"George Dishman" wrote in message
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[Reposting as this seems to have got lost]

"greywolf42" wrote in message
. ..
"George Dishman" wrote in message
...

"greywolf42" wrote in message
. ..
"George Dishman" wrote in message
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"greywolf42" wrote in message
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{snip higher levels}

I'm going to do some major snipping and some rearranging
too, these posts are becoming entirely swamped by side
issues. I guess you may feel I have altered the context
but it's difficult to avoid if this is to be a readable
reply.

Fair enough. I've had to do this from time to time with other posts, and
posters.

First let's get the topic clear. You said:

We aren't discussing the "Hubble Law". Of course the Hubble Law is
linear with distance at a given time! That's because it assumes
linearity!

and you give this reference:

Here is the initial exchange, from

http://www.google.com/groups?selm=pD...ewsgroup s.co
m

Phillip Helbig:
"Hubble's Law says that recession velocity is proportional to distance."

greywolf42:
"The 'Hubble's law' to which you are referring is a theoretical
construct. Hubble's data connects distance with redshift -- not with
recession velocity."

However, that is only part of the exchange: Here is the whole
quote from Lars and Phillip:

"Phillip Helbig---remove CLOTHES to reply"

wrote in message ...
In article , "Lars Wahlin"
writes:

A few years ago data from the Ia Supernova Cosmology Project found
that Hubble's law is not linear but changes in a nonlinear fashion
at
large distances, i.e. The universe is accelerating.

This is just plain wrong. Hubble's Law says that recession velocity
is proportional to distance.

To me it is clear that Lars was referring to the
relationship between observed redshift and distance,
which is non-linear, while Phillip is clearly referring
to the relationship between recession speed and distance
at a particluar epoch which is linear as discussed below.

So when you say "We aren't discussing the 'Hubble Law'.",
I have to disagree,

That's one heck of a roundabout and turbid way of "clarifying" the topic!

What exactly are you disagreeing with? Do you understand the difference
between Hubble's data and the "Hubble law?"

and when you say

But the use of terminology is an
irrelevant issue. Let's get back to physics.

I also think what started this is that Lars and Phillip
were talking about different relationships, though each
might consider it to be "The Hubble Law".

That was part of my point, thanks.

The Hubble Law is explicitly theoretical, not observational. I was
attempting to clarify.

Now you also said above "Of course the Hubble Law is linear
with distance at a given time! That's because it assumes
linearity!" and you seem to confirm that opinion he

The Hubble distance is a theoretical derivative number. One starts with
the
observed local value of the redshift-distance relation. Then one assumes
that the r-d relation is explicitly linear -- this is called the "Hubble
Law." ...

Again you seem to be implying linearity is purely an
assumption.

Actually, I've stated so explicitly, several times. I'm not simply implying
it.

They specifically address whether the universe is
homogenous and isotropic which leads to linearity
as a function of distance at a common time.

That is an incorrect conclusion. As noted before, a steady-state
universe could be both homegenous and isotropic, and STILL
not have a linear function of redshift versus distance
(at common time). The linear assumption is a completely
separate assumption, limited to the Big Bang theory.

You were correct when you said "One starts with the
observed local value of the redshift-distance relation."
but the assumption is that this is due to expansion over
local scales.

It doesn't matter what ad hoc explanation you make to back up the linear
assumption. The assumption of a linear relationship is still an assumption.

If the universe is homogenous then you can
imagine a slice through the universe at a given epoch to
be tiled with regions all similar to the local area we
can observe and linearity of velocity with distance then
follows if the universe is homogeneous and isotropic but
ONLY at a given epoch, i.e. over a surface of uniform
cosmic age. I'm sure you follow, the logic is trivial.

The assumption *is* trivial.

Linearity itself is therefore not an assumption but a
consequence of the cosmological principle plus the
observed linearity at small scales.

Uh, no. The assumption came first. Then the "cosmological principle" was
built upon the edifice of the linear assumption. You can see the linear
assumption explicitly in Hubble's original graph. Velocity versus distance.
When Hubble's data was redshift vs. distance.

Incidentally, in a homegenous and isotropic steady-state
universe, the relationship between speed and distance is
still linear but with a constant of proportionality with
the value zero.

Only if you assume the Big-bang relationship, that redshift is ever and
always only due to doppler shift or expansion.

Let's try this with math, instead of words. As we seem to be talking
past one another. For the moment, let us ignore possible changes
with time.

I agree, that's a sensible approach.

The standard Hubble Law is of the form:
V = H D
Where D is the distance in Mpc, V is the recessional velocity in kps,
and the Hubble constant is given in units of kps/Mpc. This equation
is explicitly linear. "H" is assumed to be constant throughout the
universe.

You cited this page

http://www.astro.ufl.edu/~guzman/ast...project01.html

but ignored this fundamental definition:

"The Hubble constant H_0 is the constant of proportionality
between recession speed v and distance d in the expanding
Universe;

v = H_0 d

The subscripted "0" refers to the present epoch because in
general H changes with time."

I did not ignore it.

Since you obviously read the page and quoted parts, I again
get the impression you deliberately ignored this definition
since it clearly repeats what I have been pointing out to
you all along.

It was irrelevant to the issue at hand. The addition of this wrinkle
affects the mathematical issue not at all.

Now let us convert this back to approximate redshift units
(approximations are fine, because the value of H is not claimed to
better precision than about +- 20%) -- since the data is
all in redshift ... not velocity:
delta lambda / lambda = H' D.

Since delta lambda over lambda is dimensionless, the units for H' would
be Mpc^-1. Where H' = H / c. [The conversion (at least at resolvable
Cepheid distance) is straightforward doppler effect:
delta lambda / lamda = v / c.]

Both equations are the same observable effect. Both are explicitly
linear, as written. Now, let us examine a simple exponential version:

delta lambda / lambda = 1 - exp(-mu D)
delta lambda / lambda = mu D + (mu D)^2 / 2 - ......

At near distances (like those of resolvable Cepheid stars), there is no
way to distinguish the linear from the exponential change. At
substantial distances (like those of the newer supernovae data),
however, the higher order terms in the approximation are no
longer negligible.

The variation of H(t) with t is also no longer negligible.

Only if you assume that H is always linear. Sure, you can make this
assumption. But it's not the only one available.

So, I could as easily use:
delta lambda / lambda = 1 - exp(-H' D)

No, instead of the constant value H', you need to use
H(t) and integrate the effect over the lookback time.

Only if the value changes with time. Which isn't the only option.

The converse (finding the time from z) is mentioned
in equation (29) of:
http://www.astro.ufl.edu/~guzman/ast...project01.html

Yes, I know. But this is irrelevant to the point under discussion.

In the case of the linear assumptions, the supernovae data must be
addressed through an additional, ad hoc, cosmological term.
In the case of the exponential fit, no additional cosmological term is
needed.

That is not true,

On the contrary, it is explicitly true. This is called "dark energy" or the
"cosmological constant."

you are oversimplifying by ignoring
the variation of H(t) at high redshift.

I did not "ignore" your assumption of time-dependence. Because it is used
solely to get around the linear distance dependence that I am discussing.

This produces
non-linearity even when there is a linear relationship
with distance at any given epoch.

Yes. But the fact that you can arbitrarily add an ad hoc time-dependence to
a linear term; does not mean that a non-linear term is just as valid.

Why do you avoid acknowledging that a nonlinear term is even conceivable?

--
greywolf42
ubi dubium ibi libertas
{remove planet for e-mail}