Information Dark Energy Matches DESI and DES Measured Parameters But Has Not Reached a Maximum Yet

1. Introduction

John Wheeler [1] considered information to be more fundamental than matter and used his slogan “It from Bit” to emphasize his view that all things physical are information-theoretic in origin. Rolf Landauer [2,3] showed that “Information is Physical”, as each bit of information has a minimum equivalent energy of k_B T ln 2 at temperature, T. Landauer’s principle has now been proven experimentally for both classical bits and quantum qu-bits [4,5,6,7], and, linking information to thermodynamics, provides an important solution to the problem of Maxwell’s Demon [8].

A “Foundational Principle” proposed by Anton Zeilinger [9] considers the attributes of all particles correspond to elemental systems, at their most fundamental level. These are effectively simple 1bit, ‘yes’/’no’, results of experimental inquiry. In a simple universe without star formation, the Landauer equivalent energy of a Zeilinger elemental particle attribute has been shown [10] to be defined exactly the same as, and with the same value as, the characteristic energy of a cosmological constant [11]. This implies a strong similarity between information energy and dark energy.

Landauer equivalent energy has been shown to contribute a significant Information Dark Energy, IDE [12,13,14,15,16,17] to the universe. The information energy contribution of each phenomena is proportional to the product NT, where N is the information/entropy bit total and T is the representative temperature. Figure 1 summarizes published entropy estimates and typical temperatures for the main entropy components to the universe [17,18,19,20]. The strongest energy contributions are provided by stellar heated gas and the intra-cluster gas within galaxy clusters, with total information energy approaching 10⁷⁰ J, the total mc² equivalent energy of the 10⁵³ kg baryons in the universe.

A key test for any proposed source of dark energy is to compare the predicted dark energy density variation over the history of the universe with the measured dark energy density history. The latest results of the Dark Energy Spectrograph Instrument, DESI, [21] and the Dark Energy Survey, DES, [22] provide evidence for a dynamic dark energy, evidence that excludes the previously assumed cosmological constant at a level >3.5 σ.

A previous IDE publication [17] only compared IDE against the initial data from DESI and DES, whereas the present work applies the updated final published DESI and DES results. Two versions of the relation between temperature and mass of astrophysical objects are applied here; one direct from a survey of galaxy and galaxy cluster measurements, and one derived from the DESI and DES results, assuming the IDE source of dark energy. The IDE prediction based on a survey of Stellar Mass Density, SMD, measurements is compared against the DES and DESI results. Also the SMD variation required for IDE to explain the DES and DESI results is compared against the SMD measurements. We find a limitation in the commonly used parameterisation of the dark energy equation of state parameter at low redshifts. Proceeding without parameterisation leads us to discover a different IDE history and identifies a future measurement that enables IDE to be falsified.

2. Information Dark Energy

2.1. Temperature Dependence on Object Mass

The Landauer equivalent energy of each bit in any system is solely dependent on the system’s temperature. We find that the typical temperatures, T, of several astrophysical structures ranging in mass, M, from 0.1 solar mass, M_⊙, to >10¹⁵ M_⊙ , each follow a similar relation. The results of a large survey of measured galaxy, galaxy group, and galaxy cluster temperatures [23] have been shown, on average, to closely follow the T α M^p relation over the range <10¹² M_⊙ to >10¹⁵ M_⊙ with an overall value of p=0.606, illustrated in Figure 2 A & 2B. In Figure 2 C we see that the temperature of stars, from 10^-1 M_⊙ to 3 x 10¹ M_⊙ in the seven main sequence spectral classes also follow the same relation, with the same value of p. In this work the SMD measurements [24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51] used previously is extended to include measurements at universe scale sizes, a<0.2, [52,53] where a is related to redshift, z, by a=1/(1+z), and today a=1.

2.2. CPL ω₀-ω_a Parameters

The time variation of the dark energy density is proportional to a^-3(1+ω) , where ω is the dark energy equation of state parameter. Any dynamic form of dark energy requires a time varying value of ω(a). In order to account for a dynamic dark energy most experimental measurements of dark energy have adopted the common CPL [54] parameters, ω₀ and ω_a , where ω(a)= ω₀ + (1-a) ω_a . The most recent DES and DESI dark energy results show ω₀ > -1 and ω_a < 0, significantly different from a cosmological constant ( ω₀ = -1, ω_a =0 ). It is therefore convenient to initially predict the IDE source of dark energy also in terms of CPL parameters.

2.3. Predicted IDE in ω₀ – ω_a Space

The SMD is a universe-wide average mass density that changed with time and is thus a function of universe scale size: SMD(a). Then we expect that IDE(a), being proportional to temperature, to vary as IDE(a) ∝ SMD(a)^p. At the same time each CPL combination of ω₀ - ω_a values defines a curve of dark energy density as a function of scale size, IDE(a) α a^{-3(1 + ω(a))} . Then, assuming an IDE source of dark energy, the p value transfers the ω(a) curve of each ω₀-_-ω_a combination to a SMD(a) curve whose measure of fit to the measured SMD(a) values [24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53] is given by a R² coefficient of determination. The coefficient R² is defined as R²=1-(RSS/TSS), where RSS is the residual sum of squares, and TSS is the total sum of squares.

There are four DESI and DES dark energy derived ω₀ - ω_a combinations listed in Table 1. DESI combines Baryon Acoustic Oscillation, BAO, and Cosmic Microwave Background, CMB, measurements with three different sets of supernova measurements, Union3, DESY5, and Pantheon plus, to provide three ω₀ -ω_a values [21]. The most constrained DES ω₀ -ω_a value is listed for a combination of BAO, Supernovae, SN, and CMB, measurements [22].

In Figure 3. the value of R² is plotted as a function of p for each of the four experimentally derived DESI and DES ω₀ - ω_a values. The grey line is the average R² of the four values. Shown below are the values of p determined from the temperatures and masses of galaxies and clusters using X-ray and lensing measurements [23]. Figure 3. enabled us to select two data driven values of p to better illustrate the similarity of dark energy history to star formation history. The value p=0.606 was determined solely by the overall p value of actual mass and temperature measurements of galaxy and clusters (Figure 2 ) [23]. The value p=0.461 is determined from the maximum R² of the average of all four DES and DESI measurements, assuming IDE (Figure 3). In Figure 3 the value p=0.606 coincides with the maximum of the single DES measurement, and the value p=0.461 coincides with the maximum of the DESI BAO+CMB+Union3 measurement.

For all combinations of ω₀ and ω_a values in steps of 0.01, over the wide range +0.9 to -3.0, we found the best fit to SMD(a) measurements (maximum R² ) listed in Table 1 for the two chosen values of p. Both of these ω₀ - ω_a combinations lie well within the range determined by DESI and DES. These maximum R², best fit values are listed in Table 2, together with the R² values for the four DES & DESI ω₀ - ω_a values. The DES R² values at p=0.606 and the DESI/Union3 at p=0.461 are high at 0.947, close to the maximum of 0.953 over all ω₀ - ω_a space for fits to this survey of SMD(a) measurements.

In Figure 4 we illustrate the clear similarity between star/structure formation and dark energy histories. For each of the two selected values of p we provide 2 plots. The right-hand plot shows the four DESI and DES 68% and 95% likelihood plots as a function of ω₀ and ω_a . Superimposed are the >0.90 and >0.94 R² contours of IDE over all ω₀–ω_a space derived solely from the survey of measured SMD(a). The location of centre points of DESI and DES plots (listed in Table 1) are highlighted by small coloured circles together with the point of maximum R² of the IDE plot. The left-hand plots illustrate how well the curves corresponding to those highlighted ω₀ - ω_a values fit the SMD measurements.

Figure 4 (Left-hand plots) and the R² values in Table 2 show that the DES most constrained result and the DESI BAO+CMB+Union 3 result both make a good fit to the SMD(a) measurement survey with curves closely following the highest R² red curve. Moreover, the range in ω₀ – ω_a space that fits SMD(a) measurements with R² >0.94 makes a good fit to the DESI and DES dark energy measurements (Figure 4. Right-hand plots). This suggests that the dark energy history is directly related to the history of astrophysical structure temperature and formation, as we expect for an IDE source. Figure 5 provides an interpretation of predicted IDE in terms of the DES & DESI measurements based on a plot used previously [55] to interpret DESI results.

Both the IDE prediction and DESI/DES measurements are consistent with a dark energy that was phantom (increasing energy density) throughout the majority of universe history. Using CPL, it appears that both IDE predicted and DESI/DES measured energy densities are now starting to decrease after recently reaching a maximum. The DESI/DES measurements are both consistent with a maximum located in the range 0.2 < z < 0.6. Figure 4 (right-hand side) illustrates a difference of the predicted IDE contours in ω₀ - ω_a space between the two selected values of p. Figure 5 (thick black line) further illustrates this by showing how the maximum R² point moves over a wider range of p values. The predicted maximum IDE energy density remains around z~0.2 over this wide range of p values, close to the time of the DES/DESI measured maximum dark energy density. The application of CPL parameters has shown that IDE has a history that closely matches the dark energy history determined by the DES and DESI instrument consortia.

2.4. A Different IDE History When Independent of CPL Parameter Limitations

While the use of CPL parameters has facilitated comparisons between different experiments, and between experiments and theory, there is no reason to expect that the actual dark energy can be fully described over a wide redshift range by any particular CPL ω₀ - ω_a combination. After a search over a range of ω₀ - ω_a values the red curve in Figure 4. (left-hand side plots) makes the overall best CPL fit (after the T α SMD^p transformation) to the SMD(a) data. The majority of SMD(a) data are located at redshifts z>0.2, beyond the maximum, but much closer to z=0 the red curves for both values of p clearly lie below SMD data points.

In contrast, the polynomial curve fit to SMD(a) (grey curve) fits data across a wider range of redshifts. That fit just shows the rate of star formation reducing as z→ 0, but arriving at z=0 just before reaching a constant level of stellar mass density, i.e. without exhibiting a maximum. Other surveys of SMD measurements [56,57,58] are all consistent with the grey line polynomial fit to the SMD data in Figure 4, and similarly the highest SMD value to date also occurs at z=0. The polynomial fit to all SMD data points, grey curves in Figure 4, is given by Eq(1):

log₁₀ (SMD(a)) = 0.19556 b³ - 2.5267 b² + 0.21949 b + 8.6446

(1)

where SMD is in units of M_⊙ Mpc^-3, and b = log₁₀ ( universe scale size, a ).

Here we apply the value p=0.606 from the direct measurements of galaxy, and galaxy cluster temperatures and masses [23] in order to avoid any dependence on CPL. Then as IDE(a) α SMD(a)^0.606 and IDE(a) α a^{-3(1 + ω(a))}, the IDE equation of state parameter is described in terms of redshift, z, by Eq (2), (continuous black curves in Figure 6A & B):

ω_ide(z) = - 0.002703 z³ + 0.04409 z² - 0.3194 z - 1.074

(2)

Figure 6A shows the polynomial fit to SMD(a) provides an IDE that exhibits a primarily phantom behaviour, ω< -1, with values of ω_ide similar to both the DESI/DES measured CPL, and IDE predicted CPL values over a wide range of redshifts, z≳0.5. However, at lower redshifts, z≲0.5, the polynomial fitting derived IDE differs significantly by tending towards a constant dark energy density (ω_ide= -1.07 at z=0) and has not yet reached a period of decreasing energy density. This clearly contrasts with the CPL parameters describing DESI/DES results and the CPL description of SMD data in Figure 6A which all cross the ω=-1 (maximum dark energy), changing from increasing (ω<-1) to decreasing (ω>-1) energy density while z>0.

This difference between polynomial and CPL fitting is primarily due to CPL only being able to provide a limited set of ω(a) curves. It is therefore possible that this effective bias of CPL has lead to a false impression, not just with IDE but perhaps also with the dark energy measurements. In order to illustrate this aspect of CPL we checked to see how CPL parameters would describe the ω_ide(z) curve of equation 2 by taking 280 values of ω_ide(z) distributed evenly in the range z=0 to z=7. This set of values was tested against all ω₀ and ω_a combinations, each in the range +1 to -3. The R²>0.80 and R²>0.85 contours are shown in Figure 5 by the dashed and continuous black lines respectively. The maximum of R²=0.86 occurred at ω₀=-0.77, ω_a =-1.33. Not surprisingly, these values from CPL fit to the polynomial are similar to the best CPL, ω₀=-0.76, ω_a =-1.29, when fitting CPL directly to the SMD data in Figure 4 (fitting to SMD^0.606). The ω(z) curve corresponding to the maximum ω₀ – ω_a combination (black dashed line in Figure 6B) closely follows the other CPL derived histories in exhibiting a recent maximum dark energy density while equation 2 without parameterisation clearly has yet to reach a maximum (black continuous line in Figure 6A & 6B). This clear difference illustrates the limits of using CPL parameters.

The use of CPL parameters has lead to interpreting DES and DESI data as exhibiting a maximum dark energy density, 0.2<z<0.6. Further work has shown [59] a correction to supernova data due to progenitor age bias may be required (dashed coloured lines in Figure 6A) that would even further emphasize the present deceleration period, but again derived through the use of CPL.

The ω_ide(z) of equation 2 is derived directly from the stellar mass density measurements assuming p=0.606. This value of p is the overall value found in the review [23] of observed halo mass-temperature relations for galaxies, groups, and clusters, and differs from the theoretical expected value of p=2/3. The effect of changing p to 0.5 and 0.7 is illustrated in Figure 6B, showing that the precise value of p is not too critical. The ω_ide(z=0) values are similar at -1.061, -1.074 and -1.086, for p=5, 6.06, and 7 respectively. The main differences occur at higher z values where the weaker effects of dark energy are less significant in the presence of the much higher matter energy densities at that time. The ω_ide(z) behaviour of IDE is similar to the dark energy freezing/thawing models [60,61] that describe the dark energy equation of state parameter in terms of generally moving either towards or away from ω=-1, rather than crossing ω=-1.

2.5. Information Dark Energy Is Falsifiable

As the IDE proposed explanation of dark energy differs significantly from the usual sources considered (see, for example, review [62]), it is important to identify a dark energy characteristic predicted by IDE that will be significantly different to the present consensus. A future measurement of such a feature then enables IDE to be falsified. The polynomial curve fit to all SMD(a) measurements is more consistent with the actual maximum IDE energy density occurring in the near future. This prediction that dark energy density has not yet passed a maximum provides a means by which IDE can be falsified when a detailed measurement of dark energy ω_ide(z) is eventually obtained, independent of the above CPL parameter limitations.

3. Discussion

The expanding universe has lead to overall matter and radiation energy densities naturally falling as a^-3 and a^-4 respectively. In contrast, both the stellar mass density and the dark energy density have increased over time, in a way that is consistent with a dark energy source directly related to star and structure formation. Within the uncertainties of entropy estimates (Figure 1), IDE could be strong enough to account for the present dark energy density. Future dark energy experiments should be able to compare measured ω(z) values directly against the ω_ide(z) of equation 2, without applying any parameterisation.

While IDE fits a CPL description over redshifts, z>0.2, Figure 4 & 6 show that the predicted behaviour of IDE over all redshifts, including z<0.2, is better described by a polynomial curve fit. The different behaviour of IDE at low redshifts, yet to cross ω=-1, provides a clear prediction that allows IDE to be falsified. The significant difference between the ω_ide(z) of equation 2 and the ω(z) from the CPL parameterisation of equation 2 (continuous and dashed black lines in Figure 6B, respectively) illustrates the limitation of CPL parameterisation. Then it is possible that the existing interpretation of DESI/DES data as providing evidence for a recent dark energy maximum may also be purely the direct result of CPL bias. The actual dark energy equation of state parameter could follow a redshift dependence closer to equation 2, regardless of the source of dark energy.

The dark energy epoch of accelerating expansion is considered [63,64] responsible for the observed reduction in star and structure formation rate, clearly evident in the SMD(a) data in Figure 4. If IDE, driven by star formation, is the main source of dark energy it appears to have fed back via accelerating expansion and has begun to effectively limit itself. Star formation is thought to be already within 5% of the maximum possible [65]. After maximum star formation IDE energy density will fall, with deceleration and eventually contraction leading to either a “big crunch” end to the universe, or a continuously cyclic “bouncing universe” [66].

The potential of IDE to solve several problems of cosmology has been covered before [17], and nine aspects are listed briefly again here for completeness, and summarised in Table 3, for comparison against other proposed dark side explanations:

• In Figure 1 information/entropy estimates show that IDE might account for the value of the present dark energy density.
• Figure 4, Figure 5 and Figure 6 show that the history of IDE is consistent with the latest DESI and DES results describing the history of dark energy when all use CPL parameterisation.
• Again using CPL, Figure 5 shows IDE fits the latest dark energy data much better than Ʌ. Then the Cosmological Constant problem could be resolved by assuming Ʌ=0, a more likely value [67].
• The most likely time for intelligent beings to have evolved is around maximum star formation and thus maximum IDE source of dark energy. An IDE source of dark energy naturally resolves the “Why now?” Cosmological Coincidence problem.
• The primarily phantom time history of IDE is similar to several dark energy histories [68,69,70] that have been shown capable of resolving both H₀ and σ₈ tensions. Over a range of redshifts, 1<z<5, predicted IDE ω values, -1.3>ω>-1.9, in Figure 6, centre around ω~-1.6, a condition that can account for the H₀ tension [71] .
• The latest dark matter search has still not registered a single confirmed dark matter particle detection[72]. As the information energy of matter is clumped in astrophysical structures at a similar energy density to matter, IDE may account for some dark matter attributed effects. General relativity shows that space-time will be equally distorted by mass and an equivalent energy. This equivalence has also been stated specifically as a ‘mass-energy-information equivalence principle’ [73].
• The location of baryons in galaxies has been shown to fully specify the location of dark matter attributed effects [74,75], as to be expected if IDE accounts for such effects.
• Early emerging massive galaxies and clusters of galaxies require a non-linear accelerating structure formation. This would be expected from increasing local IDE matter effects but not with the linear hierarchical structure formation of ɅCDM [76].
• Equation 2. and Figure 6B predict a difference by which IDE can be falsified.

4. Summary

This work emphasizes simplicity, wielding Occam’s razor, and naturalness, relying on mostly proven physics. The approach has been primarily driven by data and indicates a strong connection between stellar mass density and dark energy histories. We have shown that IDE predicts a dark energy history that matches the dark energy measurements of DESI and DES with a clear overlap in ω₀ – ω_a space. But the use of CPL parameters, with the associated limited range of ω(a) functions fails to model IDE accurately at low reshifts (z<0.2). The predicted different behaviour of IDE at those low redshifts, yet to reach a maximum dark energy density, provides a means of falsification. IDE has the potential ability to resolve many dark side problems and tensions.