More on time, look back and otherwise t in R(t) starts from beginning of big bang Everywhere in universe starts “aging” simultaneously Observational consequences =>
An object at “high” redshift = 1+z = 1.5 (for example) R(t 0 )/R(t) = 1.5 where t = age of universe when the light we are just seeing now left the object If this is galaxy, and we assume all galaxies formed at the same t in the universe, then The galaxy we observe at 1+z =1.5 is younger than ours (we are at t = t 0 ) and must be younger than other galaxies formed at the same time but seen at lower 1+z This assumes an expanding universe and a finite speed for light
General consequence: observing to higher and higher 1+z is like time travel to a younger and younger part of the universe. Look back to 1+z =1000 see universe just beginning to form “clumps” which will evolve into galaxies and clusters of galaxies (see slide 6 for age versus 1+z) As universe ages and expands, the part of the brick wall we were observing now has aged and becomes transparent => the material just behind it becomes the new brick wall!
We see more and more of Universe as time goes on until acceleration makes effective redshift too high for us to see the light The part behind the brick wall in this model was younger. This is why when it ages, it evolves to the t of the brick wall and becomes the new brick wall. Since time itself is moving forward, R(t 0 ) is now larger and so the 1+z we observe the brick wall at is less and the universe is cooling off!
Behind brick wall brick wall Can’t see this because behind the brick wall Can see OK, from brick wall Think of needing to lead a moving target to hit it with a gun
Another Example: Assume we have 3 planets in row separated by 25 light years each: A 25 lt yrs B 25 lt yrs C Assume A and C can watch a patch of ground continuosly B sends out a signal at the speed of light to plant an acorn. Somebody at both of these patches has an acorn the signal. Assume C is Earth and the signal is received in 1908
Assume also people on both A and C plant the acorn when they receive the message In 1908 we will still see on A bare ground because it takes it takes 50 years for light to get here, so we won’t see the acorn planting until 1958. Now you, are born circa 1981-1984 when you see the tree on A to be 23-26 years old while your tree is 76-73 years old! => The tree on A looks younger. => Look at high 1+z, look far away, see younger (on average) galaxies (assumes galaxies aren’t forming now
Now the “standard fare” Einstein’s model of the universe geometry is defined by mass
Prologue: => Tests! Assume General Relativity works
Prologue (cont): Deflection predicted for star light as it passes close to the sun with “ordinary” physics gives wrong answer by factor of 2 [Equate energy to mass and calculate orbit using Newton’s law of gravity gives wrong answer] GR gets it right
Prologue (cont): No deflection if no gravity Deflection due to gravity, angle twice as large due to GR
Prologue (cont): Use a solar eclipse! Very first try may have “fudged’ results, but truly verified and it does give the factor of two larger effect predicted by GR.
Prologue (cont): Another effect: The motion of Mercury about the sun is affected by GR in a noticeable way because Mercury is so close to the sun. The effect is called the “precession of the perihelion.” The direction of its axis changes at the rate of 43 ’’ arc seconds)/century Mercury Sun
Prologue (cont): => GR seems pretty good Cosmological Principle: The Universe is homogenous and isotropic, i.e. everywhere is the universe is the same as everywhere else and all directions are the same Move on:
Distance measure The path of light (a geodesic = the most direct = shortest) Define a “proper time” to travel between two points is calculated from d = (cdt) 2 (dx 2 + dy 2 + dz 2 ) = 0 for light Never mind the books comments of the LONGEST path
Roberston-Walker Metric Use d = 0 and space-time curvature and get: (cdt) 2 = R 2 (t)(dr 2 /(1-kr 2 ) R(t) is the scale factor of universe If we take r = 0 to be here, then r not = 0 is the coordinate of a point relative to us and R(t)r is the distance (not corrected for rapid [ near c = 0.1 c] velocity of recession
If know R(t) as function of t and can calculate distance. The time we use is the time at R since the big bang see slide 6 for relationship between 1+z (R(to)/R(t)) and t
Comments on math and notation “d” in front of symbol indicates that this is a small (enough) change in that variable that if something is changing as variable (or thing), the change is negligible over that small change. Also we some times write this with a “dot” above the variable instead as in R
Comments on notation continued Example: velocity = ds/dt = change in distance with respect to time or v = s nd if there is no acceleration, then v = s/t or vt = s or velocity x time = distance Expanding surface, take into account the geometry of the surface and the time!
Co-moving coordinates and Redshift R(10 43 sec)r = 10 43 x c at the beginning R(today)r= 13 billion years x c = 1.2 10 28 cm
Co-moving coordinates and Redshift So, R(to)r/R(10 43 sec)r = 15x10 9 x 3.17x 10 7 /10 43 to get all in seconds, and remember the co-moving r’s cancel out = about 5 x 10 60
k is the curvature that we’ve used before: R(t) is the scale factor for this surface. little r only tells us relative distances R(t) sets the scale we see something “bad” happens when k is equal to and r = 1! => Key features and concepts:
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