I Hubble Parameter as function of time in universe models

AI Thread Summary
The discussion revolves around the Hubble parameter (H) as a function of time in various cosmological models, particularly the L-CDM model and scenarios with different values of Lambda (Λ) and curvature (k). Participants explore how to generate graphs for these models and share tips on using a specific calculator to manipulate parameters effectively. There are inquiries about displaying specific curves and the possibility of simulating recollapse scenarios, with some limitations noted regarding the calculator's functionality. Additionally, mathematical expressions for scale factor and cosmological time in a closed matter-only universe are provided, along with insights into the behavior of the Hubble parameter over time. The conversation emphasizes the complexity of modeling different cosmological scenarios and the need for further exploration of parameter settings.
timmdeeg
Gold Member
Messages
1,538
Reaction score
342
This graph shows ##H## as a function of time related to the L-CDM model. Do we (@Jorrie) have similar graphs e.g. for ##\Lambda=0##; ##k=-1## critical, ##\Lambda=0##; ##k=0## open, ##\Lambda=0##; ##k=+1## closed?

That would be great, thanks in advance.

1668953461789.png
 
Last edited:
Space news on Phys.org
Not precisely, but you can get close enough for all practical purposes by playing around with the input parameters and output options. E.g. ##\Lambda = 0.0000001##, set the output scaling to Normalized, select Chart and set hor and vert scales appropriately:
1669026724844.png

For Open and Closed cases, you play around with ##\Omega##. I have used the http://jorrie.epizy.com/docs/index.html?i=1 version, which has more liberal range limits than the approved Github version.
 
Ah, great, thanks for your advise!

One question, how can I show only one of these curves?
1669049449361.png
 
Last edited:
Just go to 'Column definition and selection'. I usually click 'none' and then select the two or three that I need. The default selections are just to give an idea of how it works.
 
Got it, thanks.
 
@Jorrie is there a way for the calculator to show recollapse? I can't seem to get there no matter how I fiddle with the parameters.
 
Lightcone8 does not allow for high Omega or very small Lambda, so I'm not sure that collapse can happen with these limitations. Any small Lambda may quickly become dominant again.
As a matter of fact it seems to crash if I set Lambda to 0.001 and Omega to 1.5. Will have to investigate that.

I recall that I have previously simulated a zero lambda situation with collapse on older, less accurate versions, but it will take some searching to find that.
 
timmdeeg said:
This graph shows ##H## as a function of time related to the L-CDM model. Do we (@Jorrie) have similar graphs e.g. for ##\Lambda=0##; ##k=-1## critical, ##\Lambda=0##; ##k=0## open, ##\Lambda=0##; ##k=+1## closed?

Bandersnatch said:
@Jorrie is there a way for the calculator to show recollapse?

In a closed matter-only (dust) FLRW univers, parametric expessions for the scale factor ##a## and cosmological time ##t## as functions of conformal time ##η## are (from Ryden)
$$\begin{align}
a\left(\eta\right) &= \frac{1}{2} \frac{\Omega_0}{\Omega_0 - 1} \left( 1 - \cos\eta \right) \\
t\left(\eta\right) &= \frac{1}{2H_0} \frac{\Omega_0}{\left( \Omega_0 - 1 \right)^{3/2}} \left( \eta - \sin\eta \right),
\end{align}$$
with ##0<\eta<2\pi##, and with ##\Omega_0>1## the present density relative to critical density.

The Hubble parameter is given by (with abuse of notation)
$$H\left(\eta\right) = \frac{1}{a} \frac{da}{dt} = \frac{1}{a}\frac{\frac{da}{d\eta}}{\frac{dt}{d\eta}} = \frac{2H_0 \left( \Omega_0 - 1 \right)^{3/2}}{\Omega_0} \frac{\sin\eta}{\left( 1 - \cos\eta \right)^2}.$$
Ii is easy to put ##\eta##, ##t\left(\eta\right)## , and ##H\left(\eta\right)## into three columns of a spreadsheet, and to use these to plot ##H\left(\eta\right)## versus ##t\left(\eta\right)## for ##0<\eta<2\pi##.
 
  • Like
Likes Bandersnatch and timmdeeg
George Jones said:
Ii is easy to
Yeah, but that requires ME to do some work, instead of somebody else ;)

For those interested, here's the graph for ##\Omega_0=1.5## and ##H_0=67.74##
1669666403563.png

And the spreadsheet

(make a copy if you want to change the parameters)

The behaviour tracks what Jorrie's calc outputs for early periods, so it's probably typed in alright.
The switcheroo towards collapse happens around 100 Gyrs for 1.5x critical density; for 2x density it's about 45 Gyrs; 800 Gyrs for 1.1 - which are the time scales I wanted to get a sense of.
 
  • Like
Likes George Jones

Similar threads

Back
Top