Is ΛCDM Holographic?

 $\Lambda$CDM and Holographic Dark Energy

 

Holographic dark energy (HDE) starts from a simple thought: the energy inside a region of size $L$ shouldn't exceed that of a black hole of the same size. Saturate that bound, up to a dimensionless number $c$ (not the speed of light), and

$$\rho_{\rm DE}=3M_p^2\frac{c^2}{L^2}, \qquad\text{i.e.}\qquad c=HL\sqrt{\Omega_{\rm DE}}.$$

Units $\hbar=k=1$, $M_p^2=1/8\pi G$, prime $=d/d\ln a$, flat universe, $\Omega_X=\rho_X/3M_p^2H^2$.

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Li's Model

In Li's 2004 model, $L$ is the future event horizon, the farthest a light ray sent today will ever get:

$$L_e=a\int_t^\infty\frac{dt'}{a(t')}, \qquad \dot L_e=HL_e-1.$$

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

And this is where the problems begin. With $c=1$ and $\Omega_{\rm DE,0}=0.6889$, $w_0=-0.887$, while Planck 2018 $w_0=-1.03\pm0.03$. Does not match observations! And any constant $c<1$ gives $w<-1$ (phantom) and a future turnaround in $H$ (for $c=0.8$, at $z\simeq-0.33$).

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So... Why Not Let $c$ Vary?

Put $\rho=3M_p^2c(a)^2/L_e^2$ into energy conservation $\rho'=-3(1+w)\rho$ and use $\dot L_e=HL_e-1$:

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

Li's formula is the special case $c'=0$. Demand ΛCDM, $w=-1$:

$$\boxed{c'=c-\sqrt{\Omega_\Lambda}}$$

For a true cosmological constant $\rho_\Lambda=3M_p^2H_\Lambda^2$, with $H_\Lambda=\sqrt{\Lambda/3}$ the Hubble rate the universe ends up at, compare with the HDE form and read off

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

which solves the equation above. So $c<1\not\Rightarrow w<-1$ once $c$ evolves. With Planck numbers $c_0=0.952$, $c\propto a$ early on, $c\to1$ late. (The ODE alone allows $c=H_\Lambda L_e+Ka$; the bounds below kill both signs of $K$, so the solution is unique.)

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What Is $c^2$, Physically?

Inside the horizon: energy $E_\Lambda=\frac{4\pi}{3}L_e^3\rho_\Lambda=H_\Lambda^2L_e^3/2G$; black-hole mass of that radius $M_{\rm BH}=L_e/2G$; horizon entropy $S_e=\pi L_e^2/G$; final de Sitter entropy $S_{\rm dS}=\pi/GH_\Lambda^2$. All ratios collapse to one number:

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

$c^2$ is how full the black-hole energy bound is, and equally how full the final entropy budget is. Today: 91%.

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Bounds

Horizon can't shrink (aka, in LCDM the event horizon grows monotonicity towards its dS limit): $\dot L_e\ge0\iff HL_e\ge1\iff c^2\ge\Omega_\Lambda$. Energy can't beat a black hole: $c^2\le1$.

$$\boxed{\Omega_\Lambda\le c^2\le1, \qquad c\to1}$$

or $S_H\le S_e\le S_{\rm dS}$. Never violated in ΛCDM. Note these are equivalent to $\dot H\le0$ and $H\ge H_\Lambda$: positive matter energy plus a de Sitter endpoint. True, but not something ΛCDM could have failed.

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How the Horizon Fills Up

Late on, let one component $\rho_x\propto a^{-n}$, $n=3(1+w_x)$, remain, with $r=\rho_x/\rho_\Lambda\ll1$. Friedmann gives $H\simeq H_\Lambda(1+r/2)$. From $\dot L_e=HL_e-1$, exactly $\dot c=Hc-H_\Lambda$; write $c=1-y$ and linearise: $\dot y\simeq H_\Lambda(y-r/2)$ with $\dot r=-nH_\Lambda r$. The solution that converges is $y=r/2(n+1)$:

$$\boxed{1-c\simeq\frac{1}{2(n+1)}\frac{\rho_x}{\rho_\Lambda}, \qquad 1-\frac{S_e}{S_{\rm dS}}\simeq\frac{1}{n+1}\frac{\rho_x}{\rho_\Lambda}}$$

For matter ($n=3$): $\frac18$ and $\frac14$. Several components add linearly. Next order: $1-c\simeq\frac{r}{2(n+1)}-\frac{3r^2}{8(2n+1)}$, which already gives $1-c_0=0.047$ today (exact 0.048). All confirmed numerically for $n=1$–$6$.

The exact version is one line:

$$1-c=a\int_a^\infty\frac{da'}{a'^2}\left(1-\frac{H_\Lambda}{H(a')}\right).$$

The entropy deficit is the integrated future departure of $H$ from $H_\Lambda$. That is why it knows $w_x$: $n$ sets how fast that departure dies. The event horizon is defined by the future, so of course it records it.

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Or, Take the Hubble Radius

With $L=H^{-1}$, $\rho_{\rm DE}=3M_p^2c(a)^2H^2$, so immediately $\boxed{\Omega_{\rm DE}=c(a)^2}$, and $\boxed{c(a)=\sqrt{\Omega_\Lambda(a)}}$ reproduces ΛCDM. This was pointed out by Lixin Xu in 2009. (Constant $c$ here gives $w=0$ in the matter era, Hsu 2004, which is why Li went to the event horizon.)

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The HDE Family

Four general types (Wang and Li 2017): (1) the original future-horizon version, which needs varying $c(a)$; (2) other length scales, i.e. generalised HDE; (3) extended Hubble scales, the Hubble-radius route above being one, Ricci another; (4) dark-sector interaction.

Nojiri and Odintsov (2006, covariant 2017) took type 2 to its limit: the cutoff is any function $L_{\rm IR}(L_p,\dot L_p,\ldots,L_e,\dot L_e,\ldots,a)$. Any expansion history can be produced this way, and Tsallis, Rényi, Barrow entropic models are all members (Nojiri, Odintsov & Paul 2021). Where is ΛCDM? Write $\rho_\Lambda=3M_p^2/L_{\rm IR}^2$ with $L_{\rm IR}=L_e/c$:

$$\boxed{L_{\rm IR}=\frac{L_e}{H_\Lambda L_e}=\frac1{H_\Lambda}}$$

the constant cutoff, the most trivial member of the family. "$c$ runs" and "the cutoff doesn't" are the same sentence.

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Dead Ends

$c=1$ forever. Saturates the bound, but gives $w_0\simeq-0.89$, not $\Lambda$CDM.

Deriving $c^2=S_e/S_{\rm dS}$ from Clausius. Since

$$T_e,dS_e=\frac{dL_e}{G}, \qquad dE_\Lambda=\frac{3c^2}{2G},dL_e,$$

$dE_\Lambda=T_e,dS_e$ holds only at $c^2=2/3$. A dynamical event horizon is not generally an equilibrium thermodynamic system.

Ignoring the boundary condition. The ODE permits $c=H_\Lambda L_e+Ka$; the event-horizon condition $L_e/a\to0$ forces $K=0$.

Claiming to predict $\Lambda$. The construction takes $H_\Lambda$ as input and rewrites $\Lambda$CDM in HDE variables. It does not determine $\Lambda$ or solve its radiative-instability problem.

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Wrap

Future-event-horizon HDE does not derive $\Lambda$. Once a positive cosmological constant and its de Sitter endpoint are supplied, however, $\Lambda$CDM has the unique event-horizon representation

$$c(a)=H_\Lambda L_e(a), \qquad c^2=\frac{E_\Lambda}{M_{\rm BH}} =\frac{S_e}{S_{\rm dS}},$$

subject to the future event-horizon boundary condition. The apparent running of $c$ is therefore not new dynamics: it is the geometric record, encoded nonlocally by the event horizon, of the remaining departure from the final de Sitter state.

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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).