Scatter diagrams are now at http://www.rescalingsymmetry.com/evi...supernova.html
They show plots of magnitude against redshift for supernova, and explain the apparently accelerating universe, without the need for dark energy.
John Hunter.
Scatter diagrams are now at http://www.rescalingsymmetry.com/evi...supernova.html
They show plots of magnitude against redshift for supernova, and explain the apparently accelerating universe, without the need for dark energy.
John Hunter.
Where did you get that 1+z=e^{2Ht}, where H in your paper is H_{o}/2? That is the same thing as found for tired light theory in a static universe, which is the first thing I asked about on this forum, and asked you about as well, which would explain the apparent acceleration of expansion, as you say. I have recently found that, indeed, a constant rate of expansion with a constant speed for light does generate that same formula as well, although there are other problems involved with the expansion theory itself. But this is definitely the formula that should be used either way, and so redshift doesn't increase in direct proportion to distance, as formerly believed, but with 1+z=e^{(Hot)}=e^{(Hod/c)}. Did you come up with that formula yourself or is there a link to a reference you could provide?
{EDIT- Actually, on second thought, I'm not sure that e^{(Hot)} and e^{(Hod/c)} would be the same thing in a non-static universe. Probably not. e^{(Hod/c)}, however, would be the same either way. (But if you are finding 't' with t=d/c, then they are the same)]
Dear grav,
Yes, sorry I was in a rush yesterday. (well actually it depends which way we are measuring t, back from now, or forward)
(1+z) = exp(Ht) usually,but in rescaling theory (1+z) = exp(2Ht) which is [exp(Ht)]^2. (www.rescalingsymmetry.com/redshift_of_light.html)
Here is more explanation:
In paper "what do we really know about cosmic acceleration"
http://arxiv.org/PS_cache/astro-ph/pdf/0512/0512586.pdf
equation (5) L is changed to L/(1+z) , due to rescaling
equation (6) d = (1+z)*integral(0 to z ) changed to d= sqrt(1+z)*integral(0 to sqrt(1+z) - 1)
and H changed to H/2
q(z) is always -1 from the rescaling theory.
so all this combined means the same equations can be used as in the paper, if luminosity distance d is swapped to
d = 2(1+z)[sqrt(1+z) -1].
And this gives the scatter diagrams in http://www.rescalingsymmetry.com/evi...supernova.html
All the best,
John Hunter.
Um, I'm hate to do this, but I think we have a problem. Perlmutter found that the expansion seems to be accelerating, right? What are the relationships for this? Do they match your diagrams? The reason I'm asking is that I've been thinking about the f=f_{o}*e^{Ht} formula (when t=d/c directly) and suddenly realized that this would indicate, at least when compared to a direct redshift to distance ratio, that the expansion must appear exponentially greater the further back in time we go. That would mean that the expansion gives the appearance of decelerating, not accelerating. As well as this, any model for expansion would also appear to decelerate as well, when compared to the same redshift to distance ratio. Could this really be what Permutter meant or are we back to the drawing boards, trying to find some kind of repulsive energy in space? What do you think?
Dear grav,
We are getting confused about the sign of t.
exp(Ht) means a rescaling which increases with time, with t measured in the usual way, i.e. increasing into the future.
The confusion probably arises due to R(t) being quoted usually as proportional to 1/(1+z), so a higher z means a smaller scale factor for the universe. Although in the rescaling theory R(t) is proportional to 1/sqrt(1+z).
All the best,
John Hunter.
It is confusing, but has nothing to do with the sign of t. z+1=e^{Ht} is the formula for a constant rate of expansion and a constant speed of light during that expansion. Now, if we compare this to a constant redshift to distance ratio, then it will appear to decelerate. The reason is that during the time of travel of the light, the wavelengths expand exponentially. So for a very large distance, meaning a greater time has passed since emission, much more expansion has occured than for smaller distances, implying that expansion was greater in the past. And only zero expansion, where galaxies are only moving away from each other inertially in static space, would produce a direct redshift to distance ratio.
If, however, we are comparing to the z+1=e^{Ht} formula to begin with, which is obviously not a direct redshift to distance ratio, then anything "smaller" than this would be seen as an accelerated expansion. That includes your formula. Tired light theory would exactly match it, and would therefore not appear to accelerate or decelerate in comparison.
Okay, so in comparison to e^{Ht}, your formula would indeed be seen as an acceleration, since it is expanding faster for lesser times than the original model would be, and tired light would only match it. But a speed of light that increases with expansion is the only one that allows for a Big Bang singularity and eliminates internal inconsistencies for expansion, as far as I can tell. I have a couple of questions for you, though. First, I thought your model was based on e^{2Ht} a while back, where H=H_{0}/2, the same as for regular expansion. Has it progressed since then?
Second, your link shows the current model as being d=(1+z)*In(1+z)*c/H. Is this correct? By recalculating, where 1+z is the ratio of the expanded wavelength to the emitted one, equal to the expanded distance between the source and observer, de (distance when emitted) and dr (distance when received), and He (value of H when emitted) and Hr (value of H when received), we get
dr=(dr/de)*In(dr/de)*c/Hr
de*Hr/c=In(dr/de)
e^{de*Hr/c}=dr/de=z+1
This could be e^{dr*Hr/c}=z+1 or e^{de*He/c}=z+1, since dr*Hr=de*He, but should not be e^{de*Hr/c}=z+1. Where did you get that?
As I said, I have found that only a speed of light that varies directly with expansion holds up to scrutiny so far. For a constant rate of expansion, this becomes z+1=1/(1-Hd/c). I would like to tear into your formula, d=2(1+z)[sqrt(1+z)-1]*c/H, if I may, and see how it holds up, as well as what kind of expansion it would imply. Is H in this formula equal to H_{0} or H_{0}/2? Also, would you happen to to know what formula Permutter is comparing to when he says that expansion appears to be accelerating? Could you post some of the numerical values of the distances and redshifts of the galaxies you are using in your graphs?
Dear grav,
In the formula d = 2(1+z)[sqrt(1+z) - 1]*c/H The H is the normal Hubble constant, (this is why the 2 appears, because the rescaling constant would be H/2).
When cosmologists tested Big Bang theory - and came up with the need for dark energy, I believe they used d = (1+z)ln(1+z)*c/H, which comes from formulae on a reference at the bottom of http://www.rescalingsymmetry.com/evi...supernova.html
or search with google for "what do we really know about cosmic"...., for Michael Turners paper.
As can be seen from the scatter diagrams, the new formula is a much better match, and dosn't require dark energy.
-----------------
There are two ways to look at a rescaling universe
i) It is static, in which case d and c can be regarded as constant. (this is often a very useful shortcut, static in scale size, although the Big Bang still occured)
ii) Every physcal quantity and constant is rescaling, which should work too, but gets complicated, in the formula you gave, c would need to rescale too, as you mentioned.
All the best,
John Hunter.
The calculations I made in my last post were according to the current distances between galaxies, not the observed distance, which is what we would measure, and what I'm sure you would be using. So I will have to find the formulas for those values and recalculate accordingly. So far, though, I have found that the only cosmology model that allows for a Big Bang singularity at T=0 with a constant rate of expansion is one where the local measure for the speed of light scales directly with the expansion, so that it increases in proportion to distance scales, precisely in line with your ideas. Also, the time of transmission of the light is t=de/ce=dr/cr, so that space would appear static when measured by the speed of light and distance at emission or reception, also in line with what you are presenting. However, for a speed of light that scales in direct proportion to the universe, the field of view scales directly as well, so the distance we observe is exactly the same as the distance when the light was emitted, not the current distance. So it would not appear static when comparing the observed distance for when the light was emitted to the current speed of light as measured locally. This might throw things off a bit when measuring for H as well.
For an accelerating universe, we would have 2a*(dr-de)=vr^2-ve^2 for constant acceleration. At T=0, de=0 and we will consider ve to be zero as well, so that we now have 2a*dr=vr^2. The acceleration and velocity are both vector quantities, so must be in direct proportion to distance, where a/dr and vr/dr are constant at any particular point in time. Dividing both sides by (2*dr)^2, we now get a/2dr=(vr/2dr)^2. If we make J=a/d, then Jr/2=(Hr/2)^2, which kinda sorta' seems to fall in line with what you are using for H/2 (although I have to say it is confusing when you express it that way), and would now include J/2 for acceleration as well.
Ignoring the values as they are observed for now, until I can work them out further, your formula becomes
d=2(1+z)[sqrt(1+z)-1]*(c/H)
(H/c)*d=2(d/d)[sqrt(1+z)-1]
H*d/2c+1=sqrt(1+z)
(H*d/2c+1)^2=1+z
(H*d/2c)^2+H*d/c+1=1+z
(H*d/2c)^2+H*d/c+1=dr/de
and since J=H^2/2 and if d/c=t, then
(H^2/2)*t^2/2+H*t+1=dr/de
J*t^2/2+H*t+1=dr/de
de*J*t^2/2+de*H*t+de=dr
a*t^2/2+v*t+de=dr
So according to this quick (non-exact) run-through, your formula doesn't just look like an acceleration, it is an acceleration, since the formula for acceleration is dr=a*t^2/2+ve*t+de, precisely what yours looks to be. I will have to run this through some more to be sure, though.
Well, I've just rerun my calculations according to what we actually observe, which is de for the distance to a galaxy, He for a measure of the Hubble constant through a comparison of the redshift as observed from that distance, and cr for the locally measured current value for the speed of light, and for a constant rate of acceleration for expansion and a speed of light that scales directly in proportion, I come up with
dr=a*t^2/2+ve*t+de
dr/de=Je*t^2/2+He*t+1
z+1=Je*(de/ce)^2/2+He*(de/ce)+1
z+1=(He^2/2)*(de/ce)^2/2+He*de/ce+1
z+1=(He*de/2ce)^2+He*de/ce+1
z+1=[(He*de/2ce)+1]^2
sqrt(z+1)=He*de/2ce+1
He*de/2ce+1=sqrt(z+1)
He*de/2ce=sqrt(z+1)-1
de=2*[sqrt(z+1)-1]*ce/He
where ce=cr*(de/dr), so
de=2*[sqrt(z+1)-1]*(cr*de/dr)/He
de=2*[sqrt(z+1)-1]*(cr/He)/(z+1)
and since d=de, H=He, and c=cr for the observed values, then
d=2*[sqrt(z+1)-1]*(c/H)/(z+1)
which is almost exactly the same as what you have, but not quite. It appears that last value for z+1 should be divided into it, not multiplied. I was also going to mention that before with what I was getting for the formula for the current model using the observed values, but wanted to make sure first. It's still a very good match, though. But try running a comparison using this formula and see if it doesn't make a better match.
Dear grav,
the formula which gives the best match to the scatter diagrams is
d= 2(1+z)sqrt[(1+z) - 1]. This formula was also derived from rescaling theory, by replacing (1+z) with sqrt(1+z), z with sqrt(1+z) - 1, and H with H/2.
It's all in http://www.rescalingsymmetry.com/evi...supernova.html
All the best,
John Hunter.
Well, the formulas are almost an exact match for a constant acceleration, so I'm sure there's something to it, but maybe I'm overlooking something in the calculations. For instance, at the end of that last post, I used d=de and H=He for the observed values, but those are the observed values for a constant rate of expansion, not for acceleration, so I probably shouldn't have done that. Maybe for a constant acceleration, they would each vary by a factor of z+1 instead. Then the two formulas would be a perfect match. I'll now need to find the field of view for a constant acceleration to find out. I'll keep working on it.
In the meantime, I can't tell exactly what formula you are replacing or how you are replacing it with the information you give. Could you post a step by step analysis of how you go about doing this, like the ones I did? Also, with each step, could you tell why each value is replaced by the ones you are using?
Last edited by grav; 2006-Dec-19 at 01:18 PM.
Dear grav,
In Professor Turners paper "what do we really know about cosmic acceleration"
http://arxiv.org/abs/astro-ph/0512586
For rescaling:
For formula (5), If we keep this the same, we can change d to incorporate all changes. Luminosity has m^2 for its length dimension, so should rescale as exp(2Ht). http://www.rescalingsymmetry.com/evi...supernova.html
exp(2Ht) is 1/(1+z) as a high value of z looks back into the past, when the scale factor was smaller. To include this in d means multiplying d by a factor sqrt(1+z).
Then in equation (6) for d, (1+z) replaced by sqrt(1+z) [these first two changes end up with (1+z) in (6) unchanged]
The limits on the integral 0 and z, changed to 0 and sqrt(1+z) - 1,
H(u) in (6) is derived from (4), for rescaling H is constant but of value H/2 (q(z) = -1 (constant) in (4))
according to the authors dln(1+u) in (4) is the same as 1/(1+u)du.
So for rescaling all this combined into d gives d = 2(1+z)[sqrt(1+z) - 1]*c/H
-----------------------------------
For Big Bang with no dark energy put q(z) = 0 in (4), which gives (4) as H(0)(1+z) and
then from (6) d = (1+z)ln(1+z)*c/H
(H is always Hubbles constant, in the above, not shown in their formulae, also misprint on (8) should end in dln(1+v))
It's not so complicated as it looks really, when q(z) is inserted as a simple number. Deriving q(z) from the supernova data, assuming Big Bang, is complicated, and has got modern cosmologists into a mess, ending up with the need for dark energy, which varies with time.
The new formula gives an excellent match , with rescaling as the only assumption, no dark energy required.
John Hunter.
P.S to use the formula to get mu, d converted to megaparsecs (to include c/H and convert to megaparsecs multiply by 4203 for an H of 71.4km/s/Mpc)
Last edited by john hunter; 2006-Dec-19 at 05:38 PM.
I had forgotten. t=de/ce=dr/cr for an accelerating universe as well as one with constant expansion when c increases in direct proportion, so the field of view is the same either way. So as far as I can tell, this is still d=de and H=He for observed values. The formula I came up with, then, should be correct, at least for a constant acceleration, which appears to come very close to matching yours, so I would guess that's what it should be, but I'll keep going over it to be sure.
<<<<<>>>>>
Let me see if I can guess what you did. You took the straight-up distance to redshift formula, Hd/c=z, which becomes d=z*c/H, and figured the distance rescales with (z+1), for d=(z+1)*z*c/H, similar to equation 6 in the Shapiro and Turner paper. Then for the value of z, you tried sqrt(z+1) instead of (z+1), which becomes sqrt(z+1)-1 instead of (z+1)-1=z, so the formula becomes d=(z+1)*[sqrt(z+1)-1]*c/H. But for small z, sqrt(z+1)-1 approximates z/2, so H must changed to be H/2 to maintain the z/H ratio for small distances, and the formula is now d=2*(z+1)*[sqrt(z+1)-1]*c/H. Is that about right?
Dear grav,
yes, its the kind of thing, but...
i) There are reasons for the changes, from rescaling theory the scale factor R(t) = exp(Ht), but 1+z = exp(2Ht), (t = time since transmission of photon).
This is why R(t) depends on sqrt(1+z) , and H is replaced with H/2.
It's described in www.rescalingsymmetry.com/redshift_of_light.html
ii) Also the (1+z) at the beginning of the formula seems unchanged due to two changes.
(1+z) should be changed to sqrt(1+z), and luminosity also depends on 1/(1+z) in rescaling theory, for flux = L/4*pi*d^2, then another sqrt(1+z) included in d.
This results in sqrt(1+z)*sqrt(1+z) = (1+z) at the beginning of the formula.
All the best,
John Hunter.
The formula d=2*[sqrt(z+1)-1]*(c/H)/(z+1) is for the observed value of H=He, but I'm thinking that we don't use the observed value for H any more than we do for c, so H would be the locally measured value in the same way c is. I found this for the constant rate of expansion, but failed to carry it over for an accelerating one for some reason. For constant expansion, H=Hr, but this is because Hr would be the same for all galaxies at any distance for some particular point in time, but might be different for an accelerating universe. This wouldn't bring things any closer to what you have, though. Actually, it takes it further away, dividing the formula by another factor or two of z+1 (I'll have to run some more calculations to find out precisely). In any case, the formula you have does seem to fit the curve best. By the looks of it, for the galaxies you have graphed, it looks like it could stand to go a little to the left, even, but dividing by factors of z+1 would only put it further to the right, or down some for smaller observed distances. I'm also wondering now how we measure these distances according to luminosity to begin with and if the redshift and any relativistic effects have already been accounted for with this.
Dear grav,
Just a quick reply for now,
Ned Wright (UCLA) has done some analysis on this, and is discussing the new formula.
All the best,
John Hunter.
I was just looking through this site and noticed that the angular distance as described there is 1/(1+z)^2 smaller than the luminosity distance. If that is the case, then as my distance is defined geometrically, and so would be the angular distance, and yours are found through a luminosity-distance relationship, I'm sure, which is (1+z)^2 greater, then the two equations we've come up with are now exactly the same. Mine is still found through real acceleration, though. You seem to be saying that the acceleration isn't real with yours, but that the velocity-redshift relationship should really be redefined by z'=2*[sqrt(1+z)-1], where z here is what we might normally expect for a direct redshift to distance formula (constant H), so z' remains the same locally, but becomes smaller with greater distance, so that it only appears to be acceleration, but is not. Since the equations for each come out to be exactly the same, that could indeed be the case. I'm not sure how to determine that for sure, though. Any ideas?
By the way, if this is to be an apparent acceleration, and not a real one, then that would also mean that the velocity to redshift relationship would be redefined in accordance to the formula for z' as well, and have some effects on the workings of relativity, since the redshift can be thought of in terms of time dilation. I guess I'll work on this next. What do you think about it?
Dear grav,
that's interesting.
I've recently done a chi-squared test for the formula d=2(1+z)[sqrt(1+z) - 1]*c/H and got a value of:
183.82 for 182 degrees of freedom.
It shows that the formula can't be ruled out, it corresponds to a z of only 0.12 on a normal distribution, which could easily happen by chance.
Good luck with your research, please let me know what you come up with.
John Hunter.