Master's students
Resources
The plan (for the moment) is that together with cosupervisors Dr. Yallup and Dr. Durakovic, I will leave in this section resources (mostly as links, but perhaps with some downloadable content) which are mentioned during our meetings.Theoretician's toolbox
There is some computational infrastructure that you will want to establish, for all the projects and to varying degrees. The sooner you can get this over with, the better!LaTeX2e
 I would strongly advise you to write your \(\LaTeX\) report, and your notes, using the REVTeX 4.2e class. What is it? In their own words: "REVTeX 4.2 is a set of macro packages designed to be used with LaTeX2e and is wellsuited for preparing manuscripts for submission to the journals of the American Physical Society (APS) and American Institute of Physics (AIP)". Generally speaking, APS journals are the superior category of journals, though there are some other good ones associated with other continents. Try to keep to doublecolumn format wherever you can.
 If you wish to compile locally, I would recommend using TeX Live and latexmk from within terminal. If you prefer a browser, use Overleaf, which is (sadly) becoming more and more popular. As students, you may be able to get Overleaf Professional features by linking with your CRSID.
 There is a well defined standard for the inclusion of code in scientific papers. To understand this, look into (and use) the listings package in your \(\LaTeX\) source.
I've prepared a more indepth template for use in the final reports. You can download the , the , and the . This template is not a style class, but a working setup which has been carefully tuned to emulate the style of a published APS manuscript.
You do not need to use the template if you enjoy typesetting and have already put a lot of love and care into your \(\LaTeX\) ecosystem. However, please at least read the text of the pdf so as to preempt a lot of avoidable/repetitive editing when the time comes for me to read your report. If you don't like the conventions or have some improvements, let me know!
Git
 As your project develops (computational projects, or those who are compiling \(\LaTeX\) locally), I would strongly recommend you to use Git and Github to versioncontrol and share it, so that I/we can look at your work. For those not acquainted with the terminal, git can be an awkward learning curve, but the rewards are lifechanging. There are many great introductions to git on the internet.
Mathematica
 You can get a license e.g. here. All Cambridge University students are entitled to a license, if you are encountering any bureaucratic difficulties in this regard, email me.
Maple
 Maple is another body of proprietary computer algebra software. There is no Universitywide license for this, but if you really (really) want to try it then email me and we can discuss.
xAct
 For abstract (and component) tensor and spinor calculus (actually for most physically motivated representations of the Lorentz group), use xAct.
 For help with xAct, email me, or try the xAct Group. The latter is actually very responsive, so don't be afraid to try!
 There are some great resources and example code for xAct in this Git repository.

Some lessons learned from a notebook that was throwing an error (see right).
Firstly, you will almost never need to use TagSetDelayed, as below:
Instead, it is better to use MakeRule and AutomaticRules as per:n /: n[a] n[a] := 1
This may seem longwinded, but it is much safer when extending to general indices! Next, the syntax for DefMetric is not quite right. It takes some close attention to the documentation, but you need a tuple to define the pre and postfix covariant derivative notation:AutomaticRules[n,MakeRule[{n[a] n[a],1},MetricOn>All,ContractMetrics>True]];
DefMetric[1, \[Gamma][a, b], D, {";","D"}, PrintAs > "\[Gamma]", InducedFrom > {g, n}]
 If you define a scalar such as Phi[], i.e. call DefTensor without index specifications, you must still use the empty square brackets whenever you use your scalar in a tensor expression. There may be some rare scenarios where you will wish to refer to the xTensor head directly.
 Likewise, covariant derivatives always have the structure CovD_[Indices__]@xTensor_, and you must remember both the indices and the action on some tensor expression (expressed either using @ or []).
 A very useful set of tools is provided by the xTras package. This is oriented towards field theory. In particular, if you have a complicated expression and ToCanonical is not properly sweeping up all constant symbols into prefactors of unique tensor terms, you should try the CollectTensors function.
Nested Sampling (NS)
 Nested Sampling is a numerical integration algorithm whose development has been driven in large part by our group here in Cambridge. Whilst it is a general numerical technique it has become synonymous with calculation of Bayesian evidences. An excellent introduction into what evidences are and why they are interesting can be found in the incomparable textbook on inference by David Mackay inference.org.uk. Two specific chapters to read are 28  the "why" behind model comparison, and 29  the "how" numerical evaluation of Bayesian inference takes place. This book predates Nested Sampling, but it is hard to look past Mackay when it comes to introducing the lineage of the technique.
 The canonical reference for NS is still the original paper by John Skilling available at projecteuclid.org. Whilst this remains a source of inspiration for the NS developer, a more modern community review article has been compiled recently that is perhaps a better starting point arXiv:2205.15570. This review does a good job of establishing the provenance of the technique in fundamental physics, the pointers towards NS as a partition function calculator is perhaps of most interest to readers of this page.
 Our group has contributed heavily to the technical landscape of NS. MultiNest arXiv:0809.3437 is a pillar of the NS community, often synonymous with the technique itself  sampling from contracting ellipsoids is pleasingly intuitively geometric  and for many years was the most complete and popular implementation of NS. PolyChord is the groups current focus in terms of development, being a "next generation" NS algorithm, specifically aiming to push NS to higher dimensional applications than previously considered. The repository of PolyChord git:PolyChordLite is a good starting point with example code to get a hands on sense. NS is nothing without an analysis built on top of it, and the python package anesthetic git:anesthetic is what we use to drive a lot of our work, again the code repository contains some useful starting worked examples.
Modified Newtonian Dynamics (MoND)
 Introduction to the specific new relativistic MoND (RMoND) addressed by this project arXiv:2007.00082.
 Quite an old study of the sphericallysymmetric field equations of a TensorVectorScalar (TeVeS) theory arXiv:0502122.
 McVittie by Kaloper (black holes embedded in FRW spacetimes) arXiv:1003.4777.
 Reduced Lagrangia in highly symmetric spacetimes, and dangers of abbreviating the path integral, are discussed in arXiv:1811.10291 and references therein.
 Various other recorded talks to be shared by email, with one embedded below.
Geometric algebra
 For many, many geometric algebra resources, see the GEOMETRY website as maintained by Chris Doran.
 For the encoding of Grassmann calculus within geometric algebra, see this quasiseminal paper.
 For the origin of the \(\mathrm{det}(g)^{1/2}\) on page 16 of that paper, see below.
 How to construct fermions in two Euclidean dimensions? This is an initial target for the lattice implementation of matter fields on dynamical triangulations. There are some indications that once could simply recycle the even subalgebra formulation of Doran and Lasenby. If one does this, how can the Grassmann formulation be accommodated? For some remarks, see below.
 The problem of combining Grassmannodd variables with the spin structure within one geometric algebra does not actually seem to be so hard. The scalar components can be thought of as inner products of the nonorthonormal Grassmann basis (two basis vectors for each lattice site or node of the dual lattice), with arbitrary vectors \(a\) and \(b\). Care must be taken in constructing the `spin structure part' of the Lagrangian, so that the order of the vectors is reflected in the order of the \((\partial_a\wedge\partial_b)\) prefactor. It should then be possible to extend this setup to nonGaussian theories via \((\partial_a\wedge\partial_b\wedge\partial_c\wedge ...)\) etc. It may still be possible to recover the Grassmannodd expressions naturally from the order of operations, but I don't think it can be done if the components are scalars as in the minimal left ideal formulation: since scalars commute with all other grades, their order will be eventually lost whenever bilinear invariants are expanded. Note that the setup to the right is actually simpler than that which I proposed in the meeting on Thursday 2nd February 2023, in that the wedge product does not appear between the spinors.
Perturbative quantum gravity
 Luca Buoninfante's thesis Ghost and singularity free theories of gravity, also 'Xived at arXiv:1610.08744, provides a great pedagogical introduction to the connection between a QFT propagator and the resulting "particle spectrum", and explains why this line of thinking requires a systematic approach (spinprojection operators, SPOs) when the classical field theory contains some d.o.frich bosonic fields (commuting tensors with lots of components, as we have in electromagnetism's vector gauge potential \(A^\mu\)). It is slightly unnecessary to use these techniques for QED, but it becomes helpful to use them for gravity whose gauge fields \(h_{\mu\nu}\equiv g_{\mu\nu}\eta_{\mu\nu}\) produces waves with only two polarisations (not ten). The modified gravity theories in this project have many more than ten d.o.f in their tensor components, so the concept of a "particle spectrum" becomes mandatory.
 A fairly old (and wellknown) treatment of the particle spectrum of torsion gravity arXiv:1411.5613. This is mostly for context, since it covers a broader category of theories to those addressed in the project (socalled parityviolating). However, the underlying method used by Karananas is the same as the one which we will be using.
 Initial application by our group of the SPO method to torsionful theories in arXiv:1812.02675. In principle, the whole methodology we are using for these projects is contained within this paper, but be careful because it is extremely terse!
 A followup paper to arXiv:1812.02675 is given in arXiv:1910.14197. This paper provides some extra insights into the information which the propagator can provide about renormalisability. For the moment, it should be only of contextual interest: in gravity we take unitarity to have primacy, and renormalisability (which we do not see at first glance in GR) is a bonus.
 A final paper in the YunCherng Lin series is found at arXiv:2005.02228. This paper extends beyond torsion gravity into a related set of models comprising Weyl gauge theory (WGT). There is no accommodation in the PSALTer code for WGT, so extensions to this area within the project will be quite speculative.
 For more information about WGT and its extensions, see arXiv:1510.06699. This paper is really more for background reading, but it serves as a crosscheck on our conventions for the gaugetheoretic formulation of gravity.
 Speaking of our conventions, we mostly adhere to the setup in Blagojevic's excellent book. For the most part, you won't need anything from beyond the first three chapters. The critical level of understanding, so as to be able to connect with the QFT literature, is as follows. You should be able to expand some Lagrangian \(\mathcal{L}=b\mathcal{L}(A^{ij}_{\ \ \ \mu},h^i_{\ \mu})\), constructed from the RiemannCartan and torsion tensors and various other ingredients (such as derivatives), to quadratic order around the vacuum. You should also be satisfied that the whole is invariant under translational and Lorentzrotational gauge transformations.
 A very useful resource which will combine much of the above literature is YunCherng's Ph.D. thesis. The PSALTer code uses very different methods to those described in this implementation (which is a complicated search over root systems), but the background physics will be very useful.
 The PSALTer code itself, when a studentfriendly version is finished later this term, will be available for download here. For the moment, you can in which I've performed the particle spectrum analysis.
 A Lentterm update to the last item above: a studentfriendly version of PSALTer is now ready and functional. However the research programme has since developed into a broader collaboration and in order to protect the IP I'll be moving the source code over from GitHub to GitLab soon. To use PSALTer in the interim, please ask in person. For the moment, you can refer to the output below of the analysis of EinsteinCartan gravity and general relativity.
 Not that it has to do with particle spectra, but someone was interested in accelerated expansion. For those of us that think it has something to do with a cosmological constant, and who are wondering why the value of such a constant is so small, check out the review by Jerome Martin at arXiv:1205.3365.
 One of the more recent particle spectrum analyses in the space of metric affine gauge theory (the superset of nonmetric, Weyl and Poincaré gauge theories of gravity in which the conncetion is entriely general), this gives a very thorough breakdown of the spinparity decomposition for the general threeindex tensor in four dimensions arXiv:1912.01023.
 The initial stages of the particle spectrum analysis assume a momentumspace representation for the Lagrangian, which should be checked when developing `short cuts' for the removal of surface terms. This also connects with the claim that the Lagrangian operator in \(\mathcal{L}=\frac{1}{2}\hat{\zeta}^\text{T}(k)\hat{\mathcal{O}}(k)\hat{\zeta}(k)\) must be Hermitian.
Nonperturbative quantum gravity
 Tim Budd's excellent online course on Monte Carlo techniques should serve as an introduction to how lattice (Q)FTs are implemented on computers. Not all of the applications are to quantum systems, so be careful! However, there is coverage of criticality which is vital in lattice QFT, and even Euclidean dynamical triangulations (which are intended to be quantum), with some quickstart python implementations of the latter.
 The Dynamical Triangulations course, also by Tim Budd, for PITP, gives a far more indepth introduction to random geometries. For those working with triangulation models, I can strongly recommend watching both of the lectures. There are also some C++ and Mathematica resources from this course: be careful however, since the C++ implementation has some "black box" matter coupled to it, while the Mathematica contains some depreciated syntax. I've fixed the notebook, and if you then you should be able to make some simple embedding visualisations such as that displayed here.
 A punchy "white paper" by Burda, Jurkiewicz and Krzywicki, which sets out a plan for putting fermions on random lattices, can be found at arXiv:9907013.
 A paper by the same authors, with much more detail, can be found at arXiv:9905015.
 A more indepth discussion of how to implement fermions and Ising systems coupled to two dimensional gravity can be found in arXiv:0110063.
 Massless MajoranaWilson fermions coupled to quantum gravity in two Euclidean dimensions are studied in arXiv:0107015. I'm happy that this is a standard formulation of EDT.
 Massless MajoranaWilson fermions coupled to quantum gravity in two Lorentzian dimensions are studied in arXiv:0306033. I've not confirmed that this is standard CDT.
 A summary of the above two papers can also be found at this Acta. Phys. Polon. B article.
 Something to note about the above implementations (and introductions to conformal field theories) is the restriction to fermions which are symmetric under charge conjugation (Majorana spinors). Majorana spinors will be a pain to work with when including gauge symmetries, so can we use Dirac spinors? To investigate this just at the level of holomorphic/antiholomorphic field equations, see below.
 A different (and far more recent) branch of the literature in which CDT and some motivated internal gauge symmetries are considered in two dimensions arXiv:2010.15714.
 The thorough introduction to CFTs from Champs, Cordes et Phénomènes Critiques can be found at arXiv:9108028.
 The conformal algebra in two dimensions has an infinite number of generators. This is quite strange, and indeed when we are doing physics we tend to throw away all the generators which produce singularities at either the north or south pole of the Riemann sphere. In higher dimensions \(d>2\) we obtain a finite algebra simply by looking at infinitessimal coordinate transformations, and the critical observation is that third derivatives of the transformation vector vanish. To recover that result, see below.
 Tong's lectures on statistical field theory can be found here.
 Why restrict to \(N\)spheres? CDT is implemented on a \(2+1\) torus in arXiv:1305.4702.
Cosmological perturbation theory
 Gaugetheoretic formulation of the torsion condensate arXiv:2003.02690. This is mostly for context, but with fair winds and following seas we might be able to begin to explore perturbation theory directly in the gauge field framework.
 Scalartensor formulation of torsionful gauge theories, which works at the background level (but how well does it work at the level of perturbations?) arXiv:2006.03581.
 Pedagogical introduction to cosmological perturbation theory found in James Fergusson's Part III Cosmology notes.
 A slightly more indepth exploration of perturbation theory, with discussion of extra d.o.f, can be found in the "Helsinki" notes.
 A fairly uptodate review of scalartensor theories, for contextual interest, can be found at arXiv:1901.08690. Note that this review covers most of the usual cases of scalar fields which may/may not augment Einstein's theory: the scalartensor models in these projects have a unique motivation which is not covered here! That said, Israel Quiros (who is a very interesting researcher) has also worked with torsionful models in the past.
 A more principled method for extracting extra scalar d.o.f from torsionful theories, with a cosmological/inflationary application, is presented in arXiv:1904.03545. The main difference here is the guarantee that no extra nonscalar torsion d.o.f are present, which could show up in the perturbation theory.
 An attempt to study cosmological perturbation theory using the tetrad and spin connection can be found at arXiv:1601.03943. In this work, the focus is on the scalar and pseudoscalar parts of the torsion tensor which Tsamparlis first demonstrated to be consistent features of the background cosmology, roughly defined as \(h\sim T{^\mu_{\ \ 0\mu}}\) and \(f\sim \epsilon^0_{\ \ \mu\nu\sigma}T{^{\mu\nu\sigma}}\). Accordingly, the analysis performed in this paper is really quite restrictive (and I'm not even sure if it is physically meaningful), but the Lu and Chee approach is at least worth a read.
 It appears that the most concerted (and most recent) work on the cosmological perturbation theory of torsionful gravity has been through the lens of the teleparallel theory. In teleparallel theory, the gravitational Lagrangian density is \(\mathcal{L}=\frac{1}{2}M_{\text{Pl}}^2\mathbb{T}\) where \(\mathbb{T}=\frac{1}{4}T_{ijk}T^{ijk}+\frac{1}{2}T_{ijk}T^{jik}T^{i}_{\ \ ji}T_{k}^{\ \ jk}\) is a specific combination of quadratic torsion invariants. For teleparallel gravity (and the natural extension to \(f(\mathbb{T})\) gravity) to function like GR, there are some nuanced conditions that have to be applied to the spin connection. It is within the context of these conditions that the cosmological perturbation theory has been explored in papers such as arXiv:1911.06064, arXiv:2001.10015 and arXiv:2110.12332. These were skimmed from a halfhour on the arXiv, so I would strongly recommend you make a very thorough literature review of possible approaches to the SVT decomposition in the gaugetheoretic formulation: other authors may have devised some really neat tricks that could save you time!
 When your code starts to become successful, you will want to implement background values of the scalar fields \(\phi\) and \(\psi\) which are commensurate with known exact solutions to the cosmological field equations of a given theory. Depending on your project abstract, you might be looking into different theories, but the candidate of most interest to our group is the constant torsion emergent gravity (CTEG), in which the pseudoscalar torsion \(\psi\) adopts a constant background value: the torsion condensate or correspondence solution. Whilst the value of the condensate is well known, as far as I can recall none of the papers on CTEG actually quote the value of \(\phi\) on the background. It turns out after a computation that \(\phi\propto H\) as the Universe evolves with Hubble number \(H\), but that the constant of proportionality varies depending on which kind of matter is dominating the cosmic fluid. To show this, and for inspiration to those who are working on exact solutions to other modified gravity theories (I'm thinking of MoND here!), there is prepared a short Wolfram script with .pdf output below. You can also .
 In tandem with the Wolfram script above, some parts of the calculation are easiest done by hand. These are shown below, and at the end in the red boxes you can see a summary of the background solutions for the CTEG.
Logistics
Below is some logistical information about the structure of research projects, including relevant deadlines. Much of this is transplanted from the various departmental websites (which you should check regularly for updates), but in places where I have my own deadlines/ammendments I will indicate in bold, underlined and italic script.
Quasiregular Friday meetings
Research group meetings for (confirmed) new members, and members who joined as summer students in recent months, will be held in slots from 09:0014:30 on Fridays (with later slots for those who really can't make the morning). These will be inperson by default, but a Zoom room will be open during that period. Book these below.
Note that whilst both astrophysicists and physicists are expected to focus more on projects during Lent term, the degree to which I can expand my calendar to match will be very limited. Therefore, I'd strongly advise that we make the most of meetings during Michaelmas.
M.Sci. and M.Ast. students for Part III Astrophysics (applications CLOSED)
Michaelmas Term
An electronic PDF copy interim progress report, length no more than 1,000 words, bearing the signature(s) of the main supervisor(s) and second supervisor, must be uploaded to the Part III/MASt Astrophysics Moodle site no later than 12:00 Friday 2nd December 2022 (the last day of Michaelmas Full Term). The report should be produced with LaTeX, or an equivalent textprocessing package and may contain material that can be incorporated in the final project report. The interim report must indicate the progress made so far and show preliminary results. It should also give a clear indication of the project aims and a detailed plan of how these aims will be achieved. This is particularly important where the results of the project depend on data that has yet to be analysed. There is no need for the interim report to reiterate the material given in the Project Handbook. The interim reports do not constitute part of the formal assessment but are regarded as an essential part of the monitoring procedure. The Course Coordinator will assess these reports and provide feedback to students and supervisors.
Lent Term
Practice oral presentations, consisting of a 20minute talk followed by up to 10 minutes of questions, to an audience of Part III Astrophysics students, Project Supervisors and the Project Coordinator will be given on the last Tuesday, Wednesday, Thursday, and Friday of Lent Term (14th, 15th, 16th and 17th March 2023). A final timetable for the presentations will be provided by email during the previous week. This practice presentation is not formally assessed but offers the opportunity to become familiar with the format of the presentation, to be assessed by the Part III Examiners in the Easter Term. Students are encouraged to attend the practice talks of their peers which will help strengthen their presentation techniques.
Easter Term
A draft of the final project report, generated with LaTeX or an equivalent textprocessing package, should be handed to the Project Supervisor no later than Monday 24th April 2023. The last Supervision, to discuss the draft report, should take place no later than Monday 1st May 2023.
I will not be available on Monday 1st, so our final supervisions will be our usual meeting slots on Friday 28th April.
An electronic PDF copy of the final project report must be uploaded to the Part III/MASt Astrophysics Moodle site no later than 12:00 BST Monday 8th May 2023. Late submissions are very strongly discouraged because you will be left with insufficient time to properly revise for the written examinations. In circumstances in which it is unavoidable you must seek permission in advance and then any late submissions must be submitted via your college Tutor with an accompanying letter of explanation from the Tutor. Your University Examination Number must NOT appear anywhere in the report or on the cover sheet.
Register interest
The projects which were advertised can be found in the IoA project booklet.
The IoA has provided the results of their student allocation algorithm. All projects are now allocated.
M.Sci. and MASt students for Part III Physics (applications CLOSED)
To see a list of abstracts on offer, go to the Teaching Information System.
In response to popular demand, one new project was added, to be cosupervised by Dr. Amel Durakovic.
Progress reports
Students will be asked to complete two progress reports. At the end of the Michaelmas Term, you must submit an Initial Report (one copy; between 4 and 6 A4 pages in length). This Initial Report should describe the project in your own words, putting the physics into context (including references to the relevant literature) and describing the goals of the project; it must also include a project plan. This report should be electronically signed by both you and your supervisor to indicate his or her agreement with the plan and should be uploaded onto the TIS by Friday 2 December 2022 by the student. A copy of the Initial Report will be retained by the Undergraduate Office and forwarded to the assessor in Easter Term – failure to submit an Initial Report will result in the loss of 5% of the available project marks.
The second report is a simple “tick box” form, which can be downloaded from the TIS during week three of the Lent term. This will invite you to report any problems with your project, and to confirm that a presentation has been scheduled. The form should be uploaded to the TIS by Wednesday 8 February 2023. The second report will not form part of any assessment but will allow any problems to be identified by Professor Hirst well before the time the project has to be handed in.
It is very important that students bring any unforeseen delays or other problems with their projects to Professor Hirst’s attention at the earliest possible opportunity. The earlier such problems are addressed, the more chance there is of taking suitable remedial action.
The project writeup
The project should usually be presented in the style of a paper published in a scientific journal. The main text (excluding appendices and abstract) should be concise (20–30 pages, 5000 words maximum (excluding references)). The text should describe and explain the main features of the project, the methods used, results, discussion and conclusions, and should be properly referenced. Detailed measurement records, calculations, programs, etc. should be included as appendices. In addition, there must be an abstract of at most 500 words.
This final writeup is an important part of the project and must be the student's own work. A lecture on Project report writing will be given on Tuesday 24 January 2023 at 4:00pm in the Small Lecture Theatre. Once the majority of the research work has been completed, the student and supervisor should discuss the general structure and content of the report before writing is started. Thereafter, the student must write the final report without advice on the report from the supervisor, although discussion of the scientific results is allowed during this period. A set of handy tips and information is given in the booklet entitled Keeping Laboratory Notes and Writing Formal Reports.
Submission of the project
The deadline for submission of the project is:
4:00 pm on the third Monday of Easter Full Term (15 May 2023)
A request for a delay in the handin date of your project report due to illness must go through your Director of Studies and then be agreed by the Applications Committee. Treat this deadline like you would an exam date.
The project report should be submitted as a pdf file to the TIS before the submission deadline. A link to your online notebook should also be added to the first page of your report before uploading the project.
To preserve anonymity when your project is looked at by the Part III examiners, your name must not appear on the project report itself. You should ensure that your candidate number appears on the first page of your project, together with the title of the project and your supervisor’s name.
Your project report should contain the following statement on the first page of the project: Except where specific reference is made to the work of others, this work is original and has not been already submitted either wholly or in part to satisfy any degree requirement at this or any other university.
Register interest
To apply to a project, please email me a copy of your CV, with some indication of your undergraduate exam record, and book a slot to discuss from the appointments below (you don't need to be "invited" to meet, just turn up!):
Initial allocations were made on Friday 14th and Sunday 16th October. Final allocations were made on Tuesday 18th October. All projects are now allocated.