This alternative that you mention is very imaginative, but It poses several problems. For example, if all matter is contracting, galaxies and clusters or any large scale structure would eventually ...

I think the most intuitive way of understanding it is using $\textbf{Gauss Law}$ for the physical part and $\textbf{Gauss Theorem}$ for the mathematical part. If you integrate the divergence of the ...

I would say that the only quantum fluctuations that are relevant for the formation of structures (such as stars or galaxies) are those of the very early universe. In particular, they play a key role ...

BAO experiments essentially measure two quantities, one parallel to the line-of-sight \begin{equation} \beta_{\parallel} = H(z_{\rm eff}) r_s (z_{\ast}) \end{equation} and the other perpendicular to ...

For the answer to the question a), just use a Taylor expansion in the parameter $x= R/d \ll 1$, so that \begin{equation} (1-x^2)^{3/2} \simeq 1- \frac{3}{2} x^2 \end{equation} and then you obtain a ...

I would say no, since the solution that you found is stationary, it cannot be propagating. For example, in one dimension, a propagating wave needs to have the functional form $\Psi = \Psi (x-ct)$, ...

I see some mistakes: First of all, according to your definition of $H_{\Lambda}$, in the right-hand side of your equation you should have $H_{\Lambda}^{2}$ and not $H_{\Lambda}$. When you solve by ...

Note that the reason why you have bubbles is that you have many repeated (i.e. dummy) indices. In the first diagram for example, i and j label the two ($k=2$) lines joined by $-V_{ij}$, but they also ...

I would say it is partially correct. Dark energy is, as you said, anything that accelerates the expansion of the universe. One can show that any component with an Equation of state parameter ...

Remember that for a 1/2 spin particle the only states we have are spin up $|\uparrow >$ and spin down $|\downarrow >$, or a general linear combination of both:  \chi = \cos{(\beta/2)} e^{-i\...

I'm not going to enter in why fundamental constants such as the electric charge or the mass of the proton are fine-tuned to favour life in the universe, since it requires a long explanation and I'm ...

What we tipically call the metric is $g_{\mu \nu}$, whereas the inverse metric is called $g^{\mu \nu}$ and it is defined through $g^{\mu \alpha} g_{\alpha \nu} = \delta^{\mu}_{\nu}$. The metric ...

The electric field $E$ is the field we apply, what we express with the first Maxwell equation is that its sources must come from the total density charge $\rho$. In a material, there will be some ...