Homework Answers
The Friedman equation is given by
If we define the Hubble parameter
Then, the Friedman equation is given by
Now for the present day flat space geometry of the universe,
And so, critical density of the universe is given by
Now, for the present day, we use the present day value of the
Hubble parameter
So, the present day critical density is
So, putting the present day values,
we get
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The Friedmann equation describes the evolution in the scale factor a of our universe. () =...
3. The Cosmological constant When formulating general relativity, Einstein believed that the Universe was static. In or...
3. The Cosmological constant When formulating general relativity, Einstein believed that the Universe was static. In order to arrange a static Universe, he proposed a change to the equations, something he would later famously call his "greatest blunder". That was the introduction of a cosmological constant, A, to the Friedmann equation (1) (a) (3 points) By referring to the Friedmann equation 1), derive an equation for a static Universe when k -0 (b) (8 points) Last week you derived an...
2. The Friedmann equation 2 0 2 Cl tells us how the reduced scale factor ä changes with time. Fro...
If. Good Handwriting!! Thank You
2. The Friedmann equation 2 0 2 Cl tells us how the reduced scale factor ä changes with time. From astro- nomical observations we know we are in a flat, matter-Lemaitre universe where S2c-0, ΩΤη-0.3089. Or-5 x 10-5( 0) and Ωυ-0.6911. a) Using the 'second form' of the Friedmann equation show, for the current epoch, äo > 0. Thus, the expansion of the universe is accelerating (b) Is there any future time (any possible à...
In a flat Universe that contains only a fluid with equation of state P = wpc,...
In a flat Universe that contains only a fluid with equation of state P = wpc, where w = -1/6, the evolution of the scale factor, R, with cosmic time, t, is given by R(t) = R, (1)", where R = R(to), and to is the present time: to = 3H, t = 0 corresponding to the Big Bang. The Hubble constant is H. = 67 kms-'Mpc-!. In this model of the Universe: 1. How does the Hubble parameter scale...
2. The acceleration equation We have derived in lectures the Friedmann and fluid equations, that describe...
2. The acceleration equation We have derived in lectures the Friedmann and fluid equations, that describe the evolution of simple cosmologies. The Friedmann equation is usually given (adopting c 1) as where a(t) is the scale factor of the universe, p is the mass (energy) density, and k is constant. (a) (5 points) The simplest (although spectacularly uninteresting) cosmology has zero mass-energy, i.e. ρ 0. Solve the Friedmann equation under this condition to obtain an expression for at(t). Describe the...
Newtonian Cosmology 1. In class, we solved the Friedmann equation for the critical case, where the...
Newtonian Cosmology 1. In class, we solved the Friedmann equation for the critical case, where the constant of integration was set to k 0; this resulted in the Einstein-de Sitter model, where a ox t2/3 Now, let us consider the closed case (k 1), where the universe starts with a Big Bang, reaches a maximum expansion, turns around, and eventually ends in a Big Crunch. For the closed model, it is convenient to write the Friedmann equation as follows: 8T...
The Hubble Constant, the density of the universe, and the cosmological constant are all vital to the evolution of our universe. Briefly discuss why each is so important -and- what would happen if we t...
The Hubble Constant, the density of the universe, and the cosmological constant are all vital to the evolution of our universe. Briefly discuss why each is so important -and- what would happen if we tweaked each one up or down a little. Are there other constants that are equally important? If so, which and why?
Why are the Hubble Constant, the Cosmological Constant, and the density parameter so important to cosmology and to you? Consider expansion rate and age. What would be different about our universe if t...
Why are the Hubble Constant, the Cosmological Constant, and the density parameter so important to cosmology and to you? Consider expansion rate and age. What would be different about our universe if the Hubble Constant were doubled from its current value? How do these constants factor into the geometry of spacetime? How have these constants changed throughout the lifetime of the universe? Are they still changing? Cite some evidence to support your claims.
Why are the Hubble Constant, the Cosmological Constant, and the density parameter so important to cosmology and to you? Consider expansion rate and age. What would be different about our universe if t...
Why are the Hubble Constant, the Cosmological Constant, and the density parameter so important to cosmology and to you? Consider expansion rate and age. What would be different about our universe if the Hubble Constant were doubled from its current value? How do these constants factor into the geometry of spacetime? How have these constants changed throughout the lifetime of the universe? Are they still changing? Cite some evidence to support your claims. Please type answer, last persons handwriting was...
2. Imagine a type of "phantom energy" with w<-1. (a) Write down the Friedmann equation for...
2. Imagine a type of "phantom energy" with w<-1. (a) Write down the Friedmann equation for a flat universe containing matter and phantom energy 1/(3 ) (b) Show that the matter-phantom equality scale factor is amp = (1- 0) (c) In the phantom energy dominated epoch show that a → at a finite time trip-to = 200-311+wl This is called a "big rip". Hint: integrate over the time range from to to trip. (d) Current data is consistent with w...
Take h = 0.7. Orad,0 = 0.0m,0-0.3, and ΩΛ,0 0.7. (a) Use the formula for how...
Take h = 0.7. Orad,0 = 0.0m,0-0.3, and ΩΛ,0 0.7. (a) Use the formula for how the Hubble parameter H(t)-à/a changes with scale factor and express it as a function of redshift. b) How fast was the universe expanding at z-0.1, z 0.4, z 0.7, z -2, 2-5, and z = 1100? (in units of km/s/Mpc)? (c) Suppose instead that there is the same amount of matter but no dark energy (i.e. no cosmological constant). What would the expansion rates...
3. The Cosmological constant When formulating general relativity, Einstein believed that the Universe was static. In order to arrange a static Universe, he proposed a change to the equations, something he would later famously call his "greatest blunder". That was the introduction of a cosmological constant, A, to the Friedmann equation (1) (a) (3 points) By referring to the Friedmann equation 1), derive an equation for a static Universe when k -0 (b) (8 points) Last week you derived an...
If. Good Handwriting!! Thank You
2. The Friedmann equation 2 0 2 Cl tells us how the reduced scale factor ä changes with time. From astro- nomical observations we know we are in a flat, matter-Lemaitre universe where S2c-0, ΩΤη-0.3089. Or-5 x 10-5( 0) and Ωυ-0.6911. a) Using the 'second form' of the Friedmann equation show, for the current epoch, äo > 0. Thus, the expansion of the universe is accelerating (b) Is there any future time (any possible à...
In a flat Universe that contains only a fluid with equation of state P = wpc, where w = -1/6, the evolution of the scale factor, R, with cosmic time, t, is given by R(t) = R, (1)", where R = R(to), and to is the present time: to = 3H, t = 0 corresponding to the Big Bang. The Hubble constant is H. = 67 kms-'Mpc-!. In this model of the Universe: 1. How does the Hubble parameter scale...
2. The acceleration equation We have derived in lectures the Friedmann and fluid equations, that describe the evolution of simple cosmologies. The Friedmann equation is usually given (adopting c 1) as where a(t) is the scale factor of the universe, p is the mass (energy) density, and k is constant. (a) (5 points) The simplest (although spectacularly uninteresting) cosmology has zero mass-energy, i.e. ρ 0. Solve the Friedmann equation under this condition to obtain an expression for at(t). Describe the...
Newtonian Cosmology 1. In class, we solved the Friedmann equation for the critical case, where the constant of integration was set to k 0; this resulted in the Einstein-de Sitter model, where a ox t2/3 Now, let us consider the closed case (k 1), where the universe starts with a Big Bang, reaches a maximum expansion, turns around, and eventually ends in a Big Crunch. For the closed model, it is convenient to write the Friedmann equation as follows: 8T...
2. Imagine a type of "phantom energy" with w<-1. (a) Write down the Friedmann equation for a flat universe containing matter and phantom energy 1/(3 ) (b) Show that the matter-phantom equality scale factor is amp = (1- 0) (c) In the phantom energy dominated epoch show that a → at a finite time trip-to = 200-311+wl This is called a "big rip". Hint: integrate over the time range from to to trip. (d) Current data is consistent with w...
Take h = 0.7. Orad,0 = 0.0m,0-0.3, and ΩΛ,0 0.7. (a) Use the formula for how the Hubble parameter H(t)-à/a changes with scale factor and express it as a function of redshift. b) How fast was the universe expanding at z-0.1, z 0.4, z 0.7, z -2, 2-5, and z = 1100? (in units of km/s/Mpc)? (c) Suppose instead that there is the same amount of matter but no dark energy (i.e. no cosmological constant). What would the expansion rates...
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