34
P. K. Suresh
Table 1 Estimated value of the tensor-to-scalar ratio for the quadratic and quartic inflation models
for various values of their respective HDO parameter
Quadratic
Quartic
α m
r
α λ
r
0
0.133
0
0.267
−0.0015
0.0725
−0.050
0.0756
−0.0017
0.0644
−0.053
0.0641
−0.0019
0.0563
−0.061
0.0336
−0.0021
0.0481
−0.072
0.0084
quartic chaotic inflationary models with quantum gravity cannot be ruled out with the
current estimated tensor-to-scalar ratio obtained from various CMB observations.
Further, note that most of the values of the HDO parameter of the chaotic inflation
model presented in the table are less than the upper limit of the effective field theory
estimate.
5 Quantum Gravity Effect on the BB Mode Angular
Spectrum of CMB
In this section, we study the quantum gravity effect on the BB mode angular power
spectrum of the CMB through the chaotic inflationary models with BK15 and Planck
2018 joint data.
The primordial gravitational waves are responsible for the B-mode polarization
of the CMB [30, 31]. The resulting BB mode correlation angular power spectrum
can be computed with following expression [32, 33]:
C
B B
l
= (4π)
2
dk k
2 P T (k)
×
τ 0
0
dτ g(τ )h k (τ )
2 j
l (x) +
4 j l (x)
x
2
,
where g(τ ) = κ
e
−κ is the visibility function and κ
is the differential optical depth,
j l is the spherical Bessel function, and x = k(τ 0 − τ ).
The unlensed and lensed BB mode correlation angular spectrum of the CMB with
quantum gravity effect for the quadratic and quartic chaotic inflation is obtained. For
this, the optical depth is taken as κ = 0.08, the tensor pivot wave number is taken as
k 0 = 0.002 Mpc
−1 , and the scalar pivot is taken as k 0 = 0.002 Mpc
−1 . The obtained
spectra are normalized with COBE.
The unlensed BB mode correlation angular spectrum for various values of the
HDO parameter for the quadratic and quartic inflation, respectively, is shown in
P. K. Suresh
Table 1 Estimated value of the tensor-to-scalar ratio for the quadratic and quartic inflation models
for various values of their respective HDO parameter
Quadratic
Quartic
α m
r
α λ
r
0
0.133
0
0.267
−0.0015
0.0725
−0.050
0.0756
−0.0017
0.0644
−0.053
0.0641
−0.0019
0.0563
−0.061
0.0336
−0.0021
0.0481
−0.072
0.0084
quartic chaotic inflationary models with quantum gravity cannot be ruled out with the
current estimated tensor-to-scalar ratio obtained from various CMB observations.
Further, note that most of the values of the HDO parameter of the chaotic inflation
model presented in the table are less than the upper limit of the effective field theory
estimate.
5 Quantum Gravity Effect on the BB Mode Angular
Spectrum of CMB
In this section, we study the quantum gravity effect on the BB mode angular power
spectrum of the CMB through the chaotic inflationary models with BK15 and Planck
2018 joint data.
The primordial gravitational waves are responsible for the B-mode polarization
of the CMB [30, 31]. The resulting BB mode correlation angular power spectrum
can be computed with following expression [32, 33]:
C
B B
l
= (4π)
2
dk k
2 P T (k)
×
τ 0
0
dτ g(τ )h k (τ )
2 j
l (x) +
4 j l (x)
x
2
,
where g(τ ) = κ
e
−κ is the visibility function and κ
is the differential optical depth,
j l is the spherical Bessel function, and x = k(τ 0 − τ ).
The unlensed and lensed BB mode correlation angular spectrum of the CMB with
quantum gravity effect for the quadratic and quartic chaotic inflation is obtained. For
this, the optical depth is taken as κ = 0.08, the tensor pivot wave number is taken as
k 0 = 0.002 Mpc
−1 , and the scalar pivot is taken as k 0 = 0.002 Mpc
−1 . The obtained
spectra are normalized with COBE.
The unlensed BB mode correlation angular spectrum for various values of the
HDO parameter for the quadratic and quartic inflation, respectively, is shown in