Viscosity in modified gravity

A bulk viscosity is introduced in the formalism of modified gravity. It is shown that, on the basis of a natural scaling law for the viscosity, a simple solution can be found for quantities such as the Hubble parameter and the energy density. These solutions may incorporate a viscosity-induced Big Rip singularity. By introducing a phase transition in the cosmic fluid, the future singularity can nevertheless in principle be avoided.

We consider the case when ζ is satisfying a scaling law, reducing in the Einstein case to a form 23 proportional to the Hubble parameter. It turns out that this scaling law is quite useful. We survey 24 first earlier developments along this line, extracting material largely from our earlier papers [9][10][11]. 25 Thereafter, as a novel development we investigate how the occurrence of a phase transition can change 26 the development of the universe, especially in the later stages approaching the future singularity. (It may 27 here appear natural to relate such a phase transition with the onset of a turbulent state of motion.) It is 28 shown that such a transition may in principle be enough to prevent the singularity to occur at all. This  The action in modified gravity is conventionally written in the general form where κ 2 = 8πG, and where L matter is the matter Lagrangian. The equations of motion are where T matter µν is the energy-momentum tensor corresponding to L matter . 33 We shall however in the following not consider the general case, but limit ourselves to the special form where We assume the spatially flat FRW metric and put the cosmological constant Λ = 0. In comoving coordinates, the components of the four-velocity U µ are U 0 = 1, U i = 0. Introducing the projection tensor h µν = g µν + U µ U ν we have for the energy- wherep is the effective pressurep = p − 3Hζ.
The scalar expansion is θ = 3ȧ/a = 3H, with H the Hubble parameter. The shear viscosity is here 39 omitted.

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The equations of motion following from the above action are The equation of state for the fluid is written as fluids imply the inequality ρ + 3p ≤ 0, thus breaking the strong energy condition. 46 We now consider the (00) component of Eq. (7), observing that R 00 = −3ä/a and R = 6(Ḣ + 2H 2 ). With T matter 00 = ρ we obtain An important property of (9) is that the four-divergence of the LHS is equal to zero, ∇ ν T matter This is as in Einstein's gravity, meaning that conservation of energy-momentum follows from the field equations. The energy conservation equation becomeṡ Differentiating (9) with respect to t and insertingρ from (10), we get Inserting R = 6(Ḣ +2H 2 ) we see that this equation for H(t) is quite complicated. We shall be interested in solutions related to the future singularity, and make therefore the ansatz where H 0 is the Hubble parameter at present time, and B a non dimensional constant. If a future 47 singularity is to happen, B must be positive.

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Before closing this section, it is desirable to comment on stability issues for our ansatz (3) for the modified Lagrangian. A theory of modified gravity should admit an asymptotically flat, static spherically symmetric solution. Now, we expect that the expression (2) for the complete action, with (3) inserted, will not be the full solution. It is reasonable to expect that the modified part will contain also other terms so that (2) makes up only a part of the complete action. Nevertheless, it is of interest to ask to what extent (3), when take separately, will behave with respect to the stability requirements. In the Solar system, far from the sources, it is known that R ≈ 10 −61 eV 2 ; it corresponds to one one hydrogen atom per cubic centimeter. [Note that 1 eV= 5.068 ×10 4 cm −1 .] On a planet, R = R b ≈ 10 −38 eV 2 , whereas the average curvature in the universe is R ≈ 10 −66 eV 2 . According to the stability analysis of Elizalde et al. [22], the stability condition for matter is In our case, this means merely that the exponent α in the expression (3) has to be greater than one. The 49 stability condition on α is quite modest.

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It is now mathematically simplifying, and physically instructive, to focus on special cases.

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As mentioned above, Einstein's gravity corresponds to f 0 = 1 and α = 1. It means that the 54 Lagrangian is linear in R. It is natural to consider this case as a reference case before embarking on 55 the nonlinear general situation. 56 We first have to adopt a definite form for the bulk viscosity. The simplest choice would be to put ζ = constant. There are however reasons to assume a slightly more complicated form, namely to put ζ proportional to the Hubble parameter H. This is physically natural, in view of the large fluid velocities expected near the future singularity. Such violent conditions should correspond to an increased value of ζ. We shall take ζ to be proportional to the scalar expansion, θ = 3H, τ E being the proportionality constant in the Einstein theory. An important property of this particular form, shown in Ref.
[23], is that if τ E is sufficiently large to satisfy the condition then a Big Rip singularity is encountered after a finite time t. Even if the universe starts out from the 57 quintessence region (−1 < w < −1/3) or (0 < γ < 2/3), the presence of a sufficiently large bulk 58 viscosity will drive it into the phantom region (w < −1) and thereafter inevitably into the Big Rip 59 singularity.

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From the governing equations we get where ρ 0 is the present (t = 0) value of the energy density. The time dependent value ρ E for the energy density according to the Einstein theory becomes We ought here to mention that other forms for the bulk viscosity, more complicated than the form (14) 61 above, have been suggested. One possibility is that ζ, in addition to the term proportional to H, contains 62 also a term proportional toä/a. Consider now the modified gravity fluid, for which f 0 and α are arbitrary constants. As before, we look for solutions satisfying the ansatz (12). It turns out that such solutions exist, if we model the bulk viscosity ζ α according to the following scaling law [10,26], We see that this scaling fits nicely with our results from the preceding subsection: if α = 1, our previous form (14) follows. The time-dependent factors in (11) drop out, and we get the following equation determining B, This equation is in general complicated. Let us consider α = 2, γ = 0 as a typical example (recall that γ = 0 corresponds to a vacuum fluid). Then we obtain from (19) (τ α → τ 2 ), If the LHS is drawn as a function of B, it is seen that there is a local maximum at B = −4/3 and a local negative minimum at B = 0, irrespective of the value of τ 2 . For all positive τ 2 there is thus one single positive root. This root is viscosity-induced, and leads to the Big Rip singularity. When τ 2 increases from zero, there is a parameter region in which there are three real roots. Assume this region, and introduce an angle φ ∈ [0, 180 0 ] such that Then the actual value of the root can be expressed as For instance, if we choose φ = 120 0 , the positive solution becomes B = 0.3547. According to Eq. (12) this gives the following Big Rip time We may also note the general relation for B following from the energy conservation equation (10), when ρ → ρ α , p → p α , ζ → ζ α , Here we used and for simplicity we used the same initial conditions at t = 0 for the modified fluid as for the Einstein 65 fluid, ρ 0α = ρ 0E ≡ ρ 0 , and H 0α = H 0E ≡ H 0 .

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In the preceding we have surveyed bulk viscosity-induced generalizations of modified gravity, following essentially the earlier treatments in Refs. [10,11,26]. Our intention in the following, as a new contribution, will be to discuss the flexibility that the above model possesses with respect to sudden changes in the time development (we will refer to it as phase transitions) in the late universe. The main point is the different solutions for B in the governing equation (11) that are possible when the scaling ansatz (18) is inserted. We obtain the following algebraic equation for B, for definiteness still assuming α = 2, This equation generalizes (20) to the case of nonvanishing γ.

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Consider the following scenario: the universe starts out from present time t = 0 and follows the equations of modified gravity, with a τ 2 -induced bulk viscosity corresponding to a positive value of B. That means, the universe develops according to with X = 1 − BH 0 t. The universe thus enfaces a future singularity at large times. Let now, at a fixed 69 time that we shall call t * , there be a phase transition in the cosmic fluid implying that the effect from 70 τ 2 goes away. It means that the further development of the fluid will be determined by the γ-dependent 71 roots of Eq. (26) when τ 2 = 0. There are three roots: 72 1) The first is B = 0. This is the de Sitter case, corresponding to where H * and ρ * follow from (27) when t = t * . By assuming that |γ| ≪ 1, which is of main physical 73 interest, we see that it is easy to determine the remaining two roots. One of them is 74 2) B = −2. This means The accelerated expansion is accordingly reversed at t = t * , and the density goes smoothly to zero at 75 large times.

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3) The third root is B = −3γ/4, which yields The sign of γ is important here. If the equation-of-state parameter w lies in the quintessence region, 77 w > −1 (γ > 0), then the density of the universe will go to zero for large times, like for the case 2) 78 above. By contrast, in the phantom region w < −1 (γ < 0), the universe will actually move towards a 79 Big Rip again, although very weakly so.

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Finally, it is of interest to compare the above results with those obtained in ordinary viscous cosmology when the universe, similarly as above, is thought to undergo a phase transition at a definite time t * . Such an investigation was recently carried out in Ref.
[13] (the one-component case treated in Sect. VI). Consider the following model: the universe starts from t = 0 as an ordinary viscous fluid with a constant bulk viscosity, and develops according to the Friedmann equations. Assume that the universe is in the phantom region, γ < 0. It follows that in the initial period 0 < t < t * , where t c means the "viscosity time", According to these equations the universe develops towards a Big Rip. Now, after t = t * we imagine an era for which γ turb = 1 + w turb > 0 and an equation of state of the form Here the subscript "turb" refers to our association in [13] of the transition at t = t * into an era dominated 81 by turbulence.

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Then, for t > t * , This means a dilution of the density again, at large times. The Big Rip may thus be avoided, as a result 83 of a phase transition in the cosmic fluid. We see that in this sense the behavior is similar in the two cases, 84 modified or ordinary, gravity.  singularities. This is thus a Big Rip scenario induced by the bulk viscosity.

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In Sect. 4 it was discussed how the future singularity can nevertheless in principle be avoided, if one 107 allows for a future phase transition in the cosmic fluid where the influence from viscosity goes to zero.

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Modified, or conventional, cosmology behave in this sense essentially in the same way.