Andrew,

Try following link and the references and other links given there in.

https://en.wikipedia.org/wiki/The_Big_Bang_Theory

Dear Virkam

I am asking a serious question while you point me to a joke. I find references under the link insufficient to resolve my problems so may be you know the email  dr Sheldon Cooper who may be able to point me to a right direction :)

I have no time to go through thousands of scattered articles discussing various aspects of BB. I am expecting there exist one consistent presentation which shows all the necessary equations, assumptions constraints methods of solution. How does continuous model resulting from general relativity of space cope with discontinuities caused by particles and clusters of particles.How can you solve unthinkable number of differential equations. How do you kow the temperature when there is no particles?  Do all these equation actually have solutions? Where is the book or article summarising the model?

“I want the full list of equations initial conditions constraints all in one as it should be in a presentation of a mathematical model from start to finish.”

I’m afraid you wouldn’t find any answer to this requirement for the present. Even more if you ask for “a succinct mathematical model” (in your summarized question).

A definite mathematical model of Big Bang theory would rest on a merging of Quantum Mechanics with General Relativity. In spite of big efforts in this direction, such a theory of “Quantum Gravitation” doesn’t exist (yet ?). Although there are interesting results and progress in several theoretical attempts like String Theory, Loop Quantum Gravity, etc.

The present state-of-art of the physical formulations of what could have happened in the early universe can be found in books like Weinberg’s “Cosmology” (as quoted above by Robert J. Low) or Viatcheslav Mukhanov’s “Physical Formulations of Cosmology” (especially Chapter 4 “The very early universe” pages 131 to 226).

If you want to start having a more qualitative view of the issue (with some equations anyway !), I suggest you following links :

1) The article of Wikipedia on the Big Bang (with plenty of references) :

https://en.wikipedia.org/wiki/Big_Bang

2) A synthetic video description (as in a nutshell) :

https://www.youtube.com/watch?v=wNDGgL73ihY

3) A series of 30 synthetic and clear video lessons given by Michel van Biezen : to start with the first lesson :

https://www.youtube.com/watch?v=x42fbMT9h8c

(the further 29 lessons can then be found there).

https://en.wikipedia.org/wiki/Big_Bang

https://www.youtube.com/watch?v=wNDGgL73ihY

https://www.youtube.com/watch?v=x42fbMT9h8c

Thank you for the response and putting my question in a perspective

I know the qualitative sources. I am after more mathematical systematic approach but in a condensed apprachable form. I think the source pointed by R.Low is a good start.

Please search for article "God before the Big Bang" that is located in research gate.

give me your comments, advice and critiques.  Looking forward to your reply

Respectfully yours

Changsoo Shin

p.s. my major is waveform inversion. 

Dear Changsoo

I am not who down votes for the opinions which are not in the mainstream however I find your approach highly speculative and built on arbitrary associations.

I generally do not see introduction of deity to science as science is exactly what deity is not ,even if it exists.

On the other hand I acknowledge that everything can be a subject of scientific inquiry as long as it is impartial and open for criticism.

Religion and science in history did not mix very well and they still do not as we see on many examples.

On the other hand I do not see any explanation for existence of something, even though as implied by Big Bang model everything extrapolates back to nothing. No science or logic can remove this mystery. The deity does not explain it either. It would require a deity to exist and be very persuasive to convince me otherwise.

Andrew, don't confuse cosmology with cosmogeny. Scientific theories are of course based on observation so for cosmology they must start in the present and work back to infer the state of the universe in earlier times. That process is limited by our knowledge so you won't find any accepted picture of the origin.

To answer your original question, the succinct equations would be those of Friedmann combined with the equation of state for the contents of the universe, radiation, matter and dark energy, as summarised in the link.

https://en.wikipedia.org/wiki/Equation_of_state_(cosmology)#FLRW_equations_and_the_equation_of_state

Andrew,

I think that you will have trouble finding one that is correct.  The theory is not so how could any model or theory that starts with it be any good?

I don't think mathematical modelling of a process is sufficient to verify the scientific nature of that process. Mathematics is certainly a science, but not necessarily it is a science of the nature. It is in actuality a science of the thought. The science of the nature, that we call "physics" is based certainly on mathematical modelling, that represents one of the two fundamentals of the scientific method. The second fundamentals is the observation and the verification of the natural reality through the experimental method. As per these considerations any theory of the Big Bang, that is the object of this discussion, can be considered a philosophic theory or a theory of the knowledge, even if it is modelled mathematically, but never ever it can be considered a scientific theory. In fact even if it could be possible to theorize a perfect system of equations in order to describe the theory of the Big Bang it not would be possible then to verify the exactness of that system of equations. The only thing that would be possible to do, and that at present it is done in Laboratories of Research, is to test to simulate the Big Bang. The simulation nevertheless isn' t able to guarantee the authenticity and the real validity of a theory. These are limits of science. If you are interested in a simple mathematical formulation of the Big Bang links that have been proposed in preceding comments can be useful, if you instead search for references that guarantee the scientific validity of the theory of the Big Bang I think they don't exist and never will exist.

      A fantastic book is:

      The three first minutes of the Universe: a modern view of the origen of the universe,  of Steven Weinberg

The latest version of the standard (mathematical) model of Big Bang which is generally accepted by majority of scientists is the LambdaCDM model. Lambda is the cosmological constant in general relativity and CDM stands for cold dark matter. 

Andrew,

If you know the mainstream tools in case of BB, do not waste the time… To find alternative resolution, is hard… I have one: philosophical essay in Hungarian (I can you send shorted explanation)… You have to concentrate to your resolution… it must be matter-energy function… To be original you have to have a totally new interpretation of gravity (I have an interpretation too present in research gate, lead to expanded understandings)… Another point an good equation will give an interpretation which will negate the BB…

As I observed from before made question (questions) that you have a good and great resolution!

Best Regards. LA.H    

Dear Daniel

Thanks for your answer. I already have the book which is indeed well written and provides some overview of the problem.

"The first three minutes" which I have is not the book I am after. That is what I call Genesis style. It is a collection of revealed truth, mixed with history of science decorated at the end with some equation magically popping out. This book is good to take things on faith not to resolve fundamental questions. It only starts from t=0.01 nd talks about infinite temperature. How could that be that a t=0 we have an infinite temperature which implies infinite energy and there is no particles yet to do the temperature. That is why I am looking for better presentations.

AW: How could that be that a t=0 we have an infinite temperature ...

Why do you imaigine we can say that the temperature was infinite? The "free lunch" hypothesis says energy was zero and one of the current problems at the cutting edge of cosmology is "reheating" or perhaps "pre-heating" of the universe towards the end of inflation so any value of temperature is speculative or at best wide bounds.

Dear George,

From the book of S.Weinberg:

"Unfortunately I cannot start the film at zero time and infinite temperature. Above a threshold temperature of fifteenhundred thousand million degrees Kelvin (1.5 X 1012° K), the universe would contain large numbers of the particles known as pi mesons, which weigh about one-seventh as much as a nuclear particle. (See table 1 on page 163). Unlike the electrons, positrons, muons and neutrinos, the pi mesons interact very strongly with each other and with nuclear particles -in fact, the continual exchange of pi mesons among nuclear particles is responsible for most of the attractive force which holds atomic nuclei together. The presence of large numbers of such strongly interacting particles makes it extraordinarily difficult to calculate the behaviour of matter at super-high temperatures, so to avoid such difficult mathematical problems I will start the story in this chapter at about one-hundredth of a second after the beginning, when the temperature had cooled to a mere hundred thousand million degrees Kelvin "

We see that the close to t=0 the hotter it gets an infinite temperature is stated at t=0

Andrew,

The first chapter of Mukhanov's book (see link) can also be seen as a succinct mathematical model of Big Bang theory.

http://www.astro.caltech.edu/~george/ay21/readings/Mukhanov_PhysFoundCosm.pdf

AW: From the book of S. Weinberg:

Note first that he published that in 1977, before COBE had made the first measurements of the CMB and before inflation had been thought of to explain the thermal uniformity observed by COBE.

AW: to avoid such difficult mathematical problems

As you quote, Weinberg there says that it was a "mathematical problem" caused by extrapolating classical theory into a quantum regime. This is still the boundary beyond which our theories can give no answers.

AW: We see that the close to t=0 the hotter it gets an infinite temperature is stated at t=0.

No, we see that Weinberg noted that the mathematics in 1977 could not address the conditions at t=0. That is still the case.

 Dear Daniele Sasso

I agree that mathematical modelling alone is not sufficient to fully explain a process without its proper physical semantics and experimental confirmation. Also the idea of its limits is important.

It is good people try to answer challenging question but when something is not complete it should not go to the public as revealed truth, because the revealed truth explains nothing.

I am involved in simulations for all my life and think I can always tell the difference between what it gives us to what things are.

Dear George,

The way an average reader understands Weinberg's sentence is that he " cannot start the film at zero time and infinite temperature" because it is too cumbersome to explain to an average reader due to mathematical problems. But infinite temperature is mentioned there for a reason, otherwise why would he mention it? I would say we cannot start the film from t=0 because we do not exactly know what happened.

I am trying to find out how one can claim the universe starting from nothing. What exactly is the equation what kind of initial condition at t=0 how it is solved and how this solution turns into partices and how many particles? I cannot believe one equation would do it all. How such solution reveals all physical laws that must operate after particles are formed. 

Thanks for this Wikipedia reference it is a step in the right direction.

AW: But infinite temperature is mentioned there for a reason, otherwise why would he mention it?

There is a simple relationship between time and temperature in the early universe when classical physics can be applied (i.e. not back to t=0) which is that the temperature generally falls as the inverse square root of the age. Extrapolating that back would give infinite temperature at t=0 but that's not valid. There are plateaus in the temperature where there are phase changes, the Hagedorn Temperature and nucleogenesis for example, but the general slope is the inverse root.

I've added some links (rather old) with basic introductions to recent ideas. From the 1998 paper:

• In modern versions of inflationary cosmology there is no pre-inflationary hot stage, the Universe initially expands quasi-exponentially in a vacuum-like state without temperature. During inflation, all energy is contained in the inflaton field Φ which is slowly rolling down to the minimum of its effective potential V (Φ). When chaotic inflation ends at Φ~Mp, the inflaton field begins to oscillate near the minimum of V(Φ) with a very large amplitude Φ~1/10Mp. In this scenario all the particles constituting the Universe are created due to the interaction with the oscillating inflaton field. Gradually, the energy of inflaton oscillations is transferred into energy of the ultra-relativistic particles. Eventually created particles come to a state of thermal equilibrium at some temperature Tr, which is called the reheating temperature. The transitional stage between inflation and the standard Big Bang is called the period of reheating after inflation.

http://hyperphysics.phy-astr.gsu.edu/hbase/astro/expand.html#c2

https://en.wikipedia.org/wiki/Zero-energy_universe

http://cds.cern.ch/record/346052/files/9802221.pdf

http://www.desy.de/~westphal/workshop_seminar_fall_2010/reheating.pdf

@Andrew Wutke,

in the absence of a scientific truth based on paradigms of the nature, for a few questions people are free to accept a religious or revealed or supernatural truth. I don't see contradictions that there are instead when a non scientific truth contrasts with a scientific truth. But science isn't able to cover all exigencies of the human mind and when science aims to give answers that it cannot give, then this pseudo-science becomes ridiculous and tedious.

Andrew,

What you seem to be asking for is a description of how our current theory of gravity incorporates what is referred to as the t=0 moment. You might try “The large scale structure of space-time” by Hawking and Ellis. By making some “reasonable” assumptions about the energy content of the stuff of the universe, plus time reversibility, the text shows how the concept of curved spacetime breaks down at some point in the past (the t=0 time). This simply means that you cannot use the spacetime concept to further understand what’s going on and must replace it with something more fundamental. For decades we have interpreted the moment of breakdown as the initial moment of time (the creation of the universe) but there is no fundamental theoretical reason why you need to do that. You could just as well imagine a continuation in time of the universe in a way that we currently cannot describe. There are epistemological reasons why one might prefer a continuation through the t=0 moment and a number of recent frameworks incorporate this.

Thanks David

I am indeed interested in the initial moment more than in other phases.

DG: For decades we have interpreted the moment of breakdown as the initial moment of time ..

That's not quite correct, we have always taken the extrapolation of the scale factor to zero as a fiducial start time but generally the breakdown of classical maths has been taken as being approximately a Planck time (~10-43s) later than that.

Inflation theories generally suggest that inflation finished around 10-32s but most say it lasted for more than 10-32s duration so the start gets pushed back beyond the fiducial start from classical theory and can even be infinite. There is no theory that combines QM and GR in the pre-Planck era so you will not get a definitive answer.

Andrew, for the actual transition from 'nothing' to 'something', there is no currently accepted theory at all and as far as I know very few credible attempts. The only paper that springs to mind is attached.

https://arxiv.org/abs/1111.0301

Thank you George and Robert

I can accept difficulties and the lack of near 0 theory. Unfortunately the picture of the theory in the public domain is quite clear: you can start from nothing and end up with galaxies. That has awaken my interest in the mater trying to find out how that could be imagined and modeled. What equations and initial conditions?

From some isolated citation some incredible densities of matter are postulated near t=0 (what matter?) 1099 kg/m3 at t=10-43s. I am thinking, the closer you go to 0 the higher density and time must "flow" slower and slower so in fact t=0 becomes t=infinity if we label our current position as t=0. That brings us closer to the concept of eternity rather to a definite beginning.

AW: time must "flow" slower

Slower as measured against what? Thinking that way is a dead end unless you invent additional time-like dimensions, time is the independent variable against which everything else is measured.

The attached paper is quite old now but probably still a reasonable introduction. Have a read of the first paragraph.

http://cds.cern.ch/record/346052/files/9802221.pdf

Dear George

Without reading the paper. The laws of physics should be independent on the choice of coordinates. I may chose then t=0 xyz=0 to be always where I am and to see the time of bb nothing more special then the time of departure of my local bus. But of course some special event tajes place there. Imagining processes evolving at usual rates then at high densities of matter progressively slower from my point of view, and minus infinity time is imaginable while always zero now seems ok too.

But this is of course my speculation. I believe the numbers you attach to time are arbitrary but some are more convenient than others. whether we chose to make time positive or negative is of no difference to laws of physics. Its just a convention. Negative time means as much as negative spatial coordinate. But the unique  sequences take place without regards what we think of time.

Of course coordinates make little difference, we can call the start of the universe t=0 or say now is t=0 and the start was at -13.8 billion years, it makes no difference. However, some laws are easier to express in some coordinates which is why science usually measures temperature in Kelvin when Celsius differs only in its origin.

The scale factor of the radiation era was proportional to the square root of time, but only if you set t=0 corresponding to a(t)=0, otherwise you have to write it as a(t-t0) where t0 is just a constant. The physics remains the same, and scaling time by having varying units has no more impact than that.

On the other hand, there is a significant difference between a(t)=√(t/t0) versus a(t)=et/tc using the same definition of time, where tc is a characteristic duration and t0 is the age now. The former has a finite past while the latter is infinite. That is a real difference in the physics.

Following is a near t = 0 theory. It is based on the idea that time is not linear, flowing in one direction, but periodic in nature. The reality is based on wavelengths and periods of particle wave. Linear time is imaginary. At planck time the frequency of a particle is inverse of planck time. This is the smallest it can get for the Planck energy level. If particle gets more energy than Planck energy, the particle wave will collapse leaving no mass gap. This results in a perfectly motionless state of energy which we describe as vacuum. And the process is reversible.

Article Periodic quantum gravity and cosmology

A rigorous mathematics describing The Big Bang Theory and its modification was displayed in our book:

The Modified Big Bang Theory.

In our time-density hyperbolic evolution equation of the universe, time and density are dependent. A very small time is struggled by a very high density. Such a mixture did prevent the scale factor to blow at t=0.

Preprint The Modified Big Bang Theory