Fundamental Physics, the Swampland of Effective Field Theory and Early. . .
413
3 Constraints from Fundamental Physics
The evolution of the very early universe should be described by the best available
theory which describes physics at the highest energies. There is evidence that all
forces of nature might unify at high energies. They must be described quantum
mechanically. The best candidate for such a quantum theory is superstring theory.
Superstring theory is based on the assumption that the basic building blocks of
nature are not point particle, but rather elementary strings.
The quantum theory of point particles is quantum field theory. There is a huge
landscape of quantum field theories: any number of space-time dimensions and
fields is allowed, and any shape of field potentials. Superstring theory is very
restrictive. The number of space-time dimensions is fixed, and the string interactions
are constrained. At low energies, the physics emerging from superstring theory
should be describable by an effective field theory.
However, the set of effective field theories compatible with string theory is
constrained by what are known as the swampland criteria. Only theories consistent
with these criteria are admissible. The vast number of field theories are not—they
are said to lie in the swampland (see [23] for a review). Note that at the moment
these criteria are not proven—they are educated guesses.
The first swampland criterion [24] is that the field range over which a given
effective field theory is valid is constrained by Δϕ < O(1)m pl . The second
condition [25] is that, for a scalar field which is rolling and which dominates the
energy density of the universe, the potential cannot be too flat:
V
V
> c 1 m
−1
pl ,
(2)
where c 1 is a constant of order one. This condition is opposite to what is required for
simple slow-roll inflation models. Hence, it appears that cosmological inflation is in
tension with superstring theory. A corollary of the second swampland condition
is that a cosmological phase dominated by a positive cosmological constant is
not possible. Hence, Dark Energy cannot be a cosmological constant [26]. Scalar
field models of Dark Energy are, however, consistent with (but constrained by) the
swampland conditions [26, 27].
In light of the tension between inflationary cosmology and the principles of
string theory it appears that we may need a new paradigm of early universe
cosmology. Such a paradigm should be based on the key new degrees of freedom
and symmetries which differentiate string theory from point particle theories. New
degrees of freedom include the string oscillatory and winding modes. Let us for
simplicity consider the background space to be toroidal. Strings on this space have
momentum modes whose energies are quantized in units of 1/R, where R is the
radius of the torus, winding modes whose energies are quantized in units of R, and
an tower of oscillatory modes whose energies are independent of R. Point particles
only have momentum modes. If we consider a box of strings in thermal equilibrium
413
3 Constraints from Fundamental Physics
The evolution of the very early universe should be described by the best available
theory which describes physics at the highest energies. There is evidence that all
forces of nature might unify at high energies. They must be described quantum
mechanically. The best candidate for such a quantum theory is superstring theory.
Superstring theory is based on the assumption that the basic building blocks of
nature are not point particle, but rather elementary strings.
The quantum theory of point particles is quantum field theory. There is a huge
landscape of quantum field theories: any number of space-time dimensions and
fields is allowed, and any shape of field potentials. Superstring theory is very
restrictive. The number of space-time dimensions is fixed, and the string interactions
are constrained. At low energies, the physics emerging from superstring theory
should be describable by an effective field theory.
However, the set of effective field theories compatible with string theory is
constrained by what are known as the swampland criteria. Only theories consistent
with these criteria are admissible. The vast number of field theories are not—they
are said to lie in the swampland (see [23] for a review). Note that at the moment
these criteria are not proven—they are educated guesses.
The first swampland criterion [24] is that the field range over which a given
effective field theory is valid is constrained by Δϕ < O(1)m pl . The second
condition [25] is that, for a scalar field which is rolling and which dominates the
energy density of the universe, the potential cannot be too flat:
V
V
> c 1 m
−1
pl ,
(2)
where c 1 is a constant of order one. This condition is opposite to what is required for
simple slow-roll inflation models. Hence, it appears that cosmological inflation is in
tension with superstring theory. A corollary of the second swampland condition
is that a cosmological phase dominated by a positive cosmological constant is
not possible. Hence, Dark Energy cannot be a cosmological constant [26]. Scalar
field models of Dark Energy are, however, consistent with (but constrained by) the
swampland conditions [26, 27].
In light of the tension between inflationary cosmology and the principles of
string theory it appears that we may need a new paradigm of early universe
cosmology. Such a paradigm should be based on the key new degrees of freedom
and symmetries which differentiate string theory from point particle theories. New
degrees of freedom include the string oscillatory and winding modes. Let us for
simplicity consider the background space to be toroidal. Strings on this space have
momentum modes whose energies are quantized in units of 1/R, where R is the
radius of the torus, winding modes whose energies are quantized in units of R, and
an tower of oscillatory modes whose energies are independent of R. Point particles
only have momentum modes. If we consider a box of strings in thermal equilibrium