# Holographic Dark Information Energy: Predicted Dark Energy Measurement

## Abstract

**:**

## 1. Introduction

^{3}where a is the universe scale size (size relative to today, a = 1, and defined in terms of redshift, z, by a = 1/(1+z) ). For a constant energy density the dark energy equation of state parameter, w, the ratio of pressure to energy per unit volume, must take the value w = −1, since energy densities vary as a

^{−3(1+w)}. Ideally we wish to find a model that satisfies these two requirements without recourse to exotic or unproven physics.

## 2. The HDIE Model

_{B}T ln2 of heat per erased bit to be dissipated into the surrounding environment. This dissipated heat increases the thermodynamic entropy of the surrounding environment to compensate for the loss of degrees of freedom and comply with the 2nd law. Information is not destroyed as the “erased” information is now effectively contained in the extra degrees of freedom created in the surrounding environment.

^{22}bits of stored digital data and we have the technological capacity to process a total of around 3 × 10

^{19}instructions per second in general-purpose computers [24]. We can assume that the main information erasure that occurs during the process of computing is caused by overwriting the processor’s instruction register, when each new instruction is read from memory. In this way mankind erases some 10

^{21}bits each second. At room temperature this rate of information erasure will generate a world-wide total of only 3W! This is insignificant, around ~10

^{−11}of the total electronic heat dissipation (ohmic and inductive heating, etc) of the world’s ~10

^{9}computer systems that each dissipate ~10

^{2}W. Similarly, erasing the 10

^{22}bit sum total of all man-made stored digital data would generate a world-wide total of just 3J.

^{2}energy equivalence for mass, differs in that it defines an inequality, specifying a minimum equivalent energy. Experimental measurements [26] show that energy from information erasure clearly tends towards the Landauer information energy in the limit. Then, as the aim of this work is primarily to account for the high dark energy value, we assume here a direct equivalence for simplicity.

_{B}T ln2 that depends on the quantity of information (or entropy), N bits, associated with the component and on the component’s temperature, T. In this way, we can consider the energy represented by information in the cosmos without requiring, or even identifying, processes whereby information may be actually “erased”.

#### 2.1. Stellar Heated Gas and Dust

_{B}T ln2, of the universe is then primarily determined by stellar heated gas and dust, the appropriate temperature, T, will be the average temperature of baryons in the universe. Figure 1a plots average baryon temperature, T, data and the fraction of baryons in stars, f, deduced from a wide literature survey of integrated stellar density measurements, extending the earlier HDIE work [20]. Data symbols and measurement source references are listed here: open circle [34]; open squares [35]; filled rectangles [36]; diamonds [37]; upside down triangles [38]; normal triangles [39]; crosses [40]; circles with dot [41]; filled circles [42] and blue line [43].

^{6}K, which, together with the estimate of N ~ 10

^{86}from surveys [20,30,31], provides a quantitative estimate of the present HDIE energy value within an order of magnitude of the observed dark energy, and effectively satisfies dark energy requirement 1. Note that this is dependant on our estimate of N for stellar heated gas and dust, only accurate to a couple of orders of magnitude. We find that, despite the very low bit equivalent energy, information energy can provide a significant contribution on cosmic scales, primarily because the universe’s mass has remained constant, while both the quantity of information (entropy), N, increased continually and the average baryon temperature, T, also increased with increasing star formation.

**Figure 1.**Plotted against log of universe scale size, a, and redshift, z, are: (

**a**) Upper Panel: Log plot of measured stellar densities (various symbols and blue line: see text for measurement sources) and resultant average baryon temperature, T, and the fraction of all baryons in stars, f. Red lines- best power law fits to data points are a

^{+0.98±0.1}for z < 1, and a

^{+2.8±0.3}for z > 1. (

**b**) Middle Panel: Log plot of energy density contributions: red continuous line, HDIE energy density corresponding to the red line fit in the upper panel; dashed red line, cosmological constant; blue line, mass; solid black line, total for HDIE case; dashed black line, total for the case of a cosmological constant; data symbols, energy densities derived from recent Hubble parameter measurements normalised to Hubble constant measurement at a=1; grey continuous line, the gedanken experiment considered in Section 3.2. (

**c**) Lower Panel: Linear plot of relative differences in total energy, and in Hubble parameter, between the HDIE model and a cosmological constant. The resolving thresholds of three next generation space and ground-based measurements are shown for comparison in green together with the error bar of the existing measurement at z = 2.3 (see Section 2.4).

^{+0.98±0.1}. Then, assuming the baryon information bit number, N, scales as a

^{+2}from the holographic principle, total HDIE energy for z < 1 scales as a

^{+2.98 ± 0.1}. Since universe volume increases as a

^{3}, there is a nearly constant HDIE energy density at z < 1. The HDIE equation of state parameter lies in the narrow range −0.96 > w

_{HDIE}> −1.03 which includes the specific value, w

_{DE}= −1 and thus satisfies dark energy requirement 2. This present work improves on the earlier HDIE work [20] significantly reducing the error in the temperature gradient at z<1 from what was effectively a

^{+0.98 ± 0.17}to a

^{+0.98 ± 0.10}.

#### 2.2. Dependence on the Holographic Principle

^{+0.98±0.1}at z < 1, Figure 1(a)] has to be combined with the, as yet unproven, holographic principle relating information content to bounding area, N α a

^{+2}.

^{+2}relation to all objects, including those well below their maximum entropy, remains an attractive but unproven hypothesis, and thus accounts for the main speculative aspect of the HDIE model.

^{+0.98 ± 0.1}at z < 1, closely centered round the specific T α a

^{+1}relation required to satisfy requirement 2, together with our ability above to satisfy requirement 1, provide significant support for the HDIE model. Accordingly, we continue our work below by considering HDIE primarily from a phenomenological point of view, and limit ourselves to only employing the main proposition of the holographic principle: i.e., that N α a

^{+2}.

#### 2.3. Dark Energy Predictions for z > 1

^{+2.8 ± 0.3}. Clearly HDIE energy density was increasing in this earlier period up to the point around z ~ 1, after which there has been a near constant value that we have shown can account for both dark energy requirements from that time onwards.

^{−3}(blue line), the resulting HDIE energy density contribution with the above assumption (red line), and a cosmological constant for comparison (red dashed line). The a

^{+2.8 ± 0.3}temperature dependance, z > 1, corresponds to an HDIE energy density gradient of a

^{+1.8 ± 0.3}when information bit quantity, N, is again assumed to scale as a

^{2}from the Holographic principle. Then the mean a

^{+1.8}HDIE energy density variation corresponds to an equation of state w

_{HDIE}= −1.6 for z > 1.

^{+1.8 ± 0.3}, with ±1sigma upper and lower bounds from the fit to data in Figure 1a, corresponds to a relative difference in total energy, ΔE/E, in Figure 1c that peaks at −5.2 ± 1.0% near z ~1.6−1.7. Although there is a clear change in gradient around z ~ 1 evident in the data plotted in Figure 1a, our fitting to gradients that change precisely at z = 1 may provide an overemphasized sharp transition in ΔE/E at z = 1. However, this transition should not significantly affect the size or the location of the predicted negative peak in ΔE/E at z ~1.7. At earlier times, z > 4, the higher mass density swamps any difference between HDIE and a cosmological constant. Later, as the mass density falls ΔE/E begins to reflect the difference in the energy densities of the two dark energy components, peaking at z ~ 1.7 as HDIE energy density rapidly increases as a

^{+1.8 ± 0.3}towards z ~ 1, after which time there is no difference between models.

_{0}, is the fundamental ratio between the recessional velocities of objects in the universe and their distance from us today, H

_{0}is just the present value of the more general Hubble parameter, H. The Hubble parameter, H, varies with changes in universe expansion rate over time and is therefore a function of universe scale factor, a. Since total energy density, E, is proportional to H

^{2}, (from the Friedmann equation, [48]) the HDIE model thus predicts that the Hubble parameter around z ~ 1.7 should be 2.6 ± 0.5% less than that expected for a cosmological constant explanation for dark energy.

#### 2.4. Measurement Capabilities of Next Generation Instruments

_{HDIE}= −1 for z < 1; and w

_{HDIE}~−1.6 for z >1. Now it is usual when designing these experiments to characterise any dynamic equation of state parameter, w(a), by a smoothly varying two parameter model, typically given as: w(a) = w

_{p}+ w

_{a}(1 − a) , where w

_{p}is the present value, the early value was w

_{p}+ w

_{a}, and the mid-point transition occurs at a = 0.5, or z = 1. The experimental figure of merit is then determined by how small the error ellipse is in the w

_{p}− w

_{a}plane. For example, the ESA Euclid measurement accuracies equivalent to 1 sigma error are expected to be 0.02 in w

_{p}, and 0.1 in w

_{a}up to z ~ 2 [55]. These accuracies are clearly sufficient to falsify HDIE where the nearest equivalent parameter values are w

_{p}= −1 and w

_{a}= −0.6, as compared to the cosmological constant values of w

_{p}= −1 and w

_{a}= 0. The expected difference between HDIE and the cosmological constant is small and peaks over a narrow predicted range of redshifts. A more appropriate way to identify the signature of HDIE is by a determination of the Hubble parameter, H, or total energy density as a function of scale factor, a, or redshift, z (as in Figure 1b,c). The HDIE model is therefore falsifiable [60], since a failure to observe this specific signature would clearly exclude this model.

#### 2.5. Characteristic Energy

_{B}T ln2, depends solely on temperature, T. Today, some 10% of the baryons are located in stars at temperatures ~2 × 10

^{7}K with characteristic energies ~10

^{3}eV. As the remaining 90% of baryons exist at very much lower temperatures the average baryon temperature of baryons is ~2 × 10

^{6}K, one tenth of stellar temperatures, and corresponds to an average characteristic energy ~10

^{2}eV.

^{2}= σT’

^{4}, where ρ is the universe total mass density (including dark matter), and σ the radiation constant. Substituting σ and k

_{B}by their definitions in terms of fundamental constants, we obtain the characteristic bit energy, E

_{char}= k

_{B}T’ ln2 = (15ρћ

^{3}c

^{5}/π

^{2})

^{1/4}ln2.

^{−3}eV. Note that we expect a temperature around 10 times the temperature of the cosmic microwave background, CMB, since present matter energy density is ~10

^{4}times the CMB energy density.

^{+1}with increasing star formation while the cosmological constant characteristic energy always falls as ρ

^{+1/4}, or a

^{−3/4}, with the cooling universe majority. Then it is difficult to see how the cosmological constant, with a characteristic energy falling as a

^{−3/4}, can produce a total energy that increases as a

^{+3}as required for a constant dark energy density. Similarly, the relics of the big bang, CMB, relic neutrinos, and relic gravitons, all exhibit decaying temperatures, and thus decaying characteristic energies, effectively discounting them from making a significant information energy contribution to a constant dark energy density.

^{+1}and, with total bit number, N, increasing as a

^{+2}by the holographic principle, provides the required a

^{+3}total energy variation.

## 3. Gedanken Experiment

#### 3.1. Hypothetical Computer Simulation

_{p}= 1.6 × 10

^{−35}m. Intergalactic baryons in the present universe, size l

_{u}~10

^{27}m, then require an accuracy of one part in 6 × 10

^{61}(~2

^{205}) and hence require 205 bits per spatial parameter. Similarly baryons located in giant molecular clouds, size l

_{gmc}~10

^{18}m, and baryons located in typical stars, e.g. the sun size l

_{s}~10

^{9}m, require ~175 bits and ~145 bits respectively per spatial parameter.

_{2}(a) up to their present 205 bits per parameter. The average number of bits per parameter per baryon, n

_{av}, is a function of scale size, a, and is then given by:

_{av}(a) = (1 − f(1)) log

_{2}(a l

_{u}/l

_{p}) + (f(1) − f(a)) log

_{2}(l

_{gmc}/l

_{p}) + f(a) log

_{2}(l

_{s}/l

_{p})

_{av}increased with intergalactic baryons but reached a peak value of 200.03 bits at a ~ 0.32, but then decreased due to increasing star formation to today’s value of 199.02 bits, almost exactly one bit below the peak value. This one bit loss can be simply explained by the 10% of baryons that formed stars lost 30 bits per spatial parameter, contributing a loss of 3 bits to n

_{av}, while the 90% intergalactic baryons only added 2 bits to n

_{av}between a = 1/4 and the present, a = 1.

_{av}by a further 1 bit. Then we could effectively compensate for our loss of one bit in n

_{av}and satisfy the 2nd law again. Interestingly, dark energy has indeed doubled the size of the universe, exactly as we require, since it has increased the energy density by a factor of four, corresponding to a doubling of the Hubble parameter. Figure 1b grey continuous line, uses the above relation and assumptions to show the minimum variation in total energy density that is required to ensure there is no decrease in the amount of information needed as input to our simple computer simulation during this period. This variation can be seen to lie close to that deduced from the effects of dark energy (whether due to HDIE or a cosmological constant).

_{av}. If we had used a different minimum resolution, for example the Fermi length, 10

^{−15}m, the above bit numbers would be 66 bits less but, without the doubling of universe size from dark energy, there would still have been the same reduction of 1 bit in n

_{av}. So, although there is considerable uncertainty in absolute quantity of information required for our simulation, we can reasonably say that the doubling in universe size due to dark energy was just what was required to ensure that the amount of information needed as input to our computer simulation did not decrease.

^{18}m) as the starting points for star formation. For comparison, at the two extremes of starting point either side, we would have obtained a value close to two bits drop in n

_{av}if we had considered star formation as starting all the way from the parent galaxies (size ~10

^{21}m). Alternatively we would obtain a value close to zero change in n

_{av}if we had considered that star formation only started much later, at the final pre-stellar stage of proto-stellar nebula (size ~10

^{15}m). However, given the typical star formation sequence and the timescale considered in Figure 1, it seems most reasonable to consider pre-existing giant molecular clouds as the effective beginning points for star formation.

#### 3.2. Algorithmic Information Content

^{80}baryons in the universe requires ~10

^{3}bits, giving a total baryon simulation requirement ~10

^{83}bits. This value is not very dependent on simulation resolution, whether at Planck or Fermi lengths. Then, by analogy to algorithmic information content, we see that this simulation requirement is, as expected, less than the above N~10

^{86}bits of HDIE, because significant structure, or non-randomness, exists today in the form of galaxies, stars etc.

^{5}years, the time of decoupling, the irregularities that eventually formed today's galaxies and stars were still insignificant at the level of 10

^{−5}, as evidenced in CMB today. Significant structure only began with the first stars at ~10

^{8}years. Between 10

^{8}years (a ~ 10

^{−2}) and today, 1.37 × 10

^{10}years (a = 1), the required simulation information increased at the rate of log

_{2}a, while holographic information increased at the faster rate of a

^{2}, to explain the above difference of ~10

^{3}between these two entropies today.

## 4. Implications for the cosmos

^{2}for the energy density equivalence of the matter contribution to the universe total energy density budget.

^{+0.98±0.1}so closely follows a

^{+1}since z ~ 1 to provide the near constant HDIE energy density, −0.96 < w

_{HDIE}< −1.03. If star formation had continued to proceed at the earlier faster rate, then it would have continued the steep a

^{+2.8 ± 0.3}average baryon temperature increase after z ~ 1. This would have increased HDIE dark energy well above its present value, lead to much greater acceleration and greater expansion, but in turn, would have resulted in much less star formation. It would appear that since z ~ 1 there has been a balance, or feedback, between expansion acceleration and star formation. This has naturally maintained the star formation rate close to a

^{+1}for a constant dark energy density. Note that the reduced rate of star and structure formation starting at z~1 was previously attributed to the onset of acceleration [67]. Thus HDIE can provide a natural explanation for the reason why w

_{DE}= −1 since z ~1.

^{−2}, and, assuming the total information, N, continues to follow the Holographic principle as a

^{+2}, provides a limiting average baryon temperature, T, variation of a

^{−1}. Thus, acceleration due to HDIE will continue, providing T does not fall off more steeply than a

^{−1}. Computer simulations of future average baryon temperatures, T, up to a = 200 [68], predict a levelling off of T since f(a) is limited by definition to f(a) < 1, with a slow eventual fall as star formation ceases, but falling less steeply than the threshold gradient of a

^{−1}. Thus acceleration should continue, until at least the universe has increased in size by a factor of 200.

## 5. Summary

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**MDPI and ACS Style**

Gough, M.P. Holographic Dark Information Energy: Predicted Dark Energy Measurement. *Entropy* **2013**, *15*, 1135-1151.
https://doi.org/10.3390/e15031135

**AMA Style**

Gough MP. Holographic Dark Information Energy: Predicted Dark Energy Measurement. *Entropy*. 2013; 15(3):1135-1151.
https://doi.org/10.3390/e15031135

**Chicago/Turabian Style**

Gough, Michael Paul. 2013. "Holographic Dark Information Energy: Predicted Dark Energy Measurement" *Entropy* 15, no. 3: 1135-1151.
https://doi.org/10.3390/e15031135