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Using these results finally we compute the following inflationary observables which is very very useful to further compute the inflationary observables from multi-

Figure 147 Using these results finally we compute the following inflationary observables which is very very useful to further compute the inflationary observables from multi-

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In this case the total effective potential of tachyonic fields Vz(T) can be recast as Consequently in such a prescription | Eq (2.16) and | iq (2.17) can be re-written as:
In this case the total effective potential of tachyonic fields Vz(T) can be recast as Consequently in such a prescription | Eq (2.16) and | iq (2.17) can be re-written as:
Here the potential satisfies the following criteria: and for multi tachyonic and assisted case the total effective potential is given by:
Here the potential satisfies the following criteria: and for multi tachyonic and assisted case the total effective potential is given by:
and at these points the value of the total effective potential is computed as: .3 Model III: Exponential potential-Type I For single field case the third model of tachyonic potential is given by [81]:
and at these points the value of the total effective potential is computed as: .3 Model III: Exponential potential-Type I For single field case the third model of tachyonic potential is given by [81]:
and for multi tachyonic and assisted case the total effective potential is given by: Here the potential satisfies the following criteria: For single field case the first model of tachyonic potential is given by [84, 124]: .5 Model V: Inverse power-law potential
and for multi tachyonic and assisted case the total effective potential is given by: Here the potential satisfies the following criteria: For single field case the first model of tachyonic potential is given by [84, 124]: .5 Model V: Inverse power-law potential
and at these points the value of the total effective potential is computed as: Cosmological dynamics from GTachyon .1 Unperturbed evolution
and at these points the value of the total effective potential is computed as: Cosmological dynamics from GTachyon .1 Unperturbed evolution
ie Similarly for the generalized situation comparing Eq (4.7) and Eq (4.8) the density p and pressure p for tachyonic field can be computed as: Further comparing Eq (4.6) and Eq (4.8) the density p and pressure p for tachyonic field can be computed as:
ie Similarly for the generalized situation comparing Eq (4.7) and Eq (4.8) the density p and pressure p for tachyonic field can be computed as: Further comparing Eq (4.6) and Eq (4.8) the density p and pressure p for tachyonic field can be computed as:
and for the generalized gq 4 1/2 case we have:
and for the generalized gq 4 1/2 case we have:
For both of the cases we fix the boundary condition in such a way that the tachy- onic field T satisfy the constraint: T(t = 0) = T(0) = 0. Further using Eq (4.17) and Eq (5.328) we get the following constraint condition for the cosmological time dependent potential V(t): integrating Eq (4.41) we get the following solutions for the tachyonic field:
For both of the cases we fix the boundary condition in such a way that the tachy- onic field T satisfy the constraint: T(t = 0) = T(0) = 0. Further using Eq (4.17) and Eq (5.328) we get the following constraint condition for the cosmological time dependent potential V(t): integrating Eq (4.41) we get the following solutions for the tachyonic field:
es q (4.50) and Eq (4.51) can Hence using the redefinition, the potential as stated in ] be re-expressed as: Using this approximation one can use the following expansion:
es q (4.50) and Eq (4.51) can Hence using the redefinition, the potential as stated in ] be re-expressed as: Using this approximation one can use the following expansion:
and finally for the usual tachyonic case the potential V (7) can be approximated and by inverting Eq (4.60) the associated time scale can be computed as:
and finally for the usual tachyonic case the potential V (7) can be approximated and by inverting Eq (4.60) the associated time scale can be computed as:
ie Hence using the solution obtained for tachyonic field as stated in Eq (4.42) we
ie Hence using the solution obtained for tachyonic field as stated in Eq (4.42) we
Here the potential satisfy the following criteria: and for the generalized case it is fixed by: - At T = 0 for single field tachyonic potential:
Here the potential satisfy the following criteria: and for the generalized case it is fixed by: - At T = 0 for single field tachyonic potential:
It is important to note for multi tachyonic case that: and at these points the value of the total effective potential is computed nae and at these points the value of the total effective potential is computed
It is important to note for multi tachyonic case that: and at these points the value of the total effective potential is computed nae and at these points the value of the total effective potential is computed
Next using Eq (4.17) Hubble parameter can be expressed in terms of the usual tachyonic potential V(7’) as:
Next using Eq (4.17) Hubble parameter can be expressed in terms of the usual tachyonic potential V(7’) as:
Let us study the limiting situation from the expression obtained for Hubble parameter explicitly: and for the generalized case we have:
Let us study the limiting situation from the expression obtained for Hubble parameter explicitly: and for the generalized case we have:
eS) In this limiting situation the solution for the scale factor a(t) from Eq (4.109)
eS) In this limiting situation the solution for the scale factor a(t) from Eq (4.109)
ie In this limiting situation the solution for the scale factor a(t) from Eq (4.114) can be expressed as: and for the generalized case we have:
ie In this limiting situation the solution for the scale factor a(t) from Eq (4.114) can be expressed as: and for the generalized case we have:
9.1.2 Analysis using Slow-roll formalism H t t ere our prime objective is to define slow-roll parameters for tachyon inflation in terms of the Hubble parameter and the single field tachyonic inflationary potential. Using the slow-roll approximation one can expand various cosmological observables in terms of small dynamical quantities derived from the appropriate derivatives of he Hubble parameter and of the inflationary potential. To start with here we use he horizon-flow parameters based on derivatives of Hubble parameter with respect to the number of e-foldings N, defined as: value of q as:
9.1.2 Analysis using Slow-roll formalism H t t ere our prime objective is to define slow-roll parameters for tachyon inflation in terms of the Hubble parameter and the single field tachyonic inflationary potential. Using the slow-roll approximation one can expand various cosmological observables in terms of small dynamical quantities derived from the appropriate derivatives of he Hubble parameter and of the inflationary potential. To start with here we use he horizon-flow parameters based on derivatives of Hubble parameter with respect to the number of e-foldings N, defined as: value of q as:
from the first three Hubble slow-roll parameter, which can be depicted as:
from the first three Hubble slow-roll parameter, which can be depicted as:
Our next objective is to express the Hubble slow-roll parameters in therms of the tachyon potential dependent slow-roll parameters. To serve this purpose let us start with writing the expression for the derivatives of the potential in terms of the Hubble slow-roll parameters. Allowing upto the second order contribution in Hubble slow- roll parameters we get [84]:
Our next objective is to express the Hubble slow-roll parameters in therms of the tachyon potential dependent slow-roll parameters. To serve this purpose let us start with writing the expression for the derivatives of the potential in terms of the Hubble slow-roll parameters. Allowing upto the second order contribution in Hubble slow- roll parameters we get [84]:
where the potential dependent slow-roll parameters ey,ny,&?,0% are defined as
where the potential dependent slow-roll parameters ey,ny,&?,0% are defined as
tachyonic inflation by rescaling with the appropriate powers of a’V(T): ie Further using Eq (5.39)-Eq (5.46) we get the following operator identity for tachyonic inflation:
tachyonic inflation by rescaling with the appropriate powers of a’V(T): ie Further using Eq (5.39)-Eq (5.46) we get the following operator identity for tachyonic inflation:
where we use the following consistency conditions for re-scaled potential dependent slow-roll parameters:
where we use the following consistency conditions for re-scaled potential dependent slow-roll parameters:
Additionally, it is important to note that in ] tachyonic string field theoretic model with ] Eq (5.65), the two gauge invariant metric perturbations ®(t,k) and W(t,k) are equal in the context of minimally coupled Hinstein gravity sector. In ] iq (5.62) and Eq (5.63) the perturbed energy density dp and pressure Op are given by:
Additionally, it is important to note that in ] tachyonic string field theoretic model with ] Eq (5.65), the two gauge invariant metric perturbations ®(t,k) and W(t,k) are equal in the context of minimally coupled Hinstein gravity sector. In ] iq (5.62) and Eq (5.63) the perturbed energy density dp and pressure Op are given by:
Further we will perform the following steps throughout the next part of the compu- Similarly after the variation of the tachyoinic field equation motion we get the fol-
Further we will perform the following steps throughout the next part of the compu- Similarly after the variation of the tachyoinic field equation motion we get the fol-
Further replacing the factor aH in Eq (5.89), we finally get the following simplified expression:
Further replacing the factor aH in Eq (5.89), we finally get the following simplified expression:
and similarly the primordial power spectrum for scalar modes at any arbitrary mo- mentum scale k can be written for both AV and BD with any arbitrary q as:
and similarly the primordial power spectrum for scalar modes at any arbitrary mo- mentum scale k can be written for both AV and BD with any arbitrary q as:
where the effective sound speed cg is given by:
where the effective sound speed cg is given by:
Now starting from the expression for primordial power spectrum for scalar modes
Now starting from the expression for primordial power spectrum for scalar modes
mentum scale k can be written for both AV and BD with any arbitrary q as:
mentum scale k can be written for both AV and BD with any arbitrary q as:
and for both AV and BD with any arbitrary q as: |. We start with the Laurent expansion of the Gamma function:
and for both AV and BD with any arbitrary q as: |. We start with the Laurent expansion of the Gamma function:
Using these approximations the primordial scalar power spectrum can be ex- pressed as: 4. Next one can compute the scalar spectral tilt (ns) of the primordial scalar
Using these approximations the primordial scalar power spectrum can be ex- pressed as: 4. Next one can compute the scalar spectral tilt (ns) of the primordial scalar
6. Finally, one can also compute the running of the running of scalar spectral tilt
6. Finally, one can also compute the running of the running of scalar spectral tilt
slow-roll parameters one can recast this consistency condition as:
slow-roll parameters one can recast this consistency condition as:
which can be treated as another slow-roll parameter in the present context. One can also recast the slow-roll parameter S in terms of the inflationary observables as:
which can be treated as another slow-roll parameter in the present context. One can also recast the slow-roll parameter S in terms of the inflationary observables as:
4. For any value of q including g = 1/2 we need to compute the following integral:
4. For any value of q including g = 1/2 we need to compute the following integral:
1.8 Semi analytical study and Cosmological parameter estimation
1.8 Semi analytical study and Cosmological parameter estimation
Figure 1. Variation of the Inverse cosh potential V(T)/X with field T’/Tp in dimensionless were ccd Next using specified form of the potential the potential dependent slow-roll pa-
Figure 1. Variation of the Inverse cosh potential V(T)/X with field T’/Tp in dimensionless were ccd Next using specified form of the potential the potential dependent slow-roll pa-
Figure 2. Behaviour of the tensor-to-scalar ratio r with respect to 2(a) the scalar spec- tral index n¢ and 2(b) the parameter g for Inverse cosh potential. The purple and blue coloured line represent the upper bound of tensor-to-scalar ratio allowed by Planck+ BI- CEP2+Keck Array joint constraint and only Planck 2015 data respectively. For both the figures red, green, brown, orange colored curve represent gq = 1/2, g = 1, q = 3/2 and q = 2 respectively. The cyan color shaded region bounded by two vertical black coloured lines in 2(a) represent the Planck 2c allowed region and the rest of the light gray shaded region is showing the lo allowed range, which is at present disfavoured by the Planck data and Planck+ BICEP2+Keck Array joint constraint. From 2(a) and 2(b), it is also observed that, within 50 < N, < 70 the Inverse cosh potential is favoured only for the characteristic index 1/2 < q < 2, by Planck 2015 data and Planck+ BICEP2+Keck Array joint analysis. In 2(b), we have explicitly shown that in r — g plane the observationally favoured lower bound for the characteristic index is q > 1/2.
Figure 2. Behaviour of the tensor-to-scalar ratio r with respect to 2(a) the scalar spec- tral index n¢ and 2(b) the parameter g for Inverse cosh potential. The purple and blue coloured line represent the upper bound of tensor-to-scalar ratio allowed by Planck+ BI- CEP2+Keck Array joint constraint and only Planck 2015 data respectively. For both the figures red, green, brown, orange colored curve represent gq = 1/2, g = 1, q = 3/2 and q = 2 respectively. The cyan color shaded region bounded by two vertical black coloured lines in 2(a) represent the Planck 2c allowed region and the rest of the light gray shaded region is showing the lo allowed range, which is at present disfavoured by the Planck data and Planck+ BICEP2+Keck Array joint constraint. From 2(a) and 2(b), it is also observed that, within 50 < N, < 70 the Inverse cosh potential is favoured only for the characteristic index 1/2 < q < 2, by Planck 2015 data and Planck+ BICEP2+Keck Array joint analysis. In 2(b), we have explicitly shown that in r — g plane the observationally favoured lower bound for the characteristic index is q > 1/2.
Figure 3. Variation of the 3(a) scalar power spectrum A¢ vs scalar spectral index ne, 3(b) scalar power spectrum A¢ vs the stringy parameter g and 3(c) scalar spectral tilt n¢ vs the stringy parameter g. The purple and blue coloured line represent the upper and lower bound allowed by WMAP-+Planck 2015 data respectively. The green dotted region bounded by two vertical black coloured lines represent the Planck 20 allowed region and the rest of the light gray shaded region is disfavoured by the Planck+WMAP constraint.
Figure 3. Variation of the 3(a) scalar power spectrum A¢ vs scalar spectral index ne, 3(b) scalar power spectrum A¢ vs the stringy parameter g and 3(c) scalar spectral tilt n¢ vs the stringy parameter g. The purple and blue coloured line represent the upper and lower bound allowed by WMAP-+Planck 2015 data respectively. The green dotted region bounded by two vertical black coloured lines represent the Planck 20 allowed region and the rest of the light gray shaded region is disfavoured by the Planck+WMAP constraint.
Figure 4. Behaviour of the 4(a) running of the scalar spectral tilt ag and 4(b) running of the running of the scalar spectral tilt k¢ with respect to the scalar spectral index n¢ for Inverse cosh potential with g = 88. For both the figures red, green, brown, orange colored curve represent g = 1/2, q = 1, q = 3/2 and q = 2 respectively. The cyan color shaded region bounded by two vertical black coloured lines in 4(a) and 4(a) represent the Planck 20 allowed region and the rest of the light gray shaded region is showing the lo allowed range, which is at present disfavoured by the Planck data and Planck+ BIC] Array joint constraint. From 4(a) and 4(b), it is also observed that, within 50 < N, < 70 the Inverse cosh potential is favoured for the characteristic index 1/2 < q < 2, 2015 data and Planck+ BICEP2+Keck Array joint analysis. fH} P2+Keck by Planck
Figure 4. Behaviour of the 4(a) running of the scalar spectral tilt ag and 4(b) running of the running of the scalar spectral tilt k¢ with respect to the scalar spectral index n¢ for Inverse cosh potential with g = 88. For both the figures red, green, brown, orange colored curve represent g = 1/2, q = 1, q = 3/2 and q = 2 respectively. The cyan color shaded region bounded by two vertical black coloured lines in 4(a) and 4(a) represent the Planck 20 allowed region and the rest of the light gray shaded region is showing the lo allowed range, which is at present disfavoured by the Planck data and Planck+ BIC] Array joint constraint. From 4(a) and 4(b), it is also observed that, within 50 < N, < 70 the Inverse cosh potential is favoured for the characteristic index 1/2 < q < 2, 2015 data and Planck+ BICEP2+Keck Array joint analysis. fH} P2+Keck by Planck
.et us now discuss the general constraints on the parameters of tachyonic strin; hheory including the factor g and on the parameters appearing in the expression fo nverse cosh potential. In Fig. (2(a)) and Fig. (2(b)), we have shown the behavio f the tensor-to-scalar ratio r with respect to the scalar spectral index n¢ and th nodel parameter g for Inverse cosh potential respectively. In both the figures th surple and blue coloured line represent t he upper bound of tensor-to-scalar rati: lowed by Planck+ BICEP2+Keck Array joint constraint and only Planck 2015 dat: espectively. For both the figures red, gree y= 1/2, qd = 1, gd = 3/2 and q = 2 res n, brown, orange colored curve represen pectively. The cyan color shaded regio! younded by two vertical black coloured lines in Fig. (2(a)) represent the Plancl 2g allowed region and the rest of the lig ht gray shaded region is showing the 1 ullowed range, which is at present disfavoured by the Planck 2015 data and Planck- 3ICEP2+Keck Array joint constraint. The rest of the region is completely rule
.et us now discuss the general constraints on the parameters of tachyonic strin; hheory including the factor g and on the parameters appearing in the expression fo nverse cosh potential. In Fig. (2(a)) and Fig. (2(b)), we have shown the behavio f the tensor-to-scalar ratio r with respect to the scalar spectral index n¢ and th nodel parameter g for Inverse cosh potential respectively. In both the figures th surple and blue coloured line represent t he upper bound of tensor-to-scalar rati: lowed by Planck+ BICEP2+Keck Array joint constraint and only Planck 2015 dat: espectively. For both the figures red, gree y= 1/2, qd = 1, gd = 3/2 and q = 2 res n, brown, orange colored curve represen pectively. The cyan color shaded regio! younded by two vertical black coloured lines in Fig. (2(a)) represent the Plancl 2g allowed region and the rest of the lig ht gray shaded region is showing the 1 ullowed range, which is at present disfavoured by the Planck 2015 data and Planck- 3ICEP2+Keck Array joint constraint. The rest of the region is completely rule
Figure 5. Variation of the Logarithmic potential V(T)/A with field T’/To in dimensionless
Figure 5. Variation of the Logarithmic potential V(T)/A with field T’/To in dimensionless
Next we compute the number of e-foldings from this model:
Next we compute the number of e-foldings from this model:
Also the field excursion can be computed as:
Also the field excursion can be computed as:
‘igure 6. Behaviour of the tensor-to-scalar ratio r with respect to 6(a) the scalar spec- ‘al index ng and 6(b) the parameter g for Logarithmic potential. The purple and blue sloured line represent the upper bound of tensor-to-scalar ratio allowed by Planck+ BI- EP2+Keck Array joint constraint and only Planck 2015 data respectively. For both the gures red, green, brown, orange colored curve represent gq = 1/2, q = 1, q = 3/2 and = 2 respectively. The cyan color shaded region bounded by two vertical black coloured nes in 6(a) represent the Planck 2c allowed region and the rest of the light gray shaded gion is showing nd Planck+ BIC] the lo allowed range, which is at present disfavoured by the Planck data EP2+Keck Array joint constraint. Fr om 6(a) and 6(b), it is also observed at, within 50 < N, < 70 the Logarithmic potential is favoured for the characteristic index /2 <q < 2, by Planck 2015 data and Planck+ BIC] EP2+Keck Array joint analysis. In (a), we have explicitly shown that in r—ne¢ plane the observationally favoured lower bound x the characteristic index is q > 1/2.
‘igure 6. Behaviour of the tensor-to-scalar ratio r with respect to 6(a) the scalar spec- ‘al index ng and 6(b) the parameter g for Logarithmic potential. The purple and blue sloured line represent the upper bound of tensor-to-scalar ratio allowed by Planck+ BI- EP2+Keck Array joint constraint and only Planck 2015 data respectively. For both the gures red, green, brown, orange colored curve represent gq = 1/2, q = 1, q = 3/2 and = 2 respectively. The cyan color shaded region bounded by two vertical black coloured nes in 6(a) represent the Planck 2c allowed region and the rest of the light gray shaded gion is showing nd Planck+ BIC] the lo allowed range, which is at present disfavoured by the Planck data EP2+Keck Array joint constraint. Fr om 6(a) and 6(b), it is also observed at, within 50 < N, < 70 the Logarithmic potential is favoured for the characteristic index /2 <q < 2, by Planck 2015 data and Planck+ BIC] EP2+Keck Array joint analysis. In (a), we have explicitly shown that in r—ne¢ plane the observationally favoured lower bound x the characteristic index is q > 1/2.
Figure 7. Variation of the 7(a) scalar power spectrum A¢ vs scalar spectral index ne, 7(b) scalar power spectrum A¢ vs the stringy parameter g and 7(c) scalar spectral tilt n¢ vs the stringy parameter g. The purple and blue coloured line represent the upper and lower bound allowed by WMAP-+Planck 2015 data respectively. The green dotted region bounded by two vertical black coloured lines represent the Planck 20 allowed region and the rest of the light gray shaded region is disfavoured by the Planck+WMAP constraint.
Figure 7. Variation of the 7(a) scalar power spectrum A¢ vs scalar spectral index ne, 7(b) scalar power spectrum A¢ vs the stringy parameter g and 7(c) scalar spectral tilt n¢ vs the stringy parameter g. The purple and blue coloured line represent the upper and lower bound allowed by WMAP-+Planck 2015 data respectively. The green dotted region bounded by two vertical black coloured lines represent the Planck 20 allowed region and the rest of the light gray shaded region is disfavoured by the Planck+WMAP constraint.
Figure 8. Variation of the Exponential potential-Type I V(T)/A with field T/T in di- mensionless units. Next using specified form of the potential the potential dependent slow-roll pa- For single field case the third model of tachyonic potential is given by:
Figure 8. Variation of the Exponential potential-Type I V(T)/A with field T/T in di- mensionless units. Next using specified form of the potential the potential dependent slow-roll pa- For single field case the third model of tachyonic potential is given by:
Figure 9. In 9(a) the behaviour of the tensor-to-scalar ratio r with respect to the scalar spec blue BIC] tral index n¢ and 9(b) the parameter g for | coloured line represent the upper bound Exponential potential-Type I. The purple and of tensor-to-scalar ratio allowed by Planck+ EP2+Keck Array joint constraint and only Planck 2015 data respectively. For both the figures red, green, brown, orange colored curve represent q = 1/2, q=1, q = 3/2 and q = 2 respectively. The cyan color shaded region bounded by two vertical black coloured lines in 9(a) represent the Planck 20 allowed region is showing the lo allowed range, which region and the rest of the light gray shaded is at present disfavoured by the Planck data and Planck+ BICEP2+Keck Array joint constraint. In 9(a), we have explicitly shown that in r — ne plane the observationally favoured lower bound for the characteristic index is q > 1/2. Variation of the scalar power spectrum A¢ vs scalar spectral index ng is shown in 9(b). The purple and blue coloured line represent the upper and lower bound allowed by WMAP-+ Planck 2015 data respectively. The green dotted region bounded by two vertical black coloured lines represent the Planck 2c allowed region and the rest of the light gray shaded region is disfavoured by the Planck+WMAP constraint. From 9(a) and 9(b), it is also observed that, within 50 < N, < 70 the Exponential potential-Type I is favoured for the characteristic index 1 < q < 2, by Planck 2015 data and Planck+ BICEP2+Keck Array joint analysis. Figure 9. In 9(a) the behaviour of the tensor-to-scalar ratio r with respect to the scalar
Figure 9. In 9(a) the behaviour of the tensor-to-scalar ratio r with respect to the scalar spec blue BIC] tral index n¢ and 9(b) the parameter g for | coloured line represent the upper bound Exponential potential-Type I. The purple and of tensor-to-scalar ratio allowed by Planck+ EP2+Keck Array joint constraint and only Planck 2015 data respectively. For both the figures red, green, brown, orange colored curve represent q = 1/2, q=1, q = 3/2 and q = 2 respectively. The cyan color shaded region bounded by two vertical black coloured lines in 9(a) represent the Planck 20 allowed region is showing the lo allowed range, which region and the rest of the light gray shaded is at present disfavoured by the Planck data and Planck+ BICEP2+Keck Array joint constraint. In 9(a), we have explicitly shown that in r — ne plane the observationally favoured lower bound for the characteristic index is q > 1/2. Variation of the scalar power spectrum A¢ vs scalar spectral index ng is shown in 9(b). The purple and blue coloured line represent the upper and lower bound allowed by WMAP-+ Planck 2015 data respectively. The green dotted region bounded by two vertical black coloured lines represent the Planck 2c allowed region and the rest of the light gray shaded region is disfavoured by the Planck+WMAP constraint. From 9(a) and 9(b), it is also observed that, within 50 < N, < 70 the Exponential potential-Type I is favoured for the characteristic index 1 < q < 2, by Planck 2015 data and Planck+ BICEP2+Keck Array joint analysis. Figure 9. In 9(a) the behaviour of the tensor-to-scalar ratio r with respect to the scalar
Figure 10. Behaviour of the 10(a) running of the scalar spectral tilt a¢ and 10(b) running of the running of the scalar spectral tilt K¢ with respect to the scalar spectral index n¢ or Exponential potential-Type I. For both the figures red colored curve represent for any value of g respectively. The cyan color shaded region bounded by two vertical black coloured ines in 10(a) and 10(a) represent the Planck 20 allowed region and the rest of the light gray shaded region is showing the lo allowed range, which is at present disfavoured by the Planck data and Planck+ BICEP2+Keck Array joint constraint. From 10(a) and 10(b), it is also observed that, within 50 < N, < 70 the Exponential potential-Type I is favoured or the characteristic index 1 < q < 2, by Planck 2015 data and Planck+ BICEP2+Keck Array joint analysis.
Figure 10. Behaviour of the 10(a) running of the scalar spectral tilt a¢ and 10(b) running of the running of the scalar spectral tilt K¢ with respect to the scalar spectral index n¢ or Exponential potential-Type I. For both the figures red colored curve represent for any value of g respectively. The cyan color shaded region bounded by two vertical black coloured ines in 10(a) and 10(a) represent the Planck 20 allowed region and the rest of the light gray shaded region is showing the lo allowed range, which is at present disfavoured by the Planck data and Planck+ BICEP2+Keck Array joint constraint. From 10(a) and 10(b), it is also observed that, within 50 < N, < 70 the Exponential potential-Type I is favoured or the characteristic index 1 < q < 2, by Planck 2015 data and Planck+ BICEP2+Keck Array joint analysis.
Let us now discuss the general constraints on the parameters of tachyonic string theory including the factor gq and on the parameters appearing in the expression for Exponential potential-Type I. In Fig. (9(a)), we have shown the behavior of the tensor-to-scalar ratio r with respect to the scalar spectral index n¢ for be Exponen- tial potential-Type I respectively. In both the figures the purple and blue coloured
Let us now discuss the general constraints on the parameters of tachyonic string theory including the factor gq and on the parameters appearing in the expression for Exponential potential-Type I. In Fig. (9(a)), we have shown the behavior of the tensor-to-scalar ratio r with respect to the scalar spectral index n¢ for be Exponen- tial potential-Type I respectively. In both the figures the purple and blue coloured
where A characterize the scale of inflation and To is the Fig. (11) we have depicted the symmetric behavior of the | I with respect to scaled field coordinate T/T in dimensio parameter of the model. In Exponential potential-Type nless units. In this case the tachyon field started rolling down from the top hight of t origin and take part in inflationary dynamics. a 2 ‘ eye 1 ef ae he potential situated at the
where A characterize the scale of inflation and To is the Fig. (11) we have depicted the symmetric behavior of the | I with respect to scaled field coordinate T/T in dimensio parameter of the model. In Exponential potential-Type nless units. In this case the tachyon field started rolling down from the top hight of t origin and take part in inflationary dynamics. a 2 ‘ eye 1 ef ae he potential situated at the
Figure 11. Variation of the Exponential potential-Type I] V(T)/A with field T/To in dimensionless units. Next we compute the number of e-foldings from this model *:
Figure 11. Variation of the Exponential potential-Type I] V(T)/A with field T/To in dimensionless units. Next we compute the number of e-foldings from this model *:
Also the field excursion can be computed as:
Also the field excursion can be computed as:
Figure 12. Behaviour of the tensor-to-scalar ratio r with respect to 12(a) the scalar spectral index n¢ and 12(b) the parameter g for blue coloured line represent the upper bound of tensor-to-scalar ratio allowed by BIC] HP2+Keck Array joint constraint and only Planck 2015 data respectively. For Exponential potential-Type I. The purple and Planck+ both the figures red, green, brown, orange colored curve represent g = 1/2, q= 1, q = 3/2 and q = 2 respectively. The cyan color shaded region bounded by two vertical black coloured 12(a) represent the Planck 2c allowed region and the rest of the light gray shade lines in d region is showing the lo allowed range, which is at present disfavoured by the Planck data and Planck+ BIC] that, within 50 < N, < 70 the Logarithmic potential is favoured for the characteris 1/2 <q < 2, by Planck 2015 data and Planck+ BICEP2+Keck Array joint analysis. EP2+Keck Array joint constraint. From 12(a) and 12(b), it is also observed tic index Figure 12. Behaviour of the tensor-to-scalar ratio r with respect to 12(a) the scalar spectral
Figure 12. Behaviour of the tensor-to-scalar ratio r with respect to 12(a) the scalar spectral index n¢ and 12(b) the parameter g for blue coloured line represent the upper bound of tensor-to-scalar ratio allowed by BIC] HP2+Keck Array joint constraint and only Planck 2015 data respectively. For Exponential potential-Type I. The purple and Planck+ both the figures red, green, brown, orange colored curve represent g = 1/2, q= 1, q = 3/2 and q = 2 respectively. The cyan color shaded region bounded by two vertical black coloured 12(a) represent the Planck 2c allowed region and the rest of the light gray shade lines in d region is showing the lo allowed range, which is at present disfavoured by the Planck data and Planck+ BIC] that, within 50 < N, < 70 the Logarithmic potential is favoured for the characteris 1/2 <q < 2, by Planck 2015 data and Planck+ BICEP2+Keck Array joint analysis. EP2+Keck Array joint constraint. From 12(a) and 12(b), it is also observed tic index Figure 12. Behaviour of the tensor-to-scalar ratio r with respect to 12(a) the scalar spectral
Figure 13. Variation of the 13(a) scalar power spectrum A¢ vs scalar spectral index ne, 13(b) scalar power spectrum A¢ vs the stringy parameter g and 13(c) scalar spectral tilt ne vs the stringy parameter g. The purple and blue coloured line represent the upper and lower bound allowed by WMAP-+Planck 2015 data respectively. The green dotted region bounded by two vertical black coloured lines represent the Planck 20 allowed region and the rest of the light gray shaded region is disfavoured by the Planck+WMAP constraint.
Figure 13. Variation of the 13(a) scalar power spectrum A¢ vs scalar spectral index ne, 13(b) scalar power spectrum A¢ vs the stringy parameter g and 13(c) scalar spectral tilt ne vs the stringy parameter g. The purple and blue coloured line represent the upper and lower bound allowed by WMAP-+Planck 2015 data respectively. The green dotted region bounded by two vertical black coloured lines represent the Planck 20 allowed region and the rest of the light gray shaded region is disfavoured by the Planck+WMAP constraint.
Figure 14. Behaviour of the 14(a) running of the scalar spectral tilt ag and 14( b) running of the running of the scalar spectral tilt k¢ with respect to the scalar spectral index n¢ fo! Inverse cosh potential with g = 88. For both the figures red, green, brown, orange colored curve represent g = 1/2, q = 1, q = 3/2 and q = 2 respectively. The cyan color shaded region bounded by two vertical black coloured lines in 14(a) and 14(a) represent the Planck 20 allowed region and the rest of the light gray shaded region is showing the lo allowed range, which is at present disfavoured by the Planck data and Planck+ BIC] Array joint constraint. From 14(a) and 14(b), it is also observed that, within 50 the Inverse cosh potential is favoured for the characteristic index 1/2 < q < 2, 2015 data and Planck+ BICEP2+Keck Array joint analysis. H}P2+Keck < N, < 7 by Planck
Figure 14. Behaviour of the 14(a) running of the scalar spectral tilt ag and 14( b) running of the running of the scalar spectral tilt k¢ with respect to the scalar spectral index n¢ fo! Inverse cosh potential with g = 88. For both the figures red, green, brown, orange colored curve represent g = 1/2, q = 1, q = 3/2 and q = 2 respectively. The cyan color shaded region bounded by two vertical black coloured lines in 14(a) and 14(a) represent the Planck 20 allowed region and the rest of the light gray shaded region is showing the lo allowed range, which is at present disfavoured by the Planck data and Planck+ BIC] Array joint constraint. From 14(a) and 14(b), it is also observed that, within 50 the Inverse cosh potential is favoured for the characteristic index 1/2 < q < 2, 2015 data and Planck+ BICEP2+Keck Array joint analysis. H}P2+Keck < N, < 7 by Planck
For inverse cosh potential we get the following consistency relations:
For inverse cosh potential we get the following consistency relations:
Figure 15. Variation of the Inverse power-law potential V(T)/A with field T/T in dimen- sionless units. Figure 15. Variation of the Inverse power-law potential V(T)/A with field T/T in dimen- where A characterize the scale of inflation and 7p is the parameter of the model. In Fig. (15) we have depicted the symmetric behavior of the Inverse power-law potential with respect to scaled field coordinate T/7o in dimensionless units. In this case the tachyon field started rolling down from the top hight of the potential situated at the origin and take part in inflationary dynamics. Next using specified form of the potential the potential dependent slow-roll pa-
Figure 15. Variation of the Inverse power-law potential V(T)/A with field T/T in dimen- sionless units. Figure 15. Variation of the Inverse power-law potential V(T)/A with field T/T in dimen- where A characterize the scale of inflation and 7p is the parameter of the model. In Fig. (15) we have depicted the symmetric behavior of the Inverse power-law potential with respect to scaled field coordinate T/7o in dimensionless units. In this case the tachyon field started rolling down from the top hight of the potential situated at the origin and take part in inflationary dynamics. Next using specified form of the potential the potential dependent slow-roll pa-
Figure 16. In 16(a) the behaviour of the tensor-to-scalar ratio r with respect to the scalar spectral index n¢ and 16(b) the parameter g for Inverse power-law potential. The purple and blue coloured line represent the upper bound of tensor-to-scalar ratio allowed by Planck+ BICEP2+Keck Array joint constraint and only Planck 2015 data respectively. For both the figures red, green, brown, orange colored curve represent gq = 1/2, q=1, q = 3/2 and q = 2 respectively. The cyan color shaded region bounded by two vertical black coloured lines in 16(a) represent the Planck 2c allowed region and the rest of the light gray shaded region is showing the lo allowed range, which is at present disfavoured by the Planck data and Planck+ BICEP2+Keck Array joint constraint. In 16(a), we have explicitly shown that in r — ng plane the observationally disfavoured value of the characteristic index is q = 1/2. Variation of the scalar power spectrum A¢ vs scalar spectral index n¢ is shown in 16(b). The purple and blue coloured line represent the upper and lower bound allowed by WMAP-+Planck 2015 data respectively. The green dotted region bounded by two vertical black coloured lines represent the Planck 20 allowed region and the rest of the light gray shaded region is disfavoured by the Planck+WMAP constraint. In 16(b), we have explicitly shown that in A¢ — n¢ plane the observationally favoured value of the characteristic index is g= 1 and q = 3/2. From 16(a) and 16(b), it is also observed that, within 50 < N, < 70 the Inverse power-law potential is favoured for the characteristic index 3/2 < q < 1, by Planck 2015 data and Planck+ BICEP2+Keck Array joint analysis. Figure 16. In 16(a) the behaviour of the tensor-to-scalar ratio r with respect to the scalar
Figure 16. In 16(a) the behaviour of the tensor-to-scalar ratio r with respect to the scalar spectral index n¢ and 16(b) the parameter g for Inverse power-law potential. The purple and blue coloured line represent the upper bound of tensor-to-scalar ratio allowed by Planck+ BICEP2+Keck Array joint constraint and only Planck 2015 data respectively. For both the figures red, green, brown, orange colored curve represent gq = 1/2, q=1, q = 3/2 and q = 2 respectively. The cyan color shaded region bounded by two vertical black coloured lines in 16(a) represent the Planck 2c allowed region and the rest of the light gray shaded region is showing the lo allowed range, which is at present disfavoured by the Planck data and Planck+ BICEP2+Keck Array joint constraint. In 16(a), we have explicitly shown that in r — ng plane the observationally disfavoured value of the characteristic index is q = 1/2. Variation of the scalar power spectrum A¢ vs scalar spectral index n¢ is shown in 16(b). The purple and blue coloured line represent the upper and lower bound allowed by WMAP-+Planck 2015 data respectively. The green dotted region bounded by two vertical black coloured lines represent the Planck 20 allowed region and the rest of the light gray shaded region is disfavoured by the Planck+WMAP constraint. In 16(b), we have explicitly shown that in A¢ — n¢ plane the observationally favoured value of the characteristic index is g= 1 and q = 3/2. From 16(a) and 16(b), it is also observed that, within 50 < N, < 70 the Inverse power-law potential is favoured for the characteristic index 3/2 < q < 1, by Planck 2015 data and Planck+ BICEP2+Keck Array joint analysis. Figure 16. In 16(a) the behaviour of the tensor-to-scalar ratio r with respect to the scalar
where 7 is the arbitrary directional unit vector in CMB map. Here V;,,(7) are the spherical harmonics which are chosen to be the basis of the harmonic expansion of the temperature anisotropy and F and B polarization. Further using the inflationary power spectra at any momentum scale k:
where 7 is the arbitrary directional unit vector in CMB map. Here V;,,(7) are the spherical harmonics which are chosen to be the basis of the harmonic expansion of the temperature anisotropy and F and B polarization. Further using the inflationary power spectra at any momentum scale k:
where ‘ = d/dT; and the potential dependent slow-roll parameters ey, ny, &?-, 0} are defined as: which is exactly similar to the expression for the slow-roll parameter as appearing in the context of single field slow-roll inflationary models. However, for the sake of clarity here we introduce new sets of potential dependent slow-roll parameters for tachyonic inflation by rescaling with the appropriate powers of a’V(T):
where ‘ = d/dT; and the potential dependent slow-roll parameters ey, ny, &?-, 0} are defined as: which is exactly similar to the expression for the slow-roll parameter as appearing in the context of single field slow-roll inflationary models. However, for the sake of clarity here we introduce new sets of potential dependent slow-roll parameters for tachyonic inflation by rescaling with the appropriate powers of a’V(T):
where we use the following consistency conditions for re-scaled potential dependent slow-roll parameters:
where we use the following consistency conditions for re-scaled potential dependent slow-roll parameters:
where Tj ena characterizes the tachyonic field value at the end of inflation t = t.,q for all 7 fields participating in assisted inflation. As in the case of assisted inflation all the M fields are identical then one can re-express the number of e-foldings as: In terms of the slow-roll parameters, the number of e-foldings can be re-expressec the M fields are identical then one can re-express the number otf e-foldings as:
where Tj ena characterizes the tachyonic field value at the end of inflation t = t.,q for all 7 fields participating in assisted inflation. As in the case of assisted inflation all the M fields are identical then one can re-express the number of e-foldings as: In terms of the slow-roll parameters, the number of e-foldings can be re-expressec the M fields are identical then one can re-express the number otf e-foldings as:
Similarly after the variation of the tachyoinic field equation motion we get the fol- lowing expressions for the perturbed equation of motion:
Similarly after the variation of the tachyoinic field equation motion we get the fol- lowing expressions for the perturbed equation of motion:
Further we will perform the following steps through out the next part of the compu- big PS acy @
Further we will perform the following steps through out the next part of the compu- big PS acy @
e Next one can compute the running of the scalar spectral tilt (ag) of the pri- Hence using the result in Eq (5.408) we get the following simplified expression for the primordial scalar power spectrum:
e Next one can compute the running of the scalar spectral tilt (ag) of the pri- Hence using the result in Eq (5.408) we get the following simplified expression for the primordial scalar power spectrum:
In this subsection we will not derive the results for assisted inflation as it is exactly same as derived in the case of single tachyonic inflation. But we will state the results
In this subsection we will not derive the results for assisted inflation as it is exactly same as derived in the case of single tachyonic inflation. But we will state the results
which can be treated as another slow-roll parameter in the present context. One can also recast the slow-roll parameter S in terms of the inflationary observables as: 5. Next the running of the sound speed cg can be written in terms of slow-rol Naramatara aac:
which can be treated as another slow-roll parameter in the present context. One can also recast the slow-roll parameter S in terms of the inflationary observables as: 5. Next the running of the sound speed cg can be written in terms of slow-rol Naramatara aac:
2. Next using Eq (5.172) we can write the following integral equation: where the tensor-to-scalar ratio r is function of k or N.
2. Next using Eq (5.172) we can write the following integral equation: where the tensor-to-scalar ratio r is function of k or N.
4. For any value of qg including gq = 1/2 we need to compute the following integral:
4. For any value of qg including gq = 1/2 we need to compute the following integral:
the field excursion for AV can be re-written as:
the field excursion for AV can be re-written as:
Table 1. Comparison between the field excursion obtained from all the tachyonic poten- tials from single field and assisted inflationary framework. Here we define yo = To/Mp, where My, = 2.43 x 10'8 GeV. The numerics clearly implies that the assisted tachyonic inflationary framework push the field excursion value to sub-Planckian value for large M with large amount, compared to the value obtained from single inflationary setup. Tech- nically this implies the doing effective field theory (EFT) with assisted framework is more safer compared to single field case, as it involves an additional parameter M. Table 1. Comparison between the field excursion obtained from all the tachyonic poten-
Table 1. Comparison between the field excursion obtained from all the tachyonic poten- tials from single field and assisted inflationary framework. Here we define yo = To/Mp, where My, = 2.43 x 10'8 GeV. The numerics clearly implies that the assisted tachyonic inflationary framework push the field excursion value to sub-Planckian value for large M with large amount, compared to the value obtained from single inflationary setup. Tech- nically this implies the doing effective field theory (EFT) with assisted framework is more safer compared to single field case, as it involves an additional parameter M. Table 1. Comparison between the field excursion obtained from all the tachyonic poten-
However, for the sake of clarity here we introduce new sets of potential dependent slow-roll parameters for multi tachyonic inflation by rescaling with the appropriate
However, for the sake of clarity here we introduce new sets of potential dependent slow-roll parameters for multi tachyonic inflation by rescaling with the appropriate
Figure 26. Diagrammatic representation of trajectories and perturbation decompositions in the field space. Here it is explicitly shown that any perturbation including field perturba- tions can be decomposed to a curvature component “curve” and an isocurvature component “G30”,
Figure 26. Diagrammatic representation of trajectories and perturbation decompositions in the field space. Here it is explicitly shown that any perturbation including field perturba- tions can be decomposed to a curvature component “curve” and an isocurvature component “G30”,
When the potential is sum-separable, the derivatives of N can be simplified into — FFP. ee
When the potential is sum-separable, the derivatives of N can be simplified into — FFP. ee
where Y and © represent in general time-dependent dimensionless functions in cos- mological perturbation theory of multi-fields. Now further integrating Eqs. (5.539) over the specified time scale one can obtain the expression for the generalized form of the transfer matrix which relate the curvature and entropy perturbations gener- ated during the situation where a given fluctuating mode is stretched outside the expansion (Hubble) scale during the epoch of inflation (k = aH = k, = kemp) to the curvature and entropy perturbations at later time via the following simplified form of the matrix equation as:
where Y and © represent in general time-dependent dimensionless functions in cos- mological perturbation theory of multi-fields. Now further integrating Eqs. (5.539) over the specified time scale one can obtain the expression for the generalized form of the transfer matrix which relate the curvature and entropy perturbations gener- ated during the situation where a given fluctuating mode is stretched outside the expansion (Hubble) scale during the epoch of inflation (k = aH = k, = kemp) to the curvature and entropy perturbations at later time via the following simplified form of the matrix equation as:
dial power spectra at the beginning point of the radiation-dominated epoch as:
dial power spectra at the beginning point of the radiation-dominated epoch as:
Additionally, it is important to mention here that, although the overall amplitude the generalized transfer functions T, cs and Tss are dependent upon the evolution © Oo fter horizon crossing via reheating phenomena and into the radiation dominated poch, the spectral tilts of the resulting primordial perturbation spectra can be finally xpressed solely in terms of the slow-roll parameters at horizon crossing during the Oo Oo O poch of inflation and also in terms of the correlation angle 6. ‘urther using Eqs. (5.545-5.547), the spectral tilts of the primordial power spec- 7
Additionally, it is important to mention here that, although the overall amplitude the generalized transfer functions T, cs and Tss are dependent upon the evolution © Oo fter horizon crossing via reheating phenomena and into the radiation dominated poch, the spectral tilts of the resulting primordial perturbation spectra can be finally xpressed solely in terms of the slow-roll parameters at horizon crossing during the Oo Oo O poch of inflation and also in terms of the correlation angle 6. ‘urther using Eqs. (5.545-5.547), the spectral tilts of the primordial power spec- 7
Also the total contribution in the slow-roll is expressed through the reduced slow-roll parameters: potential dependent slow-roll parameters for 7 th species are computed as:
Also the total contribution in the slow-roll is expressed through the reduced slow-roll parameters: potential dependent slow-roll parameters for 7 th species are computed as:
Additionally here we introduce a new factor g; for each non identical 7 number of multi tachyonic fields, defined as:
Additionally here we introduce a new factor g; for each non identical 7 number of multi tachyonic fields, defined as:
Further using the condition to end inflation: Next we compute the number of e-foldings from this model:
Further using the condition to end inflation: Next we compute the number of e-foldings from this model:
In this paper, we have explored various cosmological consequences from GTachyonic field. We start with the basic introduction of tachyons in the context of non-BPS string theory, where we also introduce the GTachyon field, in presence of which the tachyon action is getting modified and one can quantify the amount of the mod- ification via a superscript g instead of 1/2. This modification exactly mimics the
In this paper, we have explored various cosmological consequences from GTachyonic field. We start with the basic introduction of tachyons in the context of non-BPS string theory, where we also introduce the GTachyon field, in presence of which the tachyon action is getting modified and one can quantify the amount of the mod- ification via a superscript g instead of 1/2. This modification exactly mimics the
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