Cosmology and the String SwamplandCumrun Vafa
Harvard University

Strings 2018
June 25, 2018

Based on two joint papers:

The plan for this talk is:-Why do we believe in the existence of dS space?
-A new swampland condition:
Lower boundfor c>0-Cosmological Implication of two swampland
criteria for past, present and future:,Why dS?
Because we live in one!

Why not rolling scalar potentials (quintessence)?

The scalar would typically couple to some matterfields and its rolling would lead to observable effects.
But, e.g., from z=1 till now

Also the coupling of the scalar field to matter would
give rise to a new `fifth force’ which would be
detectable at astrophysical distance. The idea that thescalar field should couple to something is natural instring theory.
The fifth force would lead to apparent violations ofequivalence principle.
The existence of fifth force is strongly bounded basedon astrophysical observations, making this ratherimplausible.

Not a good argument:
The scalar field should couple strongly to SOME
fields but not necessarily visible matter fields
making their detection more difficult:

The scalar field could couple to DM.
So the rolling scalar anticipates DM!

But there is another strange feature of quintessence
models:Not only(in Planck units)
But also for the quintessence models not to be incontradiction with observational bounds on w weneed

Sounds like double fine tuning unless we cannaturally haveUnlike AdS, constructing dS vacua (even meta-stableones) in string theory seems very difficult.
Despite heroic efforts (see in particular KKLT) theseattempts are at the level of proposed scenarios
rather than rigorous constructions and there are anumber of criticisms leveled against them.
To construct dS vacua one needs to do something
exotic. For example:
Maldacena-Nunez no go theorem:in the limit of supergravity (i.e. when not too muchcurvature in Planck units) no dS in M-theory!

So let us dare to ask: What if there are no criticalpoints of V with positive value?
If so, the next natural question is how close we can get
gradient of V to zero? Could it be that there is auniversal bound:This cannot be. For example in a supersymmetrictheory with 0 cosmological constant, we can consider amassive fieldand go arbitrarily close to 0.

One can instead consider a bound:And a natural choice for f is:Where c>0 and order 1 in Planck units. In other words
for a universal cPreliminary checks:
is trivially satisfied with V<0. So this iscompatible with susy and known critical points of V for
AdS.
Also it is trivially satisfied for known supersymmetricexamples with zero cosmological constant (say for type II
string theory on Calabi-Yau threefolds).
For example the existence of moduli is compatible withit.
How about V>0? We can start from a supersymmetriccase with 0 cosmological constant and deform.

If we deform it by going away from 0 cosmologicalconstant, it is equivalent to giving vev to fields, and toleading order this is the same as changing V to

This is consistent with the bound
When one recalls that the effective field theory is
expected to break down for large values ofOne can consider M-theory in supergravity limit:

and consider compactifying to d dimensions on anarbitrary 7 manifolds with arbitrary flux. We get an
effective potential
which is a function of
infinitely many scalars (which parameterize all possible
internal metrics and fluxes on the 7-manifold). It is
hard to believe but its true that for arbitrary metricand flux there is no critical point of V with V>0, as M-N
no-go theorem shows (in supergravity approximation).

The M-N no-go theorem can be strengthened:
If we compactify M-theory to d dimensions, one caneasily show, using volume rescaling, that in supergravity
limit

which is realized forIf we assume the Strong Energy Condition (SEC), or more
precisely for compactifications respecting that,And for compactifications respecting Null Energy Condition(NEC) with zero or negative average scalar curvature

Where D=11,10 and d is the dimension we compactify to.
For example D=10, d=4:

Other examples:
Heterotic O(16)xO(16) strings. Non-susy, no tachyons in
10 d, and at weak coupling has positive cosmologicalconstant:

Also extending an argument of[Hertzberg,Kachru,Taylor,Tegmark] and [Wrase,Zagermann]:

So the upshot is that the conjecture is not unreasonable withc of order 1 in Planck units.
This makes quintessence more natural with |V’| of order V.

Cosmological Implications
We now turn to cosmological implications of two
swampland criteria:,with[Ooguri,V]being close to 1 in Planck units.

We divide the discussion to past, present and future.

Past
Early universe: Inflation has some tension with bothcriteria. The constant c is related to the slow rollparameter

The current observational bounds on the B-mode lead(for textbook inflation models) toand. However the textbook models of inflation
when combined with spectral tilt gets ruled out.
Among the more favored inflationary models, theplateau models, one finds the bound.

Moreover, the number of e-fold being greater than
60 leads (for plateau models) toagain in mild tension with swampland criteria(This tension has already been noted in theliterature).

Present
Present epoch: The swampland criteria only allow
quintessence models. Quite remarkably it can be shownthat the current observational bounds are compatible
with both criteria as long as c < 0.6 :Also the initial condition for z=1 is not fine tuned due to
tracking behaviour (except for the value of dark energy).

One finds that there is a universal bound on thevalue of (1+w) today. It predicts that it is bigger
thanFuture
If we lived in dS space, the lifetime of the universe,
before there is a phase transition can be arbitrarily
large and typically has nothing to do with the timescale set by the dark energy:

This leaves a puzzle: Why current age is related toFutureFutureThe two swampland criteria can be used to show thatin a time of order of Hubble time, the universe will
undergo a phase transition: Either we get a tower oflight modes, or accelerating expansion will stop.
It can be shown that this will happen in N Hubble times

The basic idea is that the current kinetic term for therolling field is away from 0 and that the rolling cannot
exceedwithout undergoing phase transition.

This uses the relation

Observational Consequences
1-More accurate measurements of w(z):Is (1+w) significantly different from 0 as we predict?

2-Dark sector couplings have been changing over timeas they presumably couple to the quintessence fields.
Observational consequences (apparent violation ofequivalence principle in the dark sector) may be
detectable.

Conclusion
It seems not unreasonable to believe dS is notrealizable in a quantum theory of gravity. This
motivates a new swampland criterion putting abound on the slope of V in terms of V.
This together with another swampland criterion(bound on range of fields) leads to-Some tension with inflation-Present epoch must be based on quintessence-The universe is about to undergo a phase transitionin O(1) Hubble time. Either a tower of lights states
emerge or cosmic acceleration stops.