Geometric Precipices in String Cosmology

Geometric Geometric Precipices in Precipices in

String CosmologyString CosmologyNemanja Kaloper, UC Nemanja Kaloper, UC

DavisDavis

Based on: arXiv:0712.1820, with S. Watson

OverviewOverview Stringy gas cosmology: a (very!) brief reviewStringy gas cosmology: a (very!) brief review Dilaton sector: a conserved discrete charge Dilaton sector: a conserved discrete charge Singular precipicesSingular precipices

Nemanja Kaloper, UC DavisNemanja Kaloper, UC Davis

An ensemble of stringy excitations propagating in the An ensemble of stringy excitations propagating in the background FRW geometry, governed by the light string modesbackground FRW geometry, governed by the light string modes

Truncate it on Truncate it on The effective 1D actionThe effective 1D action

Claim: on homogeneus and isotropic backgrounds this may be Claim: on homogeneus and isotropic backgrounds this may be a good description even for very small universe thanks to T-a good description even for very small universe thanks to T-duality!duality!

“… “… not expected on inhomogeneous/anisotropic backgrounds, not expected on inhomogeneous/anisotropic backgrounds, but maybe those could be ignored …”but maybe those could be ignored …”

Note that after duality revolution the isotropic string Note that after duality revolution the isotropic string cosmologies may be viewed as anisotropic worlds in M-theory: cosmologies may be viewed as anisotropic worlds in M-theory: warning flag!warning flag!

Idea of Stringy Gas CosmologyIdea of Stringy Gas Cosmology

Brandenberger & Vafa; Tseytlin & Vafa, `91; …


Matter sector yields the sources Matter sector yields the sources

Direct variation yields, for single fluid in the Direct variation yields, for single fluid in the time gaugetime gauge

where the reduced dilaton and string coupling arewhere the reduced dilaton and string coupling are

These equations are in fact exactly integrable. But These equations are in fact exactly integrable. But instead of writing solutions, let us understand instead of writing solutions, let us understand them…them…

Field Equations and Phase Field Equations and Phase SpaceSpace

Tseytlin-Vafa ProposalTseytlin-Vafa Proposal


Dilaton crucial: at high energies Dilaton crucial: at high energies ~ M~ MS S , not , not stabilized - theory is T duality invariant to start stabilized - theory is T duality invariant to start with.with.

The sign of is The sign of is constantconstant if if energy is positive - a conserved discrete charge!energy is positive - a conserved discrete charge!

Single fluid cosmologies, with the required Single fluid cosmologies, with the required asymptotics, classified by the arrow of time and asymptotics, classified by the arrow of time and the dilaton sign.the dilaton sign.

Four classes of solutions, flowing to/fro the Four classes of solutions, flowing to/fro the special solutions - evolutionary saddle points. special solutions - evolutionary saddle points. They are (+) and (-) branches (signs of ), just They are (+) and (-) branches (signs of ), just like in PBB.like in PBB.

Phase space analysis is most revealing.Phase space analysis is most revealing.

What’s Really Going On?What’s Really Going On?

Hagedorn gasHagedorn gas

Momentum mode gasMomentum mode gas

Winding mode gasWinding mode gas


The stringy gas cosmology wants to link different The stringy gas cosmology wants to link different solutions and avoid singularity; eg. of Tseytlin-solutions and avoid singularity; eg. of Tseytlin-Vafa: Vafa:

Hagedorn Hagedorn II Momentum 1. Momentum 1. These are on These are on different branchesdifferent branches: (+) and (-)!: (+) and (-)! For matching to occur, dilaton velocity must For matching to occur, dilaton velocity must

reverse: there must be region where the `egg reverse: there must be region where the `egg function’ vanishes! function’ vanishes!

For this to happen we must ensure:For this to happen we must ensure: Negative energy density somewhereNegative energy density somewhere Collision between the trajectory and this regionCollision between the trajectory and this region Subsequent escape without further collisionSubsequent escape without further collision

No-go & Null EnergyNo-go & Null Energy

Negative energy density somewhere: put in a potential which dips below zero for some values of the dilaton. But that in itself is NOT ENOUGH! What is required is something like:

Excluded interior

e < 0


Consider evolution of Consider evolution of ee along a trajectory: along a trajectory:

(+/-) refers to the branch on which this evolves.(+/-) refers to the branch on which this evolves. Assume that a trajectory hits the egg, and Assume that a trajectory hits the egg, and

escapes and integrate between the hit and the escapes and integrate between the hit and the escape time:escape time:

The 3rd term on the LHS is the negative area The 3rd term on the LHS is the negative area between the curve and the horizontal axis. between the curve and the horizontal axis. Rewrite asRewrite as

Same idea as in NK, Madden, Olive, 1995.


If you start on (+) branch:If you start on (+) branch:

LHS positive definite: to have the transition from + to - LHS positive definite: to have the transition from + to - must get RHS to be negative.must get RHS to be negative.

If you start on (-) branch:If you start on (-) branch:

Branch change from - to + can occur with normal Branch change from - to + can occur with normal matter.matter.

But then all the - branch solutions are past-singular But then all the - branch solutions are past-singular and + ones are future singular.and + ones are future singular.

ThisThis excludes excludes the transitions proposed by Tseytlin and the transitions proposed by Tseytlin and Vafa, and their applications to phenomenology by Vafa, and their applications to phenomenology by Brandenberger et al. This also excludes the loitering Brandenberger et al. This also excludes the loitering universe idea of Brandenberger et al, for positive universe idea of Brandenberger et al, for positive energy.energy.


If energy is strictly positive, it is impossible to If energy is strictly positive, it is impossible to smoothly connect different branches.smoothly connect different branches.

Impossible to remove the singularity from the Impossible to remove the singularity from the EFT picture EFT picture UNLESSUNLESS one violates the null one violates the null energy condition. Only then + to - transitions, energy condition. Only then + to - transitions, (ie. evolution between saddle points) occur - (ie. evolution between saddle points) occur - and perhaps might be pushed past them by T-and perhaps might be pushed past them by T-duality.duality.

In the standard framework where null energy In the standard framework where null energy condition is maintained such evolution condition is maintained such evolution does notdoes not happen and singularity is present somewhere.happen and singularity is present somewhere.

A variant of the A variant of the Hawking-Penrose singularity Hawking-Penrose singularity theoremtheorem, which one expects in the Einstein , which one expects in the Einstein frame!frame!


Thus: those claims about getting Thus: those claims about getting interesting cosmological perturbations interesting cosmological perturbations from stringy gases are not reliable.from stringy gases are not reliable.

So: are these frameworks completely So: are these frameworks completely useless?useless? Transitions from - to + may occur if Transitions from - to + may occur if

were allowed. May be of some were allowed. May be of some interest, to probe the perturbations in a interest, to probe the perturbations in a small universe limit, using T-duality.small universe limit, using T-duality.

Perturbations will break T-duality, but Perturbations will break T-duality, but presumably weakly. Is there a way to presumably weakly. Is there a way to explore perturbations at the `trans-explore perturbations at the `trans-planckian’ (well, really `trans-stringy’) planckian’ (well, really `trans-stringy’) scales using the T-dual?scales using the T-dual?

So what do the So what do the phenomenological applications phenomenological applications of stringy gases in cosmology of stringy gases in cosmology

- to date - look like?- to date - look like?

In Lieu of a Summary…In Lieu of a Summary…

It’s about nothing!

I still don’t know what the idea is!

… I think you may have something there...

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ЕЩЕ МНОГО ЛЕТ!

Geometric Precipices in String Cosmology

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