The Cosmological Constant and Technical Naturalness

Sunny Itzhaki

hep-th/0604190 + work to appear

The CC problem used to be a simple problem

~

One loop

~ g

The CC problem used to be a simple problem… to state:

Why is the CC vanishing? The CC term is a relevant term that receives large quantum corrections:

Lorentz invariance + dim. analysis give

Cut off scale

The goal was to find a way to get 0.

The CC problem seems to have little to do with particle physics and more to do withdeep issues in quantum gravity.

• Old proposals to solve the CC problem:

1- Hawking’s wave function of the universe (most likely to have vanishing CC).

2- Coleman’s wormholes.

…

• Particle physics does not help much:

SUSY, that helps in a similar problem with the Higgs mass, gives at best

which is times larger than the total energy density in the universe.

604 10~)( TeV ~

6010

Now we are trying to explain a small and positive CC

•The bad news is that the CC problem evolved into three problems:

1- Why is the CC so small (in particle physics units)?

2- If so small why not zero?

3- Why now? Roughly when galaxies were formed the CC is of the order of the matter energy density in the universe.

• The bad news is that the CC problem evolved into three problems:

1- Why is the CC so small (in particle physics units)?

2- If so small why not zero?

3- Why now? Roughly when galaxies were formed the CC is of the order of the matter energy density in the universe.

• The good news is that we have a scale, to work with.

For example:

1- (…, Banks, …)

303 10~10 eV

Loop corrections to the CC are within the inflation range

In TeV SUSY theories the natural range of the vacuum energy is

Gauge mediation models Gravity mediation

Which is within the inflation range

• This assumption does not solve the CC problem, but if true it changes its nature:

CC and Inflation

The question is now: why is the ratio of the vacuum energy during inflationto the current vacuum energy so large and yet not infinite?

Why it is :

and not

Can be

The question is: why is the ratio of the vacuum energy during inflationto the current vacuum energy so large and yet not infinite?

Why it is :

and not

The good thing is that we have to do it only once:

Temperature driven phase transitions (like the EW or QCD)will not change the vacuum energy.

Out line

• Abbott’s model (85). The empty universe problem.

• The hep-th/0604190 model.

The cold universe problem.

• Solving the cold universe problem.

Abbott’s Model (85)

The action is

Instantons induce a potential:

When we have the symmetry

is technically natural. (similar to the mass of the electron)

• Small M is natural.

The renormalizedCC term

Also at the quantum level the potential looks like:

• In quantum mechanics the local minima are on equal footing.

• Here the situation is more interesting:

Hawking temperature in de-Sitter is .

• For in effect there are no local minima.

• For we have tunneling. The decay rate is Most of the time at small CC.

Regardless of we end up with a small CC

Regardless of we end up with a small CC

BUT we also end up with an empty universe.

This is known as the emptiness problem that appears also in other approaches tothe CC problem.

The hep-th/0604190 model

Let’s modify Abbott’s model in the following way:

The relaxation action is a simpler version of Abbott’s action

where .

Much like in Abbott’s case the vacuum energy is reduced slowly.

The challenge is to evade the emptiness problem by converting the potential energyinto kinetic energy.

is designed to fix that while making sure that the vacuum energy at the end of inflation is small.

That is makes

sure that we have

and not

We take

The potential is designed to have the following properties:

and

Now the dynamics is more interesting:

The effective mass is

Slow roll approximation.

(This is where it is important that )

Now the dynamics is more interesting:

The effective mass is

There is a phase transition: V

For

At the critical vacuum energy

an instability is developed and

acquires an expectation value.

The end result is a flat spacewith plenty of kinetic energy.

There are a couple of bounds on the we can get this way:

1- Energy conservation:

2- Vacuum energy can be converted to kinetic energy only when

the slow-roll approximation is not valid:

In our case (1) and (2) are the same so an upper bound on

is

What about quantum correction?

Claim:

The present value of the vacuum energy + Technical naturalness of the model ___________________________________

• The upper bound on the reheating temperature is at the TeV scale.

• SUSY is broken at around the TeV scale.

What about quantum correction?

Let’s consider the simplest potential

Quantum corrections to give non-vanishing vacuum energy, , that should be at most .

Because of the relevant term it is hard to control these quantum corrections without SUSY.

With SUSY we have

When SUSY is broken these corrections are enhanced:

• Gravity always mediates SUSY breaking from one sector to the other:

So the picture is:

Hidden sector

TeV SUSY

sector

Gravity mediation

(N) MSSM

Gauge mediation

Hidden sector

TeV SUSY

sector

Gravity mediation

(N) MSSM

Gauge mediation

So far we talked about: vacuum energy kinetic energy.

Hidden sector

TeV SUSY

sector

Gravity mediation

(N) MSSM

Gauge mediation

So far we talked about: vacuum energy kinetic energy.

we should have: vacuum energy kinetic energy SM heat.

Re-heating coupling

Can we re-heat without spoiling the naturalness of the model?

Direct approach:

Take

We need

BUT then:

A resolution:

Take instead

This includes

So

Looks like the exact same situation/problem as before.

But actually

So works fine.

Conclusion:

We have considered a model that:

• Appears to be consistent with basic exp. requirements (TeV SUSY and inflation).

• CC is technically natural.

Can the model be embedded in string theory?