Growth of Dark Matter Perturbations during Kination

I Introduction

There are no direct observational probes of the period between the end of inflation and the beginning of Big Bang Nucleosynthesis (BBN), and as a result, our understanding of this period is severely limited. Unfortunately, this ignorance hinders our ability to understand both baryogenesis and dark matter production (Giudice et al. 2001, e.g.). There is hope that the spectrum of gravitational waves generated prior to BBN could probe this era, but these probes either require futuristic gravitational wave detectors Boyle and Steinhardt 2008; Easther and Lim 2006; Easther et al. 2008; Amin et al. 2014; Giblin and Thrane 2014 or a network of cosmic strings Cui et al. 2018. The matter power spectrum provides another way to probe this era. For example, an early-matter-dominated era (EMDE) prior to BBN enhances the small-scale matter power spectrum and increases the abundance of microhalos Erickcek and Sigurdson 2011; Erickcek 2015. These microhalos enhance the dark matter annihilation rate by several orders of magnitude, depending on the cutoff in the small-scale matter power spectrum. These boosted annihilation rates are sufficient to bring some EMDE scenarios with otherwise undetectable dark matter particles into tension with Fermi-LAT observations of dwarf spheroidal galaxies Erickcek 2015; Erickcek et al. 2016.

Another possibility is that there was a period of kination between the end of inflation and the beginning of BBN, during which the Universe was dominated by a fast-rolling scalar field (a kinaton) Spokoiny 1993; Joyce 1997; Ferreira and Joyce 1998. Kination was initially proposed as a post-inflationary model that allows the Universe to transition to radiation domination even if the inflaton does not fully decay into radiation Spokoiny 1993. Kination also facilitates baryogenesis Joyce 1997, and the kinaton can mimic the effects of a cosmological constant if its potential energy becomes dominant at very late times Ferreira and Joyce 1998; Peebles and Vilenkin 1999; Dimopoulos and Valle 2002; Dimopoulos 2003; Chung et al. 2007.

In Ref. Redmond and Erickcek 2017, we explored how the dark matter density evolves if it is thermally produced during an era of kination, and we derived analytic expressions for the dark matter relic abundance; see also Refs. Profumo and Ullio 2003; Pallis 2005; Pallis 2006a; Pallis 2006b; Gomez et al. 2009; Lola et al. 2009; Pallis 2010; D’Eramo et al. 2017; Visinelli 2017; D’Eramo et al. 2018. To obtain the observed dark matter relic abundance, dark matter that is thermally produced during an era of kination requires larger annihilation cross sections than dark matter that is thermally produced during radiation domination. Using recent observational limits on dark matter annihilations within dwarf spheroidal galaxies Ackermann et al. 2015 and the Galactic Center Lefranc and Moulin 2016, we were able to place tight constraints on the dark matter mass and the temperature at which kinaton-radiation equality occurs, provided that the dark matter reaches thermal equilibrium during an era of kination Redmond and Erickcek 2017.

In this work, we study what effect kination has on the growth of dark matter density perturbations. If kination enhances the growth of dark matter density perturbations, the resulting small-scale structure would increase the dark matter annihilation rate. This boost to the annihilation rate would place even tighter constraints on scenarios where dark matter reaches thermal equilibrium during an era of kination. If the growth of perturbations during kination amplifies the dark matter annihilation rate by a factor of 10, then dark matter that is thermally produced during kination and annihilates via the $b\overline{b}$, $\tau^{+}\tau^{-}$, or $W^{+}W^{-}$ annihilation channels will be ruled out Redmond and Erickcek 2017.

First, we numerically determine the evolution of cosmological perturbations during an era of kination. Surprisingly, we find that dark matter density perturbations grow linearly with the scale factor for perturbation modes that enter the horizon during kination. To better understand this linear growth, we derive analytic expressions for the evolution of the gravitational potential $\Phi$ and fractional dark matter density perturbation $\delta_{\chi}$, not only during an era of kination, but also for scenarios where the dominant component of the Universe has a generic equation-of-state parameter $w$. We determine that once a mode enters the horizon, the gravitational potential drops sharply and then oscillates with a decaying amplitude if the dominant energy density has ${w>0}$. In addition, if ${w>1/3}$, then ${\delta_{\chi}\propto a^{3w/2-1/2}}$, where $a$ is the scale factor. Therefore, if a perturbation mode enters the horizon during an era of kination $(w=1)$, then $\delta_{\chi}$ grows linearly with the scale factor. This growth leaves an imprint on the matter power spectrum. We determine that for modes that enter the horizon during an era of kination, ${\delta_{\chi}/\Phi_{0}\propto k^{1/2}}$, where k is the comoving wave number, and $\Phi_{0}$ is the value of the gravitational potential on superhorizon scales during kination.

Our perturbation analysis is applicable for scenarios in which dark matter does and does not reach thermal equilibrium during an era of kination. References Redmond and Erickcek 2017; D’Eramo et al. 2017 determined that if dark matter reaches thermal equilibrium during an era of kination, annihilations do not cease until after the Universe becomes radiation dominated. We determine that these “relentless” annihilations do not significantly influence the evolution of $\delta_{\chi}$ after a mode has entered the horizon. Since dark matter annihilation cannot lead to deviations from adiabaticity on superhorizon scales Weinberg 2003; Weinberg 2004a; Weinberg 2004b, “relentless” annihilation has a minimal effect on the matter power spectrum.

In Section II, we present the evolution equations that govern density and velocity perturbations. In Sections III.1 and III.2, we derive analytic expressions for the evolution of the gravitational potential and dark matter density perturbations, respectively. In Section IV, we determine how the matter power spectrum scales with wave number following an era of kination. In Section V, we summarize our results and discuss their implications. The appendices detail the derivation of the perturbation evolution equations and their initial conditions. Natural units $(\hbar=c=k_{B}=1)$ are used throughout this work.

II Perturbation Evolution

We consider a three-fluid model consisting of dark matter, radiation, and the kinaton. The kinaton is a fast-rolling scalar field: ${w\equiv P_{\phi}/\rho_{\phi}\simeq 1}$, where $P_{\phi}$ is the kinaton pressure and $\rho_{\phi}$ is the kinaton energy density. We assume that the dark matter is composed of Majorana particles and that the kinaton does not decay nor otherwise interact with radiation or dark matter. However, dark matter and radiation are thermally coupled via pair production and annihilation. Therefore, the equations for $\rho_{\phi}$, the radiation energy density $\rho_{r}$, and the dark matter number density $n_{\chi}$ are

where $K_{2}(z)$ is a modified Bessel function of the second kind. Equation (3) matches Eq. (2) to within $0.1\%$ for $m_{\chi}/T\gtrsim 6$. In addition, when evaluating ${\langle E_{\chi}\rangle}$, we make the approximation that ${\langle E_{\chi}\rangle\simeq\sqrt{m_{\chi}^{2}+(3.151\,T)^{2}}}$, which matches ${\rho_{\chi}/n_{\chi}}$ to within $10\%$.

We numerically solve Eq. (4) for various $k$ values starting well before each mode enters the horizon and after the dark matter becomes nonrelativistic ${(m_{\chi}/T\gtrsim 3)}$. For any given $k$ mode, we assume that the perturbations are adiabatic before horizon entry. References Weinberg 2004a; Weinberg 2004b demonstrated that perturbations that are initially adiabatic remain adiabatic before horizon entry even in the presence of energy exchange between fluids. This implies that the initial perturbations are all directly related to the initial gravitational potential $\Phi_{0}$: see Appendix B.

Once a mode enters the horizon, the dark matter density perturbation experiences a kick from the decaying gravitational potential (see Figure 3). After the kick, the dark matter density perturbations grow linearly with the scale factor until kinaton-radiation equality, after which they grow logarithmically. The evolution of dark matter density perturbations is oddly similar during an era of kination and matter domination, in spite of the fact that the pressure of the kinaton forces $\Phi$ to evolve very differently during an era of kination. In the following sections, we analytically solve for the evolution of $\Phi$ and $\delta_{\chi}$ in order to determine the physical mechanism behind the linear growth of $\delta_{\chi}$ during kination.

III Analytic Expressions

To understand the evolution of $\delta_{\chi}$, we must first understand the evolution of $\Phi$. To do so, we compare how $\Phi$ evolves for modes that enter the horizon during various eras. To form a single differential equation for $\Phi$, we start with the time-time and space-space components of the perturbed Einstein equations:

where a dot represents differentiation with respect to conformal time, and $\delta\rho$ and $\delta P$ are the dominant fluid’s energy density and pressure perturbations. Assuming that the dominant fluid has a constant equation of state, ${\delta P=w\,\delta\rho}$. Combining Eq. (6) with the second Friedmann equation yields a second-order differential equation for $\Phi$ that is dependent on $w$:

where $\tau$ is the conformal time. The solution to Eq. (7) for ${w>0}$ is

where $b=1/2-3(1+w)/(3w+1)$, $C_{1}$ and $C_{2}$ are integration constants, $J_{b}$ is a Bessel function of the first kind, and $Y_{b}$ is a Bessel function of the second kind. Conformal time and the scale factor are related via $w$: since ${H(a)=H_{1}a^{-3(1+w)/2}}$, ${\tau=[H_{1}(3w+1)/2]^{-1}\,a^{(3w+1)/2}}$. Using this relation, Eq. (8) is rewritten in terms of the scale factor, and $C_{1}$ and $C_{2}$ are determined by demanding that $\Phi\rightarrow\Phi_{0}$ as $a\rightarrow 0$:

III.2 δχ\delta_{\chi} Evolution

If a perturbation mode enters the horizon during an era of kination, we have seen numerically that the dark matter density perturbation experiences linear growth. By deriving an analytic expression for $\delta_{\chi}$ we will gain an understanding of where this linear growth originates. In the limit of kinaton domination, Eqs. (4c) and (4d) can be rewritten as

From these equations, we derive a single second order differential equation for $\delta_{\chi}$:

where ′ denotes a derivative with respect to $b$. Given the homogeneous solutions $C_{1}$ and $C_{2}a$, the Green’s function for an era of kination is $(a-b)$. Therefore, the particular solution (PS) is

The particular solution and its derivative equal zero at ${a=0}$. If we neglect the effects of dark matter annihilations, the adiabatic initial condition for $\delta_{\chi}$ requires ${\delta_{\chi}=\Phi_{0}}$ and ${\delta_{\chi}^{\prime}(a)=0}$ at ${a=0}$, which implies that ${C_{2}=0}$ and ${C_{1}=\Phi_{0}}$. Combining the homogeneous and particular solution produces the final analytic expression for $\delta_{\chi}$:

Combining these equations results in a second-order differential equation for $\delta_{\chi}$:

The homogeneous equation corresponding to Eq. (20) is ${\delta^{\prime\prime}_{\chi}+\frac{3}{2}(1-w)\frac{\delta^{\prime}_{\chi}}{a}=0}$. If $w\neq 1/3$, the homogeneous solution is

whereas if $w=1/3$, the homogeneous solution is

We showed in Figure 2 that the evolution of $\Phi$ is qualitatively the same for perturbation modes that enter the horizon when the dominant component of the Universe has $w>0$. Therefore, the source term will also be qualitatively the same for these scenarios. Since the integral of the source term is constant at late times, the Green’s function produces a similar functional form for the evolution of $\delta_{\chi}$ compared to the homogeneous solution. Therefore, if a perturbation mode enters the horizon and ${w>0}$ and ${w\neq 1/3}$, then the Green’s function demands

where $A$ and $B$ are integration constants. Overall, the logarithmic growth experienced by subhorizon matter perturbations during radiation domination is a by-product of the homogeneous solution to Eq. (20). Similarly, it is the homogeneous solution that leads to the linear growth of subhorizon matter perturbations during an era of kination.

The different $\delta_{\chi}$ growth rates can be attributed to the motion of the dark matter particles. We saw in Figure 1 that once a mode enters the horizon, the dark matter density perturbation experiences a kick from the decaying gravitational potential. This kick causes the dark matter particles to drift toward overdense areas, even after $\Phi\rightarrow 0$. The comoving displacement of massive particles is

The fact that dark matter particles are drifting toward initially overdense regions does not necessarily mean structure will form during an era of kination. In other words, it is still uncertain how $\delta_{\chi}$ will evolve in the non-linear regime. One possibility is that dark matter particles are moving fast enough that, instead of collapsing and forming structure, they pass by each other and overdense regions becomes underdense. Collapse may still be possible, however, if local areas of matter domination persist long enough to halt the motion of particles through the overdense region. Further investigation is required to determine how $\delta_{\chi}$ evolves in the non-linear regime during an era of kination or radiation domination. However, we can be certain that modes that remain linear until matter-radiation equality will form halos. In addition, these halos form earlier than they would if the Universe had been radiation dominated when the relevant scales entered the horizon, due to the enhanced growth of $\delta_{\chi}$ during kination.

IV The Matter Power Spectrum

V Conclusion

Acknowledgments

We thank M. Sten Delos and Chris Hirata for insightful and helpful discussions. This work was supported by NSF Grant No. PHY-1417446.

Appendix A Derivation of the Perturbation Equations

The perturbation evolution equations are derived by perturbing the covariant form of the energy-transfer equations given in Eq. (1). We follow the same approach as that outlined in Refs. Erickcek and Sigurdson 2011; Barenboim and Rasero 2014; Fan et al. 2014; Erickcek 2015. The kinaton, dark matter, and radiation all behave as perfect fluids with energy momentum tensors

where $i$ represents the individual fluids. In the absence of spatial variations,

where a dot represents differentiation with respect to proper time. Using Eq. (31) and Eq. (1), the covariant energy exchange for this three-fluid model is

Equation (33) is different than the definition of $L_{\nu}$ in Ref. Erickcek 2015. We have corrected the expression for $L_{\nu}$ to account for the fact that, while in thermal equilibrium, the dark matter is pair-produced with the same velocity as the radiation. This change introduces coupling terms between $\theta_{\chi}$ and $\theta_{r}$ that are only relevant while pair production is important.

Next, we evaluate Eq. (30) using the perturbed Friedmann-Robertson-Walker (FRW) metric

Taking into account first-order perturbations, the ${\mu=0}$ component of Eq. (30) requires that each fluid obey the equation

where $w_{i}$ is the equation of state parameter for a given fluid, $\theta_{i}\equiv a\partial_{j}v^{j}$ is the divergence of the fluid’s physical velocity, and $Q^{(i),(0)}_{0}$ and $Q^{(i),(1)}_{0}$ are the zeroth-order and first-order components of $Q_{0}^{(i)}$ for each fluid. The divergence of the spatial component of Eq. (30) requires that each fluid obey the equation

Applying Eqs. (38) and (39) to the kinaton (${w_{k}=1}$), dark matter (${w_{\chi}=0}$), and radiation (${w_{r}=1/3}$) yields the following perturbation equations

The perturbed time-time component of the Einstein equation yields

Equation (40) assumes that the scalar field does not interact with either the dark matter or radiation and also assumes that the dark matter is created solely from thermal production.

Appendix B Initial Conditions

We first solve for the evolution of the kinaton perturbations and the gravitational potential during an era of kination for superhorizon modes. Simplifying Eqs. (4a), (4b), and (5) yields:

Since the number of relativistic particles created or destroyed from dark matter annihilations is not sufficient to influence the evolution of $\rho_{r}$, the interaction between dark matter and radiation will not influence the evolution of radiation perturbations. Evaluating Eqs. (4e) and (4f) in the superhorizon limit, while ignoring the effects of dark matter annihilations, results in

For freeze-in scenarios, the initial condition for $\delta_{\chi}$ is more difficult to determine from the perturbation equations. We therefore choose the freeze-in initial condition for $\delta_{\chi}$ to ensure that ${\delta_{i}(\rho_{i}/\rho^{\prime}_{i})}$ is the same for all fluids. Equations (44b) and (45a) already imply that the perturbations to the kinaton and radiation are adiabatic. To solve for the initial condition for $\delta_{\chi}$ we set ${\delta_{\chi}(\rho_{\chi}/\rho^{\prime}_{\chi})=\delta_{\phi}(\rho_{\phi}/\rho^{\prime}_{\phi})}$. Since ${\rho_{\phi}\propto a^{-6}}$,

Finally, since adiabatic perturbations require $\theta$ to be the same for all fluids Weinberg 2003,

References