From Wikipedia, the
free encyclopedia (http://en.wikipedia.org/wiki/Hubble's_law)
Hubble's law describes the
observation in physical
cosmology
that the velocity at which various galaxies are receding from the Earth is proportional to their distance from us.^{[1]} The law was first formulated by Edwin Hubble in 1929^{[2]} after nearly a decade of observations. The recession velocity of the
objects was inferred from their redshifts, many measured much earlier by Vesto Slipher (1917) and related to velocity by
him.^{[3]} It is considered the first
observational basis for the expanding space paradigm and today serves as one of the pieces of evidence most often
cited in support of the Big Bang model.
The law is often expressed by the
equation V = H_{0}D, with H_{0} the constant of proportionality (the
Hubble constant) between the distance D to a galaxy and its velocity V. The SI unit of H_{0} is s^{-1} but it is most frequently
quoted in (km/s)/Mpc, thus giving the speed in km/s of a
galaxy one Megaparsec away. The reciprocal of H_{0} is the Hubble
time.
The Particle Data Group documents quote a "best modern
value" of the Hubble constant as H_{0} = 72 (km/s)/Mpc (± 10%). This
value comes from the use of type Ia supernovae (which give relative distances to
about 5%) along with data from Cepheid variables gathered by the Hubble Space Telescope. The value from the WMAP survey is
71 km/s per Megaparsec.
The Hubble parameter has the dimensions of inverse time, so a
Hubble time t_{H} may be obtained by inverting the present value of the
Hubble parameter.
وهذه الأرقام لقيمة ثابت هابل قريبة جدا من رقم آية القسم بمواقع النجوم: ]فَلا أُقْسِمُ
بِمَوَاقِعِ النُّجُومِ * وَإِنَّهُ لَقَسَمٌ لَوْ تَعْلَمُونَ عَظِيمٌ * إِنَّهُ
لَقُرْآنٌ كَرِيمٌ [ (الواقعة الآيات 75-77).
Locations
and settings of the stars: God says: [Furthermore I
swear by The locations of the Stars,- And that is indeed A mighty
adjuration If ye but knew,- That is indeed A Qur-an most honourable]
(Surah. 56 verses. 75-77).
I
swear by The locations of
the Stars: The word locations
(positions) of the Stars comes approximately in the middle of this verse;
whose number is 75. Thus, 74 + 0.5 = 74.5 is almost equal to the
most recent observational determination of H_{0} = 72 (km/s)/Mpc (± 10%).
This is one of the miraculous aspects of this Qur'anic verse.
This most recent accurate determination, H_{0} = 72 (km/s)/Mpc (± 10%), agree closely with an
earlier measurement of H_{0} = 72 ± 8 km/s/Mpc obtained in 2001 also by
the HST.^{[5]} In August
Discovery
A decade before Hubble made his
observations, a number of physicists and mathematicians had established a consistent theory
of the relationship between space and time by using Einstein's field equations of general
relativity.
Applying the most general
principles to
the nature of the universe yielded a dynamic solution that conflicted with the
then prevailing notion of a static universe.
) إني عند النبي صلى الله
عليه وسلم إذ جاءه قوم من بني تميم ، فقال: (اقبلوا البشرى يا بني تميم) . قالوا :
بشرتنا فأعطنا ، فدخل ناس من أهل اليمن ، فقال : ( اقبلوا البشرى يا أهل اليمن ،
إذ لم يقبلها بنو تميم ) . قالوا: قبلنا، جئناك لنتفقه في الدين، ولنسألك عن أول
هذا الأمر ما كان، قال: ( كان الله ولم يكن شيء قبله، وكان عرشه على الماء، ثم خلق السماوات والأرض،
وكتب في الذكر كل شيء ). ثم أتاني رجل فقال : يا عمران أدرك ناقتك فقد ذهبت ،
فانطلقت أطلبها ، فإذا السراب ينقطع دونها ، وايم الله لوددت أنها قد ذهبت ولم أقم
. ) (الراوي: عمران بن
حصين المحدث:
البخاري - المصدر:
صحيح البخاري - الصفحة أو الرقم: 7418، خلاصة حكم المحدث:
[صحيح]).
(أتيت رسول الله
صلى الله عليه وسلم فعقلت ناقتي بالباب ، فدخلت ، فأتاه نفر من أهل اليمن فقال :
اقبلوها يا أهل اليمن إذا لم يقبلها إخوانكم بني تميم ، فقالوا : قبلنا يا رسول
الله ، أتيناك لنتفقه في الدين ، ونسألك عن أول هذا الأمر كيف كان ؟ قال : كان
الله ولم يكن شيء غيره ، وكان عرشه على الماء ، ثم كتب جل
ثناؤه في الذكر كل شيء ، ثم خلق السموات والأرض ، ثم أتاني فقال : أدرك ناقتك فقد
ذهبت ، فخرجت فوجدتها ينقطع دونها السراب ، وايم الله لوددت أني تركتها ) (الراوي: عمران بن حصين المحدث: أبو نعيم - المصدر: حلية الأولياء - الصفحة أو الرقم: 8/285، خلاصة حكم المحدث: صحيح متفق عليه [أي:بين العلماء]
Cepheid variable
stars outside of the Milky Way
Edwin Hubble did most of his
professional astronomical observing work at Mount Wilson Observatory, the world's most powerful telescope at the time. His
observations of Cepheid variable stars in spiral nebulae enabled him to calculate the
distances to these objects. Surprisingly, these objects were discovered to be
at distances which placed them well outside the Milky Way. They continued to be called
"nebulae" and it was only gradually that the term
"galaxies" took over.
Combining redshifts
with distance measurements
Fit of redshift velocities to Hubble's law; patterned after
William C. Keel (2007). The Road to Galaxy Formation. Berlin: Springer published in
association with Praxis Pub., Chichester, UK. ISBN 3540725342. http://books.google.com/books?id=BUgJGypUYF0C&pg=PA8. Various estimates for the Hubble constant exist. The
HST Key H_{0} Group fitted type Ia supernovae for redshifts
between 0.01 and 0.1 to find that H_{0} = 71 ± 2 (statistical) ± 6
(systematic) km s^{−1}Mpc^{−1},^{[9]} while Sandage et al. find H_{0} = 62.3 ± 1.3 (statistical) ± 5 (systematic) km s^{−1}Mpc^{−1}.^{[10]}
In order to measure the
Hubble constant, all one needs a distance and a redshift to a galaxy that is
distant enough that its peculiar velocity does not matter. Measuring redshifts
for galaxies is easy, but measuring distances is hard.
Measuring distances to
galaxies and stars is hard. This is
clearly indicated by the same Qur'anic verse:
[Furthermore I swear by The locations of the
Stars,- And that is indeed A mighty adjuration If ye but knew,- That
is indeed A Qur-an most honourable] (Surah. 56 verses. 75-77).
The Hubble constant is
therefore not easy to measure, and it is not surprising that there is
controversy about its value. In fact, there are generally two schools of
thought: one group likes a Hubble constant around 55 (Read more: http://www.faqs.org/faqs/astronomy/faq/part8/section-12.html#ixzz0mV1AYbnT).
للمزيد أنظر : Hubbles constant .
God says: [Furthermore I swear by The locations of the Stars, -
And that is indeed A mighty adjuration If ye but knew, - That is indeed A
Qur-an most honourable] (Surah. 56 verses. 75-77).
Locations
(positions) of the Stars is mentioned by this Surah; whose number (56).
This surah has a total of 96 verses.
Thus, 55 + (75/96) = 55.78. This
numbers is extremely close to this alternative value of H_{0} = 55. This is also another miraculous aspect of
this Qur'anic verse.
The parameters that appear in
Hubble’s law: velocities and distances, are not directly measured. In reality
we determine, say, a supernova brightness, which provides information about its
distance, and the redshift z = ∆λ/λ of its spectrum of radiation. Hubble correlated
brightness and parameter z.
Combining his measurements of galaxy
distances with Vesto
Slipher's
measurements of the redshifts associated with the galaxies, Hubble
discovered a rough proportionality between redshift of an object and its
distance. Though there was considerable scatter (now known to be caused by peculiar velocities), Hubble was able to plot a trend
line from the 46 galaxies he studied and obtain a
value for the Hubble constant of 500 km/s/Mpc (much higher than the currently accepted value due to errors
in his distance calibrations). (See cosmic distance ladder for details.)
At the time of discovery and
development of Hubble’s law it was acceptable to explain redshift phenomenon as
a Doppler
shift in the
context of special relativity, and use the Doppler formula to associate
redshift z with velocity. Today the
velocity-distance relationship of Hubble's law is viewed as a theoretical
result with velocity to be connected with observed redshift not by the Doppler
effect, but by a cosmological model relating recessional velocity to the
expansion of the universe. Even for small z the velocity entering the Hubble law is no longer
interpreted as a Doppler effect, although at small z the
velocity-redshift relation for both interpretations is the same.
In 1958, the first good estimate of H_{0}, 75 km/s/Mpc, was published by Allan Sandage^{[11]}, but it would be decades before a
consensus was achieved.
Hubble Diagram
Hubble's Law can be easily depicted
in a "Hubble Diagram" in which the velocity (assumed approximately
proportional to the redshift) of an object is plotted with respect to its
distance from the observer.^{[12]} A straight line of positive slope on
this diagram is the visual depiction of Hubble's Law.
The cosmological
constant abandoned
Main article: Cosmological constant
After Hubble's discovery was
published, Albert
Einstein
abandoned his work on the cosmological constant (which he had designed to allow for a static solution to his
equations). He later termed this work his "greatest blunder" since
the assumption of a static universe had prevented him from predicting the
expanding universe. Einstein made a famous trip to Mount Wilson in 1931 to
thank Hubble for providing the observational basis for modern cosmology. However,
the cosmological constant has regained attention in recent decades as a
hypothesis for dark
energy. This is indicated by the
verse:
- (وَالسَّمَاءَ بَنَيْنَاهَا بِأَيْيدٍ
وَإِنَّا لَمُوسِعُونَ _{*} وَالأرْضَ فَرَشْنَاهَا فَنِعْمَ الْمَاهِدُونَ) ]
48-47 الذاريات[.
"
We have built The Sama - Firmament - with might, We indeed Have vast
power; to create the vastness of Space and continue to expand it * And We have spread out Ardh
- Ground; interior or lower part of the Universe; the dark matter holding the
galaxies -: How excellently We do spread out!" (Surah No. 51, verse 47- 48).
(أَأَنْتُمْ أَشَدُّ خَلْقًا أَمْ السَّمَاءُ بَنَاهَا _{*} رَفَعَ سَمْكَهَا
فَسَوَّاهَا)
[النّازعات 27-28]
[27] What! Are ye the more difficult to create or the Samaa
(Firmaments) (above)? (Allah) hath constructed it: [28] On high hath He raised
its canopy, and He hath given it order and perfection. ) (S. 79, V. 27)
ALLAH (GOD) have built The
Sama - Firmament - with might, He indeed Have vast power; to create its
vastness and continue to expand it, and on high hath He raised its canopy. This gives rise to the so called negative
pressure (Dark energy); since Sama is a solid construction (roof) with no
cracks, and completely covers and surrounds the universe. This is clearly indicated by the verses:
- (أَلَمْ تَر أَنَّ اللَّهَ سَخَّرَ لَكُمْ مَا فِي الْأَرْضِ
وَالْفُلْكَ تَجْرِي فِي الْبَحْرِ بِأَمْرِهِ وَيُمْسِكُ السَّمَاءَ أَنْ تَقَعَ
عَلَى الْأَرْضِ إِلَّا بِإِذْنِهِ إِنَّ اللَّهَ بِالنَّاسِ لَرَءُوفٌ رَحِيمٌ)
سورة الحج آية رقم 65 .
) Seest thou not that Allah has made subject to you (men) all
that is on the earth, and the ships that sail through the sea by His command?
He withholds the Sama –
Firmament ; sky - (not rain as in yusuf Ali) from falling on the Ardh
except by His leave: for Allah is Most Kind and Most Merciful to man. ) (S. 22, V. 65)
Interpretation
A variety of possible recessional
velocity vs. redshift functions including the simple linear relation V = cz; a variety of possible shapes from
theories related to general relativity; and a curve that does not permit speeds
faster than light in accordance with special relativity. All curves are linear
at low redshifts. See Davis and Lineweaver.^{[13]}
The discovery of the linear
relationship between redshift and distance, coupled with a
supposed linear relation between recessional
velocity and
redshift, yields a straightforward mathematical expression for Hubble's Law as
follows:
where
Hubble's law is considered a
fundamental relation between recessional velocity and distance. However, the
relation between recessional velocity and redshift depends on the cosmological
model adopted, and is not established except for small redshifts.
For distances D larger than
the radius of the Hubble sphere r_{HS} , objects
recede at a rate faster than the speed of light:^{[14]}
In as much as the Hubble
"constant" is not constant at all, but varies with time in a manner
dictated by the choice of cosmological model, the radius of the Hubble sphere
may increase or decrease over various time intervals. The subscript '0' indicates
the value of the Hubble constant today.^{[15]}
Redshift velocity and
recessional velocity
Redshift can be measured by determining the
wavelength of a known transition, such as hydrogen α-lines for distant
quasars, and finding the fractional shift compared to a stationary reference.
Thus redshift is a quantity unambiguous for experimental observation. The
relation of redshift to recessional velocity is another matter. For an
extensive discussion, see Harrison.^{[16]}
Redshift velocity
The redshift z often is
described as a redshift velocity, which is the recessional velocity that
would produce the same redshift if it were caused by a linear Doppler effect (which, however, is not the case, as
the shift is caused in part by a cosmological expansion of space, and because the velocities involved are too large to
use a non-relativistic formula for Doppler shift). This redshift velocity can
easily exceed the speed of light.^{[17]} In other words, to determine the
redshift velocity v_{rs}, the relation:
is used.^{[18]}^{[19]} That is, there is no fundamental
difference between redshift velocity and redshift: they are rigidly
proportional, and not related by any theoretical reasoning. The motivation
behind the "redshift velocity" terminology is that the redshift
velocity agrees with the velocity from a low-velocity simplification of the
so-called Fizeau-Doppler formula^{[20]}
Here, λ_{o}, λ_{e} are the observed and emitted
wavelengths respectively. The "redshift velocity" V_{rs} is not so simply related to real
velocity at larger velocities, however, and this terminology leads to confusion
if interpreted as a real velocity. Next, the connection between redshift or
redshift velocity and recessional velocity is discussed. This discussion is
based on Sartori.^{[21]}
Recessional velocity
Suppose R(t) is called the scale factor of the universe, and increases as the universe expands in a
manner that depends upon the cosmological
model
selected. Its meaning is that all measured distances D(t) between co-moving points increase
proportionally to R. (The co-moving points are not
moving relative to each other except as a result of the expansion of space.) In
other words:
where t_{0} is some reference time. If light is
emitted from a galaxy at time t_{e} and received by us at t_{0}, it is red shifted due to the expansion of space, and
this redshift z is simply:
Suppose a galaxy is at distance D, and this distance changes with time
at a rate d_{t}D . We call this rate of recession the
"recession velocity" v_{r}:
We now define the Hubble constant as
and discover the Hubble law:
From this perspective, Hubble's law
is a fundamental relation between (i) the recessional velocity contributed by
the expansion of space and (ii) the distance to an object; the connection
between redshift and distance is a crutch used to connect Hubble's law with
observations. This law can be related to redshift z approximately by making a Taylor series expansion:
If the distance is not too large, all
other complications of the model become small corrections and the time interval
is simply the distance divided by the speed of light:
or
According to this approach, the
relation cz = v_{r} is an approximation valid at low redshifts,
to be replaced by a relation at large redshifts that is model-dependent. See velocity-redshift figure.
Observability of
parameters
Strictly speaking, neither V nor D in the formula are directly
observable, because they are properties now of a galaxy, whereas our
observations refer to the galaxy in the past, at the time that the light we
currently see left it.
For relatively nearby galaxies (redshift z much less than unity), V and D will not have changed
much, and v can be estimated using the formula V = zc where c is the speed
of light.
This gives the empirical relation found by Hubble.
For distant galaxies, V (or D) cannot be calculated from z
without specifying a detailed model for how H changes with time. The redshift is
not even directly related to the recession velocity at the time the light set
out, but it does have a simple interpretation: (1+z) is the factor by which the universe
has expanded while the photon was traveling towards the observer.
Expansion velocity vs
relative velocity
In using Hubble's law to determine
distances, only the velocity due to the expansion of the universe can be used.
Since gravitationally interacting galaxies move relative to each other
independent of the expansion of the universe, these relative velocities, called
peculiar
velocities,
need to be accounted for in the application of Hubble's law.
The Finger of God effect is one result of this
phenomenon discovered in 1938 by Benjamin Kenneally. In systems that are gravitationally
bound, such
as galaxies or our planetary system, the expansion of space is a much weaker
effect than the attractive force of gravity.
Idealized Hubble's
Law
The mathematical derivation of an
idealized Hubble's Law for a uniformly expanding universe is a fairly
elementary theorem of geometry in 3-dimensional Cartesian/Newtonian coordinate space, which, considered as a metric space, is entirely homogeneous and isotropic (properties do not vary with location or direction). Simply
stated the theorem is this:
Any two points
which are moving away from the origin, each along straight lines and with speed
proportional to distance from the origin, will be moving away from each other
with a speed proportional to their distance apart.
In fact this applies to non-Cartesian
spaces as long as they are locally homogeneous and isotropic; specifically to
the negatively- and positively-curved spaces frequently considered as cosmological
models (see shape of the universe).
An observation stemming from this
theorem is that seeing objects recede from us on Earth is not an indication
that Earth is near to a center from which the expansion is occurring, but
rather that every observer in an expanding universe will see objects
receding from them.
The ‘ultimate fate’
and age of the universe
The age and ultimate fate of the universe can be determined by measuring the
Hubble constant today and extrapolating with the observed value of the
deceleration parameter, uniquely characterized by values of density parameters
(Ω_{M} and Ω_{Λ}). A "closed universe" with
Ω_{M}
> 1 and Ω_{Λ} = 0 comes to an end in a Big Crunch and is considerably younger than its
Hubble age.
(وَمَا قَدَرُوا اللَّهَ
حَقَّ قَدْرِهِ وَالْأَرْضُ جَمِيعًا قَبْضَتُهُ يَوْمَ الْقِيَامَةِ
وَالسَّمَاوَاتُ مَطْوِيَّاتٌ بِيَمِينِهِ سُبْحَانَهُ وَتَعَالَى عَمَّا
يُشْرِكُونَ) [الزمر 67]. الْأَرْضُ هنا تعني الأرضون
السّبع، حيث ستعود يوم القيامة مجموعة ورتقاً. وكذلك السماوات مطويات بيد الجبار سبحانه.
"No just estimate have they made of
Allah, such as is due to Him: The Ardhean (lower part of universe; mostly dark
matter holding galaxies) will be coupled, His handful on the Day of Judgment,
and the heavens will be rolled up in His right hand: Glory to Him! High is He
above the Partners they attribute to Him!: (Al-Zumar 39:67).
An "open universe" with Ω_{M} ≤ 1 and Ω_{Λ} = 0 expands forever and has an age that
is closer to its Hubble age. For the accelerating universe with nonzero Ω_{Λ} that we inhabit, the age of the universe is coincidentally
very close to the Hubble age.
The value of the Hubble parameter
changes over time either increasing or decreasing depending on the sign of the
so-called deceleration parameter q which is defined by
In a universe with a deceleration
parameter equal to zero, it follows that H = 1/t,
where t is the time since the Big Bang. A
non-zero, time-dependent value of q simply requires integration of the Friedmann equations backwards
from the present time to the time when the comoving horizon size was zero.
It was long thought that q was positive, indicating that the
expansion is slowing down due to gravitational attraction. This would imply an
age of the universe less than 1/H (which is about 14 billion years). For instance, a value for q of 1/2 (once favoured by most
theorists) would give the age of the universe as 2/(3H). The discovery in 1998 that q is apparently negative means that
the universe could actually be older than 1/H. However, estimates of the age
of the universe are very close to 1/H.
Olbers' paradox
Main article: Olbers' paradox
The expansion of space summarized by
the Big Bang interpretation of Hubble's Law is relevant to the old conundrum
known as Olbers'
paradox: if
the universe were infinite, static, and filled with a uniform
distribution of stars (notice that this also requires an
infinite number of stars), then every line of sight in the sky would end on a
star, and the sky would be as bright as the surface of a star. However,
the night sky is largely dark. Since the 1600s, astronomers and other thinkers
have proposed many possible ways to resolve this paradox, but the currently
accepted resolution depends in part upon the Big Bang theory and in part upon the Hubble expansion. In a universe that exists for a
finite amount of time, only the light of finitely many stars has had a chance
to reach us yet, and the paradox is resolved. Additionally, in an expanding
universe distant objects recede from us, which causes the light
emanating from them to be redshifted and diminished in brightness.
Although both effects contribute, the redshift is the less important of the
two.^{[22]}
This is among various aspects
indicated by Prophets Hadith (saying):
(سمعت
عليا وسئل عن { فَلَا أُقْسِم بِالْخُنَّسِ الْجَوَارِ الْكُنَّس (التكوير
15-16)}
فقال : هِيَ النُّجُوم تَخْنَس بِالنَّهَارِ وَتَظْهَر (وتكنس) بِاللَّيْلِ)
(الراوي: خالد بن
عرعرة - خلاصة الدرجة: إسناده جيد - المحدث: ابن كثير - المصدر: تفسير القرآن - الصفحة أو الرقم 8/359 )
(So verily I call to witness the Stars, that recede. Constantly moving, and accreating. And the Night as it gets Darker) (S. 81; V. 15-17)
Determining the
Hubble constant
The value of the Hubble constant is
estimated by measuring the redshift of distant galaxies and then determining the distances to the same galaxies (by some other method than Hubble's
law). Uncertainties in the physical assumptions used to determine these
distances have caused varying estimates of the Hubble constant. For most of the
second half of the 20th century the value of H_{0} was estimated to be between 50 and
90 (km/s)/Mpc.
Disputes over
Hubble's constant
“ |
Astrophysicists are always wrong,
but never in doubt. ... RP Kirshner^{[23]} |
” |
The
value of the Hubble constant was the topic of a long and rather bitter
controversy between Gérard de Vaucouleurs who claimed the value was around 100 and Allan Sandage who claimed the value was near 50.^{[24]}
In
This difference was partially
resolved with the introduction of the ΛCDM model of the universe in the late 1990s.
The ΛCDM model
With the ΛCDM model observations of high-redshift
clusters at X-ray and microwave wavelengths using the Sunyaev-Zel'dovich effect, measurements of anisotropies in the cosmic microwave background radiation, and optical surveys all gave a
value of around 70 for the constant.^{[}^{citation needed}^{]}
Using Hubble space
telescope data
The Hubble Key Project (led by Dr.
Wendy L. Freedman, Carnegie Observatories) used the Hubble space telescope to establish the most precise optical determination in May
2001^{[25]} of 72 ± 8 (km/s)/Mpc, consistent with a measurement of H_{0} based upon Sunyaev-Zel'dovich effect observations of many galaxy clusters having a similar accuracy.
Using WMAP data
The most precise cosmic microwave background radiation determinations were 71 ± 4 (km/s)/Mpc, by WMAP in 2003, and 70.4 ^{+1.5}_{−1.6} (km/s)/Mpc, for measurements up to
2006.^{[26]} The five year release from WMAP in
2008 finds 71.9 ^{+2.6}_{−2.7} (km/s)/Mpc.[1]
These values arise from fitting a
combination of WMAP and other cosmological data to the simplest version of the
ΛCDM model. If the data is fitted with more general versions, H_{0} tends to be smaller and more
uncertain: typically around 67 ± 4 (km/s)/Mpc although some models allow values near 63 (km/s)/Mpc.^{[27]}
The number of the Surah
Using Chandra X-ray
Observatory data
In August 2006, using NASA's Chandra X-ray Observatory, a team from NASA's Marshall Space Flight Center (MSFC) found the Hubble constant to
be 77 (km/s)/Mpc, with an uncertainty of
about 15%.^{[28]} The consistency of the measurements
from all these methods lends support to both the measured value of H_{0} and the ΛCDM model.
Acceleration of the
expansion
A value for q measured from standard candle observations of Type Ia supernovae, which was determined in 1998 to be
negative, surprised many astronomers with the implication that the expansion of
the universe is currently "accelerating" (although the Hubble factor
is still decreasing with time; see the articles on dark energy and the ΛCDM model).
Derivation of the
Hubble parameter
Start with the Friedmann
equation:
where H is the Hubble parameter, a is the scale factor, G is the gravitational constant, k is the
normalised spatial curvature of the universe and equal to −1, 0, or +1,
and Λ is the cosmological constant.
Matter-dominated
universe (with a cosmological constant)
If the universe is matter-dominated, then the mass density of the
universe ρ can just be taken to include matter
so
where is the density of matter today. We know for
nonrelativistic particles that their mass density decreases proportional to the
inverse volume of the universe so the equation above must be true. We can also
define (see density
parameter for
Ω_{m})
so ρ = ρ_{c}Ω_{m}
/ a^{3}. Also, by definition,
and
where the subscript nought refers to
the values today, and a_{0} = 1. Substituting all of this in into the Friedman
equation at the start of this section and replacing a with a = 1 / (1 + z) gives
Matter- and dark
energy-dominated universe
If the universe is both matter-dominated and dark energy-dominated, then the above equation
for the Hubble parameter will also be a function of the equation of state of dark energy. So now:
ρ = ρ_{m}(a)
+ ρ_{de}(a),
where ρ_{de} is the mass density of the dark
energy. By definition an equation of state in cosmology is P = wρc^{2}, and if we substitute this into the
fluid equation, which describes how the mass density of the universe evolves
with time,
If w is constant,
Therefore for dark energy with a
constant equation of state w, .
If we substitute this into the Friedman equation in a similar way as before,
but this time set k = 0 which is assuming we live in a spatially flat
universe, (see Shape of the Universe)
If dark energy does not have a
constant equation-of-state w, then
and to solve this we must parametrize
w(a), for example if w(a) = w_{0} + w_{a}(1
− a),
giving
Units derived from
the Hubble constant
Hubble time
The Hubble constant H_{0} has units of inverse time, i.e. H_{0} ~ 2.29×10^{−18} s^{−1}. “Hubble time” is defined as 1 / H_{0}. The value of Hubble time in the standard cosmological model is 4.35×10^{17} s or 13.8 billion years. (Liddle 2003, p. 57) The phrase
"expansion timescale" means "Hubble time".[2]. If the value of H_{0} were to stay constant, a naive
interpretion of the Hubble time is that it is the time taken for the universe
to increase in size by a factor of e (because the solution of dx/dt = xH_{0} is x = s_{0}exp(H_{0}t), where s_{0} is the size of some feature at some
arbitrary initial condition t = 0). However, over long periods of time the dynamics are complicated by general
relativity, dark energy, inflation, etc., as explained above.
Inflation, and the so
called dark energy are indicated by the Qur'anic verses:
(ثُمَّ اسْتَوَى إِلَى
السَّمَاءِ وَهِيَ دُخَانٌ فَقَالَ لَهَا وَلِلْأَرْضِ ائْتِيَا طَوْعًا أَوْ
كَرْهًا قَالَتَا أَتَيْنَا طَائِعِينَ * فَقَضَاهُنَّ سَبْعَ سَمَاوَاتٍ فِي يَوْمَيْنِ) [فصّلت آية 11-12
].
Allâh says: "Moreover, He comprehended in His design the Sama
(upper part of universe), and it had been smoke: He said to it and to Ardh
(lower - interior - part of the Universe; not earth): 'Come ye, willingly or
unwillingly.' They said: 'We do come, in willing obedience'. So He completed
them as seven firmaments in two Days (periods) …" (Surah 41, Verses
11-12).
"And We indeed Have vast power; to expand it". This interprets as: ALLAH constructs Sama via expansion ([i]). ALLAH create and elevate Sama
with vast force and power, and We (ALLAH) are able to expand it as We desire ([ii]). We are able to expand, as We expand its
construction ([iii]).
- (وَالسَّمَاءَ بَنَيْنَاهَا بِأَيْيدٍ
وَإِنَّا لَمُوسِعُونَ _{*} وَالأرْضَ فَرَشْنَاهَا فَنِعْمَ الْمَاهِدُونَ) ]
48-47 الذّاريات[.
"
We have built The Sama - Firmament - with might, We indeed Have vast
power; to create the vastness of Space and continue to expand it * And We have spread out Ardh
- Ground; interior or lower part of the Universe; the dark matter holding the
galaxies -: How excellently We do spread out!" (Surah No. 51, verse 47- 48).
(أَأَنْتُمْ أَشَدُّ
خَلْقًا أَمْ السَّمَاءُ بَنَاهَا _{*} رَفَعَ سَمْكَهَا فَسَوَّاهَا) [النّازعات 27-28]
[27] What! Are ye the more difficult to create or the Samaa
(Firmaments) (above)? (Allah) hath constructed it: [28] On high hath He raised
its canopy, and He hath given it order and perfection.
Hubble length
The Hubble length is a unit of
distance in cosmology, defined as c
/ H_{0}
- the speed of light multiplied by the Hubble time. It is equivalent to 4228 million parsecs or 13.8 billion light years. (The numerical value of the Hubble
length in light years is, by definition, equal to that of the Hubble time in
years.)
Hubble volume
The Hubble volume is sometimes
defined as a volume of the universe with a comoving size of c / H_{0}. The exact definition varies: it is
sometimes defined as the volume of a sphere with radius c / H_{0}, or alternatively, a cube of side c / H_{0}. Some cosmologists even use the term
Hubble volume to refer to the volume of the observable
universe,
although this has a radius approximately three times larger.
Notes
1. ^ Peter Coles, ed (2001). Routledge Critical Dictionary of the
New Cosmology. Routledge. p. 202. ISBN 0203164571. http://books.google.com/books?id=BgNGWVr5yhIC&pg=PA202.
2. ^ Hubble, Edwin, "A Relation between Distance and
Radial Velocity among Extra-Galactic Nebulae" (1929) Proceedings of the
National Academy of Sciences of the United States of America, Volume 15,
Issue 3, pp. 168-173 (Full article, PDF)
3. ^ Malcolm S Longair (2006). The Cosmic Century. Cambridge University Press.
p. 109. ISBN 0521474361. http://books.google.com/books?id=z0vlYHQZHJcC&pg=RA2-PA109.
4. ^ "Refined Hubble Constant Narrows
Possible Explanations for Dark Energy". 2009-05-09. http://hubblesite.org/newscenter/archive/releases/2009/08/full/. Retrieved 2009-05-09.
5. ^ W. L. Freedman, B. F. Madore, B. K.
Gibson, L. Ferrarese, D. D. Kelson, S. Sakai, J. R. Mould, R. C. Kennicutt,
Jr., H. C. Ford, J. A. Graham, J. P. Huchra, S. M. G. Hughes, G. D.
Illingworth, L. M. Macri, P. B. Stetson (2001). "Final Results from the Hubble Space
Telescope Key Project to Measure the Hubble Constant". The Astrophysical Journal 553 (1): 47–72. doi:10.1086/320638. http://adsabs.harvard.edu/cgi-bin/nph-bib_query?2001ApJ...553...47F.
6. ^ "Chandra Confirms the Hubble
Constant".
2006-08-08. http://www.universetoday.com/2006/08/08/chandra-confirms-the-hubble-constant/. Retrieved 2007-03-07.
7. ^ "WMAP's Universe". NASA. http://wmap.gsfc.nasa.gov/universe/uni_expansion.html.
8. ^ Friedman, A. (1922), "Über
die Krümmung des Raumes", Zeitschrift für Physik 10 (1): 377–386, doi:10.1007/BF01332580. (English translation: "On the
Curvature of Space", General Relativity and Gravitation 31 (12):
1991–2000, 1999, doi:10.1023/A:1026751225741.)
9. ^ Wendy L Freeman et al.
(2001). "Final Results from the Hubble
Space Telescope Key Project to Measure the Hubble Constant". Astrophys J 553: 47–72. doi:10.1086/320638. http://arxiv.org/abs/astro-ph/0012376v1.
10. ^ Steven Weinberg (2008). Cosmology. Oxford University Press.
p. 28. ISBN 0198526822. http://books.google.com/books?id=nqQZdg020fsC&pg=PA28.
11. ^ Sandage, A. R. (May,1958).
"Current Problems in the Extragalactic Distance Scale.". Astrophysical Journal 127 (3): 513–526. doi:10.1086/146483. Bibcode: 1958ApJ...127..513S.
12. ^ R. P. Kirshner, Hubble's Diagram and
Cosmic Expansion, Online Article
13. ^ Tamara M. Davis, Charles H.
Lineweaver (2000). "Superluminal Recessional
Velocities". ArXiv preprint. http://arxiv.org/abs/astro-ph/0011070v2.
14. ^ It is argued that such motion is
compatible with special relativity because every observer is the center of
their own Hubble sphere, and the objects moving faster than the speed of light
are therefore outside the reach of any inertial frame of reference. See TM
Davis & CH Linewater (2003). "Expanding Confusion: common misconceptions of
cosmological horizons and the superluminal expansion of the universe". ArXiv preprint. http://arxiv.org/abs/astro-ph/0310808v2.
15. ^ William C. Keel (2007). The Road to Galaxy Formation (2 ed.). Springer. p. 7. ISBN 3540725342. http://books.google.com/books?id=BUgJGypUYF0C&pg=PA7.
16. ^ Edward Harrison (1992). "The redshift-distance and
velocity-distance laws". Astrophysical Journal, Part 1 403: 28–31. doi:10.1086/172179. http://adsabs.harvard.edu/abs/1993ApJ...403...28H. . A pdf file can be found here.
17. ^ MS Madsen (1995). The Dynamic Cosmos. CRC Press. p. 35. ISBN 0412623005. http://books.google.com/books?id=_2GeJxVvyFMC&pg=PA35.
18. ^ Avishai Dekel, J. P. Ostriker
(1999). Formation of Structure in the
Universe.
Cambridge University Press. p. 164. ISBN 0521586321. http://books.google.com/books?id=yAroX6tx-l0C&pg=PA164.
19. ^ Thanu Padmanabhan (1993). Structure formation in the universe. Cambridge University Press.
p. 58. ISBN 0521424860. http://books.google.com/books?id=AJlOVBRZJtIC&pg=PA58.
20. ^ Leo Sartori (1996). Understanding
Relativity. University of California Press. p. 163, Appendix 5B. ISBN 0520200292.
21. ^ Leo Sartori (1996). [0520200292 Understanding
Relativity]. University of California Press. pp. 304–305. 0520200292.
22. ^ S. I. Chase, Olbers' Paradox, entry in the Physics FAQ; see also I. Asimov, "The Black of Night", in Asimov
on Astronomy (Doubleday, 1974), ISBN 0-385-04111-X.
23. ^ Quoted by RP Kirshner
24. ^ Dennis Overbye, Lonely Hearts of the Cosmos: The
Scientific Quest for the Secret of the Universe, Harper-Collins (1991), ISBN 0-06-015964-2 & ISBN 0-330-29585-3 (finalist, Nation Book Critics
Circle Award for non-fiction). Second edition (with new afterword), Back Bay,
1999. Gives an account of the history of the dispute and rivalries.
25. ^ W. L. Freedman, B. F. Madore, B. K. Gibson,
L. Ferrarese, D. D. Kelson, S. Sakai, J. R. Mould, R. C. Kennicutt, Jr., H. C.
Ford, J. A. Graham, J. P. Huchra, S. M. G. Hughes, G. D. Illingworth, L. M.
Macri, P. B. Stetson (2001). "Final Results from the Hubble
Space Telescope Key Project to Measure the Hubble Constant". The Astrophysical Journal 553 (1): 47–72. doi:10.1086/320638. http://adsabs.harvard.edu/cgi-bin/nph-bib_query?2001ApJ...553...47F. . Preprint available here.
26. ^ D. N. Spergel et al. (2007).
"Three-year Wilkinson Microwave Anisotropy Probe (WMAP) Observations:
Implications for Cosmology". Astrophysical Journal Supplement Series
170: 377–408. doi:10.1086/513700. ;
available online at LAMBDA
27. ^ Results for H_{0} and
other cosmological parameters obtained by fitting a variety of models to
several combinations of WMAP and other data are available at the NASA's LAMBDA website.
28. ^ Chandra independently determines
Hubble constant in Spaceflight Now
References
[i] ) See the following references:-
- Ibn Attiyeh
al-Andalusi, Abi Mohammed (546 Hijri) al-Muharur al-Wajeez fi Tafseer
al-Kettab al-Aziz (1413 Hijri-1993) (The Editing Summary in the Interpretation
of the Glorious Quran) Vol., 5:181.
- Abi Al-Abbas, Shehab ed-Din (1994) al-Dar
al-Masoun fi Oloum al-Kettab al-Kaknoun.
- Abu Hayan, (654-754 Hijri) An-Nahr
al-Madd, vol. 5: Part 5: 244.
- Ibn Attiyeh al-Andalusi, (546 Hijri), (1413
Hijri-1993), vol. 5:181.
- Shehab ed-Din (1994), Part 6: 192.
- Abu Hayan,
(654-754 Hijri) al-Bahr al-Muheet,
Part 9: 560
[ii] ) See the following references:-
- as-Sammurgandi,
Abi al-Layeth Nasser bin Mohammed (1993) Bahr al-Oulum (The Sea of
Knowledge)
- al-Jouzi, Abi al-Faruj Jamal ed-Din (1987) Zad
al-Maseer fi Elm at-Tafseer (The Provision of Walk in the Science of
Interpretation) Beirut, Dar al-Fikr, 8 Parts, Part 7: 212.
- al-Kasimi, Mohammed Jamal (1332 Hijri-
1914) Mahasen at-Ta’weel (The Advantages of Paraphrase), Dar al-Fiker
(1978), vol. 9, Part 2: 202-03.
- al-Khateeb, 1970, vol. 4: 529-39.
- al-Zamakhshari, 538 Hijri, vol. 4: 20.
- al-Razi,
1208, vol. 4:
227
[iii] ) See the following references:-
- al-Kasimi,
Mohammed Jamal (1332 Hijri- 1914) Mahasen at-Ta’weel (The Advantages of
Paraphrase), Dar al-Fiker (1978), vol. 9, Part 2: 202-03.
- al-Maourdi, Tasneef Abi al-Hasan
al-Basri (364-450 Hijri) Revised and commented on by as-Siyyed bin
Abdulraheem. Al-Nukat wal Oyoun: Tafseer
al-Maourdi (Secrets - details - and the Eyes: al-Maourdi’s Interpretation).
- al-Nasseri, 1985, Part 6: 93.