Cosmological Principle-Homogenous and Isotropic

[Savin Beniwal]

Hi all

I have questions regarding the Cosmological Principle that usually we
study universe is SPATIALLY homogeneous and isotropic(around every
point) at large scale (>150MPC). Here homogenous means--> No special
location and Isotropic means-->No special point. Also, this was
confirmed by Hubble in 1929 that if distances are expanding (or
contracting), the speed must be proportional to distance – Hubble’s Law
is inevitable.

But my questions is that if there were a proportionality relation
between velocity and square of distance rather than a linear relation
between r and v. Even then can we understand the homogenous and
isotropic concept from Hubble's law under this nonlinear relation?

Thank you for your reply and discussion.


With Regards!!!
----Savin(Darshan) Beniwal

[Phillip Helbig (undress to reply)]

Post by Savin Beniwal
I have questions regarding the Cosmological Principle that usually we
study universe is SPATIALLY homogeneous and isotropic(around every
point) at large scale (>150MPC). Here homogenous means--> No special
location and Isotropic means-->No special point. Also, this was
confirmed by Hubble in 1929 that if distances are expanding (or
contracting), the speed must be proportional to distance =E2=80=93 Hubble=

's Law

Post by Savin Beniwal
is inevitable.
But my questions is that if there were a proportionality relation
between velocity and square of distance rather than a linear relation
between r and v. Even then can we understand the homogenous and
isotropic concept from Hubble's law under this nonlinear relation?

No.

Say you are at the origin, at distance 1 velocity is 1, at distance 2
velocity is 4, at distance 3 velocity is 9, and so on. For an observer
at distance 1, your distance 2 is just 1 unit of distance away, but its
speed relative to the observer at 1 is 3 (4-1), whereas it should be 1
if the distance is 1.

In short, homogeneity and isotropy demand a linear velocity--distance
law, since otherwise homogeneity and isotropy couldn't persist. (Note
that this is purely kinematics, no dynamics, hence this does not depend
on general relativity in any way.)

[Lawrence Crowell]

Post by Savin Beniwal
Hi all
I have questions regarding the Cosmological Principle that usually we
study universe is SPATIALLY homogeneous and isotropic(around every
point) at large scale (>150MPC). Here homogenous means--> No special
location and Isotropic means-->No special point. Also, this was
confirmed by Hubble in 1929 that if distances are expanding (or
contracting), the speed must be proportional to distance =E2=80=93 Hubble's Law
is inevitable.
But my questions is that if there were a proportionality relation
between velocity and square of distance rather than a linear relation
between r and v. Even then can we understand the homogenous and
isotropic concept from Hubble's law under this nonlinear relation?
Thank you for your reply and discussion.
With Regards!!!
----Savin(Darshan) Beniwal

The scale factor in FLRW cosmology expands as a(t) ~ a_0 exp(tH)
where H is the Hubble factor. Now take the derivative of this to
get

da/dt = Ha.

The actual distance is the scale factor times the "ruler" with some
unit distance x so the distance d is d = xa and with v = x dx/dt
we have v = Hd. That is the standard Hubble rule. However, in this
case d is based on an expanding scale and this lacks linearity, so
for d_0 = xa_0 we have

v = Hd_0exp(tH).

The time t = d_0/c and now Taylor expand

v = Hd_0 + (Hd)^2/c + 1/2(Hd)^3/c^2 + ... .

The rule v = Hd_0 is the linear rule that Hubble found. This is how
the expansion for sufficiently large distances, usually with z > 1,
is nonlinear.

[Phillip Helbig (undress to reply)]

Post by Lawrence Crowell
The scale factor in FLRW cosmology expands as a(t) ~ a_0 exp(tH)
where H is the Hubble factor.

No, not in general. FLRW means Friedmann-Lemaitre-Robertson-Walker.
Actually, Robertson-Walker is enough to answer the question: by
definition these are homogeneous and isotropic models, for which the
only possbible velocity-distance relation is a linear one. The
Friedmann-Lemaitre models are based on general relativity and have
ordinary matter and the cosmological constant as components (whereby the
possibility that one or both of these is zero is also covered). The
expansion law depends on the amounts of these components; in general,
the relative amounts also change with time. Exponential expansion is
the case ONLY for no matter and a cosmological constant (and holds for
all times). It is true that this is APPROXIMATELY true in OUR universe
NOW (and in the future will become more and more true, since as the
matter thins out due to expansion and the cosmological constant is,
errm, constant, asymptotically our universe will approach the so-called
de Sitter model of exponential expansion with (just) a cosmological
constant).

Post by Lawrence Crowell
Now take the derivative of this to
get
da/dt = Ha.

Yes, but the velocity is ALWAYS EXACTLY proportional to the distance in
ANY FLRW model.

Post by Lawrence Crowell
The actual distance is the scale factor times the "ruler" with some
unit distance x so the distance d is d = xa and with v = x dx/dt
we have v = Hd. That is the standard Hubble rule. However, in this
case d is based on an expanding scale and this lacks linearity, so
for d_0 = xa_0 we have
v = Hd_0exp(tH).
The time t = d_0/c and now Taylor expand
v = Hd_0 + (Hd)^2/c + 1/2(Hd)^3/c^2 + ... .
The rule v = Hd_0 is the linear rule that Hubble found. This is how
the expansion for sufficiently large distances, usually with z > 1,
is nonlinear.

No. This is not even wrong. The velocity is always exactly
proportional to the distance, but this regards the proper distance and
its derivative. Edward Harrison devoted an entire chapter in his
excellent cosmology textbook to this:

@BOOK { EHarrison81a ,
AUTHOR = "Edward R. Harrison",
TITLE = "Cosmology, the Science of the Universe",
PUBLISHER = CUP,
YEAR = "1981",
ADDRESS = "Cambridge (UK)"
}

(Note that there is also a second edition, from 2000 I believe.) He
also wrote a paper detailing this:

@ARTICLE { EHarrison93a ,
AUTHOR = "Edward R. Harrison",
TITLE = "The Redshift-Distance and Velocity-Distance
Laws",
JOURNAL = APJ,
YEAR = "1993",
VOLUME = "403",
NUMBER = "1",
PAGES = "28",
MONTH = jan
}

Even professional astronomers get it wrong, as I pointed out here:

http://www.astro.multivax.de:8000/helbig/research/publications/info/a_formula_for_confusion.html

[Lawrence Crowell]

to reply) wrote:
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Xref: number.nntp.giganews.com sci.physics.research:95052

Technically you are right, and I was referring to de Sitter spacetime. I
was blurring the two sort of intentionally to show how Hubble law as a
linear law is "deformed" into a nonlinear one.

As for below, the nonlinear situation is what gives the departure from
13.8 billion years in time and 46 billion years in radius out to the CMB
with z = 1100.

Post by Phillip Helbig (undress to reply)

Post by Lawrence Crowell
Now take the derivative of this to
get
da/dt = Ha.

Yes, but the velocity is ALWAYS EXACTLY proportional to the distance in
ANY FLRW model.

Post by Lawrence Crowell
The actual distance is the scale factor times the "ruler" with some
unit distance x so the distance d is d = xa and with v = x dx/dt
we have v = Hd. That is the standard Hubble rule. However, in this
case d is based on an expanding scale and this lacks linearity, so
for d_0 = xa_0 we have
v = Hd_0exp(tH).
The time t = d_0/c and now Taylor expand
v = Hd_0 + (Hd)^2/c + 1/2(Hd)^3/c^2 + ... .
The rule v = Hd_0 is the linear rule that Hubble found. This is how
the expansion for sufficiently large distances, usually with z > 1,
is nonlinear.

No. This is not even wrong. The velocity is always exactly
proportional to the distance, but this regards the proper distance and
its derivative. Edward Harrison devoted an entire chapter in his
@BOOK { EHarrison81a ,
AUTHOR = "Edward R. Harrison",
TITLE = "Cosmology, the Science of the Universe",
PUBLISHER = CUP,
YEAR = "1981",
ADDRESS = "Cambridge (UK)"
}
(Note that there is also a second edition, from 2000 I believe.) He
@ARTICLE { EHarrison93a ,
AUTHOR = "Edward R. Harrison",
TITLE = "The Redshift-Distance and Velocity-Distance
Laws",
JOURNAL = APJ,
YEAR = "1993",
VOLUME = "403",
NUMBER = "1",
PAGES = "28",
MONTH = jan
}
http://www.astro.multivax.de:8000/helbig/research/publications/info/a_formula_for_confusion.html

[Savin Beniwal]

Post by Phillip Helbig (undress to reply)

Post by Savin Beniwal
I have questions regarding the Cosmological Principle that usually we
study universe is SPATIALLY homogeneous and isotropic(around every
point) at large scale (>150MPC). Here homogenous means--> No special
location and Isotropic means-->No special point. Also, this was
confirmed by Hubble in 1929 that if distances are expanding (or
contracting), the speed must be proportional to distance =E2=80=93 Hubble=

's Law

Post by Savin Beniwal
is inevitable.
But my questions is that if there were a proportionality relation
between velocity and square of distance rather than a linear relation
between r and v. Even then can we understand the homogenous and
isotropic concept from Hubble's law under this nonlinear relation?

No.
Say you are at the origin, at distance 1 velocity is 1, at distance 2
velocity is 4, at distance 3 velocity is 9, and so on. For an observer
at distance 1, your distance 2 is just 1 unit of distance away, but its
speed relative to the observer at 1 is 3 (4-1), whereas it should be 1
if the distance is 1.

Is this way correct to add/subtract velocities as velocities of
galaxy is about the speed of light? If we consider the velocity of
galaxy is equal or greater then speed of light (Not a surprise at
all), even then there will be a linear relation between velocity
and distance as Hubble stated?

[[Mod. note -- No. -- jt]]

Post by Phillip Helbig (undress to reply)
In short, homogeneity and isotropy demand a linear velocity--distance
law, since otherwise homogeneity and isotropy couldn't persist. (Note
that this is purely kinematics, no dynamics, hence this does not depend
on general relativity in any way.)