Mass loss and longevity of gravitationally bound oscillating scalar lumps (oscillatons) in D-dimensions

Mass loss and longevity of gravitationally bound oscillating scalar lumps (oscillatons) in -dimensions

Gyula Fodor, Péter Forgács, Márk Mezei MTA RMKI, H-1525 Budapest 114, P.O.Box 49, Hungary,
LMPT, CNRS-UMR 6083, Université de Tours, Parc de Grandmont, 37200 Tours, France
Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139,USA
Institute for Theoretical Physics, Eötvös University, H-1117 Budapest, Pázmány Péter sétány 1/A, Hungary,
July 16, 2019
Abstract

Spherically symmetric oscillatons (also referred to as oscillating soliton stars) i.e. gravitationally bound oscillating scalar lumps are considered in theories containing a massive self-interacting real scalar field coupled to Einstein’s gravity in dimensional spacetimes. Oscillations are known to decay by emitting scalar radiation with a characteristic time scale which is, however, extremely long, it can be comparable even to the lifetime of our universe. In the limit when the central density (or amplitude) of the oscillaton tends to zero (small-amplitude limit) a method is introduced to compute the transcendentally small amplitude of the outgoing waves. The results are illustrated in detail on the simplest case, a single massive free scalar field coupled to gravity.

preprint: MIT-CTP 4104

I Introduction

Numerical simulations of Seidel and SuenSeidel1 () have revealed that spatially localized, extremely long living, oscillating configurations evolve from quite general initial data in the spherically symmetric sector of Einstein’s gravity coupled to a a free, massive real Klein-Gordon field. For example, they observed that initially Gaussian pulses evolve quickly into configurations which appear to be time-periodic. It has been already noted in Ref.Seidel1 (), that the resulting objects may not be strictly time-periodic, rather they may evolve on a secular time scale many orders of magnitude longer than the observed oscillation period. These interesting objects were first baptized ”oscillating soliton stars” in Ref. Seidel1 (), but somewhat later the same objects have been referred to as ”oscillatons” by the same authors Seidel2 (). This latter name has been by now widely adopted, and we shall also stick to its usage throughout this paper. It has been observed in the numerical simulations of Ref. Seidel1 () that oscillatons are stable during the time evolution. Moreover it has been argued in Ref.Seidel2 () that oscillatons do form in physical processes through a dissipationless gravitational cooling mechanism, making them of great physical importance. For example oscillatons would be good candidates for dark matter in our Universe.

On the other hand, stimulated by the seminal work of Dashen, Hasslacher and Neveu in the one-dimensional -theory Dashen (), numerical simulations have revealed that in an impressive number of scalar field theories spatially localized structures –oscillons– form from generic initial data which become very closely time periodic, and live for very long times BogMak2 (); CopelGM95 (); Chris (); PietteZakr98 (); Honda (); Hindmarsh-Salmi06 (); SafTra (); fggirs (); sicilia2 (). These objects oscillate nearly periodically in time, resembling “true” (i.e. time-periodic) breathers. An oscillon possesses a “radiative” tail outside of its core region where its energy is leaking continuously in form of (scalar) radiation. Therefore a simple approximate physical picture of a sufficiently small-amplitude oscillon is the that of a “true” breather whose frequency is increasing on a secular time scale since the amplitude of the outgoing radiation is much smaller than that of the core. It has been shown in Refs. FFGR (), FFHL (), that slowly radiating oscillons can be well described by a special class of exactly time-periodic “quasibreathers” (QB). Being time periodic, QBs are easier to describe mathematically by ordinary Fourier analysis than the long time asymptotics of oscillons. A QB possesses a localized core in space (just like true breathers) which approximates that of the corresponding oscillon very well, but in addition it has a standing wave tail whose amplitude is minimized. This is a physically motivated condition, which heuristically singles out “the” solution approximating a true breather as well as possible, for which this amplitude would be identically zero. The amplitude of the standing wave tail of a QB is closely related to that of the oscillon radiation, therefore its computation is of prime interest. Roughly speaking “half” of the standing wave tail corresponds to incoming radiation from spatial infinity. It is the incoming radiation that maintains the time periodicity of the QBs by compensating the energy loss through the outgoing waves. In a series of papers FFHL (); FFHM (); moredim () a method has been developed to compute the leading part of the exponentially suppressed tail amplitude of QBs, in a large class of scalar theories in various dimensions, in the limit when the QB core amplitude is small. Although oscillons continuously loose energy through radiation, many of them are remarkably stable. The longevity and the ubiquity of oscillons make them of potentially great physical interest Kolb:1993hw (); Khlopov (); Broadhead:2005hn (); Gleiser:2007ts (); Borsanyi2 (). Quite importantly oscillons also appear in the course of time evolution when other fields, e.g. vector fields are present Farhi05 (); Graham07a (); Graham07b (). There is little doubt that oscillons and oscillatons are closely related objects.

The basic physical mechanism for the anti-intuitively slow radiation of oscillons is that the lowest frequency mode of the scalar field is trapped below the mass threshold and only the higher frequency modes are coupled to the continuum.

In this paper we generalize the method of Refs. FFHL (); FFHM (); moredim () to compute the mass loss of spherically symmetric oscillations induced by scalar radiation in the limit of small oscillaton amplitudes, , in dimensional spacetimes. These methods have been succesfully applied to -dimensional scalar field theories coupled to a dilaton field dilaton (). Numerous similarities exist between coupling a theory to a dilaton field and to gravitation: the field configurations are of order and the lowest order equations determining the profiles are the Schrödinger-Newton equations. The stability pattern is also analogous. Despite these similarities between the dilaton and the gravitational theory there are some technical and even some conceptual differences. Since there is no timelike Killing vector neither for oscillatons nor for the corresponding QBs, already the very definition of mass and mass loss is less obvious than in flat spacetime. Another conceptual issue is that the spacetime of a time-periodic QB is not asymptotically flat, which is related to the fact that the “total mass” of a QB is infinite. In the case of spherical symmetry considered in this paper a suitable local mass function is the Misner-Sharp energy and the mass loss can be defined with aid of the Kodama vector. The issue of the precise asymptotics of spacetimes can be sidestepped in the limit by considering only a restricted, approximatively flat spacetime region containing the core of the QB (having a size of order ) and part of its oscillating tail. We find that to leading order in the expansion the oscillaton core is determined by the -dimensional analogues of the Schrödinger-Newton equations Ruffini (); Friedberg (); Ferrell (); Moroz (); Tod () independently of the self-interaction potential. It turns out that exponentially localized oscillatons exist for . These findings show a striking similarity to dilaton-scalar theories as found in Ref. dilaton (). In the case of spherically symmetric oscillatons no gravitational radiation is expected due to Birkhoff’s theorem. The mass loss of spherically symmetric oscillatons is entirely due to scalar radiation.

The following simple formula gives the mass loss of a small-amplitude oscillaton in spatial dimensions:

(1)

where denotes the mass of the scalar field, is a -dependent constant, while depends on both and the self-interaction scalar potential. The numerical values of in the Einstein-Klein-Gordon (EKG) theory for spatial dimensions are given in Table 6. We also compute and tabulate the most important physical properties of oscillatons in the EKG theory (their mass as a function of time, their radii). We would like to stress, that the method is applicable for oscillatons in scalar theories with any self-interaction potential developable into power series.

In the seminal work of Don N. Page Page () both the classical and quantum decay rate of oscillatons has been considered for the case of free massive scalars in the EKG theory (for ). We agree with the overall qualitative picture of the oscillaton’s mass loss found in Ref.Page (), however, there are also some differences in the quantitative results. For example, the amplitude of the outgoing wave (related to ) found by our method differs significantly from that of Ref.Page (). The main source of this discrepancy is due to the fact that this amplitude is given by an infinite series in the expansion, where all terms contribute by the same order, whereas in the estimate of Ref.Page () only the lowest order term in this series has been used. Our methods which are based on the work of Segur-Kruskal SK () avoid this difficulty altogether, moreover for the class of self-interaction potentials containing only even powers of the scalar field, , the radiation amplitude can be computed analytically using Borel summation.

We now give a lightning review on previous results scattered in the literature on oscillatons in 3+1 dimensions. For a given scalar field mass, , there is a one-parameter family of oscillatons, parametrized, for example, by the central amplitude of the field, . As increases from small values, the mass of the oscillaton, , is getting larger, while the radius of the configuration decreases. For a critical value of the central amplitude, , a maximal mass configuration is reached. Oscillatons with central amplitudes are unstable Seidel1 (). This behavior is both qualitatively and quantitatively very similar to that of boson stars Ruffini (); Feinblum (); Kaup (), and also to the behavior of white dwarfs and neutron stars Harrison (). For reviews of the vast literature on boson stars see for example, Refs. Jetzer () and SchunkMielke (). In Refs. Hawley (); Hawleyphd () a one-parameter family of oscillaton-type solutions in an Einstein-scalar theory with two massive, real scalar fields has been presented, which are essentially transitional states between boson stars and oscillatons.

The interaction of weak gravity axion field oscillatons with white dwarfs and neutron stars have been discussed in Iwazaki1 (); Iwazaki2 (), proposing a possible mechanism for gamma ray bursts Iwazaki3 (). Since for very low mass scalar fields oscillatons may be extremely heavy, it has been suggested that they may be the central object of galaxies MatosGuzman (), or form the dark matter galactic halos Alcubgalactic (); Susperregi (); Guzman1 (); Guzman2 (); Hernandez (); Guzman3 (); Bernal ().

Qualitatively good results for various properties of oscillatons has been obtained by Ureña-López Lopez (), truncating the Fourier mode decomposition of the field equations at as low order as , where is the fundamental frequency. Then the space and time dependence of the scalar field separates as . Oscillatons with nontrivial self-interaction potentials have also been studied in Lopez (), indicating that similarly to boson stars, the maximal mass can be significantly larger than in the Klein-Gordon case.

The Fourier mode equations have been studied in LopezMatos () up to orders . The obtained value of the maximal mass by this higher order truncation is in Planck units. For small-amplitude nearly Minkowskian configurations spatial derivatives are also small, and in Ref. LopezMatos () (and independently in Kichena2 ()) it has been demonstrated that such nearly flat oscillatons can be described by a pair of coupled differential equations, the so called time independent the Schrödinger-Newton equations Ruffini (); Friedberg (); Ferrell (); Moroz (); Tod (). These equations also describe the weak gravity limit of boson stars. For quantum mechanical motivations leading to the Schrödinger-Newton equations see Diosi (); Penrose ().

The time evolution of perturbed oscillatons has been investigated in detail by Alcub (). For each mass smaller than the maximal oscillaton mass there are two oscillaton configurations. The one with the larger radius is a stable S-branch oscillaton, and the other is an unstable U-branch oscillaton. Moderately perturbed S-branch oscillatons vibrate with a low frequency corresponding to a quasinormal mode. Perturbed U-branch oscillatons collapse to black holes if the perturbation increases their mass, otherwise they migrate to an S-branch oscillaton. Actually, U-branch oscillatons turn out to be the critical solutions for type I critical collapse of massive scalar fields Brady (). Corresponding apparently periodic objects also form in the critical collapse of massive vector fields Garfinkle ().

There are also excited state oscillatons, indexed by the nodes of the scalar field. The instability and the decay of excited state oscillatons into black holes or S-branch oscillatons is described in Balak (). The evolution of oscillatons on a full 3D grid has been also performed in Balak (), calculating the emitted gravitational radiation. Since gravity theories are equivalent to ordinary general relativity coupled to a real scalar field, oscillatons naturally form in these theories as well Obregon (). The geodesics around oscillatons has been investigated in Becerril ().

The plan of the paper is the following. In Section II the general formalism concerning a classical real scalar field coupled to gravitation in dimensional, spherically symmetric spacetimes is set up. In subsection II.3 the coupled Einstein-scalar equations are explicited in a spatially conformally flat coordinate system. In Section III the small-amplitude expansion is presented and is carried out in detail. In subsection III.3 it is shown that in leading order one obtains the Schrödinger-Newton eqs. in dimensions. In subsection III.4 the next to leading order results are given. Subsection III.5 contains an analysis of the singularities in the complexified radial variable. In Section IV the proper mass resp. the total mass of the QB core is evaluated in subsection IV.1 resp. subsection IV.2. In subsection IV.4 a conjecture for a criterion of oscillaton stability is formulated. In Section V the Fourier analysis of the field equations is related to the small-amplitude expansion, and the amplitude of the standing wave tail of the QB is determined using Borel summation techniques. In subsection V.6 the mass loss rate of oscillatons in the EKG theory is computed for and for various values of the mass of the scalar field.

Ii Scalar field on curved background

ii.1 Field equations

We consider a real scalar field with a self-interaction potential in a dimensional curved spacetime with metric . We use Planck units with . For a free field with mass the potential is . The total Lagrangian density is

(2)

where the Einstein Lagrangian density is , and the Lagrangian density belonging to the scalar field is

(3)

Variation of the action with respect to yields the wave equation

(4)

while variation with respect to yields Einstein equations

(5)

where the stress-energy tensor is

(6)

If then, by definition, the Einstein tensor is traceless, and from the trace of the Einstein equations it follows that . Hence we assume that .

We shall assume that the self-interaction potential, , has a minimum at , and expand its derivative as

(7)

where are constants. In order to get rid of the factors in the equations we introduce a rescaled scalar field and potential by

(8)

Then

(9)

with

(10)

The mass of the field is . If the pair and solves the field equations with a potential , then and , for any positive constant , is a solution with a rescaled potential . It is sufficient to study the problem with potentials satisfying , since the solutions corresponding to an arbitrary potential can be obtained from the solutions with an appropriate potential with by applying the transformation

(11)

To simplify the expressions, unless explicitly stated, in the following we assume .

ii.2 Spherically symmetric dimensional spacetime

We consider a spherically symmetric dimensional spacetime with coordinates . The metric can be chosen diagonal with components

(12)

where , and are functions of temporal coordinate and radial coordinate . The nonvanishing components of the Einstein tensor and the form of the wave equation are given in Appendix A.

A natural radius function, , can be defined in terms of the area of the symmetry spheres in general spherically symmetric spacetimes. In the metric (12) it is simply

(13)

The Kodama vector Kodama (); Hayward () is defined then by

(14)

where is the volume form in the plane. Choosing the orientation such that makes future pointing, with nonvanishing components

(15)

It can be checked that, in general, the Kodama vector is divergence free, . Since contracting with the Einstein tensor, , the current

(16)

is also divergence free, , it defines a conserved charge. Integrating on a constant hypersurface with a future oriented unit normal vector , the conserved charge is

(17)

It is possible to show Kodama (); Hayward (), that agrees with the Misner-Sharp energy (or local mass) function MisnerSharp (), which can be defined for arbitrary dimensions by

(18)

It can be checked by a lengthy calculation, that the derivative of the mass function is

(19)

For the radial derivative follows that

(20)

which, comparing with (17), gives . Since for large the function tends to the total mass, this relation will be important when calculating the mass loss rate caused by the scalar radiation in Section V.6. The time derivative of the mass function is

(21)

This equation is according to the expectation, that, because of the spherical symmetry, the mass loss is caused only by the outward energy current of the massive scalar field. If at large distances the metric becomes asymptotically Minkowskian, , and , then using (6) and (8),

(22)

ii.3 Spatially conformally flat coordinate system

The diffeomorphism freedom of the general spherically symmetric time-dependent metric form (12) can be fixed in various ways. The most obvious choice is the use of Schwarzschild area coordinates by setting . However, as it was pointed out by Don N. Page in Page (), for the oscillaton problem it is more instructive to use the spatially conformally flat coordinate system defined by

(23)

even if some expressions are becoming longer by this choice. As we will see in Sec. III and in Appendix B, inside the oscillaton the spheres described by constant Schwarzschild coordinates are oscillating with much larger amplitude than the constant spheres in the conformally flat coordinate system. In both coordinates, when the functions and tend to , the spacetime approaches the flat Minkowskian metric.

In the spatially conformally flat coordinate system the Einstein equations take the form

(24)
(25)
(26)
(27)

The right hand sides are equal to , , and , respectively. The wave equation is then

(28)

Iii Small-amplitude expansion

The small-amplitude expansion procedure has been applied successfully to describe the core region of one-dimensional flat background oscillons in scalar theory Dashen (); SK (); Kichenassamy (). Later it has been generalized for dimensional spherically symmetric systems in FFHL (), and to a scalar-dilaton system in dilaton (). In this section we generalize the method for the case when the scalar field is coupled to gravity.

iii.1 Choice of coordinates

We are looking for spatially localized bounded solutions of the field equations (5) for which is small and the metric is close to flat Minkowskian. We use the spatially conformally flat coordinate system defined by (23). It turns out, that under this approximation, all configurations that remain bounded as time passes are necessarily periodically oscillating in time. We expect that similarly to flat background oscillons, the smaller the amplitude of an oscillaton is, the larger its spatial extent becomes. Numerical simulation of oscillatons clearly support this expectation. Therefore, we introduce a new radial coordinate by

(29)

where denotes the small-amplitude parameter. We expand and the metric functions in powers of as

(30)
(31)
(32)

Since we intend to use asymptotically Minkowskian coordinates, where far from the oscillaton measures the proper time and the radial distances, we look for functions , and that tend to zero when . One could initially include odd powers of into the expansions (30)-(32), however, it can be shown by the method presented below, that the coefficients of those terms necessarily vanish when we are looking for configurations that remain bounded in time.

The frequency of the oscillaton also depends on its amplitude. Similarly to the flat background case we expect that the smaller the amplitude is, the closer the frequency becomes to the threshold . Numerical simulations also show this. Hence we introduce a rescaled time coordinate by

(33)

and expand the square of the dependent factor as

(34)

It is possible to allow odd powers of into the expansion of , but the coefficients of those terms turn out to be zero when solving the equations arising from the small-amplitude expansion. There is a considerable freedom in choosing different parametrizations of the small-amplitude states, changing the actual form of the function . The physical parameter is not but the frequency of the periodic states that will be given by . Similarly to the dilaton model in dilaton (), we will show, that for spatial dimensions the parametrization of the small-amplitude states can be fixed by setting .

iii.2 Leading order results

The field equations we solve are the Einstein equations (24)-(27), together with the wave equation (28), using the spatially conformally flat coordinate system . The results of the corresponding calculations in Schwarzschild area coordinates are presented in Appendix B. Since we look for spatially slowly varying configurations with an dependent frequency, we apply the expansion in and coordinates. This can be achieved by replacing the time and space derivatives as

(35)

and substituting .

From the components of the field equations follows that

(36)

where three new functions, , and are introduced, depending only on . From the part of (26) it follows that is a constant. Then by a shift in the time coordinate we set

(37)

This shows that the scalar field oscillates simultaneously, with the same phase at all radii.

The component of the field equations yield that

(38)
(39)
(40)

where , and are three new functions of . If , from the equations also follows that

(41)

and that the functions and are determined by the coupled differential equations

(42)
(43)

If then , and there are no nontrivial localized regular solutions for and , so we assume from now. We note that at all orders terms can be absorbed by a small shift in the time coordinate. After this, no terms appear in the expansion, resulting in the time reflection symmetry at .

Since we have already set , equations (42) and (43) do not depend on the coefficients of the potential . To order the functions , and are the same for any potential. This means that the leading order small-amplitude behavior of oscillatons is always the same as for the Klein-Gordon case.

iii.3 Schrödinger-Newton equations

Introducing the functions and by

(44)

equations (42) and (43) can be written into the form which is called the time-independent Schrödinger-Newton (SN) equations in the literature Ruffini (); Friedberg (); Ferrell (); Moroz (); Tod ():

(45)
(46)

Equations (45) and (46) have the scaling invariance

(47)

If the SN equations have a family of solutions with tending to zero exponentially as , and tending to a constant as

(48)

The solutions are indexed by the number of nodes of . The nodeless solution corresponds to the lowest energy and most stable oscillaton. We use the scaling freedom (47) to make the nodeless solution unique by setting . At the same time we change the parametrization by requiring

(49)

ensuring that the limiting value of vanishes. Then for large

(50)

with only exponentially decaying corrections. Going to higher orders, it can be shown that one can always make the choice for , thereby fixing the parametrization, and setting

(51)

For the explicit form of the asymptotically decaying solutions are known

(52)

where is any constant. In this case, since both and tend to zero at infinity, we have no method yet to fix the value of in (52). Moreover, in order to ensure that tends to zero at infinity we have to set

(53)

For there are no solutions of the SN equations representing localized configurations Choquard ().

Motivated by the asymptotic behavior of , if it is useful to introduce the variables

(54)

In dimensions these variables tend exponentially to the earlier introduced constants

(55)

Then the SN equations can be written into the equivalent form

(56)
(57)
(58)

which is more appropriate for finding high precision numerical solutions. Equation (56) will turn out to be useful when integrating the mass-energy density in Section IV.1 in order to determine the proper mass.

iii.4 Higher order expansion

From the components of the field equations follows the time dependence of ,

(59)

where and are functions of . The functions and are determined by the coupled equations

(60)
(61)

We look for the unique solution for which both and tend to zero as . For the function goes to zero exponentially, while for large

(62)

where is defined in (48), and and are some constants. If and are solutions of (60) and (61), then for any constant

(63)
(64)

are also solutions. This family of solutions is generated by the scaling freedom (47) of the SN equations. If we have any solution of (60) and (61) then by choosing appropriately we can get another solution for which in (62).

The equation for is

(65)

For large the function tends to zero as

(66)

The part of is determined by

(67)

We remind the reader, that for the choice , is natural, while for necessarily . For a Klein-Gordon field in the only nonvanishing coefficient is .

Summarizing the results, the scalar field and the metric components up to order are

(68)
(69)
(70)

Going to higher orders, the expressions get rather complicated. However, it can be seen that for symmetric potentials, when , the scalar field contains only components with odd , while and only contains even Fourier components.

Some of the higher order expressions simplifies considerably when considering symmetric potentials with . Because the first radiating mode proportional to emerges at order in in symmetric potentials, we present its higher order expression for the symmetric case

(71)

where is a function of determined by lengthy differential equations arising at higher orders.

For the Klein-Gordon case in spatial dimensions we plot the numerically obtained functions , , , , and on Figs. 1 and 2.

Figure 1: The exponentially decaying functions , , and for the small-amplitude expansion of the Klein-Gordon oscillaton in the case.
Figure 2: The functions , , and for the Klein-Gordon system. These functions tend to zero according to a power law for .

Equations (68)-(70) determine a one-parameter family of oscillating configurations depending on the parameter . This family solves the field equations with a scalar field mass . By applying the rescaling (11) to the and coordinates, we can obtain one-parameter families of solutions with any scalar mass .

To order, the metric is static. This is the biggest advantage of the spatially conformally flat coordinate system over the Schwarzschild area coordinates . In the Schwarzschild system the constant observers “feel” an order small oscillation in the metric (see Appendix B). The magnitude of the acceleration of the constant observers in the general metric (12) is

(72)

which has an order oscillating component when using Schwarzschild coordinates, while in spatially conformally flat coordinates the temporal change in the acceleration is only of order .

The function is equal to to order in the conformally flat coordinates. This motivates the metric form choice

(73)

which has been employed for the case in Page ().

iii.5 Singularities on the complex plane

As we will see in Section V, in order to determine the energy loss of oscillatons it is advantageous to extend the functions , and to the complex plane. In the small-amplitude expansion formalism the extension of the coefficient functions , and have symmetrically positioned poles along the imaginary axis, induced by the poles of the SN equations. We consider the closest pair of singularities, located at , since these will provide the dominant contribution to the energy loss. The numerically determined location of the pole for the spatial dimensions where there is an exponentially localized core is

(74)
(75)
(76)

The leading order behavior of the functions near the poles can be determined analytically, even if the solution of the SN equations is only known numerically on the real axis. Let us measure distances from the upper singularity by a coordinate defined as

(77)

Close to the pole we can expand the SN equations, and obtain that and have the same behavior,

(78)

even though they clearly differ on the real axis. We note that for there are logarithmic terms in the expansion of and , starting with terms proportional to . According to (41) and (44), the expression (78) determines the parts of , and near the pole.

Substituting into (60), (61), (65) and (67), the order contributions , , and can also be determined around the pole. We give the results for the Klein-Gordon case, when for :

(79)
(80)
(81)
(82)

The constant can only be determined from the specific behavior of the functions on the real axis, namely from the requirement of the exponential decay of for large real .

Iv Proper and total mass

iv.1 Proper mass

In this subsection we present the calculation of the proper mass , which is usually obtained by the integral of the mass-energy density over a spatial slice of the corresponding spacetime. In the next subsection the calculation of the total mass will be performed, by investigating the asymptotic behavior of the metric components. The difference defines the gravitational binding energy, which is expected to be positive.

The mass-energy density is , where the unit timelike vector has the components . In terms of the rescaled scalar field ,

(83)

The total proper mass in the metric (12) is defined by the dimensional volume integral

(84)

Applying this for the small-amplitude expansion of oscillatons in spatially conformally flat coordinates, and using that and , we can write

(85)

Using (41) and (43), for the proper mass we obtain

(86)

where

(87)