Collapsing haloes and expanding voids: can modelling them help us understand the dynamics of our Universe?

Title: A novel approach to cosmological non-linearities as an effective fluid

Authors: Leonardo Giani, Rodrigo von Marttens, Ryan Camilleri

First author institution: The School of Mathematics and Physics, the University of Queensland, Australia

Status: Available on ArXiv [submitted to Physical Review Letters]

A fundamental assumption of modern cosmology is the cosmological principle. This statement essentially says that, on average, over a large enough patch of the Universe, it is homogeneous (it looks the same from anywhere) and isotropic (it looks the same in all directions). In order to describe how the Universe changes over time (since the Universe is expanding), cosmologists use something called the Friedmann-Lemaître-Robertson-Walker (FLRW) metric. In General Relativity, a metric allows one to describe the distance between two points when gravity curves spacetime. The FLRW metric allows cosmologists to determine the distance light travels from galaxies and how much their wavelength stretches over these distances (called a redshift) based on how quickly or slowly the Universe has expanded at different times and how much energy density it contains. Additionally, the Friedmann equations are important to describe how the amount of matter, radiation and even dark energy (an unknown force or energy that drives the accelerated expansion of the Universe) impacts this expansion rate.

Currently, cosmologists believe that tiny quantum fluctuations seeded the initial variations in the density of matter across the Universe, that led to the present day distribution of over-densities of matter (where you might find lots of gas and stars) and under-densities of matter (like voids) in space. These over and under-densities seeded the growth of visible structures like clusters of galaxies and the stars and planets they contain. Technically, the growth of these structures could impact how the Universe expands over time as local regions are not really homogeneous or isotropic; the modelling for this can be complex. This is referred to as backreaction and is not really accounted for in the FLRW metric. It is debated among cosmologists how much backreaction impacts our measurements when we use the FLRW metric to calculate distances to objects in the Universe; some research finds the impact of backreaction is negligible, while others have stated it may be responsible for the accelerated expansion of the Universe (i.e. that this could explain away the need for dark energy to explain our observations).

A model to account for backreaction

In today’s bite, we look at a model which attempts to deal with this backreaction. Essentially, the authors of today’s paper add two new parameters to the standard Friedmann equations used to describe an expanding Universe. Normally, a single parameter called is used to describe all the matter energy density in the Universe. However, in this model, some of this matter is accounted for by the two new parameters, which account for matter within regions of space that are inhomogeneous and lead to backreaction in the Universe. One of these parameters can be thought of as the matter due to collapsing regions of matter (we will call them collapsing bubbles in this bite), and the other parameter due to expanding voids (which we will call expanding bubbles). Normally, cosmologists treat matter as ‘dust’, which is just a way of saying that matter does not exert any pressure. However, the expanding bubbles and collapsing bubbles have an associated pressure in the modified Friedmann equations. The collapsing bubbles exert positive pressure while the expanding bubbles exert a kind of ‘negative pressure’. Additionally, the authors modify the continuity equation (which is basically an equation that states mass in the Universe is conserved) to account for these parameters.

Interestingly, the parameters of the modified Friedmann equations here can be mapped into a set of equations called the ‘scalar-averaged backreaction equations’. These equations were developed by a theorist in earlier work in order to describe the average evolution of a inhomogeneous volume of space with its own expansion rate. The individual bubbles in this work are modelled as spheres, which allows them to be treated as their own ‘mini’ FLRW universes. This allows a mapping between the description of the model with bubbles in this work and the backreaction equations developed earlier.

Next the authors determine how the new matter parameters impact the expansion rate of the Universe overall. It turns out that this can be expressed in terms of the net amount of the different kinds of matter (dust, expanding bubbles, and collapsing bubbles) with some mathematics in a complex way. Additionally, the net amount of the matter in the bubbles depends on the relative fraction of matter in bubbles compared to the total matter density and the size of each bubble, parameterized by minimum radii and for collapsing and expanding bubbles respectively. 

How much matter creates backreaction?

The next step is to work out, how many of these bubbles are there? The authors use a well-established formalism to determine the number of bubbles of different masses that should exist. Then, they need to determine which of these bubbles can be expected to contribute to the backreaction based on their physical sizes; this sets how many bubbles there are of different densities and, thus, the overall density of matter contained in expanding or collapsing bubbles. Any collapsing bubbles of a density that is too great have likely already become gravitationally stable (virialized) through collapse at an earlier time in the Universe’s history, so these densities can’t be included in the calculation. Collapsing bubbles of too low a density, however (those which are not non-linear), will also not contribute to the backreaction. In contrast, for expanding bubbles, the density needs to be sufficiently low that they are expanding in such a way they also do not simply expand with the background (the rest of the Universe) but not so under-dense that they are also virialized (also gravitationally stable). In other words, the bubbles need to be both non-linear (practically meaning here that they do not expand with the background) by having a sufficiently different density to the average of the Universe, but also not already virialized. Figure 1 shows a prediction for the fraction of energy density in expanding bubbles and collapsing bubbles as a function of their minimum possible size and , respectively. These minimum radii are effectively the unknowns that tell us how much matter density sources backreaction.

Fits with data

The authors then go on to test what current data indicates for the parameters and . They test on data from the Dark Energy Survey Year 5 release, which can be used to measure the expansion rate of the Universe, and data from the Sloan Digital Sky Survey and the 6-degree Field galaxy Survey. Essentially this data captures information about the variance of the matter density and growth rate of structures in the Universe. The results for fits to the data are shown in Figure 2. The green region shows a part of the 2D parameter space that, if the data preferred, might suggest that backreaction could explain some issues in modern cosmology.

Summary

The authors of this work have shown a new way to attempt to parameterize and measure the effects of backreaction in the Universe. Interestingly, it has been proposed that understanding backreaction better by introducing the parameters and to improve the modelling could solve some issues in cosmology. One example is the Hubble tension, which refers to the mismatch in measurements of the present-day universal expansion rate from different cosmological datasets. The modelling derived in this work and the demonstrated parameter fits may help improve future analyses of data to constrain cosmological parameters and the expansion history of our Universe.

Edited by: Sowkhya Shanbhog

Featured image credit: A close-up image of the surface of a soap bubble. By KarlGaf, CCA by 4, via Wikimedia Commons.

Disclaimer: The author of this article is a member of the same institution as the first and third author of the paper featured in this bite, but has made no contribution to the presented work.

• About the Author

About Abbé Whitford

I am a third year PhD student at the University of Queensland, studying Large Scale Structure cosmology with galaxy clustering and peculiar velocities, and using Large Scale Structure to measure the properties of neutrinos.

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