Help! How to convert "acceleration of expansion" to SI units?

Does anybody know how to convert the value of
"acceleration of expansion" given in units that are
possibly used by astronomers and specified as
[h^3/Mpc^3/yr] into SI units (in this case 1/s/s)?

It seems that dH/dt, where H is "Hubble's constant"
[1/s] and "t" is "cosmic time" [s] should be related
somehow to what astronomers understand as
"acceleration of expansion", just what is this
relation?

I suspect that "h" in those astronomical units isn't
neither "hour" nor "Planck's constant" but I can't
be sure. I suspect also that there is speed of light
implied somewhere, and possibly some other
constant, so if anyone knows please explain
how "acceleration of expansion" is understood
in astronomy. It might benefit a lot of physicists.

I asked already one astronomer from the team
that made the measurements of the "acceleration
of expansion" but he wasn't sure about the value
in s^(-2), nor had time to find out, so he advised
me to search the internet, which I already did
without any luck. I found hundreds of articles
about "accelerating expansion" and how
important it is but none providing any definition
of "accelerating expansion" in astronomy. I have
the number for it, which is (1.4 +/-0.5)E-4, and
only finding what it means is tough.

-- Jim

Jim Jastrzebski wrote:
Does anybody know how to convert the value of
"acceleration of expansion" given in units that are
possibly used by astronomers and specified as
[h^3/Mpc^3/yr] into SI units (in this case 1/s/s)?

It seems that dH/dt, where H is "Hubble's constant"
[1/s] and "t" is "cosmic time" [s] should be related
somehow to what astronomers understand as
"acceleration of expansion", just what is this
relation?

I suspect that "h" in those astronomical units isn't
neither "hour" nor "Planck's constant" but I can't
be sure. [[...]]

Astronomers conventionally use "h" to mean the Hubble constant in
units of 100 km/sec/Mpc, i.e. H / (100 km/sec/Mpc). I believe
recent experimental values for the Hubble constant are between
70 and 75 km/sec/Mpc, i.e. h is between 0.70 and 0.75, but I have
not closely followed progress in this area.

To learn how many meters there are in a megaparsec (Mpc), there are
two basic methods:
* The quick-n-dirty way is to look it up in a book, eg
Allen "Astrophysical Quantities" is the standard reference
book astronomers use for this sort of thing. Many good undergraduate
astronomy textbooks also have tables of astronomical units.
* Work it out: 1 parsec is defined the distance at which 1 AU
subtends an angular size of 1 arc second, and 1 AU is defined as
the semimajor axis of the Earth's orbit, about 149e6 kilometers.
So 1 parsec
= 149e6 kilometers * (number of arc seconds in a radian)
There are 3600 arc seconds in a degree, and pi radians = 180 degrees,
so you should be able to work out the rest from there.

ciao,

--
-- "Jonathan Thornburg (remove -animal to reply)"
Max-Planck-Institut fuer Gravitationsphysik (Albert-Einstein-Institut),
Golm, Germany, "Old Europe" http://www.aei.mpg.de/~jthorn/home.html
"Washing one's hands of the conflict between the powerful and the
powerless means to side with the powerful, not to be neutral."
-- quote by Freire / poster by Oxfam

In article ,
Jim Jastrzebski wrote:
Does anybody know how to convert the value of
"acceleration of expansion" given in units that are
possibly used by astronomers and specified as
[h^3/Mpc^3/yr] into SI units (in this case 1/s/s)?

The main thing you have to know is that h here stands for the dimensionless
quantity

h = H / (100 km/(s Mpc)).

This is used because lots of things scale with the value of the Hubble
parameter, which isn't as well-known as we'd like. Measurements of H
nowadays give numbers around 70 km/(s Mpc), so h is about 0.7.

The expression h^3/Mpc^3/yr looks to me like it means (h^3/Mpc^3)/yr,
or h^3/(Mpc^3 yr). That's not dimensionally equivalent to 1/time^2
On the other hand, if it's supposed to be h^3/(Mpc^3/yr), or
h^3 yr/Mpc^3, and if we're allowed to toss around factors
of c at will, then they are equivalent. According to the miraculous
Unix units tool,

: % units
: 1948 units, 71 prefixes, 28 functions
:
: You have: Mpc^(-3) yr c^3
: You want: s^(-2)
: * 2.8940442e-35
: / 3.4553722e+34
: You have:
: %

I didn't include the h's there, so toss in an extra factor of 0.7^3.

I suspect that "h" in those astronomical units isn't
neither "hour" nor "Planck's constant" but I can't
be sure.

This particular notation for the Hubble parameter is quite common,
but still most articles that use it state its meaning explicitly.
If you're reading an article that used it without explaining it,
I'd say the author was careless. (On the other hand, if you're
looking at, say, slides from a talk, it wouldn't be at all surprising
for the symbol h to be used in this way without explanation.)

I suspect also that there is speed of light
implied somewhere, and possibly some other
constant, so if anyone knows please explain
how "acceleration of expansion" is understood
in astronomy. It might benefit a lot of physicists.

There's not a standard definition for a number called "the acceleration
of the Universe." If there were, I'd guess that the most likely choice
would be a'' / a, where a(t) is the scale factor and primes denote
time derivatives. This has units of time^(-2) as you suspected.

Historically, the number cosmologists have most often talked about
for characterizing the acceleration was the "deceleration parameter,"
denoted q. From memory (I'm not near my textbooks at the moment),
this is defined to be

q = - (a a'') / (a'^2)

The name "deceleration parameter" and the minus sign in the definition
are because everyone expected the Universe to be decelerating. The
deceleration parameter is a dimensionless quantity.

By the way, it's tempting to use dH/dt as a measure of acceleration,
but you have to resist the temptation! The reason is that H
varies with time, even if there is no acceleration. In a Universe
that expands at a steady rate (any given galaxy's speed with respect
to any other galaxy remaining constant in time), H is inversely
proportional to t (because H is speed over distance, and the
distances are increasing with time). In such a Universe, dH/dt
is negative, even though there is no acceleration.



I asked already one astronomer from the team
that made the measurements of the "acceleration
of expansion" but he wasn't sure about the value
in s^(-2), nor had time to find out, so he advised
me to search the internet, which I already did
without any luck. I found hundreds of articles
about "accelerating expansion" and how
important it is but none providing any definition
of "accelerating expansion" in astronomy. I have
the number for it, which is (1.4 +/-0.5)E-4, and
only finding what it means is tough.

Where did you get this number? I'm surprised that anyone
in the field would quote such a number without explaining
precisely what was meant by it.

-Ted

--
[E-mail me at , as opposed to .]

"Jonathan Thornburg" wrote in message
...
Jim Jastrzebski wrote:
Does anybody know how to convert the value of
"acceleration of expansion" given in units that are
possibly used by astronomers and specified as
[h^3/Mpc^3/yr] into SI units (in this case 1/s/s)?
the rest snipped for brevty

Astronomers conventionally use "h" to mean the Hubble constant in
units of 100 km/sec/Mpc, i.e. H / (100 km/sec/Mpc). I believe
recent experimental values for the Hubble constant are between
70 and 75 km/sec/Mpc, i.e. h is between 0.70 and 0.75, but I have
not closely followed progress in this area.

Since H = 72+/-8 km/s/Mpc (Freedman at al. 2001), it sets
h = 0.72 and leaves only the rest of the units to interpret.

To learn how many meters there are in a megaparsec (Mpc),
there are two basic methods:

snip for brevty

I know of course how to convert Mpc to meters and year
to seconds (approximately at least, which year?) but I don't
understand how those units express "acceleration of expansion"
to convert them to s^(-2) since for the time being the result
comes out in m^(-3)s^(-1). Am I to multiply the result by
c^3 and then take square root of the result to get s^(-2)?
Why? What's the theory behind it?

-- Jim

wrote in message
...
In article ,
Jim Jastrzebski wrote:
Does anybody know how to convert the value of
"acceleration of expansion" given in units that are
possibly used by astronomers and specified as
[h^3/Mpc^3/yr] into SI units (in this case 1/s/s)?

The main thing you have to know is that h here stands
for the dimensionless quantity

h = H / (100 km/(s Mpc)).

OK.

snip for brevity

I suspect also that there is speed of light
implied somewhere, and possibly some other
constant, so if anyone knows please explain
how "acceleration of expansion" is understood
in astronomy.

There's not a standard definition for a number called "the acceleration
of the Universe." If there were, I'd guess that the most likely choice
would be a'' / a, where a(t) is the scale factor and primes denote
time derivatives. This has units of time^(-2) as you suspected.

Historically, the number cosmologists have most often talked about
for characterizing the acceleration was the "deceleration parameter,"
denoted q. From memory (I'm not near my textbooks at the moment),
this is defined to be

q = - (a a'') / (a'^2)

The name "deceleration parameter" and the minus sign in the definition
are because everyone expected the Universe to be decelerating. The
deceleration parameter is a dimensionless quantity.

By the way, it's tempting to use dH/dt as a measure of acceleration,
but you have to resist the temptation! The reason is that H
varies with time, even if there is no acceleration. In a Universe
that expands at a steady rate (any given galaxy's speed with respect
to any other galaxy remaining constant in time), H is inversely
proportional to t (because H is speed over distance, and the
distances are increasing with time). In such a Universe, dH/dt
is negative, even though there is no acceleration.

I meant dH/dt as a "real rate of expansion", already adjusted
for the nonlinearity of the *observed* Hubble's redshift. I.e.
the difference between actually observed H(t) and H(t)
that would be observed for uniformly expanding universe.

I asked already one astronomer from the team
that made the measurements of the "acceleration
of expansion" but he wasn't sure about the value
in s^(-2), nor had time to find out, so he advised
me to search the internet, which I already did
without any luck. I found hundreds of articles
about "accelerating expansion" and how
important it is but none providing any definition
of "accelerating expansion" in astronomy. I have
the number for it, which is (1.4 +/-0.5)E-4, and
only finding what it means is tough.

Where did you get this number? I'm surprised that anyone
in the field would quote such a number without explaining
precisely what was meant by it.

I got the number from http://arxiv.org/abs/astro-ph/0305008

They say: "Consequently, these measurements not only provide
another quantitative confirmation of the importance of dark
energy, but also constitute a powerful qualitative test for the
cosmological origin of cosmic acceleration. We find a rate for
SN Ia of 1.4+/-0.5E-04 h^3/Mpc^3/yr at a mean redshift of 0.5."

Actually, they don't say that the rate that they find is the rate
of "cosmic acceleration". It might be some other "rate" obvious
to astronomers because of the units, and I just got mislead by
the previous sentence that ends in "cosmic acceleration".
Would it be some "rate of 'dark energy'". Does it makes
sense to astronmers?

My problem is that I think that I showed that physics of
Einsteinian gravity predicts the relation between Hubble
constant and the "cosmic acceleration" dH/dt (the "real"
observed acceleration, after the mentioned above
adjustment for the visual effects). The relation is to be
dH/dt = (H^2)/2. So the value H = 70 km/s/Mpc, it
implies dH/dt = 2.5E-36 /s/s.

Obviously I'd like to know how close to the actual
observations this prediction is. I was told that dH/dt
can't be predicted theoretically while I maintain that
it is a necessary conclusion of Einstein's theory and
orthogonal to the existence of any "dark energy".

-- Jim

-Ted

Jim Jastrzebski wrote:
http://arxiv.org/abs/astro-ph/0305008

They say: "Consequently, these measurements not only provide
another quantitative confirmation of the importance of dark
energy, but also constitute a powerful qualitative test for the
cosmological origin of cosmic acceleration. We find a rate for
SN Ia of 1.4+/-0.5E-04 h^3/Mpc^3/yr at a mean redshift of 0.5."

Actually, they don't say that the rate that they find is the rate
of "cosmic acceleration". It might be some other "rate" obvious
to astronomers because of the units, and I just got mislead by
the previous sentence that ends in "cosmic acceleration".

I'm afraid so.

Reading astro-ph/0305008's abstract, I'm fairly sure that what they
mean is that the rate at which SN Ia occurred at z=0.5
[i.e. at places/times in the history of the universe
such that the supernove's light now reaches us with
redshift z=0.5]
is (1.4 +/- 0.5) * 10^{-4} * h^3 supernova per cubic megaparsec per year.

Someone in this thread quoted h = 0.72 or so, which makes h^3 = 0.37,
so astro-ph/0305008 is saying that (again, at places/times from which
light now reaches us with redshift z=0.5) SN Ia occured at a rate of
(0.52 +/- 0.18) * 10^{-4} supernova per cubic megaparsec per year.

ciao,

--
-- "Jonathan Thornburg (remove -animal to reply)"
Max-Planck-Institut fuer Gravitationsphysik (Albert-Einstein-Institut),
Golm, Germany, "Old Europe" http://www.aei.mpg.de/~jthorn/home.html
"Washing one's hands of the conflict between the powerful and the
powerless means to side with the powerful, not to be neutral."
-- quote by Freire / poster by Oxfam

Jim Jastrzebski wrote:
cosmological origin of cosmic acceleration. We find a rate for
SN Ia of 1.4+/-0.5E-04 h^3/Mpc^3/yr at a mean redshift of 0.5."

This simply is the rate with which SN Ia occur in a given volume
at a mean redshift of 0.5. It has nothing to do with what you
call "rate of expansion". I'm not really sure what you mean by
this anyway.

-- Jo:rg

--
Fortune cookie of the day:
Machine-Independent, adj.:
Does not run on any existing machine.

"Jonathan Thornburg" wrote in message
...
Jim Jastrzebski wrote:
http://arxiv.org/abs/astro-ph/0305008

snipped for brevity

Actually, they don't say that the rate that they find is the rate
of "cosmic acceleration". It might be some other "rate" obvious
to astronomers because of the units, and I just got mislead by
the previous sentence that ends in "cosmic acceleration".

I'm afraid so.

Reading astro-ph/0305008's abstract, I'm fairly sure that what they
mean is that the rate at which SN Ia occurred at z=0.5
[i.e. at places/times in the history of the universe
such that the supernove's light now reaches us with
redshift z=0.5]
is (1.4 +/- 0.5) * 10^{-4} * h^3 supernova per cubic
megaparsec per year.

That's a real bummer :-( but if it is known that the expansion
of space is accelerating there should be some data that say
how much it is accelerating (otherwise how would we even
know that there is an acceleration?)

Does anyone here know a link to such data, or just the
number and its error bar, possibly in units of s^(-2)?

-- Jim