1. Introduction
Recent years have witnessed an increased interest in bulk-viscous properties in the cosmic fluid. From a hydrodynamicist’s point of view, it is almost surprising that this surge of interest has not occurred earlier. As is known, viscous effects are quite ubiquitous in ordinary hydrodynamics, and one should not expect the cosmic fluid to be an exception in that respect. There are in general two viscosity coefficients, the shear viscosity
and the bulk viscosity
, corresponding to first-order deviation from thermal equilibrium, although normally, the shear viscosity is omitted in cosmology because of the assumption about the spatial isotropy of the fluid. However, what happens if one takes away the assumption about isotropy? We demonstrate that in the Kasner universe, shear viscosity seems unphysical. Hence, we are left with bulk-viscous modifications to the equation of state. Moreover, we present in this paper a striking similarity between the results obtained for the bulk viscosity of the late Universe [
1] based on a much-used phenomenological approach (
) and that calculated by Husdal and others (see below) for the early lepton era.
We will assume a spatially flat Friedmann–Lemaître–Robertson–Walker universe, where the metric is:
and the energy-momentum tensor of the whole fluid is:
with
as the projection tensor. The scalar expansion is
. In comoving coordinates, the components of the fluid four-velocity are
.
The total energy density
of the cosmic fluid is taken to be composed of several parts,
where subscripts
, and
refer to dark energy, dark matter, baryons, and radiation, respectively. Experiments show that the dark sector amounts to about 95% of the total energy content [
2]. We will define
as the sum of the dark matter and baryons,
The contribution from radiation is negligible. Defining the usual density parameters
with
the critical density, we have:
At present (subscript 0), the Planck experiment finds ([
2], Table 2).
, summing up to unity. It is also useful to note that
km s
Mpc
s
,
kg m
.
Consider now the Friedmann equations:
The obvious task is to figure out how to model the bulk viscosity (although not a focus in the present work, we also remark that a positive, non-vanishing bulk viscosity will generate entropy [
3]). One obvious choice—and the one made in this paper—is to model the bulk viscosity as a function of the total energy density,
. In [
1,
4], we advocated the power-law form:
with
a constant. Preference, although not a very strong one, was given to the case
, in agreement with several other investigators having compared with experiments.
Our motivations for undertaking the present investigation are the following:
(i) The phenomenological approach above was based on redshifts up to about
. It is of interest to make a big jump in the cosmological scale, back to the lepton-photon universe, characterized by temperatures between
K and
K, where the Universe was populated by photons, neutrinos, electrons, and their antiparticles. Furthermore, under such circumstances, a bulk viscosity appears, explained in kinetic terms as a result of the imbalance between the free paths of neutrinos and the other particles. The maximum bulk viscosity occurs at the time of neutrino decoupling,
K. We will base our analysis on the recent papers of Husdal et al. [
5,
6,
7]. Moreover, we will show that a bold extrapolation of the formula (
8) with
back to this very early instant brings surprisingly good agreement with the kinetic theory based result for
. It becomes suggestive to assume that Formula (
8) holds for very longer times back than what is so far justified from observations. In other words, there is apparently a kind of symmetry between the early-time cosmology and the present-day cosmology as far as the bulk viscosity is concerned.
(ii) Some effort ought to be made to clarify the reasons why there are apparently conflicting statements in the recent literature. The extensive analysis of Yang et al. [
8] favors the form
with
, thus of a generalized Chaplygin form, and quite different from what we stated above. As one might expect, this discrepancy is rooted in differences in the initial formalism. We consider this theme in some detail in
Section 3.
(iii) The common omission of the shear viscosity in cosmology is not quite trivial, all the time that the shear viscosity is the dominant viscosity in ordinary fluid mechanics. One might suspect that even a slight anisotropy in the cosmic fluid could easily compensate for the bulk viscosity. We consider this point in
Section 4, choosing the anisotropic Kasner universe as an example. It actually turns out that the Kasner model is not easily compatible with a shear viscosity. On the other hand, the model admits a bulk viscosity without any problems, in the degenerate case of spatial isotropy.
Readers interested in review articles on viscous cosmology may consult [
3,
9,
10,
11,
12]. By now, the literature is rich with contributions on the topic of viscous cosmology. The contributions include investigations of the early universe [
13,
14,
15,
16], the late universe [
17,
18,
19], the phantom divide [
20,
21,
22,
23,
24,
25], models for the dark sector [
26,
27,
28,
29], and others [
30,
31,
32,
33,
34,
35,
36,
37,
38,
39].
2. Possible Relationship to the Bulk Viscosity in the Lepton-Photon Epoch
On the basis of
measured for different moderate values of
z (
), we estimated in [
1,
4] the present bulk viscosity
to lie in the interval:
Although we refer the interested reader to the sources for detailed explanations, we will in the following clarify under what assumptions these results were obtained. First of all, we made use of the ansatz (
8), testing three different values for the exponent
,
. The equation of state was assumed in the simple form
with
a constant. In view of the dominance of the dark energy component of the fluid, we took the value of
w to lie closely to
when we investigated a one-component fluid model [
4]. Mathematically, this means:
From the 2015 Planck data [
40],
. It corresponds to:
From the second Friedmann Equation (
7) together with the ansatz (
8), we see that one may introduce an effective pressure
, with an effective equation of state parameter:
Thus, if , is in general a function of .
We know, however, that the cosmic fluid is not actually a one-component fluid, but rather consists of many different constituents. As mentioned earlier, the total energy density is assumed to be composed by a dark energy component
and a matter component
satisfying
. Denoting the homogeneous solution corresponding to
by
, we can decompose:
with
. Moreover, for simplicity, we assume that the viscosity can be associated with the fluid as a whole. Going beyond a phenomenological approach, this seems to imply that the viscosity directly or indirectly is sourced by the interaction of the various components of the phenomenological one-component fluid. Consider [
41,
42] for interesting discussions. Furthermore, a bulk viscosity associated with the overall fluid would be most accurate if the dark sector interacts with the constituents of the standard model. Associating the viscosity with the fluid as a whole enables us to write:
where
is determined from Friedmann’s equations. It is useful to define the auxiliary quantity:
whose value is about unity in astronomical units (note the subscript
on
B; this subscript was not used in our previous works, but should have been there in order to indicate that this is a present-day value). We introduce the common notation
and give the final formulas for the most actual option
only. Then,
We have thus delineated the assumptions that led us to the result (
9), by comparison with the observations of
. This is also largely in agreement with earlier investigators [
31,
43].
Let us now go back to the early Universe, the lepton-photon era, in which case the viscosity coefficients have to be calculated by kinetic theory. There are two factors that are important for the calculation, namely the state of the system (we assume it to be a pure lepton-photon mixture) and then the transport equations for the fluid. The free mean paths for the neutrinos are much larger than those of the electromagnetically interacting particles, thus building up a temperature difference between the fluid components. The electromagnetic particles will cool somewhat faster than the neutrinos. Both coefficients
and
can be evaluated via the Chapman–Enskog approximation [
5,
6,
7,
44,
45], here given for
only,
The
n is the particle density;
are the coefficients for particle
k for the linearized relativistic Boltzmann equation; and
are known state parameters. As mentioned above, the most significant instant is that of neutrino decoupling,
K, at which Husdal obtained [
6]:
This value of
can be compared with that obtained from the expression (
8) extrapolated back to the instant of neutrino decoupling. The relation between
and
T in the early Universe is [
7] (in geometric units):
with
denoting the effective degrees of freedom at temperature
T.
Table 1 shows the resulting estimates for
for the three actual parameter values
. What is apparent is that, by choosing
and also by taking
equal to the logarithmic mean of the interval (
9), i.e.,
we find practically the same value of
as with (
8).
We find this coincidence striking. Of course, it is not a proof for the extended applicability of the formula (
8), but we think it deserves attention. It suggests that the formula can be used beyond the interval where it was originally constructed. Furthermore, it is notable that the parameter value
turns out to be the favorable choice, in agreement with other analyses as mentioned earlier.
3. Discussion
We shall show that the results obtained in this paper are actually not surprising at all, when considered from a phenomenological point of view, where the fluid has an overall viscosity. Consider the Bianchi identity for the overall cosmological fluid
:
This equation may be rewritten as:
where we have defined:
Thus,
r is the ratio between the viscous pressure
and the equilibrium pressure
. In the Eckart formalism, the viscous pressure exerted by
should remain a first order modification to the equilibrium pressure
throughout the history of the Universe. This translates into the requirement that
. Then, under the weak assumption that
is a monotonic function, we find:
Detailed analysis of the dynamical behavior of the cosmological fluid could of course reveal small variations in the ratio
r, so this relationship represents an approximation. Moreover, the variation in
r could very well prove to be important in a variety of contexts (Such as that of understanding the (microscopic) mechanism that gives rise to the viscosity, or structure formation), but in this phenomenological analysis, we are nevertheless more interested in large-scale variations. Thus, putting
translates into
. For a one-component fluid
, this is, by the application of the first Friedmann equation, just the same as ansatz (
8). As such, our result is trivial, yet obviously worth mentioning, considering the wide application of other functional forms for the viscosity, also in the case of one-component fluids. The fact that the results of Husdal actually agree with our own calculations, as inferred from supernova observations, shows that our theoretical prejudice seems to be confirmed by observation. This is, of course, not a trivial point. In more general terms, consider a viscosity
and an equilibrium pressure
. Then, the ratio
r defined in (
24) becomes:
The choice
now gives the condition (
25). One may also observe that any
will cause the ratio to grow if
decreases. Consequently, unless
asymptotically approaches a constant value, the ratio must therefore eventually grow out of bounds of the first-order thermodynamic (Eckart) formalism.
Comparison with the Result Obtained by Yang et al.
As mentioned in the Introduction, Yang et al. [
8] suggested a model where the effective pressure
of the dark fluid
is:
where
w is the equation of state parameter,
and
m are parameters of the theory, and
is the total energy density. The above equation makes sense only if
and
w are (respectively) the effective pressure and equation of state parameter for the unified dark fluid only, and not the overall fluid
. We therefore henceforth make this assumption. Thus, interpreting their results, we reach the conclusion that they worked with a different theory from our own. While they attributed the viscosity to the dark fluid only, we attribute the viscosity to the overall fluid. Perhaps more importantly, we fix the dark matter and dark energy components at present when obtaining our estimates for the present-day viscosity. As far as we understand, this is different from Yang et al., who (more appropriately) avoided such an a priori fixing of parameters. To sum up, the discrepancy derives from the difference in the theories. Without further examination, one may not conclude that the two approaches are in disagreement, per say. We shall repeat from the preceding subsection, however, that
r in Equation (
26) will eventually grow out of bounds for
, since
. Hence, a viscosity
cannot be considered as a bulk-viscous modification in the ordinary sense, unless
. As a general dynamical modification to a homogeneous equation of state, the results of Yang et al. seem however to be valid.
4. The Viscous Kasner Universe
It is of interest to consider the anisotropic universe. The reason why the Universe is usually considered to be spatially isotropic is that observation strongly indicates such behavior. However, there is also evidence, in cosmology, as well as in ordinary fluid mechanics, that the shear viscosity grossly dominates over the bulk viscosity in magnitude. Thus, it might be possible that the combination of a slight anisotropy with a dominant shear viscosity leads to physically detectable consequences after all. This is the motivation for the analysis in the present section. We will focus on the anisotropic Kasner universe, as a typical example of an anisotropic space. It belongs to Bianchi-Type I. An introduction to this model can be found, for instance, in [
46].
Consider the Kasner universe:
where the three
are constants, in the original formulation, which refers to a vacuum. Einstein’s equations can be written:
and the non-vanishing Christoffel symbols are (no sum over
i):
Allowing for both shear and bulk viscosities, we write the energy-momentum tensor as:
where the scalar expansion
is the trace of the expansion tensor:
and
is the shear tensor:
Defining the numbers
S and
Q as:
we can then write:
With
being the expansion factors of the metric, the directional Hubble parameters become
, and the average Hubble parameter becomes:
Let now
constant be the thermodynamic parameter,
With
, we can then write the Einstein Equation (
29) as:
The basic formalism given here is as in our earlier works [
47,
48]. We are thus considering an isotropic fluid in an anisotropic space (what we mean by this is that the same fluid in an isotropic background would not possess shear viscosity). The fluid itself is modeled as a usual fluid with density
and scalar pressure
p. Our model is thus different from one in which the fluid is taken to have anisotropic properties; cf., for instance, [
49]. We will assume that the physical properties of the fluid are given at some initial time called
and investigate if these initial conditions are compatible with constant values of
S and
Q. In the vacuum case (no fluid at all), one has
[
46]. We expect the Kasner model to be appropriate for the early Universe and will naturally choose the instant of neutrino decoupling,
K,
s, as mentioned above.
We consider the development of the fluid from
onwards; the initial energy and pressure being
and
. It is notable that the governing Equation (
38) actually fix the later time dependence to be:
whereby we obtain a time independent set of equations,
(note that in fundamental units where the basic unit is
cm, one has
cm
,
cm
). It is also convenient to note the following equations derived from those above,
The basic formalism outlined so far is essentially as in our earlier papers [
47,
48]. We assume now that the physical quantities
are given at
and investigate if these initial conditions lead to acceptable values for the coefficients
in an anisotropic Kasner universe. From Equation (
41), it follows that if the three
are to be unequal, the multiplying factor of
has to be zero,
This is thus an algebraic restriction on the
. If the fluid is nonviscous, then
, in accordance with the original Kasner model in a vacuum. Once
S is known,
Q follows at once from Equation (
43) as:
Note that the bulk viscosity does not appear in the last two equations. This is as should be expected physically: anisotropy is caused by the shear only.
Now, going over to dimensional units, we first note the useful relations
m kg
, 1 MeV
J m
. In the radiation dominated era:
With
Pa s, we then obtain from Equation (
44):
The energy density in this region can be roughly estimated from
, where
is the radiation constant:
Here, the degeneracy factor is omitted (cf., for instance, [
13]). Then,
J m
. Alternatively, we may use the equation
to get somewhat more accurately:
or
J m
. With the latter values, we calculate:
This is a non-acceptable result, as Q is a sum of quadratic numbers. We conclude that the Kasner model does not appear to be compatible with a shear viscosity.
There is also another reason why anisotropy (
) is problematic in the Kasner model. From Equation (
38), it follows that if
, which is the case for the usual fluids, the combination
becomes negative. Although negative viscosities are occasionally considered in cosmology (cf. for instance, [
50]), such a case is physically not very natural. This point was first noticed by Cataldo and Campo [
51].
It is worthwhile to notice that the isotropic Kasner geometry easily allows for a viscosity. That means only the bulk viscosity comes into question, as this viscosity concept goes along with spatial isotropy. Let us consider this case in some more detail, defining
a by
. From Equation (
41), we obtain, still in dimensional units,
This equation shows that if the physical quantities
and
, as well as the bulk viscosity
are given at
, then the isotropic version of the Kasner metric (
28) is determined.
There is finally one exceptional case that should be noticed, namely a Zel’dovich fluid for which
, the velocity of sound being equal to
c. In that case,
The metric turns out to be determined by the bulk viscosity as the only physical parameter; the actual value of
being irrelevant. If the fluid is nonviscous,
, then
. This corresponds to the metric coefficients in Equation (
28) being
.