The Fate of Dense Scalar Stars

TL;DR

This paper investigates the long-term behavior of scalar stars, such as axion stars, using full general relativity simulations, revealing conditions under which they are stable or collapse into black holes, with implications for cosmology and gravitational waves.

Contribution

It provides the first comprehensive numerical analysis of scalar star fate including gravity, showing how different potentials influence stability and collapse, and discusses cosmological consequences.

Findings

KKLT potentials lead to meta-stable scalar stars.

Alpha-attractor potentials can cause collapse to black holes.

Different potentials imply varied gravitational wave signatures.

Abstract

Long-lived pseudo-solitonic objects, known as oscillons/oscillatons, which we collectively call real scalar stars, are ubiquitous in early Universe cosmology of scalar field theories. Typical examples are axions stars and moduli stars. Using numerical simulations in full general relativity to include the effects of gravity, we study the fate of real scalar stars and find that depending on the scalar potential they are either meta-stable or collapse to black holes. In particular we find that for KKLT potentials the configurations are meta-stable despite the asymmetry of the potential, consistently with the results from lattice simulations that do not include gravitational effects. For $\alpha$-attractor potentials collapse to black holes is possible in a region of the parameter space where scalar stars would instead seem to be meta-stable or even disperse without including gravity. Each case gives rise to different cosmological implications which may affect the stochastic spectrum of gravitational waves.

Tables

Table 1. Table 1: Classification of complex and real scalar stars. Here and in the rest of the paper we denote by “ G = 0 𝐺 0 G=0 ” the cases in which gravity effects are negligible (where G 𝐺 G is the Newton’s constant), and by “ G = 1 𝐺 1 G=1 ” the cases in which gravity effects are important.

Scalar	$G=0$	$G=1$
Complex	Q-Balls Global $U\left(1\right)$	Mini-Boson Stars	Boson Stars
		weak self-interactions	strong self-interactions
Real	Oscillons	Oscillatons
	attractive self-interactions

Table 2. Table 2: Summary of the results. The capital letters stand for: D = Dispersion, MS = Meta-Stable, C = Collapse to a black hole. For completeness, we show the comparison with the free-field case: the stability in that case is achieved only if the configuration lies on the stability curve, as explained in  UrenaLopez:2001tw .

Model	Parameters	$M=2.6M_p^2/m$		$M=2.9M_p^2/m$		$M=3.0M_p^2/m$
		$G=0$	$G=1$	$G=0$	$G=1$	$G=0$	$G=1$
Free-Field	$\times$	D	MS	D	MS	D	MS
	$\Lambda /M_p=0.14$	MS	MS	MS	MS	MS	MS
T-models	$\Lambda /M_p=0.17$	MS	MS	MS	C	MS	C
	$\Lambda /M_p=0.2$	MS	MS	MS	C	MS	C
E-models	$\Lambda /M_p=1$	D	MS	D	C	D	C
	$\Lambda /M_p=2$	D	MS	D	MS	D	C
KKLT	$\left|W_0\right|={10}^{-2}$	MS	MS	MS	MS	MS	MS

Equations

$$f\gtrsim {\left(\frac{m}{\text{TeV}}\right)}^{5/6}\text{Hz},$$

$$\varphi (t,\mathbf{x})=\Phi (t)\mathcal{F}(t,\mathbf{x}),$$

$${\Omega }_{\text{GW}}\propto {\mathcal{A}}^4,$$

$${\tau }_{\text{BH}}\sim \frac{M^3}{M_p^4}\simeq {\left(\frac{M_p}{m}\right)}^2\times m^{-1},$$

$$V$$

$$\equiv m^2{\Lambda }^2\left[\frac{1}{2}{\left(\frac{\varphi }{\Lambda }\right)}^2+\frac{1}{3!}\frac{\lambda \Lambda }{m^2}{\left(\frac{\varphi }{\Lambda }\right)}^3+\frac{1}{4!}\frac{g{\Lambda }^2}{m^2}{\left(\frac{\varphi }{\Lambda }\right)}^4\right].$$

$$\lambda \sim m\left(\frac{m}{\Lambda }\right),g\sim {\left(\frac{m}{\Lambda }\right)}^2.$$

$$\frac{5}{3}{\left(\frac{\lambda }{m}\right)}^2-g>0.$$

$$M\simeq \frac{m}{g}\simeq \frac{{\Lambda }^2}{m},R\simeq \frac{1}{m}.$$

$$\frac{R_s}{R}\approx {\left(\frac{\Lambda }{M_p}\right)}^2,$$

$$V=V_0{\tanh }^{2n}\left(\frac{\varphi }{\Lambda }\right),$$

$$r\approx \frac{2}{N_e^2}\frac{{\Lambda }^2}{M_p^2},$$

$${\mathcal{A}}_s=\frac{H^2}{8{\pi }^2ϵ}\sim 2\times 10^{-9}$$

$$V=V_0{\left(\frac{\varphi }{\Lambda }\right)}^{2n}\left\{1-\frac{2n}{3}\frac{{\varphi }^2}{{\Lambda }^2}+\frac{7n+10n^2}{45}\frac{{\varphi }^4}{{\Lambda }^4}+\mathcal{O}({\varphi }^6)\right\}$$

$$V=V_0{\left(1-e^{-\frac{\varphi }{\Lambda }}\right)}^2.$$

$$r\approx \frac{8}{N_e^2}{\left(\frac{\Lambda }{M_p}\right)}^2\text{and}{\mathcal{A}}_s\approx \frac{N_e^2}{12{\pi }^2}\frac{V_0}{{\Lambda }^2{M}_p^2}.$$

$$\frac{V}{V_0}={\left(\frac{\varphi }{\Lambda }\right)}^2-{\left(\frac{\varphi }{\Lambda }\right)}^3+\frac{7}{12}{\left(\frac{\varphi }{\Lambda }\right)}^4+\mathcal{O}\left({\left(\frac{\varphi }{\Lambda }\right)}^6\right)$$

$$K\supset -3\ln \left(T+\overline{T}\right),W=W_0+A{e}^{-a_{N}T},$$

$$\mathcal{L}\supset \frac{3}{4}{\left(\frac{\partial \tau }{\tau }\right)}^2=\frac{1}{2}(\partial \varphi {)}^2,$$

$$\frac{\varphi }{M_p}=\sqrt{\frac{3}{2}}\ln \tau .$$

$$V=\frac{V_0}{6{\tau }^2}\left[a{A}^2(3+a\tau )e^{-2a\tau }-3aA{W}_0{e}^{-a\tau }\right]+V_{\mathrm{dS}},$$

$$W_0=-\frac{1}{3}A{e}^{-a\bar{\tau }}(5-2a\bar{\tau }).$$

$$\begin{array}{rl}V=& \frac{a^3{A}^2{e}^{-2a\bar{\tau }}}{12{\bar{\tau }}^2}\{(3+2a\bar{\tau })(\tau -\bar{\tau }{)}^2+\left(\frac{-19}{3}a-\frac{6}{\bar{\tau }}-2a^2\bar{\tau }\right)(\tau -\bar{\tau }{)}^3\\ & +\left(\frac{59}{12}{a}^2+\frac{9}{{\bar{\tau }}^2}+\frac{32a}{3\bar{\tau }}+\frac{7a^3\bar{\tau }}{6}\right)(\tau -\bar{\tau }{)}^4+....\}.\end{array}$$

$$d{s}^2=-{\alpha }^{2}d{t}^2+{\gamma }_{ij}(d{x}^i+{\beta }^{i}dt)(d{x}^j+{\beta }^{j}dt),$$

Full text

1 Introduction

The discovery of the Higgs field Aad:2012tfa ; Chatrchyan:2012xdj represents the first observation of a fundamental scalar field111Even though the possibility that the Higgs is not a fundamental particle is not ruled out yet, see Csaki:2015hcd for a recent review.. However, there are many reasons to believe that the Higgs is not the only fundamental scalar degree of freedom existing in nature. In fact, most extensions of the Standard Model of particle physics (SM) provide for the existence of scalar fields at various energy scales. Some examples include:

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The existence of a pseudo-scalar field, the QCD axion, is implied by one of the best motivated solutions to the strong CP problem of QCD, via the Peccei-Quinn mechanism Peccei:1977hh , see Marsh:2015xka for a review.

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At least one scalar field is required in the simplest models of inflation, which is the currently most accepted extension of $\Lambda$CDM: it describes an accelerated expansion of the Universe during its early stages due to the potential energy stored in a scalar field that is undergoing a slow-roll motion, see e.g. Baumann:2009ds for a review.

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Any supersymmetric extension of the SM includes scalar fields with masses above the electroweak scale, see e.g. Martin:1997ns ; Weinberg:2000cr .

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String theory predicts the existence of several gravitationally coupled scalar fields, called moduli, whose vacuum expectation values parametrize the size and shape of the extra-dimensions required for the consistency of the theory Ibanez:2012zz . String theory also predicts the existence of a plethora of axion-like particles, not necessarily related to the strong CP problem of QCD Arvanitaki:2009fg ; Cicoli:2012sz .

• •

Very recently four-dimensional extensions of the SM (not including gravity) modelled as fundamental QFT have been considered Litim:2014uca . They are based on the UV completion mechanism called asymptotic safety, characterized by the presence of a non trivial UV fixed point of the renormalization group which allows to overcome problems such as the presence of Landau poles in the theory. Perturbative analyses of interacting non-Abelian gauge fields, fermions and scalars in the Veneziano limit for different gauge groups and matter representations have shown that generically several fundamental scalar fields are required for the UV fixed point to exist.

The classical equations of motion of scalar fields admit (meta-)stable localized solutions222The energy density associated to a (pseudo-)solitonic solution goes rapidly to zero far away from the centre of the (pseudo-)soliton. While solitons are classically stable due to some conservation law, pseudo-solitons can be long-lived but eventually decay., also known as (pseudo-)solitons Weinberg:1996kr ; Rubakov:2002fi ; Manton:2004tk ; Tong:2005un . The rapid dispersion of such localized solutions is avoided due to the non-linearity of the equations of motion, giving rise to stable or long-lived compact objects that, if formed during the early Universe, can leave observable signatures in the form of a stochastic background of Gravitational Waves (GWs). The detection of such GW background would provide valuable information about the pre-Big Bang Nucleosynthesis (BBN) era Kusenko:2008zm ; Dolgov:2011cq ; Amin:2011hj ; Amin:2014eta ; Amin:2013ika ; Amin:2010jq ; Antusch:2016con ; Antusch:2017flz . In some scenarios, long-lived (pseudo-)solitons can survive till the current time, either providing a natural candidate for dark matter Kusenko:1997ad ; Kusenko:1997si ; Kusenko:2001vu ; Olle:2019kbo or a source of distinct signatures if a fraction of the dark matter is in the form of compact objects, see e.g. Bai:2016wpg ; Braaten:2016dlp ; Levkov:2016rkk ; Eby:2017xaw ; Hui:2016ltb ; Desjacques:2017fmf .

There exist a plethora of distinct (pseudo-)solitonic compact objects, whose differences can be traced back to the mechanism that guarantees their (meta-)stability, see Krippendorf:2018tei for a recent review. If this is ensured by the conservation of a topological charge, the resulting compact object is a topological soliton Manton:2004tk . Examples of this class are kinks, vortices and skyrmions. If the stability is given by the conservation of a Noether charge, then the compact object is a non-topological soliton Lee:1991ax : some of the simplest representatives of this class are summarized in Tab. 1. We collectively refer to the objects appearing in Tab. 1, i.e. non-topological solitons composed by a single scalar field, as scalar stars.

A complex scalar field whose Lagrangian obeys a global $U(1)$ symmetry333E.g. if the complex field $\phi$ has canonical kinetic terms and scalar potential $V=\frac{1}{2}m^{2}|\phi|^{2}+\frac{g}{4!}|\phi|^{4}$. gives rise to non-topological solitons called boson stars Jetzer:1991jr ; the global $U(1)$ charge guarantees their stability. In the regime in which gravity is negligible the same system leads to non-topological solitons such as Q-balls Coleman:1985ki : they can be formed if the attractive self-interactions of the complex scalar field compensate for the gradient energy. Contrary to fermion stars (like neutron stars) for which the mass of the compact object is $M_{f}\sim M_{p}^{3}/m^{2}$ (where $M_{p}$ is the reduced Planck mass and $m$ the mass of the corresponding particle), typical boson stars have a much smaller mass $M_{b}\sim M_{p}^{2}/m$ and are therefore sometimes called mini-boson stars Jetzer:1991jr . If the self-interactions are strong enough, i.e. if $gM_{p}^{2}/m^{2}\gg 1$ (the coupling $g$ is defined in the scalar potential in footnote 3), the star can be as heavy as the corresponding fermion star and then it is simply called boson star Colpi:1986ye .

In the case of real scalar fields, there is no global symmetry to guarantee the stability of the compact objects, but these can be long-lived due to approximate symmetries Mukaida:2016hwd . The object in question in this case is a pseudo-soliton, whose main representative is given by an oscillaton. Typical examples of oscillatons are axion stars444Axion stars are expected to be produced in the unbroken PQ scenario Kolb:1993zz ; Kolb:1993hw ; Hogan:1988mp ; Visinelli:2017ooc and for this reason they are particularly well motivated.. Oscillons Seidel:1991zh ; Kolb:1993hw ; Alcubierre:2003sx ; UrenaLopez:2001tw ; UrenaLopez:2002gx ; UrenaLopez:2012zz belong to the same class as oscillatons, but they are typically restricted to the regime in which the role of gravity is negligible. Their stability is provided by attractive non-linear interactions, rather than gravity: their existence requires the scalar potential to be shallower than quadratic at least on one side around the minimum. Oscillons are formed during a preheating-like stage in a wide variety of inflationary models Amin:2011hj ; Amin:2014eta ; Amin:2013ika ; Amin:2010jq ; Antusch:2016con ; Antusch:2017flz , as well as in string moduli potentials as a consequence of field displacement Antusch:2017flz . We collectively refer to pseudo-solitonic objects arising from a single real scalar field, i.e. the bottom line of Tab. 1, as real scalar stars (RSSs). The goal in this paper is to explore the importance of gravity for the stability of RSSs, studying their dynamics at the border between the oscillon and oscillaton regimes.

The fate of RSSs is the subject of the analysis in the present paper. It is known that in the free-field case the solutions are characterized by a stable and an unstable branch UrenaLopez:2002gx : perturbed configurations belonging to the latter either collapse to black holes or they migrate to the stable branch, depending on the sign of the perturbation. In the interacting case, studying the equilibrium configurations is challenging both from the analytic and from the numerical points of view UrenaLopez:2012zz . It is, however, possible to address this question performing numerical studies that include the effects of General Relativity (GR) and the full non-linear dynamics of the self-interactions555See Cotner:2018vug for an alternative approach..

An example of such a study is Helfer:2016ljl , where the authors considered stability of RSSs in the specific case of an axion potential. As expected they found that, depending on the parameters of the model, axions stars can be stable, disperse or collapse to black holes. Interestingly, non-linear interactions are crucial for a proper understanding of the dynamics of these objects: even if the initial radius of the star is much larger than the corresponding Schwarzschild radius, the interplay of non-linear interactions and gravity can drive the star to collapse or to dispersion.