- #1
Salah93
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Homework Statement
The aim is to find a solution for the scale factor in a Robertson Walker Metric with a scalar field and a Lagrange multiplier.
Homework Equations
I have this action
S=-\frac{1}{2}\int d^{4}x\sqrt{-g}[R(g_{\mu\nu})+\lambda(g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-1)+L_{m}]
In conformal for Robertson Walker metric ds^{2}=a^{2}(\eta)(d\eta^{2}-\delta_{ij}dx^{i}dx^{j})
where \phi is a scalar field (that depends only on conformal time ), \lambda a Lagrange multiplier, L_m lagrangian matter that I set to 0 and R the Ricci Scalar.So the Ricci Scalar is R=-6\frac{a''}{a^{3}} ()' is derivative with respect to conformal time (I found the value from the value that one finds for R from universal proper time, that is R=-6(\frac{\ddot a}{a}+\frac{\dot a^2}{a^2}) )
So the action is
S=-\frac{1}{2}\int d^{4}xa^{4}[-6\frac{a''}{a^{3}}+\lambda(a^{-2}\eta^{00}\partial_{0}\phi\partial_{0}\phi-1)]=-\frac{1}{2}lim_{V\rightarrow\infty}V\int d\eta[-6\frac{a''}{a^{3}}+\lambda(a^{-2}\phi'^{2}-1)]a^{4}
The Attempt at a Solution
I have the various L=L_{V}=-\frac{1}{2}V[-6\frac{a''}{a^{3}}+\lambda(a^{-2}\phi'^{2}-1)]a^{4} so because there is only the multiplicative factor V they all lead to the same eq. of motion. So I can omit the limit and take V=1. So the action is now
S=-\frac{1}{2}\int d\eta[-6\frac{a''}{a^{3}}+\lambda(a^{-2}\phi'^{2}-1)]a^{4}=-\frac{1}{2}\int d\eta[-6aa^{''}+\lambda a^{4}(a^{-2}\phi'^{2}-1)]
And without considering surface elements the action become
S=-\frac{1}{2}\int d\eta[6a'^{2}+\lambda a^{4}(a^{-2}\phi'^{2}-1)]
Now I want to find eq. of motion:
Variation w.r.t a gives:
6a''-\frac{1}{2}[\frac{\partial(\lambda a^{4})}{\partial a}(a^{-2}\phi'^{2}-1)-a\lambda\phi'^{2}2]=0
W.r.t \lambda:
\frac{\phi^{'2}}{a^{2}}-1=0\rightarrow\phi'^{2}=a^{2}
W.r.t \phi:
\frac{d}{d\eta}(2\phi'\lambda a^{2})=0
\phi'\lambda a^{2}=\lambda a^{3}=\eta+K\rightarrow\lambda=\frac{\eta+K}{a^{3}}
K is a real constantAnd so in the end I have:
6a^{''}+\eta+K=0
Integrating
a'(\eta)=-\frac{\eta^{2}}{12}-\frac{K\eta}{6}+C
And again:
a(\eta)=-\frac{\eta^{3}}{36}-\frac{K\eta^{2}}{12}+C\eta+B
B and C real constants.
Now, if \eta goes to +\infty is a problem because the scale factor becomes negative. And so the only possibility for this case in a flat Robertson Walker metric is that I have(setting B=0) a(\eta)=0 and then after a finite interval again a(\eta)=0 becuase then it will be negative and I think there isn't a physical meaning for negative scale factor. So, where are the errors? Because I can't interpret well this result.
If I imagine it as a 1-D motion of a particle I can study without any problems the system. But here I am studying the evolution of a particular case of the universe. How can I interpret negative scale factor(if all above is correct)?
I hope that the question is written well. Thank you