CMB anomalies and the effects of local features of the inflaton potential
Eur. Phys. J. C
CMB anomalies and the effects of local features of the inflaton potential
Alexander Gallego Cadavid 2 3 4
Antonio Enea Romano 1 2 3
Stefano Gariazzo 0 1 5
0 INFN, Sezione di Torino , Via P. Giuria 1, 10125 Turin , Italy
1 Department of Physics, University of Torino , Via P. Giuria 1, 10125 Turin , Italy
2 Yukawa Institute for Theoretical Physics, Kyoto University , Kyoto , Japan
3 Instituto de Fisica, Universidad de Antioquia , A.A.1226 Medellin , Colombia
4 ICRANet , Piazza della Repubblica 10, 65122 Pescara , Italy
5 Instituto de Física Corpuscular (CSICUniversitat de València) , Paterna, Valencia , Spain
Recent analysis of the WMAP and Planck data have shown the presence of a dip and a bump in the spectrum of primordial perturbations at the scales k = 0.002 Mpc−1 and k = 0.0035 Mpc−1, respectively. We analyze for the first time the effects of a local feature in the inflaton potential to explain the observed deviations from scale invariance in the primordial spectrum. We perform a bestfit analysis of the cosmic microwave background (CMB) radiation temperature and polarization data. The effects of the features can improve the agreement with observational data respect to the featureless model. The bestfit local feature affects the primordial curvature spectrum mainly in the region of the bump, leaving the spectrum unaffected on other scales.

There are important observational motivations to study
modifications of the inflaton potential, like the observed deviations
of the spectrum of primordial curvature perturbations from a
powerlaw spectrum [1–21]. In Refs. [1–13] the authors study
the effects of analyzing the cosmic microwave background
(CMB) radiation using a free function for the spectrum of
primordial scalar perturbations, i.e., they do not consider the
usual powerlaw spectrum predicted by most of the simplest
inflationary models [14,15,22]. For example, the primordial
spectrum can be parametrized with wavelets [4,5], linear
interpolation [6–8], interpolating spline functions [11–13],
among other methods [1,16].
Some interesting evidence of these deviations were given
in [1,16] where it was used a method based on a
piecewise cubic Hermite interpolating polynomial (PCHIP) for
the primordial power spectrum. This analysis showed that the
spectrum of primordial perturbations can be approximated
with a power law in the range of values 0.007 Mpc−1 <
k < 0.2 Mpc−1, while in the range 0.001 Mpc−1 < k <
0.0035 Mpc−1 there are a dip and a bump at k = 0.002
Mpc−1 and k = 0.0035 Mpc−1, with a statistical
significance of about 2σ and 1σ , respectively. Similar results
were reported in several other analyses [1–3,14,23–34] using
different techniques and both the WMAP [35] and the
Planck [36,37] measurements. In this paper we study how
local features of the inflaton potential can model this type of
local glitches of the spectrum of primordial curvature
perturbations. We also study the effects of these features on the
primordial tensorial perturbation spectrum.
Features of the inflaton potential can affect the evolution
of primordial curvature perturbations [1,38–59] and
consequently generate a variation in the amplitude of the
spectrum and bispectrum [38–46,56]. This can provide a better fit
of the observational data in the regions where the spectrum
shows some deviations from a power law [1–3,40,41,43–
46,51,52,60–62]. In this paper we perform a bestfit analysis
of the CMB radiation temperature and polarization [63] data
and we study the effects of a local feature of the inflation
potential which affects the primordial curvature spectrum in
the region of the bump.
2 Local features
We consider a single scalar field minimally coupled to gravity
with a standard kinetic term according to the action
S =
d4x √−g 21 M P2l R − 21 gμν ∂μφ∂ν φ − V (φ) , (1)
where MPl = (8π G)−1/2 is the reduced Planck mass and
gμν is the flat FLRW metric. The Friedmann equation and
the equation of motion of the inflaton are obtained from the
variation of the action with respect to the metric and the scalar
field, respectively,
H 2 ≡
a 2
˙
a
= 3M P2l
where H is the Hubble parameter, and dots and ∂φ denote
derivatives with respect to time and scalar field, respectively.
The slowroll parameters are defined
We consider a potential energy given by [39]
where V0(φ) is the featureless potential and VF corresponds
to a step symmetrically dumped by an even power negative
exponential factor. In this paper we will consider the case of
a quadratic inflaton potential,
The tensortoscalar ratio for a monomial potential φn is r ≈
16n/(4Ne + n), where Ne is the number of efolds before
the end of inflation [14, 15]. In the case of quadratic inflation
r ≈ 0.16 for Ne ≈ 50, which is not in good agreement with
observational data. Our analysis confirms this when we fit
data without the feature. We will show later that the effects
Fig. 1 The numerically computed slowroll parameters and η around
the feature time t0 for λ = −4 × 10−12, σ = 0.05, and k0 =
1.2 × 10−3 (blue), λ = −10−12, σ = 0.05, and k0 = 1.3 × 10−3
(red), λ = −10−11, σ = 0.04, and k0 = 1.3 × 10−3 (green), and
λ = −1.5 × 10−11, σ = 0.04, and k0 = 1.2 × 10−3 (orange). The
dashed lines correspond to the featureless slowroll parameters
of local features improve the agreement with CMB data but
not enough to get a χ 2 as low as the one of other inflationary
models with lower values of r .
This type of modification of the slowroll potential is
called a local feature (LF) [39] which differs from the branch
feature (BF) [39, 56], since the potential is symmetric with
respect to the location of the feature and it is only affected in
a limited range of the scalar field value. Due to this the
spectrum and bispectrum are only modified in a narrow range of
scales, in contrast to the BF in which there are differences in
the power spectrum between large and small scale which are
absent in the case of LF. In some cases the step in the spectrum
due to a BF can be very small, and the difference between
large and small scale effects would not make BF
observationally distinguishable from LF. Nevertheless in general the
oscillation patterns produce in the spectrum by a single BF
would be different because a single LF can be considered as
the combination of two appropriate BF [39].
In this paper we use the local type effect of these features to
model phenomenologically local glitches of the primordial
scalar spectrum on the scales k = 0.002 Mpc−1 and k =
0.0035 Mpc−1 [1], and to study their effects on the primordial
tensor spectrum, without affecting other scales.
The effects of the feature on the slowroll parameters are
shown in Fig. 1, where we can see that there are oscillations of
the slowroll parameters around the feature time t0, defined as
φ0 = φ (t0) [39]. The magnitude of the potential modification
is controlled by the parameter λ, as its effect is such that larger
value of λ give larger values of the slowroll parameters. The
size of the range of field values where the potential is affected
by the feature is determined by the parameter σ and the
slowroll parameters are smaller for larger σ . We define k0 as the
scale exiting the horizon at the feature time t0, k0 = −1/τ0,
where τ0 is the value of conformal time corresponding to
t0. Oscillations occur around k0, and their location can be
controlled by changing φ0. We adopt a system of units in
which c = h¯ = MPl = 1.
3 Spectrum of curvature tensor perturbations
In order to study the curvature perturbations we expand
perturbatively the action with respect to the background FLRW
solution. The second order action for scalar perturbations in
the comoving gauge takes the form [64]
S2 =
The equation for curvature perturbations ζ obtained from the
Lagrange equations is
Fig. 2 The power spectrum of primordial curvature perturbations Pζ
is plotted for λ = −4 × 10−12, σ = 0.05, and k0 = 1.2 × 10−3 (blue),
λ = −10−12, σ = 0.05, and k0 = 1.3 × 10−3 (red), λ = −10−11, σ =
0.04, and k0 = 1.3 × 10−3 (green), and λ = −1.5 × 10−11, σ = 0.04,
and k0 = 1.2 × 10−3 (orange). The dashed lines correspond to the
featureless spectrum
Fig. 3 The power spectrum of primordial tensor perturbations Ph is
plotted for λ = −4 × 10−12, σ = 0.05, and k0 = 1.2 × 10−3 (blue),
λ = −10−12, σ = 0.05, and k0 = 1.3 × 10−3 (red), λ = −10−11, σ =
0.04, and k0 = 1.3 × 10−3 (green), and λ = −1.5 × 10−11, σ = 0.04,
and k0 = 1.2 × 10−3 (orange). The dashed lines correspond to the
featureless spectrum. The plot on the right corresponds to a zoom of
the left plot. As can be seen, the effects of the different features on the
spectrum Ph are rather small and the spectra of the models with features
are difficult to distinguish from the featureless model spectrum
where k is the comoving wave number, z ≡ a√2 , and
primes denote a derivative with respect to the conformal time.
The twopoint function of curvature perturbations is
The effects of the features on the primordial scalar spectrum
are plotted in Fig. 2 for different values of the parameters
λ, σ , and k0 [39]. The spectrum of primordial curvature
perturbations has oscillations around k0, whose amplitude is
larger for larger λ since the latter controls the magnitude of
the potential modification. The amplitude of the spectrum
oscillations is larger for smaller σ , because in this case the
change in the potential is more abrupt and consequently the
slowroll parameters are larger.
The equation for tensor perturbations can be derived in a
way similar to the case of scalar perturbations, giving
where again k is the comoving wave number. The power
spectrum of tensor perturbations is obtained from the
twopoint function as
Ph (k) ≡
from which the tensortoscalar ratio can be defined as the
ratio between the spectrum of tensor and scalar perturbations
as
The effects of the features on the primordial tensor spectrum
are plotted in Fig. 3 for different values of the parameters
λ, σ , and k0. These effects are not very significant and in
fact the observational data analysis we will present in the
rest of the paper confirms that local features affect mainly
the curvature spectrum.
4 Effects of local features on the CMB spectrum
In Fig. 4 we show the effects of local features on the
temperature (TT) CMB spectrum. Since we are considering a feature
of local type, as theoretically expected, the spectrum is not
affected on scales sufficiently far from k0. Branch features
[39] could on the contrary also introduce a step in the power
spectrum, modifying it also on scales far from k0, and for this
reason LF are more appropriate to model local deviations of
the spectrum.
The main effects produced by the LF appear between =
10 and = 100 in the TT spectrum. They correspond to the
wiggles of the primordial scalar fluctuations shown in Fig. 2.
The class of LF we consider allows one to fit the small bump
at 40 better than the dip at 20 in the CMB spectrum.
The impact of the LF on the BB spectrum is much smaller,
Table 1 Constraints on the cosmological parameters and χ 2 for the
model with and without feature. All the constraints are given at 1σ
confidence level. The lower limits on the feature parameters correspond
to the limits we used as a prior. The bestfit values are the values inside
curly brackets. We separately report the different contributions to the
χ 2 (Planck low , Planck high and from the priors on the nuisance
parameters) and the total
[0.05, 1.23) {1.12}
5.3 +−13..11
since, as discussed previously, the effect of the feature on the
primordial tensorial perturbations spectrum is negligible.
4.1 The observational data analysis method
To study the effects produced by local features on the CMB
spectrum, we modified the Boltzmann code CAMB [65]
that computes the theoretical spectra and the corresponding
Markov chain monte carlo (MCMC) code CosmoMC [66] in
Fig. 5 A comparison between the model with and without features
is given for the parameters H0, ωb, ωc, θ , τ and ln(1010 As ). All the
results are obtained considering the Planck low +high data
combinaorder to use a nonstandard power spectrum for the
primordial curvature perturbations.
As a base model we considered the standard
parameterization of the CDM model for the evolution of the universe,
which includes four parameters: the current energy density
of baryons and of cold dark matter (CDM) bh2 and ch2,
the ratio between the sound horizon and the angular diameter
distance at decoupling θ , and the optical depth to
reionization τ .
tion. As can be seen the effects of the feature on the estimation of these
noninflationary cosmological parameters is negligible
The parameterization of the primordial power spectra is
modified to take into account the presence of the local
feature. To see the effects of the feature, we compare the results
obtained in the featureless model with the ones obtained
when a local feature is added. The comparison of the effects
of LF of different inflationary potentials is left for future
studies.
The data sets that we use to test the LF are taken from
the last release from the Planck Collaboration [37] for the
Fig. 7 The numerically computed spectrum of the primordial
curvature fluctuations Pζ and of the tensor perturbations Ph are plotted for
the bestfit values in Table 1: λ = −1.12 × 10−12, σ = 0.053, and
k0 = 1.13 × 10−3 (blue). On the left, the red lines correspond to the
bestfit reconstructed primordial power spectrum from Ref. [16]. The
dashed lines correspond to the featureless spectrum
temperature and Emode polarization modes. We consider
the temperature and polarization power spectra in the range
2 ≤ ≤ 29 (low ) and only the temperature power
spectrum at higher multipoles, 30 ≤ ≤ 2500 (high ). Since
the polarization spectra at high multipoles are still under
discussion and some residual systematics were detected by
the Planck Collaboration [67, 68], we do not include the
full polarization spectra obtained by Planck. Moreover, we
Fig. 8 The 1, 2 and 3σ constraints obtained from observational data
analysis are plotted for the primordial curvature perturbations spectrum
for the model with local features. The spectrum for the featureless model
is plotted with a red line
do not include the data on the BB spectrum as obtained
from the Bicep2/Keck Collaboration [69], because the
baseline inflationary model that we consider (φ2) cannot
reproduce the small amount of primordial tensor modes that are
observed after cleaning the Bicep2/Keck data using the
polarized dust emission obtained by the high frequency maps by
Planck [70].
5 Results
The results of the data fitting analysis are reported in Table 1
and in Figs. 5, 6, 7 and 8.
In Table 1 we show the bestfit values written inside
brackets and the 1σ constraints of the parameters. It should be
noted that the bounds we obtain are more stringent than the
Planck ones because ns is not a free parameter. Fixing the
value of scalar spectral index reduces the confidence ranges
for the others parameters, and consequently our bounds are
smaller. If we had left free the potential of the inflaton in
a generic monomial form V0 ∼ φn, then we could have
obtained larger bounds as in the Planck team analysis where
ns is a free parameter. This could be done in a future work,
but it goes beyond the scope of the present paper.
Comparing the results obtained with and without feature
we can see that the presence of the LF has no impact on
the background cosmological parameters. This is clear from
the marginalized 1D and 2D plots in Fig. 5. The effect of
the feature is evident around the location of the bump of the
CMB temperature spectrum (see Fig. 4), and it corresponds
to an improvement of the total χ 2. As reported in Table 1, the
improvement comes from the χ 2 of the low Planck
likelihood. Our analysis cannot be compared with the Planck
results [68,70], we are assuming the φ2 inflationary model
instead of using a phenomenological approach with
independent ns and r . Quadratic inflation corresponds to high
values of r which are not in agreement with the Planck
bestfit model obtained using ns and r as independent parameters.
The effects of the feature improve the χ 2 with respect to the
featureless φ2 case, but this improvement is not large enough
to make it competitive with other models. Nevertheless, the
same LF could be applied to other inflationary scenarios to
produce an analogous improvement of the χ 2. The analyses
of the effects of the LF for inflationary models that are in
better agreement with the observed CMB spectra are left for
future studies.
In Fig. 6 we show the 1D marginalized posterior
distributions and the correlations between the feature parameters.
From the correlation plot between λ and k0 we can see that
the size of the feature can be larger if the feature is located
at a smaller wavemode k0. This is because the CMB
temperature spectrum does not allow any wiggles above 60,
thus limiting the amplitude of the feature. The 2D plots for
the parameter σ seem to show that there is no lower bound
on it. This is not in tension with the 1σ constraints on the σ
parameter reported in Table 1, because of volume effects that
occur in the Bayesian marginalization procedure. The
preference for a nonminimum value of σ is mild, indeed there
is no lower bound at 2σ confidence level.
The constraints on the primordial scalar spectrum are
shown in Figs. 7 and 8. In the left panel of Fig. 7 we compare
the bestfit primordial power spectrum of scalar
perturbations obtained in our analysis (blue) and the reconstructed
one from Ref. [16]. The comparison underlines how a local
feature can reproduce the behavior of the primordial
spectrum, but further studies, which will be presented in some
future work, on the feature potential are required in order to
obtain a perfect agreement. The right panel of the same Fig. 7
shows that the effect of the feature is very small in the tensor
spectrum. In Fig. 8 we plot the marginalized constraints on
the primordial scalar spectrum. The 1, 2, and 3σ bands refer
to the model with LF, while the solid black line shows the
corresponding bestfit spectrum, computed from the entire set
of primordial spectra obtained from the MCMC scan. The
red dashed line shows the spectrum obtained for the same
cosmological parameters but without the feature. From the
figures we can note that the effects that the LF brings about
are more important for the scalar spectrum, while they are
negligible for the tensor spectrum. For this reason, we do not
show the same plot as for Fig. 8 for the tensor spectrum.
6 Conclusions
We have studied the effects of local features in the inflaton
potential on the spectra of primordial curvature perturbations
and their impact on the temperature anisotropies of the CMB.
In order to study the effects on the CMB spectrum we have
modified the CAMB and CosmoMC codes in order to use a
nonstandard powerlaw power spectrum for the primordial
perturbations, to take into account the presence of the local
feature. We have performed a bestfit analysis of CMB
temperature and polarization data from Planck. We have found
no significant effects on cosmological parameters related to
the propagation of CMB photons after decoupling, while LF
improve the fit of the CMB temperature and polarization data.
We have also confirmed the theoretical expectation that local
features do not affect the primordial power spectrum at scales
far from the characteristic scale k0, which leaves the horizon
around the feature time.
In the future it will be interesting to analyze the effects
of local features in order to explain other deviations of the
CMB spectrum, such as for example the anomalies occurring
around l ≈ 800 [2]. It will also be important to study the
effects of LF in inflationary models with different featureless
V0 potentials, and to compare them to the effects of branch
features.
Acknowledgements This work was supported by the European Union
(European Social Fund, ESF) and Greek national funds under the
ARISTEIA II? Action. Part of the work of SG was supported by the
Theoretical Astroparticle Physics research Grant No. 2012CPPYP7 under the
Program PRIN 2012 funded by the Ministero dell’Istruzione,
Università e della Ricerca (MIUR), and in part is also supported by the Spanish
Grants FPA201458183P, Multidark CSD200900064 and
SEV20140398 (MINECO), and PROMETEOII/2014/084 (Generalitat
Valenciana). The work of AGC was supported by the Colombian Department
of Science, Technology, and Innovation COLCIENCIAS research Grant
No. 6172013. AGC acknowledges the partial support from the
International Center for Relativistic Astrophysics Network ICRANet. For
part of the calculations we used the Cloud infrastructure of the Centro
di Calcolo in the Torino section of INFN. AER work was supported
by the Dedicacion exclusica and Sostenibilidad programs at UDEA,
the UDEA CODI project 20154044 and 201610945, and Colciencias
mobility program.
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