Using a MCMC code that implements the Brans-Dicke's model with all the modified equations of field.

Finally, I get this curve that represents the scale factor versus cosmic time:

enter image description here

with the following expression:

enter image description here

and we get finally:

enter image description here

What should I do for priors in my MCMC code to produce an older universe since on this attached ?, the universe is young compared to the LCDM model.

Have I got to play with $tc$ or constants $C1$ and $C2$ in this expression :

C1 and C2


All I want is to have an older universe (at least -10 Gy from the present).

Any remark/track/clue are welcome.


  1. Some precisions for @buzz about the first answer that I got.

Yes, $\Omega_k\equiv\Omega_K$ and $\Omega_{de}=\Omega_{\Lambda}$ ($de$ for Dark Energy).

You are right when you say that I didn't take into account of radiation $\Omega_{r}$. Actually, if I include it in the computation of $\Omega_k=1-\Omega_{m}-\Omega_{\Lambda}-\Omega_{r}, this doesn't seem to have an impact on the young age of universe that I get by computing scale factor $a(t)$.

In my MCMC code, I have put the following priors on cosmological parameters :

variables = {$Omega_m$,\Omega_k,H0,\Phi_0,d\Phi_0,\Omega_{BD}};
priors={{0.27,0.33},{-0.01,0.01}, {64,76}, {0,4}, {0., 0.01}, {0,1000}};

I get the follwing estimations after an MCMC run :

{Ωm, Ωk, Ωde, H0, ϕ0, dϕ0, ωBD, tc} -> {0.270023, 0.00311401, 0.726863, 64.0442, 2.66395, 0.000218655, 907.375, 0.449043}
  1. Intuitively, by making decrease the Hubble constant $H_0$ (let say $H_0=60$, I make universe older with the following curve :

enter image description here

I manage to reach over the 8 Gyr instead of 6 in my first attempt.

This coould be explained by the fact that age of universe is classical kind of :

$t_{age}=\dfrac{1}{H_0}$ , so if I take a lower $H_0$, I will have an older universe.

But the problem is that $H_0$ is larger (with Planck, $H_0=67 km/s/Mpc$ and with Riess, $H0=72 km/s/Mpc$).

How to find a way into Brans-Dicke's model to compensate the low estimation of $H_0=64$ that I get with MCMC by another parameter (Maybe "$t_c$" or $\omega_{BD}$) ? The ideal would be to have an estimation close to Planck and have in the same time an older universe

Edit 2:

I have got an interesting result (by playing on H0, omega_BD, phi0 and dphi0) :

$$ \Omega_r=10^{-4};\\ \Omega_m=0.270023;\\ \Omega_{de}=\Omega_{\Lambda}=0.726863;\\ \Omega_k=1.0-\Omega_m-\Omega_{de}-\Omega_r;\\ H0=67;\\ \Phi_0=0.66395\\ tc=0.0001;\\ $$

scale factor curve

you can see a first part with decellerated expansion and after an accelerated expansion. Now, I have to make universe older (only 4 Gyrs on my figure) but I don't know how to make it older :

I must notify this result is a fine-tuned result by playing with all parameters


1 Answer 1


I have been working on a similar problem.


I would like to help your problem, but I do not understand your symbols.

I get Ω_m and H_0, but I'm not sure about Ω_de and Ω_k. I guess Ω_de is what I call Ω_Lambda. Regarding Ω_k, this is a more accurate value. See


for the value Ω_k = 8.24 x 10^-5. I am also interested where you got

Ω_m = 0.240303

since the value in the reference above is 0.27 .

You seem to ignore Ω_r (radiation density).

You also seem to ignore the Friedmann equation in

https://en.wikipedia.org/wiki/Friedmann_equations#Detailed_derivation .

If you can clarify my confusion, I will try to help you.

  • $\begingroup$ Thanks for your answer. I have put in Edit morre precisons on what I have done. For the scale factor, I use Runge-Kutta 4 applied on modified Friedmann's equations of Brans-Dicke's models (slightly different from LCDM but implying $\omega_{BD}$ and $\phi$ potential). Best regards $\endgroup$
    – ACat
    May 29 at 5:57
  • $\begingroup$ I apologize @ACat . I looked at en.wikipedia.org/wiki/Brans%E2%80%93Dicke_theory. I found that I am completely not able to understand the related mathematics. I would appreciate it you posted a comment with references for the values you included in your post: Ω_m = 0.240303, ΩΛ=0.726863, Ωr=10−4. Also, the equation t = 1/H_0 is wrong. The correct equation is derived from en.wikipedia.org/wiki/Friedmann_equations#Detailed_derivation $\endgroup$
    – Buzz
    May 30 at 18:49
  • $\begingroup$ The correct equation under the above heading is the integral (it has the form below). t H_0 = INTEGRAL(0,a) SQRT("sum of four Omega terms") da. $\endgroup$
    – Buzz
    May 30 at 18:58
  • $\begingroup$ The SQRT should be 1/SQRT. $\endgroup$
    – Buzz
    May 30 at 23:38

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