This week's CQW is a guest post by Phillip Helbig:

Read the question then, without calculating or estimating anything, first make a quick guess as to the result. Then work out the result (an order of magnitude or two is close enough).

Neglect gravity and other types of interactions and imagine the entire observable universe being compressed into a ball (rather like a movie of the expanding universe played in reverse). There are about a hundred billion stars per galaxy on average, and at least a hundred billion galaxies in the observable universe. Or think of the Hubble Deep Field, which has the angular size of a small rice grain held at arm's length, full of galaxies - and this was intentionally chosen because it was apparently empty! The sky is about 25 million times larger.

What is the size of the ball when the density is equal to the density of nuclear matter?

# cosmology question of the week

CQW is an educational resource for theoretical physics and astrophysics, field theory, relativity and cosmology. we post a new question every wednesday for students to get and for teachers to stay in shape.

## Wednesday, November 25, 2015

## Wednesday, November 18, 2015

### let it rip!

what property of dark energy would lead to an infinite scale factor in a finite time in the future? would a cosmological constant be able to do this?

## Wednesday, November 11, 2015

### no big bang

can you construct a cosmological model with an asymptotically constant scale factor in the past, such that the age of the universe would be infinite and no big bang would have occurred? what would be a stable construction?

## Wednesday, November 4, 2015

### vacuum fluctuations

what's wrong with this argument? a moving observer would see quantum fluctuations in e.g. the electromagnetic field blueshifted from the forward and redshifted from the backward direction and would be able to determine the velocity by the amplitude of the motion dipole.

## Wednesday, October 28, 2015

### conditions for cosmic horizons

This week's CQW is a guest post by Phillip Helbig:

An important concept in classical cosmology (for the purposes of this post defined as cosmology within the framework of Friedmann-LemaĆ®tre cosmological models) is that of horizons. The particle horizon is the spherical surface from which radiation from the big bang is just reaching us now or, alternatively, due to symmetry, the sphere which is now just reached by radiation emitted from our location at the big bang. This is the same as the "observable universe". The event horizon is the sphere beyond which radiation emitted now will never reach us or, alternatively, due to symmetry, the sphere which will ever just be reached by radiation emitted from our location now. (Note for experts: I am discussing spatial horizons defined at the current cosmic time. In the literature, one also finds discussion of horizons defined in space-time. Somewhat confusing is when the particle horizon is described as a two-dimensional surface in space and the event horizon as a three dimensional surface in space-time. There is no inconsistency, but one must be careful when comparing different papers on this topic.)

The particle horizon exists if the integral

\begin{equation}

\int_{0}^{t_0} \frac{\mathrm{d}t}{R(t)}

\end{equation}

is finite; $t$ is cosmic time ($t=0$ corresponds to the big bang if there is one), $t_0$ is the current epoch, and $R$ is the scale factor.

The event horizon exists if the integral

\begin{equation}

\int_{t_0}^{\infty} \frac{\mathrm{d}t}{R(t)}

\end{equation}

is finite. One can of course work out $R(t)$ for given values of the cosmological parameters $\Omega$ (density parameter) and $\Lambda$ (cosmological constant), compute the integral, and check whether it is finite in order to determine whether the corresponding horizon exists for the cosmological model in question.

However, there are simple qualitative descriptions which give the necessary and sufficient conditions for each type of horizon. What are they?

An important concept in classical cosmology (for the purposes of this post defined as cosmology within the framework of Friedmann-LemaĆ®tre cosmological models) is that of horizons. The particle horizon is the spherical surface from which radiation from the big bang is just reaching us now or, alternatively, due to symmetry, the sphere which is now just reached by radiation emitted from our location at the big bang. This is the same as the "observable universe". The event horizon is the sphere beyond which radiation emitted now will never reach us or, alternatively, due to symmetry, the sphere which will ever just be reached by radiation emitted from our location now. (Note for experts: I am discussing spatial horizons defined at the current cosmic time. In the literature, one also finds discussion of horizons defined in space-time. Somewhat confusing is when the particle horizon is described as a two-dimensional surface in space and the event horizon as a three dimensional surface in space-time. There is no inconsistency, but one must be careful when comparing different papers on this topic.)

The particle horizon exists if the integral

\begin{equation}

\int_{0}^{t_0} \frac{\mathrm{d}t}{R(t)}

\end{equation}

is finite; $t$ is cosmic time ($t=0$ corresponds to the big bang if there is one), $t_0$ is the current epoch, and $R$ is the scale factor.

The event horizon exists if the integral

\begin{equation}

\int_{t_0}^{\infty} \frac{\mathrm{d}t}{R(t)}

\end{equation}

is finite. One can of course work out $R(t)$ for given values of the cosmological parameters $\Omega$ (density parameter) and $\Lambda$ (cosmological constant), compute the integral, and check whether it is finite in order to determine whether the corresponding horizon exists for the cosmological model in question.

However, there are simple qualitative descriptions which give the necessary and sufficient conditions for each type of horizon. What are they?

## Wednesday, October 21, 2015

### falling faster

can freely falling bodies overtake each other on the same trajectory? what would Newton and what would Einstein give as a reason?

## Wednesday, October 14, 2015

### cosmologies without beginning

what would be observable consequences in a cosmological model where $a=0$ is not reached in a finite past? could you tweak the cosmological parameters in such a way that there's no CMB or no BBN?

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