Copy-Horizon Cosmology from Quantum Information Copy Time: An Operational Infrared Cutoff for Holographic Dark Energy with Testable Signatures

Mohamed Sacha

doi:10.20944/preprints202601.0639.v1

Submitted:

06 January 2026

Posted:

09 January 2026

Read the latest preprint version here

Abstract

We present a strict, non-circular formulation of a “copy-horizon” infrared (IR) scale Lcopy(t) defined operationally from a quantum-information copy time τ_copy(L, t) by the single criterion τ_copy(L_copy(t), t) = H−1(t). The definition requires only mild locality/monotonicity assumptions and does not postulate an a priori cosmological IR cutoff (such as the future event horizon). We then combine this operational IR scale with the Cohen–Kaplan–Nelson (CKN) gravitational collapse bound to obtain the L−2 energy-density scaling as a consistency constraint, and we formulate “saturation” as a falsifiable mechanism with a severe inequality c_Q(z) ≤ 1. We derive the minimal background consequence w_Q(t) = −1 + (2/3)(L̇_copy/(HL_copy)) and show how a hydrodynamic realization of τcopy yields rigid consistency relations linking expansion, growth, and transient time-of-flight observables.

Keywords:

quantum information

;

operational infrared cutoff

;

holographic dark energy

;

cosmology

;

non-circular derivation

;

falsifiable signatures

Subject:

Physical Sciences - Theoretical Physics

Scope and Non-Circularity

This paper isolates a single operational input—the copy time

τ_{c o p y} (L, t)

—and derives its minimal cosmological consequences without introducing an independent cosmological infrared cutoff by fiat. In particular, we do not define

L_{c o p y}

as the future event horizon. The event-horizon cutoff is used only as a comparator in a dedicated discussion paragraph.

Operational Definition of $τ_{c o p y}$

Let

A

and

B

be spatially separated regions of a quantum many-body system. At

t = 0

a local perturbation is applied in

A

(e.g. by a unitary

U_{A}

), yielding two global states

ρ_{0} (t)

and

ρ_{1} (t)

at time

t

with reduced states

ρ_{0, B} (t)

and

ρ_{1, B} (t)

on

B

. Given a measurement class

M_{B}

accessible to the receiver, define the optimal distinguishability advantage

Δ_{M_{B}} (t; L) \equiv \underset{M \in M_{B}}{s u p} |T r [M (ρ_{1, B} (t) - ρ_{0, B} (t))]| .

For unrestricted

M_{B}

this reduces to the Helstrom distinguishability associated with the trace distance . Fix

0 < ε < 1

. The copy time is

τ_{c o p y} (L, t) \equiv i n f \{τ \geq 0 : Δ_{M_{B}} (τ; L, t) \geq ε\} .

The functional form of

τ_{c o p y}

is determined by microscopic dynamics and the choice of

M_{B}

; it is not part of the definition. For a complete phenomenology, concrete realizations must be provided (see Sec. 8).

Brick 1: Defining $L_{c o p y} (t)$ with no Model Parameters

We work in a spatially flat FLRW background with Hubble rate

H (t) = \dot{a} / a

. We define the copy-horizon scale

L_{c o p y} (t)

as the largest physical separation for which the copy certificate is achievable within one Hubble time:

τ_{c o p y} (L_{c o p y} (t), t) = H^{- 1} (t) .

Assuming

\partial_{L} τ_{c o p y} \geq 0

and mild regularity, Eq. [eq:brick1] defines

L_{c o p y} (t)

uniquely (locally in time). Differentiating yields the implicit evolution equation

\partial_{L} τ_{c o p y} (L_{c o p y}, t) {\dot{L}}_{c o p y} + \partial_{t} τ_{c o p y} (L_{c o p y}, t) = - \frac{\dot{H}}{H^{2}} .

This step is non-circular:

L_{c o p y}

is determined by an operational time criterion, not by an assumed

w (z)

.

Illustrative realizations of

τ_{c o p y} (L)

(ballistic and diffusive) and the definition

τ_{c o p y} (L_{c o p y}) = H^{- 1}

. The intersection determines

L_{c o p y} (t)

without assuming a cosmological cutoff.

Brick 2: $L^{- 2}$ Scaling from Gravitational Consistency

The CKN bound constrains any effective description in a region of size

L

: the energy

E \sim ρ L^{3}

must not exceed the mass of a black hole of the same size,

M_{B H} (L) \sim M_{P}^{2} L

. Hence

ρ L^{3} ≲ M_{P}^{2} L \Rightarrow ρ ≲ M_{P}^{2} L^{- 2} .

Equation [eq:CKN] is not a dark-energy model; it is a consistency constraint. The same

L^{- 2}

scaling underlies the holographic dark energy (HDE) literature, but here it is anchored to the operational scale

L = L_{c o p y} (t)

rather than an a priori cosmological horizon choice .

Saturation as Mechanism and a Severe Inequality

We define the copy-horizon component

Q

by evaluating the CKN-consistent scaling at

L = L_{c o p y} (t)

and allow for a (dimensionless) saturation parameter

c_{Q}

:

ρ_{Q} (t) = 3 c_{Q}^{2} M_{P}^{2} L_{c o p y} {(t)}^{- 2} .

A severe, falsifiable interpretation is

0 < c_{Q} (z) \leq 1 for all tested z,

where

c_{Q} > 1

would signal a violation of the gravitational consistency bound at the operational IR scale. In the Supplement we provide an explicit attractor toy-model in which saturation emerges as a stable fixed point.

Logical flow of the strict QICT program: operational

τ_{c o p y}

\to

definition of

L_{c o p y}

\to

CKN bound

\to

saturation mechanism and testable consistency conditions.

Brick 3: Fixing $c_{Q}$ from $Ω_{Q, 0}$

Define

ρ_{c} (t) = 3 M_{P}^{2} H {(t)}^{2}

and

Ω_{Q} (t) = ρ_{Q} / ρ_{c}

. At

t = t_{0}

(today),

ρ_{Q, 0} = 3 M_{P}^{2} H_{0}^{2} Ω_{Q, 0}

. Equating with Eq. [eq:rhoQ] gives

c_{Q}^{2} = Ω_{Q, 0} (H_{0} L_{c o p y} (t_{0}))^{2},

so once

L_{c o p y} (t_{0})

is obtained from Eq. [eq:brick1], the amplitude is fixed with no multi-parameter cosmological fit.

Minimal Cosmological Consequence: $w_{Q}$ from Conservation

Assuming no background energy exchange between

Q

and matter,

ρ_{Q}

obeys

{\dot{ρ}}_{Q} + 3 H (1 + w_{Q}) ρ_{Q} = 0 .

Since

ρ_{Q} \propto L_{c o p y}^{- 2}

, we have

{\dot{ρ}}_{Q} / ρ_{Q} = - 2 {\dot{L}}_{c o p y} / L_{c o p y}

, hence

w_{Q} (t) = - 1 + \frac{2}{3} \frac{{\dot{L}}_{c o p y} (t)}{H (t) L_{c o p y} (t)} .

Thus

w_{Q}

is a consequence once

L_{c o p y} (t)

is determined from

τ_{c o p y}

.

Concrete Realizations of $τ_{c o p y}$ and Rigid Consistency Relations

Ballistic (Causal) Lower Bound

Locality implies a light-cone constraint (Lieb–Robinson or relativity) : no measurement on

B

can have

Δ > 0

before

t < L / v_{L R}

. Therefore

τ_{c o p y} (L, t) \geq L / v_{L R} (t) .

Diffusive Hydrodynamic Theorem (Constructive)

Assume a conserved density

n (x, t)

coupled to the perturbation and a hydrodynamic regime

\partial_{t} n = D (t) \nabla^{2} n

with slowly varying

D (t)

. For an observable

N_{B} = \int_{B} n (x) d^{3} x

in the measurement class, the mean response at distance

L

is controlled by the diffusive Green’s function. Using a severe SNR/Helstrom bound (e.g. via Cauchy–Schwarz and Pinsker-type inequalities) , one obtains asymptotically

τ_{c o p y} (L, t) = \frac{L^{2}}{4 D (t)} [1 + O (l o g L)],

up to subleading logarithmic terms that depend on thresholding and region geometry.

A Hard “Triangle” Test

If the diffusive closure [eq:diffusive] applies on the cosmological effective description, Eq. [eq:brick1] yields

L_{c o p y} (t) = 2 \sqrt{\frac{D (t)}{H (t)}} .

Consequently, [eq:wQ] becomes a rigid relation between

w_{Q}

and the log-derivatives of

D

and

H

. The strict program is then to confront a single inferred

L_{c o p y} (z)

(and

c_{Q} (z)

) against multiple datasets: expansion

H (z)

(BAO/SNe/CMB), growth

f σ_{8} (z)

and lensing, and an independent transient observable (e.g. energy-dependent time-of-flight dispersion

δ t (E, z)

if present). The joint compatibility is summarized schematically in Figure 3.

A hard observational test: the same

L_{c o p y} (z)

and

c_{Q} (z) \leq 1

must be compatible with expansion, growth/lensing, and an independent transient observable (if predicted) within the same microphysical closure.

Figure 1. Logical flow of the strict QICT construction: operational copy-time $\rightarrow$ copy-horizon scale $\rightarrow$ gravitational consistency bound $\rightarrow$ saturation hypothesis.

Figure 2. Examples of copy-time scalings (ballistic and diffusive) and the operational definition $\tau_{\mathrm{copy}}(L_{\mathrm{copy}})=H^{-1}$.

Figure 3. Cross-observable consistency concept linking expansion $H(z)$, equation of state $w_Q(z)$, and an independent transient timing observable.

Minimal Real-Data Sanity Check (Cosmic Chronometers)

To demonstrate that the strict framework can be confronted with actual expansion-rate measurements without introducing many phenomenological degrees of freedom, we present a deliberately rigid closure example: (i) the diffusive scaling [eq:diffusive] with constant effective diffusivity

D (z) = D_{0}

, (ii) maximal saturation

c_{Q} = 1

motivated by the max-capacity attractor picture in the Supplement, and (iii) spatial flatness with radiation neglected for

z ≲ 2

.

In this limit, the background expansion becomes parameter-free once

(H_{0}, Ω_{m 0})

are fixed. Writing

E (z) = H (z) / H_{0}

and

Ω_{Q 0} = 1 - Ω_{m 0}

, the Friedmann equation reduces to a cubic relation

E {(z)}^{3} - E (z) Ω_{m 0} {(1 + z)}^{3} - Ω_{Q 0} = 0 .

We plot the corresponding prediction (normalized to the Planck 2018 baseline values ) against a 32-point cosmic-chronometer compilation in Figure 4. This is not advertised as a definitive global fit, but as a minimal, falsifiable closure demonstration that can be systematically upgraded (time-dependent

D (z)

, inclusion of radiation/curvature, joint BAO/SNe/CMB likelihoods).

Minimal real-data sanity check: cosmic-chronometer

H (z)

points compared to the rigid constant-

D

diffusive closure with maximal saturation

c_{Q} = 1

(no fitted parameters in this illustration; the normalization uses Planck 2018 values ).

Figure 4. Minimal quantitative illustration using a public cosmic-chronometer $H(z)$ compilation: data versus an illustrative QICT closure curve (not a full likelihood analysis).

Figure 5. Derived $c_Q(z)$ from the minimal $H(z)$ illustration, displaying the strict consistency bound $c_Q(z)\le 1$ over the tested redshift range.

Figure 6. Illustrative (mock) evolution of the QICT equation-of-state $w_Q(z)$ implied by the copy-horizon closure.

Figure 7. Illustrative (mock) evolution of the saturation parameter $c_Q(z)$ showing the falsifiable bound $c_Q(z)\le 1$ under an attractor mechanism.

Relation to Standard HDE Cutoffs (Comparator Only)

Standard holographic dark energy often adopts a future event-horizon cutoff

L_{H D E} \equiv R_{h}

. In contrast, the present work defines

L_{c o p y}

operationally by Eq. [eq:brick1]. The event-horizon choice may be treated as a comparator model, or potentially recovered as a special case under additional assumptions on the functional dependence of

τ_{c o p y} (L, t)

; we do not assume such an identification in the main derivation.

Conclusions

The strict QICT copy-horizon program is characterized by (i) an operational definition of

τ_{c o p y}

and

L_{c o p y}

, (ii) a gravitational consistency bound implying

L^{- 2}

scaling, and (iii) falsifiable saturation conditions such as

c_{Q} (z) \leq 1

and multi-observable consistency relations. The accompanying Supplement and derivation package provide additional proofs, sector separation statements, and reproducible scripts.

References

C. W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, 1976). [CrossRef]
S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory (North-Holland, 1982). [CrossRef]
M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2010).
E. H. Lieb and D. W. Robinson, “The finite group velocity of quantum spin systems,” Commun. Math. Phys. 28, 251–257 (1972). [CrossRef]
Cohen, D. Kaplan, and A. Nelson, “Effective field theory, black holes, and the cosmological constant,” Phys. Rev. Lett. 82, 4971 (1999). [CrossRef]
M. Li, “A model of holographic dark energy,” Phys. Lett. B 603, 1–5 (2004).
N. Aghanim et al. (Planck Collaboration), “Planck 2018 results. VI. Cosmological parameters,” Astron. Astrophys. 641, A6 (2020). [CrossRef]
M. Moresco, R. Jimenez, L. Verde, A. Cimatti, and L. Pozzetti, “Setting the Stage for Cosmic Chronometers. II. Impact of Stellar Population Synthesis Models Systematics and Full Covariance Matrix,” Astrophys. J. 898, 82 (2020); arXiv:2003.07362. [CrossRef]
Mehrabi, Cosmic_chronometer_data (GitHub repository), file HzTable_MM_BC32.txt (accessed 2026-01-06).
Csiszár and J. Körner, Information Theory: Coding Theorems for Discrete Memoryless Systems (Akadémiai Kiadó, 1981). (See Pinsker-type inequalities.).

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Copy-Horizon Cosmology from Quantum Information Copy Time: An Operational Infrared Cutoff for Holographic Dark Energy with Testable Signatures

Abstract

Keywords:

Subject:

Scope and Non-Circularity

Operational Definition of $τ_{c o p y}$

Brick 1: Defining $L_{c o p y} (t)$ with no Model Parameters

Brick 2: $L^{- 2}$ Scaling from Gravitational Consistency

Saturation as Mechanism and a Severe Inequality

Brick 3: Fixing $c_{Q}$ from $Ω_{Q, 0}$

Minimal Cosmological Consequence: $w_{Q}$ from Conservation

Concrete Realizations of $τ_{c o p y}$ and Rigid Consistency Relations

Ballistic (Causal) Lower Bound

Diffusive Hydrodynamic Theorem (Constructive)

A Hard “Triangle” Test

Minimal Real-Data Sanity Check (Cosmic Chronometers)

Relation to Standard HDE Cutoffs (Comparator Only)

Conclusions

References

MDPI Initiatives

Important Links

Subscribe

Copy-Horizon Cosmology from Quantum Information Copy Time: An Operational Infrared Cutoff for Holographic Dark Energy with Testable Signatures

Abstract

Keywords:

Subject:

Scope and Non-Circularity

Operational Definition of τ c o p y

Brick 1: Defining L c o p y t with no Model Parameters

Brick 2: L − 2 Scaling from Gravitational Consistency

Saturation as Mechanism and a Severe Inequality

Brick 3: Fixing c Q from Ω Q , 0

Minimal Cosmological Consequence: w Q from Conservation

Concrete Realizations of τ c o p y and Rigid Consistency Relations

Ballistic (Causal) Lower Bound

Diffusive Hydrodynamic Theorem (Constructive)

A Hard “Triangle” Test

Minimal Real-Data Sanity Check (Cosmic Chronometers)

Relation to Standard HDE Cutoffs (Comparator Only)

Conclusions

References

MDPI Initiatives

Important Links

Subscribe

Operational Definition of $τ_{c o p y}$

Brick 1: Defining $L_{c o p y} (t)$ with no Model Parameters

Brick 2: $L^{- 2}$ Scaling from Gravitational Consistency

Brick 3: Fixing $c_{Q}$ from $Ω_{Q, 0}$

Minimal Cosmological Consequence: $w_{Q}$ from Conservation

Concrete Realizations of $τ_{c o p y}$ and Rigid Consistency Relations