On the Holographic Principle in a Radiation Dominated Universe

Erik Verlinde

RECORD

On the Holographic Principle in a Radiation Dominated Universe

Erik Verlinde

High Energy Physics - Theory · Submitted 17 Aug 2000

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim A new purely holographic bound is postulated which restricts the sub-extensive entropy associated with the Casimir energy.

how it is stated · asserted 0.56 read the record →

Comment (0) Read on arXiv ↗

Cite ▾

Claim paper

The compiled record

The paper's own sentences, arranged. 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.

Open the compiled record

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,552 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.20 · human participants 0.01 · controlled comparison 0.08 · randomized assignment 0.01 · states quantitative result 0.72 · compares with prior results 0.81 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

The holographic principle is studied in the context of a $n+1$ dimensional radiation dominated closed Friedman-Robertson-Walker (FRW) universe. The radiation is represented by a conformal field theory with a large central charge. Following recent ideas on holography, it is argued that the entropy density in the early universe is bounded by a multiple of the Hubble constant. The entropy of the CFT is expressed in terms of the energy and the Casimir energy via a universal Cardy formula that is valid for all dimensions. A new purely holographic bound is postulated which restricts the sub-extensive entropy associated with the Casimir energy. Unlike the Hubble bound, the new bound remains valid throughout the cosmological evolution. When the new bound is saturated the Friedman equation exactly coincides with the universal Cardy formula, and the temperature is uniquely fixed in terms of the Hubble parameter and its time-derivative.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The holographic principle is studied in the context of a $n+1$ dimensional radiation dominated closed Friedman-Robertson-Walker (FRW) universe.	background 0.60	0.68	0.03	asserted 0.97
s1	The radiation is represented by a conformal field theory with a large central charge.	method 0.81	0.20	0.02	asserted 1.00
s2	Following recent ideas on holography, it is argued that the entropy density in the early universe is bounded by a multiple of the Hubble constant.	result 0.55	0.52	0.28	asserted 0.48
s3	The entropy of the CFT is expressed in terms of the energy and the Casimir energy via a universal Cardy formula that is valid for all dimensions.	method 0.73	0.34	0.07	asserted 1.00
s4	A new purely holographic bound is postulated which restricts the sub-extensive entropy associated with the Casimir energy.	contribution 0.98	0.77	0.31	asserted 0.56
s5	Unlike the Hubble bound, the new bound remains valid throughout the cosmological evolution.	result 0.89	0.75	0.24	asserted 1.00
s6	When the new bound is saturated the Friedman equation exactly coincides with the universal Cardy formula, and the temperature is uniquely fixed in terms of the Hubble parameter and its time-derivative.	result 0.99	0.81	0.36	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.98 · evidence from proof or derivation 0.66.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper

Original abstract

The holographic principle is studied in the context of a $n+1$ dimensional radiation dominated closed Friedman-Robertson-Walker (FRW) universe. The radiation is represented by a conformal field theory with a large central charge. Following recent ideas on holography, it is argued that the entropy density in the early universe is bounded by a multiple of the Hubble constant. The entropy of the CFT is expressed in terms of the energy and the Casimir energy via a universal Cardy formula that is valid for all dimensions. A new purely holographic bound is postulated which restricts the sub-extensive entropy associated with the Casimir energy. Unlike the Hubble bound, the new bound remains valid throughout the cosmological evolution. When the new bound is saturated the Friedman equation exactly coincides with the universal Cardy formula, and the temperature is uniquely fixed in terms of the Hubble parameter and its time-derivative.

Human review

How Pith checks papers → · Full audit of this paper →

References

Forward citations

Cited by 23 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Universality of the microcanonical entropy at large spin
hep-th 2025-05 conditional novelty 8.0 of 10

For any twist interval at or above (c-1)/12, the large-spin operator density matches the vacuum-block prediction with relative error vanishing faster than any power, using a twist window that may shrink as J^{-1/4+epsilon}.
Emergent Gravity and the Dark Universe
hep-th 2016-11 conditional novelty 8.0 of 10

Dark matter phenomena follow from an elastic response of de Sitter entropy to baryonic matter, yielding the Tully-Fisher scaling with a0/6 and no parameter fitted to galaxy data.
Universal Asymptotics for High Energy CFT Data
hep-th 2023-06 conditional novelty 7.0 of 10

A thermal effective action plus a conjectural hot-spot rule yields universal asymptotics for high-energy CFT densities and heavy OPE coefficients.
Finite-cutoff Holographic Thermodynamics
hep-th 2025-07 conditional novelty 6.0 of 10

Finite-cutoff holography yields a thermodynamic duality between T^2-deformed CFTs and Schwarzschild-AdS black holes with a Dirichlet wall, including a teardrop coexistence curve with up to three states at one temperature.
Cardy Entropy of Charged and Rotating Asymptotically AdS and Lifshitz Solutions with a Generalized Chern-Simons term
hep-th 2025-06 conditional novelty 6.0 of 10

A generalized Cardy-like formula using the soliton mass as vacuum energy reproduces the entropy of charged rotating AdS and Lifshitz black holes in three dimensions.

Show 18 more citing papers

The Cardy Formula from Goldstone Bosons
hep-th 2019-07 unverdicted novelty 6.0 of 10

Derives the Cardy formula for 2D CFTs from the Schwarzian action of pseudo Goldstone bosons under anomalous conformal symmetry breaking, without modular invariance.
Cosmology at the end of the world
hep-th 2019-07 accept novelty 6.0 of 10

A charged AdS-Reissner-Nordstrom black hole with a near-critical-tension end-of-the-world brane supports both a sensible dual CFT state and a long-lived quasi-bound graviton, giving a holographic braneworld cosmology ...
Mass, Entropy and Holography in Asymptotically de Sitter Spaces
hep-th 2001-10 conditional novelty 6.0 of 10

De Sitter black holes and spinning conical defects all weigh less than empty de Sitter space, and a Cardy-style state count reproduces their horizon entropy.
Entropy bounds, Geroch process, and the sign of deformation parameter
hep-th 2026-07 conditional novelty 5.0 of 10

Negative GUP deformation relaxes the Bekenstein entropy bound and positive deformation tightens it, derived via the Geroch process in both 3+1 and 2+1 dimensions.
Holographic pressure and volume for black holes
hep-th 2026-02 unverdicted novelty 5.0 of 10

Introduces a holographic pressure and volume for static spherically symmetric black holes via quasi-local thermodynamics, showing large black holes become extensive in the large-system limit while small ones do not.
Modular forms and a generalized Cardy formula in higher dimensions
hep-th 2015-08 conditional novelty 5.0 of 10

For a conformal field theory on a d-dimensional spatial torus, the high-energy density of states is determined solely by the Casimir energy density on a parallel-plate geometry, generalizing the two-dimensional Cardy formula.
Tachyon condensation and off-shell gravity/gauge duality
hep-th 2006-04 reject novelty 5.0 of 10

The paper proposes matching off-shell tachyon condensation to off-shell AdS black holes and, speculatively, to a ten-dimensional Schwarzschild black hole in a cavity.
Unifying phantom inflation with late-time acceleration: scalar phantom-non-phantom transition model and generalized holographic dark energy
hep-th 2005-06 conditional novelty 5.0 of 10

A sign-changing scalar kinetic coupling lets a single expansion history contain phantom inflation, a standard intermediate era, and phantom late-time acceleration, and generalized holographic cutoffs can be tuned to r...
First Law of Thermodynamics and Friedmann Equations of Friedmann-Robertson-Walker Universe
hep-th 2005-01 conditional novelty 5.0 of 10

Applying the first law of thermodynamics to the apparent horizon reproduces the Friedmann equations for any spatial curvature and for Gauss-Bonnet and Lovelock gravity.
Hamilton-Jacobi Approach to Inflationary Scenarios through Extended Entropies: An Observational Perspective
astro-ph.CO 2026-05 unverdicted novelty 4.0 of 10

Hamilton-Jacobi analysis of slow-roll inflation with non-Bekenstein-Hawking entropies yields fitted entropy parameters (δ≈1.1-1.2, α∼10^{-14}, K∼10^{-17}) consistent with ns and r data.
On the Field Theoretical Description of an Alternative Model to Generalized Chaplygin Gas and its Thermodynamic Behaviour
gr-qc 2024-12 reject novelty 4.0 of 10

The authors reconstruct quintessence, k-essence, and DBI scalar-field potentials for the sinc dark-energy fluid, but their thermodynamic phase transition is built from arbitrary constants and the model is disfavored b...
Three puzzles in cosmology
hep-th 2020-09 conditional novelty 4.0 of 10

The paper speculates that 'vecro' vacuum fluctuations, already proposed for black holes, can resolve the cosmological constant, information, and early-universe puzzles.
Emergent Gravity in a Holographic Universe
hep-th 2019-08 conditional novelty 3.0 of 10

Causal diamonds in symmetric spacetimes obey a thermodynamic first law with negative temperature, and the Einstein equations can be recast as an entropy equilibrium condition, with a long-string CFT picture for non-Ad...
Phase transitions for the topological de Sitter spaces and Schwarzschild-de Sitter black hole
hep-th 2006-03 conditional novelty 3.0 of 10

The hyperbolic topological de Sitter space can make a Hawking-Page transition at T=3/(2πl); the Schwarzschild-de Sitter black hole shows a tentative Nariai transition at T=0.
Phase transition between the BTZ black hole and AdS space
gr-qc 2006-03 conditional novelty 3.0 of 10

A conical singularity mass connects thermal AdS space and the BTZ black hole through a free-energy crossing, and a separate transition to the massless BTZ black hole is assumed.
Viscous Cosmology for Early- and Late-Time Universe
gr-qc 2017-06 conditional novelty 2.0 of 10

Bulk viscosity, treated as a free function of density or Hubble rate, can reproduce inflation, late-time acceleration, phantom crossing, and bounces, but the review offers no microphysical derivation of the viscosity itself.
Black holes, cosmological solutions, future singularities, and their thermodynamical properties in modified gravity theories
gr-qc 2012-07 conditional novelty 2.0 of 10

The review maps black-hole thermodynamics, cosmological reconstruction, and entropy bounds in f(R) gravity, concluding that the proposed entropy bound is not universal near future singularities.
Dark energy cosmology: the equivalent description via different theoretical models and cosmography tests
gr-qc 2012-05 conditional novelty 2.0 of 10

The paper reviews how many dark energy and modified-gravity models can be recast as equivalent fluid or scalar-field descriptions and can be tuned to fit current cosmological data.

How later papers used it

68 later papers cite it

Evidence: substantial. 66 of the citing papers share no author with it, and 8 of those carry a confident label from Pith's citation classifier.

Pith read 24 of the 130 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

2 later papers by other authors are listed below: 1 used its method or tool and 1 compared its own results with it.

Uses its method or tool 1 of 2 papers with this label

Used a holographic method provided in this paper to give the temperature, entropy, charge and chemical potential on the brane from the corresponding thermodynamic quantities of an AdS dilaton black hole.

Brane universe and holography in spacetime of charged AdS dilaton black hole (2019) · 1.00 · paraphrased: the licence does not allow quoting

Compares with it 1 of 2 papers with this label

Found the same scaling by identifying the energy in the density of states with the black hole mass, which coincides with this paper's formula once the Casimir energy is expressed in terms of the entropy.

Entanglement spectra from holography (2024) · 0.96 · paraphrased: the licence does not allow quoting

How the literature uses it

68 arXiv papers cite this in 130 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 84 sentences without a confident label

Builds on it 2 papers

This section is devoted to consideration of the Cardy-Verlinde formula for more general scenarios than those considered in previous works (see [ref] - [ref] ).

Cardy-Verlinde formula in FRW Universe with inhomogeneous generalized fluid and dynamical entropy bounds near the future singularity · Generalization of Cardy-Verlinde formula in FRW Universe for various types of fluids · 0.83
This section is devoted to consideration of the Cardy-Verlinde formula for more general scenarios than those considered in previous works (see [ref] ).

On Friedmann-Lema\^{\i}tre-Robertson-Walker cosmologies in non-standard gravity · Generalization of Cardy-Verlinde formula in FLRW Universe for various types of fluids · 0.91

Uses its method or tool 2 papers

By using a holographic method provided in [ref] , the temperature, entropy, charge and chemical potential on brane could be given from the corresponding thermodynamic quantities of AdS dilaton black hole.

Brane universe and holography in spacetime of charged AdS dilaton black hole · Conclusion and Discussion · 1.00
By substituting Eqs. ([ref])–([ref]) into the Friedmann equation shown above, we have [ref] the Cardy-Verlinde holographic formula of the Friedmann equation [ref]

Dark energy cosmology: the equivalent description via different theoretical models and cosmography tests · The Hubble entropy in the holographic principle · 0.85

Uses its data or values 1 paper

The physical energy of the brane universe was given by the author of Ref. [ref] :

Holographic complexity growth in a FLRW universe · Summary of Results and Discussion · 0.86

Compares with it 2 papers

This result agrees with the previous findings for the case of CFT spectra in two (see [ref] ) and higher dimensions [ref] .

Entanglement spectra from holography · Discussion of the results · 0.96
This is significantly different from the earlier works [ref] , where the CFT lives on the brane.

On holographic braneworld cosmology · Introduction · 1.00

Cites it as support 1 paper

Second, a connection between zero extrinsic curvature and low entropy is suggested by Verlinde's [ref] observation that Cardy's formula for the entropy of a conformal field theory can reproduce the Friedmann equation.

Initial Conditions for Bubble Universes · Section · 0.99

Cites it as background 32 papers

In Ref. [ref] , Verlinde proposed a new universal bound on cosmology based on a restriction of the Casimir energy $E_C$; cf. his entropy formula ([ref]).

Cardy-Verlinde formula in FRW Universe with inhomogeneous generalized fluid and dynamical entropy bounds near the future singularity · On the cosmological bounds near future singularities · 0.93
The Cardy-Verlinde formula recently proposed by Verlinde [ref] , relates the entropy of a certain CFT to its total energy and Casimir energy in arbitrary dimensions.

Self-gravitational corrections to the Cardy-Verlinde formula of charged BTZ black hole · Introduction · 0.98
An interesting and puzzling (due to its generality, including flat spacetime solutions for which there is no concrete holographic proposal) result of Verlinde [ref] is that the entropy of AdS black holes, with a field theory dual, can be put in a form similar to Cardy formula of two-dimensional CFTs.

Holography of AdS Hairy Black Holes and Cardy-Verlinde Formula · Introduction · 0.84

Discussion

Know something about this paper? Leave a comment, or highlight any passage above to comment on it.