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, 0809.1441AA, Sousa 0712.3460

de Alwis th/0506266, th/0506267Binetruy, Dvali, Kallosh, Van Proeyen th/0402046Choi, Falkowski, Nilles, Olechovski th/0411066Gallego, Serone 0812.0369, 0904.2537Brizi, 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

from Urrestilla, Mukherjee, Liddle Bevis, Hindmarsh, Kunz 0803.2059

WMAP

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

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

Einstein-Hilbert action with matter and cosmological constant

For inflation need much larger vev of the potential

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.

SUSY breaking and accelerated expansion

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

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

|F|2------------------------------

-

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

D=4, N=1 SUGRA with neutral scalars - cont.

Configurations with zero F-terms are automatically criticalpoints (*) of the potential

(*) not necessarily minima

Invisible moduli:

How decoupled can a scalar field be?

The problem:

This works in global SUSYbut 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 H0 (constant) they can be truncated,

Seff( L) = S (H0 , L )Can be consistent with supersymmetry,

(consistent truncation) But the vev is not usually constant

(both coincide if T and V are separable)

H = H0 solution of the e.o.m. ?

H = H0

i.e.

OK at one vacuum L= L0 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 LBut for consistent decoupling the solution must be

Some compactifications/critical points satisfy this property, others don’t.

Only then can we be sure that the low energy effectiveaction with H fixed at H0 is correct

Two obvious solutions: AA, Hardeman, Sousa 0809.1441-The LHS is independent of L

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:

Suppose there is a constant solution H 0 to DHW = 0

then

either

or

No-scale models with W indep of L are OKCanonical Kahler functions K ~ H H + … (or quadratic) also OK

(this is the decouplingansatz for global susy)

We can quantify how much this condition of L-independenceis violated in specific compactifications, e.g. type IIB KKLT and LVS (large volume scenarios) Balasubramanian Berglund Conlon Quevedo th/0502058

~ O(1 - 10) for KKLT

KKLT: ignore K

~ O(10-6) for LVS(with string loop corrections, ~ vol -2/3)

LVS: ignore Wnp

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) for global SUSYK= K(H) + K (L) , W = W(H) W(L) 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 rollbased 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 AdS maxima uplift to stable minima in dS for high V/m3/2

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, 0809.1441AA, Sousa 0712.3460

de Alwis th/0506266, th/0506267Binetruy, Dvali, Kallosh, Van Proeyen th/0402046Choi, Falkowski, Nilles, Olechovski th/0411066Gallego, Serone 0812.0369, 0904.2537Brizi, Gomez-Reino, Scrucca 0904.0370

AA, Hardeman, Palma in preparation

A necessary condition for (meta)stable de SitterVacua 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 superpotentialie those fields can have positive mass squared

A necessary condition for (meta)stability in the sgoldstino direction is

New dS vacua without uplifting

Stability of invisible moduli

Local AdS maximabefore uplifting

Local AdS minimabefore upliftingSaddle points

before uplifting

STABLEafter uplifting

A one-modulus toy model

F-terms of light,susy breaking sector

(depends on W)

F-terms of heavy SUSY sector

AA, Sousa 0712.3460 AA, Hardeman, Sousa 0809.1441

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)

.