CMB anomalies and the effects of local features of the inflaton potential
Alexander Gallego Cadavid, Antonio Enea Romano, Stefano Gariazzo
CCMB anomalies and the effects of local features of the inflatonpotential
Alexander Gallego Cadavid , , , Antonio Enea Romano , , , Stefano Gariazzo , , Yukawa Institute for Theoretical Physics, Kyoto University, Japan Department of Physics, University of Torino,Via P. Giuria 1, I–10125 Torino, Italy INFN, Sezione di Torino, Via P. Giuria 1, I–10125 Torino, Italy Instituto de F´ısica Corpuscular (CSIC-Universitat de Val`encia), Paterna (Valencia), Spain ICRANet, Piazza della Repubblica 10, I–65122 Pescara Instituto de Fisica, Universidad de Antioquia, A.A.1226, Medellin, Colombia
Abstract
Recent analysis of the WMAP and Planck data have shown the presence of a dip and a bumpin the spectrum of primordial perturbations at the scales k = 0 .
002 Mpc − and k = 0 . − respectively. We analyze for the first time the effects a local feature in the inflaton potential toexplain 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 featurelessmodel. The best fit local feature affects the primordial curvature spectrum mainly in the region ofthe bump, leaving the spectrum unaffected on other scales. a r X i v : . [ a s t r o - ph . C O ] A p r . INTRODUCTION There are important observational motivations to study modifications of the inflaton poten-tial, like the observed deviations of the spectrum of primordial curvature perturbations from apower law spectrum [1–21]. In Refs. [1–13] the authors study the effects of analizing the cosmicmicrowave background (CMB) radiation using a free function for the spectrum of primordialscalar perturbations, i.e., they do not consider the usual power law spectrum predicted by mostof the simplest inflationary models [14, 15, 22]. For example, the primordial spectrum canbe 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 amethod based on a piecewise cubic Hermite interpolating polynomial (PCHIP) for the primor-dial power spectrum. This analysis showed that the spectrum of primordial perturbations canbe approximated with a power law in the range of values 0 .
007 Mpc − < k < . − whilein the range 0 .
001 Mpc − < k < . − there are a dip and a bump at k = 0 .
002 Mpc − and k = 0 . − , with a statistical significance of about 2 σ and 1 σ , respectively. Similarresults were reported in several other analyses [1–3, 14, 23–34] using different techniques andboth the WMAP [35] and Planck [36, 37] measurements. In this paper we study how localfeatures of the inflaton potential can model this type of local glitches of the spectrum of pri-mordial curvature perturbations. We also study the effects of these features on the primordialtensorial perturbation spectrum.Features of the inflaton potential can affect the evolution of primordial curvature perturba-tions [1, 38–59] and consequently generate a variation in the amplitude of the spectrum andbispectrum [38–46, 56]. This can provide a better fit of the observational data in the regionswhere 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 best fit analysis of the CMB radiation temperature and polarizationdata and we study the effects of a local feature of the inflation potential which affects theprimordial curvature spectrum in the region of the bump.2 I. LOCAL FEATURES
We consider a single scalar field minimally coupled to gravity with a standard kinetic termaccording to the action S = (cid:90) d x √− g (cid:20) M P l R − g µν ∂ µ φ∂ ν φ − V ( φ ) (cid:21) , (1)where M P l = (8 πG ) − / is the reduced Planck mass and g µν is the flat F LRW metric. TheFriedmann equation and the equation of motion of the inflaton are obtained from the variationof the action with respect to the metric and the scalar field respectively H ≡ (cid:18) ˙ aa (cid:19) = 13 M P l (cid:18)
12 ˙ φ + V ( φ ) (cid:19) , (2)¨ φ + 3 H ˙ φ + ∂ φ V = 0 , (3)where H is the Hubble parameter, and dots and ∂ φ denote derivatives with respect to time andscalar field respectively. The slow-roll parameters are defined (cid:15) ≡ − ˙ HH , η ≡ ˙ (cid:15)(cid:15)H . (4)We consider a potential energy given by [39] V ( φ ) = V ( φ ) + V F ( φ ) , (5) V F ( φ ) = λe − ( φ − φ σ ) , (6)where V ( φ ) is the featureless potential and V F corresponds to a step symmetrically dumpedby an even power negative exponential factor. In this paper we will consider the case of aquadratic inflaton potential V ( φ ) = 12 m φ . (7)The tensor-to-scalar ratio for a monomial potential φ n is r ≈ n/ (4 N e + n ), where N e is thenumber of e -folds before the end of inflation [14, 15]. In the case of quadratic inflation r ≈ . N e ≈
50, which is not in good agreement with observational data. Our analysis confirmsthis when we fit data without the feature. We will show later that the effects of local featuresimprove the agreement with CMB data but not enough to get a χ as low as the one of otherinflationary models with lower values of r .This type of modification of the slow-roll potential is called local feature (LF) [39] whichdiffers from the branch feature (BF) [39, 56] since the potential is symmetric with respect to3he 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, incontrast to the BF in which there are differences in the power spectrum between large andsmall scale which are absent in the case of LF. In some cases the step in the spectrum dueto a BF can be very small, and the difference between large and small scale effects wouldnot make BF observationally distinguishable from LF. Nevertheless in general the oscillationpatterns produce in the spectrum by a single BF would be different because a single LF can beconsidered as the combination of two appropriate BF [39].In this paper we use the local type effect of these features to model phenomenologi-cally local glitches of the primordial scalar spectrum on the scales k = 0 .
002 Mpc − and k = 0 . − [1], and to study their effects on the primordial tensor spectrum, with-out affecting other scales.The effects of the feature on the slow-roll parameters are shown in Fig. 1, where we cansee that there are oscillations of the slow-roll parameters around the feature time t , definedas φ = φ ( t ) [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 slow-roll parameters. Thesize of the range of field values where the potential is affected by the feature is determined bythe parameter σ and the slow-roll parameters are smaller for larger σ . We define k as thescale exiting the horizon at the feature time t , k = − /τ , where τ is the value of conformaltime corresponding to t . Oscillations occur around k , and their location can be controlled bychanging φ . We adopt a system of units in which c = ¯ h = M P l = 1.
III. SPECTRUM OF CURVATURE TENSOR PERTURBATIONS
In order to study the curvature perturbations we expand perturbatively the action withrespect to the background
F LRW solution. The second order action for scalar perturbationsin the comoving gauge takes the form [63] S = (cid:90) dtd x (cid:104) a (cid:15) ˙ ζ − a(cid:15) ( ∂ζ ) (cid:105) . (8)The equation for curvature perturbations ζ obtained from the Lagrange equations is ∂∂t (cid:32) a (cid:15) ∂ζ∂t (cid:33) − a(cid:15)δ ij ∂ ζ∂x i ∂x j = 0 . (9)4 .6 0.8 1.0 1.2 1.4 1.6 1.8 tt Ε tt (cid:45) (cid:45) Η Figure 1: The numerically computed slow-roll parameters (cid:15) and η around the feature time t for λ = − × − , σ = 0 .
05, and k = 1 . × − (blue), λ = − − , σ = 0 .
05, and k = 1 . × − (red), λ = − − , σ = 0 .
04, and k = 1 . × − (green), and λ = − . × − , σ = 0 .
04, and k = 1 . × − (orange). The dashed lines correspond to the featureless slow roll parameters. Taking the Fourier transform and using conformal time dτ ≡ dt/a we get ζ (cid:48)(cid:48) k + 2 z (cid:48) z ζ (cid:48) k + k ζ k = 0 , (10)where k is the comoving wave number, z ≡ a √ (cid:15) , and primes denote a derivative with respectto the conformal time. The two-point function of curvature perturbations is (cid:68) ˆ ζ ( (cid:126)k , t ) ˆ ζ ( (cid:126)k , t ) (cid:69) ≡ (2 π ) π k P ζ ( k ) δ (3) ( (cid:126)k + (cid:126)k ) , (11)where the power spectrum of curvature perturbations is defined as P ζ ( k ) ≡ k π | ζ k | . (12)The effects of the features on the primordial scalar spectrum are plotted in Fig. 2 for differentvalues of the parameters λ, σ , and k [39]. The spectrum of primordial curvature perturbationshas oscillations around k , whose amplitude is larger for larger λ since the latter controls themagnitude of the potential modification. The amplitude of the spectrum oscillations is largerfor smaller σ , because in this case the change in the potential is more abrupt and consequentlythe slow-roll parameters are larger.The equation for tensor perturbations can be derived in a way similar to the case of scalarperturbations, giving h (cid:48)(cid:48) k + 2 a (cid:48) a h (cid:48) k + k h k = 0 , (13)5 (cid:180) (cid:45) k Mpc (cid:45) (cid:180) (cid:45) P Ζ Figure 2: The power spectrum of primordial curvature perturbations P ζ is plotted for λ = − × − , σ = 0 .
05, and k = 1 . × − (blue), λ = − − , σ = 0 .
05, and k = 1 . × − (red), λ = − − , σ = 0 .
04, and k = 1 . × − (green), and λ = − . × − , σ = 0 .
04, and k = 1 . × − (orange). The dashed lines correspond to the featureless spectrum. where again k is the comoving wave number. The power spectrum of tensor perturbations isobtained from the two-point function as P h ( k ) ≡ k π | h k | , (14)from which the tensor-to-scalar ratio can be defined as the ratio between the spectrum of tensorand scalar perturbations as r ≡ P h P ζ . (15)The effects of the features on the primordial tensor spectrum are plotted in Fig. 3 for differentvalues of the parameters λ, σ , and k . These effects are not very significant and in fact theobservational data analysis we will present in the rest of the paper confirms that local featuresaffect mainly the curvature spectrum. 6 (cid:180) (cid:45) k Mpc (cid:45) (cid:180) (cid:45) P h k Mpc (cid:45) (cid:180) (cid:45) P h Figure 3: The power spectrum of primordial tensor perturbations P h is plotted for λ = − × − , σ =0 .
05, and k = 1 . × − (blue), λ = − − , σ = 0 .
05, and k = 1 . × − (red), λ = − − , σ =0 .
04, and k = 1 . × − (green), and λ = − . × − , σ = 0 .
04, and k = 1 . × − (orange). Thedashed lines correspond to the featureless spectrum. The plot on the right corresponds to a zoom ofthe left plot. As it can be seen the effects of the different features on the spectrum P h are rather smalland the spectra of the models with features are difficult to distinguish from the featureless modelspectrum. IV. 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 isnot affected on scales sufficiently far from k . Branch features [39] could on the contrary alsointroduce a step in the power spectrum, modifying it also on scales far from k , and for thisreason LF are more appropriate to model local deviations of the spectrum.The main effects produced by the LF appear between (cid:96) = 10 and (cid:96) = 100 in the TTspectrum. They correspond to the wiggles of the primordial scalar fluctuations shown in Fig. 2.The class of LF we consider allows to fit the small bump at (cid:96) (cid:39)
40 better than the dip at (cid:96) (cid:39)
20 in the CMB spectrum. The impact of the LF on the BB spectrum is much smaller,since, as discussed previously, the effect of the feature on the primordial tensorial perturbationsspectrum is negligible.
A. The observational data analysis method
To study the effects produced by local features on the CMB spectrum, we modified theBoltzmann code
CAMB [66] that computes the theoretical spectra and the corresponding MarkovChain Monte Carlo (MCMC) code
CosmoMC [67] in order to use a non-standard power spectrum7 ‘ D TT ‘ [ µ K ] featurelessfeaturePlanck2015 ‘ ∆ D TT ‘ / D TT ‘ Figure 4: The D T Tl = (cid:96) ( (cid:96) + 1) C T T(cid:96) / (2 π ) spectrum in units of µK is plotted as a function of themultipole l . We compare the best-fit obtained using the inflationary model without feature (red line)to the one obtained introducing the local feature (black line). In the lower panel we plot the relativedifference with respect to the featureless case. The data points are from the 2015 Planck release [37].The cosmological parameters used to compute the two spectra are reported in Tab. I. for the primordial curvature perturbations.As a base model we considered the standard parameterization of the ΛCDM model for theevolution of the universe, that includes four parameters: the current energy density of baryonsand of Cold Dark Matter (CDM) Ω b h and Ω c h , the ratio between the sound horizon and theangular diameter distance at decoupling θ , and the optical depth to reionization τ .The parameterization of the primordial power spectra is modified to take into account thepresence of the local feature. To see the effects of the feature, we compare the results obtainedin the featureless model with the ones obtained when a local feature is added. The comparisonof 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 Col-8aboration [37] for the temperature and E-mode polarization modes. We consider the tempera-ture and polarization power spectra in the range 2 ≤ (cid:96) ≤
29 (low- (cid:96) ) and only the temperaturepower spectrum at higher multipoles, 30 ≤ (cid:96) ≤ (cid:96) ). Since the polarization spec-tra at high multipoles are still under discussion and some residual systematics were detectedby the Planck Collaboration [68, 69], we do not include the full polarization spectra obtainedby Planck. Moreover, we do not include the data on the BB spectrum as obtained from theBicep2/Keck collaboration [70], because the baseline inflationary model that we consider ( φ )cannot reproduce the small amount of primordial tensor modes that are observed after cleaningthe Bicep2/Keck data using the polarized dust emission obtained by the high frequency mapsby Planck [65]. V. RESULTS
The results of the data fitting analysis are reported in Tab. I and in Figs. 5 to 8.In Tab. I we show the best-fit values written inside brackets and the 1 σ constraints of theparameters. It should be noted that the bounds we obtain are more stringent than the Planckones because n s is not a free parameter. Fixing the value of scalar spectral index reduces theconfidence ranges for the others parameters, and consequently our bounds are smaller. If wehad left free the potential of the inflaton in a generic monomial form V ∼ φ n , then we couldhave obtained larger bounds as in the Planck team analysis where n s is a free parameter. Thiscould 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 presenceof the LF has no impact on the background cosmological parameters. This is clear from themarginalized 1D and 2D plots in Fig. 5. The effect of the feature is evident around the location ofthe bump of the CMB temperature spectrum (see Fig. 4), and it corresponds to an improvementof the total χ . As reported in Tab. I, the improvement comes from the χ of the low- (cid:96) Plancklikelihood. Our analysis cannot be compared with the Planck results [65, 69], we are assumingthe φ inflationary model instead of using a phenomenological approach with independent n s and r . Quadratic inflation corresponds to high values of r which are not in agreement withthe Planck best-fit model obtained using n s and r as independent parameters. The effects ofthe feature improve the χ with respect to the featureless φ case, but this improvement is notlarge enough to make it competitive with other models. Nevertheless, the same LF could beapplied to other inflationary scenarios to produce an analogous improvement of the χ . The9 arameter with feature featureless10 ω b . ± .
019 2 . ± . ω c . ± .
001 0 . ± . θ . ± . . ± . τ . ± .
015 0 . ± . A s ) 3 . ± .
031 3 . ± . H . ± . . ± . − λ [0 . , . { . } -10 σ . +1 . − . -10 k [1 . , . { . } - χ (cid:96) χ (cid:96) χ χ tot χ for the model with and without feature. Allthe constraints are given at 1 σ confidence level. The lower limits on the feature parameters correspondto the limits we used as a prior. The best fit are values inside curly brackets. We separately reportthe different contributions to the χ (Planck low- (cid:96) , Planck high- (cid:96) and from the priors on the nuisanceparameters) and the total. analyses of the effects of the LF for inflationary models that are in better agreement with theobserved CMB spectra are left for future studies.In Fig. 6 we show the 1D marginalized posterior distributions and the correlations betweenthe feature parameters. From the correlation plot between λ and k we can see that the sizeof the feature can be larger if the feature is located at a smaller wavemode k . This is becausethe CMB temperature spectrum does not allow any wiggles above (cid:96) (cid:39)
60, thus limiting theamplitude of the feature. The 2D plots for the parameter σ seem to show that there is no lowerbound on it. This is not in tension with the 1 σ constraints on the σ parameter reported inTab. I, because of volume effects that occur in the Bayesian marginalization procedure. Thepreference for a non-minimum value of σ is mild, indeed there is no lower bound at 2 σ confidencelevel. 10 .94 3.00 3.06 3.12 3.18 ln(10 A s ) ω b ω c θ M C τ
65 66 67 68 H l n ( A s ) ω b ω c θ MC τ featurefeatureless Figure 5: A comparison between the model with and without features is given for the parameters H , ω b , ω c , θ , τ and ln(10 A s ). All the results are obtained considering the Planck low- (cid:96) +high (cid:96) datacombination. As can be seen the effects of the feature on the estimation of these non inflationarycosmological parameters is negligible. The constraints on the primordial scalar spectrum are shown in Figs. 7 and 8. In the leftpanel of Fig. 7 we compare the best-fit primordial power spectrum of scalar perturbationsobtained in our analysis (blue) and the reconstructed one from Ref. [16]. The comparisonunderlines how a local feature can reproduce the behaviour of the primordial spectrum, but11 .00110.00120.00130.0014 k σ − − λ k σ Figure 6: The results of the data fitting analysis for the parameters λ , σ and k are shown for themodel with local features. further studies, which will be presented in some future work, on the feature potential arerequired in order to obtain a perfect agreement. The right panel of the same Fig. 7 shows thatthe effect of the feature is very small in the tensor spectrum. In Fig. 8 we plot the marginalizedconstraints on the primordial scalar spectrum. The 1, 2, and 3 σ bands refer to the model withLF, while the solid black line shows the corresponding best-fit spectrum, computed from theentire set of primordial spectra obtained from the MCMC scan. The red dashed line shows12 (cid:180) (cid:45) k Mpc (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) P Ζ (cid:180) (cid:45) k Mpc (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) P h Figure 7: The numerically computed spectrum of the primordial curvature fluctuations P ζ and of thetensor perturbations P h are plotted for the best fit values in Tab. I: λ = − . × − , σ = 0 . k = 1 . × − (blue). On the left, the red lines correspond to the best-fit reconstructed primordialpower spectrum from Ref. [16]. The dashed lines correspond to the featureless spectrum. -4 -3 -2 -1 k [Mpc − ] x P ζ ( k ) feature Figure 8: The 1, 2 and 3 σ constraints obtained from observational data analysis are plotted for theprimordial curvature perturbations spectrum for the model with local features. The spectrum for thefeatureless model is plotted with a red line. VI. CONCLUSIONS
We have studied the effects of local features in the inflaton potential on the spectra ofprimordial curvature perturbations and their impact on the temperature anisotropies of theCMB. In order to study the effects on the CMB spectrum we have modified the
CAMB and
CosmoMC codes in order to use a non-standard power-law power spectrum for the primordialperturbations, to take into account the presence of the local feature. We have performed abest fit analysis of CMB temperature and polarization data from Planck. We have found nosignificant effects on cosmological parameters related to the propagation of CMB photons afterdecoupling, while LF improve the fit of the CMB temperature and polarization data. Wehave also confirmed the theoretical expectation that local features do not affect the primordialpower spectrum at scales far from the characteristic scale k , which leaves the horizon aroundthe feature time.In the future it will be interesting to analyze the effects of local features in order to explainother 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 withdifferent featureless V potentials, and to compare them to the effects of branch features. Acknowledgments
This work was supported by the European Union (European Social Fund, ESF) and Greeknational funds under the ARISTEIA II Action. Part of the work of S.G. was supported by theTheoretical Astroparticle Physics research Grant No. 2012CPPYP7 under the Program PRIN2012 funded by the Ministero dell’Istruzione, Universit`a e della Ricerca (MIUR), and in part isalso supported by the Spanish grants FPA2014-58183-P, Multidark CSD2009-00064 and SEV-2014-0398 (MINECO), and PROMETEOII/2014/084 (Generalitat Valenciana). The work ofA.G.C. was supported by the Colombian Department of Science, Technology, and InnovationCOLCIENCIAS research Grant No. 617-2013. A.G.C. acknowledges the partial support from14he International Center for Relativistic Astrophysics Network ICRANet. For part of thecalculations we used the Cloud infrastructure of the Centro di Calcolo in the Torino section ofINFN. AER work was supported by the Dedicacion exclusica and Sostenibilidad programs atUDEA, the UDEA CODI project 2015-4044 and 2016-10945, and Colciencias mobility program. [1] S. Gariazzo, C. Giunti, and M. Laveder, JCAP , 023 (2015), arXiv:1412.7405.[2] D. K. Hazra, A. Shafieloo, and T. Souradeep, JCAP , 011 (2014), arXiv:1406.4827.[3] P. Hunt and S. Sarkar, JCAP , 025 (2014), arXiv:1308.2317.[4] P. Mukherjee and Y. Wang, Astrophys. J. , 38 (2003), arXiv:astro-ph/0301058.[5] P. Mukherjee and Y. Wang, Astrophys. J. , 1 (2003), arXiv:astro-ph/0303211.[6] S. L. Bridle, A. M. Lewis, J. Weller, and G. Efstathiou, Mon. Not. Roy. Astron. Soc. , L72(2003), arXiv:astro-ph/0302306.[7] D. K. Hazra, A. Shafieloo, and G. F. Smoot, JCAP , 035 (2013), arXiv:1310.3038.[8] J. A. Vazquez, M. Bridges, Y.-Z. Ma, and M. P. Hobson, JCAP , 001 (2013),arXiv:1303.4014.[9] S. Hannestad, Phys. Rev.
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