Large Primordial Fluctuations in
Gravitational Waves from Phase Transitions

Here, we relate the anisotropies in the CMB and GWB to the independent primordial fluctuations $\zeta_{\phi}$ and $\zeta_{\chi}$²²2A more relevant quantity to measure isocurvature is ${\cal S}_{\chi}=\zeta_{\chi}-\zeta_{\phi}$. However, for $\zeta_{\chi}\gg\zeta_{\phi}$, which is of interest to us in this work, we can simply take ${\cal S}_{\chi}\approx\zeta_{\chi}$.,

$\displaystyle\zeta_{\phi}(k)=\frac{H_{\rm inf}\delta\phi(k)}{\dot{\phi}_{0}},\quad\zeta_{\chi}(k)=\frac{2\delta\chi(k)}{3\chi_{0}}\,,$

(2.1)

where $\delta\phi\sim\delta\chi\sim H_{\rm inf}$, and the expressions are evaluated at the horizon exit of the co-moving mode $k$. The fluctuations in GWB will be dominated by the PT sector (reheated by $\chi$), $\delta_{\rm GW}\sim\delta_{\rm PT}\sim\zeta_{\chi}$. More precisely, the anisotropies can be written in terms of the gauge-invariant fluctuations as [12, 20]

$\displaystyle\delta_{\rm GW}=4\zeta_{\chi}\left(1-\frac{4}{3}f_{\rm PT}\right)-\frac{16}{3}\zeta_{\phi}\,,$

(2.2)

where

$\displaystyle f_{\rm PT}\equiv\frac{\rho_{\rm PT}}{\rho_{\rm total}}$

(2.3)

is the fraction of energy density in the PT sector compared to the total energy density. Note that $\zeta_{\phi}\sim 10^{-5}$ sets the minimum $\delta_{\rm GW}$. This comes from the Sachs-Wolfe (SW) effect, which is dominated by the fluctuations of the non-PT sector. A small $\zeta_{\chi}<10^{-5}$ would be swamped by the SW contribution, leaving the GWB to be predominantly adiabatic. On the other hand, large isocurvature in GWB coming from $\zeta_{\chi}>10^{-5}$, would clearly imply the existence of a new light quantum field during inflation. Hence, we will focus on the range $\zeta_{\chi}>10^{-5}$.

The fluctuations in the SM plasma (and hence in the CMB and matter distribution) after $t_{\rm dec}$ are a weighted superposition of the fluctuations in the PT sector and those in the decay products of the non-PT sector, $\delta_{\gamma}\sim\delta_{\rm nPT}(t_{\rm dec})+f_{\rm PT}\delta_{\rm PT}$. In terms of the primordial fluctuations,

$\displaystyle\delta_{\gamma}\approx-\frac{4}{5}\zeta_{\phi}-\frac{4}{5}f_{\rm PT}\zeta_{\chi}.$

(2.4)

As expected, the contribution of $\zeta_{\chi}$ to CMB can be kept small by choosing sufficiently small $f_{\rm PT}$. From the CMB observations [21], $\delta_{\gamma}\simeq 3.6\times 10^{-5}$, giving a constraint

$\displaystyle f_{\rm PT}\,\zeta_{\chi}\leq 4.5\times 10^{-5}\,.$

(2.5)

Therefore, since we are interested in $\zeta_{\chi}\gg 10^{-5}$, the PT sector must be sufficiently subdominant, $f_{\rm PT}\ll 1$. This leads to the simplification of Eq. (2.2),

$$\delta_{\rm GW}\approx 4\zeta_{\chi}.$$

(2.6)

Additionally, if $\zeta_{\chi}$ contains qualitative features which are not observed in CMB, the constraint on $f_{\rm PT}\,\zeta_{\chi}$ will be even stronger, requiring yet smaller $f_{\rm PT}$.

It is known that the GW production is suppressed when the PT sector constitutes a smaller fraction $f_{\rm PT}$ of the total energy density at the time of the phase transition. In terms of the fraction of energy in gravitational waves compared to the critical energy density at the time of production $t_{\rm PT}$, the suppression goes as

$\displaystyle\Omega_{\rm GW}(t_{\rm PT})\equiv\frac{\rho_{\rm GW}(t_{\rm PT})}{\rho_{\rm total}(t_{\rm PT})}\propto f_{\rm PT}^{2}$

(2.7)

for dominant sources of gravitational wave production, such as bubble collisions and acoustic waves [22, 23]. The proportionality constant is determined by the microphysics of the phase transition. Heuristically, Einstein equation for the gravitational radiation takes the form $\omega_{*}^{2}h_{\rm GW}\sim\rho_{\rm PT}/M_{\text{pl}}^{2}$, where $\omega_{*}$ is the typical GW frequency at production, $h_{\rm GW}$ is the amplitude of the metric perturbation corresponding to the gravitational wave, and $M_{\text{pl}}^{2}=(8\pi G)^{-1}$ is the reduced Planck mass. Since the energy density in GW goes as $\rho_{\rm GW}\sim\omega_{*}^{2}M_{\text{pl}}^{2}h_{\rm GW}^{2}$, it follows that $\Omega_{\rm GW}(t_{\rm PT})\sim f_{\rm PT}^{2}H^{2}/\omega_{*}^{2}$. Time dependence is set by the $H$ scale so that $\omega_{*}/H$ is determined by the microphysics, independent of cosmological abundances³³3There are milder dependencies on $f_{\rm PT}$ which we ignore since we are tracking leading order effects., leading to Eq. (2.7).

We specialize to the plausible and relatively tractable scenario in which the gravitational waves are produced dominantly from bubble wall collisions in the thin wall regime⁴⁴4In the case of significant coupling between the PT plasma and bubble walls, the majority of latent heat can be transferred to the plasma making sound waves the dominant source of gravitational waves. However, it turns out that for the parameters that we choose as benchmark, i.e. $\alpha_{\rm PT}\sim 1$ and $\tilde{\beta}\sim 10$, the sound wave contribution will be similar in strength to the estimate in Eq. (2.10) even if all the latent heat is transferred to the kinetic energy of the plasma [24].. The energy density in the gravitational waves dilutes like radiation after production. Then the GWB spectrum today is given by [1]

$\displaystyle\Omega_{\rm GW}^{(0){\rm bub}}h^{2}$

$\displaystyle\approx 1.67\times 10^{-5}\tilde{\beta}^{-2}f_{\rm PT}^{2}\,\left(\frac{\kappa\alpha_{\rm PT}}{1+f_{\rm PT}\alpha_{\rm PT}}\right)^{2}\left(\frac{100}{g_{*}(t_{\rm PT})}\right)^{1/3}\left(\frac{0.11v_{w}^{3}}{0.42+v_{w}^{2}}\right)S_{\rm bub}(\omega),$

(2.8)

where the frequency dependence is

$\displaystyle S_{\rm bub}(\omega)$

$\displaystyle=\frac{3.8(\omega/\omega_{\rm peak})^{2.8}}{1+2.8(\omega/\omega_{\rm peak})^{3.8}}\,.$

(2.9)

Here, $\alpha_{\rm PT}$ is the ratio of the latent heat released compared to the energy density of the PT plasma before the phase transition, and $\kappa$ is the efficiency of the latent heat transfer to bubble walls. We will take $\alpha\sim 1$, such that the efficiency $\kappa\sim 1$, and the bubble wall velocity $v_{w}\approx 1$. The peak frequency today $\omega_{\rm peak}$ can be estimated by redshifting the typical frequency at the time of phase transition $\omega_{*}$, given by $\omega_{*}\approx 0.23\,H(t_{\rm PT})\,\tilde{\beta}$. We choose $\tilde{\beta}=10$ for all our benchmarks. $g_{*}$ corresponds to the effective number of relativistic degrees of freedom, and we take $g_{*}(t_{\rm PT})\sim 100$ ⁵⁵5In reality, $g_{*}(t_{\rm PT})>100$ depending on the exact extension of the SM undergoing the phase transition. However, this is a small effect considering the $g_{*}^{-1/3}$ dependence in Eq. (2.8).. With this choice of parameters, the expression in Eq. (2.8) at the peak frequency is simplified to

$\displaystyle\Omega_{\rm GW}^{(0)}h^{2}\approx$

$\displaystyle\,1.3\times 10^{-8}f_{\rm PT}^{2}\,,$

(2.10)

where we have taken $f_{\rm PT}\ll 1$.

Following Eq. (2.6), the size of the fluctuations in the GWB, which is relevant for detection, is

$\displaystyle\delta\Omega_{\rm GW}^{(0)}h^{2}\approx\Omega_{\rm GW}^{(0)}h^{2}\times 4\zeta_{\chi}\approx 5\times 10^{-8}f_{\rm PT}^{2}\zeta_{\chi}\,.$

(2.11)

The constraint of Eq. (2.5) gives us an upper bound on the inhomogeneities of the GWB,

$\displaystyle\delta\Omega_{\rm GW}^{(0)}h^{2}\leq 2.2\times 10^{-12}f_{\rm PT}\,.$

(2.12)

Note that despite relative anisotropy $\delta\rho_{\rm GW}/\rho_{\rm GW}\sim\zeta_{\chi}$ being large $>10^{-5}$, the absolute signal strength of the fluctuations is suppressed by one power of $f_{\rm PT}$, making it more difficult to detect.

In this model with eMD, the PT sector dominates the energy density at the phase transition, $\rho_{\rm total}(t_{\rm PT})\approx\rho_{\rm PT}(t_{\rm PT})$ and $\hat{f}_{\rm PT}(t_{\rm PT})\approx 1$. Hence, in Eq. (2.8) for GW production at the time of the transition, $f_{\rm PT}^{2}$ is not a suppression.

However, the period of eMD dilutes GW radiation because of the differential redshifting of matter and GW radiation, such that

$\displaystyle\frac{\rho_{\rm GW}(t_{\rm dec})}{\rho_{\rm total}(t_{\rm dec})}=\hat{f}_{\rm PT}(t_{\rm dec})\frac{\rho_{\rm GW}(t_{\rm PT})}{\rho_{\rm total}(t_{\rm PT})}\,.$

(3.2)

This, along with the same microphysical parameters as in section 2.2 for the phase transition, gives the fractional energy density in GWB today as

$\displaystyle\Omega_{\rm GW}^{(0)}h^{2}\approx$

$\displaystyle\,3.2\times 10^{-9}f_{\rm PT}\,.$

(3.3)

Note that the GW signal in this model, despite the dilution from eMD, has only one power of $f_{\rm PT}$ suppression, compared to the quadratic suppression in Eq. (2.7).

The anisotropy of the GWB can again be related to the fluctuations of $\chi$,

$\displaystyle\delta_{\rm GW}\approx-\frac{4}{3}\zeta_{\chi}\,.$

(3.4)

The factor in front of $\zeta_{\chi}$ is smaller compared to Eq. (2.6) because the SW contribution, which also comes from the PT sector in this case, is anti-correlated with the inherent inhomogeneities of the GWB. The expression for the anisotropies of the CMB for all relevant scales (i.e., the scales that re-enter much after $t_{\rm dec}$) remains the same as Eq. (2.4). This is shown explicitly in section 4.1. Then, even in the eMD model, we have

$\displaystyle\delta_{\gamma}\approx-\frac{4}{5}\zeta_{\phi}-\frac{4}{5}f_{\rm PT}\zeta_{\chi}\,.$

(3.5)

The inhomogeneity in the GWB is

$\displaystyle\delta\Omega_{\rm GW}^{(0)}h^{2}\approx\Omega_{\rm GW}^{(0)}h^{2}\times\frac{4}{3}\zeta_{\chi}\approx 4\times 10^{-9}f_{\rm PT}\zeta_{\chi}\,.$

(3.6)

We must still satisfy the constraint in Eq. (2.5). Saturating it gives

$\displaystyle\delta\Omega_{\rm GW}^{(0)}h^{2}\leq 1.8\times 10^{-13}\,,$

(3.7)

which is now independent of $f_{\rm PT}$, as opposed to Eq. (2.12). The possibility that the relative anisotropies in the GWB are large while the absolute size of the anisoptropic signal is unsuppressed by $f_{\rm PT}$, is the main result of our paper.

The frequency spectrum of the GWB will be slightly shifted compared to the radiation-dominated scenario due to the modified cosmological history. At production, the typical frequency of GWB for parameters given in section 2.2 is [1]

$\displaystyle\omega_{*}\approx 0.23\tilde{\beta}H(t_{\rm PT}).$

(3.8)

Red-shifting this frequency gives the peak frequency of the spectrum today

	$\displaystyle\omega_{\rm peak}$	$\displaystyle=\omega_{*}\left(\frac{a_{\rm PT}}{a_{0}}\right)$
		$\displaystyle\approx 0.23\tilde{\beta}\left(\frac{a_{\rm PT}}{a_{0}}\right)\left(\frac{H(t_{\rm PT})}{H(t_{\rm md})}\right)\left(\frac{H(t_{\rm md})}{H(t_{\rm dec})}\right)\left(\frac{H(t_{\rm dec})}{H(t_{eq})}\right)\left(\frac{H(t_{eq})}{H(t_{0})}\right)H(t_{0}).$		(3.9)

Using $H\propto a^{-3/2}$ during matter dominance, while $H\propto a^{-2}$ during radiation dominance,

	$\displaystyle\omega_{\rm peak}\approx$	$\displaystyle\,0.23\tilde{\beta}\left(\frac{a_{\rm PT}}{a_{0}}\right)\left(\frac{a_{\rm md}}{a_{PT}}\right)^{2}\left(\frac{a_{\rm dec}}{a_{\rm md}}\right)^{3/2}\left(\frac{a_{eq}}{a_{\rm dec}}\right)^{2}\left(\frac{a_{0}}{a_{eq}}\right)^{3/2}H(t_{0})$
	$\displaystyle\approx$	$\displaystyle\,2.8\times 10^{-17}\tilde{\beta}\left(\frac{a_{eq}}{a_{\rm PT}}\right)f_{\rm PT}^{1/2}\,\,{\rm Hz},$		(3.10)

where we have used the fact that the redshift at matter-radiation equality in the standard cosmological history $z_{eq}\sim a_{eq}^{-1}\sim 3100$, $H(t_{0})\sim 2.2\times 10^{-18}$ Hz, and $a_{\rm md}/a_{\rm dec}\approx f_{\rm PT}$ from Eq. (3.1). Now,

$\displaystyle\frac{a_{eq}}{a_{\rm PT}}=\frac{T_{\rm PT}(t_{\rm PT})}{T_{\rm SM}(t_{eq}))}f_{\rm PT}^{-1/4}\left(\frac{g_{*}(t_{\rm md})}{g_{*}(t_{\rm dec})}\right)^{1/4}.$

(3.11)

Then with $T_{\rm SM}(t_{eq})\sim 1$ eV, $g_{*}(t_{\rm md})\sim g_{*}(t_{\rm dec})\sim 100$, we get

$\displaystyle\omega_{\rm peak}\approx 2.8\times 10^{-5}\tilde{\beta}f_{\rm PT}^{1/4}\left(\frac{T_{\rm PT}(t_{\rm PT})}{\rm TeV}\right)\,{\rm Hz}.$

(3.12)

Note that the dependence on $f_{\rm PT}$ is very mild, less than an order of magnitude in the range that we will be led to consider. For $\tilde{\beta}=10$ and $f_{\rm PT}=4.5\times 10^{-3}$, a phase transition at $T_{\rm PT}(t_{\rm PT})\approx 69$ TeV gives $\omega_{\rm peak}\approx 5$ mHz, which falls within the sensitivity of LISA [3], BBO [25], and DECIGO [26]. Choosing $T_{\rm PT}(t_{\rm md})\approx 5.8$ TeV and $T_{\rm SM}(t_{\rm dec})\approx 100$ GeV gives the correct duration of eMD. For larger $f_{\rm PT}=4.5\times 10^{-2}$, a smaller $T_{\rm PT}(t_{\rm PT})\approx 39$ TeV gives the same peak frequency. In this case, $T_{\rm PT}(t_{\rm md})\approx 1$ TeV and $T_{\rm SM}(t_{\rm dec})\approx 100$ GeV produces required duration of eMD. Our choice of $T_{\rm SM}(t_{\rm dec})\approx 100$ GeV here is motivated by the considerations discussed in section 4.2.1. In principle, this temperature could be smaller depending on the model of DM.

Our modified early cosmology reverts to standard cosmology at $t_{\rm dec}$, where we will take $T_{\rm SM}(t_{\rm dec})\gtrsim 100$ GeV. We want to ensure that the observed fluctuation modes of the CMB and LSS would be unaffected by the modified cosmology before $t_{\rm dec}$, and this is relatively straightforward to analyze because they are all superhorizon prior to $t_{\rm dec}$, given the high $T_{\rm SM}(t_{\rm dec})$.

We use the convention where the perturbed FLRW metric is given by [27],

$\displaystyle g_{00}=-1-2\Psi,\quad g_{ij}=a^{2}(t)\delta_{ij}[1+2\Phi],\quad g_{0i}=0\,.$

(4.1)

In the absence of anisotropic stress, $\Phi=-\Psi$, and the relevant Einstein equation is

$\displaystyle\eta\Phi^{\prime}+\Phi+\frac{k^{2}\eta^{2}}{3}\Phi$

$\displaystyle=\frac{1}{2}\sum_{i}f_{i}\delta_{i}\,,$

(4.2)

where ^′ denotes partial derivative with respect to the conformal time $\eta$ and $k$ is the co-moving momentum. $f_{i}\equiv\rho_{i}/\rho_{\rm total}$ is the fraction of the energy density in the $i^{\rm th}$ sector and $\delta_{i}\equiv\delta\rho_{i}/\rho_{i}$ is the fractional density perturbation. When a certain mode is superhorizon, the $k^{2}$-term in Eq. (4.2) can be neglected, giving

$\displaystyle\eta\Phi^{\prime}+\Phi=\frac{1}{2}\sum_{i}f_{i}\delta_{i}\,.$

(4.3)

The gauge invariant fluctuation of a non-interacting fluid is conserved on superhorizon scales. For fluid $i$, it is given by

$\displaystyle\zeta_{i}=\Phi+\frac{\delta_{i}}{3(1+w_{i})}\,,$

(4.4)

where $w_{i}$ is its equation of state. Then $\zeta_{\rm nPT}=\zeta_{\phi}$ and $\zeta_{\rm PT}=\zeta_{\chi}$ remain constant until horizon re-entry. This also dictates the superhorizon evolution of radiation and matter density perturbations,

$\displaystyle\delta^{\prime}_{\rm rad}=-4\Phi^{\prime},\quad\delta^{\prime}_{\rm mat}=-3\Phi^{\prime}\,.$

(4.5)

Using eqs. (4.3) and (4.5), we numerically evaluated the evolution of $\Phi$ for $\zeta_{\chi}\approx 10^{-3}$ and $f_{\rm PT}\approx 10^{-2}$, shown in figure 3. While $\Phi$ is larger during eRD when the total curvature perturbation is dominated by $\zeta_{\chi}$, it becomes small again after eMD following $\zeta_{\phi}$⁶⁶6A similar conclusion was found in [28]. They consider a model with two curvatons where one of the curvatons with larger perturbations dominates the energy density during an earlier era of radiation domination.. The fluctuation in the CMB can be written in terms of $\zeta_{\phi}$ and $\Phi$ as $\delta_{\gamma}\approx 4(\zeta_{\phi}+f_{\rm PT}\zeta_{\chi}-2\Phi)$. We have chosen $\zeta_{\phi}$ such that $(\zeta_{\phi}+f_{\rm PT}\zeta_{\chi})\approx 4.5\times 10^{-5}$. From figure 3, $\Phi\approx 3\times 10^{-5}$ for the superhorizon modes during eRD. After matter-radiation equality, this will drop by a factor of ($9/10$) such that $\Phi\approx 2.7\times 10^{-5}$. Then $\delta_{\gamma}\approx 4(\zeta_{\phi}+f_{\rm PT}\zeta_{\chi}-2\Phi)\approx 3.6\times 10^{-5}$, which is consistent with the observed value in the CMB. This shows that the anisotropy in CMB (and in other maps of adiabatic perturbations) remains small at $10^{-5}$ even in our modified scenario with PT sector domination during eRD.

In this section, we focus on the evolution of perturbations for modes that re-enter the horizon during eRD and eMD. This will be relevant for the constraints on $\zeta_{\chi}$ on small scales, which can indirectly affect CMB or LSS.

The gradient terms that we neglected for superhorizon evolution become important as the mode becomes subhorizon. The evolution of $\Phi$ is

	$\displaystyle\eta\Phi^{\prime}+\Phi+\frac{k^{2}\eta^{2}}{3}\Phi$	$\displaystyle=\frac{1}{2}\sum_{i}f_{i}\delta_{i}$		(4.6)
	$\displaystyle\delta^{\prime\prime}_{\rm rad}+\frac{k^{3}}{3}\delta_{\rm rad}$	$\displaystyle=-4\Phi^{\prime\prime}+\frac{4k^{2}}{3}\Phi$
	$\displaystyle\delta^{\prime\prime}_{\rm mat}+\frac{1}{\eta}\delta^{\prime}_{\rm mat}$	$\displaystyle=-3\Phi^{\prime\prime}+k^{2}\Phi-\frac{3}{\eta}\Phi^{\prime}\,.$

Consider a co-moving mode re-entering during eRD, $k_{\rm eRD}$. Just before its horizon re-entry, $\Phi(k_{\rm eRD}\eta\lesssim 1)\approx(2/3)\zeta_{\chi}$ since the PT sector is dominant. Now $\zeta_{\rm nPT}=\Phi+\delta_{\rm nPT}/4\sim 10^{-5}$ is conserved on superhorizon scales. Then for $\zeta_{\chi}\gg\zeta_{\phi}$, $\delta_{\rm nPT}(k_{\rm eRD}\eta<1)\approx-4\Phi\approx-(8/3)\zeta_{\chi}$ using eqs. (4.3) and (4.5).

After horizon re-entry at $\eta_{\rm eRD}\sim 1/k_{\rm eRD}$, perturbations follow eqs. 4.6. Below, we describe their qualitative behavior. The fluctuations in the radiation of the PT sector, $\delta_{\rm PT}$, oscillates in time with a constant amplitude throughout eRD and eMD. The behaviour of $\Phi$ and $\delta_{\rm nPT}$, however, changes from eRD to eMD. During eRD, $|\Phi|\propto\eta^{-2}$ decreases, while the amplitude of $\delta_{\rm nPT}$ remains constant when it is radiation dominated and increases logarithmically $\sim\log(a)$ when it becomes non-relativistic. This logarithmic growth is insignificant if $\epsilon$ is close to 1, which we take to be the case. The linear growth during eMD, however, is significant when $\delta_{\rm nPT}\propto a$, and consequently $\Phi$ becomes constant. In summary,

$\displaystyle|\delta_{\rm nPT}(k_{\rm eRD})|$

$\displaystyle\approx\begin{cases}\dfrac{8}{3}\zeta_{\chi}&\eta_{\rm eRD}\leq\eta<\eta_{\rm md}\vspace{0.3em}\\ 2\dfrac{\eta^{2}}{\eta_{\rm md}^{2}}\zeta_{\chi}&\eta_{\rm md}\leq\eta<\eta_{\rm dec}\vspace{0.3em}\\ \dfrac{8}{3}\dfrac{\eta_{\rm dec}^{2}}{\eta_{\rm md}^{2}}\zeta_{\chi}&\eta_{\rm dec}\leq\eta\,,\end{cases}$

(4.7)

where we have used adiabatic relation $\delta_{\rm rad}/4=\delta_{\rm mat}/3$ at the transitions between radiation to matter dominance in the non-PT sector, and $a\propto\eta^{2}$ during MD. The evolution of $\Phi$ is

$\displaystyle|\Phi(k_{\rm eRD})|$

$\displaystyle\approx\begin{cases}\dfrac{2}{3}\dfrac{\eta_{\rm eRD}^{2}}{\eta^{2}}\zeta_{\chi}&\eta_{\rm eRD}\leq\eta<\eta_{\rm md}\vspace{0.3em}\\ \dfrac{3}{5}\dfrac{\eta_{\rm eRD}^{2}}{\eta_{\rm md}^{2}}\zeta_{\chi}&\eta_{\rm md}\leq\eta<\eta_{\rm dec}\vspace{0.3em}\\ \dfrac{2}{3}\dfrac{\eta_{\rm eRD}^{2}\eta_{\rm dec}^{2}}{\eta_{\rm md}^{2}\eta^{2}}\zeta_{\chi}&\eta_{\rm dec}\leq\eta\,,\end{cases}$

(4.8)

where we have used the fact that $\Phi$ decreases by a factor of ($9/10$) when the dominant density perturbations transition from RD to MD, and increases by ($10/9$) during the transition from MD to RD.

Now let us similarly analyse the evolution of perturbations for a co-moving mode $k_{\rm eMD}$ that re-enters during eMD. Since energy density in the PT sector with large perturbations drops during eMD, $\Phi(k_{\rm eMD})$ decreases even when the mode is superhorizon, as can be seen in figure 3. Using superhorizon conservation of $\zeta_{\rm nPT}$, we see that $\delta_{\rm nPT}\sim-\Phi$ also decreases. After horizon re-entry, however, $\delta_{\rm nPT}\propto a$ increases while $\Phi$ remains constant till the end of the eMD era. Then

$\displaystyle|\delta_{\rm nPT}(k_{\rm eMD})|$

$\displaystyle\approx\begin{cases}2\dfrac{(\eta_{\rm md}\eta)^{2}}{\eta_{\rm eMD}^{4}}\zeta_{\chi}&\eta_{\rm eMD}\leq\eta<\eta_{\rm dec}\vspace{0.3em}\\ \dfrac{8}{3}\dfrac{(\eta_{\rm md}\eta_{\rm dec})^{2}}{\eta_{\rm eMD}^{4}}\zeta_{\chi}&\eta_{\rm dec}\leq\eta\,,\end{cases}$

(4.9)

and the evolution of $\Phi$ is

$\displaystyle|\Phi(k_{\rm eMD})|$

$\displaystyle\approx\begin{cases}\dfrac{4\hat{f}_{\rm PT}}{5+\hat{f}_{\rm PT}}\zeta_{\chi}+\dfrac{3(1-\hat{f}_{\rm PT})}{5+\hat{f}_{\rm PT}}\zeta_{\phi}\approx\dfrac{4\eta_{\rm md}^{2}}{5\eta_{\rm eMD}^{2}}\zeta_{\chi}+\dfrac{3}{5}\zeta_{\phi}&\eta_{\rm eMD}\leq\eta<\eta_{\rm dec}\vspace{0.3em}\\ \dfrac{\eta_{\rm dec}^{2}}{\eta^{2}}\left[\dfrac{4f_{\rm PT}}{5}\zeta_{\chi}+\dfrac{4}{5}\zeta_{\phi}\right]\approx\dfrac{\eta_{\rm dec}^{2}}{\eta^{2}}\left(\dfrac{4}{5}\dfrac{\eta_{\rm md}^{2}}{\eta_{\rm eMD}^{2}}\zeta_{\chi}+\dfrac{4}{5}\zeta_{\phi}\right)&\eta_{\rm dec}\leq\eta\,.\end{cases}$

(4.10)

Here, we have used $f_{\rm PT}\ll 1$. The results of this section will be useful while analysing constraints on our eMD model, which we discuss next.