section and in SI Table 2. The numerical stability in the calculation of has been addressed in the SI.

5

with and a proportionality and a scaling parameter, respectively; and with being one of the three cosmological models selected above (see SI Sect. 3). Using the fitting results, we defined a function (Methods) from where we calculated two constants equivalent to the Hubble one as,

6

We found that CDM was the only model producing two results compatible with current accepted values. These two Hubble constants are,

7

8

as is in units of . Let’s highlight that is associated with the filling of the space of the dust grains forming the fractal limit set, as measured by , while relates to the filling of the space of the larger structures formed by the fractal motifs, as measured by . These distinct ways of determining the Hubble constant open the door to an innovative explanation of the fractal origins of the Hubble tension, as measures based on the CMB experiments are associated with a description of the early Universe at ⁵, where the grained detail of the anisotropic fractal determines the result. Meanwhile, measures based on local distances should not be influenced by such a grained detail but by the even courser profile of the structure given by fractal motifs -- in this analogical context, structure formation could be associated with lacunar motifs, and the anisotropies observed in galaxy clustering along the cosmic web could reflect variations in motif distribution. Remarkably, is compatible with mostly all the determinations of based on CMB, e.g., according to⁵, the most widely cited prediction from Planck in a flat CDM model is Km/s/Mpc³¹, which is in the range of . Also, within the margin of error, the obtained value for approximates certain estimations of the Hubble constant based on local universe measures, e.g., ³², ³³, ³⁴, ³⁵, ³⁶ and ³⁷, where for simplicity we have intentionally omitted the reported errors. It is interesting to note that the EDE and EDEP models are able to produce “acceptable” values just for the case: Km/s/Mpc and Km/s/Mpc, respectively; yielding undervalued results for : Km/s/Mpc and Km/s/Mpc, respectively (see SI Table 2). This situation may be indicating that these models are limited in their description at shorter redshifts, but are approximately capturing features of the early universe, particularly in the EDEP model case -- where a similar parameter to that of CDM was found (SI, Sect. 2). In our framework this result means that the EDE and EDEP models are not able to describe the motif generated space.

Discussion

As the qubit population evolves through a critical state, the system exhibits phase structure across all scales. These phases may be interpreted as reflecting the connectivity architecture of quantum gates, motivating the conjecture that, at criticality, a quantum circuit whose population dynamics follows the cascade described in Fig. 1 approaches maximal computational complexity. For sufficiently large connectivity (K > > 1), such a circuit would exhibit computational capabilities approaching the theoretical limits imposed by the Bekenstein bound¹¹, consistent with the picture of black holes as maximally complex quantum circuits¹². These computational properties align naturally with principles of reservoir computing^38–40.

Remarkably, the leading eigenvalues governing the qubit population dynamics carry coefficients whose evolution mirrors the behavior of dark energy and matter density parameters. This correspondence suggests that information processing within AdS black holes and cosmological volume growth may share common dynamical principles rooted in nonlinear dynamics at criticality^19,24. The broader relationship between information and energy has a well-established history^41–43 and remains an active area of research, with recent work establishing connections to gravity⁴⁴ and large-scale cosmic structure^45,46. The scope of the present work is intentionally circumscribed: we examine information growth within AdS black holes and establish formal parallels with cosmological expansion, without proposing a complete cosmological model. Potential extensions to other observational tensions, such as the S8 discrepancy, through the framework’s emergent dimensions ( and ) remain speculative and require detailed calculations before any stronger claims can be made -- but it is still suggestive that even with limited iterations, , is close to the ratio in the S8 tension (, taking the mean from different sources^31,47–50.))

Methods

Derivation of the logistic equation from Eq. (1)

From the Eq. (1) with , one obtains,

9

or,

10

Applying the transformations:

Equation (10) is,

11

that after omitting primes and defining the control parameter, , results in the well known logistic map:

12

An explicit example of the leading term of the eigenvalues controlling the dynamics of the qubit population at generation

Here we show the leading term at generation , called . We can grasp how the eigenvalues behave by observing the way the term is assembled (see²⁴ for the meaning of and ),

13

The nested coefficients structure a cascade as illustrated in the Fig. 1. Given the initial coefficient values for the second order term: , the linear term: , and the independent term: ; a generation rule (shown bellow) allow us to calculate the coefficients that will form the leading term of the eigenvalues at the next iteration , shown in the second row of the figure. Now, this new set of coefficients allows for the calculation of the nested coefficients of the leading term of the eigenvalues at the iteration (third row at the same figure); a full cascade is generated by iteratively applying the generation rule. On each iteration , new terms are generated.

The generation rule

The coefficients of higher order values of the leading term can be precisely obtained from the preceding terms by the following generation rule. In general, if we know and , can be obtained because, in a generation , the term , with ; generates the polynomial

14

in a next generation . We don’t need to determine the unknowns, , because currently (see details in²⁴).

Fractal filling of the space

Calling the first two non-zero coefficients, and , and the four nonzero coefficients in the next generation, , , and (Fig. 1) the full fractal structure (but with no information about the coefficients intensities as given by the generation rule²⁴) can be recreated by the rules:

15

16

17

Any motif in the left side generates in the next generation the motifs at the right side. These rules allow us to reach a higher . Let’s define an approximation to the course grained dimension as,

18

For large enough this quantity converges to the limit set’s fractal dimension, i.e., . As a calculation for a very high is not plausible, we settle for iterating until (see Fig. 5a). It can be seen that , and those increments defined by,

19

decrease with , as (Fig. 5b). For large enough, is a measure of how the coloured component of the fractal fills an embedding space that growths as . Let’s calculate the ratio, , and remark that the inverse cascade shown in Fig. 2c satisfies, and ( for the inverse cascade). Thus,

20

This function is shown at Fig. 5c, it is qualitatively similar to the evolution of the Hubble parameter ³⁰. Equation (20) describes how the colored component of the cascade growths while considering the grained detail of the geometrical object. One may also consider an even courser measure by determining the number of motifs as defined at the left side of Eqs. (15)-(17). Let’s define an approximation to a motif dimension as.

21

and

22

where, , is the number of motifs forming the fractal set at iteration , with no distinction between dissimilar motifs, we can see that . Also, let’s define their successive differences,

23

and proceed to calculate how – in the inverse cascade – the structure at the level of the motifs grows,

24

The behavior of , and can be seen in Fig. 6. Note that also shows qualitative similarity with .

Determining the Hubble constants

The analysis was restricted to the best behavior models as shown in SI Sect. 3. It is possible to find a set of parameters yielding satisfactory fits such that . The fitted curves are depicted in the left columns of the Fig. 7 and in the SI Figs. 2 and 3; the corresponding parameters values are summarized in the SI Table 2. It is satisfied that,

25

26

and we can define,

27

But we know the posterior fitting but no . To use the posterior fitting the relative rate of change of the cosmic scale factor, , is assumed to be approached by the relative rate of change of the fractal dimension. Such an assumption is based on the following observations: i) both and depend on a parameter, time in the case of and in the case of -- in the inverse cascade elapses following the time arrow in figure 2; ii) describes how physical separations grow and, in the inverse cascade, describes how distances between components of the fractal grow; iii)) both and measure a self-similar process of space growing and iv) both and measure relative rate of change.). Therefore, 

28

As raising to is equivalent to rescale , we can write,

and

29

and substituting (25) into (24) we obtain,

30

where the prime is omitted. This is the same to say that introducing into Eq. (27) translates to an effective which is a half of the original, then the one appearing in the denominator should be times the original . To calculate a constant equivalent to the Hubble one we make the ansatz,

31

where, , because it is current time in the fractal formulation. The results calculated for are summarised in the SI Table 2, and the representation of the Eq. (30) is shown in the right columns of the Fig. 7 and the SI Figs. 2 and 3. was extrapolated to using the Julia package Interpolations.jl. To calculate an error estimation for we took the mean value of the propagated error of the Eq. (30) for the points in the high fluctuating section (see SI final section).

Supplementary Information

Below is the link to the electronic supplementary material.

Author contributions

JLCF conceived and developed all aspects of this research.

Data availability

Data used for this study has been generated using Julia codes available at https://github.com/tacitadeplata/BHInfHTens.

Declarations

Competing interests

The authors declare no competing interests.