Cosmological perturbations on the Phantom brane
Abstract
We obtain a closed system of equations for scalar perturbations in a multicomponent braneworld. Our braneworld possesses a phantomlike equation of state at late times, , but no bigrip future singularity. In addition to matter and radiation, the braneworld possesses a new effective degree of freedom – the ‘Weyl fluid’ or ‘dark radiation’. Setting initial conditions on superHubble spatial scales at the epoch of radiation domination, we evolve perturbations of radiation, pressureless matter and the Weyl fluid until the present epoch. We observe a gradual decrease in the amplitude of the Weylfluid perturbations after Hubbleradius crossing, which results in a negligible effect of the Weyl fluid on the evolution of matter perturbations on spatial scales relevant for structure formation. Consequently, the quasistatic approximation of Koyama and Maartens provides a good fit to the exact results during the matterdominated epoch. We find that the latetime growth of density perturbations on the brane proceeds at a faster rate than in CDM. Additionally, the gravitational potentials and evolve differently on the brane than in CDM, for which . On the brane, by contrast, the ratio exceeds unity during the late matterdominated epoch (). These features emerge as smoking gun tests of phantom brane cosmology and allow predictions of this scenario to be tested against observations of galaxy clustering and largescale structure.
Contents
I Introduction
Cosmology, during the past two decades, has witnessed the introduction and development of several bold new theoretical ideas. One, especially radical paradigm, involves the braneworld concept. According to this paradigm (see Maartens:2010ar () for a review), our universe is a lowerdimensional hypersurface (the ‘brane’) embedded in a higherdimensional spacetime (the ‘bulk’). A new feature of the braneworld paradigm, which distinguishes it from the earlier Kaluza–Klein constructs, is that spacetime dimensions orthogonal to the brane need not be compact but could be ‘large’ ADD () and even infinite in length. In the simplest and most thoroughly investigated cosmological models, there is only one large extra dimension accessible to gravity, while all standardmodel fields are assumed to be trapped on the brane. From the viewpoint of our fourdimensional world, this manifests as a modification of gravity. In the seminal Randall–Sundrum (RS) model RS (), gravity is modified on relatively small spatial scales. Apart from other interesting applications, this model was used to provide an alternative explanation of galactic rotation curves and Xray profiles of galactic clusters without invoking the notion of dark matter Rotation curves ().
An important class of braneworld models contains the socalled ‘inducedgravity’ term in the action for the brane (it is induced by quantum corrections from the matter fields, hence the term), and modifies gravity on relatively large spatial scales. First proposed in Collins:2000yb (); DGP (); Shtanov:2000vr (), it has become known as the Dvali–Gabadadze–Porrati (DGP) model. Depending on the embedding of the brane in the bulk space, this model has two branches of cosmological solutions Deffayet:2000uy (). The ‘selfaccelerating’ branch was proposed to describe cosmology with latetime acceleration without bulk and brane cosmological constants Deffayet:2001pu (), while the ‘normal’ branch requires at least a cosmological constant on the brane (called brane tension) to accelerate cosmic expansion. The selfaccelerating branch was later shown to be plagued by the existence of ghost excitations Ghosts (). Without any additional modification, this leaves the normal branch as the only physically viable solution of this braneworld model, consistent with current cosmological observations of cosmic acceleration. It is this braneworld model that will be the subject of investigation in this paper.
As a model of dark energy, the normal braneworld branch exhibits an interesting generic feature of superacceleration which is reflected in the phantomlike effective equation of state Sahni:2002dx (); Alam_Sahni (); Lue:2004za (). Interestingly, the Phantom brane smoothly evolves to a de Sitter stage without running into a ‘BigRip’ future singularity typical of conventional phantom models. Such a phantomlike equation of state appears to be consistent with the most recent set of observations of type Ia supernovae combined with other data sets Rest:2013mwz (). The Phantom brane has a number of interesting properties: (i) it is ghostfree and is characterized by the effective equation of state , (ii) for an appropriate choice of cosmological parameters, even a spatially flat braneworld can ‘loiter’ Sahni:2004fb (), (iii) the Phantom brane possesses the remarkable property of ‘cosmic mimicry,’ wherein a highdensity braneworld exhibits the precise expansion history of CDM Sahni:2005mc (); for reviews, see Sahni:review (). Just like the Randall–Sundrum model, this braneworld model was also used as an alternative explanation of rotation curves in galaxies without dark matter Viznyuk:2007ft ().
The structure of the universe on the largest scales is spectacular, and consists of a ‘cosmic web’ of intertwining galactic superclusters separated from each other by large voids. Whereas the full description of the supercluster–void complex demands a knowledge of nonlinear gravitational clustering, useful insight into structure formation can already be gleaned from the linear (and weakly nonlinear) approximation SahniColes (). Linearized gravitational clustering in the braneworld model encounters obvious difficulties and complications connected with the existence of a large extra dimension. One has to take into account the corresponding dynamical degree of freedom and specify appropriate boundary conditions in the bulk space. In the simple case of a spatially flat brane, the extra dimension is noncompact, and one has to deal with its spatial infinity. The bulk gravitational effects then lead to a nonlocal character of the resulting equations on the brane. In spite of this difficulty, by using a very convenient Mukohyama master variable and master equation Mukohyama:2000ui (); Mukohyama:2001yp (), some progress has been made in this direction Deffayet:2002fn (); Deffayet:2004xg (); Koyama:2005kd (); Koyama:2006ef (); Sawicki:2006jj (); Song:2007wd (); Cardoso:2007xc (); Seahra:2010fj () by employing various plausible simplifying assumptions or approximations and by direct numerical integration. Most successful amongst these has been the quasistatic (QS) approximation due to Koyama and Maartens Koyama:2005kd () which is based on the assumption of slow temporal evolution of (all) fivedimensional perturbations on subHubble spatial scales, when compared with spatial gradients (for an extension into the nonlinear regime, see nonlinear ()). The behavior of perturbations on superHubble spatial scales was investigated within the scaling ansatz proposed in Sawicki:2006jj () and further developed in Song:2007wd (); Seahra:2010fj (). The validity of the quasistatic and scaling approximations was confirmed by numerical integration of the perturbation equations in five dimensions Cardoso:2007xc (); Seahra:2010fj ().
In our previous work Viznyuk:2013ywa (), we addressed the problem of scalar cosmological perturbations in a matterdominated braneworld model. We considered a marginally spatially closed (with topology ) braneworld model, in which the ‘noboundary’ smoothness conditions for the fivedimensional perturbations were set in the fourball bulk space bounded by the brane. In the limit when the spatial curvature radius of the brane was large, we were able to arrive at a closed system of equations for scalar cosmological perturbations without any simplifying assumptions.
Our approach differs from the semianalytic theory and numerical computations developed in Sawicki:2006jj (); Song:2007wd (); Cardoso:2007xc (); Seahra:2010fj (). First of all, we only require the regularity condition in the compact bulk space but do not impose any additional boundary conditions in the bulk; in particular, we do not demand the bulk perturbations to vanish on the past Cauchy horizon of the brane. At the same time, as we have shown, due to the same regularity condition, the system of equations for perturbations becomes effectively closed on the brane and does not require integration in the bulk. This is a great simplification of the theory.
In our approach, the dynamical entities that describe perturbations on the brane are the usual matter components and the socalled Weyl fluid, or dark radiation, which stems from the projection of the fivedimensional Weyl tensor onto the brane. The closed system of dynamical equations allows one to trace the behavior of matter and Weylfluid perturbations once the initial conditions for these quantities are specified. In particular, for modes well inside the Hubble radius during matter domination, the matter density perturbation evolves as [see also Eq. (146) below]
(1) 
where is the scale factor, and , , are integration constants. Apart from the usual growing mode , we observe two oscillating modes with decreasing amplitudes. These modes are induced by the dynamics of the Weyl fluid, or dark radiation. Note that, since these oscillatory modes have their origin in the bulk, they are absent in the scaling approximation of Sawicki:2006jj (); Song:2007wd (); Seahra:2010fj () or in the quasistatic approximation of Koyama:2005kd ().
Whether or not the presence of such extra modes, with dynamical origin in the bulk, can be significant for the braneworld model, depends upon the amplitudes and , and these, in turn, are determined by the primordial power spectrum of the Weyl fluid and its evolution during the radiationdominated epoch. This calls for a development of the theory of scalar cosmological perturbations for a universe filled with several components, each with an arbitrary equation of state. Such a treatment will enable one to follow the evolution of perturbations starting from deep within the radiationdominated regime all the way up to the current stage of accelerated expansion. This will be the main focus of the present paper.
Our paper is organized as follows. In the next section, we describe the background cosmological evolution of the normal branch of the braneworld model embedded in a flat fivedimensional bulk. In Sec. III, we investigate the system of equations describing scalar cosmological perturbations in this model. This system is not closed on the brane because the evolution equation for the anisotropic stress from the bulk degree of freedom projected to the brane (the socalled Weyl fluid, or dark radiation) is missing. We solve this problem by proceeding to a marginally closed braneworld and imposing the regularity conditions in the bulk, as described in Viznyuk:2013ywa (). The system of equations on the brane (in the limit of a large spatial radius) now becomes closed and can therefore be used for the analysis of perturbations. In Sec. IV, we discuss possible ways of setting initial conditions for Weylfluid perturbations and investigate the evolution of all perturbations starting from superHubble scales until the end of the radiationdominated epoch. Perturbations during matterdomination are considered in Sec. V. In Sec. VI, we present the results of numerical integration of the joint system of equations describing perturbations in radiation, pressureless matter and the Weyl fluid and compare these with CDM and with the results from the quasistatic approximation. Our results are summarized in Sec. VII.
Ii Background cosmological evolution
Our braneworld model has the action Collins:2000yb (); Shtanov:2000vr (); Sahni:2002dx ()
(2) 
where is the scalar curvature of the fivedimensional bulk, and is the scalar curvature corresponding to the induced metric on the brane. The symbol denotes the Lagrangian density of the fourdimensional matter fields whose dynamics is restricted to the brane so that they interact only with the induced metric . The quantity is the trace of the symmetric tensor of extrinsic curvature of the brane. All integrations over the bulk and brane are taken with the corresponding natural volume elements. The universal constants and play the role of the fivedimensional and fourdimensional Planck masses, respectively. The symbol denotes the bulk cosmological constant, and is the brane tension.
Action (2) leads to the following effective equation on the brane Shiromizu:1999wj (); Sahni:2005mc ():
(3) 
where
(4) 
are convenient parameters, and
(5) 
(6) 
Gravitational dynamics on the brane is not closed because of the presence of the symmetric traceless tensor in (3), which stems from the projection of the fivedimensional Weyl tensor from the bulk onto the brane. We are free to interpret this tensor as the stressenergy tensor of some effective fluid, which we call the ‘Weyl fluid’ in this article (in some works, the term ‘dark radiation’ is also used).
The tensor is not freely specifiable on the brane, but is related to the tensor through the conservation equation
(7) 
which is a consequence of the Bianchi identity applied to (3) and the law of stress–energy conservation for matter:
(8) 
In a universe consisting of several noninteracting components, the conservation law (8) is satisfied by each component separately.
The cosmological evolution of the Friedmann–Robertson–Walker (FRW) brane
(9) 
can be obtained from (3) with the following result Collins:2000yb (); Shtanov:2000vr (); Deffayet:2000uy (); Deffayet:2001pu (); Sahni:2002dx ():
(10) 
Here, is the Hubble parameter, is the energy density of matter on the brane and is a constant resulting from the presence of the symmetric traceless tensor in the field equations (3). The parameter corresponds to different spatial geometries of the maximally symmetric spatial metric .
The sign ambiguity in front of the square root in equation (10) reflects the two different ways in which the bulk can be bounded by the brane Deffayet:2000uy (); Sahni:2002dx (), resulting in two different branches of solutions. These are usually called the normal branch (lower sign) and the selfaccelerating branch (upper sign).
In what follows, we investigate the evolution of perturbations in a marginally flat () normal branch of the braneworld model embedded in flat bulk spacetime (which means , ). This setup will enable us to obtain a closed system of equations on the brane. The background cosmological equation (10) then reduces to
(11) 
or, equivalently:
(12) 
One immediately sees that, in the regime , our braneworld expands like CDM with the gravitational constant and with the combination playing the role of the cosmological constant.
Our braneworld is assumed to be filled with a multicomponent fluid, with total energy density and pressure . The usual conservation law holds for each component separately:
(13) 
where is the equation of state parameter for the component labeled by .
In terms of the cosmological parameters
(14) 
where and is the present value of the Hubble parameter, one can write the evolution equation (12) in the form
(15) 
where , and is the present value of the scale factor. The cosmological parameters are related through the equation
(16) 
For further convenience, we introduce the timedependent parameters and :
(17) 
(18) 
Then, from (12) and (13) one can derive a useful equation
(19) 
Restricting our attention for the moment to the present epoch, when the density of matter greatly exceeds that of radiation, we find
(20) 
where
(21) 
so that
(22)  
(23) 
An important feature of the Phantombrane is that, for a given value of , its expansion rate is slower than that in CDM, see figure 1. This property of our model is of special significance when one compares its observational predictions with observational data Sahni:2014ooa (). Indeed, recent measurements of the expansion rate at high redshifts using the data on baryon acoustic oscillations (BAO) indicate km/sec/Mpc at , which is below the value predicted by CDM Delubac et al. (2014). This tension is alleviated in the Phantom brane Sahni:2014ooa ().
The effective equation of state (EOS) of dark energy on the brane is given by ss06 ()
(24) 
where the prime denotes differentiation with respect to . Substituting (20) into (24) one finds that the present value of the EOS is
(25) 
Consequently, the Phantom brane possesses a phantom EOS , just as its name suggests.
Iii Evolution of scalar cosmological perturbations
iii.1 General equations for a multicomponent fluid
Scalar cosmological perturbations of the induced metric on the brane are most conveniently described by the relativistic potentials and in the longitudinal gauge:
(26) 
The components of the linearly perturbed stress–energy tensor of the component of matter can be parameterized as follows:^{1}^{1}1The spatial indices in purely spatially defined quantities (such as and ) are always raised and lowered using the spatial metric ; in particular, . The symbol denotes the covariant derivative with respect to the spatial metric , and the spatial Laplacian is .
(27) 
where , , , and describe scalar perturbations.
Although the term [see (10)], which can be treated as the homogeneous part of the Weyl fluid, or dark radiation, was set to zero in (11), perturbations of the Weyl fluid should be taken into account. Therefore, in a similar way we introduce scalar perturbations , , and of the traceless tensor :
(28) 
where .
We call and the momentum potentials for the matter components and Weyl fluid, respectively; the quantities and are their energydensity perturbations, and and are the scalar potentials for their anisotropic stresses.
In this notation, the effective field equation leads to the following system for perturbations in the Fourierspace representation with respect to comoving spatial coordinates Viznyuk:2013ywa (); Shtanov:2007dh (); Viznyuk:2012oda ():
(29) 
(30) 
(31) 
where is the comoving wavenumber and , were defined in (17), (18). From the conservation law (8) applied to each component separately, we have
(32) 
(33) 
For simplicity, we have assumed here that the anisotropic stresses of all matter components vanish: . Note that in general relativity these assumptions would lead to the equality of relativistic gravitational potentials and , but, in the braneworld, this is not the case due to the presence of the anisotropic stress of the Weyl fluid [see (31)]. The contribution from in the braneworld model cannot be ignored, but should be established from the analysis of fivedimensional perturbations in the bulk.
The pressure perturbations are usually decomposed into adiabatic and isentropic parts:
(34) 
where
(35) 
is the adiabatic sound speed, and describe the nonadiabatic pressure perturbations. In what follows, we restrict ourselves to adiabatic perturbations by setting . At the same time, the equation of state parameters can be arbitrary functions of time.
In the quasistatic approximation proposed by Koyama and Maartens in Koyama:2005kd (), there arises the following approximate relation between and :
(38) 
Relation (38) is properly justified on subHubble scales, where . As noted in Sec. I, the KoyamaMaartens relation cannot be used to study the behavior of perturbations of the Weyl fluid during the early radiative epoch on superHubble spatial scales.
A more general relation between and was derived in Viznyuk:2013ywa () in the limit of a marginally closed braneworld:
(39) 
Relation (39) accounts for the temporal evolution of the Weyl fluid, which makes it possible to trace the evolution of perturbations right from their initial values on superHubble spatial scales all the way until the present time. We shall use equation (39) in the present analysis.
To investigate perturbations of a multicomponent fluid, we introduce convenient variables
(40) 
The variables are proportional to the physical velocity potentials . It is reasonable to introduce similar variables for the Weyl fluid. The Weyl fluid, being described by a traceless effective stressenergy tensor in (3), behaves in a way rather similar to radiation. Hence, since the background density of the Weyl fluid vanishes, we use the radiation background component to define
(41) 
where and are, respectively, the energy density and pressure of radiation. Naturally this assumes the presence of radiation during all stages of cosmological expansion, which is certainly true for our universe after inflation.
In terms of the new variables (40), (41), we have the following closed system of equations on the brane:
(42) 
(43) 
(44) 
(45) 
(46) 
(47) 
(48) 
(49) 
Remarkably, equations (47)–(49) lead to a single equation for the variable :
(50) 
where
(51) 
As one can see, perturbations of all matter species influence the evolution of the Weyl fluid. In turn, perturbations of the Weyl fluid affect the gravitational potentials via (42) and (44), which influences the perturbations of matter via (45) and (46).
iii.2 Perturbations of a single fluid
In this subsection, we consider the situation when one fluid component dominates over the rest. In this case, for the dominating component we can use a singlefluid version^{2}^{2}2Some correction from subdominant matter component might be expected from the term in (56) in the superHubble regime, when spatial gradients are neglected. However, for the adiabatic modes under consideration in this paper, such corrections are absent due to appropriate initial conditions; see Eq. (78) below. of (56) and (57):^{3}^{3}3We omit the label for the dominating matter component in this subsection.
(58) 
(59) 
Remarkably, using (58), (59) and (53), one can derive a single secondorder differential equation for the variable :
(60) 
which should be supplemented with (50):
(61) 
The system of equations (III.2), (61) describes perturbations of a relativistic fluid with arbitrary equation of state and whose adiabatic sound speed is given by (35).
iii.3 The Friedmann expansion regime
An important feature of our braneworld, which distinguishes it from the Randall–Sundrum model, is that the effect of the extra dimension on cosmic expansion is usually small at early times Shtanov:2000vr (); Sahni:2002dx (). We refer to this early epoch as the Friedmann regime, since the equations of dimensional general relativity determine the course of cosmic expansion during this early time [see Eq. (11)]. Nevertheless, at the perturbative level, perturbations of the extradimensional Weyl fluid exist at all times and can never be ignored. Thus, we investigate the evolution of perturbations during early times when
(64) 
implying that the effect of the extra dimension (parameterized by the inverse length ) on background evolution is small. In this approximation, equation (11) turns into the Friedmann expansion law
(65) 
so that
(66) 
Relation (61), which describes the evolution of the Weylfluid, simplifies to
(67) 
where we have neglected terms of order with respect to unity, according to (64), (66). Perturbations of the energy density in each fluid component under this approximation can be derived from (56), (57):
(68)  
(69) 
where the variable is related to via (47).
Evolution of the gravitational potentials during the Friedmann regime is determined by [see (54) and (55)]:
(70)  
(71) 
Finally, we note that in the case of a singlecomponent fluid, we can, instead of (68) and (69), employ the earlytime version of (III.2):
(72) 
One finds that, in the formal limit of , perturbations of the Weyl fluid do not affect those of ordinary matter. Thus, we expect that perturbations of matter components at early times will behave as in general relativity. However, the approximation is too crude and does not allow control of its accuracy [in contrast to the approximation , which was used to derive (67)–(72)].
iii.4 Scaling approximation on superHubble scales
We can observe that, at the early stages of cosmological evolution, when the Friedmann approximation considered in the previous subsection is applicable, our approach matches well with the scaling ansatz considered in Sawicki:2006jj (); Song:2007wd (); Seahra:2010fj (). Indeed, in the Friedmann regime, the expansion of the brane is driven by the energy density of the dominating matter component [see (65)] which evolves by a power law in the scale factor . One can expect the existence of solution of (67) for the variable in the form of a power of as well. In such a case, we have the orderofmagnitude estimates
(73) 
On superHubble scales, where , the last term on the lefthand side of (67) can be neglected. If we also neglect the homogeneous part of solution of (67) (this condition is equivalent to that there is no sources for except the brane itself), we obtain
(74) 
where and are both related to the dominating matter component. To be more specific, in the era of matter domination, we have , , and , which results in . In the case of radiation domination, we have , [see also (90)], and . Correspondingly, in this case.^{4}^{4}4 We note that the variable is related to the master variable (projected onto the brane) via (see Viznyuk:2013ywa ()). As follows from our consideration, the master variable on the brane behaves as , where in the regime of radiation domination, and if pressureless matter dominates over radiation. We observe that the powers in the evolution law of the variable coincide with those predicted by the scaling ansatz in Sawicki:2006jj ().
Taking into account (47), we also have the orderofmagnitude estimate
(75) 
Then, we can apply (70) and (71) to establish the following relation between the gravitational potentials:
(76) 
Thus, in the regime of Friedmann expansion, which is characterized by the condition , perturbations on superHubble spatial scales are described by the approximate relation
(77) 
Relation (77) can be considered as a superhorizon counterpart of (39), because, if we assume it, we get a closed system of equations for perturbations on the brane. Remarkably, the scaling ansatz for braneworld perturbations gives the same closing relation on superHubble scales Seahra:2010fj (), which indicates the match between the two methods, at least to the leading order in a small parameter . In this sense, the scaling ansatz, which is based on the assumption of vanishing bulk master variable on the past Cauchy horizon, can be regarded as an approximate partial solution of a more general condition (39). Our condition (39) allows one to investigate the behavior of perturbations which have been originated purely in the bulk, along with perturbations originated purely on the brane. A rigorous definition of these two modes will be given in the next section.
Iv Perturbations during the radiative epoch
iv.1 Initial conditions
The primordial spectra for scalar cosmological perturbations are specified deep within the radiation domination epoch. At that time, the modes relevant to structure formation belong to superHubble spatial scales, and perturbations of pressureless matter^{5}^{5}5Pressureless matter in our investigation possesses all characteristics of cold dark matter. are decoupled from those of radiation.^{6}^{6}6Ultrarelativistic primordial plasma will be treated as an ideal radiation. Effects related to baryons and neutrino will not be considered in this work. As is well known, in general relativity, adiabatic nondecaying modes^{7}^{7}7In this paper, we do not investigate possible effects of isocurvature modes. on superHubble scales remain almost constant in time, and are related to the value of the gravitational potential as follows:^{8}^{8}8In terms of energy density contrasts this relation implies: .
(78) 
where the subscript ‘’ denotes initial values, the subscript ‘’ refers to pressureless matter, and the subscript ‘’ to radiation.
As the effects of the extra dimension in our model weaken at early times, we expect the above relations to also be valid in our braneworld during the radiative epoch (this expectation will be confirmed in the next section). We also assume that initial linear perturbations of matter and radiation are random with Gaussian statistics. In this case, they are completely characterized by the power spectrum , defined as
(79) 
Cosmological observations, interpreted within the framework of the CDM model, indicate that the initial power spectrum is nearly flat. The following parametrization is commonly used:
(80) 
where is a pivot scale which, in the Planck data analysis Ade:2013zuv (), is chosen to be , defines the normalization of the spectrum, and is its slope. The power spectrum is flat if the scalar spectral index , which is very close to the observed value Ade:2013zuv ().
We will see in the next section that perturbations of the Weyl fluid, as well as those of matter, weakly depend on time before Hubbleradius crossing. It is thus natural to assume that the initial value of the Weyl fluid is also randomly distributed with Gaussian statistics, so that
(81) 
where the primordial power spectrum for the Weyl fluid can, in principle, be different from for matter (and radiation). Setting aside the issue of generation of primordial perturbations, one can study the consequences of a general parametrization of the preceding type (80):
(82) 
where is the same pivot scale as in (80), and is the normalization of the initial power spectrum for the Weyl fluid.
In this paper we do not discuss possible mechanisms for the generation of primordial perturbations in our braneworld, and therefore cannot tell whether or not the primordial perturbations of the Weyl fluid are correlated with those of matter and radiation. For simplicity we shall assume them to be statistically independent. In this case, one can consider the evolution of two basic modes:
 Brane mode

initial perturbations of the Weyl fluid are absent:
(83)  Bulk mode

initial perturbations in matter & radiation on the brane are absent:
(84)
Here, defines the normalization of the primordial power spectrum, and is a numerical constant describing the intensity of the bulk mode. In the case of superposition of these two modes, the power spectrum of any quantity will be given by the sum of the corresponding power spectra (since these two modes are assumed to be statistically independent).
We note that, if the initial adiabatic perturbation for matter/radiation and for the Weyl fluid are not assumed to be statistically independent, say, if is proportional to