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------------------------- -------- 2334 - The Expanding Universe
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- This energy of empty space is causing the Universe to expand at an ever increasing rate. As Edwin Hubble confirmed the farther the galaxies are away from us the faster they are receding. That is, the expansion is accelerating.
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- It is a tug of war in the Universe with gravity trying to slow mass from expanding into the cosmos and the dark energy, or the “cosmological constant” greater than zero, trying to expand it more rapidly. The acceleration of expansion is winning and the effects of gravity are growing weaker as the Universe continues its journey into the unknown.
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- Einstein said that gravity can best be understood as the bending of space. The cosmological constant is simply a constant of integration that comes out in Einstein’s integration formulas.
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- The problem is you need to have some known conditions in order to solve for the integration constant. We do not have any known conditions at this point. It could be zero, or less than one, or greater than one. We have not figured this out yet.
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- It is interesting to at least take a look at Einstein’s equations even though I will not pretend to understand them.
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- It starts with R = a function of time. R is the distance, or the scale of the Universe. The first derivative of distance with respect to time is velocity of the Universe. The second derivative is acceleration.
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---------------------- The second derivative of R = 4*pi*G*R*(q+3*p/c^2)/3 + 1/3*Cc*R
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---------------------- Where G is the Gravitational Constant
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---------------------- “q” is the inertial mass density of matter
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---------------------- “p” is the pressure
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----------------- Cc is the cosmological constant, or the “ repulsive effect of the vacuum”
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---------------------- The first derivative of distance is velocity.
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----------- (The first derivative of R)^2 = 8*pi*G*q*R^2/3 - c^2/Y^2 + 1/3*Cc*R^2
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- Where Y is the radius of the curvature of the geometry of the Universe
c^2/Y^2 becomes a constant of integration. Integration of acceleration is the first derivative and is again velocity, or how fast the Universe is expanding.
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- All the cosmological formulas can be derived from these two equations, simply making different assumptions for the various parameters gets different results.
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- However, the redshift of galaxies and supernovae are indicating that the cosmological constant must be greater than one. The redshift determines the relative recession velocity along the line of sight between the observer and the light source. If all the wavelengths are 1% longer that usual, and all the frequencies are 1% lower than usual, then the source is moving away from us at 1% the speed of light. The shift to longer wavelengths is toward the red end of the spectrum, therefore a redshift. If the galaxies were moving toward us it would be a blueshift.
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- Cosmological constant greater than one means the ratio of vacuum energy, converted to mass, and the critical density is > 1. The critical density is the exact average density needed in the Universe in order to bring the expansion to a complete stop.
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- The Universe would be in a static state, neither expanding nor contracting. If the average density of mass > 1 then gravity causes the Universe to collapse. If the average density of cosmological constant > 1 then the Universe expands forever.
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- Edwin Hubble determined that the galaxies recession velocity increased in proportion to their distances from us. The constant of proportionality is know as Hubble’s constant, Ho.
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------------------------------- Velocity = Distance * Ho
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------------------ Ho = 72 km/sec/megaparsec ( one estimate astronomers are using )
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- Ho is the slope of the straight line when you plot distance versus velocity for galaxies at various distances out in space.
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- Where the straight line intersects the time of zero time and zero distance that must be the start of the Big Bang. Therefore, 1/Ho must be equal to the age of the Universe. If Ho = 72 km/sec/megaparsec, then for every megaparsec greater distance the galaxies are, they are traveling 72 km/sec faster.
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- We can calculate the age of the Universe by the reciprocal of Ho. 1/Ho = 1/72 m/sec/mps * 3.09*10^19 km/mps / 3.16*10^7 seconds/year = 13.6 billion years.
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- If the Universe is flat, the normalized cosmological constant approximately = zero, then the age of the receding galaxy can be calculated from its redshift, z.
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-------------------------- z = change in wavelength / expected wavelength
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-------------------------- % of Universe Age = 1/ (1+z)^ 3/2
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- If the redshift is close to 1, approaching the speed of light then the velocity is calculated with:
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------------ Galaxy’s receding velocity / speed of light = proportional to the redshift:
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------------------------------ V / c = z^2 + 2z / z^2 + 2z +2
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-------------------- A galaxy has been located with a redshift of 5.6. z = 5.6.
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- From these formulas its age is 5.9% the age of the Universe or 900,000 years old, since the Big Bang. It is traveling at a recession velocity of 95.5% the speed of light. This puts the distance of that galaxy at 12.97 billion lightyears, assuming a Hubble constant, Ho = 72 meters / second / megaparsec.
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----------------------- % of Universe Age = 1/ (1+z)^ 3/2
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----------------------- z+ 5.6 Age = 1/(6.6)^3/2 = 1/16.96 = 5.9%
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- If the Universe is 15 billion years old then this galaxy is 885,000 years old.
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---------------------- V / c = z^2 + 2z / z^2 + 2z +2
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---------------------- V / c = 5.6^2 + 2*5.6 / 5.6^2 + 2*5.6z +2
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---------------------- V/c = 42.56/44.46 = .955
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- This galaxy is has a receding velocity away from us at 95.5 % the speed of light.
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--------------------- Hubble’s constant = receding velocity / distance
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--------------------- Distance = velocity / Ho
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--------------------- Distance = .955 * 2.998 10^5 km/sec / 72 km/(sec * megaparsec)
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-------------------- Distance = .0398 10^5 * ( megaparsec) = .0398 10^5 * 3.262 *10^6 lightyears = .13 * 10^11
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------------------- Distance = 13 billion lightyears away.
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- If the value of the density of the Universe is less than the critical density then the Universe is open, negatively curved, and will expand forever.
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- The value for the critical density = proportional to Hubble’s constant( expanding ) and inversely proportional to the Gravitational Constant ( contracting):
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-------------------- The value for the critical density = 3 * Ho^2 / 8*pi*G
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------------------- Critical density = 9.75 * 10^-24 grams / cubic meter.
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- If the actual density is less than this the Universe will expand forever. Visible matter that we can see with the most powerful telescopes does not account for even 1% of this critical density. However, gravity’s behavior tells us the matter is there; it emits no electromagnetic radiation, so it is dark matter. It is believed that 95% of the Universe is made up of Dark Matter.
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- At 10^-33 seconds after the Big Bang rapid Inflation caused the volume of 3*10^23 centimeters to expand, inflate, doubled in size, 200 times, equal to a factor of 10^60 times in volume. This Inflation Era explains why the universe is homogeneous, the same everywhere, and why it is flat, with matter plus the energy in the vacuum of space, the cosmological constant, nearly equal to the critical density.
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- “Homogeneous” because the enormous expansion started with identical physical conditions throughout. “Flat” because it grew so large that it appears flat to us, just as the spherical Earth appears flat when we look at our horizon from our small vantage point.
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- If we normalize, divide the density of matter in the Universe by the critical density, Wm = 0.3 And, we divide the cosmological constant, the energy in the vacuum of space, by the critical density, Wv = 0.7. For a flat Universe Wm + Wv = 1.
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- To convert the density of energy to the density of matter, divide by the speed of light squared ( 9*10^16 m^2/sec^2 ). Then, ratio of density of condensing matter plus the ratio of density of expanding energy = 1.
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- Wm + Wv = 1 This equation simply says the Universe’s curvature is flat because the density of mass plus the density of dark energy ( converted to mass by E=mc^2) is equal to the critical density needed to stop the Universe’s expansion.
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- If the sum is less than 1, the mass of the Universe is less than the critical density needed, then the curvature is negative and expansion can not be stopped, in fact it speeds up until the average density thins out into nothing.
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--------------------- The value for the critical density = 3 * Ho^2 / 8*pi*G
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--------------------- Ho = 72 km/sec/megaparsec
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--------------------- critical density = 15,552 km^2/(sec^2*mps^2) / 25.13 * 6.669 * 10^-11 m^3/(sec^2*kg)
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--------------------- critical density = 92.8 * 10^11 km^2 * kg / mps^2 * m ^3
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--------------------- mps^2 = (3.0857 *10^19)^2 = 9.53 *10^38 km^2
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--------------------- critical density = 92.8 * 10^11 km^2 * kg / 9.53 *10^38 km^2 * m^3
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--------------------- critical density = 9.75 * 10^-27 kg / m^3
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--------------------- critical density = 9.75 * 10^-24 grams / m^3
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- The mass of a proton is 1.2 * 10^-24 grams
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- Therefore the critical density needed to halt the Universe expansion is approximately eight protons per cubic meter. The best vacuums produced in the lab still has 1,000’s of protons per cubic meter.
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- These density numbers are not known, but only the best estimates with today’s discoveries and analysis. Better satellites to study the Cosmic Microwave Background Radiation and better telescopes to study the Supernova explosions billions of light years away, are making more accurate measurements of these constants so that we can better understand the birth, evolution, and fate of our Universe.
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- See Review 53 “ The Farthest Galaxy in Our Universe”
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- April 14, 2019 20 380
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-------------------------- Monday, April 15, 2019 --------------------------
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