1. P-Term Cosmology A.C. Davis (with C. Burrage) 0705.1657,0707.3610

2. Outline Motivation: Inflation in String theory What is the P-term model? Problems of FI terms in supergravity Construction of P-term potentials in supergravity Cosmology Conclusions

3. Inflation and String Theory Inflation is a successful way of explaining the large scale behaviour of the universe The simplest models require a scalar field and a gently sloped potential Currently there is no explanation of what this field is, or where the potential comes from String theory is a proposed ‘theory of everything’ We would like to see how it gives rise to observed phenomena We would like to find a model of string theory inflation

4. z In type II string theories there exist ‘D-p Brane’ solutions z Open strings have their end points on the Branes located at orbifold fixed points in the compact dimensions z Inflation occurs as two brane approach and collide z Superstring theories contain additional massless ‘moduli’ fields z Need a potential for these moduli so they are stabilised Brane Inflation in String Theory

5. Moduli Stabilisation One of most developed models of Brane inflation is KKLMMT, based on methods of moduli stabilisation of KKLT etc. In warped compactifications of IIB the inflaton has too large a mass The D3/D7 model has a shift symmetry (inflaton position of D3 wrt D7) which gives a flat direction in the potential and hence a small inflaton mass

6. D3 Brane φ φ inflaton field Brane-Antibrane Inflation

7. P-term P-term model with global SU(2,2|2) superconformal gauge theory corresponds to dual gauge theory of supersymmetric D3/D7 Branes Superconformal symmetry broken to N=2 SUSY by the vev of auxiliary triplet field P of the vector multiplet Contains a triplet of constant terms which arise from a magnetic flux triplet in the D3/D7 construction The 4D effective theory of the D3/D7 model –Is N=2 SUSY –Potential is generated by the vev of the auxiliary field P –P contains a triplet of FI terms

8. Scalar potentials in Supersymmetry In N=1 SUSY –with superpotential –and D-term F, D are N=1 auxiliary fields –D-term potentials –F-term potentials Both give rise to hybrid inflation

9. Hybrid Inflation Such potentials have –Flat directions (minima) –Degenerate SUSY minima Suitable for –Inflation –Cosmic string formation

10. P-term Scalar Potentials The Super-Bogomol’nyi limit is –the theory contains a second supersymmetry (second symmetry is anti-chiral) From an N=2 view point the auxiliary field is an SU(2) triplet P –Can choose direction to look like F or D FI terms appear in both the D-term and the superpotential –is of the right form for hybrid inflation D-term F-term P-term

11. FI terms in Supergravity A U(1) gauge theory with an FI term can be coupled to supergravity only if the superpotential of the theory is invariant under the R-symmetry and transforms under the gauge symmetries An example: With a single U(1) gauge group the superpotential transforms as If the superpotential takes the simple form then the charges of the scalar fields must satisfy There is no straightforward way to generalise the SUSY formalism

12. FI terms in N=2 Supergravity Killing vector generating gauge group No. hypermultiplets Moment map containing matter fields Complex structures z Scalar potential is constructed from moment maps y Moment maps are a generalisation of concept of linear and angular momentum to symplectic manifolds y These are functions of the Killing vectors doing the gauging y FI terms would be constants in the moment maps

13. FI terms in N=2 Supergravity We can include constant terms by writing –where Right choice of Killing vectors and geometry gives rise to constant terms in the compensator e.g.: –Killing vector: –gives moment map: Constant term! SU(2) connections SU(2) compensator Hep-th/0511001, Achucarro et al

14. A P-term Model For a P-term model we want moment maps containing FI terms with an arbitrary choice of direction We gauge Killing vectors of the form –where is an SO(3) rotation amongst the b coords The corresponding moment map is

15. Truncating N=2 to N=1 SUGRA The truncation conditions are The N=1 scalar potential can be constructed from and the metric and vector kinetic matrices as

16. After truncation we want enough degrees of freedom to get an inflaton field and a matter field –The simplest choice is We choose the quaternionic geometry with metric –the truncation always imposes So the reduced K-H manifold is described by We choose the minimal special-geometry defined by the prepotential where Truncation implies Choice of Geometry

17. Gauging one Symmetry Have a superpotential but no D-term Have a D-term but no superpotential

18. Gauging one Symmetry Have a superpotential but no D-term Inflation harder to find except if when the potential has a flat direction along There is a supersymmetric minimum Have a D-term but no superpotential The potential is flat for the inflaton There is a supersymmetric minimum Cosmic strings are formed at the end of inflation

19. Gauging three symmetries We gauge one Killing vector to give the D-term, and two independent Killing vectors to give the superpotential In order to generate a group we must impose –The group is no longer Abelian –The N=1 gauge group is U(1) The N=1 scalar potential is gives an inflationary valley There is a SUSY minimum Strings at end of inflation

20. P-Strings and Semi-local Strings P-term models have string formation at the end of inflation – depending on how the symmetry has been broken from N=2 to N=1 we can get F, D or general P-strings formed. The structure is that of sugra strings, with usual Nielsen-Olesen profiles plus conditions on the gravitino. D-term strings are strictly BPS and give a relation between the gravitino potential and the FI term.

21. Semi-local strings can also form and evade the constraints on the string tension. These can arise naturally in these N=2 models depending on the manifold – (with Burrage, Esole and Sousa).

22. Conclusions P-term models arise as the 4-D effective theory of some types of Brane inflation The historical problems of embedding P-term models in supergravity can be avoided In our simple model we can make different choices of gauging which reduce to U(1) in N=1, but which give differently shaped potentials All the potentials we have looked at contain ‘inflationary valleys’ –But the FI constants are fixed –Cosmic strings are formed at the end of inflation.