REVIEW
The Holographic Dark Energy in a Non-flat Universe
Astrophysics · Submitted 12 Apr 2004 · 14 references
Pith's verdict: A competent closed-universe extension of Li's holographic dark energy, with a real falsifiable bound — but the c = 1 claim is weaker than the abstract sells, and eq. (21) has a backwards inequality sitting in the load-bearing argument.
T0 · AI review
Why Pith says this
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Referee report
The paper
The paper claims
The paper argues that the single free number in holographic dark energy must be exactly 1, because any smaller value makes the universe's entropy shrink and any larger value makes an empty universe create entropy forever.
The reading
What carries the argument
The load-bearing object is the infrared cutoff $L$, the radius of the future event horizon measured on the sphere of the horizon rather than along the radial direction. In a flat universe the two coincide and the model reduces to $ρ_Λ = 3c^2 M_p^2 R_h^{-2}$ with $H R_h = c√{Ω_Λ}$. In a curved universe the authors replace $R_h$ by $L = a r(t)$, where $r(t)$ is defined by integrating $dr/√{1-kr^2}$ out to the horizon, so that $r = k^{-1/2} sin y$ with $y = √{k}R_h/a$. The stated reason for this choice is that it keeps the dark energy density constant in a de Sitter slice with positively curved spatial sections, which the original flat-universe definition fails to do once curvature is present. The machinery then converts a thermodynamic requirement on horizon entropy into a bound on the equation of state, and hence into an observational prediction.
Extended reading notes
Core claim
The central claim is that the holographic parameter $c$ is fixed at 1 by consistency conditions rather than by fit. Two arguments carry it. First, the entropy of the horizon, $S = π M_p^2 L^2$, must not decrease; differentiating it gives $dS/dt = 2(S/L)(c√{Ω_Λ} - cos y)$, so requiring $dS/dt ≥ 0$ forces $c ≥ 1$ and simultaneously forbids $w_Λ < -1$. Second, if $c$ were strictly greater than 1, an otherwise empty universe would produce entropy indefinitely, which the authors find implausible, and observation already constrains any deviation to be small with no theoretical reason for it to be small. Combined with the modified horizon definition in a closed universe, $L = a r(t)$ with $r(t) = k^{-1/2} sin(√{k} R_h/a)$, the equation of state becomes $w_Λ = -⅓(1 + (2/c)√{Ω_Λ} cos y)$, bounded below by $w_{min} = -⅓(1 + 2√{Ω_Λ})$. For $Ω_Λ = 0.73$ today that bound is $-0.90$, a number future dark-energy measurements can confirm or eliminate.
Load-bearing premise
The closed-universe results all rest on replacing the radial horizon distance with the horizon radius measured on the sphere, a definition the authors adopt because it makes the pure dark-energy case give a constant density, and a different cutoff that also reduces to the flat case would give a different lower bound on the equation of state.
Editorial extensions
If this is right
- The model has no free parameter once $c = 1$ is adopted, so its equation of state is fixed by the measured dark-energy density rather than fitted.
- In a closed universe the equation-of-state index is bounded below by $w_{min} = -⅓(1 + 2√{Ω_Λ})$, which is about $-0.90$ for $Ω_Λ = 0.73$ today.
- Phantom dark energy with $w_Λ < -1$ is excluded, because it would require the horizon entropy to decrease.
- A consistent cosmology requires that the inflaton decay: if it never did, the Hubble scale would stay fixed and the holographic dark energy could never come to dominate, so reheating is a necessary ingredient of the model.
- In the far future the dark-energy density approaches a constant for $c = 1$ and redshifts away for $c > 1$, with $Ω_Λ$ approaching 1 from above in a closed universe.
- The open-universe case follows from the closed case by sending $k → -k$, $Ω_k → -Ω_k$ and $γ → -γ$.
Reading between the lines
- Because the model carries no adjustable parameter, a future measurement of $w_Λ$ between $-1$ and $-0.90$ would not be a mild tension to be absorbed by shifting $c$; it would count against the construction as a whole, which makes the model unusually exposed to coming surveys.
- The bound depends on today's $Ω_Λ$; a percent-level tightening of $Ω_Λ$ and of spatial curvature from galaxy surveys would sharpen or loosen the $-0.90$ figure enough to make the prediction genuinely checkable rather than merely satisfied.
- The forced-decay argument suggests a two-way link with inflation: reheating parameters inferred from the cosmic microwave background would, in this picture, be constrained from below by the requirement that dark energy eventually dominate.
- The same entropy-monotonicity step could be applied to other horizon-based cutoffs; if a different curved-space definition also reduces to the flat case but yields a different lower bound on $w_Λ$, the prediction would be a property of the chosen cutoff rather than of holography itself.
Original abstract
We study the model for holographic dark energy in a spatially closed universe, generalizing the proposal in hep-th/0403127 for a flat universe. We provide independent arguments for the choice of the parameter $c=1$ in the holographic dark energy model. On the one hand, $c$ can not be less than 1, to avoid violating the second law of thermodynamics. On the other hand, observation suggests $c$ be very close to 1, it is hard to justify a small deviation of $c$ from 1, if $c>1$.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
We study the model for holographic dark energy in a spatially closed universe, generalizing the proposal in hep-th/0403127 for a flat universe. We provide independent arguments for the choice of the parameter $c=1$ in the holographic dark energy model. On the one hand, $c$ can not be less than 1, to avoid violating the second law of thermodynamics. On the other hand, observation suggests $c$ be very close to 1, it is hard to justify a small deviation of $c$ from 1, if $c>1$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We study the model for holographic dark energy in a spatially closed universe, generalizing the proposal in hep-th/0403127 for a flat universe. | contribution 0.99 | 0.90 | 0.07 | asserted 0.96 |
| s1 | We provide independent arguments for the choice of the parameter $c=1$ in the holographic dark energy model. | contribution 0.98 | 0.91 | 0.24 | asserted 1.00 |
| s2 | On the one hand, $c$ can not be less than 1, to avoid violating the second law of thermodynamics. | result 0.47 | 0.48 | 0.14 | asserted 0.99 |
| s3 | On the other hand, observation suggests $c$ be very close to 1, it is hard to justify a small deviation of $c$ from 1, if $c>1$. | contribution 0.40 | 0.16 | 0.17 | hedged possibility 0.84 |
What was examined, and what was not
- arXiv listing abstract v3
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
References
How the literature uses it
47 arXiv papers cite this in 67 sentences
Papers citing it · 44 sentences without a confident label
Compares with it 3 papers
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In the limit $α→0$ we re-obtain the results of [ref] and those of conventional 4D calculations [ref] , although in the 5D study the interpretation and explanation of these results is fundamentally different.
Holographic Dark Energy in Braneworld Models with a Gauss-Bonnet Term in the Bulk. Interacting Behavior and the w =-1 Crossing · Discussion-Conclusions · 0.96
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Moreover, for $Δ=0$ we acquire the expression of standard holographic dark energy in closed universe [ref] .
Barrow Holographic Dark Energy in non-flat Universe · Positive spatial curvature · 0.89
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One can in general acquire either different or exactly identical 4D behavior, comparing to that obtained in conventional 4D literature [ref] .
Holographic Dark Energy in Braneworld Models with Moving Branes and the w=-1 Crossing · Introduction · 0.91
Cites it as background 18 papers
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A flat universe composed of $ρ_q$ and the cold dark matter $ρ_m$ is governed by the Friedmann equation [equation] and their continuity equations [equation] Introducing the density parameters $Ω_i=ρ_i/3m^2_pH^2$, then one finds [equation] which implies that the Friedmann equation ([ref]) can be rewritten as [equation] Considering the age of the universe $T=∫^a_0da'/H'a'$, its pressure is determined solely by Eq.([ref]) with $x=lna$ [ref] [equation] which provides the EOS [ref] [equation] Here $ω_q$ is determined by the evolution equation [equation] where $^'$ and $$ are the derivative with respect to $x$ and cosmic time $t$, respectively.
Instability of agegraphic dark energy models · Noninteracting case · 0.86
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One is the holographic dark energy model [ref] and the other is the agegraphic dark energy model [ref] .
Origin of holographic dark energy models · Introduction · 0.93
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Regardless, cosmological models exist in this class and Holographic Dark Energy (HDE) [ref] can be regarded as a prominent minimal model (see also [ref] for generalisations).
A Critique of Holographic Dark Energy · Introduction · 0.97
Discussion
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