Multi-Brid DBI Inflation

Abstract

It is shown that it is possible to extend the application of $\delta \mathcal{N}$ formalism to some special separable non-canonic cases. In this work, we extended the multi-brid idea to the multi-field separable model with a non-canonical kinetic term, mainly DBI (Dirac–Born–Infeld) action. Multi-brid stands for multi-component hybrid inflation which is based on δN formalism. To be more explicit, we considered the DBI model for two different limits, viz., speed limit and constant sound speed, as examples.

DEVELOPMENT OF \(\delta \mathcal{N}\) FORMALISM

In this section we review the approach of [37] to extend $\delta \mathcal{N}$ formalism to more general cases by using a super-potential formalism.

1.1 A.1. The Background

The Friedman equations are,

$${{H}^{2}} = \frac{{{{\kappa }^{2}}}}{3}(2{{\Sigma }_{I}}{{P}_{{I,{{X}_{I}}}}}{{X}_{I}} - {{P}_{I}}),$$

(A.1)

$$\dot {H} = - \frac{{{{\kappa }^{2}}}}{2}{{\Sigma }_{I}}{{P}_{{I,{{X}_{I}}}}}\mathop {\dot {\phi }}\nolimits_I ,$$

(A.2)

where ${{\kappa }^{2}} = 8\pi G$. We can define momentum as

$$\pi \equiv {{P}_{{I,{{X}_{I}}}}}\mathop {\dot {\phi }}\nolimits_I .$$

(A.3)

Since the Lagrangian is not singular, it is possible to obtain ${{\dot {\phi }}_{I}}$ as a function of ϕ_I and π_I, i.e.,

$$\mathop {\dot {\phi }}\nolimits_I = {{F}_{I}}({{\phi }_{I}},{{\pi }_{I}}).$$

(A.4)

In principle, it is possible to solve the equations of motion and gain π_I in terms of ϕ_I and initial momenta, c_I’s. Now we can replace ${{\dot {\phi }}_{I}}$ in energy density and express Hubble Parameter as below, c

$$H = 2W({{\phi }_{J}},{{c}_{J}}).$$

(A.5)

From (2) and (5) it is reasonable to conclude that

$${{\pi }_{I}} = - \frac{4}{{{{\kappa }^{2}}}}\frac{{\partial W}}{{\partial {{\phi }_{I}}}}.$$

(A.6)

Replacing in (1), gives a differential equation for W as,

$${{W}^{2}} = \frac{{{{\kappa }^{2}}}}{{12}}\rho \left[ {{{\phi }_{I}},\frac{{\partial W}}{{\partial {{\phi }_{I}}}}} \right].$$

(A.7)

W is called superpotential.

1.2 A.2. $\delta \mathcal{N}$ in Non-Slow-Roll Limit

The $\delta \mathcal{N}$ formalism presume the separate universe assumption. The separate universe approach states that when the physical scale of fluctuations, L, in comparison with the Hubble length, H^–1, is very big, i.e., $L \gg {{H}^{{ - 1}}}$, each region of Hubble size can be considered as a FRW universe; therefore each patch evolves independently. We define small parameter $\varepsilon$ as $\varepsilon \equiv \frac{1}{{LH}}$ and expand the equations in the order of that parameter. We employ ADM formalism,

$$\begin{gathered} d{{s}^{2}} = {{g}_{{\mu \nu }}}d{{x}^{\mu }}d{{x}^{\nu }} - \alpha d{{t}^{2}} \\ + \;{{\gamma }_{{ij}}}(d{{x}^{i}} + {{\beta }^{i}}dt)(d{{x}^{j}} + {{\beta }^{j}}dt), \\ \end{gathered} $$

(A.8)

where α and ${{\beta }^{i}}$ are lapse and shift vector, respectively. γ^ij is spatial metric which for later convenience decomposed as

$${{\gamma }_{{ij}}} = {{a}^{2}}{{e}^{{2\mathcal{R}\left( x \right)}}}{{[{{e}^{{h\left( x \right)}}}]}_{{ij}}},$$

(A.9)

${\text{tr}}[h] = 0$. As usual the extrinsic curvature is defined as below,

$$K = {{\nabla }_{\mu }}{{n}^{\mu }} = \frac{1}{{\sqrt { - g} }}{{\partial }_{\mu }}(\sqrt { - g} {{n}^{\mu }}),$$

(A.10)

the ${{n}^{\mu }}$ is unit tangent vector to time-like congruence orthogonal to t = const hyper-surfaces, n^μ = ${{\alpha }^{{ - 1}}}(1, - {{\beta }^{i}})$, replacing in (A.10) we obtain,

$$K = {{\alpha }^{{ - 1}}}[3(H + \dot {\mathcal{R}}) - {{D}_{i}}{{\beta }^{i}}].$$

(A.11)

The the expansion along these normal congruence can be interpreted as difference in number of e-folding, therefore we define the number of e-folds as below,

$${\text{N}} = \frac{1}{3}\int {\alpha Kdt} .$$

(A.12)

The derivative with respect to ${\text{N}}$ is defined as,

$${{\partial }_{{\text{N}}}} \equiv \frac{3}{{\alpha K}}{{\partial }_{t}}.$$

(A.13)

It is assumed that in the limit $\varepsilon \to 0$ we arrive at FRW. We choose the gauge in which

$${{\partial }^{i}}{{h}_{{ij}}} = 0.$$

(A.14)

In gradient expansion we have [35],

$${{\partial }_{j}}{{\beta }^{i}} = \mathcal{O}({{\varepsilon }^{2}}),$$

(A.15)

$${{\dot {h}}_{{ij}}} = \mathcal{O}({{\varepsilon }^{2}}),$$

(A.16)

and for scalar fields

$${{T}_{{ij}}} - \frac{1}{3}{{\gamma }^{{kl}}}{{T}_{{kl}}}{{\gamma }_{{ij}}} = \mathcal{O}({{\varepsilon }^{2}}).$$

(A.17)

Up to second order of ε, the Einstein equations and the field equations of the scalar fields read as [36],

$${{K}^{2}} = \frac{{{{\kappa }^{2}}}}{3}({{K}^{2}}{{P}_{{I,{{X}_{I}}}}}{{\partial }_{\mathcal{N}}}{{\varphi }_{I}}{{\partial }_{\mathcal{N}}}{{\varphi }_{I}} - 9P) + \mathcal{O}({{\varepsilon }^{2}}),$$

(A.18)

$${{\partial }_{{\text{N}}}}K = - \frac{{{{\kappa }^{2}}}}{2}K{{P}_{{I,{{X}_{I}}}}}{{\partial }_{{\text{N}}}}{{\varphi }_{I}}{{\partial }_{{\text{N}}}}{{\varphi }_{I}} + \mathcal{O}({{\varepsilon }^{2}}),$$

(A.19)

$$\begin{gathered} K{{\partial }_{{\text{N}}}}(K{{P}_{{I,{{X}_{I}}}}}{{\partial }_{{\text{N}}}}{{\phi }_{I}}) + 3{{K}^{2}}{{P}_{{I,{{X}_{I}}}}}\partial {\text{N}}{{\phi }_{I}} \\ - \;9{{P}_{{I,{{\phi }_{I}}}}} = \mathcal{O}({{\varepsilon }^{2}}), \\ \end{gathered} $$

(A.20)

and momentum constraint is as follows,

$${{\partial }_{i}}K = - \frac{{{{\kappa }^{2}}}}{2}K{{P}_{{I,{{X}_{I}}}}}{{\partial }_{{\text{N}}}}{{\phi }^{I}}{{\partial }_{i}}{{\phi }^{I}} + \mathcal{O}(a{{\varepsilon }^{3}}).$$

(A.21)

By taking the spatial derivative of (A.18) and replacing (A.19) and (A.20), we arrive at

$${{\partial }_{i}}K = - \frac{{{{\kappa }^{2}}}}{2}K{{P}_{{I,{{X}_{I}}}}}{{\partial }_{{\text{N}}}}{{\phi }_{I}}{{\partial }_{i}}{{\phi }_{I}} + {{B}_{i}} + \mathcal{O}(a{{\varepsilon }^{3}}),$$

(A.22)

where

$$\begin{gathered} {{B}_{i}} = \frac{{{{\kappa }^{2}}K}}{{2{{\partial }_{{\text{N}}}}\ln{{e}^{{3{\text{N}}}}}K}} \\ \times \,[{{\partial }_{{\text{N}}}}{{\phi }_{I}}{{\partial }_{i}}({{P}_{{I,{{X}_{I}}}}}{{\partial }_{{\text{N}}}}{{\phi }_{I}}) - {{\partial }_{{\text{N}}}}({{P}_{{I,{{X}_{I}}}}}{{\partial }_{{\text{N}}}}{{\phi }_{I}}){{\partial }_{i}}{{\phi }_{I}}]. \\ \end{gathered} $$

(A.23)

By comparing (A.21) and (A.22), it is obvious that for ensuring consistency between Hamilton and momentum constraints, we must have

$${{a}^{{ - 1}}}{{B}_{i}} = ({{\varepsilon }^{3}}).$$

(A.24)

In [37] it is shown that under attractor assumption, this condition is satisfied in more general case than slow-roll condition. With the same argument we said in the background, it is possible to expand the field equation in terms of superpotential as,

$${{P}_{{I,{{X}_{I}}}}}{{\partial }_{{\text{N}}}}{{\phi }^{I}} = - \frac{2}{{{{\kappa }^{2}}}}\frac{{\partial \ln W}}{{\partial {{\phi }^{I}}}} + \mathcal{O}(a{{\varepsilon }^{2}}),$$

(A.25)

where $K = 6W + \mathcal{O}(a{{\varepsilon }^{2}})$. In attractor regime, we can ignore the dependence of W on c_I and the leading term in (A.23) vanishes so the consistency condition is s-atisfied.

From the above discussion we have

$${{D}_{i}}{{\beta }^{i}} = \mathcal{O}({{\varepsilon }^{2}}),$$

(A.26)

so we can ignore the last term in (A.11). We define

$$\delta \mathcal{N}\left( {{{t}_{2}},{{t}_{1}};x} \right) \equiv {\text{N}}\left( {{{t}_{2}},{{t}_{1}};x} \right) - \mathcal{N}\left( {{{t}_{2}},{{t}_{1}}} \right),$$

(A.27)

in which $\mathcal{N}\left( {{{t}_{2}},{{t}_{1}}} \right)$ is e-folding number in unperturbed universe, replacing (A.11) in (A.12) and neglecting the terms of order $\mathcal{O}({{\varepsilon }^{2}})$ we arrive at

$$\delta \mathcal{N}\left( {{{t}_{2}},{{t}_{1}};x} \right) = \mathcal{R}\left( {{{t}_{2}},x} \right) - \mathcal{R}\left( {{{t}_{1}},x} \right).$$

(A.28)

At initial space-like hyper-surface ${{\Sigma }_{i}}$ the spatial curvature vanishes, i.e., $\mathcal{R}\left( {{{t}_{i}},x} \right) = 0$. We choose a constant energy density hyper-surface, Σ_f, as the final hyper-surface $\rho \left( {{{t}_{f}},x} \right) \equiv {{\rho }_{f}} = {\text{const}}$. By a appropriate choice of coordinate system, these two hyper-surfaces can be made to be constant t slices. We define the adiabatic curvature perturbation as,

$$\zeta \left( {{{t}_{f}},x} \right) \equiv \mathcal{R}\left( {{{t}_{f}},x} \right).$$

(A.29)

Setting ${{t}_{1}} = {{t}_{i}}$ and ${{t}_{2}} = {{t}_{f}}$ and using separate universe approach we arrive at the usual relation between curvature perturbation and $\delta \mathcal{N}$,

$$\zeta \left( {{{t}_{f}},x} \right) = \delta \mathcal{N}\left( {{{t}_{f}},{{t}_{i}};x} \right) \equiv \delta \mathcal{N}\left( {{{t}_{f}};\delta {{\phi }_{i}}\left( x \right)} \right),$$

(A.30)

in the above equation we assume that $\delta {{\phi }_{i}}(x) \equiv \delta \phi ({{t}_{i}},x)$ does not depend on momenta.