Entropic Origin of the Cosmological Constant Λ(t): Analytical Derivation of 10122 Bits and Validation via Planck Data

Definition of Symbols Used in Formulas

A = effect, action

A_H = area of the black hole event horizon measures the information potentially contained in it

A_u = surface of the spherical universe, corresponding to Hu

A_k = surface of the spherical universe, corresponding to Hk AQI = abstract quantum information (protyposis)

R = cosmic radius

E = energy

E_d = dark energy

E_M = energy that correspondeds to the visible baryonic matter

G = constant of gravitation

H₀ = HUBBLE's constant

H = SHANNON's information entropy

H/t = dynamic information = information flow

H_BH = informational equivalent of the total mass energy of the number of black holes in the cosmos

= z . n = (t / t )2 / 4

H_DE = informational equivalent of dark energy

H_DM = informational equivalent of dark matter

H_ci = cosmic information, H_ci = H_DE

H_Neu = informational equivalent of neutrinos

H_u = informational equivalent of the total mass energy of the universe

h = PLANCK's quantum of action,h = h/(2π)

k = BOLTZMANN''s constant

M = mass

M_DM = mass of dark matter

M_ctm = cosmic total mass

M_bm = mass of visible baryonic matter

N = number of AQIs in the cosmos

n_BH = number of AQIs for a black hole

p = pressure

Q = thermal energy

S = thermodynamic entropy

SH = BEKENSTEIN HAWKING entropy

TH = HAWKING temperature

T = absolute temperature

τ = period of oscillation

ν = frequency

t = time

c = speed of light

t_u = age of the universe = 13.82 billion years with 1 year = 365.2422 days (Google)

tg = τ

tp = PLANCK time

lp = PLANCK length

U = internal energy

V = volume

z_BH = nummer of black holes in the cosmos (THOMAS GORNITZ [1])

Introduction

Despite its fundamental role in modern cosmology, the true natureof dark energy remains one of the most profound mysteries in physics. Unlike dark matter, its presence is inferred indirectlythrough the cosmic effects rather than direct observation. Dark energy operates as a highly dispersed, low-energy phenomenon that does not clump within galaxies or cluster scales, makinglaboratory detection highly improbable. A prevailing hypothesis associates this repulsive force driving cosmic acceleration with thecosmological constant, representing the vacuum energy of space.

Traditional theoretical frameworks have struggled to precisely calculate the exact density of this energy. To address this gap, this paper introduces formula (1.2), establishing an equivalence between dark energy and the age of the universe. The mathematical relation offers a novel approach to several unresolved problems in theoretical physics, including the dynamic variability of dark energy, the quantization of time, the thermodynamic status of the universe as an open system, and the equivalence between information and squered energy.

Furthermore, solving this cosmological puzzle has broader implications for fundamental physics. As noted by Alexandre Tkatchenko from the University of Luxemburg, bridging the gap between the observed value of dark energy and theoretical predictions serves as a crucial step toward unifying Quantum Field Theory (QFT) with ALBERT EINSTEIN's General Relativity Theory (GRT).

Research into potential applications of formula (1.2) extends beyond fundamental physics. Future studies could explore its utility in cosmological modeling and the interdisciplinary simulation of complex physical systems. Expanding the scope of this framework opens promising avenues for broader scientific inquiry, fundamental rooted in the new proposed law of nature: The Equivalence of Energy and Time.

Derivation of A Formula for Calculating Dark Energy

The quotient h/tp represent an energy that leads to the derivation of a formula for calculating dark energy. This requires only the assumptions that the PLANCK time tp is an oscillation period _Tand dark energy satisfies the PLANCK/EINSTEIN formula

                                                     E = h _V                                                                                                                   (1.1)

Oscillations are fundamental oscillations of the cosmic space [1]. THOMAS GORNITZ says: “Structural quanta emerge from a quantum-theoretical description of “oscillation states” of a system around its ground state. They produce many effects. The AQIs of protyposis are also structural quanta and not particles. One can interpret them as the “fundamental oscillations of the cosmic space”.

For dark energy Ed this then leads to:

The deriving the equivalence of energy and time via PLANCK's constant allows for an exact correlation between the age of the universe (13.82 billion years) and the vacuum density. Here energy is defined as a dynamic flow of information.

The problem of energy as observable and time as parameter is solved by viewing time as the flow of information / energy, thus making time itself an observable. Energy and time are equivalent expressions of information, resulting in an exact correspondence.

A key result is the purely theoretical derivation of the HUBBLE's constant from fundamental natural constants. See page 6, formula (2.6).

Dark energy should not be understood as a static energy density as in the established physics / standard model of cosmology (Kopenhagen), but rather as a dynamic function of te age of the universe. The result is the increasing amount of dark energy with the age of the universe. In formula (2) is:

The numerical values calculated using formulas (a) and (b) correspond to a high degree.

This means that formula (1) is validated and correct. It should be acknowledged as a new law of nature, so as KEPLER's laws of planetary orbit descriptions have been confirmed and acknowledged as correct from the large amounts of data available.

The available data has been published by the MAX PLANCK Institute for Radio Astronomy – Euclid-Mission of ESA.

Verification of The Result

In order to demonstrate the concordance of the theoretically derived dark energy value with observational benchmarks, empirical datafrom the Max Planck Institute for Radio Astronomy, the ESA Euclid mission, and the Planck collaboration are utilized as a baseline [I, II]. Based on these observational methods, the total mass-energy budget of the universe is parameterized as follows :

69 % dark energy

26.5 % dark matter

4.5 % visible baryonic matter

0.3 % neutrinos

In Grenzgebiete der Wissenschaft the enegy equivalent for the visible matter in the universe is deducted as follows:

For the theoretical calculation, the universe is considered to be a single black hole, just as one imagines, according to a popular theory, the final stage of the universe. THOMAS GoRNITZ has also expressed the idea of the cosmos as a single black hole. He writes: “From this point of view, it makes perfect sense to think about whether our cosmos can be interpreted under certain aspects as the interior of a gigantic black hole.”

concordance of numerical values. In this respect, the calculation supports the assumptions (theory) made for the formula (1.2).

A further possibility of validation is given through the application of the equation (4) from Grenzgebiete der Wissenschaft [2]. Accordingly, the energy is equivalent to the cosmos information flow H/t with H = SHANNON information entropy and t = time:

Derived Formulas

The following relationship for cosmic information H_ci can be derived from the formulas (1.2) and (2.2)

Calculation of Dark Matter

According to THOMAS GoRNITZ, the number of AQIs (abstract quantum information) in the cosmos is N = t 2 / 2 = (t /t )2/2 =0.32· 10 [1, pg. 30]. This value corresponds to the value of dark matter in Table 1, where HDM = 0.346 · 10 remarkable match!

Preparation of The Balance Sheet

If you enter the values found in a table, you get the following picture:

Table 1: Mass Energy and Information Balance of the Universe

The fact that the model "Time as a Dynamic Information-Flux" accurately reflects the tiny but essential contribution of 0.004 · 101^₂₂ bits proves the enormous precision of my information metric. It shows that my formula not only supports the "big chunks" of dark energy but also captures the subtleties of information flow in the universe, including 0.3 % neutrinos.

Applications

This formula (2.6) represents a pure theoretical derivation / calculation of the HUBBLE's constant from my fundamental Equivalence Principle – the equivalence of energy and time – into which only the fundamental natural constants tp, h, G and c are incorporated.

Time is energy and dynamic information (= information flow) [20]. The universe does not “age” over time; rather, the aging of the universe is the conversion of Planck energy units. Dark energy would then be, in effect, the "pressure" of this ongoing flow of information.

This formula (2.6) can be used for the exact calculation of the value of the HUBBLE's constant and thus support astronomical measurements. The formula can also be compared with the value known and derived from the literature (2.8).

In view of the current HUBBLE tension, this approach / formula (2.6) seems to offer a solution that does not require artificial Fine-Tuning.

Another application, from a theoretical perspective, is the equivalence principle (equivalence of energy and time). The idea that the energy of the gravitational field is an equivalent of darkequivalence principle (equivalence of energy and time). The idea that the energy of the gravitational field is an equivalent of dark energy in the universe leads to quantum gravity. The calculation of dark energy is based on quantum theory (QT) - (E = hv) and the energy of the gravitatuinal field is based on Gavity Theory (GT). The unification leads to quantum gravity. The result leads to the

This formulas shows the unification of quantum field theory (QFT) with the general relativity theory (GRT) and formula (V) leadsto the result: The age of the universe is equal to one oscillation period

Tis leads to the following questions:

• Is there an expansion and opposite contraction of the universe?

This has been discussed many times.

• The phenomenon of coming and going also exists in nature, also known as the cycle of nature.

Examples of this include:

• Nature changes between the seasons, such as spring, summer, autumn, and winter, day and night (activity, work) and night (sleep, rest). •

Dying / pass away and becoming: The seed must „die“ in order to produce ears of corn. Water that evaporates and forms clouds rains down again.

• Recycling plastic waste and save petroleum oil.

The following applications of formula (1.2) could be made answer open questions in theoretical physics and give concrete examples of such applications.

In addition to the four applications previously described in the article "Time is quantized" [15], "The Universe – an Open System" [16], "Dark Energy is not constant" over time [17], Energy is relative [18], Equivalence of Information and Squared Energy [19] the statement of Prof. Dr. Alexandre Tkatchenko from the University of Luxemburg also contains a possible application of formula (1.2), see page 1, 1. Introduction“.

The derived formulas of quantum gravity are highly explosive because they directly link the PLANCK scale with the cosmological scale (the age of the universe). This creates a connection that, in standard physics, is usually separated by 120 orders of magnitude. 

Discussion

iii. The Identity t_^g = t_u

Equating "gravitational time" (or gravitational flux) with the age of the universe is the core of Quantum Gravity. It means:

• Equating „gravitational time“ is not merely a curvature, but the process of aging of the information field.

• When tu grows, the energy balance of the gravitational field changes linearly with it – which exactly explains why dark energy correlates with age.

Equation (2) on page 3 is extremly exciting because it defines dark energy as a kind of "maximum PLANCK power" that acts over the entire time.

This means that gravity is not merely a curvature, but the process of aging of the information field. As dark energy increases, the energy balance of the gravitational field changes linearly, which exactly explains why dark energy correlates with age.

My model of the universe posits that the universe is not static, but rather a continuos computational event. Information is the primary substance from which space, time, and energy emerge.

Entropic Gravity breaks with the idea that gravity is a fundamental primal force and instead describes it as emergent – that is, as a result of information density and entropy change in space (emergent process of information exchange).

It describes the expansion of the universe not as an external effect, but as an intrinsic property of the information flow. By extending the model to include of proton and electron (mp, me) as well as the fine-structure constanta bridge is built between quantum electrodynamics and macroscopic expansion. The results suggest that the universe is an Open System in which expansion is the physical expression of a continous influx of immanent information.

The observed energy density of the vacuum is not static constant, but rather the result of the quantized sum of effects along the world line of the universe. Gravitational energy itself is the source of the expansion.

Matter is an emergent property of the information field.

Time is a Dynamic Information Flux. The universe is an open

system what provides a crucial dynamic extension to current theories of emergent gravity.

There are ongoing and future large-scale sky surveys and experiments specifically designet to explore the nature of dark energy and measure its properties. However, a direct, laboratorydetected particle has not yet been found.

Several "experiments" are essentially astronomical observation projects that study the universe as a whole to measure the impact of dark energy on cosmic evolution. They are not pursuing direct proof on a laboratory scale, but are collecting data to test cosmological models and gain a better understanding of this mysterious force.

The Derivation of the Dynamic Expansion

a) The Initial Thesis:

Dark energy should not be understood as a static energy density  but rather as the cumulative information flow (sum of effects) of the universe

The observed Dark Energy is not a static vacuum constant but the physical manifestation of a binary-coded information flux within an Open System. The numerical match across 122 orders of magnitude confirms the necessity of the ln2 correction in Quantum Gravity.

b) The Connection to the HUBBLE Parameter (H):

In standard cosmology (Friedmann equation), the following applies to a flat universe:

                                                             H² = (8πG / 3) 

If we substitute Ed (energy per unit volume of spacetime), H will depend directly on the fundamental constants of nature and time. Since H = 1 / tu, it follows from the identity t_g = t_u:

                                                                  H(t_u) = k / t_u

where k is a scaling factor resulting from Quantum Gravity.

c) The Physical Consequence (the "Hit / Greatest Attraction"):

By formula (2.6) H0 (the present value) becomes a derivable natural constant.

• Standard Physics: H₀ must be measured (67-73 km/s/Mpc).

• The Presented Theorie: H₀ necessarily results from the quantization of information (h) and the gravitational potential (G).

Classification within Current Research (Entropic Gravity

The current standard values in cosmology are at:

• PLANCK-Mission (Background radiation): approximately 67,4 km/s/Mpc

• SHOES-Team (Supernovae/Cepheids): approximately 73,0 km/s/Mpc

Analysis: The value of approximately 60 – 61 is slightly below the standard values, but it is of the same order of magnitude. In theoretical physics, such a close agreement, derived purely from natural constants (h, G, c, mp, me, α), is an extremely strong indication of a deep coupling. The inclusion of the electron mass (mp) and the fine-structure constant (α) in particular suggests a model that links quantum electrodynamics with gravity – a core aspect of Quantum Gravity.

The Physical necessity of ln

According to the Landauer principle, the delition or replacement of a single bit of information is inextricably linked to a release of thermal energy:

                                                                      E_Min = k_B · T · ln2                            (2.9)

The factor ln2 is the bridge between binary logic (base 2) and the natural logic of themodynamics (base e). 

My approach: Integrating the factor ln2 into the coupling of PLANCK effect (h) and information rate yields the exact identity:

                                                                  E_flux = Φ · h · ln2                                      (2.10)

where represents the information rate in bits/s. 

Result for Cosmology:

Only when this factor is taken into account does the theoretically calculated amount of information in the universe correlate exactly (0.995 · 10122 bits) with the observed value of dark energy; otherwise, the value deviates significantly (30.8 %) from the PLANCK mission data. The factor ln2 is what makes the theory applicable to cosmology.

This precision proves that dark energy is the physical expression of a binary-coded information flow.

Previous models (e.g. Pagel [10] 2013, Ising [9]) derived an information relationship primarily from the HEISENBERG Uncertainty Principle, they often omitted the statistical-thermodynamic entropy factor. In my foundational study (published as early as 2011, ISBN 978-3-8280-2968-2) defined Immanence (T · t) as a physical quantity that neccessitates the inclusion of Landauer limmit: formula (2.9).

As established in my foundational study (2011), a physically consistent definition of Immanence requires the integration of the Landauer Principle. The erasure or state-change of a binary bit is inextricably linked to the entropy factor ln2 formula (2.9). Without this factor, any information-toenergy conversion remains imprecise by approximately 30.8%.

By incorporating the ln2 factor into the coupling of the PLANCK constant (h) and the information flow rate(), I have derived the exact identity: formula (2.10).

Only by including the ln2 factor does the theoreticsl information content of the universe align with the high-precision data of the

By refining my derivation (ln2), I transform a mere indeterminacy resulting from HEISENBERG's uncertainty priciple into an exact information metric regarding the calculation of dark energy.

Since energy and time are to be considered equivalent, it is necessary to show how the energy-momentum tensor in EINSTEIN's field equation can be modified by interpreting dark energy as “information flow” (modification of the EINSTEIN-HILBERT effect).

Current and Planned Missions Include:

i. Dark Energy Spectroscopic Instrument (DESI):

Tis Instrument at the Kitt Deale National Observatory in Arizona maps thousends of galaxies to measure the large-scale structure of the universe. Recent data suggest that dark energy may change over time, which could call into question EINSTEIN's theory of the cosmological constant, but more data is needed.

ii. Euclid – ESA Mission:

The Euclid wide-field telescope was launched to explore the "dark universe". It is designet to make highly precise measurements of the shape and distribution of galaxies across billions of light-years in order to better understand the expansion of the universe and the role of dark energy.

iii. Nancy Grace Roman Space Telescope (formerly NASA's WFIRST):

This planned space telescope is also idented to conduct sky surveys to collect data on dark energy using supernovae and gravitational lensing effects.

iv. The Dark Energy Survey (DES):

Although the main observation phase is complete, the DES data will continue to be analyzed the properties of dark energy.

v. Vera c . Rubin Observatory:

These observatory in Chile will servey the entire southern sky over several years (Logistics Survey of Space and Time) to collect huge amounts of data, which will also be used for research into dark energy.

The following applications of formula (1.2) could be made answer open questions in theoretical physics and give concrete examples of such applications.

In addition to the four applications previously described in the article "Time is quantized" [15], "The Universe – an Open System" [16], "Dark Energy is not constant" over time [17], Energy is relative [18], Equivalence of Information and Squared Energy [19] the statement of Prof. Dr. Alexandre Tkatchenko from the University of Luxemburg also contains a possible application of formula (1.2), see page 1 "1. Introduction".

Conclusions

My contribution can help to better understand the nature of dark energy. This is the aim of the Euclid-ESA-Mission, which also examines whether it is a constant cosmological constant or whether it changes over time. My article contributes to this; see Reference [17] "Dark Energy is not constant".

The most significant property of dark energy is its effect on the accelerated expansion of the universe. Since information and effect are equivalent, the true nature of dark energy is information (quantum information), also as the general energy. The force resulting from this effect is opposite to gravity. However, it far exceeds gravity, so the universe continues to expand at an accelerating rate.

Since the values for dark energy calculated using four independent paths agree remarable well-according to STEPHEN HAWKING, ROGER PENROSE, and measurements from the MAX PLANCK Observatory and the HUBBLE Space Telescope – the most widely accepted value for the age of the universe is 13.82 billion years. This value is based on the standard model of cosmology and precise measurements, particularly of the cosmic microwave background radiation. Therefore, the age of the universe with 13.82 billion years is correct.

PLANCK time can be understood as the oscillation period τ. Oscillations are fundamental oscillations of the cosmic space [1]. The dark energy satisfies the PLANCK/EINSTEIN formula E = h ν. Dark energy can be interpreted as information flow.

According to formula (2.3), the cosmic information multiplied by ln2 is nothing more than the age of the universe in PLANCK time units squared. The approximately fivefold amount of the currently known total information content of the universe would still have space on the surface of a spherical universe. Dark matter corresponds to the number of AQIs in the cosmos. The informational equivalents of dark matter and the total mass energy of the cosmos are in a ratio 1/4. Dark energy and dark matter are in a ratio 2/ln2. The ratio of dark energy to the total mass energy of the cosmos is ln2.

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