View Full Version : Natural vs Unnatural in cosmology
Serenus Zeitblom
Apr30-04, 03:09 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\nhelbig@astro.multiCLOTHESvax.de (Phillip Helbig---remove CLOTHES to reply) wrote in message news:<c66mbc\\$fau\\$2@online.de>...\n> .\n>\n> I think it\'s important to keep in mind that there is no observational\n> evidence that the dark energy (or "smooth tension" in Sean Carroll\'s\n> better terminology) is not a classical cosmological constant.\n>\n\nAnd I think it\'s even more important to keep in mind that there is\nno observational evidence that the smooth tension IS a classical\ncosmological constant! The idea that the CC is somehow "more natural"\nor "simpler" than other candidates would make sense if we had a\ntheoretical basis for the CC, but that, as you know, is notoriously\nnot the case. Things like quintessence or phantoms are, it is true,\nbadly motivated theoretically...but that does not distinguish them\nfrom the CC! The CC is like the Rolling Stones --- the fact that\nthe latter have no talent somehow seems more acceptable because they\nhave had no talent for so long. The same goes for things like\nnon-trivial topology in cosmology. There is *no* sense in which\na simply connected manifold is "more natural" than a non-SC manifold,\nand clearly there are many more non-SC manifolds than SC ones. Heck,\na [flat] torus looks a *lot* simpler and more natural than infinite R^3 to\nme! [Remember the definition of a manifold: it\'s something that locally\nresembles a piece of a torus. Right? :) ]\n\nAs far\nas I know, and I\'d love to be corrected, there are no theoretical\nindications that complicated topology is somehow "bad". So if it\nsomehow turns out that our Universe *is* simply connected, this would\nbe an amazing fact *requiring an explanation*. Same goes for the\ncosmological constant if indeed that is the smooth tension. Of course\nit is possible to construct models in which w is nearly constant for\na long time and then suddenly changes---we are going to need such\nmodels if, as lots of string theorists believe, the universe is\nheaded for a big crunch after it gets over this temporary adolescent\nperiod of acceleration......\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>helbig@astro.multiCLOTHESvax.de (Phillip Helbig---remove CLOTHES to reply) wrote in message news:<c66mbc$fau$2@online.de>...
> .
>
> I think it's important to keep in mind that there is no observational
> evidence that the dark energy (or "smooth tension" in Sean Carroll's
> better terminology) is not a classical cosmological constant.
>
And I think it's even more important to keep in mind that there is
no observational evidence that the smooth tension IS a classical
cosmological constant! The idea that the CC is somehow "more natural"
or "simpler" than other candidates would make sense if we had a
theoretical basis for the CC, but that, as you know, is notoriously
not the case. Things like quintessence or phantoms are, it is true,
badly motivated theoretically...but that does not distinguish them
from the CC! The CC is like the Rolling Stones --- the fact that
the latter have no talent somehow seems more acceptable because they
have had no talent for so long. The same goes for things like
non-trivial topology in cosmology. There is *no* sense in which
a simply connected manifold is "more natural" than a non-SC manifold,
and clearly there are many more non-SC manifolds than SC ones. Heck,
a [flat] torus looks a *lot* simpler and more natural than infinite R^3 to
me! [Remember the definition of a manifold: it's something that locally
resembles a piece of a torus. Right? :) ]
As far
as I know, and I'd love to be corrected, there are no theoretical
indications that complicated topology is somehow "bad". So if it
somehow turns out that our Universe *is* simply connected, this would
be an amazing fact *requiring an explanation*. Same goes for the
cosmological constant if indeed that is the smooth tension. Of course
it is possible to construct models in which w is nearly constant for
a long time and then suddenly changes---we are going to need such
models if, as lots of string theorists believe, the universe is
headed for a big crunch after it gets over this temporary adolescent
period of acceleration......
Phillip Helbig---remove CLOTHES to reply
May3-04, 04:52 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>In article <c7fd6c7a.0404292359.7d0ca8a@posting.google.com>,\ nserenuszeitblomphd@yahoo.com (Serenus Zeitblom) writes:\n\n> > I think it\'s important to keep in mind that there is no observational\n> > evidence that the dark energy (or "smooth tension" in Sean Carroll\'s\n> > better terminology) is not a classical cosmological constant.\n\nTwo distinct points are to be made here.\n\n> And I think it\'s even more important to keep in mind that there is\n> no observational evidence that the smooth tension IS a classical\n> cosmological constant!\n\nFirst, what I meant was that there is no evidence that it is a) varying\nwith time or b) due to any of the causes which have been put forth\nrecently.\n\n> The idea that the CC is somehow "more natural"\n> or "simpler" than other candidates would make sense if we had a\n> theoretical basis for the CC, but that, as you know, is notoriously\n> not the case.\n\nI think that depends on one\'s perspective. Of course, the universe\nshouldn\'t depend on one\'s perspective (Einstein would agree here in the\nsense that the Moon is there if we don\'t look, but I\'m not talking\nquantum mechanics now, just history of science), as some who cite\nEinstein\'s "biggest blunder" remark would have one believe (the\ncosmological constant must be a bad idea, because at first Einstein\ndidn\'t include it, and though he later did he still later regretted it).\nOn the other hand, the question "what is natural" is I think a valid\nquestion. In particle physics, it is standard practice to say that if\nNature has a degree of freedom, then it is used. If one experimentally\nfinds that this is not the case, then one has discovered a conservation\nlaw/symmetry/quantum number---and of course the burden of proof is on\nthe discoverer. In that sense, it is a non-zero cosmological constant\nwhich is natural and one exactly zero would need an explanation.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>In article <c7fd6c7a.0404292359.7d0ca8a@posting.google.com>,
serenuszeitblomphd@yahoo.com (Serenus Zeitblom) writes:
> > I think it's important to keep in mind that there is no observational
> > evidence that the dark energy (or "smooth tension" in Sean Carroll's
> > better terminology) is not a classical cosmological constant.
Two distinct points are to be made here.
> And I think it's even more important to keep in mind that there is
> no observational evidence that the smooth tension IS a classical
> cosmological constant!
First, what I meant was that there is no evidence that it is a) varying
with time or b) due to any of the causes which have been put forth
recently.
> The idea that the CC is somehow "more natural"
> or "simpler" than other candidates would make sense if we had a
> theoretical basis for the CC, but that, as you know, is notoriously
> not the case.
I think that depends on one's perspective. Of course, the universe
shouldn't depend on one's perspective (Einstein would agree here in the
sense that the Moon is there if we don't look, but I'm not talking
quantum mechanics now, just history of science), as some who cite
Einstein's "biggest blunder" remark would have one believe (the
cosmological constant must be a bad idea, because at first Einstein
didn't include it, and though he later did he still later regretted it).
On the other hand, the question "what is natural" is I think a valid
question. In particle physics, it is standard practice to say that if
Nature has a degree of freedom, then it is used. If one experimentally
finds that this is not the case, then one has discovered a conservation
law/symmetry/quantum number---and of course the burden of proof is on
the discoverer. In that sense, it is a non-zero cosmological constant
which is natural and one exactly zero would need an explanation.
ebunn@lfa221051.richmond.edu
May3-04, 04:52 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\nIn article <c7fd6c7a.0404292359.7d0ca8a@posting.google.com>,\ nSerenus Zeitblom <serenuszeitblomphd@yahoo.com> wrote:\n\n>And I think it\'s even more important to keep in mind that there is\n>no observational evidence that the smooth tension IS a classical\n>cosmological constant! The idea that the CC is somehow "more natural"\n>or "simpler" than other candidates would make sense if we had a\n>theoretical basis for the CC, but that, as you know, is notoriously\n>not the case. Things like quintessence or phantoms are, it is true,\n>badly motivated theoretically...but that does not distinguish them\n>from the CC!\n\nThis is quite true, and it\'s good to remind people of it.\nAmong our many bad habits, cosmologists tend to give too much weight\nto "naturalness" considerations, which are often hard to distinguish\nfrom considerations of taste or familiarity.\n\n>The same goes for things like\n>non-trivial topology in cosmology. There is *no* sense in which\n>a simply connected manifold is "more natural" than a non-SC manifold,\n>and clearly there are many more non-SC manifolds than SC ones. Heck,\n>a [flat] torus looks a *lot* simpler and more natural than infinite R^3 to\n>me!\n\nI think that the "naturalness" argument about topology is really more\nof a fine-tuning / coincidence argument. If spacetime is multiply\nconnected, then it has a length scale associated with it (say, the\nshortest dimension of the fundamental cell or something). That length\nscale could have been anything: it could have been many orders of\nmagnitude less than the current horizon scale, or many orders of\nmagnitude more. If the Universe turns out to have nontrivial\ntopology, *and* if that nontrivial topology turns out to be detectable\nin any imaginable experiment, then that length scale will have to be\nquite close to the current horizon scale. That seems to many people\nto be an unlikely coincidence.\n\nIn other words, it\'s not so much that multiply-connected is unnatural,\nbut that multiply-connected-on-the-same-scale-as-the-horizon is\nunnatural (because coincidental).\n\nPersonally, I think that these fine-tuning / coincidence arguments are\noften overstated. That\'s another bad habit of cosmologists, but it\'s\na somewhat different bad habit from the naturalness one discussed\nabove. On behalf of cosmologists, I\'d like to ask you to keep our\nmany vices straight!\n\nIn any case, the Universe seems to have hit us with at least one of\nthose coincidences of scale: whatever the dark energy is, its density\nsurely varies with time in a very different way from the density of\nmatter. It seems to be a coincidence that we were around at the specific\nepoch when the two were comparable in size. That\'s the sort of coincidence\nthat anti-fine-tuning folks have always used as a reason to disfavor\ncertain theories, but the Universe doesn\'t seem to have paid us\nany heed.\n\n-Ted\n\n--\n[E-mail me at name@domain.edu, as opposed to name@machine.domain.edu.]\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>In article <c7fd6c7a.0404292359.7d0ca8a@posting.google.com>,
Serenus Zeitblom <serenuszeitblomphd@yahoo.com> wrote:
>And I think it's even more important to keep in mind that there is
>no observational evidence that the smooth tension IS a classical
>cosmological constant! The idea that the CC is somehow "more natural"
>or "simpler" than other candidates would make sense if we had a
>theoretical basis for the CC, but that, as you know, is notoriously
>not the case. Things like quintessence or phantoms are, it is true,
>badly motivated theoretically...but that does not distinguish them
>from the CC!
This is quite true, and it's good to remind people of it.
Among our many bad habits, cosmologists tend to give too much weight
to "naturalness" considerations, which are often hard to distinguish
from considerations of taste or familiarity.
>The same goes for things like
>non-trivial topology in cosmology. There is *no* sense in which
>a simply connected manifold is "more natural" than a non-SC manifold,
>and clearly there are many more non-SC manifolds than SC ones. Heck,
>a [flat] torus looks a *lot* simpler and more natural than infinite R^3 to
>me!
I think that the "naturalness" argument about topology is really more
of a fine-tuning / coincidence argument. If spacetime is multiply
connected, then it has a length scale associated with it (say, the
shortest dimension of the fundamental cell or something). That length
scale could have been anything: it could have been many orders of
magnitude less than the current horizon scale, or many orders of
magnitude more. If the Universe turns out to have nontrivial
topology, *and* if that nontrivial topology turns out to be detectable
in any imaginable experiment, then that length scale will have to be
quite close to the current horizon scale. That seems to many people
to be an unlikely coincidence.
In other words, it's not so much that multiply-connected is unnatural,
but that multiply-connected-on-the-same-scale-as-the-horizon is
unnatural (because coincidental).
Personally, I think that these fine-tuning / coincidence arguments are
often overstated. That's another bad habit of cosmologists, but it's
a somewhat different bad habit from the naturalness one discussed
above. On behalf of cosmologists, I'd like to ask you to keep our
many vices straight!
In any case, the Universe seems to have hit us with at least one of
those coincidences of scale: whatever the dark energy is, its density
surely varies with time in a very different way from the density of
matter. It seems to be a coincidence that we were around at the specific
epoch when the two were comparable in size. That's the sort of coincidence
that anti-fine-tuning folks have always used as a reason to disfavor
certain theories, but the Universe doesn't seem to have paid us
any heed.
-Ted
--
[E-mail me at name@domain.edu, as opposed to name@machine.domain.edu.]
Phillip Helbig---remove CLOTHES to reply
May3-04, 05:51 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>In article <c6u2cp\\$15p\\$1@lfa222122.richmond.edu>,\nebunn@ lfa221051.richmond.edu writes:\n\n> I think that the "naturalness" argument about topology is really more\n> of a fine-tuning / coincidence argument. If spacetime is multiply\n> connected, then it has a length scale associated with it (say, the\n> shortest dimension of the fundamental cell or something).\n\nI\'m not a topologist, but...couldn\'t there be something like\nscale-invariant multiply-connectedness?\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>In article <c6u2cp$15p$1@lfa222122.richmond.edu>,
ebunn@lfa221051.richmond.edu writes:
> I think that the "naturalness" argument about topology is really more
> of a fine-tuning / coincidence argument. If spacetime is multiply
> connected, then it has a length scale associated with it (say, the
> shortest dimension of the fundamental cell or something).
I'm not a topologist, but...couldn't there be something like
scale-invariant multiply-connectedness?
ebunn@lfa221051.richmond.edu
May6-04, 07:52 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\nIn article <c75rph\\$dbg\\$1@online.de>,\nPhillip Helbig---remove CLOTHES to reply <helbig@astro.multiCLOTHESvax.de> wrote:\n\n>I\'m not a topologist, but...couldn\'t there be something like\n>scale-invariant multiply-connectedness?\n\nI don\'t see how. In fact, I don\'t really understand what such a thing\nwould mean.\n\nAs far as I know, the only way to construct a multiply-connected\ncosmological (that is, locally approximately FRW) spacetime is to take\nsome sort of fundamental cell in the covering space (i.e., a\nsimply-connected homogeneous 3-space) and identify various faces of\nthe cell. The familiar example is the 3-torus, where the cell is a\nrectangular solid and opposite faces are identified. All the others\nare variations on that theme, I believe.\n\nTo be a bit more precise, you start with your covering space, and pick\na discrete group of isometries. You then identify all points that are\nconnected by an element of that discrete group. (To get the flat\nthree-torus, your isometries are just translations by jA+kB+lC, where\nj,k,l are integers and A,B,C are fixed 3-vectors.)\n\nAnyway, that fundamental cell is a three-dimensional hunk of space (at\nany given time), so it\'s got length scales associated with it. For\ninstance, it\'s got a volume (at any given time), whose cube root gives\nyou a length scale. So I don\'t see how such a thing could be\nscale-invariant.\n\nI could be wrong, of course. If so, I hope someone will correct me.\n\n-Ted\n\n\n--\n[E-mail me at name@domain.edu, as opposed to name@machine.domain.edu.]\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>In article <c75rph$dbg$1@online.de>,
Phillip Helbig---remove CLOTHES to reply <helbig@astro.multiCLOTHESvax.de> wrote:
>I'm not a topologist, but...couldn't there be something like
>scale-invariant multiply-connectedness?
I don't see how. In fact, I don't really understand what such a thing
would mean.
As far as I know, the only way to construct a multiply-connected
cosmological (that is, locally approximately FRW) spacetime is to take
some sort of fundamental cell in the covering space (i.e., a
simply-connected homogeneous 3-space) and identify various faces of
the cell. The familiar example is the 3-torus, where the cell is a
rectangular solid and opposite faces are identified. All the others
are variations on that theme, I believe.
To be a bit more precise, you start with your covering space, and pick
a discrete group of isometries. You then identify all points that are
connected by an element of that discrete group. (To get the flat
three-torus, your isometries are just translations by jA+kB+lC, where
j,k,l are integers and A,B,C are fixed 3-vectors.)
Anyway, that fundamental cell is a three-dimensional hunk of space (at
any given time), so it's got length scales associated with it. For
instance, it's got a volume (at any given time), whose cube root gives
you a length scale. So I don't see how such a thing could be
scale-invariant.
I could be wrong, of course. If so, I hope someone will correct me.
-Ted
--
[E-mail me at name@domain.edu, as opposed to name@machine.domain.edu.]
Thomas Dent
May14-04, 04:05 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>serenuszeitblomphd@yahoo.com (Serenus Zeitblom) wrote in message\n\n>\n> And I think it\'s even more important to keep in mind that there is\n> no observational evidence that the smooth tension IS a classical\n> cosmological constant! The idea that the CC is somehow "more natural"\n> or "simpler" than other candidates would make sense if we had a\n> theoretical basis for the CC, but that, as you know, is notoriously\n> not the case.\n\nIt *is* simpler, it is a very simple model to explain, and even to\nmotivate, requires precisely 1 piece of information to completely\nspecify and is completely predictive once that is fixed.\n\nModels with fewer free parameters are in general preferred, the plain\nvanilla cosmological constant has the least of any. There is a reason\nwhy Einstein thought of it and didn\'t think of quintessence or\nphantoms or ghost condensates.\n\nPlus, we expect from QFT that a plain constant cosmological constant\nshould exist - only the order of magnitude is wrong(!)\n\n> Things like quintessence or phantoms are, it is true,\n> badly motivated theoretically...but that does not distinguish them\n> from the CC!\n\nExcept that they have more free parameters (unless the model is built\nextremely cleverly in such a way that its results don\'t depend on the\nextra parameters...).\n\n> There is *no* sense in which\n> a simply connected manifold is "more natural" than a non-SC manifold,\n> and clearly there are many more non-SC manifolds than SC ones.\n\nAgain, it requires less information to specify a given\nsimply-connected manifold than a NSC one, so the SC models have fewer\nadjustable parameters in some sense - less choice. Less choice is\ngood, until ruled out by observation.\n\n> Of course\n> it is possible to construct models in which w is nearly constant for\n> a long time and then suddenly changes---we are going to need such\n> models if, as lots of string theorists believe, the universe is\n> headed for a big crunch after it gets over this temporary adolescent\n> period of acceleration...\n\nI don\'t know about "lots of string theorists", and I don\'t know why\nthey should be in favour of a Big Crunch. Linde has a model which\nrolls over from acceleration into a crunch in a finite time, so\nperhaps you used the equation\n\n1 Linde + collaborators = lots of string theorists.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>serenuszeitblomphd@yahoo.com (Serenus Zeitblom) wrote in message
>
> And I think it's even more important to keep in mind that there is
> no observational evidence that the smooth tension IS a classical
> cosmological constant! The idea that the CC is somehow "more natural"
> or "simpler" than other candidates would make sense if we had a
> theoretical basis for the CC, but that, as you know, is notoriously
> not the case.
It *is* simpler, it is a very simple model to explain, and even to
motivate, requires precisely 1 piece of information to completely
specify and is completely predictive once that is fixed.
Models with fewer free parameters are in general preferred, the plain
vanilla cosmological constant has the least of any. There is a reason
why Einstein thought of it and didn't think of quintessence or
phantoms or ghost condensates.
Plus, we expect from QFT that a plain constant cosmological constant
should exist - only the order of magnitude is wrong(!)
> Things like quintessence or phantoms are, it is true,
> badly motivated theoretically...but that does not distinguish them
> from the CC!
Except that they have more free parameters (unless the model is built
extremely cleverly in such a way that its results don't depend on the
extra parameters...).
> There is *no* sense in which
> a simply connected manifold is "more natural" than a non-SC manifold,
> and clearly there are many more non-SC manifolds than SC ones.
Again, it requires less information to specify a given
simply-connected manifold than a NSC one, so the SC models have fewer
adjustable parameters in some sense - less choice. Less choice is
good, until ruled out by observation.
> Of course
> it is possible to construct models in which w is nearly constant for
> a long time and then suddenly changes---we are going to need such
> models if, as lots of string theorists believe, the universe is
> headed for a big crunch after it gets over this temporary adolescent
> period of acceleration...
I don't know about "lots of string theorists", and I don't know why
they should be in favour of a Big Crunch. Linde has a model which
rolls over from acceleration into a crunch in a finite time, so
perhaps you used the equation
1 Linde + collaborators = lots of string theorists.
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