1. Spherical Collapse and the Mass Function – Chameleon Dark Energy Stephen Appleby, APCTP-TUS dark energy workshop 5 th June, 2014 M. Kopp, S.A.A, I. Achitouv, J. Weller Phys. Rev. D 88, 084015

2. Talk Overview Chameleon dark energy – brief introduction Spherical collapse in General Relativity Spherical collapse with a chameleon field Conclusions

3. Scalar Fields If dark energy is dynamical, then a light scalar field can be used to model the expansion history. The scalar field must be light to give dynamical acceleration at the present time. Light scalar fields will generically produce large effects at solar system scales. Example – Brans-Dicke gravity.

4. Brans-Dicke Field Consider the weak field limit of the Einstein equations

5. Light Scalar Fields To provide a mechanism for dynamical dark energy, the scalar field must be light Light scalar fields generically introduce order unity effects in the Solar System, for ‘natural’ parameter values How can we construct a scalar field that can accelerate the expansion whilst simultaneously satisfying astrophysical tests of gravity?

6. Light Scalar Fields 1)Force the field to have no couplings -Quintessence, K-essence … 2) Couple the field to dark matter but not baryons -Coupled dark energy -Violation of equivalence principle 3) Universal coupling, introduce non-linear self interactions to suppress dynamics on small scales -Chameleon, Galileon, DGP -Self interactions can be derivative (Vainshtein) or potential (Chameleon)

7. Chameleons How the chameleon field works - (Khoury, Weltman 2004)

8. Chameleons How the chameleon field works - Effective mass of the field becomes large – field becomes non-dynamical At cosmological densities, the field can roll (Khoury, Weltman 2004)

9. Chameleons Chameleon fields are designed to reproduce General Relativity in high density surroundings (such as in the solar system) We can test for their presence on larger scales, where the average density is much lower. Galaxy clusters represent the largest bound objects in the Universe, and are an ideal test for these models

10. Galaxy Clusters What do we measure? o The number of galaxy clusters observed as a function of mass. We require a theoretical prediction for the halo mass function This can be achieved in GR using spherical collapse and the excursion set formalism

11. Excursion Set Approach At any point in space, performs a random walk as a function of smoothing scale S. Process is Markovian Boundary conditions

12. Mass Function - GR Analytic analysis – Excursion set approach o Initial density perturbation is a Gaussian random field with variance o Take a region of space containing mass M that is collapsing. This is connected with a comoving length scale by o Dark matter halos form at the peaks in the density field.

13. Collapse Threshold - GR An important quantity is the threshold of collapse In General Relativity, one can calculate analytically for a spherically symmetric collapsing over-density

14. Collapse Threshold - GR The Einstein and fluid equations describing a spherically symmetric over-density – Must input the initial density profile Initial velocity perturbation obtained by solving the constraint equation on the initial time slice

15. Collapse Threshold - GR Due to Birkhoff’s theorem, there exists an exact solution to the full non-linear collapse equations in GR Solve the non-linear collapse to the collapse time, and then evolve the linear perturbation equations to this time to obtain Solution -

16. Collapse Threshold - Chameleon For Chameleon dark energy models, Birkhoff’s theorem is not applicable! Cannot use the fully non- linear equations as in GR Must solve the modified equations

17. Collapse Threshold – Quasi- Static Approximation The full Relativistic equations are much more complicated! We have applied the quasi-static approximation This is an ansatz – in GR one can show that it is valid for the whole collapse (even on approach to the singular collapse point).

18. Collapse Threshold – Quasi- Static Approximation For non-standard cosmologies, one must be careful to check that the quasi-static approximation is valid

19. Collapse Threshold - Chameleon For Chameleon dark energy models, Birkhoff’s theorem is not applicable! Cannot use the fully non- linear equations as in GR Must solve the modified equations

20. Collapse Threshold - Chameleon Problem – we can no longer use a top-hat initial condition as in GR! Shell crossing occurs – different regions of the profile collapse at different rates, leading to a spike in the density field (Alex Borisov, Alex Borisov Bhuvnesh JainBhuvnesh Jain, Pengjie Zhang arxiv:1102.4839)

21. Collapse Threshold - Chameleon We must use a ‘well motivated’ initial condition, derived from a primordial density perturbation We begin our numerical calculations at, so we must multiply this initial condition with the late time transfer function (obtained from CAMB). We then Fourier transform to obtain

22. Collapse Threshold - Chameleon

23. Collapse Threshold

24. We perform one thousand spherical collapse runs using different initial conditions and chameleon potential parameters. We use these to construct a fitting function This converges to LCDM for

25. Mass Function We use our fitting function as a scale dependent (moving barrier) in the excursion set approach outlined earlier. The mass function now exhibits a non-trivial scale dependent deviation from GR – order unity effects! One can use our mass function to confront this class of models with data…

26. Conclusions We have considered a non-standard dark energy model – chameleons The most significant constraints on such models will arise in regions of low density – galaxy clusters! We have constructed a mass function that can be used to constrain this class of models When studying non-standard cosmologies, there are many complications o Is the initial density profile ‘natural’? o Quasi-static approximation valid? o Moving barrier – requires non-trivial extension to excursion set approach o Definition of the mass of a cluster related to filter used – complicates analysis Next – Ellipsoidal collapse? 