PHYS08200605006 D.K. Hazra - Homi Bhabha National Institute

4.3. THE NUMERICAL COMPUTATION OF THE SCALAR BI-SPECTRUM
zero contributions to the bi-spectrum can be easily calculated. The second example that
we shall discuss is the Starobinsky model described by the potential (4.3) wherein, under
certain conditions, the complete scalar bi-spectrum can be evaluated in the equilateral
limit (see Ref. [54]; in this context, also see Refs. [55]).
Let us first consider the case of power law inflation described by the scale factor (4.36)
with γ ≤ −2. In such a case, as we have seen,ǫ 1 is a constant and, hence,ǫ 2 and ǫ ′ 2 , which
involve derivatives of ǫ 1 , reduce to zero. Since the contributions due to the fourth and
the seventh terms, viz. G 4 (k) and G 7 (k), depend on ǫ ′ 2 andǫ 2 , respectively [cf. Eqs. (4.11d)
and (4.12)], these terms vanish identically in power law inflation. Note that the modesv k
given by Eq. (4.37) depend only on the combination kη. Moreover, as ǫ 1 is a constant in
power law inflation, we havef k ∝ v k /a . Under these conditions, with a simple rescaling
of the variable of integration in the expressions (4.11a), (4.11b), (4.11c), (4.11e) and (4.11f),
it is straightforward to show that, in the equilateral limit we are focusing on, the quantities
G 1 , G 2 , G 3 , G 5 and G 6 , all depend on the wavenumber as k γ+1/2 . Then, upon making
use of the asymptotic form of the modes f k , it is easy to illustrate that the corresponding
contributions to the bi-spectrum, viz. G 1 + G 3 , G 2 and G 5 + G 6 , all behave as k 2(2γ+1) .
Since the power spectrum in power law inflation is known to have the form k 2(γ+2) (see,
for example, Refs. [105]), the expression (4.34) for f eq then immediately suggests that
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the quantity will be strictly scale invariant for all γ. In fact, apart from these results,
it is also simple to establish the following relation between the different contributions:
G 5 +G 6 = −(3ǫ 1 /16)(G 1 +G 3 ), a result, which, in fact, also holds in slow roll inflation [54].
In other words, in power law inflation, it is possible to arrive at the spectral dependence
of the non-zero contributions to the bi-spectrum without having to explicitly calculate
the integrals involved. Further, one can establish that the non-Gaussianity parameterf eq
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is exactly scale independent for any value of γ. While these arguments does not help
us in determining the amplitude of the various contributions to the bi-spectrum or the
non-Gaussianity parameter, their spectral shape and the relative magnitude of the abovementioned
terms provide crucial analytical results to crosscheck our numerical code. In
Figure 4.4, we have plotted the different non-zero contributions to the bi-spectrum computed
using our numerical code and the spectral dependence we have arrived at above
analytically for two different values of γ in the case of power law inflation. We have also
indicated the relative magnitude of the first and the third and the fifth and the sixth terms
arrived at numerically. Lastly, we have also illustrated the scale independent behaviour
of the non-Gaussianity parameter f eq for both the values of γ. It is clear from the figure
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that the numerical results agree well with the results and conclusions that we arrived at
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