Perturbations of Mimetic Curvaton

The generalized multi-field mimetic inflation model will have the action form as the following Chamseddine et al. (2014); Zhang et al. (2023):

	$\displaystyle S$	$\displaystyle=\int d^{4}x\sqrt{-g}\bigg{[}\frac{M_{\text{P}}^{2}}{2}R$
		$\displaystyle-\frac{1}{2}\lambda\left(G_{ab}g^{\mu\nu}\partial_{\mu}\phi^{a}\partial_{\nu}\phi^{b}+\Omega^{2}\right)-V(\phi^{a},\lambda)\bigg{]},$		(2)

where $\lambda$ is a Lagrangian multiplier, $G_{ab}\equiv G_{ab}(\phi^{a})$ is the metric of the field space, and $\Omega$ is a constant corresponding to the typical energy scale of $\phi^{a}$. Since now we consider $\lambda$ to be an auxiliary field as well, we extend the potential to contain also $\lambda$. In FLRW metric one can obtain the Friedmann equations,

	$\displaystyle 3M_{\text{P}}^{2}H^{2}$	$\displaystyle=$	$\displaystyle V+\lambda\left(\frac{1}{2}\dot{\phi}_{a}\dot{\phi}^{a}+\frac{1}{2}\Omega^{2}\right)~{},\text{and}$		(3)
	$\displaystyle 2M_{\text{P}}^{2}\dot{H}$	$\displaystyle=$	$\displaystyle-\lambda(\dot{\phi}_{a}\dot{\phi}^{a})~{},$		(4)

where $H\equiv\dot{a}/a$ is Hubble parameter, and $\dot{\phi}_{a}\dot{\phi}^{a}\equiv G_{ab}\dot{\phi}^{a}\dot{\phi}^{b}$. Varying the action with respect to $\lambda$, one can get the constraint equation:

$\displaystyle\frac{1}{2}\dot{\phi}_{a}\dot{\phi}^{a}-\frac{1}{2}\Omega^{2}-V_{\lambda}=0~{},$

(5)

where $V_{\lambda}\equiv dV/d\lambda$. This makes the Friedmann equations (3) become:

	$\displaystyle 3M_{\text{P}}^{2}H^{2}$	$\displaystyle=$	$\displaystyle V+\lambda\left(V_{\lambda}+\Omega^{2}\right)~{},\text{and}$		(6)
	$\displaystyle 2M_{\text{P}}^{2}\dot{H}$	$\displaystyle=$	$\displaystyle-\lambda(2V_{\lambda}+\Omega^{2})~{}.$		(7)

Note that when the potential does not depend on $\lambda$, the Friedmann equations reduce to the normal one in Chamseddine et al. (2014); Zhang et al. (2023). It is also straightforward to get the equations of motion for the mimetic fields $\phi^{a}$:

$\displaystyle\tilde{D}_{t}\dot{\phi}^{a}+\left(3H+\frac{\dot{\lambda}}{\lambda}\right)\dot{\phi}^{a}+\frac{1}{\lambda}V^{,a}=0,$

(8)

where $\tilde{D}_{t}X^{a}\equiv\dot{X}^{a}+\tilde{\Gamma}^{a}_{bc}\dot{\phi}^{b}X^{c}$ with $\tilde{\Gamma}^{a}_{bc}$ is Christoffel connection of the field space and $V^{,a}\equiv G^{ab}\partial V(\phi^{c})/\partial\phi^{b}$.

The continuous equation gives the evolution of Lagrangian multiplier $\lambda$, which is

$\displaystyle(\Omega^{2}+V_{\lambda})\dot{\lambda}+6H\lambda(\frac{1}{2}\Omega^{2}+V_{\lambda})+\lambda\dot{V}_{\lambda}+\dot{V}=0~{}.$

(9)

Due to the involvement of $V_{\lambda}$, the solution of the above equation is not so obvious. However, we can still get some ansatz solutions by requiring that the system drives an inflationary period, with a matter-dominant epoch following. In the inflationary period where generally the slow-roll condition must be satisfied, it is natural to require a nearly flat potential, and $V\gg\lambda$. In this case, one has $V_{\lambda}\simeq 0$. On the other hand, in the matter-dominant epoch, one requirement of the solution is to have $V\ll\lambda\propto a^{-3}$, while $V_{\lambda}$ is nearly a constant. For the first condition, $\lambda\propto a^{-3}$, Eq. (9) is derived as:

$\displaystyle\lambda\dot{V}_{\lambda}+V_{\phi^{a}}\dot{\phi}^{a}=0~{},$

(10)

with $\dot{V}=V_{\lambda}\dot{\lambda}+V_{\phi^{a}}\dot{\phi}^{a}$ and $\dot{\lambda}=-3H\lambda$. For the second condition, $V_{\lambda}$ is a constant, we can require that the potential linearly depends on $\lambda$, which further gives constraints on the dependence of the potential on the fields: $V_{\phi^{a}}\dot{\phi}^{a}=0$. This new constraint should be taken in our following model building.

Next we study the cosmological perturbations of the system. We perturb the mimetic fields as

$\displaystyle\phi^{a}=\phi^{a}(t)+\delta\phi^{a}(t,x)~{},~{}\lambda=\lambda_{0}(t)+\delta\lambda(t,x)~{},$

(11)

and the metric in the co-moving spatial flat gauge,

$\displaystyle ds^{2}=(-N^{2}+N_{i}N^{i})dt^{2}+2N_{i}dtdx^{i}+h_{ij}dx^{i}dx^{j},$

(12)

where $N=1+n$ is the lapse function, $N_{i}=\partial_{i}B$ is the shift function and $h_{ij}$ is the metric of the spatial hyper-surface. The second order perturbed action is:

	$\displaystyle S^{(2)}$	$\displaystyle=\int d^{4}x\bigg{(}-\frac{1}{2}a^{3}\lambda\partial_{\mu}\delta\phi_{a}\partial^{\mu}\delta\phi^{a}-\frac{1}{2}\mathcal{M}^{2}_{\ ab}\delta\phi^{a}\delta\phi^{b}$
		$\displaystyle+a^{3}\delta\lambda\bigg{(}\dot{\phi}_{a}\delta\dot{\phi}^{a}-\frac{1}{2}\epsilon H\delta\phi_{a}\dot{\phi}^{a}+\frac{1}{2}G_{bc,a}\dot{\phi}^{b}\dot{\phi}^{c}\delta\phi^{a}$
		$\displaystyle+\frac{1}{2}V_{,a\lambda}\delta\phi^{a}\bigg{)}\bigg{)},$		(13)

where the mass matrix

	$\displaystyle\mathcal{M}^{2}_{\ ab}$	$\displaystyle=a^{3}\bigg{(}\frac{1}{2}V\epsilon^{2}H^{2}G_{ac}G_{bd}\dot{\phi}^{c}\dot{\phi}^{d}-\frac{1}{2}\lambda G_{cd,ab}\dot{\phi}^{c}\dot{\phi}^{d}$
		$\displaystyle+\frac{1}{2}\lambda\epsilon HG_{cd,a}G_{be}\dot{\phi}^{c}\dot{\phi}^{d}\dot{\phi}^{e}+\epsilon HV_{,a}G_{bc}\dot{\phi}^{c}+V_{,ab}\bigg{)}$
		$\displaystyle-\frac{d}{dt}\bigg{(}\frac{1}{2}a^{3}\lambda\epsilon HG_{ac}G_{bd}\dot{\phi}^{c}\dot{\phi}^{d}-a^{3}\lambda G_{bc,a}\dot{\phi}^{c}\bigg{)}~{},$		(14)

and $\epsilon\equiv-\dot{H}/H^{2}$ is the slow-roll paramter. Variation with respect to $\delta\lambda$ gives a constraint on perturbations:

	$\displaystyle\mathcal{Z}_{\delta\lambda}$	$\displaystyle=\dot{\phi}_{a}\delta\dot{\phi}^{a}-\frac{1}{2}\epsilon H\dot{\phi}_{a}\delta\phi^{a}+\frac{1}{2}G_{bc,a}\dot{\phi}^{b}\dot{\phi}^{c}\delta\phi^{a}+\frac{1}{2}V_{,a\lambda}\delta\phi^{a}$
		$\displaystyle=0,$		(15)

which is the perturbed mimetic constraint (5). The entire Hamiltonian analysis can been seen in Mansoori et al. (2022).

The equations of motion for the field-perturbations are:

	$\displaystyle\delta\ddot{\phi}^{a}-\partial_{i}\partial^{i}\delta\phi^{a}+\bigg{(}3H+\frac{\dot{\lambda}}{\lambda}\bigg{)}\delta\dot{\phi}^{a}+G^{a}_{\ b,c}\dot{\phi}^{c}\delta\dot{\phi}^{b}$
	$\displaystyle+\frac{1}{a^{3}\lambda}\mathcal{M}^{2a}_{\ \ b}\delta\phi^{b}+\dot{\phi}^{a}\frac{\delta\dot{\lambda}}{\lambda}$
	$\displaystyle+\bigg{(}\frac{1}{2}\epsilon H\dot{\phi}^{a}-\frac{1}{2}V^{,a}_{\lambda}-\frac{1}{\lambda}V^{,a}-\frac{\dot{\lambda}}{\lambda}\dot{\phi}^{a}\bigg{)}\frac{\delta\lambda}{\lambda}=0~{}.$		(16)

Substitute Eq. (16) into the time derivative of the perturbed constraint (15), one can obtain the equation of motion of $\delta\lambda$ as:

	$\displaystyle\dot{\phi}_{a}\dot{\phi}^{a}\frac{\delta\dot{\lambda}}{\lambda}+\bigg{(}\frac{1}{2}\epsilon H\dot{\phi}_{a}\dot{\phi}^{a}-\frac{1}{2}V_{,a\lambda}\dot{\phi}^{a}-\frac{1}{\lambda}V_{a}\dot{\phi}^{a}-\frac{\dot{\lambda}}{\lambda}\dot{\phi}_{a}\dot{\phi}^{a}\bigg{)}\frac{\delta\lambda}{\lambda}$
	$\displaystyle+\dot{\phi}_{a}\delta\ddot{\phi}^{a}-\dot{\mathcal{Z}}_{\delta\lambda}-\dot{\phi}_{a}\partial_{i}\partial^{i}\delta\phi^{a}+\bigg{(}3H+\frac{\dot{\lambda}}{\lambda}\bigg{)}\dot{\phi}_{a}\delta\dot{\phi}^{a}$
	$\displaystyle+G_{ab,c}\dot{\phi}^{a}\dot{\phi}^{c}\delta\dot{\phi}^{b}+\frac{1}{a^{3}\lambda}\mathcal{M}^{2}_{\ ab}\dot{\phi}^{a}\delta\phi^{b}=0~{}.$		(17)

Due to the perturbed constraint equation, the perturbation of the multiplier $\delta\lambda$ is not a DOF, and the system has the same DOF as the usual multi-field system Shen et al. (2019); Mansoori et al. (2022). However, as Mansoori et al. (2022) has found, at least in the case where $V$ is independent on $\lambda$, the perturbed mimetic constraint will combine the adiabatic and entropy modes in the following form,

$\displaystyle\dot{u}_{T}=\epsilon Hu_{T}+\dot{\Theta}u_{N}$

(18)

where $u_{T}$ and $u_{N}$ are the adiabatic perturbation and the entropy perturbation respectively and $\dot{\Theta}$ is the angular velocity of perturbations’ trajectory in the field space. This will make the kinetic term of the adiabatic perturbation disappear in the perturbed action (13), causing the seemingly non-propagation of the adiabatic perturbation.

In Zhang et al. (2023), we discuss the curvaton mechanism in the framework of mimetic gravity theory, where the adiabatic perturbation can be generated from the entropic one. We assume the first of the two fields, $\varphi$, to be the inflaton field, while the second one, $\theta$, to be the curvaton field. Technically, we chose a very specific form of the field metric $G_{ab}$:

$\displaystyle G_{ab}=\text{diag}\left\{1,6\sinh^{2}\left(\frac{\varphi}{\sqrt{6}M}\right)\right\}~{},$

(19)

where $M$ is a constant respect with the typical scale of $\varphi$. Moreover, for simplicity we consider the parameter $\lambda$ to be an algebraic multiplier, therefore the perturbation of $\lambda$ was turned off. For the potential form, we choose $V\approx\tanh^{2n}(\varphi/\sqrt{6}M)$ for inflationary epoch. Such kind of potential satisfies the slow-roll condition which is required by inflation, and gives rise to a solution of the two fields with constant-velocities: $\dot{\varphi}\simeq\pm\sqrt{2}M^{2}$, $\dot{\theta}\simeq 0$. In the matter-dominated epoch when the potential reaches its minimum, we set its form to be $V\simeq\lambda[m_{\text{eff}}^{2}\varphi^{2}+\alpha\ln(\varphi/\varphi_{1})]$, where the $\alpha$-term is a small correction to the main quadratic form and $\varphi_{1}$ is a constant. This will induce a solution of a semi-oscillating $\varphi$: $\varphi\simeq\sqrt{\sin(m_{\text{eff}}t)}$, and $\dot{\theta}\simeq(m_{\text{eff}}t)^{-1}$. Moreover, for the curvaton model, we turned off all the metric perturbations. Making use of the perturbation equations (16), one gets the solutions for $\delta\varphi$ and $\delta\theta$, respectively,

	$\displaystyle\delta\varphi\simeq\delta\theta\simeq\frac{H}{2\pi}~{},~{}~{}~{}\text{for inflationary epoch},\text{ and}$		(20)
	$\displaystyle\begin{cases}\delta\varphi&=\frac{1}{\varphi}\big{(}D_{+}e^{ik\tau}+D_{-}e^{-ik\tau}\big{)},\\ \delta\theta&=\frac{\sqrt{3}}{3Am_{\text{eff}}t}\big{(}D_{+}e^{ik\tau}+D_{-}e^{-ik\tau}\big{)},\end{cases}\text{for MD epoch}.$		(21)

When the curvaton dominates the universe or decays into the background, its perturbation will be transferred into the curvature perturbation, $\zeta\equiv-H\delta\rho/\dot{\rho}$. In detail, $\zeta$ will be dependent on both $\delta\varphi$ and $\delta\theta$. According to the calculations in Zhang et al. (2023), the $\delta\varphi$-containing term will dilute as time goes, therefore $\zeta$ will only depend on $\delta\theta$, in the form of mean square root. After some manipulations, we finally get the power spectrum of the curvature perturbation as:

$\displaystyle P_{\zeta}\equiv\frac{k^{3}}{2\phi^{2}}|\zeta|^{2}\sim r^{2}\frac{H^{2}}{2\pi^{2}M_{p}^{2}\epsilon\dot{\theta}_{i}^{2}}~{},$

(22)

where $r\equiv\rho_{\theta}/\rho_{\text{tot}}$ is the energy fraction of $\theta$, and $\dot{\theta}_{i}$ is the value of $\dot{\theta}$ at the beginning of the matter-dominated era.

In this section, first of all we give a specific case of the mimetic curvaton model, which can drive a inflationary period. For simplicity, we re-scale the choose the field-space metric to be the trivial one:

$\displaystyle G_{ab}=\text{diag}\{1,1\}~{}.$

(23)

According to the discussions in the last section, we need to choose a flat potential in order to satisfy the slow-roll condition. One of such a potential is

$\displaystyle V(\varphi,\theta,\lambda)=V(\varphi)=3\tanh^{2}\left(\frac{\varphi}{\sqrt{6\Omega}}\right)~{}.$

(24)

Therefore, according to the field equations (8), one has

$\displaystyle\ddot{\varphi}+\big{(}\Omega^{2}-\dot{\varphi}^{2}\big{)}\frac{V_{\varphi}}{\lambda\Omega^{2}}=0~{},\ \ddot{\theta}-\frac{V_{\varphi}\dot{\varphi}}{\lambda\Omega^{2}}\dot{\theta}=0~{}.$

(25)

These equations give the solution as

$\displaystyle\varphi=\varphi_{0}-\Omega t~{},\ \dot{\theta}=\dot{\theta}_{0}\exp\left(\int\frac{V_{\varphi}\dot{\varphi}}{\lambda\Omega^{2}}dt\right)\approx 0~{}.$

(26)

On the other hand, the equation for $\lambda$ becomes:

$\displaystyle\dot{\lambda}+3H\lambda-\frac{\sqrt{6}}{\Omega^{3/2}}\cosh^{-2}\left(\frac{\varphi}{\sqrt{6\Omega}}\right)\tanh\left(\frac{\varphi}{\sqrt{6\Omega}}\right)=0~{}.$

(27)

which is difficult to obtain an exact solution directly. However, we can discuss about its approximate solution during different stages in inflation. From now on we take $\Omega=1$ to briefly obtain analytical solutions in this section. Since the potential (24) with the behavior as (26) is a large-field kind of inflaton potential, it brings the $\varphi$ field from a large value to a small one. Therefore, at the early time of inflation when $\varphi\gg 1$, the last term in Eq. (27) can be neglected, giving rise to an approximate solution of $\lambda$:

$\displaystyle\lambda=\lambda_{0}e^{-3Ht}~{}.$

(28)

Meanwhile, at the late time of evolution when $\varphi\lesssim 1$, the last term in Eq. (27) cannot be neglected. Since the inflation is still not ending, we have

$\displaystyle V_{\varphi}\sim\sqrt{6}e^{-\sqrt{2/3}(\varphi_{0}-t)}~{}.$

(29)

The $\lambda$ has the solution as:

$\displaystyle\lambda=\frac{\sqrt{6}}{\sqrt{2/3}+3}e^{\sqrt{2/3}(t-\varphi_{0})}~{}.$

(30)

In order to verify with the above analysis, We also perform numerical calculations for Eqs. (25) and (27), and plot the evolution of the fields $\varphi$, $\theta$ and $\lambda$ in Fig. 1. From the plot we can see that, the inflaton field $\varphi$ evolves with nearly a constant velocity, while $\theta$ is nearly constant with a vanishing velocity. The evolution of $\varphi$ and $\theta$ also satisfies the constraint equation (5). Moreover, one can see that $\lambda$ will have a “turn-around” behavior, and the turn-around time will be determined by the specific choice of initial values and the form of the potential.

In this section we discuss about the perturbations generated during the inflationary period. For two-field system, applying the curvaton mechanism with no metric perturbation one can obtain the equations of field-perturbations as

	$\displaystyle\delta\ddot{\phi}^{a}+(3H+\frac{\dot{\lambda}}{\lambda})\delta\dot{\phi}^{a}+(\frac{k^{2}}{a^{2}}+\frac{1}{\lambda}V_{\phi^{a}\phi^{a}})\delta\phi^{a}$
	$\displaystyle-\frac{1}{2\lambda}V_{\phi^{a}\phi^{b}}\delta\phi^{b}+\frac{\delta\lambda}{\lambda}(\ddot{\phi}^{a}+3H\dot{\phi}^{a}+V_{\phi^{a}\lambda})=0~{},$		(31)

and the perturbed constraint equation becomes

$\displaystyle\dot{\varphi}\delta\dot{\varphi}+\dot{\theta}\delta\dot{\theta}-V_{\lambda\varphi}\delta\varphi-V_{\lambda\theta}\delta\theta-V_{\lambda\lambda}\delta\lambda=0~{}.$

(32)

In inflationary epoch, we have $V=V(\varphi)$, $\dot{\lambda}/H\lambda\ll 1$. Meanwhile, the background level of the fields $\varphi$ and $\theta$ satisfy Eq. (25). Then the above equations turn out to be

	$\displaystyle\delta\ddot{\varphi}+3H\delta\dot{\varphi}+\bigg{(}\frac{k^{2}}{a^{2}}+\frac{V_{\varphi\varphi}}{\lambda}\bigg{)}\delta\varphi$
	$\displaystyle+(\ddot{\varphi}+3H\dot{\varphi})\frac{\delta\lambda}{\lambda}+\dot{\varphi}\frac{\delta\dot{\lambda}}{\lambda}$	$\displaystyle=$	$\displaystyle 0~{},$		(33)
	$\displaystyle\delta\ddot{\theta}+3H\delta\dot{\theta}+\frac{k^{2}}{a^{2}}\delta\theta$	$\displaystyle=$	$\displaystyle 0~{},$		(34)
	$\displaystyle\dot{\varphi}\delta\dot{\varphi}$	$\displaystyle=$	$\displaystyle 0~{}.$		(35)

From the second and the third equation one can immediately get the solution for $\delta\varphi$ and $\delta\theta$ during inflation:

$\displaystyle\delta\varphi\sim\text{const.}\times k^{3/2}~{},~{}~{}~{}\delta\theta\simeq\frac{H}{2\pi}~{},$

(36)

where a factor as $k^{3/2}$ is added to $\delta\varphi$ since the solutions are in coordinate phase space. Thus from the first equation one can obtain

$\displaystyle\delta\lambda=-\frac{a^{-3}}{\Omega^{2}}\int a^{3}(V_{\varphi}\delta\dot{\varphi}+V_{\varphi\varphi}\dot{\varphi}\delta\varphi)dt~{}.$

(37)

Making use of the adiabatic-entropy decomposition proposed in Gordon et al. (2000); Lalak et al. (2007); Langlois and Renaux-Petel (2008), one can define the adiabatic and the entropy modes of those field-perturbations as

$\displaystyle Q_{\sigma}\equiv\frac{\dot{\varphi}}{\dot{\sigma}}\delta\varphi+\frac{\dot{\theta}}{\dot{\sigma}}\delta\theta~{},~{}Q_{s}\equiv-\frac{\dot{\theta}}{\dot{\sigma}}\delta\varphi+\frac{\dot{\varphi}}{\dot{\sigma}}\delta\theta~{}.$

(38)

where $\dot{\sigma}\equiv\sqrt{\dot{\varphi}^{2}+\dot{\theta}^{2}}$. Both $Q_{\sigma}$ and $Q_{s}$ are gauge-invariant variables. We can further define the curvature and isocurvature perturbations as:

$\displaystyle{\cal R}=\frac{H}{\dot{\sigma}}Q_{\sigma}~{},~{}~{}~{}{\cal S}=\frac{H}{\dot{\sigma}}Q_{s}~{}.$

(39)

In the inflationary epoch where $\dot{\theta}\simeq 0$, one has ${\cal R}\simeq\delta\varphi$ and ${\cal S}\simeq\delta\theta$. Moreover, from the solution of $\delta\varphi$ and $\delta\theta$ in (36), we can straightforwardly have:

$\displaystyle{\cal R}\sim\text{const.}\times k^{3/2}~{},~{}~{}~{}{\cal S}\simeq\frac{H}{2\pi}~{}.$

(40)

Along with the inflaton field $\varphi$ goes down towards the minimum value of the potential, it will finally falls into the “valley” surrounding its minimum. In order to get a period of matter-dominated epoch, we now turn on the dependence of the potential on the other fields $\theta$ and $\lambda$, and design its ansatz form to be:

$\displaystyle V(\varphi,\theta,\lambda)=\frac{1}{2}\lambda(m^{2}\varphi^{2}+m^{2}\theta^{2}-\Omega^{2})~{},$

(41)

where $m$ is the effective mass of both fields $\varphi$ and $\theta$ fields. Given that the difference in masses between these two fields does not induce any new physical processes, for simplicity we assume that they have the same mass. Since in the current work $\lambda$ is set to be less than zero, the potential is actually a concave function of $\varphi$, which is consistent with the potential form (24) in the large $\varphi$ region. Assuming

$\displaystyle\lambda\sim a^{-3}~{},$

(42)

the equation of motion for both fields (8) turns out to be

$\displaystyle\ddot{\varphi}+\frac{V_{\varphi}}{\lambda}=0~{},~{}\ddot{\theta}+\frac{V_{\theta}}{\lambda}=0~{}.$

(43)

With the potential form (41), and the initial conditions coming from the inflationary epoch (namely $\dot{\varphi}\simeq-\Omega$, $\dot{\theta}\simeq 0$), one has the solution as

$\displaystyle\varphi=-A\Omega\sin(mt)~{},~{}~{}~{}\theta=A\Omega\cos(mt)~{},$

(44)

with $A$ as the amplitude of both $\varphi$ and $\theta$.

As a consistency check, we would like to claim that such a solution is consistent with the assumption (42): With this solution, we always have $-m^{2}\varphi^{2}-m^{2}\theta^{2}+\Omega^{2}\simeq 0$, therefore $V_{\lambda}\simeq 0$ and condition (10) is always satisfied. Therefore, we can believe that Eq. (44) is the right solution we look for.

Considering the behaviors of the fields at the inflationary epoch as well, we can summarize the overall trajectory of the system in the $\varphi-\theta$ phase space: Firstly, the system falls down the potential along the $\varphi$ direction, which causes the linear reduction of $\varphi$ field, while the $\theta$ field stays tuned with no variation either. The constant velocity of $\varphi$ gives rise to a period of inflation. Then at a certain point, the $\theta$ field is triggered and obtains similar velocity as that of $\varphi$, therefore, both the two fields start to wind around the potential and maintain at a certain altitude, due to the mimetic constraint. This gives rise to matter-dominated epoch. We show the sketch plot of the trajectory of our system in Fig. 2. It is difficult to present also the dependence on $\lambda$ of the potential, but from the expression (41) one can see that in the matter-dominant epoch, the potential will be decreasing as $a^{-3}$ along with the total factor $\lambda$.

Now we analyse the evolution of perturbations during matter-dominated epoch. According to Eqs. (15) and (16) with the background solution (42) and (44), one gets the perturbation equations for this epoch to be

	$\displaystyle\delta\ddot{\varphi}+\left(\frac{k^{2}}{a^{2}}+m^{2}\right)\delta\varphi+\left(3H\delta\lambda+\delta\dot{\lambda}\right)\frac{\dot{\varphi}}{\lambda}$	$\displaystyle=$	$\displaystyle 0~{},$
	$\displaystyle\delta\ddot{\theta}+\left(\frac{k^{2}}{a^{2}}+m^{2}\right)\delta\theta+\left(3H\delta\lambda+\delta\dot{\lambda}\right)\frac{\dot{\theta}}{\lambda}$	$\displaystyle=$	$\displaystyle 0~{},$		(45)

while the constraint equation is

$\displaystyle\dot{\theta}\delta\dot{\theta}+\dot{\varphi}\delta\dot{\varphi}-m^{2}\varphi\delta\varphi-m^{2}\theta\delta\theta=0~{}.$

(46)

Thanks to the triangle functional forms of the background solution of $\varphi$ and $\theta$, some terms in Eqs. (45) get cancelled so that the dependence of $\delta\lambda$ in these two equations has a unitary form and can be removed using the constraint equation. Taking time derivative to the constraint equation (46) one gets

	$\displaystyle\dot{\theta}\delta\ddot{\theta}+\dot{\varphi}\delta\ddot{\varphi}+(\ddot{\theta}-m^{2}\theta)\delta\dot{\theta}+(\ddot{\varphi}-m^{2}\varphi)\delta\dot{\varphi}$
	$\displaystyle-m^{2}\dot{\varphi}\delta\varphi-m^{2}\dot{\theta}\delta\theta=0~{}.$		(47)

Rearrange Eqs. (45) and (47), one finally gets:

	$\displaystyle\delta\ddot{\varphi}+\left(\frac{k^{2}}{a^{2}}+m^{2}\right)\delta\varphi-\frac{\dot{\varphi}}{A^{2}m^{2}\Omega^{2}}\left[\left(\frac{k^{2}}{a^{2}}+2m^{2}\right)(\dot{\theta}\delta\theta+\dot{\varphi}\delta\varphi)+2m^{2}(\theta\delta\dot{\theta}+\varphi\delta\dot{\varphi})\right]$	$\displaystyle=$	$\displaystyle 0~{},$		(48)
	$\displaystyle\delta\ddot{\theta}+\left(\frac{k^{2}}{a^{2}}+m^{2}\right)\delta\theta-\frac{\dot{\theta}}{A^{2}m^{2}\Omega^{2}}\left[\left(\frac{k^{2}}{a^{2}}+2m^{2}\right)(\dot{\theta}\delta\theta+\dot{\varphi}\delta\varphi)+2m^{2}(\theta\delta\dot{\theta}+\varphi\delta\dot{\varphi})\right]$	$\displaystyle=$	$\displaystyle 0~{},$		(49)
	$\displaystyle\delta\dot{\lambda}+3H\delta\lambda+\frac{\lambda}{A^{2}m^{2}\Omega^{2}}\left[\left(\frac{k^{2}}{a^{2}}+2m^{2}\right)(\dot{\theta}\delta\theta+\dot{\varphi}\delta\varphi)+2m^{2}(\theta\delta\dot{\theta}+\varphi\delta\dot{\varphi})\right]$	$\displaystyle=$	$\displaystyle 0~{}.$		(50)

From the above equations one can see that, in this epoch the equations of field-perturbations actually don’t depend on $\delta\lambda$, which simplifies our calculations.

Using the definition of the adiabatic and the entropy perturbation modes given in (38), one has

	$\displaystyle Q_{\sigma}^{\prime\prime}$	$\displaystyle=$	$\displaystyle 0~{},$		(51)
	$\displaystyle Q_{s}^{\prime\prime}+\frac{\kappa^{2}}{a^{2}}Q_{s}+2Q_{\sigma}^{\prime}$	$\displaystyle=$	$\displaystyle 0~{},$		(52)

where $\prime$ denotes derivative with respect to the normalized time $x\equiv mt$, and $\kappa\equiv k/m$ is also the normalized wave-number. In matter-dominated epoch, the scale factor and the Hubble parameter are $a(x)=a_{0}x^{2/3}$ and ${\cal H}(x)=2m/(3x)$ respectively. Furthermore, from the definition of the curvature and isocurvature perturbations given in (39), one has equations as

	$\displaystyle{\cal R}^{\prime\prime}+3{\cal H}{\cal R}^{\prime}$	$\displaystyle=$	$\displaystyle 0~{},$		(53)
	$\displaystyle{\cal S}^{\prime\prime}+3{\cal H}{\cal S}^{\prime}+\frac{\kappa^{2}}{a^{2}}{\cal S}+2{\cal R}^{\prime}+3{\cal H}{\cal R}$	$\displaystyle=$	$\displaystyle 0~{}.$		(54)

Interestingly, we can see that the curvature perturbation ${\cal R}$ satisfies a homogeneous equation of motion which is seemingly not dependent on $k$. However, it is not true since it is also influenced by the initial conditions, as we will see later.

The above equations gives the solutions

	$\displaystyle Q_{\sigma}$	$\displaystyle=$	$\displaystyle C_{1}+C_{2}(x-x_{0}),$
	$\displaystyle Q_{s}$	$\displaystyle=$	$\displaystyle C_{3}L_{1}(x)+C_{4}L_{2}(x)-2C_{2}L_{3}(x)~{},$		(55)

where

	$\displaystyle L_{1}(x)$	$\displaystyle=$	$\displaystyle-3px^{1/3}\cos(3px^{1/3})+\sin(3px^{1/3}),$
	$\displaystyle L_{2}(x)$	$\displaystyle=$	$\displaystyle\cos(3px^{1/3})+3px^{1/3}\sin(3px^{1/3}),\text{ and}$
	$\displaystyle L_{3}(x)$	$\displaystyle=$	$\displaystyle-\frac{8}{81p^{6}}-\frac{4x^{2/3}}{p^{4}}+\frac{x^{4/3}}{p^{2}},$		(56)

are the general solutions for the homogeneous part of Eq. (52), and $p\equiv\kappa/a_{0}$. Here we set the initial time of this epoch to be $x=x_{0}$. When the perturbations reentered the horizon in matter-dominated epoch, one has $k_{\ast}=a(x_{\ast})H(x_{\ast})$, which gives rise to $x_{\ast}^{1/3}p=2/3$.

In order to get the exact perturbations during the matter-dominated epoch, we should use the continuity conditions to match the solutions between (55) and those of the inflationary epoch, namely Eq. (40). Here we consider two kinds of continuity conditions. One is to consider the continuity of the curvature/isocurvature perturbations themselves, namely ${\cal R}$ and ${\cal S}$. In this case, one obtains the initial conditions for the matter-dominated epoch as

	$\displaystyle\mathcal{R}_{0}$	$\displaystyle=$	$\displaystyle\delta\varphi_{0}\equiv\beta\frac{H}{2\pi}p^{3/2}~{},\quad\mathcal{R}_{0}^{\prime}=0~{},$
	$\displaystyle\mathcal{S}_{0}$	$\displaystyle=$	$\displaystyle\delta\theta_{0}=\frac{H}{2\pi}~{},\quad\mathcal{S}_{0}^{\prime}=0~{},$		(57)

where $\beta$ is a constant scaling the amplitude of $\delta\varphi$. Substituting the above conditions into Eq. (55) one can get the exact solution as

	$\displaystyle\mathcal{R}$	$\displaystyle=$	$\displaystyle\frac{2}{3x}(C_{1}+C_{2}x)\delta\theta_{0}~{},$
	$\displaystyle\mathcal{S}$	$\displaystyle=$	$\displaystyle\frac{2}{3x}[C_{3}L_{1}(x)+C_{4}L_{2}(x)-2C_{2}L_{3}(x)]\delta\theta_{0}~{},$		(58)

where

	$\displaystyle C_{1}$	$\displaystyle=$	$\displaystyle 0,\quad C_{2}=\frac{3}{2}\beta p^{3/2}~{},$
	$\displaystyle\frac{C_{3}}{x}$	$\displaystyle=$	$\displaystyle\frac{1}{\gamma^{3}}\left[\left(\frac{1}{6}-\frac{1}{2}\gamma_{0}^{2}\right)\cos(3\gamma_{0})+\frac{1}{2}\gamma_{0}\sin(3\gamma_{0})\right]$
		$\displaystyle+$	$\displaystyle\frac{\beta p^{1/2}}{\gamma^{3}}\bigg{[}\left(\frac{16}{81}-\frac{8}{9}\gamma_{0}^{2}\right)\sin(3\gamma_{0})+\bigg{(}\frac{2}{3}\gamma_{0}^{3}$
		$\displaystyle-$	$\displaystyle\frac{16}{27}\gamma_{0}\bigg{)}\cos(3\gamma_{0})\bigg{]}\propto k^{0}+\mathcal{O}(k^{3/2})~{},$
	$\displaystyle\frac{C_{4}}{x}$	$\displaystyle=$	$\displaystyle\frac{1}{\gamma^{3}}\left[\frac{1}{2}\gamma_{0}\cos{\left(3\gamma_{0}\right)}-\left(\frac{1}{6}-\frac{1}{2}\gamma_{0}^{2}\right)\sin{\left(3\gamma_{0}\right)}\right]$
		$\displaystyle+$	$\displaystyle\frac{\beta p^{3/2}}{\gamma^{3}}\bigg{[}\left(\frac{16}{81}-\frac{8}{9}\gamma_{0}^{2}\right)\cos{\left(3\gamma_{0}\right)}$
		$\displaystyle-$	$\displaystyle\left(\frac{2}{3}\gamma_{0}^{3}-\frac{16}{27}\gamma_{0}\right)\sin{\left(3\gamma_{0}\right)}\bigg{]}\propto k^{0}+\mathcal{O}\left(k^{3/2}\right)$
	$\displaystyle\frac{C_{2}}{x}L_{3}$	$\displaystyle=$	$\displaystyle\frac{\beta}{p^{3/2}}\left(\gamma-\frac{4}{9}\gamma^{-1}-\frac{8}{81}\gamma^{-3}\right)\propto k^{-3/2}~{},$		(59)

where $\gamma\equiv x^{1/3}p\in(0,2/3]$, and $\gamma_{0}=x_{0}^{1/3}p\propto k\ll 1$. For the curvature perturbation, it obviously follows $\mathcal{R}\propto k^{3/2}$ with a spectral index $n_{s}=4$, while the isocurvature perturbation ${\cal S}$ can obtain a scale-invariant spectrum by requiring $\beta\ll 1$. Furthermore, on a larger scale, due to the dominance of the $C_{3}L_{3}/x$ term, the isocurvature perturbation will have a red-tilted spectrum, characterized by $n_{s}=-2$.

Another continuous condition is to consider the continuity of the adiabatic and the entropy perturbation modes $Q_{\sigma}$ and $Q_{s}$, which can give rise to another set of initial conditions as

	$\displaystyle Q_{s}(x=x_{0})$	$\displaystyle=$	$\displaystyle-\delta\theta_{0}~{},~{}Q_{s}^{\prime}(x=x_{0})=\delta\phi_{0}=\beta p^{3/2}\delta\theta_{0}~{},$
	$\displaystyle Q_{\sigma}(x=x_{0})$	$\displaystyle=$	$\displaystyle-\delta\phi_{0}~{},~{}Q_{\sigma}^{\prime}(x=x_{0})=-\delta\theta_{0}.$		(60)

Following a similar way, we can derive the expressions for curvature and isocurvature perturbations:

	$\displaystyle\mathcal{R}$	$\displaystyle=$	$\displaystyle-\delta\theta_{0}[1+(\beta p^{3/2}-x_{0})p^{3}\gamma^{-3}]~{},$
	$\displaystyle\mathcal{S}$	$\displaystyle=$	$\displaystyle-\delta\theta_{0}(S_{1}+\beta p^{3/2}S_{2})~{},$		(61)

where

	$\displaystyle S_{1}$	$\displaystyle=$	$\displaystyle\frac{1}{p^{3}}\left(\frac{16}{81}\gamma^{-3}-\frac{16}{27}\gamma^{-2}\sin(3\gamma-3\gamma_{0})+\frac{8}{9}\gamma^{-1}-2\gamma\right)$
		$\displaystyle+$	$\displaystyle\frac{p^{2}}{x_{0}^{1/3}}\left[\frac{\cos(3\gamma-3\gamma_{0})}{\gamma^{2}}-\frac{\sin(3\gamma-3\gamma_{0})}{3\gamma^{3}}\right]$
		$\displaystyle+$	$\displaystyle\frac{x_{0}^{1/3}}{p^{2}\gamma}\bigg{[}\left(\frac{8\gamma_{0}}{9\gamma^{2}}+\frac{2\gamma_{0}^{2}}{\gamma}-\frac{16}{9\gamma}\right)\cos(3\gamma-3\gamma_{0})$
		$\displaystyle+$	$\displaystyle\left(\frac{16}{27\gamma^{2}}+\frac{8\gamma_{0}}{3\gamma}-\frac{2}{3\gamma}\right)\sin(3\gamma-3\gamma_{0})\bigg{]}~{},$
	$\displaystyle S_{2}$	$\displaystyle=$	$\displaystyle\frac{1}{9\gamma^{3}}[3(\gamma-\gamma_{0})\cos(3\gamma-3\gamma_{0})$		(62)
		$\displaystyle-$	$\displaystyle(1+3\gamma_{0}\gamma)\sin(3\gamma-3\gamma_{0})]~{}.$

From the above we can see that, under the condition $\beta p^{3/2}\ll x_{0}=\gamma_{0}p^{-3}\ll 1$, we can obtain a scale-invariant curvature perturbation $\mathcal{R}\sim k^{0}+\mathcal{O}(k^{9/2})$, while the isocurvature perturbation is red-tilted, $\mathcal{S}\sim k^{-3}$. Moreover, on small scales both curvature and isocurvature perturbations will have a blue-tilt, which can be responsible for primordial black hole generation.