I'm a bit astonished that this question has not received any answer yet ! Is there any reason for that ? After all, this diagram of the conformal horizon scale in function of the logarithm of the scale factor was also pointed out as a good teaching tool in 2008 (see "VII - Approaches to teaching dark energy" on p. 5 of enclosed Resource Letter). It would be surprising that, 8 years later, no use would have been done of this diagram to compare dark energy models in courses, lectures, discussions .... (?)    

I think the problem is that your question can only be answered by people working on the alternative theories and this site has a fairly small community that may not include those. Some of the theories also would have ill-defined curves, quintessence for example is a time-varying dark energy but there is no unique formula for that variation.

@ George,

I agree with you. But, one of the formulas for quintessence (the most popular ?) would already suffice in first instance ...

Dear Guibert, maybe you want to consider also these facts:

1) there's no sure experimental confirmation about the real meaning of Hubble's law. It predicts a redshift which increase with distance. And we experimentally notice this fact. But why does it happen?  We answer: "for sure, since the more distant a galaxy is, the more red is the shift". But other causes can't be excluded at the moment. There's not only a single cause for redshift.

2) As we know, lambda refers to dark energy. About the 70% of all energy in universe. The question is: does the simple presence of dark energy suffice to produce  gravitational interaction? (attractive or repulsive). Or we also need dynamic processes which have to concern dark energy (e.g. perturbations due to the presence of matter, as I think)?

Maybe this won't help much for the question you asked, but I want to say that we can't understand which model is the right one until we don't understand how gravity works from a quantum point of view, what dark energy and space itself are and what the true mechanism behind Hubble's law is.

At the moment it's only speculation. When you cross speculations you can see divergences or similarities, but this doesn't mean anything. It depends on what a hypothesis move from. Twenty hypothesis (theories) with similar preconditions lead to similar outputs (curves) in a graphic like that. But this doesn't mean that they're right (nor close to the truth) or to be preferred only because they're more. It's not a matter of quantity. And the right hypothesis could be NONE of those.

Dear Marco,

 I can fully understand your distrust towards all these models of dark energy. They are indeed speculative.

You write «As we know, lambda refers to dark energy ». But in fact I am unhappy even with that claim. I don’t like very much this tendency to invoke the cosmological constant Λ as soon as some unexplained new observation is encountered. It is like with the Loch Ness monster. It reappears when we are nearly convinced that it does not exist or at least that it can be forgotten. In 1917, Einstein’s picked it up to have his cosmological model based on General Relativity static. Then, in the years 1920, thanks to Friedmann, Lemaître and observations by Shipher and Hubble, Λ could be put aside. But then again, about a decade later, there was the puzzle why measurements showed that some stars were older than the Universe. A second chance was given to Λ which came back in the limelight. But, more precise measurements of the Hubble constant in the years 1950 (closer to 100 km/s/Mpc than to 500 km/s/Mpc) made Λ again unnecessary. Then there was this new observation of the years 1960 that we only observe quasars in a small redshift interval at high z. And Λ popped up once more …. And disappeared 10 years later when it was observed that the interval was not so small. We could then rest a bit of 20 years without need to invoke Λ. But, from the end of the years 1990, the cosmological constant came to light again : now it was – and it is still 19 years later - to explain the meanwhile observed acceleration of the expansion of the Universe : see Schwarzschild (1998), Riess et al. (1998), Perlmutter et al. (1999), Riess, Nugent, Schmidt et al. (2001), and their measurements on distant Ia supernovae + the WMAP and Planck mission reports on measurements of the Cosmic Microwave Background Radiation (2013 and 2015).

 So now, in 2016, the cosmological constant is firmly installed as the simplest (see Occam’s razor) explanation for the repulsive « dark » energy inferred from the Universe’s acceleration. The simplest and also compatible with all measurements. Even more, it is related to quantum vacuum energy. A nice trip for Einstein’s « greatest blunder of my life », isn’t it ?.

Therefore, in other RG questions, I tried to get more insight in the way quantum vacuum fluctuations could result in a stable constant repulsive energy (Λ should be constant in time) during a bit of billions of years, but the given answers lacked full assurance and completeness (for a non-expert).

As you see, while you speak of speculative models, I am not even convinced by the simplest and most accepted of them !

 Therefore, this question. Yes, the models are speculative, but it is the work of theoricians to put forward models in expectation that they once will be confirmed by observation (here the expected "Joint Dark Energy Mission" ?). It is then a question of efficiency for us who are not directly involved in the subject not to take too much time with things we find less convincing.

The diagram of the conformal horizon scale in function of the logarithm of the scale factor is a good solution for that. It has the great advantage to group and synthesize several models allowing to compare them. It is also given in Scholarpedia which is a kind of Wikipedia for researchers.

Therefore, it would be interesting to know in a nutshell whether :

• the curves for the different existing dark energy models have similar shapes, e.g. with VM and BW framing the preferred Λ model and the other models somewhere in between (as you suggest it) ? A by-question is whether there are other models with similar « q » slope value as the Λ model ;
• it could be possible that some models have different shapes, e.g. intersect the curve corresponding to the Λ model ? What would this then mean ? And if it is not possible, what is the (theoretical ?) reason for it ? 

So, I am convinced that any answer giving some hints on how the curves would like for a few other models than the 3 shown would trigger thinking not only for experts in cosmology, but also for researchers in other disciplines, teachers and scholars.

I took the record of lambda "popping up and disappearing" from the excellent Doctoral Thesis of Nan Li : "Cosmological Backreaction (From the local Hubble expansion rate to dark energy" (2008)

The false flat universe, needs mysterious ingredients such as dark matter, dark energy and inflation to bridge the gap between theory and observation. We did prove mathematically that the actual unverse is globally hyperbolic. The mathematical model predicts the equation of state w =- -1. The hyperbolic structure causes the accelerated expansion without needs for dark energy.

Article The Big Bang hyperbolic universe neither needs inflation nor...

Book The Hyperbolic Universe.

what about the universe if it's 4-dimensional Klein's bottle? Observer can see it as flat but it's not infinite.

Many thanks Salah and Esa for your answers. As the Conformal Hubble Scale diagram could in fact also be extended to other models than dark energy models, could you indicate how the curves for the models referred to in your answers would loook like in this diagram ? One just has to plot the conformal horizon scale (a.H)-1 on the ordinate in function of the logarithm of the scale factor (i.e. ln a)  on the abscissa for the past evolution of the universe. Both the scale factor "a" and the Hubble constant "H" are normalized in the sense that :

• a=R/R0 with "R" the scale factor in the past as obtained from the model and "R0" the present scale factor --> thus : a0= 1 at present time (with the usual practice of using subscript "0" to mean "at present time")
• H=H/H0 with "H" the Hubble parameter in the past as obtained from the model and "H0" the present Hubble parameter (e.g. following the recent Planck measurements : H0 = 67.8 km.s-1.Mpc-1) --> thus : H0=1 at present time.

Ok. I tried to approximate the bottle-universe model with quite a guesswork, added as turquoise. Still, the interpretations are different from actual accelerating universe, but I converted it in the diagram to be comparatable.