ΛCDM Model

[Anton Viola]

Really a strange topic, the ΛCDM Model ...

The Lambda-CDM, Lambda cold dark matter, or ΛCDM model is a mathematical model of the Big Bang theory with three major components: 

1. a cosmological constant denoted by lambda (Λ) associated with dark energy; 

2. the postulated cold dark matter denoted by CDM; 

3. ordinary matter.

Standard ΛCDM requires only 6 independent parameters to completely specify the cosmological model.

I am trying to understand this model in detail, but not too successful so far. Any inputs here?

Thanks in advance!

Anton

 

ΛCDM Model V0.1.xlsx

[andrew s]

I can recommend "An Introduction to Modern Cosmology" by Andrew Liddle. It should provide the background you need.

Regards Andrew 

Edited May 4 by andrew s

[Anton Viola]

Thanks! Ordered the book, looking forward!

Best,

Anton

my website

[vlaiv]

Important thing to note is that this model does not have equations like we are used to.

It can only be solved by numerical methods / via computer because it is set of differential equations without nice analytical solution.

Observational data is fed into computer program and best fit is produced. This best fit yields certain functions and numerical values for constants that we have as "solution" to the model.

 

[andrew s]

1 hour ago, vlaiv said:

Important thing to note is that this model does not have equations like we are used to.

It can only be solved by numerical methods / via computer because it is set of differential equations without nice analytical solution.

Observational data is fed into computer program and best fit is produced. This best fit yields certain functions and numerical values for constants that we have as "solution" to the model.

 

Actually,  that's the norm in physics. Even the 3 body problem in Newtonian physics has to no analytical solutions. It is also true in quantum mechanics,  fluid mechanics and solid state physics etc. etc.

Regards Andrew 

[George Jones]

15 hours ago, vlaiv said:

Important thing to note is that this model does not have equations like we are used to.

It can only be solved by numerical methods / via computer because it is set of differential equations without nice analytical solution.

Observational data is fed into computer program and best fit is produced. This best fit yields certain functions and numerical values for constants that we have as "solution" to the model.

 

Actually the differential equation for the scale factor (the Friedmann equatiion) for Lambda CDM universes has exact solution in terms of elliptic functions.

[saac]

Same process in engineering (fluid dynamics) with use of the Navier Stokes equations in analysis of turbulent flow regimes; from what I remember there are no easy analytical solutions (modelled by differential equations then cracked with lots of CPU power). Laminar flow, yes, but that's boring  

Jim 

[George Jones]

2 hours ago, saac said:

Same process in engineering (fluid dynamics) with use of the Navier Stokes equations in analysis of turbulent flow regimes; from what I remember there are no easy analytical solutions (modelled by differential equations then cracked with lots of CPU power). Laminar flow, yes, but that's boring  

Jim 

Wandering off-topic a bit, but the Navier-Stoke situation, for me, is quite interesting. Engineers and physicists are happy to solve Navier-Stokes on a computer, but mathematicians have yet to be convinced that any reasonable solutions to Navier-Stokes exist (as the mathematics of mathematicians). There is $1 million riding on this! Anyone who can prove existence (without even writing down a solution), or who can find an explicit solution, wins a Clay Millennium Prize.

The attitude of hard-nosed physicists and engineers is that mathematicians are too worried about crossing every t and dotting every i.

[saac]

4 minutes ago, George Jones said:

Wandering off-topic a bit, but the Navier-Stoke situation, for me, is quite interesting. Engineers and physicists are happy to solve Navier-Stokes on a computer, but mathematicians have yet to be convinced that any reasonable solutions to Navier-Stokes exist (as the mathematics of mathematicians). There is $1 million riding on this! Anyone who can prove existence (without even writing down a solution), or who can find an explicit solution, wins a Clay Millennium Prize.

The attitude of hard-nosed physicists and engineers is that mathematicians are too worried about crossing every t and dotting every i.

I never knew that George,  it made me laugh. The thought of engineers and physicist just cracking on with it while the Mathematicians hesitate; afterall somebody has to design the pumps, gas turbines and water park flume rides!  I'll cut the Maths dudes some slack, afterall, I suppose the dot placing is tricky on those imaginary i s .    

A million dollars you say eh, now where did I put my laplace tables  

Jim 

[andrew s]

The interplay of mathematics and physics is fascinating.  At times maths has led e.g. Riemman geometry required for general relativity and others where physics led e.g. the Dirac delta function eventually made respectable by mathematicians with via the theory of distributions.

Regards Andrew 

[saac]

2 hours ago, andrew s said:

The interplay of mathematics and physics is fascinating.  At times maths has led e.g. Riemman geometry required for general relativity and others where physics led e.g. the Dirac delta function eventually made respectable by mathematicians with via the theory of distributions.

Regards Andrew 

What actually separates the two disciplines, particularly theoretical physics and say pure mathematics?  I mean at the most professional level say in research or academia is there a particular body of knowledge or skill-set that differentiates the two disciplines. I've always thought that at that level the disciplines must really converge. Or does it simply come down to what holds the interest and hence motivates a particular researcher?  I used to think that the level of command of mathematics would be a limiting factor. But then Einstein used to make fun of his mathematical abilities yet he obviously had imagination, insight maybe a natural intuition of the physical world that allowed him to see what he needed to see. I guess this is where the value of interdisciplinary teams come in. 

Jim 

[andrew s]

I used to joke that physics is just mathematics with boundary conditions.

It's perhaps more than that. Physics is applying mathematics to an end while for mathematician it is an end in itself. 

It would, however, be hard to tell some theoretical physicists from some applied mathematician (beard length perhaps?). 

If it works physicists are content to use it even if it is not subject to a rigorous proof. As Godel proved there may be accurate theorems that can't be proved! 

Regards Andrew 

[Nik271]

Just to give another example where maths is widely used without rigorous proof: the RSA encryption algorithm commonly used to send  data securely  is based on very simple idea, that is hard to factorise integers into prime factors. There is no rigorous proof that this is indeed hard, in fact this is a special case of another million dollar question P vs NP which is about algorithm complexity.

[Anton Viola]

Well, took a few steps back and investigated theory/equations about black holes. Found an excellent resource (from 2014, but still valid ..?), which shows clearly a lack of consensus (extremely wide spread of the “expert opinions”). Summarised what I found in there attached spreadsheet. Or have I overlooked things?

 

Anton at www.astronomy-morsels.ch

Black Holes V1.0.xlsx

[George Jones]

On 09/05/2024 at 11:22, Anton Viola said:

Well, took a few steps back and investigated theory/equations about black holes. Found an excellent resource (from 2014, but still valid ..?), which shows clearly a lack of consensus (extremely wide spread of the “expert opinions”). Summarised what I found in there attached spreadsheet. Or have I overlooked things?
 

Sorry, but I think that this is a terrible resource. I only have had a look at a small portion of the spreadsheet, but I have found mistakes in 3 cells. When these mistakes are corrected, these 3 cells all take on the same value, which is a value that some other cells already have.

I started to type a detailed mathematical response in a coffee shop, but a friend came in and I didn't finish. Now I have to go home and face the Saturday expectations of my wife.

[Anton Viola]

Looking forward to your feedback! Started checking the formulaes and references myself as well.

[George Jones]

On 11/05/2024 at 08:59, George Jones said:

Sorry, but I think that this is a terrible resource. I only have had a look at a small portion of the spreadsheet, but I have found mistakes in 3 cells. When these mistakes are corrected, these 3 cells all take on the same value, which is a value that some other cells already have.

In the temperature part of Henry K.O. Norman's spreadsheet (which I have attached), the corrections changed the temperature values in 3 cells to 1.234E-08.

 

On 12/05/2024 at 13:51, Anton Viola said:

Looking forward to your feedback! Started checking the formulaes and references myself as well.

I have since looked at another of Norman's expressions for temperature, 10²⁶M⁻¹K. In line  82, Norman writes "(10) Hawking’s 1975 paper, his second temperature guess (the factor 10²⁶ is nowhere explained)."

Hawking published his 1975 paper in a high-level mathematical physics journal, and the reader is expected to be able to supply easy (for the research physicists that read this journal) results, such as the 10²⁶ factor. In the last paragraph of the third page of this paper, Hawking writes "In ordinary units this temperature is of the order of 10²⁶M⁻¹K ". In the next sentence he writes "solar mass (10³³g )". Consequently, "ordinary units" means cgs units (centimetre, gram, second), not mks units (metre, kilogram, second).

Once the correct units are used, it is fairly straightforward to derive the 10²⁶ factor. See my attached pdf for details.

Hawking SGL 0.pdf Black Hole Properties Norman.xlsx