Reconstruction of broad features in the primordial spectrum and inflaton potential from Planck
aa r X i v : . [ a s t r o - ph . C O ] J a n Reconstruction of broad features inthe primordial spectrum and inflatonpotential from Planck
Dhiraj Kumar Hazra a Arman Shafieloo a,b
George F.Smoot c,d,e a Asia Pacific Center for Theoretical Physics, Pohang, Gyeongbuk 790-784, Korea b Department of Physics, POSTECH, Pohang, Gyeongbuk 790-784, Korea c Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA d Institute for the Early Universe, Ewha Womans University Seoul, 120-750, Korea e Paris Centre for Cosmological Physics, Universite Paris Diderot, FranceE-mail: [email protected], [email protected], [email protected]
Abstract.
With the recently published Cosmic Microwave Background data fromPlanck we address the optimized binning of the primordial power spectrum. As animportant modification to the usual binning of the primordial spectrum, along withthe spectral amplitude of the bins, we allow the position of the bins also to vary. Thistechnique enables us to address the location of the possible broad physical featuresin the primordial spectrum with relatively smaller number of bins compared to theanalysis performed earlier. This approach is in fact a reconstruction method lookingfor broad features in the primordial spectrum and avoiding fitting noise in the data.Performing Markov Chain Monte Carlo analysis we present samples of the allowedprimordial spectra with broad features consistent with Planck data. To test howrealistic it is to have step-like features in primordial spectrum we revisit an inflationarymodel, proposed by A. A. Starobinsky which can address the similar features obtainedfrom the binning of the spectrum. Using the publicly available code BINGO, wenumerically calculate the local f NL for this model in equilateral and arbitrary triangularconfigurations of wavevectors and show that the obtained non-Gaussianity for thismodel is consistent with Planck results. In this paper we have also considered differentspectral tilts at different bins to identify the cosmological scale that the spectral indexneeds to have a red tilt and it is interesting to report that spectral index cannot bewell constrained up to k ≈ . − . ontents At the current status of cosmological observation, Cosmic Microwave Background(CMB) is the most precise probe of primordial perturbations for a wide range ofcosmological scales. WMAP [1] provided possible hints of large scale anomalies inthe primordial power spectrum (PPS), that could indicate features in the primordialspectrum. However, cosmic variance limited the significance of any deviation from afeature-less primordial spectrum and these outliers could be simply statistical fluctua-tions that one can expect in any random realization of the data [2]. Cosmic variancealso affects the recent Planck observations; however, since Planck [3] has detected thepossible anomalies in the same scales as has been reported in WMAP, one can arguethat these features are not due to systematics in WMAP. This motivates further in-vestigation of the features in the power spectrum indicated by the Planck data. Whilewe cannot certainly argue that these features have a real physical origin, it is still veryimportant to study different possibilities. For instance, it is possible to go beyond thethe assumption of power-law form of the primordial spectrum and test more rigorouslythe possible inflationary scenarios and other models that could produce the observedfeatures in the angular power spectrum. Our approach in this paper is to look forthe most broad features in the primordial perturbations by optimized binning of theprimordial spectrum varying the width and positions of the bins. Allowing the binsto have different width and positions can be in fact analogous to reconstruction of theprimordial spectrum given some minimal degrees of flexibility.In this approach we can be assured that we are not fitting noise and any im-provement in the likelihood should be related to the broad features of the primordial– 1 –pectrum. Direct reconstruction of the primordial spectrum from the data, which workswith large number of degrees of freedom compared to binned reconstruction, also hinttowards possible features in the data [2, 4–9], however without a proper smoothing ofthe primordial spectrum it is hard to distinguish noise fitting features from possible real outliers. From inflationary theories we know that a change in the slow roll potential ininflation can lead to features in the spectrum which can fit the data better by fittingthe outliers unaddressed by the power law spectrum [9–11] and thereby provide a bet-ter fit. Our method of binning the spectrum also hint towards the possible deviationsfrom the power law spectrum which is certainly useful in building inflationary models.In this work we start our analysis with two bins for the primordial spectrum. Wechange the amplitudes of the two bins and also the position of the bins and look for thebest likelihood to the Planck data. This would allow us to find out how well we canimprove the fit to the data considering limited number of bins compared to the power-law form of PPS. We continue the procedure by introducing more number of bins forthe form of PPS while for each bin the position and amplitude are varied. We useMonte Carlo Markov Chain (MCMC) analysis to do our analysis and estimation of theposition and amplitudes of the bins. Efforts on finding optimal binning of primordialspectrum [4] ∗ and analysis with small scale CMB data from Atacama CosmologyTelescope using 20 fixed bins in the primordial spectrum [5] have been carried outbefore. We should note here about the important advantage of our approach to theusual binning of the primordial spectrum. While we are dealing with unknowns andwe do not know if there are in fact features in the primordial spectrum or not, and ifthere are features where these features are located, binning of the primordial spectrumin an arbitrary way (like equispaced in cosmological scales) may result to dilutingstatistical significance. This can clearly happen if a possible feature is located inthe middle of a bin (rather than being in two neighboring bins) so arbitrary binningmay not necessarily find a possible feature in the data. In our approach we startwith two dynamical bins and while we introduce more number of bins we look at theimprovements in the likelihood and where these bins are located. This will provide uswith clear understanding of the problem and allow us to study the primordial spectrumstep by step adding more complication and without missing a broad feature. We shouldmention that analysis with variation of the position of the bins has been carried outbefore [12] for the purpose of model selection in the context of Bayesian interpretationwhich is different from the line of our analysis.We have also considered simple two tilt model to estimate at what scale we needto have a red tilt in the primordial spectrum. Varying the tilts in two different bins weaddress the constraint on the spectral tilt from CMB in different cosmological scales.In the next part of this paper we study how we can generate similar featuresin the PPS like two-bins model, looking at single field inflationary scenarios. Whilehaving sharp/discontinuous transitions between bins are in fact non-physical, we alsointroduce phenomenological Tanh form of the primordial spectrum and we argue thatin fact such form can be generated by a simple inflationary model. ∗ Here the optimal binning refers to the construction of the power spectrum using Fisher matrixformalism with the help of the signal to noise ratio for a corresponding observation. – 2 –part from the spectral amplitude and tilt, Planck data provides us with theconstraints on primordial non-Gaussianity [13] which indicates a Gaussian model iscompletely consistent with Planck. While we know inflationary models with featuresin the primordial spectrum can result to some considerable large non-Gaussianity itis important to find a way to address this issue in our analysis. We have considereda simple inflationary model proposed by A. A. Starobinsky [10] where the resultantprimordial spectrum is very similar to some of our main phenomenological shapesof the primordial spectrum. This would also help us to estimate the extent of non-Gaussianity (specifically the bi-spectrum) we should expect from similar shapes ofthe primordial spectrum. We use the publicly available code BI-spectra and Non-Gaussianity Operator, BINGO [14] to calculate the local f NL in equilateral limit andfor arbitrary triangular configurations of wavevectors for the modified Starobinsky-1992model of inflation.This paper is organized as follows. In the next Models and Methodology sectionwe shall discuss the modeling of the binned power spectrum which is discontinuous atthe bin positions and describe an equivalent smooth form of PPS with the introductionof a Tanh step. In the same section we shall present the Starobinsky-1992 model ofinflation and model it appropriately to act as a candidate of the binned spectrum. Theessentials of complete numerical computation of the power spectrum and bi-spectrafor a canonical scalar field model is also discussed there with an aim to apply it forthe modified Starobinsky model. In what follows in the results section the we shalldiscuss improvements in likelihood using the binned PPS and also the range of allowedvariations in the primordial spectrum with respect to the power law. We shall alsodiscuss the best fit theoretical model and the non-Gaussianities obtained. We closewith a brief discussion towards the end. In this section we shall discuss the phenomenological models we have tested, the priorson the parameters and the method of confronting them with the data. We shall alsorevisit the Starobinsky model [10] with a modification (appeared in the public codeBINGO [14]) and discuss a few essential details on the framework of the calculation of f NL . The power law spectrum is expressed by 2 parameters, the spectral amplitude A S andthe spectral index n S through, P S ( k ) = A S × (cid:20) kk (cid:21) ( n S − (2.1)with A S defined as amplitude at pivot scale k . In this analysis we have kept theposition of the bins as variables and we define them as the following.– 3 –n logarithmic scale the first bin associated to r b1 starts from a minimum wavenum-ber value k min ( k min is the minimum wavenumber for a given radiative transport kernel)up to k b1 given by: k b1 = exp[ln( k min ) + r b1 × [ln( k max ) − ln( k min )]] (2.2)To connect r b1 appropriately with the physical scales we would like to mentionthat k min ∼ × − Mpc − and k max ∼ . − for the best fit concordance modeland these values do not change notably when we change the background parameters.The logarithmic scale allows to dig out low- ℓ features more carefully since atthis region the possible anomalies are most prominent. We scan the parameter r b1 forvalues between 0 and 1 which ensures the total k-space coverage taking into account allpossible combinations. For more than 2 bins the second break in the power spectrumis located at k b2 which depends on a new parameter r b2 following equation similar toEq. 2.2 but the k min is replaced by the first bin position k b1 . The different bin positionsare given by, k b2 = exp[ln( k b1 ) + r b2 × [ln( k max ) − ln( k b1 )]] k b3 = exp[ln( k b2 ) + r b3 × [ln( k max ) − ln( k b2 )]] ..... = .....k bN = exp[ln( k b(N − ) + r bN × [ln( k max ) − ln( k b(N − )]] (2.3)Defining the bins in this way ensures that while scanning the parameter space of r b1 − bN (for N+1 bins) spanning from 0 to 1, we cover all different combinations of binswith different lengths. The amplitudes of bin b1 − bN are given by A b1 − bN which isdefined with a amplitude magnifier m b1 with respect to the amplitude of the best fitpower law spectrum from Planck. We allow a broad priors for the amplitudes ln[ m b1 ]to range from -10 to 3 to allow very small and large amplitudes for different bins. Forall the bins we also allow the overall tilt n S (same for all bins) to vary. In this paperwe call this model as model-A along with its number of bins (such as Model-A with 2bins, etc). We should note that Model-A with two bins has 2 more degrees of freedomin comparison to standard power-law case and adding each extra bin requires two moredegrees of freedom. In the previous section to avoid large number of degrees of freedom, we assumed onlyone spectral index for all bins. In this section we consider two different kind of twobins models where each bin can have its own spectral index.Model-B is similar to Model-A but we assign different spectral index for each bin.In this paper we considered Model-B with only two bins and we simply call it Model-B.Model-B has three extra degrees of freedom in comparison with standard power-lawmodel.Model-C is somehow simpler than Model-B. In Model-C we have also two binsbut there is no step between the two bins. These two bins can have different tilts– 4 –nd their amplitudes are equated at the transition ( i.e. at k b1 ). Model-C looks like abroken line while its parts are still attached. Model-C has two extra degrees of freedomin comparison with standard power-law model. Note that similar model was initiallydiscussed in [15].For model-B and model-C we have denoted the two spectral indices as n b1 and n b2 . We have allowed red and blue tilts for both bins with n b ranging from 0 to 2.These two simple phenomenological models can hint us towards any special tran-sitional behavior in the data. As we will see later, Model-C can put a clear lower boundon the scale that we necessarily need to have a red tilt in the primordial spectrum. The sharp/discontinuous transition of power between different bins looks unrealisticand non physical. However, we can have a smooth transition between bins if we usesome mathematical function such as hyperbolic tangent. Two avoid complications, wetry only to mimic Model-A with two bins by parametrizing the primordial spectrumusing Tanh, P tanhS ( k ) = P PlawS ( k ) × (cid:20) − α tanh (cid:20) k − k b1 ∆ (cid:21)(cid:21) × A Scale (2.4)where P PlawS ( k ) is the power law spectrum with amplitude fixed to its best fit valueobtained in Planck analysis [28]. k b1 is exactly same as it appeared in Eq. 2.2 and itdenotes the position of the transition. α acts as a height of the step and ∆ acts as asteepness/width of the step.To scale the base model we include another parameter A Scale (instead of A s in thestandard power-law case). In our analysis, α ranges from 1 to -1 so that it includesthe power law model corresponding to α = 0. Moreover as we did not want the powerspectrum to become negative, we ensured 1 − α tanh[ k − k b1 ∆ ] remains positive always(with the highest possible α = 1). The value α > k < k b1 and α < k < k b1 . We have allowed a verysharp transition through ∆ so as to mimic the bin results closely. The given priors onln ∆, ranging from -10 to 0 allow a wide range of possibilities with sharp as well asperfectly smooth crossovers. We have allowed variations of A Scale in logarithmic scalestoo such that ln A Scale can take values from -1 to 1.Tanh model has three extra degrees of freedom in comparison to standard power-law and 1 extra degree of freedom in comparison to Model-A with two bins.We have not taken into account the effects of tensor perturbations as they arefound to be negligible to be included in temperature spectrum analysis and we focuson the scalar sector only in this work.
It is always important to support a phenomenological model or a parametric formby a physical theoretical model. In the previous section we used Tanh function tomimic our two bin model and make a smooth transition between the two bins. In– 5 –his section we study an inflationary scenario that can result to similar tanh featurein the primordial spectrum. This is important for two reasons. First of all it showsthat such binned features are not unusual in the context of inflationary cosmology andsecond, we can calculate non-Gaussianity for inflationary scenarios and this can helpus to estimate if such binned shaped primordial spectra are in conflict with Planckestimation of non-Gaussianity or not.Primordial power spectrum originates from the generation and evolution of per-turbations during the epoch of inflation. We usually work with the simplest power lawform of primordial spectrum as this is motivated from the assumption that the scalarinflaton field ( φ ) rolls slowly down its potential V ( φ ) all the way during inflation.The background evolution of the scalar field is given by the following Klein-Gordonequation, ¨ φ + 3 H ˙ φ + d V / d φ = 0 , (2.5)where, H is the Hubble parameter (= ˙ a/a with a as scale factor appearing inFLRW metric) and a overdot (˙) refers to differentiation w.r.t. cosmic time. The slowroll of the scalar field is usually quantified with slow roll parameters ǫ i which are definedby, ǫ i +1 = d ln ǫ i / d N [ i ≥ , (2.6)where N is the number of e-folds and ǫ = − ˙ H/H . A strict slow roll would imply ǫ i ≪ e-folds during inflation.Quantum fluctuations of the inflaton φ induces the scalar perturbation which isrepresented by curvature perturbation R , the Fourier transform of which satisfies thecurvature perturbation equation † (Mukhanov-Sasaki equation of the following form)given by, R ′′ k + 2 z ′ z R ′ k + k R k = 0 , (2.7)where, z = a ˙ φ/H and k is the wavevector. To get the perturbation spectrum, inpractice we start integrating Eq. 2.7 from deep inside the Hubble radius using Bunch-Davies initial condition to the point where modes go outside the Hubble radius (super-Hubble scales) where R freezes. From the R at super-Hubble scales the scalar powerspectrum P S ( k ) which is the two point correlation of the curvature perturbation, iscalculated as P S ( k ) = ( k / π ) |R k | .Having described the perturbation equations to derive the power spectrum, weshall now discuss the main integrals to calculate the non-Gaussianities in a model ofinflation. Using Maldacena formalism [17], the bi-spectrum which is denoted by thethree point correlation function of the curvature perturbation can be described with † For more discussion see [16] and the references therein – 6 –he following function G ( k , k , k ), given by, G ( k , k , k ) ≡ X C =1 G C ( k , k , k ) ≡ M Pl X C=1 ( [ R k ( η e ) R k ( η e ) R k ( η e )] G C ( k , k , k )+ (cid:2) R ∗ k ( η e ) R ∗ k ( η e ) R ∗ k ( η e ) (cid:3) G ∗ C ( k , k , k ) ) + G ( k , k , k ) . (2.8)where, M Pl = (8 π G) − and G C ( k , k , k ) with C = (1 ,
6) are the terms appearing inthe interaction Hamiltonian and is described by the following integrals, G ( k , k , k ) = 2 i Z η e η i d η a ǫ (cid:0) R ∗ k R ′∗ k R ′∗ k + two permutations (cid:1) , (2.9) G ( k , k , k ) = − i ( k · k + two permutations) × Z η e η i d η a ǫ R ∗ k R ∗ k R ∗ k , (2.10) G ( k , k , k ) = − i Z η e η i d η a ǫ "(cid:18) k · k k (cid:19) R ∗ k R ′∗ k R ′∗ k +five permutations , (2.11) G ( k , k , k ) = i Z η e η i d η a ǫ ǫ ′ (cid:0) R ∗ k R ∗ k R ′∗ k + two permutations (cid:1) , (2.12) G ( k , k , k ) = i Z η e η i d η a ǫ "(cid:18) k · k k (cid:19) R ∗ k R ′∗ k R ′∗ k +five permutations , (2.13) G ( k , k , k ) = i Z η e η i d η a ǫ ((cid:20) k ( k · k ) k k (cid:21) R ∗ k R ′∗ k R ′∗ k +two permutations ) , (2.14)where η is the conformal time and η i and η f denotes the beginning and end of inflationrespectively. The seventh term G ( k , k , k ) appears due to a field redefinition andcan be expressed as, G ( k , k , k ) = ǫ ( η e )2 (cid:0) |R k ( η e ) | |R k ( η e ) | + two permutations (cid:1) . (2.15)– 7 –or extensive discussions on this topic see Refs. [14, 17–19]. The extent of the non-Gaussianity, denoted by the number f NL is related to G ( k , k , k ) by the relation, f NL ( k , k , k ) = −
103 (2 π ) − ( k k k ) G ( k , k , k ) × (cid:2) k P S ( k ) P S ( k ) + two permutations (cid:3) − . (2.16)For equilateral case k = k = k = k and under this condition it can be shownthat the first and the third term and the fifth and the sixth term differ by a constantmultiplicative factor and can be clubbed together as G ( k ) and G ( k ) respectively.Equipped with the theory of calculation of power spectrum and bi-spectrum fora given model of inflation we shall now discuss the model that we have presented inthis paper.In this paper we revisit a model proposed by A. A. Starobinsky [10] which hasbeen discussed before in the context of non-Gaussianity too [14, 19, 20]. The potentialin this model is linear with a break at φ = φ . V ( φ ) = (cid:26) V + A + ( φ − φ ) for φ > φ ,V + A − ( φ − φ ) for φ < φ . (2.17)where, A + / − are the two slopes of the potential and the constant V is the value ofthe potential at the transition. The potential has a discontinuity in its first derivativeand its second derivative contains a divergent delta function. Due to the break in thepotential the field fast rolls near the transition and introduces wiggles in the primordialspectrum [14, 19, 20]. As in our case, we need to represent a power spectrum that canrepresent the binned/tanh spectrum closely, we needed to smooth the transition withproper function. We smooth the transition in the following way as appeared in thepublicly available code BINGO [21]: V ( φ ) = V + 12 ( A + + A − )( φ − φ ) + 12 ( A + − A − )( φ − φ ) tanh [ α S ( φ − φ )] . (2.18)Where, α S regulates the steepness of the transition which upon assuming a very largevalue exactly reproduces Eq. 2.17. We have checked that tuning the α S it is possibleto get power spectrum similar to the Tanh model as explained in Eq. 2.4. However,given this potential we also wanted to see the best fit primordial spectrum that canbe obtained from the data. Using BINGO we have solved the background and pertur-bation equation without any slow-roll approximation and used an add-on (yet to bepublicly available) of BINGO to incorporate the analysis in CAMB [22, 23]. We havesearched for the best fit power spectrum and best fit potential parameters. Havingfound the best fit we have used the best fit values to calculate the f NL for this model inequilateral and arbitrary triangular configurations of wavevectors. We should mentionthat we have not included the tensor perturbations in our analysis as the first slow-rollparameter ǫ is O (10 − ) and the tensor to scalar ratio r ≃ ǫ is completely negligibleto affect our analysis. Along with the cosmological parameters to find the best fit wehave searched in the potential parameter space consisting of V , A + , A − , φ and α S .– 8 –he priors on the potential parameters are chosen such that they can largely cover thepower spectrum generated by the priors in Tanh model.We should mention that we have used the publicly available code CAMB [22, 23]to calculate the angular power spectrum and CosmoMC [24, 25] with Planck likeli-hood [26] to obtain bounds on the power spectrum and other cosmological parameters.For finding the best fit parameters and the likelihood we have used Powell’s BOBYQAmethod of iterative minimization [27]. We have compared and ensured that the bestfit obtained by MCMC and Powell’s method are comparable with a difference of ≃ ℓ (2-49) likelihood is calculated using the commander likelihood and the high- ℓ (50-2500) is calculated through CAMspec in four different frequency channels. Alongwith the cosmological parameter we have also marginalized our results over total 14nuisance parameters describing various foreground effects in 4 different frequencies.The priors on the cosmological parameters along with the nuisance parameters usedin all the cases are same as has been used in the Planck analysis for the base ΛCDMmodel [28]. In this section we shall present the results of our analysis. We shall first demonstratethe improvement in fit obtained for different models, following which we shall providebounds on the reconstructed primordial power spectrum. Finally we shall close withthe theoretical model that we have revisited in this paper and the non-Gaussianitiesgenerated by this model. -7-6-5-4-3-2-1 0 1 2 3 4 ∆ χ Number of bins
Binned spectrum Starobinsky modelTwo bins model-CTwo bins model-BTanh model
Figure 1 . The improvement in χ with respect to the power law ΛCDM model as a function ofnumber of bins is plotted. Using horizontal lines we have plotted the improvement obtained withdifferent models that have been used in this analysis. – 9 –o begin with, we plot the improvement in fit obtained in all the models we havetested compared to power law primordial spectrum. In Fig. 1 we have plotted thedifference of minimum χ of the model we considered and the base power law ΛCDMmodel. To obtain the values we have used Powell’s BOBYQA method as has beenmentioned earlier. Note that with 2 bins it is possible to get a better fit of about 5 overthe power law model with only two extra degrees of freedom. Here we should mentionthat most of the improvement is coming from the low- ℓ commander likelihood. We haverestricted our analysis of binned spectrum to 4 bins since considering more number ofbins did not result to a significantly better likelihood to the data despite of immensecomputational expense. For instance, considering 4 bins introduces 8 parameters todescribe the primordial spectrum (4 amplitude parameters, 3 bin position parametersand one tilt parameter) which, together with 4 background parameters and 14 nuisanceparameters makes the method to find the best fit extensively time consuming. In factwe did fit the primordial spectrum with 5 bins as well but it did not improve the fitconsiderably beyond the 4 bin case. This is probably due to two reasons. Firstly dueto large number of degrees of freedom the searching of the global minimum becomesinefficient. Another possibility is with 4 bins all the large features in the data areaddressed and to get a better fit beyond that we may need sharp changes/featuresin the primordial spectrum which can not be addressed by the binning. We shouldremind here that the models with two, three and four bins have two, four and six extradegrees of freedom in comparison to standard power-law case.In the same plot in Fig. 1 we have located the stands of other models in the∆ χ space with horizontal lines. Two bins with 2 tilts and 2 amplitudes (model-B)fitting the data better than the two bin case as expected. Model-C which has the samedegrees of freedom as two bin model-A is fitting the data slightly worse than model-Awith two bins. Surprisingly the Tanh model is fitting the data better than the 3 binswith one less degree of freedom. The modified Starobinsky model has also been ableto provide a better fit to the data by 3.5. We shall comment more on this towards theend of this section.In Fig. 2 we have plotted the best fit power spectrum and the corresponding C TT ℓ obtained for different models in our analysis. It is interesting to see that the lack ofpower at low- ℓ suggest a step-like feature in the primordial spectrum and the position ofthe step is found to be at around 2 × − Mpc − in all the cases. The Tanh parametricform of the primordial spectrum also suggests a sharp transition mimicking the modelwith two bins. The result from Model-A with three and four bins shows that evennearly zero power is allowed in very large scales (low- k region). The Model-A withfour bins shows another tiny feature near k ≃ . − which in a way indicatesthat there is no further scope of significant improvement in likelihood fitting the databy introducing larger number of bins in the form of PPS. Two bins model-B does notshow notably different behavior, however model-C which has 2 tilts and the spectralamplitude matches at the transition chooses a blue tilt in large scales and red tiltcomparable with n S = 0 .
96 at small scales. For two bin model-C the scale where thesecond bin starts k ≃ . − implies that at least from k min to k ≃ . − a blue spectral tilt is more favored. This is not evident from the model-B as the– 10 – e-093e-09 1e-05 0.0001 0.001 0.01 0.1 P S ( k ) k in Mpc -1 Power law best-fitTwo bins model-A best-fitThree bins best-fitFour bins best-fit 01000200030004000500060007000 1 10 100 1000 ℓ ( ℓ + ) C ℓ TT / π [ µ K ] ℓ Planck low-lPower law best fit Two bins model-A best fitThree bins best fitFour bins best fit P S ( k ) k in Mpc -1 Power law best-fitTwo bins model-B best-fitTwo bins model-C best-fitTanh model best-fit ℓ ( ℓ + ) C ℓ TT / π [ µ K ] ℓ Planck low-lPower law best fit Two bins model-B best fitTwo bins model-C best fitTanh model best fit
Figure 2 . The best fit power spectrum and the corresponding C ℓ ’s obtained from comparing themodels against the Planck TT spectrum. In the upper left panel we present the best fit binnedspectrum till 4 bins (in colors described in each of the plots) and the best fit power law spectrum(in black). In the upper right panel we have plotted the corresponding C ℓ ’s and the Planck low- ℓC TT ℓ data. In the inset the Planck low- ℓ region is highlighted. In the lower panel we have plottedthe remaining spectra and the corresponding C ℓ ’s that have been used in our analysis (except theStarobinsky model). The colors are described in the plot itself. amplitude of the first bin compensates for the blue tilt of the model-C. So the bestfit of all the models point to the fact that at low- k , till nearly 0 . − the powerspectrum prefers to have a dip than a bump and at high- k beyond k > .
01 the powerspectra for all the models more or less follow the power law best fit model.
Having dealt with the best fit power spectrum and the improvement in fit with thefeatured models we shall now discuss the results from MCMC where we obtain thebounds and thereby present the range of allowed power spectra.
In Fig. 3 we have plotted the results for model-A with two bins. The left plot inthe figure, which resembles the reconstructed band we have obtained in a previous– 11 – b1 l n ( A b1 / A b2 ) ln (10 × A)2.5 3 3.5 4 4.5 5 −10 −8 −6 −4 −2 0 2−2−1.5−1−0.500.511.52 ln[A b1 /A S ] l n [ A b2 / A S ] r b1 Figure 3 . For two bins, samples within 3 σ contours are plotted. In the plot appearing in the leftpanel we have plotted the logarithm of the ratio of the amplitudes of the two bins as a function ofthe bin position r b1 (as appears in Eq. 2.2). It should be noted that when the bin position is locatednear very large scales k min (corresponding to r b1 ≃ A ] where A is the mean amplitude computed from the two bins. In the right panel the plottedsamples correspond to the 3 σ allowed values of relative amplitudes of the first and the second binswith respect to the amplitude of the best fit power law PPS ( A S ). It is important to note thatwhile the amplitude of the first bin is not constrained properly, there is a little room left to playwith the amplitude of the second bin. The color corresponds to the bin position which clearly statesas the bin position nears the smallest probed scale k max the spectrum goes back to power law withln[ A b1 /A S ] = ln[ A b2 /A S ] = 0 analysis [2] shows that the ratio of the two amplitudes A b1 /A b2 is not well constrainedwhen the position of the bin is in low- k region. This is due to the fact that there isnearly zero signal around k min and A b1 can have any arbitrary value. However, as thebin position shifts to higher k -value the power spectrum with nearly equal amplitudesare supported by the data. It should be mentioned that there is a large degeneracyin the parameter space of A b1 /A b2 . For example two different spectra can have same A b1 /A b2 (say, =1) but can be drastically different than the power law amplitude. Tobreak this degeneracy we have plotted the ln[10 A ], where A = [ A b1 + A b2 ] /
2, the meanamplitude. Note that, here too, as the bin position shifts to the higher wavenumbers,the samples become closer to the power law best fit value of ln[10 A ] = 3 . σ allowed values of the two bin amplitudes m b1 = A b1 /A S and m b2 = A b2 /A S . Note that A S is the best fit amplitude of the power-law PPS from Planck data. It is clear from the plot that the amplitude of the first binis not really constrained within the prior range used but the amplitude of the secondbin is tightly constrained around the power law best fit value. Though the amplitudeof the first bin is unconstrained, the statement completely depends on the position ofthe bin. The samples in this plot has been colored with the values corresponding to– 12 – N o r m a li z ed L i k e li hood α N o r m a li z ed L i k e li hood ln ∆ b1 l n ( − α ) ln(10 × A)2.4 2.5 2.6 2.7 2.8 2.9 3 3.1 3.2
Figure 4 . For the Tanh model the 1-dimensional confidence contours of the strength of the Tanhstep α and the steepness of the step ∆(as appears in Eq. 2.4) are plotted (Upper panel). As a negativevalue of α correspond to a dip in lower- k it is clear from the contour of α that a dip is much favoredthan a bump in large scales. At the lower panel in this figure we have plotted the samples of maximumcorrections to the power law induced by the Tanh model within the 3 σ allowed values. Note thatthis plot is strikingly similar to the left plot of Fig. 3. The different colors again correspond to thedifferent values of ln(10 A ) where A is the mean amplitude. the position of the bin. As the position of the bin approaches k max , the amplitude ofthe first bin too is constrained to a narrow region near ln m b1 = 0. However, notethat here the samples gather to the left of zero which means a the data favor a dipin the amplitude of the first bin, even when the bin is positioned more than halfway( r b1 ≃ .
6) of the total k -range in log scale.In the case of Tanh model r b1 acts as a location of the step. In Fig. 4 we haveplotted the results corresponding to the Tanh model. In the upper panel of the figurewe have plotted the marginalized 1D likelihood of the α and ln ∆ (in Eq. 2.4). TheTanh step is modeled in such a way (see Eq. 2.4) so that a negative value of α denotes– 13 –he power in k < k b1 is lower in amplitude compared to the power in k > k b1 . The plotclearly reflects that a dip in the larger scales is much favored than a bump. As ln ∆is unconstrained from below it reflects that a sharp transition (say, corresponding toln ∆ ≃ − ≃ − − α ) as function of the position of the Tanh stepwithin 3 σ confidence limit. As Tanh function becomes ±
1, for a very small value of ∆,1 − α denotes the maximum change over the power law spectrum. Note that the shapeof both plots are in excellent agreement. Sparse samples in low r b1 region indicatesthat the amplitude at k < k b1 is not tightly constrained till r b1 ≃ .
6, however, notablya dip in k < k b1 is more favored. Again to break similar degeneracy explained before,we have colored the samples with ln[10 A ] where A correspond to the mean amplitude i.e. A Scale × A S . Here too, we see that, as r b1 nears 1 the spectrum merges to powerlaw best fit. Moreover we notice the constraint on the maximum allowed correctionfactor 1 − α tightens gradually with increasing r b1 . Note that each of these samplescorresponds to a reconstructed power spectrum which are in agreement with the Planckdata. In the analysis explained above for Model-A and Tanh case we could not study allowedrange of spectral indices separately at different scales since we used only one spectralindex for all bins. In this regard we worked also with two bins model-B and model-Cwhere in each bin we allow the spectral indices to vary independently. The spectralindices of the first and second bins are denoted by n b1 and n b2 respectively. In Fig. 5the results of the MCMC analysis are plotted. The results for model-B and model-Care plotted in upper and middle panels respectively. In the left panel the 3 σ contoursof n b1 and n b2 are plotted (for upper and middle panel) which shows that the tilt of thefirst bin is unconstrained within the prior range. However, the spectral index of thesecond bin is constrained tightly and it rejects a completely scale invariant spectrum(corresponding to n b2 = 1 at k > k b1 ) with more than 3 σ confidence. Model-C hasone less free parameter than Model-B and hence constrains the spectral indices withhigher CL. A clearer picture can be provided in the right panel in the same figurefor both the models. We have plotted samples from the left panel colored by theposition of the bin. Note that the samples in model-B is more sparse than model-C.For example, when the bin position approaches k max (say r b1 ≃ . − .
7, correspondingto ( ∼ . − . − )) for model-B the first bin can have both red and blue tiltas red points are distributed in both sides of n b1 = 1 which implies that a worse fitimposed by a particular value of tilt can be compensated by the amplitude of the firstbin. – 14 –
0 0.5 1 1.5 2n b1 n b2 b1 n b2 r b1
0 0.5 1 1.5 2n b1 n b2 b1 n b2 r b1
0 0.2 0.4 0.6 0.8r b1 n b1 P S ( k ) k in Mpc -1 Samples of two bin model-CPower law best fit
Figure 5 . For 2 bins model-B (upper panel) and model-C (middle panel) the constraints on the twospectral tilts are shown. In the left panel the 3 σ confidence contours of n b1 and n b2 are plotted wherewe see that while the first bin is allowed to have both red and blue tilt, the second bin must have ared tilt. For the second bin a strictly scale invariant spectrum, n b2 = 1 is not favored by the data inmore than 3 σ level. In model-C as the number of amplitude decreases, due to lack of degeneracies,both the tilts are better constrained. In the right panel (for the plots at top and middle), for boththe models the samples of n b1 and n b2 are plotted which lies within 3 σ CL. Samples are colored bythe bin position. The left plots at the bottom corresponds to confidence contours of bin position( r b1 ) and spectral index of the first bin ( n b1 ) for two bin model-C. To its right we have plotted a fewsamples of the power spectrum within 2 σ allowed range of parameters for the same model. The redstraight lines in the left plots correspond to n S = 0 . – 15 –owever, for model-C where we have only one amplitude scaling parameter wesee that we loose the freedom in the first bin. Here we identify a clear pattern. For r b1 ≤ . . ≤ r b1 ≤ . k region. Finally, for r b1 ≥ . r b1 and the n b1 for model-C with two bins are plotted in the left plot at the bottom of Fig. 5. Notethat till r b1 = 0 . k ∼ . M pc − ) the spectral tilt does not haveto be red even in 1 σ . Smaller scales beyond that the gradual convergence of contoursprovides a clearer picture of the power of the data to constrain the spectral indicatesalong the cosmological scales. For an illustration, we have also plotted a few samplesof power spectrum (in red) for model-C in the bottom right panel along with the bestfit power law spectrum(in blue). At the end of this section we shall discuss the final part of our analysis which dealswith the modified Starobinsky model to explain the step-like features in the primordialpower spectrum. As has been discussed before, we have worked with the Starobinsky-1992 model of inflation with a break in the potential wherein for our purpose, followingprevious analysis [14], we have modeled the break as a smooth transition with Tanhfunction (Eq. 2.18). In the final Fig. 6 we discuss the results obtained with thistheoretical model. In the first plot appearing in the left panel we have plotted thepotential and its first derivative corresponding to the best fit values of the potentialparameters obtained from Powell’s BOBYQA method. Note that while the change inthe tilt is not clearly visible in the potential plot, its first derivative distinctly shows thetransition. The best fit value of α S supported by the data indicates a smooth transitionas shown in the plot of d V ( φ ) / d φ . Without any smoothing, the d V ( φ ) / d φ from Eq. 2.17would be discontinuous and shall have wiggles in the primordial spectrum. We havealso plotted the behavior of the inflaton near the break in the potential in the right topplot. Here too, though a mild transition is visible, it is hard to estimate the amount offast roll. As the fast roll is quantified by the slow-roll parameters, we have plotted the ǫ as a function of N in the left middle plot. This plot clearly shows the deviation fromthe slow roll with a clear transition near N ∼ −
12. In this plot the x-axis at the topcontains the wavenumbers of the physical modes in Mpc − which leaves the Hubbleradius at the corresponding N provided in the x-axis at the bottom. It is evident thatthese modes will be affected by the fast roll and imprint the effect in the primordialspectrum P S ( k ). To its right, we have plotted the best fit power spectrum from themodel and the best fit power law spectrum. Note that modes indicated in the left plotcontain the feature in the primordial spectrum. We should mention that due to thedip in low- k we get an improvement of 3.2 in the commander likelihood for ℓ = 2 − ℓ do not get any substantial improvement ( ∼ .
4) from CAMspec.– 16 – V ( ϕ ) d V ( ϕ ) / d ϕ ϕ in M Pl ϕ ( N ) i n M P l N ǫ ( N ) N k = aH in MPc -1 P S ( k ) k in Mpc -1 Starobinsky model best fitPower law best fit | f N L ( k ) | k in Mpc -1 f NL from G f NL from G f NL from G f NL from G k / k k / k f N L ( k = k * , k , k ) Figure 6 . The best fit results obtained upon confronting the Starobinsky model with the Planckdata. In the left plot of the top panel we have plotted the best fit potential and its first derivative asfunction of the field φ . Note that both the potential and its derivatives are normalized to 1 near atthe break. In the top right panel we have plotted the behavior of the scalar field φ as a function of e-folds (N) near the break. In left plot of the middle panel we have plotted the first slow roll parameter ǫ as a function of e-folds (N) near the break in the potential where the fast roll takes place. The topaxis in the same plot indicates the modes k in Mpc − which leaves the Hubble radius around thecorresponding e-folds in the x-axis below. On to its right we have plotted the best fit scalar powerspectrum obtained from the theoretical model along with the best fit power law model. In the left ploton the lower panel we plot the absolute value of f NL ( k ) in the equilateral triangular configurations ofwavevectors as a function of k for different bi-spectrum integrals (see, Eq. 2.9-2.14) calculated fromBINGO. To its right we have plotted the f NL ( k ) from the dominant contribution G for arbitrarytriangular configurations of wavevectors. – 17 –e must highlight an important apparent contradiction that though a strictlyscale invariant spectrum is not favored by the data with 5 . σ CL as reported by Planckteam [28], our model being scale invariant at large- k is still supported by the data. Inour analysis with two bin model-B and model-C we have shown that only from high- k region, the data prefers departure from n S = 1 with more than 3 σ confidence. Howevera closer look at the theoretical best fit spectrum would reveal that there is indeed nodiscrepancy, as due to the mild bump near 0 . − which flattens out at smallscales acts as an effective red tilt and thereby it fits high- ℓ the data equally well. Inthe left panel appearing at the bottom we have plotted the absolute value of local f NL as a function of k for equilateral triangular configurations where, k = k = k = k .Note that, if the inflaton deviates from slow-roll the fourth term in the bi-spectrumintegral (Eq. 2.12) becomes dominant as it contains the factor ǫ ǫ ′ which becomes largeif a fast roll takes place. In Ref. [14] it has been demonstrated that G + G calculatedin the super-Hubble limit (with the fourth term integrated from sub-Hubble limit) isequivalent to G + G evaluated till the end of inflation. It is clear from the plot thatthe maximum f Local NL obtained from the best fit model is O (0 .
1) and which is very muchconsistent with the bounds on f Local NL obtained in Planck analysis [13]. In the finalplot of the same figure appearing in the bottom right panel we plot f Local NL for differenttriangular configurations of wavevectors. Here we have fixed k = k ∗ where k ∗ is theparticular scale where f Local NL reaches its maximum value in equilateral limit. Keeping k fixed at k ∗ we have plotted f Local NL as a function of k /k and k /k . The color-barrepresents the value of f Local NL which reaches its highest value at 0.14 for the best fitmodel and which is again certainly consistent with Planck limits. However, we shouldmention that this low value of non-Gaussianities does not throw away the possibilitiesof having a model with a different sharp wave features and with f Local NL ∼ O (1). Ourmain aim of this analysis was to locate the feature positions with the binning andpresent a viable theoretical model for that.Towards the end we would like to highlight another important fact that withStarobinsky model we find that the best fit values of cosmological parameters shiftedconsiderably from the best fit values by assuming power-law form of PPS. For examplethe best fit H and Ω m is found to be 70.3 and 0.277 respectively. This reflects how thederived cosmological parameters are degenerate with assumptions of the primordialspectrum [8]. In this paper we presented an optimally binned reconstruction of primordial powerspectrum using the recently released Planck temperature data. With up to 4 binswe addressed the locations of the possible broad features in the primordial spectrumthat could help improving the fit to the data with respect to the power law form ofthe primordial power spectrum. We realized that these broad features in fact help usto fit better the lack of power and outliers at low multiples reported by Planck [28].In this analysis we have kept the position of the bins as variables that enables us tolocate the possible broad features in the spectrum. We report that with only two– 18 –ins it is possible to get an improvement in χ by 5 compared to power law bestfit model (with only 2 extra degrees of freedom). Most of the improvement comesfrom the low- ℓ data and in fact restoring the lack of power at low- ℓ multipoles usingthe binned form of the PPS. It should be mentioned that the low- ℓ improvement in χ relaxes the bounds on the spectral tilt and we find that only at the high- ℓ (orequivalent high- k ) the spectral tilt is bounded to be nearly scale invariant. With twodifferent tilts in two bins we have identified the scales ( ∼ . − ) from which thedata constrains the spectrum to become nearly scale invariant with red tilt ( n s < k is in excellent agreement with the data.Basically our results suggest that till certain scales the amplitude and the tilt of thepower spectrum is unconstrained within a considerably large limit. Our method beingdependent of limited number of bins is unable to address high- ℓ anomalies ( if any )as the corresponding high- k space is more densely sampled compared to low- k andone needs to have relatively large number of binning to address possible small scalefeatures which is computationally expensive and not efficient due to large parameterspace. While our analysis within the context of our studies shows that the power-lawform of the primordial spectrum is relatively consistent to the Planck temperaturedata, to test rigorously whether the basic power law model is in perfect agreementwith the data or not one needs to perform a proper consistency check by simulatingmany realizations of the data in different channels as has been described in [2, 8]which is beyond the scope of this work. We shall try to revisit this issue in greaterdetail in our next work on reconstruction of primordial spectrum.We also reviewedStarobinsky-1992 model of inflaton with a minor modification proposed before andconfront this model with the data. We have chosen this specific model to study sincewe initially expected that this model may provide us with similar features as the oneswe obtained using our optimized binning approach. We indeed found that the bestfit power spectrum obtained from this model resembles in shape to the Tanh model(also our Model-A with two bins). Using the publicly available code BINGO we havefound that the f Local NL in the equilateral limit and arbitrary triangular configurations ofwavevectors are in excellent agreement with Planck bounds on local bi-spectrum. Thisexercise with an actual theoretical model basically indicates that it is possible to havea step like form of the primordial spectrum with two different amplitudes connectedby a smooth transition and generates low level of non-Gaussianities compatible withPlanck results. Acknowledgments
D.K.H and A.S wish to acknowledge support from the Korea Ministry of Education,Science and Technology, Gyeongsangbuk-Do and Pohang City for Independent JuniorResearch Groups at the Asia Pacific Center for Theoretical Physics.– 19 – eferences [1] G. Hinshaw, D. Larson, E. Komatsu, D. N. Spergel, C. L. Bennett, J. Dunkley,M. R. Nolta and M. Halpern et al. , arXiv:1212.5226 [astro-ph.CO].[2] D. K. Hazra, A. Shafieloo and T. Souradeep, JCAP , 031 (2013) [arXiv:1303.4143[astro-ph.CO]].[3] See, [4] P. Paykari and A. H. Jaffe, Astrophys. J. (2010) 1 [arXiv:0902.4399[astro-ph.CO]].[5] R. Hlozek, J. Dunkley, G. Addison, J. W. Appel, J. R. Bond, C. S. Carvalho, S. Dasand M. Devlin et al. , Astrophys. J. (2012) 90 [arXiv:1105.4887 [astro-ph.CO]].[6] A. Shafieloo and T. Souradeep, Phys. Rev. D (2004) 043523 [astro-ph/0312174].[7] S. Hannestad, Phys. Rev. D (2001) 043009 [astro-ph/0009296]; M. Tegmark andM. Zaldarriaga, Phys. Rev. D (2002) 103508 [astro-ph/0207047]; S. L. Bridle,A. M. Lewis, J. Weller and G. Efstathiou, Mon. Not. Roy. Astron. Soc. (2003)L72 [astro-ph/0302306]; P. Mukherjee and Y. Wang, Astrophys. J. (2003) 1[astro-ph/0303211]; S. Hannestad, JCAP (2004) 002 [astro-ph/0311491];D. Tocchini-Valentini, Y. Hoffman and J. Silk, Mon. Not. Roy. Astron. Soc. (2006) 1095 [astro-ph/0509478]; N. Kogo, M. Sasaki and J. ’i. Yokoyama, Prog. Theor.Phys. (2005) 555 [astro-ph/0504471]; S. M. Leach, Mon. Not. Roy. Astron. Soc. (2006) 646 [astro-ph/0506390]; A. Shafieloo, T. Souradeep, P. Manimaran,P. K. Panigrahi and R. Rangarajan, Phys. Rev. D (2007) 123502[astro-ph/0611352]; A. Shafieloo and T. Souradeep, Phys. Rev. D (2008) 023511[arXiv:0709.1944 [astro-ph]]; R. Nagata and J. ’i. Yokoyama, Phys. Rev. D (2008)123002 [arXiv:0809.4537 [astro-ph]]; R. Nagata and J. ’i. Yokoyama, Phys. Rev. D (2009) 043010 [arXiv:0812.4585 [astro-ph]]; K. Ichiki and R. Nagata, Phys. Rev. D (2009) 083002; A. Shafieloo and T. Souradeep, New J. Phys. (2011) 103024[arXiv:0901.0716 [astro-ph.CO]]; G. Nicholson and C. R. Contaldi, JCAP (2009)011 [arXiv:0903.1106 [astro-ph.CO]]; G. Nicholson, C. R. Contaldi and P. Paykari,JCAP (2010) 016 [arXiv:0909.5092 [astro-ph.CO]]; M. Bridges et al,Mon. Not. R. Astron. Soc. (2012) 050 [arXiv:1209.2147 [astro-ph.CO]]; P. Hunt and S. Sarkar,arXiv:1308.2317 [astro-ph.CO].[8] D. K. Hazra, A. Shafieloo and T. Souradeep, Phys. Rev. D , 123528 (2013)[arXiv:1303.5336 [astro-ph.CO]].[9] P. A. R. Ade et al. [Planck Collaboration], arXiv:1303.5082 [astro-ph.CO].[10] A. A. Starobinsky, JETP Lett. (1992) 489 [Pisma Zh. Eksp. Teor. Fiz. (1992)477].[11] L. A. Kofman and D. Y. .Pogosian, Phys. Lett. B (1988) 508; D. S. Salopek,J. R. Bond and J. M. Bardeen, Phys. Rev. D (1989) 1753; D. Polarski andA. A. Starobinsky, Nucl. Phys. B (1992) 623; J. A. Adams, G. G. Ross andS. Sarkar, Nucl. Phys. B (1997) 405 [hep-ph/9704286]; D. J. H. Chung, – 20 – . W. Kolb, A. Riotto and I. I. Tkachev, Phys. Rev. D (2000) 043508[hep-ph/9910437]; J. Martin, A. Riazuelo and M. Sakellariadou, Phys. Rev. D (2000) 083518 [astro-ph/9904167]; J. Martin and R. H. Brandenberger, Phys. Rev. D (2001) 123501 [hep-th/0005209]; J. Barriga, E. Gaztanaga, M. G. Santos andS. Sarkar, Mon. Not. Roy. Astron. Soc. (2001) 977 [astro-ph/0011398];J. A. Adams, B. Cresswell and R. Easther, Phys. Rev. D (2001) 123514[astro-ph/0102236]; U. H. Danielsson, Phys. Rev. D (2002) 023511[hep-th/0203198]; C. R. Contaldi, M. Peloso, L. Kofman and A. D. Linde, JCAP (2003) 002 [astro-ph/0303636]; O. Elgaroy, S. Hannestad and T. Haugboelle, JCAP (2003) 008 [astro-ph/0306229]; N. Kaloper and M. Kaplinghat, Phys. Rev. D (2003) 123522 [hep-th/0307016]; H. V. Peiris et al. [WMAP Collaboration], Astrophys.J. Suppl. (2003) 213 [astro-ph/0302225]; J. Martin and C. Ringeval, Phys. Rev. D (2004) 083515 [astro-ph/0310382]; J. Martin and C. Ringeval, Phys. Rev. D (2004) 127303 [astro-ph/0402609]; J. Martin and C. Ringeval, JCAP (2005) 007[hep-ph/0405249]; L. Sriramkumar and T. Padmanabhan, Phys. Rev. D , 103512(2005); [gr-qc/0408034]. P. Hunt and S. Sarkar, Phys. Rev. D (2004) 103518[astro-ph/0408138]; M. Bridges, A. N. Lasenby and M. P. Hobson, Mon. Not. Roy.Astron. Soc. (2006) 1123 [astro-ph/0511573]; R. Allahverdi, K. Enqvist,J. Garcia-Bellido and A. Mazumdar, Phys. Rev. Lett. (2006) 191304[hep-ph/0605035]; R. Allahverdi, K. Enqvist, J. Garcia-Bellido, A. Jokinen andA. Mazumdar, JCAP (2007) 019 [hep-ph/0610134]; J. C. Bueno Sanchez,K. Dimopoulos and D. H. Lyth, JCAP (2007) 015 [hep-ph/0608299]; L. Covi,J. Hamann, A. Melchiorri, A. Slosar and I. Sorbera, Phys. Rev. D (2006) 083509[astro-ph/0606452]; J. M. Cline and L. Hoi, JCAP (2006) 007[astro-ph/0603403]; D. N. Spergel et al. [WMAP Collaboration], Astrophys. J. Suppl. (2007) 377 [astro-ph/0603449]; P. Hunt and S. Sarkar, Phys. Rev. D (2007)123504 [arXiv:0706.2443 [astro-ph]]; J. Hamann, L. Covi, A. Melchiorri and A. Slosar,Phys. Rev. D (2007) 023503 [astro-ph/0701380]; M. Joy, V. Sahni andA. A. Starobinsky, Phys. Rev. D (2008) 023514 [arXiv:0711.1585 [astro-ph]];M. Joy, A. Shafieloo, V. Sahni and A. A. Starobinsky, JCAP (2009) 028[arXiv:0807.3334 [astro-ph]]; R. K. Jain, P. Chingangbam, J. -O. Gong,L. Sriramkumar and T. Souradeep, JCAP (2009) 009 [arXiv:0809.3915[astro-ph]]; C. Pahud, M. Kamionkowski and A. RLiddle, Phys. Rev. D (2009)083503 [arXiv:0807.0322 [astro-ph]]; R. Flauger, L. McAllister, E. Pajer, A. Westphaland G. Xu, JCAP (2010) 009 [arXiv:0907.2916 [hep-th]]; M. J. Mortonson,C. Dvorkin, H. V. Peiris and W. Hu, Phys. Rev. D (2009) 103519 [arXiv:0903.4920[astro-ph.CO]]; R. K. Jain, P. Chingangbam, L. Sriramkumar and T. Souradeep, Phys.Rev. D (2010) 023509 [arXiv:0904.2518 [astro-ph.CO]]; C. Dvorkin and W. Hu,Phys. Rev. D (2010) 023518 [arXiv:0910.2237 [astro-ph.CO]]; N. Barnaby andZ. Huang, Phys. Rev. D (2009) 126018 [arXiv:0909.0751 [astro-ph.CO]]; K. Ichiki,R. Nagata and J. ’i. Yokoyama, Phys. Rev. D (2010) 083010 [arXiv:0911.5108[astro-ph.CO]]; D. K. Hazra, M. Aich, R. K. Jain, L. Sriramkumar and T. Souradeep,JCAP (2010) 008 [arXiv:1005.2175 [astro-ph.CO]]; W. Hu, Phys. Rev. D (2011) 027303 [arXiv:1104.4500 [astro-ph.CO]]; M. Aich, D. K. Hazra, L. Sriramkumarand T. Souradeep, arXiv:1106.2798 [astro-ph.CO]; D. K. Hazra, JCAP (2013) 003[arXiv:1210.7170 [astro-ph.CO]]. H. Peiris, R. Easther and R. Flauger, arXiv:1303.2616[astro-ph.CO]. R. Easther and R. Flauger, arXiv:1308.3736 [astro-ph.CO]; M. Benetti, – 21 – rXiv:1308.6406 [astro-ph.CO]; P. D. Meerburg, D. N. Spergel and B. D. Wandelt,arXiv:1308.3704 [astro-ph.CO]; P. D. Meerburg and D. N. Spergel, arXiv:1308.3705[astro-ph.CO].[12] J. A. Vazquez, M. Bridges, M. P. Hobson and A. N. Lasenby, JCAP , 006 (2012)[arXiv:1203.1252 [astro-ph.CO]].[13] P. A. R. Ade et al. [Planck Collaboration], arXiv:1303.5084 [astro-ph.CO].[14] D. K. Hazra, L. Sriramkumar and J. Martin, JCAP , 026 (2013) [arXiv:1201.0926[astro-ph.CO]].[15] A. Blanchard, M. Douspis, M. Rowan-Robinson and S. Sarkar, Astron. Astrophys. , 35 (2003) [astro-ph/0304237].[16] L. Sriramkumar, arXiv:0904.4584 [astro-ph.CO].[17] J. Maldacena, JHEP , 013 (2003).[18] D. Seery and J. E. Lidsey, JCAP , 003 (2005); X. Chen, Phys. Rev. D , 123518(2005); X. Chen, M.-x. Huang, S. Kachru and G. Shiu, JCAP , 002 (2007);D. Langlois, S. Renaux-Petel, D. A. Steer and T. Tanaka, Phys. Rev. Lett. ,061301 (2008); Phys. Rev. D , 063523 (2008);X. Chen, Adv. Astron. , 638979(2010); X. Chen, R. Easther and E. A. Lim, JCAP , 023 (2007); JCAP , 010(2008);S. Hotchkiss and S. Sarkar, JCAP , 024 (2010); S. Hannestad,T. Haugbolle, P. R. Jarnhus and M. S. Sloth, JCAP , 001 (2010); R. Flauger andE. Pajer, JCAP , 017 (2011); P. Adshead, W. Hu, C. Dvorkin and H. V. Peiris,Phys. Rev. D , 043519 (2011); P. Adshead, C. Dvorkin, W. Hu and E. A. Lim,Phys. Rev. D , 023531 (2012);D. K. Hazra, J. Martin and L. Sriramkumar, Phys.Rev. D , 063523 (2012).[19] J. Martin and L. Sriramkumar, JCAP , 008 (2012).[20] F. Arroja, A. E. Romano and M. Sasaki, Phys. Rev. D , 123503 (2011); F. Arrojaand M. Sasaki, JCAP , 012 (2012).[21] See, ∼ sriram/bingo/bingo.html [22] See, http://camb.info/. [23] A. Lewis, A. Challinor and A. Lasenby, Astrophys. J. (2000) 473[astro-ph/9911177].[24] See, http://cosmologist.info/cosmomc/. [25] A. Lewis and S. Bridle, Phys. Rev. D (2002) 103511 [astro-ph/0205436].[26] P. A. R. Ade et al. [Planck Collaboration], arXiv:1303.5075 [astro-ph.CO].[27] M. J. D. Powell, Cambridge NA Report NA2009/06, University of Cambridge,Cambridge (2009).[28] P. A. R. Ade et al. [Planck Collaboration], arXiv:1303.5076 [astro-ph.CO].[Planck Collaboration], arXiv:1303.5076 [astro-ph.CO].