DARK ENERGY PARTICLE PHYSICS PROBLEMS

STEEN HANNESTADUNIVERSITY OF AARHUS

NBI, 27 AUGUST 2007

WHAT ARE THE PARTICLE PHYSICS PROBLEMSRELATED TO DARK ENERGY?

A WELL-KNOWN PROBLEM IN COSMOLOGY IS THAT VACUUMENERGY IS INFINITE, OR AT LEAST GIVEN BY SOME ULTRAVIOLETCUT-OFF IMPOSED ON THE THEORY

FURTHERMORE IT IS PROPORTIONAL TO VOLUME, I.E. IT DOES NOTDILUTE WITH THE COSMOLOGICAL EXPANSION.

IN PRINCIPLE THE VACUUM ENERGY PROBLEM COULD BE A QUESTIONOF RENORMALIZATION, BUT THAT REQUIRES EXTREME FINE TUNING

4~

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3

2

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THE FIELD THEORY VIEW OF DARK ENERGY

Dark energy is associated with a very low energyscale

eV 10~~ 34/1 E

Such a low energy scale is extremely hard to realise in realistic particle physics models of dark energy

Possible ”solutions” are:

A slowly rolling scalar field – Quintessence(does not solve the underlying problem)

Solutions from string theory – KKLT, holography(again not solutions strictly in field theory)

Dark energy is not a field theory prolemModified gravity – DGP, f(R), etc.

Problems in quintessence:

In order to have w ~ -1 the scalar field mass must be extremely small (i.e. the potential is extremely flat)

Why a new scalar field with characteristic energy scale 10-3 eV?

eV 10~~ 331 Hm

How can such a small mass be radiatively stable?

Here are a few examples of models which at leastseemingly avoid this problem

Coupling neutrinos to dark energy- An example of how to get around this fine tuning

The neutrino mass scale is suspiciously close to therequired quintessence energy scale

eV 1 eV 05.0 mThere are models on the market in which neutrinomasses are generated by the interaction with a newscalar field (Majoron type models)

Normally the neutrino mass comes from the breakingof a U(1) symmetry at some high energy scale

However, what if the breaking scale is put by hand tobe very low?

We assume the existence of a new scalar field interactingwith neutrinos

The scalar field is not at its global minimum, but rollingin a potential

In this case the neutrino mass becomes a dynamicalquantity, calculable from the VEV of the scalar field-> Mass varying neutrinos (Fardon, Nelson, Weiner 2003)

.....2

1

mVL

The Lagrangian of the combined system is then given by

where the neutrino mass is now given from the scalar fieldVEV

Why is this good?

The effective scalar field potential now has a contributionfrom the neutrino energy density

PVV 3eff

If chosen properly the additional neutrino contributioncan provide a minimum of the effective scalar fieldpotential

This in turn means that it is possible to have

22eff

22 H

d

Vdm

while still maintaining the slow-roll condition

Only the non-relativistic regime is important for understandingthe scenario

We will assume that the field evolves adiabatically so thatat any given point

0'''eff VV

IT IS POSSIBLE TO CONSTRUCT STABLE MODELSOF MASS VARYING NEUTRINOS.(O.E. BJAELDE ET AL. 2007)

INTERESTINGLY, THEY WILL HAVE OBSERVABLECONSEQUENCES IN THE FORM OF VARYINGEOS ETC.

IT IS POSSIBLE TO MAKE PHANTOM-LIKE DARKENERGY BY THIS METHOD

Introduction

Quintessence:

Requires an extremely small scalar mass

Technically unnatural unless protected by some symmetryRequires a new light scale 0.001eV << 1TeV where

the symmetry is broken

•Loop corrections will in general give large contributions to scalar masses Λ2

Symmetry needed!

•SUSY protects scalar masses, but is broken at a TeV SUSY masses naturally of order TeV

•Shift symmetry φ -> φ + const. forbids a mass term for φ•A symmetry breaking term Λ4f(φ/M) introduces a technically natural small mass•Successful quintessence requires a new light mass scale Λ ~ 0.001eV of explicit sym. breaking

Example: pNBG quintessenceFrieman et al. 95

Example: Neutrino mass seesaw

• The neutrino mass matrix:

• Has a small neutrino mass eigenvalue:

Different approach: Seesaw

Is it possible to get a small quintessence mass from a scalar mass seesaw?

Problems with a scalar mass seesaw:

• The neutrino seesaw mass m2/M is never small enough if m > TeV !• No chiral symmetries to protect zero’s in diagonal

However - for 8x8 matrices we do have matrices with:

• Mass eigenvalues m5/M4

• Only M’s in the diagonal

Consider a potential of the type:

where mij is 0, m or M

A CONCRETE EXAMPLE

• The Matrix can be diagonalized, yielding

where the ai’s are all of order one.

Negative eigenvalues tachyonic instabilities(for Fermions these can be cured by chiral rotation)

• Adding Yukawa’s in tachyonic directions will in general also lift the mass of the light direction

we need to be more careful!

• The light mass eigenstate 8 can be written in terms of the original interaction eigenstates:

• It turns out that

The light direction 8 has a very suppressed contribution of

2

We can lift the tachyonic directions by adding Yukawas in the 2 direction - without harming our light mass eigenstate

•Since the m ‘s break all discrete symmetries, we may worry if we can protect the zero’s in the matrix from obtaining values of order m

Brane configuration:•Suppose each of the scalar fields are quasi-localized on each their brane with wave functions in the extra dimensions proportional to

•If the branes are on top of each other, one has bilinear mixing terms of the type

•However, if the branes are geographically separated in the extra dimensions, the overlap of the wavefunctions are exponentially suppressed

when the extra dimension is integrated out, the bilinear interactions of the effective theory are exponentially suppressed

The Origin of the Matrix

• Now, assume that all the elements (m2)ij are given by M2,

• The suppression of the bilinear interaction terms is given by M2exp(-Mr) with M MGUT

• The branes can be taken to lie on top of each other in the directions where bilinear elements in the mass matrix of scale M are induced

• While the branes are separated by r 60/M in the directions where bilinear elements at the soft scale m v are induced.

• In the directions where there are zeros in the off-diagonal, the branes are separated by a distance r >> 262/M.

• Assume that there are three brane fixed points, A, B and C, in each dimension

• A and B are separated from each other by r 60/M

• C separated from A and B at a distance r >> 262/M,

• This leads to the following brane configuration for eight branes in six extra dimensions:

FINALLY, AN EXAMPLE OF A STRING INSPIREDMODEL: WHAT IS HOLOGRAPHY IN COSMOLOGY?

A WELL-KNOWN PROBLEM IN COSMOLOGY IS THAT VACUUMENERGY IS INFINITE, OR AT LEAST GIVEN BY SOME ULTRAVIOLETCUT-OFF IMPOSED ON THE THEORY

FURTHERMORE IT IS PROPORTIONAL TO VOLUME, I.E. IT DOES NOTDILUTE WITH THE COSMOLOGICAL EXPANSION

THE SECOND FACT COULD BE POINTING TO A SOLUTION TO THE PROBLEM

VS

4~

IDEA: SINCE WE KNOW THAT THE ENTROPY OF A BLACK HOLEIS PROPORTIONAL TO ITS AREA (BEKENSTEIN, HAWKING)

IT COULD BE CONJECTURED THAT THERE IS AN UPPERBOUND TO THE ENTROPY IN A GIVEN VOLUME, GIVEN BY THEENTROPY OF A BLACK HOLE OF THE SAME SIZE(BEKENSTEIN, ’THOOFT, SUSSKIND).THIS CONJECTURE IS KNOWN AS THE SPHERICAL ENTROPYBOUND

4/AS

4/AS

THIS CONJECTURE IMMEDIATELY COMES INTO CONFLICT WITHLOCAL FIELD THEORY.

WHY?

IN A GIVEN VOLUME THE ENTROPY IN Z FIELDS IS GIVEN ROUGHLYBY

4~ ZVTS

BY TAKING Z LARGE ENOUGH ONE CAN ALWAYS VIOLATE THESPHERICAL ENTROPY BOUND.IN PRACTISE THIS IS HARD, BUT IN PRINCIPLE IT CAN BE DONE.

WHERE IS THE PROBLEM? IN FIELD THEORY OR IN GRAVITATION?

THE HOLOGRAPHIC PRINCIPLE STATES THAT THE PROBLEM ISIN THE CONCEPT OF LOCAL FIELD THEORY.

THE IDEA IS THAT FOR A REGION R IN D DIMENSIONS, ANY PHYSICAL PROCESS IN R CAN BE DESCRIBED BY PROCESSESON THE D-1 DIMENSIONAL BOUNDARY.

THIS IS COMPATIBLE WITH THE ENTROPY BOUND. SINCE ENTROPYCAN ONLY SCALE AS AREA, THE ANSWER FROM LOCAL FIELDTHEORY MUST BE WRONG.

THIS WAS FIRST POINTED OUT BY ’THOOFT IN 1993 AND CALLEDDIMENSIONAL REDUCTION IN QUANTUM GRAVITY

IN 1995 IT WAS PUT INTO THE SPHERICAL ENTROPY BOUNDFORMULATION BY SUSSKIND

IN 2000 BOUSSO FORMULATED IT AS THE ”COVARIANT ENTROPY BOUND” WHICH CAN BE USED IN COSMOLOGY FOR FRW METRICS.

ITS CONSEQUENCES ARE NOT IMMEDIATELY CLEAR, BUT THE PRINCIPLE CLEARLY STATES THAT THE EVOLUTION OF ANY FRWMETRIC IS GIVEN BY ITS BOUNDARY CONDITIONS.

IN THE COVARIANT ENTROPY BOUND THIS WOULD (PRESUMABLY)MEAN ITS FUTURE EVENT HORIZON.

ONE POSSIBLE AND VERY INTERESTING CONSEQUENCE WOULD BEIF VACUUM ENERGY IS ALSO BOUNDED FROM ABOVE. THAT THISSHOULD BE TRUE DOES NOT FOLLOW FROM THE ENTROPY BOUNDBECAUSE VACUUM ENERGY IN ITSELF DOES NOT CONTAIN ENTROPY.

HOWEVER (COHEN ET AL., LI, …) A HOLOGRAPHIC BOUND ON THE VACUUM ENERGY CAN BE CONJECTURED FROM THE ASSUMPTIONTHAT THE VACUUM ENERGY IN A REGION CANNOT BE LARGER THAN THEMASS IN THE SAME REGION

WHICH REGION IS THAT? THE FUTURE EVENT HORIZON IS THE MOSTSTRAIGHTFORWARD ANSWER (BUT OF COURSE NOT NECESSARILY TRUE).

THAT LEADS TO SIMPLE PREDICTIONS FOR THE EQUATION OF STATEETC (LI 2003).

ANOTHER INTERESTING QUESTION IS COSMOLOGICAL PERTURBATIONS.

IF PHYSICS IN A VOLUME IS GIVEN BY EFFECTS ON ITS BOUNDARY ONLY,THEN PRESUMABLY SUPER-HORIZON PERTURBATIONS CANNOT CONTRIBUTE TO THE EVOLUTION OF LOCAL FLUCTUATIONS.

THIS LEADS TO A NATURAL CUT-OFF IN POWER AROUND THE PRESENTHORIZON (WHICH IS CLOSE TO THE FUTURE EVENT HORIZON), WHICHCOULD BE COMPATIBLE WITH WMAP (ENQVIST, HANNESTAD, SLOTH 2005)

CONCLUSIONS:

PLUS SIDE: MOST (BUT NOT ALL) OF THE MOREEXOTIC DARK ENERGY MODELS HAVEOBSERVABLE SIGNATURES DIFFERENTFROM LAMBDA

MINUS: TOO MANY MODELSWITHOUT ADDITIONAL INPUT FROM THEORY OR OTHER EXPERIMENTSA TIME-VARYING EOS DOES NOTNECESSARILY POINT TO A SPECIFICMODEL