Int J Theor Phys (2012) 51:2771–2778DOI 10.1007/s10773-012-1152-4

Can Neutrino Viscosity Drive the Late Time CosmicAcceleration?

Sudipta Das · Narayan Banerjee

Received: 16 November 2011 / Accepted: 1 April 2012 / Published online: 19 April 2012© Springer Science+Business Media, LLC 2012

Abstract In this paper it has been shown that the neutrino bulk viscous stresses can give riseto the late time acceleration of the universe. It is found that a number of spatially flat FRWmodels with a negative deceleration parameter can be constructed using neutrino viscosityand one of them mimics a ΛCDM model. This does not require any exotic dark energycomponent or any modification of gravity.

Keywords Dark energy · Neutrino viscosity · Accelerated expansion of the universe

1 Introduction

That the present universe is undergoing an accelerated expansion has now been firmly estab-lished. The initial indications came from the supernovae data [1–4] and were soon confirmedby many high precision observations including the WMAP [5–9]. Theoretical physics hasthus been thrust with the challenge of finding the agent, dubbed “dark energy”, which candrive this acceleration. Naturally a host of candidates appeared in the literature which canprovide this antigravity effect. However, the dark energy should become dominant only dur-ing the later stages of matter era so that nucleosynthesis in the early universe and the largescale structure formation in the matter dominated regime could proceed unhindered andmake the universe look the place where we live in now.

S. DasHarish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211 019, India

Present address:S. DasDepartment of Physics, Visva-Bharati, Santiniketan 731235, Indiae-mail: sudipta.das@visva-bharati.ac.in

N. Banerjee (�)Indian Institute of Science Education and Research-Kolkata, Mohanpur Campus, P.O. BCKV MainOffice, Nadia, West Bengal 741252, Indiae-mail: narayan@iiserkol.ac.in

2772 Int J Theor Phys (2012) 51:2771–2778

Amongst the various dark energy candidates, the cosmological constant Λ is certainlythe most talked about one. It matches different observational requirements quite efficientlyand has been known in cosmology for quite a long time for the various roles it could play.The insurmountable problem is of course that of the huge discrepancy between the requiredvalue of Λ and that predicted theoretically [10, 11].

The quintessence models, where an effective negative pressure generated by a scalarfield potential drives the acceleration, work extremely well to fit into various observationalconstraints. For a very brief review, we refer to Martin’s recent work [12]. But none of thepotentials employed for the purpose can boast of any sound theoretical motivation. Nonminimally coupled scalar field theories like Brans–Dicke theory can also be used whereeven a dark energy is not required [13], but the value of the Brans–Dicke parameter ω needsto be given a very small value contrary to the observational requirements. Scalar-tensortheories with a dark energy, particularly where an interaction between the dark energy andthe geometrical scalar field like the Brans–Dicke scalar field is allowed so as to alleviate thecoincidence problem, appear to do well [14, 15]. But the nature of the interaction is hardlywell-motivated. Also, a recent work shows that energy has to be pumped in to the dark matterfrom the dark energy sector as demanded by the second law of thermodynamics [16]. Thisis indeed counter-intuitive in view of the fact that the dark energy dominates over the darkmatter only during the later stages of evolution. Chaplygin gas models [17, 18], modifiedgravity theories [19–27] and many other models are proposed. They all have their successstories as well as failures in some way or the other. In the absence of a clear verdict in favourof a particular dark energy candidate, all of them have to be discussed seriously. There areexcellent reviews regarding different models and their relative merits [28–30].

One important general feature is that all these models use either some kind of an exoticfield or some modification of the firmly established general relativity—the effect of themodification being hardly required by other branches of well established physical theories,and the possibility of an actual physical detection of them appears to be quite a far-fetchedone.

Recently a well known sector of matter, whose existence in abundance has been firmlyestablished, namely the neutrino distribution, has been proposed as a candidate for the darkenergy [31–37]. The motivation comes from particle physics, and for a brief but comprehen-sive review we refer to [38]. However, in these models, neutrinoes are normally the sectorwhich ‘feels’ the existence of dark energy and cannot really solve the problem by itself, i.e.,without a quintessence potential. For a review of the neutrino properties in a cosmologicalcontext, we refer to [39].

In the present work, we treat the neutrinoes completely classically and show that bulkviscous stresses in the distribution can indeed do the trick.

The advantage of the neutrinoes is that they are real objects, the method of detectionbeing quite well conceived. The neutrinoes were decoupled from the background radiationquite early in the evolution when the temperature was as high as 3 × 1010 K [40]. Thusthe interaction of the neutrinoes with other forms of matter can be ignored and hence theproblem of the “direction of the flow of energy” [16] does not arise and the model becomesmuch more tractable.

Neutrino viscosity, both in the form of shear [41] or in the form of bulk viscosity [42–44]had been investigated quite a long time back for various purpose. There has been a renewedinterest in the neutrino viscosity quite recently as well [45]. The problem of dissipativeeffects like viscosity or heat conduction had been that of a parabolic transport equation,which could allow the signals to travel with super-luminal speed resulting in a violationof causality. But the extended irreversible thermodynamics, which modifies the transport

Int J Theor Phys (2012) 51:2771–2778 2773

equation by including a relaxation time and a further divergence term to avoid this problem,is now quite well understood [46] and the modified transport equation had already been quiteextensively used in cosmology [47–53].

In what follows, we employ a two-component non-interacting matter sector, one is thenormal cold dark matter and the other being a neutrino distribution. The latter is endowedwith bulk viscosity which produces a negative stress. It is shown that a very simple acceler-ated model can be constructed from this. The model looks simple as the shear viscosity isneglected in order to be consistent with the isotropic nature of the universe.

2 Field Equations and Results

Einstein equations for a spatially flat FRW universe are given by

3a2

a2= ρm + ρn, (1)

2a

a+ a2

a2= −pn − Π, (2)

where a is the scale factor for the model given by

ds2 = dt2 − a2(t)(dr2 + r2dΩ2

);ρm and ρn are the densities of the normal matter and the neutrino distribution respectivelyand an overhead dot represents a differentiation with respect to the cosmic time t . The equa-tions are written in units where 8πG = 1. Consistent with the present universe, we assumethat the normal matter sector is pressureless. The energy density of a gas is given by

ρn = nM + 3

2nkT , (3)

where n is the number density, M is the rest mass, T the temperature and k is the Boltz-mann’s constant. In what follows, an equation of state for the neutrino distribution is takenas

pn = γρn, (4)

where pn is the neutrino pressure. For a completely relativistic case where the rest massof the neutrino can be neglected in comparison with the temperature (i.e. the kinetic part),γ = 1

3 , whereas for a nonreltivistic case where the rest mass M overwhelmingly predomi-nates, γ = 0. The bulk viscous stress satisfies the causal transport equation

Π + τΠ = −3ηH − 1

2τΠ

[3H + τ

τ− η

η− T

T

], (5)

where τ is the relaxation time for the dissipative effects, η is the co-efficient of viscosityand T is the temperature. Being non-interacting amongst each other, both ρm and ρn satisfytheir own conservation equations. For ρm, the conservation equation

ρm + 3Hρm = 0

immediately integrates to yield

ρm = ρm0

a3, (6)

2774 Int J Theor Phys (2012) 51:2771–2778

Fig. 1 Plot of q vs. z for thechoice of a given by (8)

ρm0 being a constant of integration. Equations (1), (2) and (6) combine to give the conser-vation equation for ρn as

ρn + 3H(ρn + pn + Π) = 0, (7)

which indeed is not an independent equation. So, one has three unknowns, namely a, ρn andΠ whereas only two equations to solve for them. In order to close the system of equations,another equation will be required. As we require a specific dynamics for the universe, whichis accelerating at the present moment but had experienced a more sedate form of a deceler-ated expansion during a none-too-distant past in the matter dominated regime itself, we takethe form of a as

a = a0

[sinh(αt)

]2/3, (8)

as the third equation. Here a0 and α are positive constants. This simple ansatz gives therequired behaviour. For small values of ‘t ’,

a ∼ t2/3,

which indeed yields a decelerated expansion. In fact the deceleration parameter

q = − a/a

a2/a2= 1

2,

which is positive and exactly same as that for a matter dominated universe without any otherfield. For a large t ,

a ∼ e2t/3.

The universe is exponentially expanding and hence accelerated with q = −1. The evolutionof q with the redshift parameter z for the choice of a as in (8) is given in Fig. 1 whichindicates that q has a smooth transition from a positive to a negative phase. As z is given by

1 + z = a0

a,

where a0 is the present value of a, z is a measure of the epoch (in the past) that one is lookingat. The present value of z is zero.

Equation (8) can be used in (2) to obtain

pn + Π = −4α2

3, (9)

Int J Theor Phys (2012) 51:2771–2778 2775

which is a constant. Equation (7) can now be integrated to yield

ρn = ρn0

a3+ 4α2

3, (10)

where ρn0 is a constant of integration. So, now the model is solved, it allows for the observedbehaviour of q and is extremely simple.

This effectively gives rise to a ΛCDM model, as Einstein’s equations now look like

3a2

a2= ρm0 + ρn0

a3+ 4α2

3, (11)

2a

a+ a2

a2= 4α2

3. (12)

The constant value 4α2

3 of pn +Π serves the purpose of a cosmological constant. So one canhave all the virtues of a ΛCDM model, without having to write a cosmological constant byhand. An effective Λ is given by the evolution of the neutrino distribution, which is alreadyquite well understood. So the model does not require any exotic field or a modification ofgravity.

The high degree of non linearity of Einstein’s equations allows us to have other solutionsas well. So this beautiful solution is by no means a unique one. In fact there could be otherconsistent accelerated solutions for (1), (2) and (7). One example is

H = a

a= A

(1 + a−3/2

), (13)

where A is a constant. This solution also describes the present acceleration quite efficientlyas discussed by the present authors [54]. This ansatz also satisfies all the field equations.However, this will not mimic a ΛCDM model as pn + Π comes out to be evolving ratherthan being a constant. Using (13) in (2), one can write Π as

pn + Π = −3AH.

The density perturbation for the “effectively ΛCDM” model given by (8) can be studied.The linearized perturbation equation

δ + 2Hδ − 4πGρmδ = 0 (14)

for the model can be numerically integrated to yield the growth of the density perturbationas shown in Fig. 2. The density contrast δ is as usual given by

δ = ρm − ρ

ρ

where ρm is the perturbed density and ρ is the background density.The temperature profile of the neutrino distribution can also be estimated with the help

of (5) and (4). For this purpose, the coefficient of viscosity η and the relaxation time τ haveto be given in terms of quantities like the density. The popular choice of η = η0ρ

m and τ = η

ρ

where m is a constant [46], however, does not seem to work well. With these choices andfor m = 1

2 , (5) provides an expression for the neutrino temperature T as

T = T0

⎡

⎣(

X√

p(2√

X + 1√

X + p + 2X + p + 1)

(2√

p√

X + 1√

X + p + (p + 1)X + 2p)√

p

)− 1η0α

√ρn0a0

3

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Fig. 2 Plot of density contrast δ

vs. a for α = 1 which clearlyshows a growing mode for thedark matter distribution

Fig. 3 Plot of neutrinotemperature T (in the unitsT0 = 1) vs. redshift z forη = η0ρ1/2 and τ = η

ρ

×( β2 + ρ2

n0γ 2

a60

X2 + 2γβρn0a3

0X

([( a30

X+ ρn0γ

β] 4α2

3β− 1

γ (a3

0X

)1γ )2

)

×(

a30

4α2X/3+ ρn0

a30

X2

)

where X = cosech2(αt) = (a0a)3 = (1 + z)3 and p = 4α2a0

3

3ρn0. The plot of T vs. z shows that

the neutrino temperature given by this ansatz shoots to a very high value at low values of theredshift z (z ∼ 1) (see Fig. 3), whereas the neutrino temperature is known to be below thatof the cosmic microwave background radiation. Some other choices of η and τ may providea better temperature profile.

3 Conclusion

As a conclusion, one can say that a sufficient bulk viscous stress in the neutrino distributioncan potentially serve the purpose of a dark energy. This does not require any ill-motivatedscalar potential or an otherwise unwarranted modification of general relativity. However,the model is far from being complete. The amount of the bulk viscous stress has to besufficient to drive the acceleration, and the correct phenomenological connection betweenthe quantities like η, τ and ρ has to be found out so that temperature of neutrinoes has amore realistic profile. The neutrino viscosity can in fact give a wide range of accelerating

Int J Theor Phys (2012) 51:2771–2778 2777

models. One of them, the effective Λ CDM model is discussed here. But the other examplementioned also works, and leaves a possibility of finding many others within the scope of it.It has to be searched which solution is favoured from the consideration of stability as wellas that of observational bounds.

Acknowledgement Authors would like to thank Relativity and Cosmology Research Centre at JadavpurUniversity, Kolkata where part of this work has been done and the referees useful comments.

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