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Is ΛCDM Holographic?

 $\Lambda$CDM and Holographic Dark Energy   Holographic dark energy is usually written $$\rho_{\rm DE} = 3M_p^2\frac{c^2}{L^2}$$ where $c$ is a dimensionless HDE parameter and $L$ is an infrared cutoff. In Li's 2004 model, $$L=L_e \equiv a(t)\int_t^\infty\frac{dt'}{a(t')}$$ is the future event horizon. Li assumed constant $c$, giving $$w_{\rm DE} = -\frac13 \left( 1+\frac{2\sqrt{\Omega_{\rm DE}}}{c} \right).$$ However, with $c=1$ and $\Omega_{{\rm DE},0}=0.6889$, $w_0\simeq-0.887$, rather than $w=-1$.  Does not match observations! Also, constant-$c$ observational fits often give $c<1$, leading to phantom evolution, null-energy-condition violation and a turning point in $H(z)$. That conclusion assumes $c'=0$.  What about if $c$ is not constant?  Then:  $$c<1, \qquad c'=c-\sqrt{\Omega_\Lambda}>0, \qquad w=-1.$$ Therefore $\dot H = -4\pi G(\rho_m+\frac43\rho_r) \le0$, so there is no finite-redshift turning point and no phantom phase. Thus ...
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Is ΛCDM Holographic?

 $\Lambda$CDM and Holographic Dark Energy

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Holographic dark energy is usually written

$$\rho_{\rm DE} = 3M_p^2\frac{c^2}{L^2}$$

where $c$ is a dimensionless HDE parameter and $L$ is an infrared cutoff. In Li's 2004 model,

$$L=L_e \equiv a(t)\int_t^\infty\frac{dt'}{a(t')}$$

is the future event horizon. Li assumed constant $c$, giving

$$w_{\rm DE} = -\frac13 \left( 1+\frac{2\sqrt{\Omega_{\rm DE}}}{c} \right).$$

However, with $c=1$ and $\Omega_{{\rm DE},0}=0.6889$, $w_0\simeq-0.887$, rather than $w=-1$.  Does not match observations!

Also, constant-$c$ observational fits often give $c<1$, leading to phantom evolution, null-energy-condition violation and a turning point in $H(z)$. That conclusion assumes $c'=0$. 

What about if $c$ is not constant? 

Then: 

$$c<1, \qquad c'=c-\sqrt{\Omega_\Lambda}>0, \qquad w=-1.$$

Therefore $\dot H = -4\pi G(\rho_m+\frac43\rho_r) \le0$, so there is no finite-redshift turning point and no phantom phase. Thus

$$\boxed{ c<1\not\Rightarrow w<-1 }$$

when $c$ evolves.

Variable-$c$ Event-Horizon HDE

 Here, the equation of state becomes

$$\boxed{ w_{\rm DE} = -\frac13 -\frac{2\sqrt{\Omega_{\rm DE}}}{3c} -\frac23\frac{c'}c }.$$

The usual Li expression is only the special case $c'=0$. And indeed, variable-$c$ HDE has been studied a lot in the literature.   

Entropy and Black-Holes

The vacuum energy inside the event horizon is

$$E_\Lambda = \frac{4\pi}{3}L_e^3\rho_\Lambda = \frac{H_\Lambda^2L_e^3}{2G}.$$

A Schwarzschild black hole with horizon radius $L_e$ has mass $M_{\rm BH}(L_e) = L_e/(2G)$. Thus

$$\boxed{ \frac{E_\Lambda}{M_{\rm BH}} = H_\Lambda^2L_e^2 = c^2. }$$

So $c^2$ is the fractional saturation of the black-hole energy bound, not necessarily a fixed constant.

The Bekenstein–Hawking entropies of the event horizon and final de Sitter horizon are

$$S_e=\frac{\pi L_e^2}{G}, \qquad S_{\rm dS} = \frac{\pi}{G H_\Lambda^2}.$$

Consequently,

$$\boxed{ c^2(a) = \frac{E_\Lambda}{M_{\rm BH}} = \frac{S_e(a)}{S_{\rm dS}}. }$$

Using the Bekenstein bound $S_B=2\pi E_\Lambda L_e$ also gives $c^2=S_B/S_{\rm BH}$. Hence

$$\boxed{ c^2 = \frac{E_\Lambda}{M_{\rm BH}} = \frac{S_B}{S_{\rm BH}} = \frac{S_e}{S_{\rm dS}}. }$$

The HDE literature offers several motivations for the $M_p^2L^{-2}$ scaling, including energy bounds, entanglement energy, holographic gas, Casimir energy and entropic-force arguments. However, none independently fixes the coefficient without additional microscopic assumptions.

Entropy Bounds

A non-shrinking event horizon requires $\dot L_e\ge0 \iff HL_e\ge1$. Because $HL_e=c/\sqrt{\Omega_\Lambda}$, this gives $c^2\ge\Omega_\Lambda$. The black-hole energy bound gives $E_\Lambda\le M_{\rm BH} \iff c^2\le1$. Therefore,

$$\boxed{ \Omega_\Lambda(a)\le c^2(a)\le1 }$$

or equivalently $S_H\le S_e\le S_{\rm dS}$. The event horizon approaches its final de Sitter value $L_e\longrightarrow H_\Lambda^{-1}$, so $\boxed{c(a)\longrightarrow1}$.

Applying the generalized second law to cosmological event horizons is not universally automatic; published analyses find it more conditional than at the apparent horizon. For ordinary non-phantom $\Lambda$CDM, however, $L_e\ge H^{-1}$ follows directly from the monotonic decrease of $H$.

$\Lambda$CDM  

Let us consider LCDM! For a cosmological constant, $\rho_\Lambda = 3M_p^2H_\Lambda^2$ where $H_\Lambda\equiv\sqrt{\Lambda/3}$.  

$$\boxed{ \rho_\Lambda = 3M_p^2\frac{c(a)^2}{L_e(a)^2} \qquad c(a)=H_\Lambda L_e(a) }$$

is the future-event-horizon HDE representation of $\Lambda$CDM. 

 $c'=c-\sqrt{\Omega_\Lambda}$, bounds $\Omega_\Lambda\le c^2\le1$, and $w=-1$  

Dead Ends and Limitations

Setting $c=1$ at all times. This saturates the black-hole bound constantly, but gives $w_0\simeq-0.89$, not $w=-1$.

Using entropy monotonicity alone. Conditions such as $0<c^2\le1$, $c'\ge0$ do not select a unique function $c(a)$.

Claiming that $c^2=S_e/S_{\rm dS}$ follows solely from the second law. It additionally requires constant vacuum density, Bekenstein–Hawking entropy and a de Sitter endpoint.

Using $dE=TdS$ naively for the event-horizon interior. The energy inside a growing horizon includes volume and boundary-flux effects; the ordinary Clausius relation does not by itself reproduce $c^2=S_e/S_{\rm dS}$.

Treating all varying-$c$ models as equivalent. A model using $L=H_0^{-1}$ has $w=-1-\frac23c'/c$, while event-horizon HDE has the additional $L_e'$ contribution.

Imposing $\dot c^2/c^2\lesssim H$ as a theorem. This proposed slow-variation criterion is acknowledged in the literature to be debatable. The present trajectory approaches $2H$ relatively during early matter domination, even though its absolute variation tends to zero because $c^2\to0$.

Claiming to have predicted $\Lambda$. The construction determines $c(a)$ after specifying $S_{\rm dS}=3\pi/(G\Lambda)$. It does not explain why that terminal entropy or $\Lambda$  has its observed value.

Wrap

Future-event-horizon HDE need not have constant $c$. For exact $\Lambda$CDM,

$$\boxed{ c(a)=H_\Lambda L_e(a) }$$

and

$$\boxed{ c^2(a) = \frac{E_\Lambda}{M_{\rm BH}} = \frac{S_B}{S_{\rm BH}} = \frac{S_e}{S_{\rm dS}}. }$$

It follows that

$$\boxed{ \Omega_\Lambda\le c^2\le1, \qquad c'=c-\sqrt{\Omega_\Lambda}, \qquad w=-1, \qquad c\to1. }$$

So $\Lambda$CDM corresponds not to permanent saturation $c=1$, but to progressive saturation of the final de Sitter entropy and black-hole energy bounds.

The remaining problem is determining $S_{\rm dS}$, equivalently $\Lambda$, from a deeper theory.

  1. M. Li, A Model of Holographic Dark Energy (2004).
  2. N. Radicella and D. Pavón, On the c2c^2 term in the holographic formula for dark energy (2010).
  3. A. Sheykhi, S. Ghaffari and N. Roshanshah, A note on holographic dark energy with varying c^2 term (2016).
  4. S. Wang, Y. Wang and M. Li, Holographic Dark Energy (2017).
  5. M. Malekjani, M. Rezaei and I. A. Akhlaghi, Can Holographic dark energy models fit the observational data? (2018).
  6. E. Ó Colgáin and M. M. Sheikh-Jabbari, A Critique of Holographic Dark Energy (2021).
  7. S. Nojiri, S. D. Odintsov and T. Paul, Different faces of generalized holographic dark energy (2021).
  8. S. Wang and M. Li, Theoretical aspects of holographic dark energy (2023).

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