Download THE COSMOLOGICAL CONSTANT PROBLEM: HOW DOES THE

THE COSMOLOGICAL
CONSTANT PROBLEM:
HOW DOES THE VACUUM
GRAVITATE?
RICARDO ZAMBUJAL FERREIRA
OVERVIEW
Ø  Einstein’s nightmare
Ø  The Cosmological Constant
Ø  Observations
Ø  What is the problem?
Ø  Vacuum Energy
Ø  The old and new CC problem
Ø  (Some) Approaches to the Problem
Ø 
Ø 
Ø 
Ø 
Ø 
Ø 
Extra Dimensions
Anthropic Principle
Modified Gravity
Massive Gravity
Underlying Symmetries
Normalized General Relativity
Ø  Conclusion
References:
•  [Weinberg 89’, Tseylin 91’,
Bertolami 09’, Davidson
and Rubin 09’, Burgess 13’,
Kaloper and Padilla 13’,14’]
EINSTEIN’S
NIGHTMARE
Ø  Einstein’s theory of General Relativity (1916) is described by the action:
Ø  Which leads to the field equations
Space-time Geometry
Matter
Ø  At that time the Universe was thought to be static. Such an action was
thought to be incompatible with a static Universe.
EINSTEIN’S
NIGHTMARE
Ø  Einstein added an extra ingredient in 1917, the Cosmological Constant (CC) Λ
Ø  The field equations are simply changed to
Ø  While the Friedmann equations give
EINSTEIN’S
NIGHTMARE
Ø  (1929) Hubble verified that Galaxies are moving apart, hence the Universe
is not static.
Ø  From then on Λ was assumed to be zero. Einstein regretted his “mistake”
for the rest of his life.
Ø  (1999) By looking at distance/redshift of supernovae it was observed that
the Universe is, in fact, accelerating.
Ø  Therefore, the Universe has to be dominated, today, by Dark Energy which
seems to be nothing more than Λ (or something very similar to it)
OBSERVATIONS
Ø  Observed value of Λ:
Ø  How does the Energy density in the Universe
changes with the expansion:
Radiation
Curvature
Matter
Cosmological Constant
Why now?
IS IT REALLY Λ?
Ø  A field with an almost flat potential
can mimick Λ.
Ø  Constraints on Dark Energy
parameter of state (Λ corresponds to
w0=-1) show that we are, at least,
very close to a pure Λ.
[Xia, J. et al. ‘13]
WHAT IS THE PROBLEM
THEN?
VACUUM ENERGY
Ø  Physics is described by fields.
Ø  Fields are quantities defined and
quantized at each point of space.
Ø  Particles are the excitations of
these fields.
Ø  The minimal energy configuration
(the vacuum) is not zero. The
“grid” has its own “self” (vacuum)
energy.
VACUUM ENERGY
GENERAL RELATIVITY
PARTICLE PHYSICS
VACUUM ENERGY
Ø  Energy density (Hamiltonian density) a scalar field (h) is
Ø  For a weakly interacting field we can
expand around its classical vacuum
corresponding to its lowest energy:
Ø  The quantum fluctuations of the field around its minimum would then be
WHAT DO WE EXPECT THE VACUUM
ENERGY TO BE?
Ø  The vacuum energy density is a constant piece in the Lagrangian, o.g., it acts
as a CC.
Ø  The quantum contribution to vacuum energy density becomes
Ø  Where wk can be seen as the energy density stored in each quantum mode.
In flat space:
Ø  The quantum contribution to the vacuum energy diverges, the energy density
is infinite!?
CUTOFF
Ø  But in Physics we consider our theories to be effective, i.e. they are just valid
until a given size/energy. Therefore, we introduce a cutoff for the range of
scales/energy we look at
Ø  The result is similar for different fields, only the coefficients would change.
Ø  The total energy density would be the sum over all types of fields.
Ø  We expect our theories to break down, at least, at Λ=Mp where quantum gravity
effects are important.
The (Cosmological Constant) Problem
•  Old CC Problem: Why is it so small?
•  New CC Problem: Why is it not zero?
•  Why now Problem: Why the CC dominates the Universe now?
WHAT CAN WE DO?
•  Can we choose Λ with 10122 precision such that it cancels
the vacuum contribution?
•  Yes, we don’t know why the parameter is so small but we
just tune the other parameter to cancel it. Nevertheless,
this is still very unstable.
•  What if we extract the cutoff dependence?
•  Yes, we can! Although it will not help very much.
CUTOFF (IN)DEPENDENCE
•  Physical theories are understood as effective field theories
within a given range of energies. However, observations should
be independent of the specific cutoff value.
•  The dependence on the cutoff is canceled by regularization/
renormalization.
•  Why should we not expect something similar?
•  Renormalized Energy Density
CUTOFF (IN)DEPENDENCE
•  Even if we choose the lighter particle we know of (the
electron) with me=0.5 MeV we are still very far away from the
observed value
•  We could also choose a very low energy effective action and
cancel Λ there. However, it will reappear very large again once one
changes slightly the renormalization (energy) scale.
(SOME)
APPROACHES TO THE PROBLEM
Ø  Extra Dimensions
Ø  Anthropic Principle
Ø  Modified Gravity
Ø  Massive Gravity
Ø  Symmetries
Ø  Normalized General Relativity
Extra Dimensions [Arkani-Hamed, Dimopoulos, Dvali 98’]
Ø  Lorentz Invariance (LI) is only required in the 4 dimensional space.
Ø  It can be broken in the extra dimensions and therefore the Λ term
could curve space time in a different way.
Ø  Problem: Extra dimensions should, probably, have been seen already
in experiments.
Anthropic Principle [Barrow, Tipler 86’]
Ø  Supposes that the fundamental theory has a huge number of
different vacua, each of them with different values of Λ.
Ø  We live in the one which is more suitable for the existence of life.
Ø  Very hard to prove/disprove.
Modified Gravity
Ø  It looks for modifications of GR which might explain the present Dark
Energy stage.
Ø  Only addresses the new CC problem.
Massive Gravity [Fierz, Pauli 39’]
Ø  The graviton, the mediator of the gravitational force, is massive.
Ø  (Linearized) Einstein equation is modified to
Ø  A zero curvature is allowed even in the presence of Λ.
Ø  Problems:
Ø  How to make ghost-free non-linear realization of the theory
Ø  Solar System constraints
Symmetries
Impose a symmetry which forbids non-zero Λ.
Scale Invariance
Ø  If the action is invariant under scale transformations.
Ø  Then Λ is forbidden in the action.
Ø  Problems:
Ø  Einstein-Hilbert action is not scale invariant.
Ø  Quantum Gravity corrections usually break scale invariance.
Supersymmetry
Ø  For each particle there is a superpartner with spin +/- ½ and the same
mass and interaction strengths.
Ø  The vacuum is invariant under the supersymmetric generators
therefore it has 0 energy density.
Ø  Problems:
Ø  We did not find supersymmetric particles.
Ø  It has to be broken at energies >1 TeV
Normalized General Relativity (NGR)
Ø  What if the action is an intensive quantity? [Tseylin 91’, Davidson and Rubin
10’]
Ø  Einstein equations would be modified to
Ø  Λ (both the bare and the vacuum contribution) cancels in the
Einstein’s equation meaning that Λ does not gravitate
Ø  Problems:
Ø  Does not protect Λ from spontaneous symmetry breakings (SSB) like
the ElectroWeak one;
Ø  If the space-time volume is infinite some quantities become ill-defined
Ø  For big and old universes the Planck mass is radiativelly unstable
Ø  It only cancels vacuum contributions at 3-level because higher order
terms have a different dependence on the space-time volume.
The revival
Ø  A new approach was proposed [Kaloper, Padilla 13’, 14’]
Ø  Matter content couples to a rescaled metric
Ø  The parameter λ sets the hierarchy between mass scales and the
Planck scale
Ø  The boundary term σ(z) fixes matter scales as a function of the space
time volume.
Ø  Λ, λ are auxiliary fields which lead to global constraints, and µ is a
dimensionfull parameter to be set by experiments
Ø  After solving for Λ, λ one gets, similarly with NGR,
Ø  Where <Q> is the average on the space time volume
Ø  As < Λ > = Λ it does not contribute to the field equations.
Improvements
Ø  Planck mass is radiatively stable for small λ
Ø  All matter (not graviton) loops contributions scale on the same way
with λ, thus the higher order terms are also canceled in the field
equation.
Ø  Spontaneously symmetry breaking contributions are small in a large
and old Universe.
Ø  There is a residual CC from the trace of the energy momentum tensor
but it is also small for large and old Universes and maximal at turnover
regions.
Predictions:
Ø  Space-time volume has to be finite so the Universe cannot expand
forever.
Ø  The present accelerating stage should be a transient stage and
possibly small spatial curvature is required.
Ø  Deviations from Einstein equation are maximal at the turn around
point.
CONCLUSION
Ø  Biggest Fine Tunning in Physics
Ø  There are just a few proposed solutions which somehow shows how
complicated is the problem.
Ø  The majority of the proposals are still far away from a solution.
Ø  Interesting recent proposal from Kaloper and Padilla needs to be analyzed
in more detail