I'd like to know your interpretation.

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Discussion Overview

The discussion revolves around the interpretation of equations related to the Hubble parameter and its derivative, specifically in the context of cosmological models. Participants explore the implications of these equations, including conditions under which the derivative of the Hubble parameter can be positive or negative, and reference Hobson's notation from a specific textbook.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants present the equation for the Hubble parameter and its derivative, suggesting that if the second derivative of scale factor \( a \) is negative, then the derivative of the Hubble parameter \( dH/dt \) is certainly negative.
  • Others argue that if the second derivative of scale factor \( a \) is positive, then \( dH/dt \) can be either positive or negative, indicating a more complex relationship.
  • One participant challenges the correctness of the equation for \( dH/dt \), suggesting a different formulation involving superscripting, while others defend the original formulation and reference the application of the chain rule.
  • Another participant mentions a specific cosmological model from Hobson's book, suggesting it as a good analytic approximation to the universe.
  • There are corrections and acknowledgments of mistakes among participants regarding the notation and formulation of the equations.

Areas of Agreement / Disagreement

Participants express disagreement regarding the correctness of the equation for \( dH/dt \), with some asserting it is correct and others claiming it is incorrect. There is no consensus on the interpretation of the equations or the implications for cosmological models.

Contextual Notes

Some participants note a lack of clarity regarding Hobson's notation, indicating that it may not be widely recognized outside the context of the referenced textbook. Additionally, there are discussions about the need for careful notation, particularly regarding superscripting in mathematical expressions.

EhsanZ
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With Hobson's notation:

H=(da/dt)/a

dH/dt = ((d2a/dt2)/a) - ((da/dt)/a)2

If ((d2a/dt2)/a) is negative, (dH/dt) is certainly negative.
But if ((d2a/dt2)/a) is positive, (dH/dt) can be positive or negative.

What is your interpretation?
 
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dH/dt = ((d2a/dt2)/a) - ((da/dt)/a)2

Above is incorrect. Should be:

dH/dt = (d2a/dt2)/a - (da/dt)/a2

Your second term is wrong.
 
No, it is correct.
I think you're forgetting the application of the chain rule.

I also think EhsanZ's interpretation is correct, although I do not know what "Hobson's notation" is.
I couldn't find it with google, except for in this thread. ;)
 
Actually I think the only thing wrong with second eq. is a lack of superscripting of the 2s:

dH/dt = ((d2a/dt2)/a) - ((da/dt)/a)2
 
EhsanZ said:
With Hobson's notation:

H=(da/dt)/a

dH/dt = ((d2a/dt2)/a) - ((da/dt)/a)2

If ((d2a/dt2)/a) is negative, (dH/dt) is certainly negative.
But if ((d2a/dt2)/a) is positive, (dH/dt) can be positive or negative.

What is your interpretation?

[itex]d^2 a/dt^2 > 0[/itex] and [itex]dH/dt < 0[/itex] means that the expansion of the universe is accelerating while the Hubble constant is decreasing. We think that this is happening today.
I like Serena said:
although I do not know what "Hobson's notation" is.
I couldn't find it with google, except for in this thread. ;)

This (I presume) refers to the notation used in the book General Relativity: An Introduction for Physicists by Hobson, Efstathiou, and Lasenby.

EhsanZ, play around with the spatially flat, matter-only Lemaitre model given on page 406 and in problem 15.23. This a good analytic approximation to our universe.
 
George is correct. If you look at the equations dispassionately, it is obvious the result can never be negative.
 
I like Serena said:
No, it is correct.
I think you're forgetting the application of the chain rule.

I also think EhsanZ's interpretation is correct, although I do not know what "Hobson's notation" is.
I couldn't find it with google, except for in this thread. ;)
You're right. My bad.
 
mathman said:
dH/dt = ((d2a/dt2)/a) - ((da/dt)/a)2

Above is incorrect. Should be:

dH/dt = (d2a/dt2)/a - (da/dt)/a2

Your second term is wrong.

No my friend! U made a mistake.
 
mathman said:
You're right. My bad.

Doesn't matter! Everybody makes mistakes.:smile:
 
  • #10
I like Serena said:
No, it is correct.
I think you're forgetting the application of the chain rule.

I also think EhsanZ's interpretation is correct, although I do not know what "Hobson's notation" is.
I couldn't find it with google, except for in this thread. ;)

I meant the notation that Hobson had used in his book named "General Relativity: An Introduction for Physicists ".
 
  • #11
BillSaltLake said:
Actually I think the only thing wrong with second eq. is a lack of superscripting of the 2s:

dH/dt = ((d2a/dt2)/a) - ((da/dt)/a)2

Yes, you're right my friend. I should've written it more carefully.
Thanks
 

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