EARLY MASS VARYING NEUTRINO DARK ENERGY: NUGGET FORMATION AND HUBBLE ANOMALY

The existence of dark matter, dark energy, and the discovery of neutrino mass have made the frontier of cosmology and particle physics both fascinating and challenging to explain. A sterile neutrino with mass $m\gtrsim 5\,{\rm keV}$ is a viable warm dark matter candidate (Dodelson & Widrow, 1994; Valdarnini et al., 1998; Boyarsky et al., 2009; Bezrukov et al., 2010; Petraki & Kusenko, 2008; Laine & Shaposhnikov, 2008; Abazajian, 2006; Irˇsič et al., 2017; Gilman et al., 2020; Nadler et al., 2020). However, a lighter fermion with $m\sim{\rm eV}$ usually falls into the same dark-matter misfortune as it free-streams until a relatively late times in cosmic evolution and erases structure on small scales. Only a tiny fraction of the dark matter abundance can be in the form of neutrinos or other light fermions which puts a stringent upper bound on neutrino mass (Hannestad et al., 2010; Bell et al., 2006; Acero & Lesgourgues, 2009). On the other hand, recent data from the MiniBooNE experiment Aguilar-Arevalo et al. (2010, 2013, 2018) might indicate the existence of light sterile neutrino states (eV - 10 eV) and also the possibility of these light states having nontrivial interactions (Dasgupta & Kopp, 2014). There have been attempts to revive the neutrino or lighter sterile neutrino as viable dark matter candidate with nontrivial cosmological histories or exotic interactions (Das & Weiner, 2011; Bjaelde & Das, 2010; Abazajian & Kusenko, 2019).

In this paper, we propose a scenario in which at some point in the radiation dominated era (RDE), a population of eV-mass decoupled fermions undergo a phase transition and clump into small dark matter nuggets due to the presence of a scalar mediated fifth force. Before the transition, the neutrino-scalar fluid behaves like early dark energy (EDE) as the field adiabatically stays at the minimum of the effective potential, $V_{\rm eff}$. Once the transition happens, fermions inside the nugget no longer free-stream and the nuggets as a whole behave like cold dark matter. Due to its interaction with a scalar $\phi$, the fermion mass is “chameleon” (Khoury & Weltman, 2004) in nature and depends on scalar vacuum expectation value. The dynamics of $\phi$ are controlled by $V_{\rm eff}$ instead of just $V(\phi)$ (Fardon et al., 2004; Das et al., 2006), and the scalar field adiabatically tracks the minima of $V_{\rm eff}$.

As the background fermion density dilutes due to the expansion of the universe, the minimum of $V_{\rm eff}$ is time-dependent and so is the fermion mass. We consider a scenario where the fermion mass is inversely proportional to the scalar vev (through the see-saw mechanism (Yanagida, 1979; Yanagida & Yoshimura, 1980; Gell-Mann et al., 1979; Mohapatra & Senjanovic, 1980)) such that at a red-shift $z_{F}$ (before matter-radiation equality), it becomes non-relativistic and the attractive scalar force starts to dominate over the free-streaming. In situations like this, it has been shown in Afshordi et al. (2005) that the effective sound speed of perturbation of the combined fluid (scalar and non-relativistic fermions) becomes imaginary following a hydro-dynamical instability which results in nugget formation. The majority of fermions within each scalar Compton volume collapse into a nugget until the Fermi pressure intervenes and balances the attractive force. From our numerical results we find that the scalar field inside the nugget is displaced in such a way that the fermion mass inside the nugget is much smaller than outside, ensuring the stability of the nuggets. Along with the nugget profile’s numerical solution, we also solve for the entire cosmological evolution for our scenario and obtain CMB angular power spectra. We find that the effects on CMB is more prominent when phase transition happens closer to MRE this in turn provides a conservative lower bound on transition redshift $z_{F}\geq 10^{5}$.

As an important aspect of the nugget formation mechanism, we explore its implication for recent Hubble tension as the neutrino-scalar fluid naturally behaves as dark energy fluid prior to nugget formation. To resolve the recent Hubble anomaly where local distance ladder measurements of $H_{0}$ disagree with the Planck measured value at $\sim 5\sigma$ (Humphreys et al., 2013; Verde et al., 2019; Wong et al., 2019), one obvious choice is to increase dark radiation before CMB. But it is a well-known fact that just increasing dark radiation content ($\Delta N_{\textrm{eff}}$) only partially resolves Hubble tension as it makes the high-$\ell$ CMB prediction deviate from Planck observation (Riess et al., 2016; Ichikawa et al., 2007; Blinov & Marques-Tavares, 2020). But instead of increasing dark radiation, the appearance of an early dark energy component before CMB has been one of the most well studied and successful avenues (Karwal & Kamionkowski, 2016; Poulin et al., 2018, 2019; Agrawal et al., 2019; Braglia et al., 2020; Chudaykin et al., 2020; Ivanov et al., 2020; Weiner et al., 2020; Hill & Baxter, 2018; Smith et al., 2020a) for solving Hubble tension though there are challenges Jedamzik et al. (2020) especially when the model is subject to recent LSS results. From the theoretical perspective, the physics of matter-radiation equality and early dark energy are completely disconnected, so that some degree of fine-tuning is needed them to appear nearly simultaneously. However in our case, the instability of the neutrino-scalar fluid naturally happens around the MRE, which occurs due to the transition of massive neutrinos from relativistic to non-relativistic. There has been recent work (Sakstein & Trodden, 2020; Niedermann & Sloth, 2019, 2020a, 2020b) where EDE has been connected to neutrino physics but in a different context than ours.

In this work, we qualitatively demonstrate that the scalar field adiabatically traces the minimum of the effective potential of neutrino-scalar interaction prior to nugget formation, thus behaving like an early DE for a short duration. This early dark energy (EDE) behaviour of the neutrino-scalar fluid increases the Hubble expansion rate in a natural way as neutrino mass controls the period of EDE. As soon as the nugget forms, neutrino decouples from the scalar field and the scalar no longer behaves like DE. We find that though the presence of EDE is natural in this scenario of nugget formation, if the entire energy goes from dark energy to dark matter (as in simple model of DM formation) it only partially solves the Hubble anomaly. This is expected as it is now a well established fact in the literature that the early dark energy phase should dilute at least like radiation or faster to resolve Hubble anomalyLin et al. (2019). But as we discuss qualitattively, it is highly possible that a major fraction of the early DE phase goes into the scalar field and provides a solution to Hubble anomly, especially through a scalar field kinetic energy dominated phase following the nugget formation, similar to Alexander & McDonough (2019). Our ongoing MCMC analysis ( work in progress ) with different potential with early mass varying neutrino dark energy model will clarify how effectively one can resolve Hubble tension in this scenario.

The plan of the paper is as follows: In Section 2, we derive the condition for instability for a quadratic potential. In Section 3, we solve for the static configuration of the scalar field and show nuggets indeed form. In Section 4, we derive the perturbation equation and numerically find for CMB angular power spectra followed by discussion of Hubble anomaly in Section 5. In Section 4.2, we study stability of DM nugget and applicable constraints on $\Delta N_{\rm eff}$ bounds given by Planck and BBN results. Finally we conclude in Section 7 and discuss future directions.

In the original mass varying neutrino model (Fardon et al., 2004) the Majorana mass term of a heavy chiral fermion in dark sector was taken to be a linear function of a scalar field $\phi$. Here we consider a more general interaction involving a function $f(\phi)$ with the following Lagrangian

$${\cal L}\supset m_{D}\psi_{1}\psi_{2}+f(\Phi)\psi_{2}\psi_{2}+V(\Phi)+\mbox{H.c.},$$

(1)

where $\psi_{1,2}$ are fermion fields, with $\psi_{2}$ corresponding to the heavier mass eigenstate. Both fermion fields are written as two component left chiral spinors, $m_{D}$ is the Dirac mass term and $V(\phi)$ is the scalar potential. Note that if $\psi_{2}$ is considerably heavier, we can integrate it out from the low energy effective theory obtaining $m(\phi)\equiv m_{\psi_{1}}=m_{D}^{2}/f(\phi)$, otherwise one can obtain the mass eigenvalues by diagonalizing the fermion mass matrix.
We adopt a simple quadratic potential $V(\phi)=m_{\phi}^{2}\phi^{2}$ and assume a Yukawa-type interaction with $f(\phi)=\tilde{\lambda}\phi$, such that the model looks like

$${\cal L}\supset m_{D}\psi_{1}\psi_{2}+\tilde{\lambda}\phi\psi_{2}\psi_{2}+V(\phi).$$

(2)

However, prior to phase transition, the dynamics of $\phi$ is controlled by an effective potential given by Fardon et al. (2004):

$$V^{\rm eff}=\rho_{\psi_{1}}+m_{\phi}^{2}\phi^{2}.$$

(3)

It is instructive to note that radiative correction to any dark energy potential is a common issue and will also be there for our quadratic potential. But there has been some effort to control it through a super symmetric theory of neutrino dark energy Fardon et al. (2006). As the first term decreases due to Hubble expansion, the minima of the potential moves to a lower value, thus increasing the mass of the lightest fermion eigenstate. In our model, $\psi_{1}$ is taken to be a sterile neutrino of mass $\sim$ eV. As the field evolves adiabatically, the field responds to the average neutrino density, relaxing at the minimum of the effective potential. In this minimum, we see that both mass of the neutrino, $m_{\psi_{1}}$ and the scalar field potential, $V(\phi)$ become functions of the neutrino density,

$$\frac{\partial V_{\textrm{eff}}}{\partial\phi}=\left(n_{\psi_{1}}+\frac{\partial V}{\partial m_{\psi_{1}}}\right)\frac{\partial m_{\psi_{1}}}{\partial\phi}=0\Rightarrow n_{\psi_{1}}=-\frac{\partial V}{\partial m_{\psi_{1}}}.$$

(4)

We have replaced $\rho_{\psi_{1}}$ with $n_{\psi_{1}}m_{\psi_{1}}$, where $n_{\psi_{1}}$ denotes the number density. The right side of the eq. (4) holds provided that $\partial m_{\psi_{1}}/\partial\phi$ doesn’t vanish. In the non-relativistic limit, there would be no pressure contribution from the neutrinos and similar to Fardon et al. (2004), we neglect the contribution of any kinetic terms of the scalar field to the energy density, which is a good approximation as long as the scalar field is uniform and tracks the minimum of its effective potential adiabatically. Therefore, we can describe the $\phi$-$\psi_{1}$ fluid by the equation of state parameter:

$$w\equiv\frac{\rm Pressure}{\text{Energy Density}}=\frac{-V}{n_{\psi_{1}}m_{\psi_{1}}+V}=-1+\frac{n_{\psi_{1}}m_{\psi_{1}}}{V_{\textrm{eff}}},$$

(5)

where we used eq. (3) in the last step. After a few steps of calculation using eqs. (4) and  (5) and the the relation between $V(\phi)$ and $m_{\psi_{1}}$ through $\phi$, we get,

	$\displaystyle n_{\psi_{1}}m_{\psi_{1}}$	$\displaystyle=$	$\displaystyle-\frac{\partial V}{\partial m_{\psi_{1}}}m_{\psi_{1}}=2m_{\phi}^{2}\left(\frac{m_{D}^{2}}{\tilde{\lambda}m_{\psi_{1}}}\right)^{2}=2V$		(6)
	$\displaystyle\Rightarrow w$	$\displaystyle=$	$\displaystyle-1+\frac{2V}{2V+V}=-\frac{1}{3}.$		(7)

For perfect fluids, the speed of sound purely arises from adiabatic perturbations in the pressure and the energy density. Hence, the adiabatic sound speed, $c_{a}^{2}$ of a fluid can be purely determined by the equation of state. The adiabatic sound speed of our fluid is given by

$$c_{a}^{2}\equiv\frac{\dot{p}}{\dot{\rho}}=w-\frac{1}{3}\frac{\dot{w}}{(1+w)H}=-\frac{1}{3}.$$

(8)

However, in imperfect fluids, dissipative processes generate entropic perturbations in the fluid and therefore we have the more general relation

$$c_{s}^{2}=\frac{\delta p}{\delta\rho}.$$

(9)

This can also be written in terms of the contribution of the adiabatic component and an additional entropic perturbation $\Gamma$ and the density fluctuation in that instantaneous time frame (Kodama & Sasaki, 1984):

$$w\Gamma=(c_{s}^{2}-c_{a}^{2})\delta.$$

(10)

As stated before, the scalar field follows the instantaneous minimum of its potential and this minimum is modulated by the cosmic expansion through the changes in the local neutrino density. So, the mass of the scalar field, $m_{\phi}$ can be much larger than the Hubble expansion rate. Consequently, the coherence length of the scalar field, $m_{\phi}^{-1}$, is much smaller than the present Hubble length. As a consequence of this Afshordi et al. (2005), combined with eq. (8), the sound speed of the neutrino-scalar fluid becomes imaginary once the neutrino becomes non-relativistic followed by instability and formation of nuggets. We have solved the linear perturbation equation (For details see Section 4) to verify this for our choice of quadratic potential. As shown in Fig.1, we compare the evolution of density perturbation of a typical mode (smaller than the range of scalar force) in the radiation dominated era.

The formation of the dense fermionic object in the presence of a scalar mediated force (eg. soliton star, fermionic Q-star) has long been an interesting research topic in different astrophysical contexts (Lee & Pang, 1987; Lynn et al., 1990). The formation of a relativistic star in chameleon theories has also drawn recent interests (Upadhye & Hu, 2009). In the context of neutrino dark energy, formation of neutrino nugget at very late time ($z$ $\sim$ few ) has been studied in Afshordi et al. (2005), while large (Mpc) stable structures of neutrinos clump in the presence of an attractive quintessence scalar has been discussed in Ref. Brouzakis & Tetradis (2006); Brouzakis et al. (2008); Pettorino et al. (2010); Wintergerst et al. (2010). As the quintessence field is extremely light and comparable to Hubble expansion rate ($m\sim H$), the range of the fifth force is very large and a neutrino clump can easily be of the order of Mpc size (Casas et al., 2016; Brouzakis et al., 2008). In all of these studies, these Mpc sizes neutrino clumps form at a very late time in cosmic evolution (around $z\sim$ few) and thus contribute only a tiny fraction to the dark matter relic density. Whereas we focus on a scalar field with much heavier mass ($m_{\phi}\sim 10^{-3}$ eV) and the formation of nuggets takes place at a much earlier redshift ($z_{F}\gtrsim 10^{6}$). We show that, as a result, these compact nuggets can, in principle, comprise the entire cold dark matter relic abundance.

In this section we first discuss the qualitative features of the nugget formation, and then numerically solve the bubble profile for the static configuration. For a detailed analysis of the nugget collapse process involving non-linear dynamics, see Narain et al. (2006). We assume that for a critical over density $\delta$, the sound speed of perturbation becomes negative and the fifth force attracts neutrinos within a Compton sphere of the scalar and finally, Fermi pressure intervenes and balances the attractive scalar force and stable nugget profile is achieved. One can in principle solve the static configuration of nugget by solving coupled Klein-Gordon equation. Taking the metric to have the form

$$ds^{2}=-A(r)dt^{2}+B(r)dx^{2}+r^{2}d\Omega^{2}$$

(11)

The gravitational mass of the system can be found from asymptotic form of $A(r)$ for $r\rightarrow\infty$ and is given by Brouzakis & Tetradis (2006)

$$M(r)=\int 4\pi r^{2}dr\left[\frac{1}{2}\left(\frac{d\phi}{dr}\right)^{2}+V(\phi(r))+\rho_{\psi_{1}}(r)\right]$$

(12)

where, the first term is the gradient energy and the other terms correspond to the $\phi$ dependent fermion mass and the scalar potential $V(\phi)$.

For our case, the size of the nuggets is very tiny and the scalar force is much stronger than gravity; thus, we can safely ignore gravity and will work with the Minkowski metric.

A general feature of the scalar field static configuration is that the scalar vev changes as a function of distance from the center of the nugget and takes an asymptotic value far away from it. The fermion number density, Fermi pressure are all functions of position through $\phi(r)$.

As the fermion mass also varies radially, we adopt the Thomas-Fermi approximation as done in Brouzakis & Tetradis (2006) to find the static configuration. In the Thomas-Fermi approximation, one assumes, at each point in space there exists a Fermi sea with local Fermi momentum $p_{F}(r)$. With an appropriate scalar potential, it has been shown in Lee & Pang (1987); Lynn et al. (1990) that a degenerate fermionic gas can be trapped in compact objects (even in the absence of gravity) which has been called as soliton star or fermion Q-star. These analyses were done with the zero temperature approximation and in this case, fermions can be modeled as ordinary nuclei. To get an estimate of the nugget radius, we will also work with zero temperature approximation. Soon, we will see that our numerical solution has similarities with these previous works where fermions are extremely light inside the nugget and become increasingly more massive as one moves towards the nugget wall. This ensures positive binding energy and thus the stability of the nuggets.

Dark Matter nugget formation from instability in neutrino is a unique phenomenon and it may have some interesting effect on both CMB and linear structure formation. In this section, we explore these effects if we let this phenomenon to account for the entire DM density in the universe. To find the effect of the nugget formation in CMB and linear structure formation, we need to solve for the linear as well as perturbation equations for the whole cosmic history. We adopt the generalized dark matter(GDM) formalism describe our interacting $\phi$-$\psi_{1}$ fluid. In this formalism, the background equation of the fluid is parameterized by its equation of state, $w_{\phi-\psi_{1}}$ which we take to be a function of the redshift $z$. At early times, since we assume the fluid to be relativistic, the coupling between neutrino and the scalar field can be ignored. So, we take $w_{\phi-\psi_{1}}\sim\frac{1}{3}$ at high redshifts. Then, as the neutrinos become non-relativistic around a critical redshift $z_{F}$, an EDE like phase appears when the scalar field relaxes into the effective minima followed by an intermediate transition period where the effective equation of state goes below zero to $w_{\phi-\psi_{1}}\sim-\frac{1}{3}$. During this period, sound speed square, $c_{s}^{2}$ also becomes negative and we have instability in the fluid. Immediately after that, phase transition happens as a result of the instability and the fluid transitions into DM nugget state, where $w_{\phi-\psi_{1}}=0$.

The perturbations equations for our fluid in synchronous gauge from the GDM formalism are given by:

	$\displaystyle\dot{\delta}$	$\displaystyle=$	$\displaystyle-(1+w)\left(\theta+\frac{\dot{h}}{2}\right)-3(c_{s}^{2}-w)H\delta$		(27)
	$\displaystyle\dot{\theta}$	$\displaystyle=$	$\displaystyle-(1-3c_{s}^{2})H\theta+\frac{c_{s}^{2}k^{2}}{1+w}\delta$		(28)

To solve the background and the perturbation equations, we modify the CLASS Boltzmann code (Blas et al., 2011) to replace the CDM component with an extra fluid component which describes our fluid through $w_{\phi-\psi_{1}}$ and $c_{s}^{2}$. In Fig.1, we indeed see that around $z_{F}$, the density fluctuations of the $\phi$-$\psi_{1}$ fluid shoots up to a huge value compared to those of the other components such CDM and neutrinos from standard $\Lambda$CDM scenario. In Fig. 3(right), we plot the fractional energy density of the dominant components of the universe at different times against redshift $z$ for the case $z_{F}=10^{5}$. We see that in early times, radiation is the dominant component as usual but a tiny yet significant portion of the universe ($\sim 10^{-2}$) consists of $\phi$-$\psi_{1}$ as dark radiation. Then around $z_{F}$, $\phi$-$\psi_{1}$ transitions from radiation into matter like state and $\Omega_{r}$ starts to fall while $\Omega_{\phi-\psi_{1}}$ rises. The Matter-radiation equality happens at $z\sim 3400$. The DM nugget energy density dominates the universe until $\Omega_{\Lambda}$ crosses $\Omega_{\phi-\psi_{1}}$ at late times ($z\sim 10^{-1}$). The value of $\Omega_{\phi-\psi_{1}}$ today comes out to be $\sim 0.2637$ which is equal to the dark matter density today according to $\Lambda$CDM.

In Fig.3(left), we plot the CMB temperature anisotropy spectrum along with the residuals from Planck $\Lambda$CDM data Ade et al. (2016) for three different critical redshifts, $z_{F}=10^{5},10^{5.5},10^{4.8}$. We see that the later the critical redshift is, the more it deviates from $\Lambda$CDM . For the case $z_{F}=10^{4.8}$, the residuals go beyond the error bar set by Planck. The residual plot is also dependent on the extent of the transition period which we haven’t considered in this paper. A proper MCMC analysis is needed to find a suitable $z_{F}$ and the extent of the transition period which is left for future work.

In this section, we discuss the recent Hubble anomaly in the context of our model mainly for two reasons. The first reason is- before the nugget formation, the neutrino-scalar fluid naturally goes through short early dark energy domination as explained before. Given that the early DE model has been proposed as one most prominent solution of Hubble anomaly, we explore if our model can be realized as a successful candidate for EDE scenario arising from (sterile) neutrino-scalar interaction.

The second reason is, in a generic EDE model, from the theoretical perspective, the physics of matter-radiation equality and early dark energy are completely disconnected - which requires fine-tuning in order for them to appear nearly simultaneously. Our model is based on physics where scalar mass is of the order of neutrino mass scale (unlike in Sakstein & Trodden (2020), where scalar mass is tuned to an extraordinarily small value, like in the case of quintessence theories) as proposed originally in the theories of neutrino dark energy and the similarity of neutrino mass and MRE scale naturally gives early dark energy phase prior to MRE. In a neutrino-scalar interaction scenario, the background equation for the neutrinos is given by

$$\dot{\rho}_{\psi_{1}}+3H(\rho_{\psi_{1}}+p_{\psi_{1}})=\frac{dln(m_{\psi_{1}})}{d\phi}\dot{\phi}(\rho_{\psi_{1}}-3p_{\psi_{1}})$$

(34)

where the right-hand side is the interaction term. We see that when neutrinos are relativistic, the interaction term is negligible since $(\rho_{\psi_{1}}-3p_{\psi_{1}})\sim 0$. Therefore, the interaction is only prominent when the neutrinos become non-relativistic around $z_{F}$ and its equation of state drops below $w<-1/3$ ( the exact value depends on the potential) and the fluid starts to behave like dark energy. This early phase of DE is only active for a short duration(see right panel of Fig. 9) followed by its disappearance as nugget formation takes place. The excess dark energy gives a local boost to local Hubble expansion rate and the sound horizon for acoustic waves in the photon-baryon fluid, $r_{s}$ is given by

$$r_{s}=\int_{z_{eq}}^{\infty}\frac{c_{s}}{H(z)}dz.$$

(35)

Since H(z) in the EDE scenario was higher than that of $\Lambda$CDM for a short period of time, it would mean that $r_{s}$ is reduced compared to the $\Lambda$CDM model. The angular diameter distance to the last scattering surface is given by

$$d_{A}=\frac{r_{s}}{\theta_{s}},$$

(36)

where $\theta_{s}$ is the angular scale of the sound horizon at matter-radiation equality. This angular scale is fixed as it is solely determined by CMB observation and hence its value is not sensitive to any model. This implies that $d_{A}$ also gets reduced in this scenario. Since $d_{A}$ is inversely proportional to the Hubble constant, this means $H_{0}$ is higher than it is in $\Lambda$CDM.

Here we do a simplest case MCMC analysis for a neutrino scalar fluid with a quadratic potential. We assume after the phase transition, the kination period dominates and most of the energy of neutrino-EDE dilutes fast as $1/a^{6}$. Our MCMC analysis indeed shows that this phase transition indeed relaxes the Hubble anomaly.

Data from MiniBooNE experiment might indicate the existence of light sterile neutrino states of mass (sub-eV to 10 eV). Though this lighter sterile neutrino has relevant properties, in general, light eV-mass sterile fermions are not viable cold dark matter (CDM) candidate due to its excessive free-streaming. Here in this paper

• •

  We realize a scenario of nugget formation with $\rm{eV}$ sterile neutrino in radiation dominated era close to MRE. We analytically show that the condition of instability is generic even with quadratic potential in RDE when sterile neutrino turns non-relativistic followed by nugget formation. Our scenario provides a complete new way of producing a cold dark matter nuggets from light eV sterile neutrino. This dark matter can contribute to a considerable fraction of DM density today depending on model paramters.

• •

  Using Thomas Fermi approximation and solving coupled Klein Gordon equation numerically, we find the static configuration of the scalar field which allows us to calculate nugget mass, radius of nugget and dark matter density. The detailed nugget profile was never solved before in neutrino DE model. The mass of the nugget in our two examples are $\sim 10^{6}$ eV and $\sim 10^{29}$ eV and the corresponding redshift of nugget formation are $z_{F}\sim 0.86\times 10^{5}$ and $6\times 10^{7}$ respectively. For these formation redshifts , we show that the entire dark matter density of the university can arise from this heavy neutrino nuggets!

• •

  We use quadratic scalar potential for the scalar field as it keeps the radiative correction to the potential under control as illustrated in super-symmetric theories of neutrino dark energy (Fardon et al., 2006). Also we use most general Yukawa type coupling in dark sector as proposed in original neutrino dark energy model.

• •

  This natural phase transition prior to MRE in Early neutrino dark energy has strong implications for recent Hubble tension. This model of EDE does not require a fine-tuned dark energy scale yet it can comprise around 10 percent of total energy budget around MRE ( as needed to solve Hubble anomaly (Poulin et al., 2016, 2019)). This early dark energy decays faster than radiation as the nugget forms and scalar rolls of freely along the quadratic potential dominated by a kinetic energy phase. In our case, the DE energy injection and its decay ( due to phase transition) is entirely controlled by $\rm{eV}$ sterile neutrino mass and in principle our scenario can realise a viable early DE particle physics model which can solve Hubble tension. EDE models are still an active area of research and we feel near future LSS data will determine its fate. But as seen from initial MCMC run, our model is consistent with higher $H_{0}$ and does not make $S_{8}$ worse like EDE or ADE models as seen in figure 10. A detailed LSS data analysis is required to precisely tell how well our model performs and what kind of potential of scalar field is preferred when confronted with large scale structure data from galaxy survey or recent weak lensing measurements. This will be reported in near future.