Fundamental Physics, the Swampland of Effective Field Theory and Early. . .
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which is necessary to obtain classical perturbations at late times (third criterion).
Finally (fourth criterion), the structure formation scenario must produce a nearly
scale-invariant spectrum of primordial perturbations (see e.g. [6] for a more detailed
discussion).
Inflationary cosmology [4] is the first scenario which satisfies the four above
criteria. During the time interval t i < t < t R during which the universe undergoes
nearly exponential expansion, the horizon expands exponentially while the Hubble
radius remains almost unchanged. Since the physical length of a fixed comoving
scale also expands nearly exponentially during the period of inflation, scales which
we observe today originate inside the Hubble radius as long as the period of
inflation is sufficiently long. Fluctuations are squeezed on super-Hubble scales for a
long time, and the approximate time-translation symmetry of the inflationary phase
ensures that the spectrum of primordial fluctuations is nearly scale-invariant [5, 7].
Bouncing cosmologies provide a second scenario in which the four criteria for
a successful early universe scenario can be satisfied. In a bouncing scenario the
horizon is infinite. The Hubble radius decreases during the period of contraction and
then increases during the period of expansion. As long as the period of contraction
is comparable in length to the period of Standard Big Bang expansion, scales
which we observed today emerge from inside the Hubble radius, thus allowing a
possible causal structure formation scenario. As in inflationary cosmology, there is a
long period during which scales propagate with super-Hubble length, thus enabling
the squeezing of the fluctuations. There are (at least) three classes of bouncing
cosmologies. First, the matter bounce [8] in which there is a long phase of matterdominated contraction. Second, there is the Pre-Big-Bang scenario [9] in which
contraction is driven by a field with an equation of state w = 1, where w is the ratio
of pressure to energy density. Finally, there is the Ekpyrotic scenario [10] in which
contraction is obtained by means of a scalar field with equation of state w 1.
There is a duality between the evolution of curvature fluctuations in a matterdominated phase of contraction and in an exponentially expanding background [11].
Hence, the matter bounce automatically leads to a roughly scale-invariant spectrum
of fluctuations. There is a duality in the evolution of scalar field fluctuations between
a cosmology with Ekpyrotic contraction and one of exponential expansion [12].
Hence, it is also possible to obtain a scale-invariant spectrum of fluctuations. In
the case of the Pre-Big-Bang scenario it is possible to obtain a scale-invariant
spectrum making use of axion fields [13]. See [14] for a detailed review of
bouncing cosmologies. Ekpyrotic and Pre-Big-Bang cosmologies produce a steep
blue spectrum of primordial gravitational waves. Hence, on cosmological scales
the spectrum of primordial gravitational waves is predicted to be negligible. This
contrasts with the predictions of inflationary models which forecast a roughly scaleinvariant spectrum.
A third scenario for early universe cosmology is the emergent scenario which is
based on the assumption that the universe emerged from an initial high density state
in which matter was in global thermal equilibrium. One toy model for this is String
Gas Cosmology [15] in which it is assumed that the universe loiters for a long time
in a Hagedorn phase of a gas of fundamental strings, and there is a phase transition
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which is necessary to obtain classical perturbations at late times (third criterion).
Finally (fourth criterion), the structure formation scenario must produce a nearly
scale-invariant spectrum of primordial perturbations (see e.g. [6] for a more detailed
discussion).
Inflationary cosmology [4] is the first scenario which satisfies the four above
criteria. During the time interval t i < t < t R during which the universe undergoes
nearly exponential expansion, the horizon expands exponentially while the Hubble
radius remains almost unchanged. Since the physical length of a fixed comoving
scale also expands nearly exponentially during the period of inflation, scales which
we observe today originate inside the Hubble radius as long as the period of
inflation is sufficiently long. Fluctuations are squeezed on super-Hubble scales for a
long time, and the approximate time-translation symmetry of the inflationary phase
ensures that the spectrum of primordial fluctuations is nearly scale-invariant [5, 7].
Bouncing cosmologies provide a second scenario in which the four criteria for
a successful early universe scenario can be satisfied. In a bouncing scenario the
horizon is infinite. The Hubble radius decreases during the period of contraction and
then increases during the period of expansion. As long as the period of contraction
is comparable in length to the period of Standard Big Bang expansion, scales
which we observed today emerge from inside the Hubble radius, thus allowing a
possible causal structure formation scenario. As in inflationary cosmology, there is a
long period during which scales propagate with super-Hubble length, thus enabling
the squeezing of the fluctuations. There are (at least) three classes of bouncing
cosmologies. First, the matter bounce [8] in which there is a long phase of matterdominated contraction. Second, there is the Pre-Big-Bang scenario [9] in which
contraction is driven by a field with an equation of state w = 1, where w is the ratio
of pressure to energy density. Finally, there is the Ekpyrotic scenario [10] in which
contraction is obtained by means of a scalar field with equation of state w 1.
There is a duality between the evolution of curvature fluctuations in a matterdominated phase of contraction and in an exponentially expanding background [11].
Hence, the matter bounce automatically leads to a roughly scale-invariant spectrum
of fluctuations. There is a duality in the evolution of scalar field fluctuations between
a cosmology with Ekpyrotic contraction and one of exponential expansion [12].
Hence, it is also possible to obtain a scale-invariant spectrum of fluctuations. In
the case of the Pre-Big-Bang scenario it is possible to obtain a scale-invariant
spectrum making use of axion fields [13]. See [14] for a detailed review of
bouncing cosmologies. Ekpyrotic and Pre-Big-Bang cosmologies produce a steep
blue spectrum of primordial gravitational waves. Hence, on cosmological scales
the spectrum of primordial gravitational waves is predicted to be negligible. This
contrasts with the predictions of inflationary models which forecast a roughly scaleinvariant spectrum.
A third scenario for early universe cosmology is the emergent scenario which is
based on the assumption that the universe emerged from an initial high density state
in which matter was in global thermal equilibrium. One toy model for this is String
Gas Cosmology [15] in which it is assumed that the universe loiters for a long time
in a Hagedorn phase of a gas of fundamental strings, and there is a phase transition