###### Abstract

Allowing for the possibility of large extra dimensions, the fundamental Planck scale could be anywhere in the range

LANCS-TH/9821

hep-ph/9810320

(October 1998)

Inflation with TeV-scale gravity

needs supersymmetry

David H. Lyth

Department of Physics,

Lancaster University,

Lancaster LA1 4YB. U. K.

E-mail:

1. There is a large hierarchy, , between
the mass of the Standard Model Higgs field and
the Planck scale .
This makes it difficult to understand the existence of the Higgs field,
because in a generic field theory valid
up to the Planck scale every elementary scalar field will have
a mass of order . To be precise, the mass will be given by
, where the first term is the tree-level value and the
are one-loop contributions of order
, with the strength of the interaction.
To avoid this disaster, it is usual to invoke supersymmetry which
automatically cancels the one-loop contributions to
sufficient accuracy.^{1}^{1}1The alternatives are an accidental cancellation, perhaps to be
understood anthropically [1], or a composite Higgs such as
occurs in technicolor theories. It is difficult to construct composite
theories that are consistent with observation.

An alternative proposal [2] is to place the fundamental Planck scale in the TeV region. This makes the one-loop contributions of order and avoids fine-tuning. The low value of is achieved by invoking extra space dimensions with large compactification radius. If there are extra space dimensions with compactification radius , and is the fundamental Planck scale in spacetime dimensions, the Planck scale that we observe is given by

(1) |

Einstein or Newtonian gravity holds on scales , but is modified on smaller scales. Putting and gives , which is allowed by observation since the law of gravity is unknown on scales . One can envisage more complicated schemes, where the compactified dimensions have different radii, but in all cases the biggest dimension must be .

In this scheme, the hierarchy is replaced by the
hierarchy , which is of a different type and might be
easier to understand. Investigations reported so far
[3] seem to suggest
that the scheme is viable, with no need for supersymmetry
in the field theory that contains the Standard Model.^{2}^{2}2Other works, for example [4], have explored the possibility
of large extra dimensions within the context of supersymmetry.
As I now explain, this apparent success disappears
as soon as one tries to construct a model of inflation, that is
presumably necessary to generate structure in the Universe.
A more detailed investigation will be reported elsewhere
[5].

2. We are concerned with the cosmology of the observable Universe.
The Universe is modeled as a practically homogeneous and isotropic fluid,
with the distance between comoving fluid elements proportional to a
universal scale factor . The evolution of the scale factor is
given by the Friedmann equation, which
assuming spatial flatness is^{3}^{3}3For simplicity, I am assuming that
during inflation has its present value.
With declared fixed (the choice of energy unit) this amounts to
saying that has its present value.
The more general possibility is considered elsewhere [5].

(2) |

and the continuity equation

(3) |

Here is the Hubble parameter, is the energy density and is the pressure.

The Friedmann and continuity equations are consequences of Einstein’s field equation, and are valid provided that all relevant quantities are smoothed on a comoving distance scale . The cosmic fluid, which is the subject of cosmology, is defined at a given epoch only after such smoothing. (There is no question of ‘modifying the Friedmann equation on short distance scales’ since we are dealing with a universal scale factor.)

The history of the Universe begins at some energy density , where is the fundamental Planck scale. In particular, the potential during inflation satisfies

(4) |

The vacuum fluctuation of the inflaton field , that is supposed to be the origin of large scale structure, is generated on each comoving scale at the epoch of horizon exit . We therefore require the Hubble distance to be much bigger than the radius of the internal dimensions, . Because of Eq. (4), this is not a very severe restriction. Since is the smallest distance that makes sense in the context of quantum gravity, we must have . Then Eqs. (4) and (2) give , whereas observation requires . Irrespective of observation, Eqs. (1) and (4) require if [6]. (Indeed, they give .) The inclusion of additional, smaller dimensions only strengthens this result.

If is the inflaton mass
during inflation,^{4}^{4}4During inflation, the ‘vacuum’ for quantum field theory is defined by
the values of the inflaton and other relevant fields, which may be
different from their true vacuum values. In hybrid inflation models
[7]
a field has a coupling like , which holds it at
the origin during inflation. Afterwards, acquires its true vacuum
value, which can give the inflaton a large mass in the vacuum.
As a result, the quantum field theory containing Standard Model particles and
the inflaton could have all scalar masses of order
in the true vacuum.
But we need a sensible quantum field theory also during
inflation.
the potential is of the form

(5) |

The dots represent additional terms, which might come from a variety of sources [6] (higher powers of representing interaction terms in the tree-level potential, logarithmic terms representing loop corrections etc.). In order to generate the nearly scale-invariant primordial curvature perturbation, that is presumed to be the origin of large scale structure, one should have [6]

(6) |

while cosmological scales are leaving the horizon.

Since each term has a different dependence, and usually varies significantly over cosmological scales, there is hardly likely to be an accurate cancellation between terms. Then, discounting the possibility which would surely place the field theory out of control, the flatness condition has to be satisfied by each term individually, with dominated by the constant term . In particular, the mass has to satisfy . Remembering that this becomes [8] , or

(7) |

To summarise, taking the fundamental Planck scale to be of order removes the hierarchy between the Higgs mass and , at the expense of introducing at least the same hierarchy between the inflaton mass and . To protect the inflaton mass from quantum corrections, supersymmetry is needed, just as it was needed to protect the Higgs mass in the case .

In order to reheat the Universe, the inflaton must have
significant couplings (not necessarily tree-level) with
Standard Model particles. As a result, these particles should belong to
the same supersymmetric field theory as the inflaton.^{5}^{5}5By contrast, the Kaluza-Klein tower of scalar particles, associated
with the extra dimensions, need not be considered as part of the same
theory since they will have very weak coupling. For the same reason,
their masses are presumably not destabilized by loop corrections.

If one accepts the hypothesis of cosmological inflation, the original motivation [2] for considering is now removed, and there seems to be no reason why Nature should have chosen this value. Still, one may choose to explore that possibility, either because it will be accessible to observation in the forseeable future or because a lot of effort has been invested in it.

3. In that case, one might ask whether a viable model of inflation can be constructed. It is easy enough to write down a potential , valid during slow-roll inflation, that gives the correct curvature perturbation on COBE scales and a spectral index within the observed band . Take, for instance, the potential

(8) |

with the additional terms negligible during slow-roll inflation. Let us assume that

(9) |

where is the end of slow-roll inflation and is the epoch when COBE scales leave the horizon. Using well-known formulas [6], the COBE constraint is independently of , and . Here is the number of -folds after COBE scales leave the horizon, given (discounting thermal inflation and late-decaying particles) by

(10) |

Moreover, , so the initial assumption Eq. (9) is not very restrictive.

One might also wish to impose the constraint ; for instance this might be necessary to have control over non-renormalizable terms if they are of order , or to have control over the running of couplings and masses in a renormalizable theory. In the above example this constraint no problem, but in general it is a severe restriction as will be discussed elsewhere [5].

4. It is generally accepted that a viable cosmology should begin with an era of inflation, to set suitable initial conditions for the subsequent hot big bang and in particular to provide an origin for structure. We have argued that the inflaton mass during inflation has to satisfy , and that supersymmetry should be invoked to stabilize this mass (or the masses of scalar fields produced after inflation). A more detailed investigation [5] supports this conclusion. If one accepts it, along with the need for inflation, one concludes that there is no reason to think that Nature has chosen , however convenient such a choice might have been for the next generation of collider experiments.

## Acknowledgements

This work was initiated at CERN. I am grateful to CERN for support, and to Steve Abel, Karim Benakli, Keith Dienes, Tony Gerghetta, John March-Russell and Toni Riotto for useful discussions and correspondence.

## References

- [1] V. Agrawal, S. M. Barr, J. F. Donoghue and D. Seckel, Phys. Rev. D57 (1998) 5480.
- [2] N. Arkani-Hamed, S. Dimopoulos and G. Dvali, hep-ph/9803315, to appear in Phys. Lett. B.
- [3] I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos and G. Dvali, hep-ph/9804398, to appear in Phys. Lett. B; N. Arkani-Hamed, S. Dimopoulos and G. Dvali, hep-ph/9807344; N. Arkani-Hamed, S. Dimopoulos and J. March-Russell, hep-th/9809124; P. C. Argyres, S. Dimopoulos and J. March-Russell, hep-th/9808138; R. Sundrum, hep-ph/9807348.
- [4] K.R. Dienes, E. Dudas and T. Gherghetta, hep-ph/9803466 and hep-ph/9806292; K.R. Dienes, E. Dudas,T. Gherghetta and A. Riotto, hep-ph/9809406. G. Shiu and S. H. H. Tye, hep-th/9805157. R. Sundrum, hep-ph/9805471.
- [5] D. H. Lyth, in preparation.
- [6] D. H. Lyth and A. Riotto, hep-ph/9807278, to appear in Phys. Reports.
- [7] A. D. Linde, Phys. Lett. B259 (1991) 38.
- [8] K. Benakli and S. Davidson, hep-ph/9810280.
- [9] P. Horava and E. Witten, Nucl. Phys. B475 (1996) 94; Nucl. Phys. B460 (1996) 506.
- [10] J.D. Lykken, Phys. Rev. D54 (1996) 3693.