# 29.2 A model of the universe  (Page 5/10)

 Page 5 / 10

In an empty universe (the dashed line [link] and the coasting universe in [link] ), neither gravity nor dark energy is important enough to affect the expansion rate, which is therefore constant throughout all time.

In a universe with dark energy, the rate of the expansion will increase with time, and the expansion will continue at an ever-faster rate. Curve 4 in [link] , which represents this universe, has a complicated shape. In the beginning, when the matter is all very close together, the rate of expansion is most influenced by gravity. Dark energy appears to act only over large scales and thus becomes more important as the universe grows larger and the matter begins to thin out. In this model, at first the universe slows down, but as space stretches, the acceleration plays a greater role and the expansion speeds up.

## The cosmic tug of war

We might summarize our discussion so far by saying that a “tug of war” is going on in the universe between the forces that push everything apart and the gravitational attraction of matter, which pulls everything together. If we can determine who will win this tug of war, we will learn the ultimate fate of the universe.

The first thing we need to know is the density of the universe. Is it greater than, less than, or equal to the critical density? The critical density today depends on the value of the expansion rate today, H 0 . If the Hubble constant    is around 20 kilometers/second per million light-years, the critical density is about 10 –26 kg/m 3 . Let’s see how this value compares with the actual density of the universe.

## Critical density of the universe

As we discussed, the critical density is that combination of matter and energy that brings the universe coasting to a stop at time infinity. Einstein’s equations lead to the following expression for the critical density ( ρ crit ):

${\rho }_{\text{crit}}=\frac{3{H}^{2}}{8\text{π}G}$

where H is the Hubble constant and G is the universal constant of gravity (6.67 × 10 –11 Nm 2 /kg 2 ).

## Solution

Let’s substitute our values and see what we get. Take an H = 22 km/s per million light-years. We need to convert both km and light-years into meters for consistency. A million light-years = 10 6 × 9.5 × 10 15 m = 9.5 × 10 21 m. And 22 km/s = 2.2 × 10 4 m/s. That makes H = 2.3 ×10 –18 /s and H 2 = 5.36 × 10 –36 /s 2 . So,

${\rho }_{\text{crit}}=\frac{3\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}5.36\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{–36}}{8\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}3.14\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}6.67\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{–11}}=9.6\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{–27}\phantom{\rule{0.2em}{0ex}}{\text{kg/m}}^{3}$

which we can round off to the 10 –26 kg/m 3 . (To make the units work out, you have to know that N , the unit of force, is the same as kg × m/s 2 .)

Now we can compare densities we measure in the universe to this critical value. Note that density is mass per unit volume, but energy has an equivalent mass of m = E / c 2 (from Einstein’s equation E = mc 2 ).

## Check your learning

1. A single grain of dust has a mass of about 1.1 × 10 –13 kg. If the average mass-energy density of space is equal to the critical density on average, how much space would be required to produce a total mass-energy equal to a dust grain?
2. If the Hubble constant were twice what it actually is, how much would the critical density be?

## Answer:

a. In this case, the average mass-energy in a volume V of space is E = ρ crit V . Thus, for space with critical density, we require that

$V=\frac{{E}_{\text{grain}}}{{\rho }_{\text{crit}}}=\frac{1.1\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{–13}\phantom{\rule{0.2em}{0ex}}\text{kg}}{9.6\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{–26}\phantom{\rule{0.2em}{0ex}}{\text{kg/m}}^{3}}=1.15\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{12}\phantom{\rule{0.2em}{0ex}}{\text{m}}^{3}={\left(10,500\phantom{\rule{0.2em}{0ex}}\text{m}\right)}^{3}\cong {\left(10.5\phantom{\rule{0.2em}{0ex}}\text{km}\right)}^{3}$

Thus, the sides of a cube of space with mass-energy density averaging that of the critical density would need to be slightly greater than 10 km to contain the total energy equal to a single grain of dust!

b. Since the critical density goes as the square of the Hubble constant, by doubling the Hubble parameter, the critical density would increase by a factor a four. So if the Hubble constant was 44 km/s per million light-years instead of 22 km/s per million light-years, the critical density would be ${\rho }_{\text{crit}}=4\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}9.6\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{–27}\phantom{\rule{0.2em}{0ex}}{\text{kg/m}}^{3}=3.8\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{–26}\phantom{\rule{0.2em}{0ex}}{\text{kg/m}}^{3}.$

#### Questions & Answers

what is steral evolution (star evolution
meron Reply
hi
Nisha
when a star's mass changes over time
Joe
hi
Joe
hi
Lisette
Hii
Asim
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Nisha
star prouduce nucluer fusion first stag ( hydroge fusion helium- carbon-neon-oxygen-silicon- last fusion iron hevy star mass not handle iron blast super nova
Nisha
what is the worm hole theory
Nisha Reply
it states that you can travel back through time by this. It's like folding a paper into half and making two points, and there states that you can bend spacetime but according to Einstein's equation that it can break easily and only really small particles can pass through it.
Dark
dark dream what is dark matter
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Dark
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Nisha
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Nisha
In radio astronomy, a fast radio burst (FRB) is a transient radiopulse of length ranging from a fraction of a millisecond to a few milliseconds, caused by some high-energy astrophysical process not yet identified. We can say that this can be they way to at least know if there's other civilizations~
Dark
~out there
Dark
Simulation hypothesis it proposes that the earth, the universe and the reality you are in is just an artificial simulation or like you are living in a computer simulation.
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Nisha
why does the matter of an objects density change the fabric of space time rather than the objects volume..? (hopefully I have expressed that question properly) thanks
Daijahrel Reply
because an objects density consists of it's energy & momentum. an objects volume, for example a photon can have no density (hopefully I expressed that answer properly)
Joe
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Melaku
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Does their is any procedure of star making
Asim Reply
Hope That Helps.....
Adam
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Nisha
waad mahad santnihiin dhamaan waxaan doonayaa inaan wax ka barto astronomyga waa maxay xidigis?
Abdicaroog Reply
understanding astronomy
ricardo Reply
What About Understanding Astronomy?
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define..
Janmarc
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Chaya Reply
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Sri
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Aymen
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Sri
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Asim
Probably Massachusetts Institute Of Technology.....
Adam
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Geometry, Algebra, Calculus, Trigonometry, Do Alot Of It And Often.....
Adam
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Adam
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Asim
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priyanshu
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Rajan Reply
Leonardo Da Vinci.....
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sakshi Reply
A Cosmological Theory Based On The Exsistence Of Cosmic Vibrating Strings.....
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Asim
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Asim
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Adam
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Adam
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Asim
A White Dwarf Is A Low Mass Star That Has Exhausted All It's Central Nuclear Fuel And Lost It's Outer Layers As A Planetary Nebula.....
Adam
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Nisha
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Nisha
Well First It's Not A Theory, It's A Hypothesis, The Great Filter Hypothesis States That What Ever Causes Dead Matter To Under Go Abiogenesis(Start Of Life) Must Be Very Hard And Very Rare In Our Universe, Because So Far We Only See 1 Civilisation That Made Through All The Requirements For.....
Adam
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priyanshu Reply
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Gospel
How did ASTRONOMY helped you understand yourself ? 🤔
Hussain Reply
why do you want to study ASTRONOMY?
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It is one of the natural sciences and therefore worth exploring. You are a part of the universe and it is a part of you. The sky is my classroom. (student of cosmology, Oxford Uni)
SuperNova
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Memo
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Carla
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Carla
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Hussain
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Dark
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Dark
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Da
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Hussain
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Valerian Reply
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Mostak Reply
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Shaikh Reply
10 to the power 11
Wwe
testing
Ahmad
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Shaikh
if u can count all the sands on the beaches and on the deserts then u will know how much it will be.......some said more than 1,000,000,000 galaxies are there in our universe...
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Shaikh
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Adam
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Caleb
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Ariana
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Abdullah
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Mostak
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Ismaee
we didn't actually find the exact number but may be it is around 100 billion
Smridhi
why this universe is expanding?
ATUL
Science Does Not Concern It Self With Why Questions, Science Is Geared For How And What Questions.....
Adam
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Adam
According to the universal law of cause and effect .any phenomenon that occur in this universe should have any reason , nothing happen without any reason.
ATUL
Can you tell me the expanding speed of this universe.
ATUL
yes that would be a great question to have answered what is the expanding speed of the universe?
Daijahrel
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SuperNova
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Adam
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Gospel
Because We Dont See Any Object In Space As It Is, We See It As It Was..... Hope This Clears That Up For You.....
Adam
And One Day We Might Have A Telescope That Will Show Us The Big Bang.....
Adam
ok....... that will be nice one
Gospel
ok....... that will be nice one
Gospel
Travel at light speed in a spaceship at 186,000 miles every SECOND. At this speed it will take you 100,000 years just to cross our Milky Way galaxy, which is just a dot in the sky. Next stop, the Andromeda galaxy, after 2.5 million years, still in the Local Group, Virgo Supercluster, still a dot.
SuperNova
Pretty Much.....
Adam
there are nearly 200 million I suppose.
Amit

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