COSMOLOGY The Universe – what is its age and origin?

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REVIEW (SUMMARY)

¢ Oppenheimer Volkhoff limit: upper limit to mass of neutron star remnant more than 1.4 𝑀↓⊙    à neutron degeneracy

¢ Supernova à extremely dense neutron star ¢  If mass above 2 – 3 𝑀↓⊙ , the star collapses into black

hole ¢ Chandrasakhar limit – upper mass limit of star

remnant less than 1.4 𝑀↓⊙  à electron degeneracy ¢ Outer layers blow off – planetary nebulae ¢ Core collapses further – white dwarf

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NEWTON’S UNIVERSE

Newton thought the Universe to be ¢  Infinite ¢ Uniform ¢  static

¢  If this were true, the night sky should be bright – where ever you look should be a star

CGR\Why is it Dark at Night_ - YouTube [720p].mp4 ©

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OLBER’S PARADOX

¢  In a given volume of space imagine an average number of stars N with density 𝜌

¢  Imagine stars in a thin shell of thickness d at a distance R

¢  Volume of shell: 4𝜋𝑅↑2 𝑑  𝑎𝑛𝑑  # of stars N in shell = n  4𝜋𝑅↑2 𝑑

¢  But brightness 𝑏= 𝐿/4𝜋𝑅↑2   ¢  Volume of shell increases

with  𝑅↑2 , but luminosity decreases with 1/𝑅↑2  

¢  Total brightness we receive from any shell should be

¢  𝐵=𝑁𝑏 ¢  𝐵=𝑛4𝜋𝑅↑2 𝑑× 𝐿/4𝜋𝑅↑2  =𝑛𝑑𝐿

R

Thickness 𝑑 #Stars ∞𝑅↑2 

∞1/𝑅↑2  

¢  Overall brightness: add up all shells

¢  Since assumed Universe is infinite, there are an infinite # of shells

¢  There should be an infinite bright night sky

Amount of light received

from shell does not

depend upon how far away

shell is

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POSSIBLE SOLUTIONS

1.  Universe not infinite or it is not uniform à but there is no edge à it must be infinite

2.  Light is absorbed by something à s.th. Should warm up and radiate energy à not observed

3.  Not static à we know it is expanding from red and blue shift à it is expanding now, but what does it imply about the past?

Is the Universe

Infinite and Uniform?

Has all light reached us?

Is the Universe

static?

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¢ Universe was created ~ 13 – 15 billion years ago

¢ We only could see light that was less than 15 billion ly away

¢  If we receive light from a finite # of stars, the night sky would be dark

Has the known

Universe changed?

Big Bang

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BIG BANG MODEL

¢ Doppler effect causes redshift for sources moving away àlonger λ à lower f

¢ Light sources are compared in their spectrum against the spectrum of a stationary hydrogen source

¢ Galaxies show redshift à they move away from each other à they must have been much closer together in the past

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AT THE BEGINNING…

¢ …there was a point of infinite density called singularity

Big Bang Explosion

•  Time and space was created some 10 – 20 billion years ago

•  It is the space in which galaxies are situated that expands

•  Balloon analogy, but à no outside à no center

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QUESTIONS

¢ What is outside the Universe? ¢ What was there before the Big Bang?

Questions are meaningless – they can’t be asked

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REDSHIFT

¢  Due to photons traveling through space

¢  Space expands causing the λ of photons to expand

¢  The longer a photon travels, the more its λ will increase

¢  The photons originating from the furthest distance will be redshifted most

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COSMIC BACKGROUND RADIATION CMB

¢  Isotropic – the same in all directions

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DICKE AND PEEBLES IN 1960 Proposal: CGR\Cosmic Origins clip_ Arno Penzias and the Expanding Universe - YouTube [720p].mp4 ¢  During early Universe sufficient

high temperature to produce helium by fusion (24% of the Universe is helium)

¢  High energy photons were produced ¢  They have a BB spectrum

corresponding to the temperature at that time

¢  As Universe expanded and cooled, photon spectrum would also cool and max λ would change in accordance with Wien’s Law

¢  Today they should have a max λ corresponding to 3K

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PENZIAS AND WILSON - 1965

•  No matter where they pointed their aerial, it picked up a steady continuous background radiation

•  All space is filled with radiation corresponding to a BB spectrum of 2.76K

•  Evidence for the Bing Bang Model

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BIG BANG MODEL

¢ Space is expanding and within space distribution of galaxies is not uniform

¢ Universe is not uniform or static ¢ Resolution to Olber’s paradox

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EXAMPLE ¢  This question is about the Big Bang model

and red-shift. ¢  (a) Describe what is meant by the Big Bang

model. Solution ¢  space and time originated from a single point in a large explosion

¢  (b) In the 1960s, Penzias and Wilson discovered a uniform cosmic background radiation (CMB) in the microwave region of the electromagnetic spectrum.

(i) Explain how the CMB is consistent with the Big Bang model. Solution ¢  temperature of the universe immediately after the Big Bang was very high;

as it expanded it cooled down; the wavelength of the CMB corresponds to a temperature consistent with this cooling down

¢  (ii) State why the red-shift of light from galaxies supports the Big Bang model.

¢  indicates that the universe is expanding;

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EXAMPLE

This question is about Olbers’ paradox. ¢  (a) Newton assumed that the universe is static and

that the stars are uniformly distributed. State one further assumption of the Newtonian universe.

Solution ¢  universe is infinite

¢  (b) Explain how Newton’s assumptions led to Olbers’ paradox.

Solution ¢  number of stars in shell increases as 𝑅↑2 

intensity decreases with 1/𝑅↑2  ;  hence brightness of shell is constant; adding all shells to infinity; sky would be uniformly bright;

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HUBBLE’S LAW (1929)

¢ Estimating the distance to galaxies, Hubble showed that the recession speed is proportional to the distance of galaxies to Earth

¢ Hubble’s Law 𝑣=   𝐻↓0 ×𝑑

¢  v = recession speed ¢  d = distance to galaxy ¢  𝐻↓0  = Hubble’s constant

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VALUE FOR 𝐻↓0  ¢ An accurate value for Hubble’s constant is

difficult to measure ¢ Units are important:

v = km 𝑠↑−1  d = Mpc 𝐻↓0 =𝑘𝑚𝑠↑−1 𝑀𝑝𝑐↑−1 

¢ From the gradient of the graph of Hubble’s data the original value for 𝐻↓0 = 500 𝑘𝑚𝑠↑−1 𝑀𝑝𝑐↑−1 

¢ Modern values state 𝐻↓0 = 70 𝑘𝑚𝑠↑−1 𝑀𝑝𝑐↑−1  Still contains uncertainty

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NEWEST VALUES FOR 𝐻↓0 

December 2012 ¢ NASA’s Wilkinson microwave

Anisotropy Probe (WMAP) gave a value of

𝐻↓0 = (69.32±0.80) 𝑘𝑚𝑠↑−1 𝑀𝑝𝑐↑−1 

March 2013 ¢ Planck mission provided a

value of 𝐻↓0 = (67.80±0.77) 𝑘𝑚𝑠↑−1 𝑀𝑝𝑐↑−1 

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EXAMPLE

¢ The value of the Hubble constant 𝐻↓0  is accepted to be in the range of 60 𝑘𝑚𝑠↑−1 𝑀𝑝𝑐↑−1  to 90 𝑘𝑚𝑠↑−1 𝑀𝑝𝑐↑−1 . A) state and explain why it is difficult to determine a precise value for 𝐻↓0 

Solution ¢ The further galaxies away, the

more difficult it is to determine the distance accurately

¢ You need a standard candle to accurately measure its luminosity

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EXAMPLE

¢ B) State one reason why it would be desirable to have a precise value of 𝐻↓0 

Solution ¢ We could get an accurate value of the rate of

expansion of the Universe ¢ We could get an accurate value of the distance to

the galaxy We could find an accurate value for

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LIMITATIONS OF HUBBLE’S LAW

¢ Standard candle ¢ Up to 60Mpc

¢  250Mpc < d < 900Mpc

¢  60Mpc < d < 250Mpc

¢  900Mpc < d < 1000Mpc

Cepheid Variable Red and blue super giants

Brightest globular clusters Supernova

Remember: as d  ↑  error in measurement ↑

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AGE OF THE UNIVERSE AND 𝐻↓0  ¢ Assume the Universe is expanding at constant

rate. Two galaxies are separated by a distance d. How long did it take to get this distance apart?

Solution ¢ Let the age or expansion time of the

Universe be T, the Hubble time will be 𝑇= 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒/𝑠𝑝𝑒𝑒𝑑 

¢ T = 𝑑/𝐻↓0 𝑑    ¢  𝑇= 1/𝐻↓0     ¢ This means the Universe has an age changing

depending on 𝐻↓0 

d

But v = 𝐻↓0 𝑑

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CALCULATING THE AGE OF THE UNIVERSE

¢ Assume 𝐻↓0 =70𝑘𝑚𝑠↑−1 𝑀𝑝𝑐↑−1 . ¢  𝑇= 1/70𝑘𝑚𝑠↑−1 𝑀𝑝𝑐↑−1  

¢  𝑇= 10↑6 𝑥9.46𝑥10↑15 𝑥3.26/70𝑥10↑3  

¢  𝑇= 3.1𝑥10↑22 /70𝑥10↑3   = 4.4  𝑥10↑17 𝑠                                                                    ~1.4𝑥10↑10 𝑦𝑒𝑎𝑟𝑠  

1ly=9.46x10↑15 𝑚 1pc = 3.26ly

14 billion years

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ASSUMPTION

¢ Galaxies and Earth are moving at constant speed of light c WRT each other

¢ There is nothing in their way to slow them down

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HUBBLE TIME

¢ Assume à galaxies have traveled at constant speed

¢  If rate of expansion is slowing down, actual age of the Universe would be less than the Hubble time

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THE ACCELERATING UNIVERSE ¢ To trace expansion history of the Universe, a

supernova needs to be used as standard candle ¢ A 1990 supernova was used to find distance for

about 1000Mpc ¢  It was found that at redshifts of 0.5 the

supernova was fainter than expected for a constant expansion

¢ Conclusion: it must accelerate as it moves further away

¢ Energy density that acts in opposition to gravity causes expansion to accelerate

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BUT…..

¢ ….Supernovae at redshifts between 0.5 and 1 were decelerating

¢ Whether Universe will expand forever or will collapse is something that cosmologists are trying to answer in terms of dark energy

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REDSHIFT: LENGTHENING OF WAVELENGTH

¢ Relative motion of source and observer – Doppler effect

¢ Relativistic time dilation

¢ A photon losing energy as it climbs out a gravitational field

What can cause redshift?

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COSMOLOGICAL REDSHIFT Z

Only used for redshift caused by overall expansion of space ¢  𝑧= λ↓𝑜𝑏 − λ↓𝑟𝑒𝑠𝑡 /λ↓𝑟𝑒𝑠𝑡   ¢  𝑧= ∆λ/λ↓0  ~ 𝑣/𝑐  v= velocity of source c = speed of light

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EXAMPLE

¢ The wavelength of the blue line in the spectrum of atomic hydrogen as measured in the lab on Earth is 486nm. In the spectrum from a distant galaxy the same blue line measures 498nm. Estimate the recession speed of the galaxy from Earth

Solution ¢  ∆λ=498  −486=12  𝑛𝑚 ¢  ∆λ/λ↓0  = 𝑣/𝑐  ¢  𝑣= ∆λ/λ↓0  ×𝑐= 12/486 ×3.0× 10↑8 =7407407𝑚𝑠↑−1  ¢  𝑑= 𝑣/𝐻↓0   ¢  𝑑= 7407𝑘𝑚𝑠↑−1 /65 =113.95~114𝑀𝑝𝑐

Assume 𝐻↓0 =65𝑘𝑚𝑠↑−1 𝑀𝑝𝑐↑−1 

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THE COSMIC SCALE FACTOR R

¢ As the Universe expands, all distances are stretched by the cosmic scale factor R

¢  If radiation was λ↓0  when emitted, but λ  when  it  was  detected,  the  cosmic  scale  factor  would  have  changed  from   𝑅↓0  to R

¢ Space has stretched by an amount ∆𝑅   in the same time λ  has  stretched  by  ∆λ

¢ Hubble’s law holds because space is expanding, not galaxies receding

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SCALE FACTOR R

¢  𝑧= ∆λ/λ = ∆𝑅/𝑅 = 𝑅− 𝑅↓0 /𝑅↓0  = 𝑅/𝑅↓0  −1 ¢  𝑧= 𝑅/𝑅↓0  −1 ¢ R = present scale factor ¢  𝑅↓0  = scale factor when radiation was emitted ¢  𝑅/𝑅↓0   = factor by which Universe has expanded

since time presently observable radiation was emitted

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EXAMPLE

¢ For a Quasar of redshift 4, we are seeing it as it was when the Universe was how big?

¢ Solution ¢  𝑧+1=𝑅/𝑅↓0   ¢  1/𝑧+1 =   𝑅↓0 /𝑅  ¢  𝑅↓0 = 1/4+1 𝑅=   1/5 𝑅

The Universe was 1/5  of its present value

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EXAMPLE ¢  A distant Quasar is detected to have a redshift value z of 5.6

a) find the speed of the Quasar WRT Earth b) estimate the ratio of its current size to the Universe’s size when the Quasar emitted the photons that were detected

Solution ¢  a) 𝑧= ∆λ/λ↓0  = 𝑣/𝑐 =5.6

v = 5.6c ¢  b) 𝑅/𝑅↓0  −1=5.6 ¢  𝑅/𝑅↓0  =5.6+1=6.6 ¢  𝑅↓0 /𝑅 = 1/6.6  ¢  𝑅↓0 =0.15𝑅

The Universe

was 15% of its current

size

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