The Copernican principle is a guiding foundation of cosmology. In short, it states that we are not in a privileged place in the Universe. A “random” observer will see the same Universe that we do. The cosmological standard model does satisfy this principle in space: at this moment, any other observer in the Universe should see the same Universe as we do (at large scales). Just like us, they see a smooth distribution of galaxies and a smooth CMB sky, with similar small anisotropies. However, we do live at a privileged time: in the history of the Universe, we just happen to be at the time when the dark energy density starts dominating over the dark matter density. This is known as the “coincidence” problem, and has been much discussed and agonized over. Here is a graphical description:
Today is very, very near where the two lines cross (redshift=0 is today; redshift=1,000 is where the CMB is generated; the Big Bang is at redshift=infinity). You can’t even see the crossing on the main plot; you need to go to the inset to see the incredibly rapid change at redshift=1. Last week at the Yukawa Institute workshop John Moffat was advocating calling the standard model “anti-Copernican” because of this fine-tuning. He has been wanting to take matters one step further: if we are willing to break the Copernican principle in time, why not seriously consider breaking it in space instead? More on this later.
The Copernican principle is one of those weird things in science that is a mix of science and aesthetics. It can’t be written down as an equation. And its application is often subject to the eye of the beholder. For example, the plot above looks like a problem because we’ve used redshift on the x-axis to represent time. There are physically motivated reasons to use this, as it relates to the size of the Universe, and is thus a proxy for many relevant physical processes. If instead we label time the way we normally measure it (as in, on your wristwatch, if you happened to have been around since the Big Bang), you get something that looks much more reasonable:
We’re no longer at a special time, and the coincidence problem vanishes. The Universe has been dark-energy dominated for billions of years, and we’re nowhere near the special crossing point. So which plot is right?
June 30th, 2010 at 7:50 am
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June 30th, 2010 at 8:56 am
To me, it is obvious that the second plot is right.
June 30th, 2010 at 9:34 am
Is it just me or is the top graph unintelligible?
June 30th, 2010 at 9:44 am
Doesn’t the first plot have our special time hard-coded in, because redshift is a measure of time which depends on an objects distance from us in space (and therefor time)? This seems like complaining that parallax measurements are anti-copernican because they are so tiny for objects far away from our “special” location. Of course they are, the scale is dependent upon our location! The case of redshift isn’t so simple as that, but only because the relationship between the scale and our position is more complicated.
But my understanding of the weird behavior of scales in an expanding spacetime isn’t perfect, so maybe I’m missing something…
June 30th, 2010 at 9:54 am
My preference is to plot redshift (really the scale factor) on a log scale — since the energy density and scale factor are typically power laws — and to include the future as well as the past, so as to be a little more fair. Also, time should flow from left to right, as everyone knows.
http://nedwww.ipac.caltech.edu/level5/Carroll2/Carroll5.html
June 30th, 2010 at 10:24 am
Yeah, don’t you know that in astronomy everything MUST be graphed on a log-log plot
June 30th, 2010 at 10:36 am
“…if we are willing to break the Copernican principle in time, why not seriously consider breaking it in space instead?” Time is space or X (with a superscript) in Einstein notation. Easy to demonstrate. One good example is to start with e^2 = p^2 + (m^2)(c^4) and take it to the point showing t = (1/c^2)* Integral (E/p)dx = X and is a function of energy, momentum and distance. Some interesting insights but nevertheless time = space. And for a bound particle it is quantized. Perhaps neither plot is right.
June 30th, 2010 at 11:18 am
“which plot is right?”
They’re equivalent. The bottom one is more readable though. The nonsense is injected once we start talking about the probability distribution that observers are “drawn from”. It’s a silly notion really, observers aren’t placed by a random number generator.
In general, for these kinds of puzzling scenarios, I advocate Radford Neal’s approach as outlined in this paper (http://arxiv.org/abs/math/0608592). It doesn’t make these puzzles easy, but I reckon if we all took it seriously we’d at least improve the conversation.
June 30th, 2010 at 12:08 pm
Weird. The cross-over point appears to be exactly at the time we believe the Earth to have formed! Coincidence?
June 30th, 2010 at 1:05 pm
Wait, on the bottom chart, 0 years is the present day? Oh that is confusing. Timelines should go from left to right.
Ross does make an interesting observation. The crossover happened when the solar system formed. What’s going on there?
June 30th, 2010 at 3:40 pm
This paper
“Through the looking glass: why the `cosmic horizon’ is not a horizon”
http://adsabs.harvard.edu/abs/2010MNRAS.404.1633V
shows that the time-averaged value of the deceleration parameter,
CoI statement – I’m an author on the paper
June 30th, 2010 at 4:18 pm
Commenter #9 is correct. The crossover point in years is within eyeball error of the time of the solar system’s formation (4567 million years for the oldest condensates, with 50-110 million years for the terrestrial planets to accrete (ending with the moon-forming impact) after that). If you think the most unique thing about the Earth* is the large cogenetic moon, then coincidence hunters can feast on the similar timing of the lunar formation and the crossover in dark energies and matter.
* Hopefully Kepler will confirm or deny this supposition in the next few years.
June 30th, 2010 at 10:22 pm
I was asked this question a year ago in class, and I still don’t understand it. In both plots, our point in time looks equally privileged. Why does it matter if there is this long tail, obscuring the cross over point ,or if we look at a graph which basically looks like an ‘x’. On both graphs, there is clearly a point on that graph which sees a universe you might not see at another point in the graph. (On the redshift graph, there is a point which lives in a dark energy dominated universe, and another point somewhere else which sees a dark matter dominated. This is also true in the time graph, and in Sean’s log plot.)
I don’t understand the ‘issue’.
Cosmology works because the universe is isotropic/homogeneous over distances of >>100MPc, therefore the universe must be ‘smooth’ if you look at times of >>100megayears. It is not smooth whichever graph you look at, therefore there is a problem? Does this mean, we just need to exist for a few hundred billion more years, then the universe over time will be homogeneous/isotropic if you look at times >> a few billion years.
July 1st, 2010 at 4:29 am
It seems to me whichever plot you lot at, there is something special about the time-scale at which we observe the universe. We live at a(t) = a_0 and t=t_0, and in both cases Omega_m ~ Omega_L requires a(t)/a_0 ~ O(1), t/t_0~ O(1). That the second plot doesn’t look like much of a coincidence is entirely due to the fact that it hasn’t been continued into the future. If one continued to some arbitary time in the far future, our particular time would again look pretty special.
In both cases, it would be better to plot log(a) or log(t) on the x-axis as, in a sense, this doesn’t require one to pick a preferred scale for a or t. Doing this, and ploting both into the past and into the future, the coincidence is present in both plots.
Another way of looking at this coincidence is that t_L = Lambda^(-1/2) is O(t_0) (specifically t_L ~ 9.7 Gyrs, t_0 ~ 13.7 Gyrs). Anthropic limits say that t_L can’t been too small (t_L >~ 0.7 Gyrs or so) but say nothing about it being much larger than t_0, the question of why t_L ~ t_0 as opposed to t_L >> t_0 remains unanswered.
For this to be answered and t_L >> t_0 discarded as, in some sense, improbable one would have to have some insight into the prior distribution of Lambda in a multiverse / landscape setting, which, at present, we don’t really. If Lambda is uniformly distributed then the problem is not too bad, if it is uniformly distributed in log-space, however, then it is bad.
July 1st, 2010 at 6:11 am
Okay, I clearly blew it with the direction of time. My apologies to everyone.
@KC (#3): The top graph is intelligible because the transition happens too quickly. Does the inset make any more sense? It has the same axis labels.
@Daniel Berdine (#4): I am plotting density with respect to critical density, as a function of the size of the Universe. For the simplest cosmological model (Einstein-deSitter, with the matter density equal to the critical density), the dark matter density = 1 for all time, and the dark energy density = 0 for all time. For our Universe, the plot is more complicated. There is genuine information in these plots.
July 1st, 2010 at 6:13 am
@Sean: Yes, I should have plotted it with time going the other way. I guess that’s one of the first things you learn when you write the book on time? I didn’t plot it into the future just because I haven’t generalize my codes to do this (yet). But I should.
Plotting it log(redshift) looks qualitatively like the redshift plot, so I left it as is to keep it simple. I guess the question is: why is energy density the important quantity for the “coincidence in time” plot, and not lookback time (which I would claim is a more relevant quantity for most astrophysical processes)? Is it just personal predilection, or is there something more profound I’m missing?
July 1st, 2010 at 6:14 am
@Ross Collins and @Lab Lemming: Yes, coincidence, as far as I can tell. I can’t see any reason the processes governing the formation of the Earth would even know about the global value of the dark energy. It has absolutely no effect on the local physics. But, still, suggestive.
@Cusp: Interesting. Coincidence wherever one looks. But that’s one of the problems with this whole enterprise. If you look hard enough, you’ll probably find something “surprising”.
@Brian: The point where the densities are equal is a privileged point. It signals a qualitative change in the behavior of the Universe (deceleration to acceleration). How “close” you are to this point on these graphs is what people get excited about.
July 1st, 2010 at 6:31 am
@daniel (#17c) — the switch from deceleration to acceleration doesn’t quite happen when the densities are equal… q = Om/2 – Ol.
July 1st, 2010 at 6:38 am
Ooh, forgot we can latex: $latex q = frac{Omega_M}{2} – Omega_Lambda $
July 1st, 2010 at 11:06 am
I think it’s at least as interesting to consider that we’re at the point in time when there’s been just sufficient time for stellar evolution to generate heavy elements and enough time has passed for these elements to self-organize into beings (i.e. us) who can look back at the universe and question things like universal evolutionary inflection points. I don’t think the universe has been around long enough for a large sampling of other self-aware entities to have evolved.
July 1st, 2010 at 12:30 pm
Copernican principle doesn’t apply to time, we have Big Bang, recombination and so on.
Also I don’t see any coincidence here, the crossing happened 4.5 billion years ago, that’s certainly not now.
The graph linked to by Sean is a textbook example of a deceptive graph as it manages to hide from the viewer the fact that the supposed coincidence is off by 1/3 of the age of the Universe.
July 1st, 2010 at 1:39 pm
>>@Cusp: Interesting. Coincidence wherever one looks. But that’s one of the problems with this whole enterprise. If you look hard enough, you’ll probably find something “surprising”.
I agree – I have been thinking about this a bit more – but haven’t got very far (curse you teaching!!). If I find anything, I’ll report back
July 1st, 2010 at 1:47 pm
Daniel, the past is routinely plotted to the right; see figures 5-7 of the following paper as an easily available open access example:
http://conferences.minerals.nt.gov.au/cabsproceedings/Final_papers/P24_Haines_Wingate.pdf
July 1st, 2010 at 2:17 pm
Clearly, the top plot is “righter” than the bottom one. There is no reason that our selection of Hubble Time which is on the order of billions of years should be what we measure the Early Universe to be.
However, I think the “Coincidence Problem” is better posed as follows: why was matter domination in our universe so short? It really only happened over a few decades in redshift/scale factor space compared to radiation domination (from inflation to just before recombination) or dark energy domination (which extends into the unknowable future). If you assume a dark energy dominated universe until heat death, in fact, the brief period of time that matter dominates the universe is anthropically fascinating. We can only exist in that part of the universe that is relatively close to matter domination.
July 1st, 2010 at 3:09 pm
@Cusp
You should go on sabbatical.
July 1st, 2010 at 3:39 pm
Josh: “However, I think the “Coincidence Problem” is better posed as follows: why was matter domination in our universe so short?…”
No, it’s not better, it’s even more absurd, matter domination lasted 8.8 billion years – 65% of the age of the Universe, dark energy so far 4.5 billion years and radiation only 0.4 billion years.
July 1st, 2010 at 3:58 pm
And besides whose to say how long these periods should have lasted?
The *real* problem is that we understand neither dark matter nor dark energy – 95% of the energy content of the Universe is a mystery.
July 1st, 2010 at 4:59 pm
>> The *real* problem is that we understand neither dark matter nor dark energy – 95% of the energy content of the Universe is a mystery.
This is just not true – we know a lot about both dark matter and dark energy – we know that dark matter interacts gravitaitonally, but weakly with the other forces, and we know dark energy has an equation of state which is causing acceleration. We do know things about them.
As for sabbatical – 2 weeks and counting!!
July 1st, 2010 at 5:00 pm
@Josh
You can ask the same questions about the dominance of free quarks before the condensation of baryons or the equivalence of forces before gravity separated out, and I think the reason is the same: the effects of changes in density of various forms of matter/energy as a result of expansion. And there’s always going to be one form that dominates for longer than all others if expansion continues or accelerates.
July 2nd, 2010 at 4:07 am
Cusp: “This is just not true – we know a lot about both dark matter and dark energy – we know that dark matter interacts gravitaitonally, but weakly with the other forces, and we know dark energy has an equation of state which is causing acceleration. We do know things about them.”
Not true? Haha, you must be a cosmologist. I like how you started with “we know a lot about both dark matter and dark energy” but after coming up with basically one property for each one ended up with “we do know things about them.” Yes, we do know things about them, but it’s certainly not “a lot” by any stretch of the word, it’s next to nothing, we know a lot about normal matter.
July 2nd, 2010 at 6:29 am
I’ve never been all that impressed by this particular problem – the timescale for the formation of intelligent life (assuming we’re a typical example – very Copernican, no?) is a few billon years, which is about how far we are in time from the matter/dark energy transition, and is a pretty sizeable fraction of the age of the Universe.
Anyway, if any civilization intelligent enough to know what dark energy is is unlikely to have formed until a pretty recent redshift, the top plot doesn’t seem to me to be very indicative of a coincidence problem.
July 2nd, 2010 at 8:04 am
for us silly people, why is this plotted backwards???
July 2nd, 2010 at 12:42 pm
>> but it’s certainly not “a lot”
Well – it is a lot – we know dark matter is not normal matter, black holes, neutrinos, 3K bananas etc – we know it interacts gravitationally, but does not interact via EM, weak, strong – we know that it doesn’t decay with a limit on cross sections into electrons, photons, VW Beetles from limits on the background. We know it does cluster on certain scales, which tells is that if it is a particle it must be moving sub-revalistically (so it’s cold), we have stringent limits on how much hold dark matter there must be. We know how much there is from a gas profiles in clusters, gravitaitonal lensing, the dynamics of dwarf galaxies, the expansion of the cosmos, and we know what shapes it clusters into, with the shapes of halos from dynamics and gravitational lensing. I could go on – but we do know a lot.
Yes, I’m a cosmologist (and proud of it). I am read and contribute to the literature. I am quite aware we have not dissected dark matter in the lab, but it does not mean we don’t know anything about it. We have not touched the centre of the Sun, but can use observations and physics to tell us what is going on there. Dark matter is no different.
July 2nd, 2010 at 3:41 pm
I’ve gotten in this argument before. When two intelligent individuals have varying opinions on what constitutes “a lot.” In my opinion we do know a lot about dark matter and dark energy, however it’s obviously not enough. Cusp, everything you listed is simply us working towards understand the basic properties of dark matter, right?
Is the importance of our dark matter and dark energy research to provide classification of these different forms of energy in our Universe? It all comes down to being able to make testable predictions, does it not? As far as I know, as a scientific community, we still don’t know enough about the mathematics behind dark matter and dark energy to make correct predictions based on theory. Which in my mind is the essence of “understanding the Universe.” Basically, we know a lot about dark energy and dark matter, but still not enough.
July 2nd, 2010 at 4:39 pm
>> As far as I know, as a scientific community, we still don’t know enough about the mathematics behind dark matter and dark energy to make correct predictions based on theory.
But this is not true – we do know enough to make a lot of predictions – we know that the gravitational influence of dark matter is – and we know that dark energy causes accelleration of expansion – and we are now developing tests to see if dark energy is clustered etc –
As I said – we know a lot.
July 2nd, 2010 at 5:58 pm
Coming at it from a big-picture cosmology point of view, if you were to tell us what dark matter was tomorrow and confirm all of our current measurements about it were correct but give us the exact values instead of our upper/lower limits, you would not change our model of the universe much at all. A self-interaction cross-section of 10^-8 cm^2/g versus 10^-6 cm^2/g wouldn’t have much of an effects on the structure of the universe. So while identifying the particle(s) that is dark matter will have a big impact on particle physics, etc, from the cosmology point of view it is pretty much solved, leaving only minor tweaks needed once we have better data. (To put it more bluntly: our N-body models look quite a bit like the current universe, barring some messy baryonic physics, and changing the dark matter properties slightly won’t really change any of this enough to matter).
George: it’s a convention in cosmology. Current age of the universe is given as 0 (as about the only thing you could get most cosmologists to agree on 40 years ago was that now = now), and time is plotted as “lookback time” so the past is positive, the future is negative.
July 2nd, 2010 at 6:13 pm
Where else on this plot could we be? How much sooner could the universe have created a comfortable place where cosmologists could evolve? (And we can’t expect to be here many millions of years hence, much less billions.) Our location in time seems much less a random thing than our location in space.
July 2nd, 2010 at 11:15 pm
Historically, it seems to me that when people start talking about the importance of “coincidences” (especially when not everyone by any stretch of the imagination believes they exist) or “big numbers” etc in physics it has always led to very little, in terms of the advancement of the science. Which is not to say the arguments aren’t philosophically interesting (and divisive) just that they aren’t very important. I see no compelling reason that suggests that this case is any different. At this Golden Age (so far) in the history of cosmology aren’t there a host of more interesting problems? (and yes, “interesting” is definitely in the eye of the beholder, and people are of course free to work on whatever problems they wish to….But you damn well better be sure you have tenure first.)
July 3rd, 2010 at 6:31 am
Cusp, I understand it’s uncomfortable to admit that despite the best efforts of cosmologists like you we still know next to nothing about dark matter and dark energy, but that’s the current state of affairs when one wants to be objective.
As I said we know a lot about normal matter, compare that to what we know about dark matter, dark matter may very well be just as complex or even more so, we have a real particle zoo in SM so there is absolutely no reason why dark matter should be made of just one particle (if it’s made of particles at all), there could be tens or hundreds of them, there may be complex novel forces acting on them, and so on and on, the possibilities are endless.
The same could hold for dark energy, it’s quite possible there is a vast and complex structure hidden behind those simplistic terms. Our understanding of dark sector is comparable to our understanding of normal matter at the end of 19 century before electron was discovered (actually it’s even worse then that since we knew much more about normal matter even then).
Daug2: “…To put it more bluntly: our N-body models look quite a bit like the current universe, barring some messy baryonic physics, and changing the dark matter properties slightly won’t really change any of this enough to matter”
They look quite a bit like current universe because they were made to look like current universe, that doesn’t mean they actually capture the properties of the universe. The fact that the Sun looks quite a bit like a bright yellow sphere doesn’t mean that a bright yellow sphere is actually a good model of the Sun. Just as this model completely omits the structure and composition of the Sun so our current models may completely omit the structure and composition of the dark sector.
July 3rd, 2010 at 8:03 am
dark matter: carbon
dark energy: matte black springs
problem solved
July 3rd, 2010 at 9:43 am
[...] Science and philosophical considerations Cosmic variance has an interesting article about the balance between dark matter and dark energy: are we currently living in an exceptional period? It depends on whether you measure time by red shifts or by “the clock”. [...]
July 3rd, 2010 at 11:21 am
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July 3rd, 2010 at 1:07 pm
As Ross Collins & LabLemming point out there is something hauntingly anthropic about the crossover time, almost exactly dating the origin of the Solar System/Earth:
T = 1/c{sqrt(Lambda)} ~9.1 Gy post-BB (4.6 Gyr ago)
One normally thinks of the SS/E as environmental physics, rather than fundamental physics, as Lambda is supposed to be. Yet as Weinberg showed, w/out our special-valued CC, the milky way galaxy , & therefore the solar system, and earth would probably not be here. Thus there exists a transition between fundamental & environmental physics, perhaps similar to that between quantum & classical physics.
July 3rd, 2010 at 3:15 pm
>> next to nothing about dark matter and dark energy
Again, this is clearly untrue. We know a lot about both. Yes, the possibilities are endless – but it is no different for “normal” matter. If there are other forces acting on dark matter (or even the dark matter zoo) they must be quite subtle.
Anyway – you’re quite clearly locked into your opinion – so no point continuing the discussion.
July 4th, 2010 at 2:40 am
Cusp, you haven’t offered any convincing arguments so why should I change my opinion? Yes, we know dark matter interacts only gravitationally and clusters, we know it doesn’t decay in a way we could detect so far, and we have a few parameters estimated. That’s it. When it comes to dark energy we know even less. So again I fail to see how that is a lot, that is barely anything to me. And no, insights like “dark matter is not bananas” don’t count.
I have one last question for you, if we know a lot about dark matter and dark energy as you say then what adjective do you propose to use for what we know about normal matter?
I hope you agree that we know way more about normal matter – we know what it’s made of, how it interacts, how it decays, how it combines into more complex entities, we have whole disciplines dealing with it’s properties on certain scales like chemistry, biology and so on, the amount of knowledge accumulated is so large it’s impossible for one person to comprehend it all. To me all this means we know a lot about normal matter, so when I compare that amount of knowledge to what we know about dark matter and dark energy I can only say we know next to nothing about them. I think that’s a justified statement but you insist that what we know about dark matter and dark energy is already a lot, obviously you are using a very different measure, but that leaves said question – what adjective are you going to use when describing what we know about normal matter so that this huge difference in our level of understanding is properly communicated?
July 4th, 2010 at 10:10 am
This topic is no doubt profoundly interesting, but completely lost on me because I don’t know what a redshift graph is. I read this blog because you guys do a wonderful job of explaining complex physics to a (relative) layman like myself; I’m a computer scientist and studying to be a mathematician, not a physicist, though I like physics. Can you post another post and give it to us “from the beginning” a little more? That’s be great.
July 4th, 2010 at 8:21 pm
Although I am no expert on this, I know that certain effects from events long past are always apparent, and some appear as if they are happening now. This is obviously true when you look at light from a distant galaxy but it is also true when you look at an object falling into a black hole. Although the object falls in quickly, its image appears to be stuck at the event horizon, redshifting until it effectively becomes redder and darker, and creeping ever more slowly toward the horizon. In such a case, we observers always see the object around the crossover point of the event horizon. I wonder by analogy if such an apparent effect is at work here.
July 5th, 2010 at 12:54 am
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July 5th, 2010 at 2:27 am
Which plot is right? But ‘right’ for what purpose? Considering this question should make clear why the more general question (is there a coincidence?) produces such a diversity of (reasonable sounding) answers.
If you wish to see whether an observer who randomly and uniformly selects a single redshift to inspect (over the range 0,1000) is likely to find the two energy densities comparable, the first plot seems more sensible.
But, if you wish to show whether an observer who randomly and uniformly selects a single time to inspect (over present-13.7Ga) is likely to find the two energy densities comparable, the second plot seems more sensible.
Are either of these good strategies for placing observers within the universe? Or, what is a plausible distribution for observers to be drawn from? I’m sure you guys all know more cosmology and biology than I do, but I reckon there’s little probability density at z >> 1. In which case the first plot is not so helpful. Seems to me it might be more interesting to speculate about the future of the universe (are we around unusually early for such a long-lived place?).
How can we judge if there is a coincidence between any two events (such as comparable energy densities and appearance of intelligent observers) if we only consider the distribution of one? If we look at one in redshift (or time) we should do the same to the other to judge coincidence, right? Not a proper answer, but hopefully helpful to the discussion.
(Now, I must read the paper by Radford Neal suggested by Brendon J. Brewer, above…)
PS: What happens if the y-axis is also on a log (or logit) scale?
July 5th, 2010 at 3:51 am
“The Copernican principle is a guiding foundation of cosmology. In short, it states that we are not in a privileged place in the Universe.”
I doubt Copernicus would have agreed with the principle that bears his name.
July 5th, 2010 at 9:02 pm
[...] Entonces, ¿qué argumento es el correcto? Autor: Daniel Holz Fecha Original: 30 de junio de 2010 Enlace Original Articulos [...]
July 7th, 2010 at 6:56 am
I think this is just the anthropic principle all over again.
Constants are tweaked as to allow life on Earth, we live in a time dominated by dark energy, blah blah… so what? We’re bound to find coincidences, if we look hard enough. When we meet a number of advanced alien civilizations, we’ll ask if they have any better idea
July 8th, 2010 at 8:52 am
I think the redshifted version is more likely, since the redshift is more accurate than time.
July 13th, 2010 at 7:55 am
In discussing naturalness of real-life data, or physical parameters, there is an empirical principle, called the Benford’s Law (see the wikipedia article about this), which implies that real-life data are typically distributed uniformly in the logarithmic scale of the variable, rather than the linear scale. This law has been found to apply to a wide variety of data sets, including electricity bills, street addresses, stock prices, population numbers, death rates, lengths of rivers, physical and mathematical constants, and processes described by power laws. So in this sense, both the plots are not quite direct to look at the naturalness problem. However, plotted against ln(a) and ln(t), the first plot is still coincidental and the second one becomes coincidental.
July 29th, 2010 at 10:27 am
[...] I discussed in a previous post, John Moffat argues that we shouldn’t be any more disturbed by this model than the standard [...]
August 13th, 2010 at 2:33 am
Like it has been said, if we look hard enough, we’re bound to find -coincidences; hey, we exist now! Now, isn’t that a coincidence? And I’m sitting on a chair: not any chair, but THIS chair under my butt… Is that a coincidence or what?
I think, in these kinds of discussion, one of the first things the authors should do is to DEFINE coincidence, so that we all know what we are talking about, and so that we can talk about the same things.
And, oh yes, I am the center of the Universe – well, one of the centers, anyway