Aiswarya. A and Minu Joy arXiv:1809.08572v2 [astro-ph.CO ... · Aiswarya. Aa;b and Minu Joya a Dept. of Physics, Alphonsa College, Pala 686574, India b School of Pure and Applied

Constraints on Tensor to Scalar Ratio using WKB

Approximation

Aiswarya. Aa,b and Minu Joya

a Dept. of Physics, Alphonsa College, Pala 686574, Indiab School of Pure and Applied Physics, Mahatma Gandhi University, Kottayam

686560, India

Abstract. Using Wenzel-Kramers-Brillouin (WKB) approximation the scalar

and tensor power spectra are obtained. Scale invariant spectra are obtained

and the spectral indices come very close to the observed data from WMAP

and Planck experiments. The advantage of this method is that, it is valid even

when slow-roll approximation fails. Constraints on the tensor to scalar ratio(r)

is also studied with the WKB Approximation. We use the Power law inflation

as the base model as it allows comparison with exact results.

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Constraints on Tensor to Scalar Ratio using WKB Approximation 2

1. Introduction

Inflationary cosmology has come a long way, starting from being the theory to explain

the shortcomings of Big Bang model [1, 2, 3] to the most accepted model with the current

observations. Success of NASA′s first CMB satellite COBE (Cosmic Background

Explorer satellite), paved way for future projects in CMB. These results gave an accurate

thermal spectrum of CMB, the first detection of primordial CMB fluctuations with

normalization at 10−5. It also gave cosmic variance limit of large angular scale power

spectrum [4]. Among the CMB missions which followed COBE, BICEP mission was

designed to measure the B-mode polarization at degree angular scales and can be

considered as the first mission to constrain the tensor-to-scalar ratio r directly from

CMB B-mode polarization. The mission gave the r constraints as

r = 0.02+0.31−0.26 or r < 0.72 at 95% C.L.

The direct measurement of r from the BB spectrum has advantage, as the

measurements of TT are limited by cosmic variance at large angular scales and the r

constraints from TT have strong parameter degeneracies such as with the scalar spectral

index ns, while the BB amplitude depends primarily on r [5].

The SPT-SZ survey helped to remove the degeneracy between ns and r by

measurements of the power spectrum from angular scales corresponding to multipole

range 650 < l < 3000, that is from the third acoustic peak through the CMB damping

tail. The constrain on scalar spectral index tightens to ns < 1.0 at 3.9σ by using SPT

and WMAP7 data [6]. The B and E mode polarization data gives constraints on the

recombination (last-scattering) and reionization epochs. The cross-correlation between

temperature anisotropies and polarization signal is a consistency check for inflationary

cosmology and helps in breaking degeneracies among the cosmological parameters [7].

Analysis of the combined BICEP2 and Keck Array data in combination with the

2015 Planck data put an upper limit r0.05 < 0.12 at 95% confidence [8]. 2018 release of

the Planck temperature, polarization and lensing data determine the spectral index of

scalar perturbations to be

ns = 0.965± 0.004

at 68% CL and find no evidence for a scale dependence of ns, either as a running or as

a running of the running. The latest Planck 95% CL upper limit on the tensor-to-scalar

ratio, r0.002 < 0.10, is further tightened by combining with the BICEP2/Keck Array

BK14 data to obtain r0.002 < 0.064 [9, 10].

In the present work we study the constraints on the tensor-to-scalar ratio r

using an alternative approximation technique, instead of slow-roll approximation, with

the Power law inflationary model. We first enumerate the impact of r on different

inflationary models and cosmological parameters, followed by a brief review on some

alternative approximation techniques. Scalar and tensor power spectra using the WKB

approximation technique are obtained. Hence we try to constrain tensor to scalar ratio

with WKB approximation and check the consistency with latest Planck observations.


2. Tensor to scalar ratio

The study of tensor to scalar ratio r gives insight into large working of the universe :-

• Primordial Gravitational waves - The tensor perturbations in the FLRW

background metric during inflation are called the primordial gravitational waves

and CMB is the only observation to validate it. The CMB sky is measured with

two observables, the temperature anisotropy and the polarisation. Primordial

gravitational waves imprints very slight variations on these variables and are

sensitive to r which is defined as [7]

r =PtPs

It contributes to the temperature anisotropies as a local quadruple since the grav-

itational wave metric is traceless. The primordial tensor power spectrum [11] is

scale invariant and the shear is impulsive at horizon entry [12].

• Energy scale of universe - r depends on the time-evolution of the inflaton field and

the relation to inflaton potential [13] is given as

r =8

M2Pl

( φH

)2

V 1/4 = 1.06× 1016( rφ

0.01

)1/4

GeV

where rφ is the tensor to scalar ratio in CMB scales [7].

• Consistency relation - r gives a test for consistency for single-field slow roll inflation

as

r = −8ntsin2∆

where sin2∆ parameterizes the ratio between the adiabatic power spectrum at

horizon-exit during inflation and the observed power spectrum giving information

about presence of multiple fields. For non-slow roll inflation, with a non-trivial cs,

it is given as

r = −ntcswhere it is seen that the non-slow roll evolution of the inflaton field is driven by a

non-canonical kinetic term.

• Inflaton Field Excursion - The inflaton field ∆φ is related to r giving the effective

number of e-folds as

Neff =

∫ N?

0

dN(r(N)

r?

) 12

where N? is the number of e-folds between the end of inflation and the horizon exit

of the CMB pivot scale. By determining, if the inflaton-field excursion was super-

planckian or not we can probe aspects of scalar field space and the UV completion

of gravity.


• Inflationary models - The shape of the inflaton potential is controlled by the slow-

roll parameters and hence measurements of r, ns, nt, αs, αt help in constraining

different inflationary models and also removes the degeneracies between the slow-

roll parameters.

3. Alternatives for Standard Slow-roll approximation

There are many models proposed which give the similar observations, which doesn’t

explicitly consider slow roll approximation for inflation but gives very similar results

without the approximation [14]. For example,

3.1. Hamilton-Jacobi formulation of inflation

Hamilton-Jacobi formulation [15, 16] is a method to rewrite the equations of motion.

It considers the scalar inflaton field itself to be time-varying but it breaks down during

the oscillatory epoch that ends inflation [17, 18].

We know the equations of motion for a spatially flat universe as

H2 =1

3M2pl

(V (φ) +1

2φ2) (1)

and

φ+ 3Hφ = −dVdφ

, (2)

where V (φ) is the potential of the scalar field. By differentiating the first equation of

motion with respect to time and by substituting in the second equation of motion, we

get

2H = − φ2

M2pl

(3)

Dividing both sides by φ gives,

φ = −2M2plH

′(φ) (4)

which gives the Freidmann-equations in the first order form,

H ′(φ)2 − 3

2M2pl

H2(φ) = − 1

2M4pl

V (φ). (5)

When H(φ) is specified it gives the corresponding potential and all the other inflationary

solutions can be derived from it. For example,

H(φ) ∝ exp(−√

1

2p

φ

Mpl

)(6)

gives the Power-law inflation. The slow-roll parameters can be written as,

εH = 3φ2

2/(V +

φ2

2

)= −dlnH

dlna(7)


ηH = − φ

Hφ= −dlnφ

dlna(8)

and inflation is precisely given as,

a > 0⇐⇒ εH < 1. (9)

3.2. Perturbed Power law inflation

With the standard Power law model the scale factor is given as, a(t) = tp where p > 1

and the slow-roll parameters are given as ε = −η = 1p. The Power-law also states that

[19] - [22],

(a) Expansion is uniformly accelerated ie ε = constant

(b) The perturbations in the inflaton field is similar to that for a massless scalar field

m2eff = 0.

The power spectrum is given as Pk = Akns−1 and spectral indexes are ns and nt = ns−1

which are constant values.

The perturbed Power law [23, 24] can be given as a slight deviation from the scale

invariant spectra. Here, the inflaton potential is estimated from the ‘reduced Hamilton

-Jacobi equation’. The corresponding slow-roll parameters are given as, ε =M2

pl

4π

(H′

H

)2

and η = −M2pl

4π

(H′′

H

). The spectral indices are then,

nt = ns − 1 =−2ε

1− ε(10)

where ε = (∂lnH∂φ

)2.

The Perturbed power-law is taken for small deviations from the uniform acceleration

in terms of ∂2lnH∂φ2

. The scalar perturbation spectrum is then obtained as,

Ps(k) =A(µT , µS)

2π

(H2π

)2 1

ε(η)(11)

and the corresponding tensor power-spectrum is,

Pt(k) =8A(µT , µS)

2π

(H2π

)2

(12)

where A(x, y) = 4yΓ2(y)(x−1/2)2y−1 , Γ(y) is the Euler gamma function where µT → µ and

µs =(µ2 − (µ− 1/2)2m

2eff

H2

)1/4

.

4. WKB approximation

The WKB approximation is a method used to get approximate solutions to linear

differential equations. Here it is applied on the Radial wave equations. It was proved

that the Balmer formula for Hydrogen atom could be derived from correct application

of the WKB approximation. Hence the method was used for cosmological perturbations


which had an analogy with the Hydrogen atom formalism. The use of this method also

gives the advantage that it can be used for both subhorizon and the superhorizon scales

and it is also an alternate method than the standard slow-roll approximation.

The non-singular solutions with initial dense de-sitter phase, first mentioned by

Starobinsky gave the tensor perturbation formulation [11]. Mukhanov and Chibisov

proposed that the spectrum of quantum fluctuation in the non-singular solution could

evolve into the present universe and derived the scalar perturbation equation [25],

followed by works of Hawking [26] and Starobinsky [27]. These scalar and tensor

perturbations formulations are reduced to a single variable [28] denoted by µS or µT ,

and the equation is similar to the Schrodinger equation [29].

d2

dη2µ+ (k2 − U(η))µ =

d2µ

dη2+ ω2(η)µ = 0 (13)

where U(η) is the potential. Then the corresponding power spectra and the spectral

indices are given as,

Ps =k3

8π2

∣∣∣µszs

∣∣∣2, PT =2k3

π2

∣∣∣µTzT

∣∣∣2 (14)

where zs = a√−aa′′a′

and zT = a. The spectral indices being,

ns − 1 =∣∣∣dlnPsdlnk

∣∣∣k=k∗

, nT =∣∣∣dlnPTdlnk

∣∣∣k=k∗

. (15)

The WKB spectra is taken as the solution for the equation,

µ′′WKB(k, η) + [ω2(k, η)−Q(k, η)]µWKB = 0 (16)

where

Q(k, η) =3

4

(ω′)2

ω− ω′′

2ω. (17)

The mode function µWKB is an approximation to the actual mode function in the limit,∣∣∣ Qω2

∣∣∣ < 1 (18)

For application of the WKB approximation, the variables are transformed to

x = ln(Hak

), u = (1− ε1)

12 e

x2µ (19)

which in turn gives the Power spectra to be,

Ps =H2

πε1M2pl

( k

aH

)3 e2ψs

(1− ε1)|ωs|(20)

PT =16H2

πM2pl

( k

aH

)3 e2ψT

(1− ε1)|ωT |(21)

Here we have seen a brief on the general cosmological perturbation equations and

the application of WKB approximation on them as was given in the work by J. Martin

and D. Schwarz (2003) [30] and the derivations of the power spectra for the scalar and

the tensor perturbations. Further we use the above method for Power law inflation as

the model is exactly solvable and hence comparable with the observation data.


5. Application of WKB approximation on Power law inflationary model

For Power law inflation, the scale factor is

a(η) = lo|η|1+β (22)

where β ≤ −2 and lo is the Hubble’s radius which is a constant when β = −2. The

corresponding Horizon flow functions are given as,

εn+1 =dln|εn|dN

(23)

for n ≥ 0. Inflation takes place when ε1 < 1. The exact power spectrum is given as

Pζ =l2pll20

f(β)k2β+4

πε1(24)

Ph =l2pll20

16f(β)k2β+4

π(25)

where lpl = m−1pl is the Planck Length and the function f(β) is given by

f(β) =1

π

[Γ(−β − 12)

2β+1

]2

(26)

where Γ is the Euler integral of the second kind.

When f(β = −2) = 1, this becomes as a special case because when β = −2 the

expression of the scalar spectrum blows up.

By applying the WKB approximation the effective frequency is given as,

ω2(η) = k2 − β(β + 1)/η2 (27)

and | Qω2 | = 1

4β(β+1). As for inflation Q

ω2 1, when β = −2 the ratio goes to 1/8 hence

satisfying the condition.

By applying the WKB approximation we get the power spectra to be

Pζ =l2pll20

1

πε1g(β)k2β+4 (28)

Ph =l2pll20

16

πg(β)k2β+4 (29)

and the function is given as,

g(β) =2e2β+1

(2β + 1)2β+2. (30)

The corresponding spectral indices are then computed to be

ns − 1 = nT = 2β + 4. (31)

The Constant-roll inflation or the Ultra-slow-roll inflationary model is also a viable

alternative to the standard slow-roll approximation. In this model, the inflationary

scenario is investigated with constant rate of roll, Φ/HΦ = −3−α (remains a constant)

[31]. The model gives experimentally viable results for α . −3. The parameter

constrains on the constant roll model and comparison with observations is explored by

Motohashi and Starobinsky [32] which gives a permitted range of r values as r . 0.07

to r 10−3.


6. WKB Power Spectra and Tensor to scalar ratio(r)

By using the WKB approximation, we first see that the effective frequencies in Eq. (27)

are given as,

ω2s = k2η2 −

(9

4+ 3ε1 +

3

2ε2

)(32)

ω2T = k2η2 −

(9

4+ 3ε1

)(33)

Then corresponding WKB spectra are derived from solving Eq. (28) & (29) with

the above frequencies and we get the scalar power spectrum as,

Pζ =H2

πε1m2pl

(18e−3)[1− 2(D+ 1)ε1−Dε2− (2ε1 + ε2)ln( kk∗

)+ O(ε2n)](34)

and the corresponding tensor power spectrum as,

Ph =16H2

πm2pl

(18e−3)[1− 2(D + 1)ε1 − 2ε1ln( kk∗

)+ O(ε2n)] (35)

with D ≡ 13− ln3 ≈ −0.765.

The Spectral indices calculated from above equations are in accordance with the

spectral indices given by slow-roll inflation but it gives the next higher order terms

which increases it’s accuracy. The spectral indices are obtained as,

ns−1 =−2ε11− ε1

− ε21− ε1

+−2ε1 − ε2

[1− 2(D + 1)ε1 −Dε2 − (2ε1 + ε2)ln( kk∗

)]+O(εn)(36)

nT =−2ε11− ε1

+−2ε1

[1− 2(D + 1)ε1 − (2ε1)ln( kk∗

)]+O(εn) (37)

With Slow-roll approximation the power spectra are [33]

Pζ =H2

πε1m2pl

[1− 2(C + 1)ε1 − Cε2 − (2ε1 + ε2)ln( kk∗

)+ O(ε2n)] (38)

and the tensor powerspectrum,

Ph =16H2

πm2pl

[1− 2(C + 1)ε1 − 2ε1ln( kk∗

)+ O(ε2n)] (39)

where C=γE + ln2 − 2 ' -0.7296 , γE '0.5772 which is the Euler constant. All the

quantities are calculated at η∗ which is when k∗ = aH(N∗). This scale k∗ is called the

Pivot scale. The spectral indices with the slow-roll approximation are given as,

ns − 1 = −2ε1 − ε2, nT = −2ε1 (40)

From this we could readily see that the spectral indices of the WKB approximation

reduces to the spectral indices of the slow-roll approximation.

The scalar and tensor power spectra are represented by using the Power law form.

For which the expressions for Hubble’s constant is

H = − H

H2=

H ′

aH2= (1 + β)η−(β+2)/lo (41)


and ε2 = 0. The power-spectra are analytically represented and by setting the pivot

scale at k∗ = 0.05Mpc−1.

The plots show the scale-invariant scalar and tensor power spectra with the slow-roll

approximation and with the WKB approximation. The power-spectra are calculated for

the β = −2.005.

Figure 1. Scalar power-spectra (1)with slow-roll approximation and (2)with

WKB approximation

It is clear that the power spectra are scale invariant with both approximation

techniques and the WKB approximation spectra shows a slight variation in the

amplitude which can be compensated by changing the amplitude terms in Eq. (34)

and (35). The corresponding spectral indices are calculated for these spectra. With

the WKB approximation it is ns − 1 = nT = 3.12177276e − 04 and with the slow roll

approximation ns − 1 = nT = 3.08732253e− 04.

The application of WKB approximation helps in determining the power spectra

even when the Slow-roll approximation is invalid and hence gives us a new frontier for

viewing inflation. The added advantage is that it gives solutions with higher accuracy

and higher order terms which is a necessity to match with the very precise observations

of WMAP and Planck.

We use CosmoMC [34], to establish the constrains on the parameters and to identify

the maximum likelihood region in the parameter space. The primordial power-spectrum

in CAMB of CosmoMC is changed to the WKB approximation power spectrum. As even

a slight change can produce large deviations in the CMB sky, we have constrained the

WKB approximation parameter to β = −2.005. The modified power spectrum is used

to then generate the CMB angular power spectrum with flat priors and using the whole


Figure 2. Tensor power spectra (1)with slow roll approximation and (2)with WKB

approximation

base-set of parameters with power law ΛCDM model.

We stay with the minimal six parameter spatially flat ΛCDM cosmological model

as our base model with base parameters as baryon density today Ωbh2, cold dark matter

density today Ωch2, amplitude of scalar power spectrum As, spectral index of scalar

power spectrum ns, tensor-to-scalar ratio r and the tilt of tensor power spectrum ntbut with WKB primordial power spectrum. This model has four free non-primordial

cosmological parameters same as the power-law baseline i.e ωb, ωc, θMC and τ . Here

we try to estimate the impact of primordial power spectrum with higher-order terms of

Hubble-flow functions, which are analogous to slow-roll parameters with emphasis on

the deviations in tensor-to-scalar ratio. We also take to consideration three neutrinos

species, with two mass-less states and a single massive neutrino of mass mν = 0.06eV.

The Planck likelihood code† (PLC/clik) and parameter chains are available from

the Planck Legacy Archive. The Planck lensing and Bicep-Keck-Planck likelihoods are

included with the cosmomc.

† 2018 Planck CMB likelihoods (except for lensing) are not published yet. The present discussion refer

to the 2015 likelihoods.


0.022 0.023

b h 2

0.11 0.115 0.12 0.125

c h 2

1.04 1.042

100 MC

0.05 0.1 0.15 3 3.1 3.2

ln (10 10 A s )

0.94 0.96 0.98 1

n s

0.78 0.8 0.82 0.84 0.86

8

0.99 1 1.01

y cal

0.996 0.998 1

c 100

2 4 6 8 10

A tSZ143

200 300 400

A PS100

64 66 68 70 72

H 0

0.66 0.7 0.74 0.26 0.3 0.34

m

0.135 0.14 0.145 0.15

m h 2

0.094 0.096 0.098

m h 3

5 10 15

z re

0 0.1 0.2

r 0.01

0.94 0.96 0.98 1

n s,0.002

0 0.05 0.1

r L=10

2 2.2 2.4

10 9 A s

0 0.2 0.4

10 9 A t

1.85 1.9 1.95

10 9 A s e -2

0 0.2 0.4

10 9 A t e -2

0.245 0.2454 0.2458

Y P

13.7 13.8 13.9

Age / Gyr

1088 1090 1092

z *

143 144 145 146 147

r *

1.04 1.042

*

0.07 0.072 0.074

r drag /D V (0.57)

0 10 20

2 prior

40 45 50 55

2 BKPLANCK

10 15 20 25

2 lensing

750 760 770 780

2 plik

800 820 840

2 CMB

Figure 3. 1D marginalized posterior distribution of the inflationary parameters

and cosmological parameters with WKB approximation


The limits and marginalized densities given in Fig. 3 are generated using the getdist

package which is part of CosmoMC, it uses the baseline likelihood which is a hybrid, by

connecting together a low-multipole likelihood with the Gaussian likelihood constructed

from the higher multipoles. We study the Bayesian factors of the model with respect

to the power-law base model and the χ2eff values with power-law model. As it is clear

the WKB power spectrum is scale invariant and the parameters are in the 95% CL of

Planck results.

0.05 0.1 0.15 0.2r

0 0.05 0.1 0.15 0.2r0.002

Figure 4. 1D marginalized posterior distribution of the Tensor to Scalar ratio

r with WKB approximation

By combining Planck temperature, low- polarization and lensing 2018 Planck results

give r0.002 < 0.10 (95% CL, Planck TT+lowE+lensing) [10]. This constraint slightly

improves on the corresponding Planck 2015, 95% CL bound, i.e., r0.002 < 0.11 [35] which

is consistent with the B-mode polarization constraint r < 0.12 (95 % CL), obtained

from a joint analysis of the BICEP2/Keck Array and Planck data. With the WKB

approximation, the upper limit on r is obtained as r < 0.1109 and r0.002 < 0.1055

(BKPlanck†+ lensing). This gives us a window into higher order corrections and better

fit to the Planck results. It is seen that the WKB best fit value of the spectral index is

ns = 0.9657 with Ho = 67.64 which are in the 95% CL of Planck observations.

7. Conclusion

Present work studied the scalar and tensor power spectra with the WKB approximation

and constrained the value of the WKB parameter β. The WKB approximation helps

in determing the power spectra even when the Slow-roll approximation is invalid, gives

the higher order corrections and the precise details of Power law model which will give

insights into paramount questions in the Inflationary model of Universe. The spectral

index values and spectra generated with WKB approximation are compared with the

† BICEP2/Keck Array and 2015 Planck joint analysis


observation data and we see that the WKB approximated power spectra could become

a viable alternative for Slow-roll approximation. Running Monte-Carlo chains with

CosmoMC and generating the plots with Getdist, the tensor to scalar ratio(r) is also

analysed by constraining the cosmological parameters with WKB approximation, setting

an upper limit at r < 0.1109. Here r is defined at the pivot scale k∗ = 0.05Mpc−1. We

also report the bounds on r0.002, the tensor to scalar ratio at k∗ = 0.002Mpc−1 as

r0.002 < 0.1055 which is consistent within 1σ with the Planck results.

8. Acknowledgments

We acknowledge the use of high performance computing system at IUCAA. MJ

acknowledges the support from U.G.C. Major research project grant, F.No. − 43 −523/2014(SR) and the Associateship of IUCAA. AA acknowledges the Junior Research

Fellowship from University Grants Commission (UGC), India. We thank the referee for

the valuable suggestions and comments.

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Aiswarya. A and Minu Joy arXiv:1809.08572v2 [astro-ph.CO ... · Aiswarya. Aa;b and Minu Joya a Dept. of Physics, Alphonsa College, Pala 686574, India b School of Pure and Applied