Fingerprints of Primordial Universe Paradigms as Features in Density Perturbations
ABSTRACT Experimentally distinguishing different primordial universe paradigms that
lead to the Big Bang model is an outstanding challenge in modern cosmology and
astrophysics. We show that a generic type of signals that exist in primordial
universe models can be used for such purpose. These signals are induced by tiny
oscillations of massive fields and manifest as features in primordial density
perturbations. They are capable of recording the time-dependence of the scale
factor of the primordial universe, and therefore provide direct evidence for
arXiv:1106.1635v1 [astro-ph.CO] 8 Jun 2011
Fingerprints of Primordial Universe Paradigms
as Features in Density Perturbations
Center for Theoretical Cosmology,
Department of Applied Mathematics and Theoretical Physics,
University of Cambridge, Cambridge CB3 0WA, UK
Experimentally distinguishing different primordial universe paradigms that lead to the Big
Bang model is an outstanding challenge in modern cosmology and astrophysics. We show
that a generic type of signals that exist in primordial universe models can be used for such
purpose. These signals are induced by tiny oscillations of massive fields and manifest as fea-
tures in primordial density perturbations. They are capable of recording the time-dependence
of the scale factor of the primordial universe, and therefore provide direct evidence for specific
Current observations have mapped out a detailed 13.7 billion years of Big Bang expansion
history for our universe. One important knowledge we learned from these observations is
that, at the very beginning of the Big Bang, there are tiny fluctuations in an otherwise very
homogeneous and isotropic background density. These fluctuations, called the primordial
density perturbations, become the seeds for the subsequent evolution of the large scale
structures. They turn out to have very special properties. They are seeded at super-horizon
scales, and are approximately scale-invariant, Gaussian and adiabatic . Understanding the
origin of these initial conditions and hence the origin of the Big Bang becomes an outstanding
challenge for the modern cosmology.
The inflation paradigm [2–4] is the leading candidate for creating such a primordial uni-
verse. The simplest inflationary scenarios not only explain why our universe is homogeneous
and isotropic, but also the properties of the density perturbations. Nonetheless, based on
current data, there exist important ambiguities and degeneracies. Competing paradigms
have been proposed to produce the same initial conditions. These alternatives include the
paradigms of cyclic universe , matter bounce [6,7] and string gas . Arguably, the infla-
tion still remains as the best available paradigm, because its generic predictions naturally fit
the data and its microscopic origin in terms of fundamental physics is promising. However,
such opinions may be subject to personal tastes; they are model-dependent and may even
evolve with time. Besides theoretical establishment for each paradigm, a universal and un-
ambiguous distinguisher is in terms of data. Namely we need to find some properties in the
density perturbations that can clearly distinguish different paradigms. Such distinguishers
are desirable even if one believes in a paradigm. In order to do so, such properties should
satisfy the following requirements. They have to be shared by all general models in one
paradigm, not just by a small subset of models; and they have to be distinctive for different
So far the primordial tensor mode is regarded as the only possible solution to this ques-
tion. They originate from gravitational waves in the primordial universe , and may be
observed in terms of polarizations in the cosmic microwave background (CMB) [10, 11].
Inflation models indeed have general predictions on the tensor mode. Namely, the tensor
modes are approximately scale-invariant with a red tilt; for some models they are observable.
Some alternative paradigms predict non-observable tensor modes, such as the cyclic universe
paradigm , or observable ones with blue-tilt, such as the string gas paradigm . However
there exist important caveats. Firstly, even for inflation, tensor modes sourced by the pri-
mordial gravitational waves are not guaranteed to be observable. While the best sensitivity
for the tensor-to-scalar ratio achievable by experiments in the near future is ∆r ∼ O(10−3),
the inflation models predict anywhere between r ∼ O(10−1), for large field models, and
r ∼ O(10−55), for small field models with TeV-scale reheating energy. Secondly, if we con-
sider more general alternatives, scale-invariant and observable tensor modes are possible.
The equation of motion obeyed by each polarization component of the tensor modes is the
same as that by the massless scalar with unit sound speed. So the tensor modes can be scale-
invariant even in non-inflationary spacetime, just as the scalar. They become observable if
the Hubble parameter is large. The matter contraction paradigm is such an example [6,7].
Therefore it is very important to look for complimentary properties in the density pertur-
bations that can be used as a model-independent distinguisher between different paradigms.
This is the main purpose of this paper.
Almost by definition, different primordial universe paradigms are distinctively character-
ized by different time-dependence in the scale factor a(t). To be explicit, we approximate
the arbitrary time-dependent scale factor as the general power-law,
a(t) = a(t0)(t/t0)p. (1)
The density perturbations are seeded at superhorizon scales, so we require that the quantum
fluctuations exit the event horizon |aτ|, where τ is the conformal time defined by dt ≡ adτ
and is related to t by aτ = t/(1 − p). To satisfy this requirement, for p > 1 we need an
expansion phase, so t runs from 0 to +∞; for 0 < p < 1 we need a contraction phase, so t
runs from −∞ to 0; for p < 0, we again need an expansion phase, so t runs from −∞ to 0.
For example, p > 1 corresponds to the inflation, p = 2/3 the matter contraction, 0 < p ≪ 1
the ekpyrotic (slowly contracting) phase, and −1 ≪ p < 0 the slowly expanding phase. τ
always runs from −∞ to 0.
Many properties of the density perturbations are convoluted consequence of this scale
factor, and this is a primary reason for the ambiguity. Perhaps the most direct way to
distinguish different paradigms would be to find some properties that can directly record the
time-dependence of the scale factor. In this paper, we show that a type of feature models
that generically exist in the primordial universe paradigms provide such opportunities.
We first describe this idea heuristically. Consider models with some small and repeated
features. Such features can be various kinds of small structures in the realistic primordial
universe models, and we will give a very general example shortly. These features generate
small and repeated beats in cosmological parameters during the evolution, which can be used
as a clock and related to the time t. In the mean while, the scale factor a and the comoving
momenta k of quantum fluctuations are related by the physical momenta by definition. The
way that the features imprint themselves on the density perturbations is through the beats
on the physical modes. Therefore the scale factor a as a function of t is translated to the
comoving momenta k as a function of features. What we can measure is features as a function
of k. Knowing that we know the inverse function of a(t). Let us now examine how this idea
may be implemented and how general it is.
In order to achieve the goal, we need to look for standard physical clocks. Such clocks
should generate repeated features with known time-dependence, although not necessarily
periodic. They should exist generically so the signals are more likely to be observed, and
they should also be associated with a set of specific patterns identifiable in observations.
The classical oscillation of massive fields are ideal for such purposes.
Massive fields with mass much larger than the horizon mass-scale 1/|aτ| exist ubiquitously
in models of primordial universe. Even when we think of single field models, in the context of
a unification theory, what we really have in mind is models with many massive modes. The
single field model is obtained as the low energy limit at or below the energy scale 1/|aτ|, after
the massive modes are integrated out. If we take into account these massive field directions,
the trajectory of the effective single field φ in this multi-dimensional field space is unlikely
a straight line. Generically it should turn from time to time. During the turning, the field
φ and the massive modes couple, and this induces the energy transfer. The massive fields
are excited and oscillate around the potential minima classically. Because generically the
fraction of energy transfer is small, the oscillation amplitudes are small but the frequency
are large. So for most purposes, these effect are averaged out and safely ignored. But for
our purpose, these small oscillations are a good candidate for the physical clock that we
are looking for. So let us examine more closely the behavior of the massive field σ in the
The equation of motion is
¨ σ + 3H ˙ σ + m2
σσ = 0 ,(2)
where the Hubble parameter H = p/t. The solution is given in terms of Bessel functions.
The asymptotic behavior of these Bessel functions at the limit mσt ≫ p2is given in terms
of sinusoidal functions, and we use these to approximate the oscillatory behavior of σ,
σ ≈ σA
sin(mσt + α) +−6p + 9p2
cos(mσt + α)
where α is a phase, and σAis the initial oscillation amplitude at t = t0. Such oscillations
induce an oscillatory component to the Hubble parameter H, because
2m2σ2+ other fields .(4)
The leading term on the right hand side of (4) does not oscillate in time because the energy
is converting between kinetic and potential energy back and forth and conserved in the
leading order. The oscillatory component for H, which we denote as Hosci, comes from the
subleading terms. Using (3), we get
sin(2mσt + 2α) .(5)
This in turn induces the oscillatory components for the parameters ǫ ≡ −˙H/H2and η ≡
˙ ǫ/(Hǫ). Again we use the subscript “osci” to denote their oscillatory components,
cos(2mσt + 2α) , (6)
˙ ηosci = −σ2
cos(2mσt + 2α) . (7)
Generally, multiple massive fields are excited sporadically along the trajectory with dif-
ferent spectra. These oscillations induce corresponding beats in the cosmological parameters.
The question now is how sensitive the observables are to these oscillations and how they can
be identified in experiments. To study this, let us turn to the subject of Bunch-Davies (BD)
vacuum and resonance mechanism.
In nearly all models of primordial universe, the quantum fluctuations start their life in a
vacuum that is mostly BD. These fluctuations later exit the event horizon and become the
seeds for the large scale structure. Their general property can be seen as follows. Consider
the fluctuations of an effectively massless scalar field, δφ(x,t), in a general time-dependent
background with the scale factor a(t),
A quantum fluctuation with comoving momentum k is within the event horizon if k > 1/|τ|.
In this limit, the equation of motion for the fluctuations approaches that in the Minkowski
spacetime limit. Along with the quantization condition,
∗− c.c. = i , (9)
the mode function in the subhorizon limit becomes
We have chosen the positive-energy mode, which corresponds to the ground state of the
Minkowski spacetime, to be the BD vacuum. The effect of the background time-dependence
is incorporated adiabatically in (10). The most important and universal property of (10) is
the oscillatory factor e−ikτ.
While the superhorizon evolution of different paradigms are very model-dependent, the
subhorizon BD-vacuum behavior is universal. This applies to both inflationary and non-
inflationary scenario, expansion and contraction universe, attractor and non-attractor evo-
lution, single field and multifield model, and curvaton and isocurvaton modes.
If the background cosmological parameters have small oscillatory components with fre-
quencies much larger than the energy scale of the event-horizon, they introduce a high
energy probe to the BD vacuum by resonating with the vacuum component that has the