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Holographic dark energy (HDE) is a cosmological model motivated as a solution to the cosmological constant problem. As summarised here the density of dark energy is (usually written with natural units, but below I've spelled out):

\begin{aligned} \rho _{DE} = \frac{3c^2}{8\pi G L^2}. C^2 \end{aligned}

Where $c$ is the (dimensionless) model parameter of the HDE which is positive and (generally) taken as constant and $C$ is the speed of light. As the Friedmann equation is $\rho_{DE}=3M_p^2H^2$ you get (in natural units) $c=HL$ with $H$ as the future Hubble horizon which equals the future event horizon, which is why Li felt the HDE parameter $c=1$. The HDE model is defined in terms of an IR cut-off $L$. The future event horizon is the common choice for $L$. This gives an EoS:

\begin{aligned} w_{DE}=-\frac{1}{3}\left( 1+\frac{2\sqrt{\varOmega _{DE}}}{c}\right) , \end{aligned}

And this is where the problems begin, because $c=1$ with say $\Omega_{DE,0}=0.6889$ gives $w_0=-0.89$ which does not match Planck $w_0=-1.03\pm0.03$

Question: If $L$ is the future event horizon, why not assume $c$ varies over time? Such that $c(t)=\sqrt\Omega_{DE}$. That would give $w =-1$. And, in the far future $c_\infty=1$.

With varying $c(t)=\sqrt\Omega_{DE}$ then the horizon in this model would be the Hubble horizon. Of course, this idea is probably not so clever, and is really just the $\Lambda$CDM model?

EDIT

Given the lack of any response to date, I've done more digging. I have found a 2018 paper Can holographic dark energy models fit the observational data, here. This confirms my suspicion that a varying $c$ term does means that the Hubble horizon is the IR cut-off $L$, and, you still get an accelerating universe. However, in that paper (and the literature cited), people have developed some esoteric models for $c(t)$, none of which seem to be as simple as just $c(t)=\sqrt\Omega_{DE}$. Maybe people just want a HDE model with a varying dark energy equation of state, not a boring constant one.

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  • $\begingroup$ I am no expert on the field, but as there are no other thoughts... From analysis of the dual problem (low pressure thermodynamics on 2D surface) there are indications that $T_{\mu\nu} \neq 0$ in deep space, as one would expect. Back to 3D bulk space, suppose $g_{\mu\nu}$ is quantized & cannot take arbitrarily small changes in values. Then $R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu} = 0$ is a diophantine equation. It is evident that this equation may not always in all cases have an integer solution. i.e. there must be some slack given, some small $T_{\mu\nu} \approx 0$, or illusory matter/energy. $\endgroup$
    – James
    Commented Aug 11 at 5:38
  • $\begingroup$ The existence of sparse hydrogen atoms, passing photons, etc. ("low pressure") in deep space means $g_{\mu\nu}$ cannot be set trivially to zero in a whole neighborhood. $\endgroup$
    – James
    Commented Aug 11 at 5:54
  • $\begingroup$ @James - Thank you. However - I am quite unclear as to the relevance of your comments to the question. $\endgroup$
    – Mr Anderson
    Commented Aug 12 at 1:12

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