Derivation of Friedmann Theory by means of Classical physics
The relevance of cosmology
On account of technology deficiency specialists were unable to inspect far space in detail before XX century. By the end of the XIX century telescope had been already invented. Due to its small accuracy the appliance could recognize form, color and trajectory of the nearest astronomical bodies: planets, coments, Moon, Sun and so on. To see things distintcly off the solar system was impossible. Nevertheless, scientists didn’t watch the world go by (figuratively). Bit by bit they had created a theory that became the first ground of rational worldview. It was called Classical physics. It was founded by Johannes Kepler, Galileo Galilei and Isaac Newton in the first half of XVII century. Universe was shown as a set of massive material bodies and actions on them in accordance with Law of universal gravitation. But still scientists were beset by the questions because a large range of things was unexplained. The next step of worldview development was Classical electrodynamics founded by James Maxwell, Hans Christian Oersted, Jean-Baptiste Biot, Felix Savart, Michael Faraday, Simeon Denis Poisson and Andre-Marie Ampere during the XIX century. Laws of charge interference modified once viewpoint: electrostatic and magnetic forces added to the gravitational. Their theory landed a hand to predict existence of electromagnetic waves that spread with a speed of light (that is electromagnetic wave). Besetting temptation of Albert Michelson and Edward Morley to check the last thing leaded to the emergence of a brand new hipotesa. By means of Michelson-Morley experiment the ideas about all-encompassing ether were broken. Hendrik Lorentz decided to look through the result of experiment outcome: the speed of light was the same at every frame of reference. Thus he derived the transformation of the physical parameters in order to clarify the outcome, – Lorentz transformation. It was a fundament of Einstein’s special theory of relativity. That turned to be a moment of a revolutional step from classical physics to Einstein’s relativistic mechanics. The accuracy of cosmological specification and predictions become better. At the same period of time the other part of physics had been developing, thermodynamics. Sadi Carnot, James Joule, Willard Gibbs and Ludwig Boltzmann were resposible for it during ХIХ-XX centuries. Rich knowledge about energy transformation appeard to be useful for theories about Universe. Afterall, the time of two crucial theories began: Quantum mechanics and General theory of relativity. We should pay tribute to the electrical science. It gives us an opportunity to look into the most appealing scene from Earth – the space.
Nowadays telescopes can recognize a particle of sand moving round the Sun. They work with all diapasons of waves. Detectors are able to trace all kinds of radiation, nuclear fragments and neutrino.
In 2016 LIGO detectors cought gravitational waves. Modern technologies let us strike up an acquaintance with unordinary astronomical events and bodies: dark holes, neutron stars, giant stars, far galaxies, clusters and superclusters of galaxies. Today is the best time to discover cosmos. It is a clue for further development of physics by two reasons: it is no longer necessary to make experiments on Earth and there are always new discoveries on the way of research.
Universe in Newton’s physics
Let there be a sphere with a center in point O. If the sphere is filled with mass isotropically in all directions, gravitational force, that would influence point A outside of the sphere, is equal to the force that would enfluence point A as if all mass is concentrated in point O.
Consequence: if point A is located inside the sphere on the distance of R2 from O, then gravitational force that would influence point A is equal to the force that would influence A as if all mass is concentrated in point O.
The law of The Universe expansion (Hubble’s law)
Let’s make an observation of The Euclidean space:
- It is isotropically filled with galaxies (white dots)
- Let`s set galaxy A as a beginning of the reference system and write down X-axis
- Mark galaxy B at X1 point
Then the distance between A and B is L=x*a(t).
α – the cosmological scale factor that depends on time. It is a parameter of world expansion.
By means of formula for L the speeds of galaxies is found as the derivative of L: v=L`=x*a`(t), where x does not change with time ( i.e, constant).
Edwin Hubble proven the relation between Redshift of far galaxies and their speeds in 1929.
|The derivative of scale factor with time|
|Speed of galaxies in x distance|
The relation between Light parameters and space expansion
Photon is one of the most abundant particle in the Universe. Current researches show that there are ten photons per one proton. Meaning that the corpuscles of light outnumber corpuscles of matter. The main properties of an electromagnetic wave are Energy (E), frequency (v), wavelength (λ) and the speed of light (c).
E=v*h, где h – постоянная Планка; v=c/λ;
Let’s write the wave of light in space: By the Hubble’s law the wavelength changes with time like that:
Then frequency v ∝ 1/a. So far as the energy is straight proportional to frequency, E ∝ 1/a. That is why the energy of each quantum of light decreases with time.
Cosmological model by Newton’s physics
Let the mass of galaxy B equals m. Let’s use the Newton’s theorem and mark that the mass of matter in radius X1 equals M. Thus we can think that point O has all the mass M. Write down the law of conservation of energy for closed system (i.e., Universe).
The first member is kinetic energy, the second is potentional.
Multiply both sides of the equation by 2:
Two times constant is some other constant, therefore const*2=const;
Divide both sides of the equation by m (as m is a constant, const/m=const):
Let’s substitute L and v taken from Hubble’s law to the equation:
Then rewrite M as ρ*V (density * volume); density is the same everywhere in our world model, volume is found like:
Eventually, the equation transforms to:
Density and α depend on time and do not depend on position. (4п/3 * G) – constant. x and, consenquently, x^2 are unic and they depend on position. The left side of the equation is proportional to x^2, so the right side should be proportional to x^2 too. Thus we can simplify the equation:
Now there is no relation of A,B positions and chosen O (reference frame). The only thing it depends on is α that changes with time.
Let’s transform the equation:
- Transfer the second member from the left side to the right
- Divide both sides by a^2
- Constant divided by a^2 should be overwritten with k
Density is in relation with time because space expands with as it flows. Imagine the region with volume that equals the unit of distance measurement in third power. Sizes change in relation with cosmological scale factor, therefore volume equals (1 length unit * a)^3. Rewrite density as a constant mass of a region divided by the volume: ρ= M/(1*a^3 ) (units of measurement are neglected). Finally, the equation looks like:
k may be both positive and negative value, minus (before k) is just a historical tradition.
Solution of Friedmann–Robertson–Walker equation
Equation works if the abundant part of matter-energy is in a state of matter (not radiation: photons, neutrino, any particles that move almost with a speed of light). Otherwise mass of a nominal region M would be able to change. For instance, photons always move with a speed of light because their rest masses are zero. Also all material objects should be preserved in nominal regions in order to keep M constant. That is why the previous equation works during Matter-dominated era. The set of solutions depends on values of unknoun parameters. The simplest option is k=0:
α may depend on time differently: in inverse ratio, in direct ratio, exponentially and so on. Let’s suppose that dependency:
α = c*tp, тогда a’=c*p*t(p-1)
Let’s make the substitution:
We can see that the power of time is two on the left side and 3p on the right side. Therefore 2=3р, р=2/3. Then a=ct^(2/3). Substitute p to the equation:
That nonlinear dependence means that space expands with acceleration. Mark that scale factor equals zero at zero point of time when density of matter was infinite.
Let’s rewrite Hubble’s constant formula:
Let’s suppose that border surfaces of a unit (with mass M) are perfectly elastic. Then the pressure, that exerts by mass M on surfaces, equals zero, because mass does not move through units.
The abundant part of matter-energy was in a state of high energy radiation before Matter-dominated era. It was clear by the evidence of orbital observations of WMAP – Wilkinson Microwave Anisotropy Probe. The spacecraft was detecting microwave light from all directions. This data was explained by the fact that the energy of high frequency radiation (born during eairly period of Universe evolution or after The Big Bang) was decreasing because of the influence of scale factor. On a small scale these microwaves isotropically filled the space. They were called Cosmic microwave background aka relic radiation. This period was named Radiation-dominated era. We has noticed the region with volume Х^3 and mass M. Now, let’s imagine that units (regions) are filled with radiation. Then the streams of light will be moving througn unit but the average amount of photons in unit is constant. As long as the wavelength is increasing with time, the average energy of photons is not a constant, it is some constant divided by scale factor: E ∝ d/a. Let’s find density of photon energy dividing the frevious equation by the volume of a unit (Х*a(t))^3, where X is supposed to be equal 1: E/V ∝ d/(a^4). This means that the density of photon matter is proportional to 1/(a^4), not 1/(a^3). Let’s change Fridman’s equation to match new conditions: 2=4*р. Then р=1/2:
Check the case when light move by X-axis in its unit, that has perfectly elastic borders. The force, that acts upon borders, equals: F=dP/dt, multiply the numerator and denominator by the speed of light: F=(dP*c)/(dt*c)=E/L. Divide both sides by the contact area of light with border: F/S=E/(L*S) => P=E/V, where E/V let’s call a density of enegry and denoteρE. Actually light is moving through all axes (X, Y, Z). Therefore general pressure on unit is: P2= ρE/3.
Dark energy-dominated era
Imagine a one-dimensional world with Newton’s law of universal gravitation and Hubble’s law. It isotropically filled with matter in the form of galaxies, clusters, stars and so on. Let us depict it as follows: All Matter in Universe is colored yellow. A, B – galaxies. Length L between A and B are increasing by Hubble’s law. Accordingly, the density of the substance will decrease. Let’s look at empty space between A and B – it is expanding.
Meaning that there are abstract objects, the space made of, appearing continuously. Their density is the same in any point of the Universe and it does not change with time. This effect is called vacuum energy or dark energy.
|α (scale factor)||P (pressure)|
Try to find out the dependence of P on a variable w: P=w*ρE
|dependence on α||P (pressure)||w (some variable )|
|ρ ∝ 1/(a^3)||0||0|
|ρ ∝ 1/a^4||ρE/3||1/3|
Firstly, let’s define w for energy density that does not change with time. Rewrite equation where the enegry of a unit is decreasing while volume is increasing: dE=-PdV. If E=ρV (ρ-energy density), then ρdV+Vdρ= > ρdV+Vdρ=-PdV = > Vdρ=-(P+ρ)dV. So far as P=w*ρE, Vdρ=-(w+1)ρdV. Transform the equation: dρ/ρ=-(w+1)dV/V. dIn(f)=df/f – logarithmic derivative. Then: dIn(ρ)=-(w+1)*dIn(V)= >In(ρ)=-(w+1)*In(V)+C=> => In(ρ)=In(1/V^(w+1) ) +C=> ρ ∝ 1/V^(w+1), V=X^3 * a^3 = 1*a^3 => ρ ∝ 1/a^(3(w+1)). If ρ is constant, w=-1 (a0 =1); P=-ρE. What means that energy density is negative value because pressure is positive on account of world expanding. Rewrite Fridman’s equation:
8п*ρ*G = Λ, где Λ – cosmological constant; H= √(Λ/3)
a`=a√(Λ/3). Let’s find intergral of a’. The only function does not change after differentiation is exponent: a ∝ exp(Ht). Period of Universe evolution with exponential dependence of scale factor from time is called Dark energy-dominated era.
Every one of third equations makes sence in any period of universe evolution. But practical influence will be different for each period. We has found out that there are tree things that affect on world expansion: relatively static matter, radiation and dark matter. In the beginning density of radiation energy had the highest value among others. That is why a ∝ t^(1/2) had the most efficient influence. Over time wavelengths of radiation were decreasing by Hubble’s law (energy and density of energy too). Then matter energy density became the highest. After matter pieces being moved from each other its density was comparable with vacuum energy density and the most important dependence was a ∝ exp(Ht).
- Leonard Susskind – All Stanford physics lectures
- Больцман. Термодинамика и энтропия.
- Steven Weinberg. Cosmology.