Cosmological evolution of a complex scalar field with repulsive or attractive self-interaction
We study the cosmological evolution of a complex scalar field with a self-interaction potential , possibly describing self-gravitating Bose-Einstein condensates, using a fully general relativistic treatment. We generalize the hydrodynamic representation of the Klein-Gordon-Einstein equations in the weak field approximation developed in our previous paper [Suárez and Chavanis, Phys. Rev. D 92, 023510 (2015)]. We establish the general equations governing the evolution of a spatially homogeneous complex scalar field in an expanding background. We show how they can be simplified in the fast oscillation regime (equivalent to the Thomas-Fermi, or semiclassical, approximation) and derive the equation of state of the scalar field in parametric form for an arbitrary potential . We explicitly consider the case of a quartic potential with repulsive or attractive self-interaction. For repulsive self-interaction, the scalar field undergoes a stiff matter era followed by a pressureless dark matter era in the weakly self-interacting regime and a stiff matter era followed by a radiationlike era and a pressureless dark matter era in the strongly self-interacting regime. For attractive self-interaction, the scalar field undergoes an inflation era followed by a stiff matter era and a pressureless dark matter era in the weakly self-interacting regime and an inflation era followed by a cosmic stringlike era and a pressureless dark matter era in the strongly self-interacting regime (the inflation era is suggested, not demonstrated). We also find a peculiar branch on which the scalar field emerges suddenly at a nonzero scale factor with a finite energy density. At early times, it behaves as a gas of cosmic strings. At later times, it behaves as dark energy with an almost constant energy density giving rise to a de Sitter evolution. This is due to spintessence. We derive the effective cosmological constant produced by the scalar field. Throughout the paper, we analytically characterize the transition scales of the scalar field and establish the domain of validity of the fast oscillation regime. We analytically confirm and complement the important results of Li, Rindler-Daller and Shapiro [Phys. Rev. D, 89, 083536 (2014)]. We determine the phase diagram of a scalar field with repulsive or attractive self-interaction. We show that the transition between the weakly self-interacting regime and the strongly self-interacting regime depends on how the scattering length of the bosons compares with their effective Schwarzschild radius. We also constrain the parameters of the scalar field from astrophysical and cosmological observations. Numerical applications are made for ultralight bosons without self-interaction (fuzzy dark matter), for bosons with repulsive self-interaction, and for bosons with attractive self-interaction (QCD axions and ultralight axions).
pacs:98.80.-k, 95.35.+d, 95.36.+x, 98.80.Jk, 04.40.-b
There is compelling observational evidence for the existence of dark matter (DM) and dark energy (DE) in the Universe. The suggestion that DM may constitute a large part of the Universe was raised by Zwicky zwicky () in 1933. Using the virial theorem to infer the average mass of galaxies within the Coma cluster, he obtained a much higher value than the mass of luminous material. He realized therefore that some mass was “missing” to account for the observations. The existence of DM has been confirmed by more precise observations of rotation curves b1 (), gravitational lensing massey (), and hot gas in clusters b2 (). On the other hand, DE is responsible for the ongoing acceleration of the Universe revealed by the high redshift of type Ia supernovae treated as standardized candles b6 (); b10 (); b11 (). Recent observations of baryonic acoustic oscillations provide another independent support to the DE hypothesis b12 (). In both cases (DM and DE) more indirect measurements come from the Cosmic Microwave Background (CMB) and large scale structure observations smoot (); jarosik (); planck ().
The variations in the temperature of the thermal CMB radiation at K throughout the sky imply and , while the power spectrum of the spatial distributions of large scale structures gives , where is the effective curvature of spacetime, is the present energy density in the relativistic CMB radiation (photons) accompanied by the low mass neutrinos that almost homogeneously fill the space, and is the current mean energy density of nonrelativistic matter which mainly consists of baryons and nonbaryonic DM. These observations give a value of for the present DE density planck ().
One of the most fundamental problems in modern cosmology concerns the nature of DM and DE. In the last decades, various DM and DE models have been studied. The simplest model of DM consists in particles moving slowly compared to the speed of light (they are cold) and interacting very weakly with ordinary matter and electromagnetic radiation. These particles, known as weakly interacting massive particles (WIMPS), behave as dust with an equation of state (EOS) parameter b1i (); b1j (); b1k (). They may correspond to supersymmetric (SUSY) particles susy (). On the other hand, the simplest manner to explain the accelerated expansion of the Universe is to introduce a cosmological constant in the Einstein equations b25 (). In that case, the value of the energy density stored in the cosmological constant represents the DE.
The standard model of cosmological structure formation in the Universe is known as the cold dark matter model with a cosmological constant (CDM) peebles1 (); b1b (); b1c (); b1a (). Cosmological observations at large scales support the CDM model with a high precision.
However, this model has some problems at small (galactic) scales for the case of DM kroupa (); b1e (); b1f (); b1d (). In particular, it predicts that DM halos should be cuspy nfw () while observations reveal that they have a flat core observations (). On the other hand, the CDM model predicts an over-abundance of small-scale structures (subhalos/satellites), much more than what is observed around the Milky Way satellites (). These problems are referred to as the “cusp problem” and “missing satellite problem”. The expression “small-scale crisis of CDM” has been coined.
Furthermore, the value of the cosmological constant assigned to DE has to face important fine tuning problems b3 (); b1h (); martin (). From the point of view of particle physics, the cosmological constant can be interpreted naturally in terms of the vacuum energy density whose scale is of the order of the Planck density . However, observationally, the cosmological constant is of the order of the present value of the Hubble parameter squared, , which corresponds to a dark energy density . The Planck density and the cosmological density differ from each other by orders of magnitude. This leads to the so-called cosmological constant problem b3 (); b1h (); martin ().
Since the CDM model poses problems, some efforts have been done in trying to understand the nature of DM and DE from the framework of quantum field theory. In particle physics and string theory, scalar fields (SF) arise in a natural way as bosonic spin- particles described by the Klein-Gordon (KG) equation kolb (); zee (). Examples include the Higgs particle, the inflaton, the dilaton field of superstring theory, tachyons etc. SFs also arise in the Kaluza-Klein and Brans-Dicke theories b1m (). In cosmology, SFs were introduced to explain the phase of inflation in the primordial Universe linde (). SF models have then been used in cosmology in various contexts and they continue to play an important role as potential DM and DE candidates.
For example, the source of DE can be attributed to a SF. A variety of SF models have been infered for this purpose (see for example b5 (); ri (); peebles1 ()). Quintessence quintessence (); b14 (), which is the simplest case, is described by an ordinary SF minimally coupled to gravity. It generally has a density and EOS parameter that vary with time, hence making it dynamic. By contrast, a cosmological constant is static, with a fixed energy density and . Phantom fields b16 (); bigrip (); b17 (); b18 () are associated to a negative kinetic term. This strange property leads to an EOS parameter implying that the energy density increases as the Universe expands, possibly leading to a big rip. It has also been suggested that, in a class of string theories, tachyonic SF b21 () can condense and have cosmological applications. Tachyons have an interesting EOS whose parameter smoothly interpolates between and , thus behaving as DE and pressureless DM. SF models describing DE usually feature masses of the order of the current Hubble scale () b54 (); b53 ().
Concerning DM, it has been proposed that DM halos can be made of a SF
described by the Klein-Gordon-Einstein (KGE) equations (see,
e.g., b26 (); b27 (); b28 (); marshrev () for reviews and nature () for high
resolution numerical simulations showing the viability of this scenario). In
models suppose that DM is a real or complex SF
minimally coupled to gravity. This SF can be self-interacting but it does
not interact with the other particles and fields, except
gravitationally. SF that interact only with gravity could be gravitationally
produced by inflation ford (). The SF may represent the wave function of
having formed a
Bose-Einstein condensate (BEC). The KGE equations describe a relativistic
SF/BEC. General relativity is necessary to describe compact SF objects such as
stars kaup (); rb (); colpi () and neutron stars with a superfluid core
page (); b45 (). It is also necessary
cosmology to describe the phase of inflation and the evolution of the
early Universe linde (). However, in the context of DM halos, Newtonian
sufficient. The evolution of a nonrelativistic SF/BEC is described by
the Gross-Pitaevskii-Poisson (GPP) equations. There
are several models of SFDM, e.g. noninteracting (fuzzy) DM b29 (),
self-interacting DM ss (), or
b30 (); b31 (); dine (); turneraxion (); b32 ().
In quantum field theory, ultralight SFs seem unnatural but renormalization effects tend to drive these scalar masses up to the scale of a new physics. Given the present observational status of cosmology, and despite all the efforts that have been made, it is fair to say that the nature of DM and DE remains a mystery. As a result, the SF scenario is an interesting suggestion that deserves to be studied in more detail.
Instead of working directly in terms of field variables, a fluid approach can be
adopted. In the nonrelativistic case, this hydrodynamic approach was introduced
by Madelung b37 () who showed that the Schrödinger equation is
equivalent to the
Euler equations for an irrotational fluid with an additional quantum potential
arising from the finite value of and accounting for
Heisenberg’s uncertainty principle. This approach has been
generalized to the GPP equations in the context of DM halos by
bohmer (); b34 (); rindler () among others. In the relativistic case,
broglie1927a (); broglie1927b (); broglie1927c () in his
so-called pilot wave theory, showed that the KG equations are equivalent to
hydrodynamic equations including a covariant quantum
approach has been generalized to the Klein-Gordon-Poisson (KGP) and KGE
equations in the context
of DM halos by b38 (); b39 (); b40 (); b41 (); chavmatos ().
The formation of large-scale structures is an important topic of cosmology. This problem was first considered by Jeans jeans () (before the discovery of the expansion of the Universe) who studied the instability of an infinite homogeneous self-gravitating classical collisional gas (see chavjeans () for a review). This study has been generalized in the context of SF theory. The Jeans instability of an infinite homogeneous self-gravitating system in a static background was studied by khlopov () for a relativistic SF described by the generalized KGP equations, using the field representation. The same problem was studied in b34 (); b35 () for a nonrelativistic SF described by the GPP equations in the context of Newtonian cosmology, and in b40 (); b41 () for a relativistic SF described by the KGE equations, using the hydrodynamic representation.
The growth of perturbations of a relativistic real SF in an expanding Universe was considered in mul (); jcap () using the field representation. The same problem was addressed in b35 (); b40 () for a complex SF using the hydrodynamic representation. Analytical results were obtained in the (nonrelativistic) matter era where the background Universe has an Einstein-de Sitter (EdS) evolution b35 (); b40 (). The matter era is valid at sufficiently late times, after the radiation-matter equality. At earlier times, the SF affects the background evolution of the Universe so we can no more assume that the scale factor follows the EdS solution.
The classical evolution of a real SF described by the KGE equations with a potential of the form in an isotropic and homogeneous cosmology was first investigated by Turner turner () (see also the subsequent works of ford (); greene (); pv ()). He showed that the SF experiences damped oscillations but that, in average, it is equivalent to a perfect fluid with an EOS (this result is valid if we neglect particle creation due to the time variation of ). For the SF behaves as pressureless matter and for it behaves as radiation. Turner also mentioned the possibility of a stiff EOS. The cosmological evolution of a spatially homogeneous real self-interacting SF with a repulsive potential described by the KGE equations competing with baryonic matter, radiation and dark energy was considered by mul (). In this work, it is found that a real self-interacting SF displays fast oscillations and that, on the mean, it undergoes a radiationlike era followed by a matterlike era. In the noninteracting case, the SF undergoes only a matterlike era b1p (). In any case, at sufficiently late times, the SF reproduces the cosmological predictions of the standard CDM model.
The cosmological evolution a complex self-interacting SF representing BECDM has been considered by harkocosmo (); b35 () who solved the (relativistic) Friedmann equations with the EOS of the BEC derived from the (nonrelativistic) GP equation after identifying , where is the rest-mass density, with the energy density . However, as clarified in stiff (), this approach is not valid in the early Universe as it combines relativistic and nonrelativistic equations. These studies may still have interest in cosmology in a different context, as discussed in cosmopoly1 (); cosmopoly2 ().
The exact relativistic cosmological evolution of a complex self-interacting SF/BEC described by the KGE equations with a repulsive potential has been considered by Li et al. b36 () (see also the previous works of sj (); js (); spintessence (); arbeycosmo ()). In this work, the evolution of the homogeneous background is studied. It is shown that the SF undergoes three successive phases: a stiff matter era, followed by a radiationlike era (that only exists for self-interacting SFs), and finally a matterlike era similar to the one appearing in the CDM model. Another cosmological model displaying a primordial stiff matter era has been developed in stiff (). Interestingly, it leads to a completely analytical cosmological solution generalizing the EdS model and the (anti)-CDM model.
In general, the SF oscillates in time and it is not clear how these oscillations can be measured in practice because there is no direct access to field variables such as . As a result, the hydrodynamic representation of the SF may be more physical than the KG equation itself because it is easier to measure hydrodynamic variables such as the energy density , the rest-mass density , and the pressure . In our previous paper b40 (), we showed that the three phases of a relativistic SF with a repulsive potential (stiff matter, radiation and pressureless matter) could be obtained from the hydrodynamic approach in complete agreement with the field theoretic approach of Li et al. b36 ().
In the present paper, we complete and generalize our study in different directions: we formulate the problem for an arbitrary SF potential , not just for a potential; we solve the equations in the fast oscillation regime and obtain several analytical results in different asymptotic limits that complement the work of Li et al. b36 (); we consider repulsive and attractive self-interaction and show that the later can lead to very peculiar results. The case of attractive self-interaction is of considerable interest since axions, that have been proposed as a serious DM candidate, usually have an attractive self-interaction. The case of attractive self-interaction has been studied previously in b34 (); cd (); b35 (); b40 (); bectcoll (). It is shown in b35 (); b40 () that an attractive self-interaction can accelerate the growth of structures is cosmology. On the other hand, it is shown in b34 (); cd (); bectcoll () that stable DM halos with an attractive self-interaction can exist only below a maximum mass that severely constrains the parameters of the SF.
The paper is organized as follows. In Sec. II, we
introduce the KG and Friedmann equations describing the cosmological evolution
of a spatially homogeneous complex
Ii Spatially homogeneous complex SF
In our previous paper b40 (), we have derived a hydrodynamic representation of the KGE equations in an expanding background in the weak field approximation. We considered a complex SF with a quartic self-interaction potential. This study was extended to the case of an arbitrary SF potential of the form in b41 (); chavmatos (). In this section, we consider the case of a spatially homogeneous complex SF. For the clarity and the simplicity of the presentation, we assume that the Universe is only composed of a SF, although it would be straightforard to include in the formalism other components such as normal radiation, baryonic matter, and dark energy (e.g., a cosmological constant).
ii.1 The KG equation for a spatially homogeneous complex SF
The cosmological evolution of a spatially homogeneous complex SF with a self-interaction potential in a Friedmann-Lemaître-Robertson-Walker (FLRW) universe is described by the KG equation
where is the Hubble parameter and is the scale factor. The second term in Eq. (1) is the Hubble drag. The rest-mass term (third term) can be written as where is the Compton wavelength of the bosons.
The energy density and the pressure of the SF are given by
The EOS parameter is defined by .
ii.2 The Friedmann equations
This equation can also be directly obtained from the Einstein field equations and constitutes the first Friedmann equation b20e (). From this equation we deduce that, as the Universe expands, the energy density decreases when , increases when , and remains constant when . In the second case, the Universe is “phantom” b16 (). The second Friedmann equation, obtained from the Einstein field equations, writes
which constitutes the third Friedmann equation. From this equation, we deduce that the expansion of the Universe is decelerating when and accelerating when . The intermediate case, in which the scale factor increases linearly with time, corresponds to .
ii.3 Hydrodynamic representation of a spatially homogeneous complex SF
We stress that it is only in the nonrelativistic limit that has the interpretation of a rest-mass density. In the relativistic regime, does not have a clear physical interpretation but it can always be defined as a convenient notation b40 (); b41 (); chavmatos (). We write the SF in the de Broglie form
where is the pseudo rest-mass density and is the real action. The total energy of the SF (including its rest mass energy) is
Equations (10)-(12) can also be obtained from the general hydrodynamic equations derived in b40 (); b41 (); chavmatos () by considering the particular case of a spatially homogeneous SF (, , , and ). In that case, Eq. (10) is deduced from the continuity equation, Eq. (11) from the quantum Bernoulli or Hamilton-Jacobi equation, and Eq. (12) from the Einstein equations. In this connection, we note that the first two terms (the terms proportional to ) in the r.h.s. of Eq. (11) correspond to the relativistic de Broglie quantum potential
for a spatially homogeneous SF. We stress that the hydrodynamic equations (10)-(12) are equivalent to the KGE equations (1), (2) and (5). Finally, we note that the hydrodynamic equations (10)-(12) with the terms in neglected provide a TF, or semiclassical, description of relativistic SFs.
The continuity equation (10) can be rewritten as a conservation law
Therefore, the total energy of the SF is exactly given by
In the hydrodynamic representation, the energy density and the pressure of a homogeneous SF can be expressed as
ii.4 Cosmological evolution of a spatially homogeneous complex SF in the fast oscillation regime
The exact equations (10)-(12) are complicated. In the case of a quartic potential with a positive scattering length, Li et al. b36 () have identified two regimes in which these equations can be simplified. When the oscillations of the SF are slower than the Hubble expansion (), the SF is equivalent to a stiff fluid with an EOS . This approximation is valid in the early Universe. At later times, when the oscillations of the SF are faster than the Hubble expansion (), it is possible to average over the fast oscillations in order to obtain a simpler dynamics. The resulting equations can be obtained either from the field theoretic approach b36 () or from the hydrodynamic approach b40 (). We note that the equations obtained in the fast oscillation regime specifically depend on the form of the SF potential. In this section, we generalize these results to the case of an arbitrary SF potential . We use the hydrodynamic approach. The field theoretic approach is exposed in Appendix A.
The simplified equations valid in the fast oscillation regime can be obtained from Eqs. (11) and (12) by neglecting the terms involving a time derivative. Interestingly, this is equivalent to neglecting the terms in . Therefore, the fast oscillation regime is equivalent to the TF, or semiclassical, approximation where the quantum potential (arising from Heisenberg’s uncertainty principle) is neglected. In that case, we obtain
Keeping only the solution of Eq. (18) that leads to a positive total energy (the solution with a negative total energy corresponds to antibosons), we get
Equations (21) and (22) determine the evolution of the scale factor of the Universe induced by a spatially homogeneous SF in the regime where its oscillations are faster than the Hubble expansion. The energy of the SF is then given by Eq. (20).
It is not convenient to solve the differential equation (22) for the scale factor because we would need to inverse Eq. (21) in order to express as a function of in the r.h.s. of Eq. (22). Instead, it is more convenient to view as a function of , given by Eq. (21), and transform Eq. (22) into a differential equation for . Taking the logarithmic derivative of Eq. (21), we get
Substituting this expression into Eq. (22), we obtain the differential equation
For a given SF potential , this equation can be solved easily as it is just a first order differential equation for . The temporal evolution of the scale factor is then obtained by plugging the solution of Eq. (24) into Eq. (21).
In the fast oscillation regime, the energy density and the pressure are given by
Using Eq. (18), we get
The pseudo velocity of sound is
We note that the pressure of a spatially homogeneous SF in the fast oscillation regime coincides with the pseudo pressure that arises in the Euler equation obtained in the hydrodynamic representation of a complex SF b40 (); b41 (); chavmatos (), i.e. (compare Eq. (28) with Eq. (38) of b41 ()). This extends to an arbitrary SF potential the result obtained in b40 () for a quartic potential (we note that this equivalence is not true for a spatially inhomogeneous SF and for a homogeneous SF outside of the fast oscillation regime).
The Universe is accelerating () when . Introducing the total potential (see Sec. V), this condition can be rewritten as . The Universe is phantom () when or, equivalently, when . However, this condition is never realized in the fast oscillation regime because of the constraint imposed by Eq. (20).
The first equation determines the relationship between and . The second relation then determines the total potential .
Remark: From Eqs. (27) and (28), we can obtain the EOS . Solving the energy equation (4) with this EOS, we can obtain . The relation can also be obtained from Eqs. (21) and (27). We can easily check that the relations are the same. Indeed, from Eqs. (4), (27) and (28) we obtain the differential equation
This differential equation is equivalent to Eq. (21). This can be seen easily by taking the logarithmic derivative of Eq. (21) which leads to Eq. (32). This shows the consistency of our approximations.
ii.5 The nonrelativistic limit
In order to take the nonrelativistic limit of the previous equations, we need to subtract the contribution of the rest mass energy of the SF. To that purpose, we make the Klein transformation
As explained previously, it is convenient to work in terms of hydrodynamic variables. We write the wave function under the Madelung form
and introduce the energy
We also note that the energy equation (4) reduces to Eq. (38). It can be integrated into which, together with Eq. (40), leads to the EdS solution. Equations (38)-(40) can also be obtained from the general hydrodynamic equations derived in b40 (); b41 (); chavmatos () by considering the particular case of a spatially homogeneous SF in the nonrelativistic limit .
Remarks: the hydrodynamic equations (10)-(12) and (38)-(40) do not involve viscous terms because they are equivalent to the KG and GP equations. As a result, they describe a superfluid. We note that Eq. (11) for or is necessary in the relativistic case in order to have a closed system of equations (since appears explicitly in Eqs. (10) and (12)) while Eq. (39) for or is not strictly necessary in the nonrelativistic case (since does not appear in Eq . (38) and (40)).
ii.6 The quartic potential
In the case where the SF describes a BEC at zero temperature, the self-interaction potential can be written as
where is the mass of the bosons and is their scattering length (see Appendix B for other expressions of the self-interaction constant). A repulsive self-interaction corresponds to and an attractive self-interaction corresponds to . In the first case, may be interpreted as the “effective radius” of the bosons if we make an analogy with a classical hard spheres gas.
Iii The case of a quartic potential with a positive scattering length
From now on, we restrict ourselves to a SF with a quartic potential given by Eq. (42). We focus on the evolution of a homogeneous SF in the regime where its oscillations are faster than the Hubble expansion. We first consider the case of a SF with a positive scattering length corresponding to a repulsive self-interaction (the noninteracting case corresponds to ). This is the most studied case in the literature. A very nice study has been done by Li et al. b36 (). Here, we complement their study and provide more explicit analytical results.
iii.1 The basic equations
The equations of the problem are
Equation (47) gives the relation between the energy density and the pseudo rest-mass density . This is a second degree equation for . The only physically acceptable solution (the one that is positive) is
It coincides with the EOS obtained by Colpi et al. colpi () in the context of boson stars (see also b45 ()). For a noninteracting SF (), Eq. (52) reduces to meaning that a noninteracting SF behaves as pressureless matter.
iii.2 The evolution of the parameters with the scale factor
The evolution of the pseudo rest-mass density with the scale factor is plotted in Fig. 1 (in the Figures, unless otherwise specified, we use the dimensionless parameters defined in Appendix C). It starts from at and decreases to as . For :
The evolution of the energy density with the scale factor is plotted in Fig. 2. It starts from at and decreases to as . For :
The pressure is always positive. It starts from at and decreases to as . For :
The relationship between the pressure and the energy density is plotted in Fig. 3.
The evolution of the EOS parameter with the scale factor is plotted in Fig. 4. It starts from
when and decreases to as . For :
The total energy starts from at and decreases up to as . For :
iii.3 The temporal evolution of the parameters
In this section, we determine the temporal evolution of the parameters assuming that the Universe contains only the SF. For a quartic potential with , the differential equation (24) becomes
The solution of this differential equation which satisfies the condition that as is
The integral can be computed analytically:
From these equations, we can obtain the temporal evolution of the pseudo rest-mass density . Then, using Eqs. (45)-(50), we can obtain the temporal evolution of the all the parameters. The temporal evolution of the scale factor is plotted in Fig. 5. It starts from at and increases to as . We do not show the other curves because they can be easily deduced from Figs. 1, 2 and 4 since is a monotonic function of time. However, we provide below the asymptotic behaviors of all the parameters.
iii.4 The different eras
In the fast oscillation regime, a SF with a repulsive self-interaction
undergoes two distinct eras. For , the EOS
(52) reduces to Eq. (57) so the SF behaves as radiation.
The scale factor increases like . For ,
(52) reduces to Eq. (58) so the SF
behaves essentially as pressureless matter (dust) like in the EdS
In conclusion, a SF with a repulsive self-interaction behaves at early times as radiation and at late times as dust. We can estimate the transition between the radiationlike era and the matterlike era of the SF as follows. First of all, using Eqs. (45) and (49), we find that the scale factor corresponding to a value of the EOS parameter is
Interestingly, this equation provides an analytical expression of the function
, the inverse of the function plotted in Fig. 4. If
we consider that the transition between the radiationlike era and the matterlike
era of the SF corresponds to ,
This corresponds to . In order to make numerical applications here and in the following sections, it is convenient to introduce the reference scale factor defined in Appendix C. Using the expression (292) of the charge of the SF, we get
Therefore, . According to Eq. (81), we note that depends only on the ratio (see Sec. III.9). For a SF with a ratio given by Eq. (262), we get . For a SF with a ratio given by Eq. (278), we get . This analytical result is in good agreement with the numerical result of Li et al. b36 () (see their Fig. 1).
iii.5 Validity of the fast oscillation regime
The previous results are valid in the fast oscillation regime . In this section, we determine the domain of validity of this regime.
Introducing the dimensionless variables of Appendix C, this condition can be rewritten as
is a new dimensionless parameter that can be interpreted as the ratio between the effective Schwarzschild radius of the bosons (see Sec. III.8) and their scattering length . Introducing proper normalizations, we get
The dimensionless variables and are plotted as a function of in Fig. 6. Their ratio is plotted as a function of in Fig. 7. The intersection of this curve with the line determines the domain of validity of the fast oscillation regime.