^a^ainstitutetext: Center for theoretical physics, College of Physics, Sichuan University,
Chengdu, 610064, China^b^binstitutetext: The Hong Kong University of Science and Technology,
Clear Water Bay, Kowloon, Hong Kong, P.R.China^c^cinstitutetext: School of Fundamental Physics and Mathematical Sciences,
Hangzhou Institute for Advanced Study, UCAS, Hangzhou 310024, China^d^dinstitutetext: Department of Physics and Chongqing Key Laboratory for Strongly Coupled Physics,
Chongqing University, Chongqing 401331, China

Elko as an inflaton candidate

Xinglong Chen [email protected] a Cheng-Yang Lee [email protected] b Yanjiao Ma [email protected] c Haomin Rao [email protected] b Wenqi Yu [email protected] d and Siyi Zhou [email protected]

Abstract

Elko is a spin-half fermion with a two-fold Wigner degeneracy and Klein-Gordon dynamics. In this paper, we show that in a spatially flat FLRW space-time, slow-roll inflation can be initiated by the homogeneous Elko fields. The inflaton is a composite scalar field obtained by contracting the spinor field with its dual. This is possible because the background evolution as described by the Friedmann equation is completely determined by the scalar field. This approach has the advantage that we do not need to specify the initial conditions for every component of the spinor fields. We derive the equation of motion for the inflaton and also show that this solution is an attractor.

1 Introduction

Inflation is a theory of rapid spatial expansion of the early universe. It is the leading paradigm describing the early universe cosmology because it resolves the flatness, horizon, and magnetic monopole problems Guth:1980zm ; Barrow:1981pa ; Linde:1983gd ; Linde:1981mu . The slow-roll inflation driven by a real scalar field is one of the simplest models. It predicts an almost scale-invariant power spectrum which is in agreement with observations on the cosmological background radiation Mukhanov:1990me . Heavy fields may be present during inflation. Their presence leads to interesting oscillating features in the squeezed limit of non-Gaussianities as is the case in the cosmological collider physics or quasi-single field inflation Chen:2010xka ; Chen:2009we ; Chen:2009zp ; Chen:2012ge ; Noumi:2012vr ; Arkani-Hamed:2015bza ; Meerburg:2016zdz ; Chen:2016uwp ; Tong:2017iat ; Tong:2018tqf ; Alexander:2019vtb ; Lu:2019tjj ; Hook:2019zxa ; Wang:2019gbi ; Wang:2020uic ; Wang:2020ioa ; Liu:2019fag ; Lu:2021wxu ; Tong:2022cdz ; Tong:2021wai . Nowadays, there are many inflationary models. Apart slow-roll scalar inflation Martin:2013tda , one may also think about inflation driven by spinor Kinney:1995xv ; Iso:2014gka ; Pereira:2017efk ; Kumar:2018rrl ; Benisty:2019jqz ; Shokri:2021aum ; Gredat:2008qf ; Shankaranarayanan:2009sz or vector fields Golovnev:2008cf ; Koivisto:2008xf ; Chiba:2008eh ; Koh:2009ne ; Setare:2013kja ; Darabi:2014aaa .

In this work, we study the theory of Elko inflation. Elko is a massive spin-half fermionic field in the $\left(\frac{1}{2},0\right)\oplus\left(0,\frac{1}{2}\right)$ representation with double Wigner degeneracy Ahluwalia:2004ab ; Ahluwalia:2004sz ; Ahluwalia:2008xi ; Ahluwalia:2009rh ; Ahluwalia:2010zn ; Lee:2012td ; Lee:2014opa ; Lee:2015sqj ; Ahluwalia:2016rwl ; Lee:2019fni ; Ahluwalia:2019etz ; Ahluwalia:2022ttu ; Ahluwalia:2022yvk ; Ahluwalia:2023slc . Double Wigner degeneracy, refers to the fact that the fermionic states belong to the irreducible unitary representation of the extended Poincaré group. These fermionic states have four degrees of freedom, two coming from spin projections and two coming from Wigner degeneracy $n=1,-1$ . The latter is defined by the discrete symmetry transformations where parity and time-reversal map a single state to a superposition of two states labeled by $n=1$ and $n=-1$ .¹¹1For the Standard Model fermions, parity and time-reversal transformations map a single state to itself with momentum and spin-projections altered. Choosing Elko to drive inflation has many advantages, for example, its status as a candidate for dark matter means it could not only drive inflation but may also explain dark matter after inflation ends.

The idea of Elko inflation is not new Boehmer:2006qq ; Boehmer:2007dh ; Boehmer:2008rz ; Boehmer:2008ah ; Boehmer:2009aw ; Boehmer:2010ma ; Basak:2012sn ; Sadjadi:2012xyd ; HoffdaSilva:2014tth ; Pereira:2014wta ; S:2014woy ; Pereira:2014pqa ; Chang:2015ufa ; Pereira:2016emd ; Pereira:2016eez ; Pereira:2017efk ; Basak_2015 . But prior to Ahluwalia:2022yvk ; Ahluwalia:2023slc , the correct degrees of freedom were not known. As a result, these models are non-local and not Lorentz covariant in Minkowski space-time. The realization that Elko are fermionic fields with double Wigner degeneracy resolves the problems of non-locality and Lorentz violation in Minkowski space-time. In de Sitter space-time, it is shown that Elko can be consistently quantized where the quantum fields satisfy the canonical anti-commutations relations Lee:2024sbg .

Here we show that Elko can act as an inflaton using the methods in Benisty:2019jqz . We define a composite scalar field $\phi\equiv\overset{\>{}^{{}^{{{\neg}}}}}{\smash[t]{\lambda}}\lambda$ where $\lambda$ is the Elko field and $\overset{\>{}^{{}^{{{\neg}}}}}{\smash[t]{\lambda}}$ is its adjoint. To the zeroth order in perturbation, we take $\lambda$ and $\overset{\>{}^{{}^{{{\neg}}}}}{\smash[t]{\lambda}}$ to be homogeneous so $\phi$ is also homogeneous. Their perturbations will be studied in future works. In the spatially flat FLRW space-time, all components of the Elko energy-momentum tensor can be expressed in terms of $\phi$ . Therefore, by solving the equation of motion for $\phi$ (which can be derived from the equations of motion for $\lambda$ and $\overset{\>{}^{{}^{{{\neg}}}}}{\smash[t]{\lambda}}$ ) and the Friedmann equations, we can determine the evolution of the Hubble parameter and the scale factor.

By choosing the appropriate parameters for the Elko potential and numerically solving for $\phi$ , we can obtain slow-roll inflation. Compared to the existing literature, the advantage of our approach is that the equation of motion for $\phi$ is a third-order differential equation so we only need to specify three initial conditions for $\phi$ , $\dot{\phi}$ and $\ddot{\phi}$ . ²²2If we solve the equations of motion for the spinor fields which are second-order differential equations, we would have to specify a total of sixteen initial conditions for $\lambda_{\ell},\overset{\>{}^{{}^{{{\neg}}}}}{\smash[t]{\lambda}}_{\ell},\dot{\lambda}_{\ell},\dot{\overset{\>{}^{{}^{{{\neg}}}}}{\smash[t]{\lambda}}_{\ell}}$ where $\ell=1,\cdots,4$ .

The paper is organized as follows. In sec. 2, we derive the Elko energy-momentum tensor in curved space-time. In sec. 3, we study in the spatially flat FLRW space-time. We derive the equations of motion of $\phi$ and express the Elko energy-momentum tensor in terms of $\phi$ . In sec. 4, we find that with the appropriate choice of parameters, Elko can act as an inflaton. Moreover, the inflation solution is also an attractor. We give a brief conclusion and outlook in sec. 5.

2 Set up

In Minkowski space-time, the equation of motion for Elko is the spinorial Klein-Gordon equation. In curved space-time, the equation of motion for Elko is

\displaystyle\gamma^{\mu}\nabla_{\mu}(\gamma^{\nu}\nabla_{\nu}\lambda)-\frac{\partial V}{\partial\overset{\>{}^{{}^{{}^{{{\neg}}}}}}{\smash[t]{\lambda}}}=0\,,

(1)

where $\lambda$ , $\overset{\>{}^{{}^{{{\neg}}}}}{\smash[t]{\lambda}}$ are the Elko field and its dual field and $V=V(\overset{\>{}^{{}^{{{\neg}}}}}{\smash[t]{\lambda}}\lambda)$ is the interacting potential. In this paper, we work within the framework of general relativity. Therefore, as explained in Lee:2024sbg , the complete Einstein-Elko action is

	$\displaystyle S$	$\displaystyle=S_{\text{GR}}+S_{\text{Elko}}$		(2)
		$\displaystyle=\int d^{4}x\sqrt{-g}\left[\frac{1}{2}M^{2}_{p}R-g^{\mu\nu}(\nabla_{\mu}\overset{\>{}^{{}^{{{\neg}}}}}{\smash[t]{\lambda}})(\nabla_{\nu}\lambda)-\frac{1}{4}R\overset{\>{}^{{}^{{{\neg}}}}}{\smash[t]{\lambda}}\lambda-V(\overset{\>{}^{{}^{{{\neg}}}}}{\smash[t]{\lambda}}\lambda)\right]\,,$		(3)

where $M_{p}$ is the reduced Planck mass, $g^{\mu\nu}$ is the metric and $R$ is the Ricci scalar. The quadratic term $\frac{1}{4}R\overset{\>{}^{{}^{{{\neg}}}}}{\smash[t]{\lambda}}\lambda$ is necessary to ensure that Elko satisfies the spinorial Klein-Gordon equation (1).

In the analysis of spinor fields in curved spacetime, it is convenient to use the tetrad formalism. The tetrads and the metric satisfy the following relationship

\displaystyle\begin{aligned} g_{\mu\nu}&=e^{a}_{~{}\mu}e^{b}_{~{}\nu}\eta_{ab}\,,\\ \eta_{ab}&=e_{a}^{~{}\mu}e_{b}^{~{}\nu}g_{\mu\nu}\,,\end{aligned}

(4)

where the local and global coordinates are labeled by the Latin ( $a,b,\cdots$ ) and Greek ( $\mu,\nu,\cdots$ ) alphabets respectively. Both the metric and the tetrad are compatible with the connection,

\nabla_{\sigma}g_{\mu\nu}=0\,,\quad\nabla_{\sigma}{e^{a}}_{\nu}=0\,.

(5)

The Dirac matrices satisfy the anti-commutation reation $\{\gamma^{\mu},\gamma^{\nu}\}=2g^{\mu\nu}\mathbb{I}_{4}$ and their indices can be raised and lowered by the tetrad

\displaystyle\gamma^{\mu}=e_{a}^{~{}\mu}\gamma^{a}\,,\quad\gamma_{\mu}=e^{b}_{~{}\mu}\gamma_{b}\,.

(6)

The result of the covariant derivative acting on Elko and its adjoint are

	$\displaystyle\nabla_{\mu}\overset{\>{}^{{}^{{{\neg}}}}}{\smash[t]{\lambda}}$	$\displaystyle=\partial_{\mu}\overset{\>{}^{{}^{{{\neg}}}}}{\smash[t]{\lambda}}-\overset{\>{}^{{}^{{{\neg}}}}}{\smash[t]{\lambda}}\Gamma_{\mu}\,,$		(7)
	$\displaystyle\nabla_{\mu}\lambda$	$\displaystyle=\partial_{\mu}\lambda+\Gamma_{\mu}\lambda\,,$		(8)

where

\Gamma_{\mu}=\frac{i}{2}{\omega_{\mu}}^{ab}\Sigma_{ab}\,,

(9)

and

\Sigma^{ab}=-\frac{i}{4}[\gamma^{a},\gamma^{b}]

(10)

is the Lorentz generator of the spinor field, and ${\omega_{\mu}}^{ab}$ is the spin connection which has the following relationship with the Christoffel symbol $\Gamma^{\nu}_{~{}\mu\sigma}$

{\omega_{\mu}}^{ab}={e^{a}}_{\nu}\left(\partial_{\mu}e^{b\nu}+e^{b\sigma}\Gamma^{\nu}_{~{}\mu\sigma}\right)\,.

(11)

We now derive the Elko energy-momentum tensor using $S_{\text{Elko}}$ . Firstly, we vary the tetrad $\delta e^{a}{}_{\mu}$ and the spin connections $\delta\omega_{\mu ab}$ independently

\displaystyle\delta S_{\text{Elko}}=\int d^{4}x\sqrt{-g}\,\left(\sigma^{\mu\nu}e_{a\mu}\delta e^{a}_{~{}\nu}+\tau^{ab\mu}\delta\omega_{\mu ab}\right)\,,

(12)

where

	$\displaystyle\sigma^{\mu\nu}=$	$\displaystyle 2\nabla^{(\mu}\overset{\>{}^{{}^{{{\neg}}}}}{\smash[t]{\lambda}}\nabla^{\nu)}\lambda-g^{\mu\nu}\left[(\nabla^{\gamma}\overset{\>{}^{{}^{{{\neg}}}}}{\smash[t]{\lambda}})(\nabla_{\gamma}\lambda)+\frac{1}{4}R\overset{\>{}^{{}^{{{\neg}}}}}{\smash[t]{\lambda}}\lambda+V(\overset{\>{}^{{}^{{{\neg}}}}}{\smash[t]{\lambda}}\lambda)\right]$
		$\displaystyle+\frac{1}{2}\left[R^{\mu\nu}\overset{\>{}^{{}^{{{\neg}}}}}{\smash[t]{\lambda}}\lambda+(g^{\mu\nu}\nabla^{\sigma}\nabla_{\sigma}-\nabla^{\mu}\nabla^{\nu})\overset{\>{}^{{}^{{{\neg}}}}}{\smash[t]{\lambda}}\lambda\right]\,,$		(13)

and

\displaystyle\tau^{ab\mu}

\displaystyle=\frac{i}{2}\left(\overset{\>{}^{{}^{{{\neg}}}}}{\smash[t]{\lambda}}\Sigma^{ab}\nabla^{\mu}\lambda-\nabla^{\mu}\overset{\>{}^{{}^{{{\neg}}}}}{\smash[t]{\lambda}}\Sigma^{ab}\lambda\right)\,.

(14)

The parenthesis on the indices is defined as

T_{(\mu_{1}\mu_{2}\cdot\cdot\cdot\mu_{n})\rho}^{\sigma}=\frac{1}{n!}\left[T_{\mu_{1}\mu_{2}\cdot\cdot\cdot\mu_{n}\rho}^{\sigma}+\left(\text{permutations of }\mu_{1}\mu_{2}\cdot\cdot\cdot\mu_{n}\right)\right]\,.

(15)

In the above derivation for (13), we have used

	$\displaystyle\delta R$	$\displaystyle=-R^{\mu\nu}\delta g_{\mu\nu}+(\nabla^{\mu}\nabla^{\nu}-g^{\mu\nu}\nabla^{\sigma}\nabla_{\sigma})\delta g_{\mu\nu}$
		$\displaystyle=-2R^{\mu\nu}e_{a(\mu}\delta e^{a}{}_{\nu)}+2\left(\nabla^{\mu}\nabla^{\nu}-g^{\mu\nu}\nabla^{\sigma}\nabla_{\sigma}\right)\left[e_{a(\mu}\delta e^{a}{}_{\nu)}\right],$		(16)

and

\delta g_{\mu\nu}=2e_{a(\mu}\delta e^{a}{}_{\nu)}\,.

(17)

Since the tetrad and spin connection are not independent but satisfy (11), the variation of the spin connection can be expressed as the variation of the tetrad

\tau^{ab\mu}\delta\omega_{\mu ab}=-(\tau^{\mu\nu\rho}+2\tau^{\rho(\mu\nu)})\nabla_{\rho}(e_{a\mu}\delta e^{a}_{~{}\nu})\,.

(18)

Substituting (18) into (12), we obtain

\delta S_{\text{Elko}}=\int d^{4}x\sqrt{-g}\left[\sigma^{\mu\nu}+\nabla_{\rho}(\tau^{\mu\nu\rho}+2\tau^{\rho(\mu\nu)})\right]e_{\alpha\mu}\delta e^{\alpha}_{~{}\nu}\,.

(19)

Finally using (17), the Elko energy-momentum tensor is given by

$\displaystyle T^{\mu\nu}$	$\displaystyle=\frac{1}{\sqrt{-g}}\frac{\delta S_{\text{Elko}}}{e_{a(\mu}\delta e^{a}{}_{\nu)}}$
	$\displaystyle=\sigma^{(\mu\nu)}+\nabla_{\rho}\left[\tau^{(\mu\nu)\rho}+2\tau^{\rho(\mu\nu)}\right]$
	$\displaystyle=2\nabla^{(\mu}\overset{\>{}^{{}^{{{\neg}}}}}{\smash[t]{\lambda}}\nabla^{\nu)}\lambda-g^{\mu\nu}\left[(\nabla^{\gamma}\overset{\>{}^{{}^{{{\neg}}}}}{\smash[t]{\lambda}})(\nabla_{\gamma}\lambda)+\frac{1}{4}R\overset{\>{}^{{}^{{{\neg}}}}}{\smash[t]{\lambda}}\lambda+V(\overset{\>{}^{{}^{{{\neg}}}}}{\smash[t]{\lambda}}\lambda)\right]+i\nabla_{\sigma}\left[\overset{\>{}^{{}^{{{\neg}}}}}{\smash[t]{\lambda}}\Sigma^{\sigma(\mu}\nabla^{\nu)}\lambda+\nabla^{(\mu}\overset{\>{}^{{}^{{{\neg}}}}}{\smash[t]{\lambda}}\Sigma^{\nu)\sigma}\lambda\right]$
	$\displaystyle~{}~{}~{}+\frac{1}{2}\left[R^{\mu\nu}\overset{\>{}^{{}^{{{\neg}}}}}{\smash[t]{\lambda}}\lambda+(g^{\mu\nu}\nabla^{\sigma}\nabla_{\sigma}-\nabla^{\mu}\nabla^{\nu})\overset{\>{}^{{}^{{{\neg}}}}}{\smash[t]{\lambda}}\lambda\right]\,.$	(20)

3 Elko in the spatially flat FLRW space-time

We now study Elko in the spatially flat FLRW space-time with the metric

ds^{2}=-dt^{2}+a^{2}(t)\delta_{ij}dx^{i}dx^{j}\,,

(21)

where $a$ is the scale factor. Here, we only consider the zeroth order in perturbation for the Elko field and its dual so they are homogeneous $\lambda=\lambda(t)$ , $\overset{\>{}^{{}^{{{\neg}}}}}{\smash[t]{\lambda}}=\overset{\>{}^{{}^{{{\neg}}}}}{\smash[t]{\lambda}}(t)$ . By solving the equations of motion for Elko and the Friedmann equations, we can determine the background evolution. They are given by

	$\displaystyle\ddot{\overset{\>{}^{{}^{{}^{{{\neg}}}}}}{\smash[t]{\lambda}}}+3H\dot{\overset{\>{}^{{}^{{}^{{{\neg}}}}}}{\smash[t]{\lambda}}}+\left(\frac{3}{2}\dot{H}+\frac{9}{4}H^{2}\right)\overset{\>{}^{{}^{{{\neg}}}}}{\smash[t]{\lambda}}+\frac{dV}{d\lambda}$	$\displaystyle=0\,,$		(22)
	$\displaystyle\ddot{\lambda}+3H\dot{\lambda}+\left(\frac{3}{2}\dot{H}+\frac{9}{4}H^{2}\right)\lambda+\frac{dV}{d\overset{\>{}^{{}^{{}^{{{\neg}}}}}}{\smash[t]{\lambda}}}$	$\displaystyle=0\,,$		(23)

and

	$\displaystyle 3H^{2}=M^{-2}_{p}\rho\,,$		(24)
	$\displaystyle 2\dot{H}+3H^{2}=-M^{-2}_{p}p\,,$		(25)

where $\dot{f}\equiv df/dt$ and $H=\dot{a}/a$ is the Hubble parameter. In the Friedmann equations, $\rho$ and $p$ are the energy density and pressure of Elko which can be obtained from (2)

T_{00}=\rho\,,~{}~{}T_{0i}=0\,,~{}~{}T_{ij}=a^{2}p\delta_{ij}\,,

(26)

where

	$\displaystyle\rho=\dot{\overset{\>{}^{{}^{{}^{{{\neg}}}}}}{\smash[t]{\lambda}}}\dot{\lambda}+\frac{3}{2}H(\dot{\overset{\>{}^{{}^{{}^{{{\neg}}}}}}{\smash[t]{\lambda}}}\lambda+\overset{\>{}^{{}^{{{\neg}}}}}{\smash[t]{\lambda}}\dot{\lambda})+\frac{9}{4}H^{2}\overset{\>{}^{{}^{{{\neg}}}}}{\smash[t]{\lambda}}\lambda+V\,,$		(27)
	$\displaystyle p=-V+\frac{1}{2}\left(\frac{dV}{d\lambda}\lambda+\overset{\>{}^{{}^{{{\neg}}}}}{\smash[t]{\lambda}}\frac{dV}{d\overset{\>{}^{{}^{{}^{{{\neg}}}}}}{\smash[t]{\lambda}}}\right)\,.$		(28)

They satisfy the continuity equation

\dot{\rho}+3H(\rho+p)=0\,.

(29)

These equations are difficult to solve directly for the following reasons. Both $\lambda$ and $\overset{\>{}^{{}^{{{\neg}}}}}{\smash[t]{\lambda}}$ are four-component spinor fields so (22-23) constitute eight second-order differential equations. To solve them, we need to specify sixteen initial conditions for $\lambda_{\ell},\overset{\>{}^{{}^{{{\neg}}}}}{\smash[t]{\lambda}}_{\ell},\dot{\lambda}_{\ell},\dot{\overset{\>{}^{{}^{{}^{{{\neg}}}}}}{\smash[t]{\lambda}}}_{\ell}$ where $\ell=1,\cdots,4$ . It is tempting to assume that all components of $\lambda_{\ell}$ , $\overset{\>{}^{{}^{{{\neg}}}}}{\smash[t]{\lambda}}_{\ell}$ have the same initial conditions and the same for their derivatives. In this case, the number of initial conditions would be reduced to two. However, as far as we can see, there does not seem to be a good reason a priori for this assumption.

The main result of this paper is that we can study the background evolution without having to solve for $\lambda$ and $\overset{\>{}^{{}^{{{\neg}}}}}{\smash[t]{\lambda}}$ . Instead, it suffices for us to consider the dynamics of the composite scalar field

\phi\equiv\overset{\>{}^{{}^{{{\neg}}}}}{\smash[t]{\lambda}}\lambda\,.

(30)

The equation of motion for $\phi$ is derived by combining (22-23) to yield

\dddot{\phi}+9H\ddot{\phi}+(9\dot{H}+27H^{2}+4V_{\phi}+2V_{\phi\phi}\phi)\dot{\phi}+(3\ddot{H}+27H\dot{H}+27H^{3}+12HV_{\phi})\phi=0\,,

(31)

where

V_{\phi}\equiv\frac{dV}{d\phi},\quad V_{\phi\phi}\equiv\frac{d^{2}V}{d\phi^{2}}.

(32)

Using the equations of motion for $\lambda$ and $\overset{\>{}^{{}^{{{\neg}}}}}{\smash[t]{\lambda}}$ , $\rho$ and $p$ can be expressed in terms of $\phi$

	$\displaystyle\rho$	$\displaystyle=\frac{1}{2}\ddot{\phi}+3H\dot{\phi}+\left(\frac{3}{2}\dot{H}+\frac{9}{2}H^{2}+V_{\phi}\right)\phi+V\,,$		(33)
	$\displaystyle p$	$\displaystyle=-V+V_{\phi}\phi\,.$		(34)

Therefore, the background evolution is completely determined by the scalar field $\phi$ that satisfies (31).

4 Elko inflation is an attractor solution

We now solve the equation of motion for $\phi$ and the Friedmann equations given in the previous section. We take the interacting potential $V$ to be a quadratic polynomial of $\phi$ so that it is renormalizable. Therefore,

V=v_{0}+v_{1}\phi+v_{2}\phi^{2}\,.

(35)

The coefficient $v_{1}$ can be a constant or it can also be proportional to the Ricci curvature $R$ , corresponding to non-minimal coupling. As far as we are concerned, it is acceptable to take $v_{1}$ to be constant because during slow-roll inflation, $R$ is approximately constant. The last term of (35) is a quartic self-interaction of $\lambda$ so $v_{2}$ is dimensionless. Substituting (35) into (28), we obtain

p=-v_{0}+v_{2}\phi^{2}\,.

(36)

It is worthwhile to note that if $V=v_{1}\phi$ , then $p=0$ so Elko behaves like dust.

Substituting (33-34) into (24-25), we obtain two solutions for $H$

H_{\pm}=\frac{-6\dot{\phi}\pm A}{9\phi-12M_{p}^{2}}\,,

(37)

where

A=\sqrt{36\dot{\phi}^{2}-(9\phi-12M_{p}^{2})\left[\frac{3(V\phi-V_{\phi}\phi^{2})}{M_{p}^{2}}+4(V_{\phi}\phi+V)+2\ddot{\phi}\right]}\,,

(38)

and two solutions for $\dot{H}$

\dot{H}_{\pm}=\frac{4V+\ddot{\phi}}{(4M_{p}^{2}-3\phi)}-\frac{A^{2}\mp 6\dot{\phi}A}{3(4M_{p}^{2}-3\phi)^{2}}\,.

(39)

To ensure that $H$ is positive and $\dot{H}$ is negative, we need $0<\phi<\frac{4}{3}M_{p}^{2}$ (the simplest solution is $V=v_{0}$ ), so the solution $H_{+}$ can be ignored. From now onwards $H_{-}\equiv H$ .

Substituting (37) into (31) with

v_{0}=M_{p}^{4}\,,\quad v_{1}=-2M_{p}^{2}\,,\quad v_{2}=1\,,

(40)

we obtain the potential given in fig. 1. In this case, the accelerated expansion occurs when the potential energy of the Elko field, $V$ is the dominant contribution to $\rho$ .

Refer to caption — Figure 1: The potential energy $V$ with $v_{0}=M_{p}^{4},v_{1}=-2M_{p}^{2}$ , and $v_{2}=1$ , where $V|_{\phi=M_{p}^{2}}=0$ . The $\phi$ axis is in units of $M^{2}_{p}$ .

Using numerical methods, substituting (37) and (39) into (31), with the initial conditions

\phi(0)\sim 0^{+}\,,\quad\dot{\phi}(0)\sim 0^{+}\,,\quad\ddot{\phi}(0)=0\,,

(41)

we are able to numerically solve the differential equation (31) to obtain the solution as shown in fig. 2, which results in slow-roll inflation. The number of e-folds depends on the initial condition of $\phi$ . Specifically, the closer $\phi(0)$ is to $0$ , the larger the number of e-folds. In general, we find that slow-roll inflation occurs when the coefficients satisfy

v_{0}\geq 0\,,\quad v_{1}\leqslant-2v_{0}\,,\quad v_{2}=\frac{v_{1}^{2}}{4v_{0}}\,.

(42)

Next, we consider the two Hubble slow-roll parameters:

\epsilon=-\frac{\dot{H}}{H^{2}}\,,~{}\eta=\frac{d\epsilon}{dN}\,,

(43)

where $N$ is the number of e-folds. When $\epsilon<1$ , the accelerated expansion starts, and it will continue if the change of $\epsilon$ per e-fold is small, that is $\eta<1$ . Substituting (37) and (39) into (43), the slow-roll parameters become

$\displaystyle\epsilon$	$\displaystyle=\frac{-9\phi_{0}(4V+\ddot{\phi})-3(A^{2}+6A\dot{\phi})}{(6\dot{\phi}+A)^{2}}\,,$	(44)
$\displaystyle\eta$	$\displaystyle=\frac{\dot{\epsilon}}{H}=\frac{18Z^{2}}{(6\dot{\phi}+A)^{4}}-\frac{27\phi_{0}^{3}\dot{\phi}\phi V_{,\phi\phi}}{2M_{p}^{2}(6\dot{\phi}+A)^{3}}-\frac{9Z}{(6\dot{\phi}+A)^{2}}\,,$	(45)
$\displaystyle Z$	$\displaystyle=3\phi_{0}(4V+\ddot{\phi})+6A\dot{\phi}+AM_{p}^{3}\,,$	(46)

where $\phi_{0}=3\phi-4M_{p}^{2}$ . In the limit $\dot{\phi},\ddot{\phi}\rightarrow 0$ , we find $|\eta|\sim 0$ . Using numerical methods, we find that when $N<50$ , $|\epsilon|,|\eta|\sim 0$ , which can be seen from fig. 3.

Finally, we have solved the equation of motion for $\phi$ and confirmed that the solution is an attractor as shown in fig. 4. When the initial conditions for $\phi(0)$ , $\dot{\phi}(0)$ , $\ddot{\phi}(0)$ are within a certain range, the final solution is an attractor.

5 Conclusion and outlook

In this paper, we have proposed an inflationary model using the Elko fields. The inflaton is a composite scalar field given by $\phi\equiv\overset{\>{}^{{}^{{{\neg}}}}}{\smash[t]{\lambda}}\lambda$ where $\lambda$ is the Elko field and $\overset{\>{}^{{}^{{{\neg}}}}}{\smash[t]{\lambda}}$ its dual. The equation of motion of the scalar field is a third-order differential equation, which is novel from any existing inflationary models we know.

We solved the third-order differential equation numerically and found the background evolution expressed in terms of $\phi$ , $\dot{\phi}$ and $\ddot{\phi}$ . From the Poincaré section, we find that the solution describing slow-roll inflation is an attractor. There are many more interesting open problems worthy of subsequent studies. They are listed below.

It is worthwhile to investigate whether the perturbation of $\phi$ can act as a curvaton field Lyth:2001nq ; Lyth:2002my ; Lyth_2003 ; Bartolo:2002vf ; Sasaki:2006kq . This is a crucial step in the efforts of trying to understand the cosmological observables of this model. As many curvaton models suggest, they can resolve the preheating problem Bastero-Gil:2003usx ; Sainio:2012rp . With the novel equation of motion found in this work, our model of Elko inflation may predict unique features that can be tested in future experiments. However, this is a challenging task because it remains to be known whether the equations of motion for $\lambda$ and $\overset{\>{}^{{}^{{{\neg}}}}}{\smash[t]{\lambda}}$ can be combined into a single equation of motion for the perturbation of $\phi$ . This is an important question for further study.

The model proposed in this work is a model of slow-roll inflation in a quasi-dS space. However, inflation need not be limited to a quasi-de-sitter background. There are also scenarios such as a power-law inflation Galtsov:1997ub ; Tsujikawa:2000wc ; Feinstein:2002aj ; Glavan:2020zne , where the scale factor is a power-law function of the physical time. There are also alternatives to inflation scenarios, such as matter bounce Brandenberger:2016vhg ; Agrawal:2021rur ; Agrawal:2022vdg ; Lohakare:2022umj ; Agrawal:2022ppe , ekpyrotic Nojiri:2022xdo ; Paul:2022mup , and many others Gu:2021qwy ; Zhu:2015xsa ; Zhu:2015owa ; Basak:2014qea . It is tempting to think about whether Elko can be responsible for other types of cosmic evolution in our universe.

Finally, as previous works have pointed out, Elko may describe the whole cosmic history Pereira:2017efk ; Pereira:2017bvq ; Pereira:2018hir . Given the problem of Hubble tension DiValentino:2021izs ; Perivolaropoulos:2021jda ; Botke2023 ; Sakstein:2019fmf ; Niedermann:2020dwg ; Dainotti:2023yrk . There are strong motivations to go beyond the $\Lambda$ CDM model. This motivates us to investigate Elko as an alternative theory to dark energy Arun:2017uaw ; Meng:2012zza ; Wang:2024rus which may resolve the Hubble tension problem.

6 Acknowledgement

This work was supported in part by the Natural Science Foundation of China under Grant No.12347101.

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