Consistent decoupling of heavy scalars and moduli in

Consistent decoupling of heavy scalars and moduli in supergravity cosmology Ana Achúcarro (Univ. Leiden / UPV-EHU Bilbao) Marcel Grossmann 12 Paris, 16/7/ 09

This talk is about a general framework to analyse (classify) cosmologies based on N=1, d=4 SUGRA (work in progress) (*) In particular, to determine if they have (meta)stable de Sitter vacua (slow-roll) inflationary trajectories and their properties / observational signatures (*) with Sjoerd Hardeman, Gonzalo Palma, Kepa Sousa LEIDEN - HAMBURG - LAUSANNE - CERN

Covi, Gómez-Reino, Gross, Louis, Palma, Scrucca 0804. 1073, 0805. 3290, 0812. 3864 Gómez-Reino, Scrucca hep-th/0602246, th/0606273, 0706. 2785 AA, Hardeman, Sousa 0806. 4364, 0809. 1441 AA, Sousa 0712. 3460 de Alwis th/0506266, th/0506267 Binetruy, Dvali, Kallosh, Van Proeyen th/0402046 Choi, Falkowski, Nilles, Olechovski th/0411066 Gallego, Serone 0812. 0369, 0904. 2537 Brizi, Gomez-Reino, Scrucca 0904. 0370 AA, Hardeman, Palma in preparation

Accelerated expansion today And in the past (inflation)

Coming soon: CMB polarization, tensor modes, non-gaussianity? (gravitational waves, multi-field inflation, cosmic strings) Planck WMAP from Urrestilla, Mukherjee, Liddle Bevis, Hindmarsh, Kunz 0803. 2059

N=1 SUGRA in four dimensions is widely used as a low-energy approximation to (some regime of) string theory Extra dimensions ---- compactification --- moduli fields No evidence of dynamical moduli What is the effect of the acceleration on the moduli? Can they be destabilized? (yes…)

A few words about notation

Scalar fields and accelerated expansion Einstein-Hilbert action with matter and cosmological constant A scalar field with = 0 can mimic a cosmological constant: - Nearly homogeneous. - Static or very slowly varying. First string theory realization of this idea: KKLT (type IIB orientifold). Kachru, Kallosh, Linde, Trivedi 03 For inflation need much larger vev of the potential

SUSY breaking and accelerated expansion A vacuum configuration with (bosons) = constant (fermions) = 0 breaks supersymmetry iff D and/or F are non-zero Global SUSY transformation generates a massless goldstino, Schematically so a positive cosmological constant (acceleration) requires broken SUSY. Moreover, F=0 implies D=0, so first study SUSY breaking by F terms. Will ignore gauge fields for this talk (no D-terms)

SUSY/SUGRA actions are entirely determined by three functions: Focus on simplest case - neutral scalar fields, no gauge fields - need only K, W

N=1 SUGRA with neutral, scalar fields - + fermions |F|2 ---------------Kähler metric (of scalar manifold) Inverse Kähler metric Kähler-covariant derivative SUSY vacua have F ~ DW = 0, and so V < 0 (unless W= 0, Minkowski) de Sitter space and inflation require SUSY breaking

D=4, N=1 SUGRA with neutral scalars - cont. In supergravity there is a redundancy between K and W - everything is described in terms of the Kähler invariant function: Work in units MP =1

Configurations with zero F-terms are automatically critical points (*) of the potential (*) not necessarily minima

Invisible moduli: How decoupled can a scalar field be?

The problem: This works in global SUSY but is not possible in supergravity, even at tree level: gravity couples to everything SUSY restricts the form of the couplings

Decoupling ansatz in global SUSY

Gravitational coupling is not dimensionless GNewton ~ MPlanck -2 As energy increases, the gravitational interaction terms increase as (E / MPlanck ) 2 But this is not good enough for inflation, in general (decoupling ansatz does not work in SUGRA, see later)

If there is a separation of scales between heavy (H) and light (L) fields, in a given vacuum, can integrate out heavy fields Not usually consistent with local supersymmetry (∂H W = 0 vs D H W = ∂H W + ∂H K W = 0 ) Not enough for Cosmology Inflation may require trajectories in field space where the v. e. v. of the inflaton changes by > O(MPL) (e. g. if tensor modes are detected) If the heavy fields are frozen at their vev H 0 (constant) they can be truncated, Seff( L) = S (H 0 , L ) Can be consistent with supersymmetry, (consistent truncation) But the vev is not usually constant (both coincide if T and V are separable)

H = H 0 solution of the e. o. m. ? H = H 0 i. e. OK at one vacuum L= L 0 but not over a whole trajectory. H is sourced by curvature in the scalar manifold --- constrain on metric.

Study the decoupling of heavy scalars and moduli (and their superpartners!) In N=1 SUGRA subject to two explicit conditions: -The v. e. v. of the heavy fields should be unaffected by low energy phenomena (in particular, SUSY breaking) -The low energy effective action should also be N=1 SUGRA for the remaining fields (*) Binetruy Dvali Kallosh Van Proeyen 04

To decouple a heavy scalar or modulus need F-term = 0 (the whole supermultiplet decouples) (also what is found in flux compactifications) Generic solution is a relation between H and L But for consistent decoupling the solution must be Only then can we be sure that the low energy effective action with H fixed at H 0 is correct Some compactifications/critical points satisfy this property, others don’t.

Two obvious solutions: -The LHS is independent of L AA, Hardeman, Sousa 0809. 1441 Binetruy, Dvali, Kallosh, Van Proeyen th/0402046 -The LHS factorizes e. g. - after Gallego and Serone 0812. 0369 The LHS of DHW = 0 factorizes (SUSY Minkowski vacuum) Need also the condition on K.

One case that does not work: (this is the decoupling ansatz for global susy) Suppose there is a constant solution H 0 to DHW = 0 then either or No-scale models with W indep of L are OK Canonical Kahler functions K ~ H H + … (or quadratic) also OK

We can quantify how much this condition of L-independence is violated in specific compactifications, e. g. type IIB KKLT and LVS (large volume scenarios) Balasubramanian Berglund Conlon Quevedo th/0502058 KKLT: ignore K LVS: ignore Wnp ~ O(1 - 10) for KKLT ~ O(10 -6) for LVS (with string loop corrections, ~ vol -2/3) Berg Haack Pajer 0704. 0737

SUMMARY / OUTLOOK • Integrating out / truncating fields (such as stabilized moduli) is problematic in SUGRA, supersymmetry protection easily lost. BEWARE OF SUGRA /string ACTIONS WITH FEW FIELDS… • The best decoupling ansatz is K= K(H) + K (L) , W = W(H) + W(L) K= K(H) + K (L) , W = W(H) W(L) for global SUSY for SUGRA [ G = G(H) + G(L) ] (preserves SUSY critical points of H, BPS configurations in the effective action for L are really BPS, etc) SOME COMPACTIFICATIONS (LVS) DO BETTER THAN OTHERS (KKLT) • Develop tools for analysing generic N=1 SUGRA potentials for cosmology. Work in progress. Some highlights (not in this talk) Necessary condition for stability of de Sitter vacua and slow roll based on the curvature of the scalar sigma-model metric New metastable de Sitter vacua without the need for uplifting “Invisible” moduli are possible, but whether they are approximately realised in string theory is a different question. If so, Local Ad. S maxima uplift to stable minima in d. S for high V/m 3/2

Covi, Gómez-Reino, Gross, Louis, Palma, Scrucca 0804. 1073, 0805. 3290, 0812. 3864 Gómez-Reino, Scrucca hep-th/0602246, th/0606273, 0706. 2785 AA, Hardeman, Sousa 0806. 4364, 0809. 1441 AA, Sousa 0712. 3460 de Alwis th/0506266, th/0506267 Binetruy, Dvali, Kallosh, Van Proeyen th/0402046 Choi, Falkowski, Nilles, Olechovski th/0411066 Gallego, Serone 0812. 0369, 0904. 2537 Brizi, Gomez-Reino, Scrucca 0904. 0370 AA, Hardeman, Palma in preparation

A necessary condition for (meta)stable de Sitter Vacua and slow-roll inflationary trajectories

Covi et al. find a necessary condition for the existence of stable de Sitter vacua that depends only on the scalar metric: The most dangerous direction in field space is the sgoldstino (the scalar superpartner of the “fermionic Goldstone” that is generated by a broken SUSY transformation) All other directions can be made stable by tuning the superpotential ie those fields can have positive mass squared A necessary condition for (meta)stability in the sgoldstino direction is New d. S vacua without uplifting

Stability of invisible moduli

A one-modulus toy model AA, Sousa 0712. 3460 AA, Hardeman, Sousa 0809. 1441 F-terms of heavy SUSY sector F-terms of light, susy breaking sector STABLE after uplifting (depends on W) Local Ad. S maxima before uplifting Saddle points before uplifting Local Ad. S minima before uplifting

The same result holds for any number of moduli: SUSY stable (|x|>2) critical points at zero uplifting are destabilized at high uplifting (expected) Critical saddle points / maxima (|x|<1) at zero uplifting become increasingly stable with high uplifting (new) In (uplifted, non-supersymmetric) Minkowski Backgrounds all SUSY critical points are stable -or flat- along the invisible moduli directions. (same as for SUSY Minkowski vacua)