In the early universe the curvature perturbations, which later are seen as temperature fluctuations in the Cosmic Microwave Background (CMB), are initially thought to be Gaussian, but can become skewed during inflation depending on the physics of their evolution. In the last few years a lot of work has been done to both find evidence of this non-Gaussianity, and to construct physical models in which it is generated in the early universe.

In the past most of the focus has been on the 3-point function or bispectrum, and discussion of non-gaussianity has boiled down to finding bounds on the parameter . In terms of the CMB the bispectrum in essence considers whether the temperature of three points on the sky is correlated. The WMAP satellite has not seen any definitive evidence of a non-zero value for  but the Planck satellite should be able to detect a signal if it is moderately large. In this work we look beyond the bispectrum to the 4-point function or trispectrum.  For the trispectrum the correlation we attempt to measure is between four different points on the sky.

In this work we have tried to find models which generate a large value for the trispectrum during inflation. We have found some new expressions for the parameters ,  and . The last two of these parametrise two different contributions to the trispectrum. The bottom line is that it is quite difficult to find conditions where the trispectrum can be large, at least under the assumptions we made of sum- and product-separable potentials using the formalism.

In the course of searching for models which give large values to these parameters we plotted the coefficient functions which need to be large as heatmaps, following Byrnes et al (arXiv:0807.1101). In order to generate these heatmaps I relied on the combination of Python, Numpy and Matplotlib, which I have used before on Pyflation. The script I used to generate the heatmap figures in the paper is now available as a repository on Bitbucket.

PS Someone really needs to work on the Wikipedia Non-Gaussianity page!

This paper investigates the isocurvature or nonadiabatic perturbations during inflationary expansion with more than one field. We performed numerical simulations using Pyflation version 0.2 which can handle multiple field inflationary models at first order in perturbation theory.

We found that, as expected, the amount of isocurvature varied considerably for the three models we examined. We compared our results to those found using a different definition for the isocurvature, which coincides with ours during slow roll evolution on large scales. As slow roll breaks down near the end of inflation the evolution of the two different forms of isocurvature can vary dramatically.

One reason to be interested in non-adiabatic pressure during the early universe is that it can be a source for the generation of vorticity, which could be observable in the polarization of the cosmic microwave background.

• Calculating Non-adiabatic Pressure Perturbations during Multi-field Inflation
  Ian Huston, Adam J. Christopherson
  Abstract: Isocurvature perturbations naturally occur in models of inflation consisting of more than one scalar field. In this paper we calculate the spectrum of isocurvature perturbations generated at the end of inflation for three different inflationary models consisting of two canonical scalar fields. The amount of non-adiabatic pressure present at the end of inflation can have observational consequences through the generation of vorticity and subsequently the sourcing of B-mode polarisation. We compare two different definitions of isocurvature perturbations and show how these quantities evolve in different ways during inflation. Our results are calculated using the open source Pyflation numerical package which is available to download.
• Quantifying the behaviour of curvature perturbations during inflation
  Ellie Nalson, Adam J. Christopherson, Ian Huston, Karim A. Malik
  Abstract: How much does the curvature perturbation change after it leaves the horizon, and when should one evaluate the power spectrum? To answer these questions we study single field inflation models numerically, and compare the evolution of different curvature perturbations from horizon crossing to the end of inflation. In particular we calculate the number of efolds it takes for the curvature perturbation at a given wavenumber to settle down to within a given fraction of their value at the end of inflation. We find that e.g. in chaotic inflation, the amplitude of the comoving and the curvature perturbation on uniform density hypersurfaces differ by up to 180 % at horizon crossing assuming the same amplitude at the end of inflation, and that it takes approximately 3 efolds for the curvature perturbation to be within 1 % of its value at the end of inflation.

As part of the work on the first of these papers, my Pyflation code has been updated to version 0.2 and now allows multiple fields systems to be used at first order.

There seemed to be a glut of papers released last Wednesday, with the post-Thanksgiving queue being worked through and people aiming to apply for jobs in December. Apologies to those irked by the overwhelming astro-ph listing that day for my part in it!

Having heard about the Big Bang model and some of its problems from my colleague Dr David Mulryne, I was given the task of outlining how the inflationary paradigm tries to solve these problems and some of the reasons we think it is a good model of the early universe.

My slides are now available on slideshare.net. As I have mentioned before I prefer minimal text in my presentations, allowing the audience to focus on me instead of paragraphs of text on the slides.

Update 03/05/2012: Removed embedded slideshare widget to avoid having all the tracking cookies.

The speed networking sessions were organised by Inspiring Futures and included lots of people with different scientific backgrounds from nuclear engineers to bumble-bee psychologists (!). Groups of school kids were directed our way and given four minutes to quiz us about our career choices, subject areas and of course financial situations. On the whole the pupils seemed really interested and open to the different areas on offer. Some of course had already decided what career they want and would not be budged whether it was drama, history or theoretical physics they had opted for.

The sessions were ably organised by Ella Bujok who had to cope with missing school groups, the evacuation on the first day and the logistical problems of moving 60 children around a room in an orderly fashion every four minutes. She also kindly took the photos below for me to prove I didn’t just play in the Wallace and Gromit area all day. I tried to also get some pictures of the main arenas but it is hard to get a sense of the scale of the show from the few shots I managed to take. There are many better overviews in the Flickr page.

Accompanying the paper is the release of the software package used to create the results. Pyflation is a Python package which calculates the first and second order perturbation results including the source term required at each time step. It requires Numpy, Scipy and PyTables and a fairly recent version of Python. The full calculation for a large range of Fourier wavemodes takes a long time on a single CPU, but there is support for splitting the calculation into many separate jobs which can be queued on a multi-core or cluster based setup. Alternatively there is also support for calculating the second order results for a single wavemode, which significantly reduces the execution time.

The Pyflation website contains all the details and download links, installation notes and links to the relevant papers. We’ve released the package under a very liberal open source license (modified BSD license) but as citations are increasingly the oil that greases the wheels of academia we ask that anyone who uses results they have obtained with Pyflation to please cite one or more of the related papers.

• Check bib style is correct.
  If the journal you are planning to submit to has a particular house style for the bibliography it is probably worth using it in the arXiv submission.
• Check bibliography text is correct.
  Even though I think BiBTeX is the great, and much easier than preparing the bibliography by hand (which some people still do), there can be instances when a stray space or mistakenly capitalised letter appear. If you get your BiBTeX entries from SPIRES or Inspire this is mostly taken care of, but there is always the odd paper or book entry that you have typed in by hand. Check the final output in the paper, not just the .bib file contents.
• Check journal info for newer papers.
  Nowadays many works are first cited as preprints before publication in a journal and it is only right to update your reference list to include any subsequent publication information. Take care that the published version is not substantially different from the preprint as even the conclusions might have changed during the refereeing process.
• Check punctuation at end of equations.
  It is considered best practice to include equations as part of a sentence and not standalone objects. Therefore you should treat them grammatically as just another word or phrase. If you continue the sentence after the equation include a comma at the end of the last equation line using \,, which introduces a small space before the comma.
• Move bibliography from bbl to tex file.
  The arXiv help page states that BiBTeX is not run when you submit a file. The way to include your bibliography is to either include your .bbl file in the bundle you submit or to manually copy in the contents of the .bbl file and replace the \bibliography and \bibliographystyle commands in your .tex file.
• Read through all equations.
  It might not be possible to rederive all the equations but a quick look through could catch any glaring errors in transcribing long equations into TeX.
• Remove revision control info from files.
  If you use revision control to keep track of the changes to your paper (and you should!) then be careful to remove any files or tag lines with this info. For CVS you may have included a $Id$ line with the revision number and date. If you are submitting a zipped directory of files be careful to exclude any control directories such as .git or .bzr. Perhaps in the age of open science every revision of your paper should be open to all, but it is likely you don’t want the whole writing process to be scrutinized.
• Remove extraneous commands.
  Any unused or temporary commands, which might for example add labels, todo lists, highlight changes, should be removed.
• Remove unnecessary comments.
  Again any unnecessary comments in the .tex file should be removed. Don’t forget that anyone will be able to read the .tex file as well as the .dvi or .pdf files. Probably not the best idea to slag off your competitors in the comments then.
• Remove unused sections.
  In a similar vein don’t leave unused sections commented out in the .tex file.
• Spellcheck document, abstract, authors.
  Although it is not straightforward to spellcheck LaTeX files you need to make sure that any easy typos have been caught. Take special care with the authors’ names, as this might be one of the most embarrassing mistakes to make.

There are probably some more things I could add, especially about the particular submission process for the arXiv but this list has served me quite well.

I have been making quite a few plots recently for a paper which should hit the arXiv very soon. The Python plotting package Matplotlib has been indispensable in this regard, especially as I took the effort of creating a script which creates all the plots. This meant that redoing all the graphs for new results or with changed sizes etc. was a simple as rerunning the script.

Quite a few of the plots use log axes and while Matplotlib performs admirably there was one problem I had with certain plots. By default, the log plots only show tick labels for each order of magnitude. Tick labels are the numbers on the x or y axis telling you the corresponding numerical value, and when the figure is zoomed in it is possible to lose the major tick label at say 10^-4 because you only want to plot values from 0.3×10^-4 and 0.5×10^-4. Obviously this removes all sense of scale from the plot. A very mediocre solution is to just zoom out until a major tick label is back in the plot but this is obviously unsatisfactory.

I looked through the Matplotlib documentation, which has very detailed information about the API and has a lot of examples, but unfortunately didn’t address this exact point. After a bit of searching I found a useful conversation on the users mailing list which got me close but didn’t use the LaTeX labels which are really essential for publication quality graphs (in my opinion anyway!). The tick labels documentation along with the major-minor ticks example led me to the Formatter classes, especially LogFormatter and LogFormatterMathtext. This looked like the right answer but unfortunately LogFormatterMathtext writes the minor tick labels in a very unusual way. Instead of 0.3×10^-4 it only writes an exponent, so 10^-4.52.

I finally settled on extending the pyplot.LogFormatter class which controls the text for the tick labels. My subclass is as follows:

import re
import pylab

class LogFormatterTeXExponent(pylab.LogFormatter, object):
    """Extends pylab.LogFormatter to use
    tex notation for tick labels."""

    def __init__(self, *args, **kwargs):
        super(LogFormatterTeXExponent,
              self).__init__(*args, **kwargs)

    def __call__(self, *args, **kwargs):
        """Wrap call to parent class with
        change to tex notation."""
        label = super(LogFormatterTeXExponent,
                      self).__call__(*args, **kwargs)
        label = re.sub(r'e(\S)0?(\d+)',
                       r'\\times 10^{\1\2}',
                       str(label))
        label = "$" + label + "$"
        return label

It is provided as is, but there shouldn’t be too much wrong with it. One odd thing is that the LogFormatter class is an old style class, so I inherited from object to make it my subclass a new style class. This might be dangerous and cause some unexpected problems.

To use the class you can do something like the following:

import pylab
import numpy as np

fig = pylab.figure()
pylab.semilogy(np.logspace(-6,-5))
ax = fig.gca()
ax.yaxis.set_minor_formatter(
    LogFormatterTeXExponent(base=10,
     labelOnlyBase=False))
pylab.draw()

Below are three different figures showing the current default situation, the result of using LogFormatterMathtext and the result of the new class. I hope this will be of use to someone who has been struggling with this problem as I have.

As I mentioned, this came up because of a paper that is very nearly completed and should be available soon. Along with the paper we should have the long promised release of the code I have been working on which solves cosmological perturbation equations during inflation. More on that soon.

Adam joined Queen Mary as a PhD student a year after I did and it’s been great sharing an office with him over the past few years. Although we haven’t yet written a paper together we’ve thrown a lot of ideas around so hopefully we can work together on something soon. When it comes to listing the names I imagine he will be arguing for alphabetical order!

Adam will be taking up his fellowship in the University of Nottingham from October 2011.

Peter Coles has a post summarizing the current swarm of rumours, and Andy Lawrence has a poll where you can pick your poison cuts-wise. At this late stage the mainstream media has finally started picking up on the message of the Science Is Vital campaign, that cuts in research funding are a ruinous road to lower growth in the future. The Guardian and The Times have both had editorials denouncing cuts to science funding in the last few days, while the FT got in ahead of the curve. Jon Snow has been outlining the possible brain drain, and the BBC have given a platform to CASE‘s Imran Khan. But is it too late to change the headline figures? The claim is that most if not all the deals are done at least as far as headline figures go. The Department of Business Innovation and Skills, which is responsible for research funding, is rumoured to be the one holdout left. Whether this indicates a change in attitude in Whitehall or an entrenched position in the Treasury, we will find out tomorrow.

A word of warning though, the announcements tomorrow are probably going to be broad and impenetrable. Colin Talbot has produced a handy guide to what to expect. (Charlie Brooker provides some light relief in his advice to George Osbourne.) The resultant changes in the budgets of the research councils, and particularly STFC, may not be known until December, when the axe will truly start to fall.

Update: 20:00 19/10 According to this Guardian story, the science research budget may escape the deepest cuts, and retain “flat cash”, in other words an approximately 10% decrease in four years, depending on inflation.