Gravitational instabilities of the cosmic neutrino background with non-zero lepton number

JID:PLB AID:33017 /SCO Doctopic: Astrophysics and Cosmology

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Physics Letters B ••• (••••) •••–•••

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Gravitational instabilities of the cosmic neutrino background with non-zero lepton number

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Neil D. Barrie, Archil Kobakhidze

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ARC Centre of Excellence for Particle Physics at the Terascale, School of Physics, The University of Sydney, NSW 2006, Australia

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a r t i c l e

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Article history: Received 3 June 2017 Received in revised form 7 July 2017 Accepted 7 July 2017 Available online xxxx Editor: J. Hisano

We argue that a cosmic neutrino background that carries non-zero lepton charge develops gravitational instabilities. Fundamentally, these instabilities are related to the mixed gravity-lepton number anomaly. We have explicitly computed the gravitational Chern–Simons term which is generated quantummechanically in the effective action in the presence of a lepton number asymmetric neutrino background. The induced Chern–Simons term has a twofold effect: (i) gravitational waves propagating in such a neutrino background exhibit birefringent behaviour leading to an enhancement/suppression of the gravitational wave amplitudes depending on the polarisation, where the magnitude of this effect is related to the size of the lepton asymmetry; (ii) Negative energy graviton modes are induced in the high frequency regime, which leads to very fast vacuum decay producing, e.g., positive energy photons and negative energy gravitons. From the constraint on the present radiation energy density, we obtain an interesting bound on the lepton asymmetry of the universe. © 2017 Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .

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1. Introduction

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Along with the Cosmic Microwave Background radiation (CMB), the existence of the Cosmic Neutrino Background (Cν B) is an inescapable prediction of the standard hot big bang cosmology (see e.g. [1] for a review). It is assumed to be a highly homogeneous and isotropic distribution of relic neutrinos with the temperature:



45 46 47 48 49 50 51 52

Tν =

55 56 57 58



11

T γ ≈ 1.945 K ,

(1)

where T γ = 2.725 K is the temperature of the CMB today. Unlike the CMB though, the Cν B is extremely hard to detect and its properties are largely unknown. Namely, the Cν B may exhibit a neutrino–antineutrino asymmetry

53 54

4

1 3

ην α =

nνα − n¯ να nγ

=

π

2

12ζ (3)



3

ξ ξα + α2

π

 ,

(2)

for each neutrino flavour α = e , μ, τ . Here ξα = μα / T is the degeneracy parameter, μα being the chemical potential for α -neutrinos. In fact, such an asymmetry is generically expected

to be of the order of the observed baryon–antibaryon asymmetry, η B = (n B − n¯ B )/nγ ∼ 10−10 , due to the equilibration by sphalerons of lepton and baryon asymmetries in the very early universe. However, there are also models [2,3] which predict an asymmetry in the neutrino sector that are many orders of magnitude larger than η B . If so, this would have interesting cosmological implications for the QCD phase transition [4] and/or large-scale magnetic fields [5]. The most stringent bound on the neutrino asymmetry comes from the successful theory of big bang nucleosynthesis (BBN). BBN primarily constrains the electron neutrino asymmetry. However, this bound applies to all flavours, since neutrino oscillations below ∼10 MeV are sizeable enough to lead to an approximate flavour equilibrium before BBN, μe ≈ μμ ≈ μτ (≡ μν ) [6–8].1 The updated analysis presented in [9] leads to the following bound on the common degeneracy parameter:

103

|ξν |  0.049

119

(3)

In this paper, we argue that the lepton asymmetry in the active neutrino sector leads to gravitational instabilities. These instabilities originate from the gravity-lepton number chiral quantum

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105 106 107 108 109 110 111 112 113 114 115 116 117 118 120 121 122 123 124

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E-mail addresses: [email protected] (N.D. Barrie), [email protected] (A. Kobakhidze).

1

See, however, a recent analysis in [10], where a larger found to be allowed.

ηνμ ,ντ asymmetry is

http://dx.doi.org/10.1016/j.physletb.2017.07.012 0370-2693/© 2017 Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .

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N.D. Barrie, A. Kobakhidze / Physics Letters B ••• (••••) •••–•••

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where S 0 ( p ) is the usual fermion propagator in vacuum. The above modified neutrino propagator to first-order in μν is given by S ( p ) ≈ S 0 ( p ) − i μν p/ −i m γ0 γ 5 p/ −i m . The higher order terms in bμ ,

1 2 3 4

or μ, are neglected because we are only interested in the linear terms in bμ , which will result in a Chern–Simons like term. Taking this, and using the standard Feynman rules, we find that the parity odd part of the full graviton polarization tensor is:

5 6 7 8

11

14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

anomaly, which is present in the Standard Model when considering Majorana neutrinos. Indeed, in the case of Majorana neutrinos, a non-zero lepton asymmetry for active neutrinos implies an imbalance between neutrinos of left-handed chirality and antineutrinos of right-handed chirality which, as we demonstrate explicitly below, leads to the inducement of the gravitational Chern–Simons term in the effective action. This is analogous to the inducement of Chern–Simons terms in gauge theories [11]. The induced Chern–Simons term causes birefringence of gravitational waves propagating in the lepton asymmetric neutrino background, which can be sizeable for gravitational waves generated at very early times. More importantly, short-scale gravitational fluctuations exhibit negative energy modes, which lead to a rapid decay of the vacuum state, e.g., into negative energy graviton and photons. Since the graviton energy is not bounded from below, the phase space for this process is formally infinite, that is, the instability is expected to develop very rapidly. Conservatively, we introduce a comoving cut-off and compute the spectrum of produced photons as a function of neutrino chemical potential. From the constraint on the radiation energy density today, we then obtain an interesting bound on the neutrino degeneracy parameter:

ξν  2 · 10−41



1015

37 38 39 40 41 42 43 44 45 46 47 48

4/3 

Ta GeV

Mp

(4)



provided that the lepton asymmetry has been generated above √ T ∗  440 M p / GeV (here M p ≈ 2.4 · 1018 GeV is the reduced

49 50

2. Graviton polarisation tensor in the lepton asymmetric Cν B

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We calculate the inducement of the Chern–Simons like terms in the effective graviton Lagrangian through the 1-loop graviton polarization diagram depicted in Fig. 1, influenced by a lepton asymmetric neutrino background. The lepton asymmetry is enforced in /γ 5 ν = the Lagrangian through a chiral chemical potential Lμν = ν¯ b ¯ 0 γ 5 ν , for which we have considered the frame in which the μν νγ / = μν γ0 ). The neutrino propagator is altered as Cν B is at rest (b follows:

60 61 62

S ( p) =

i p / − m − b/γ 5

63 64 65

≡ S 0 ( p) +

∞  n =1

=

i p / −m

Sn ( p) ,

∞  

−ib/γ 5

n =0

i

n

72

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 + T r (γρ S 0 ( p )γμ S 1 ( p + k))

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78 80 81

(6)

μν α k εμρα 0 8π 2



1 dx

4π 2 λ

2 

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M2

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0

2

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90

k2 M2

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(1 +  )kν kσ

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+ (24x2 − 44x + 18)( − 1) M 2 ηνσ 2

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2

− 16x (1 − x) ( )k ηνσ



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− (80x − 192x + 156x − 50x + 5)( )kν kσ

(7)

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+ (μ ↔ ν ) + (ρ ↔ σ ) + (μ ↔ ν , ρ ↔ σ ) ,

(8)

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4

3

2

where M 2 = m2 − x(1 − x)k2 and the limit  → 0 has been assumed. In simplifying this result we find a divergent quantity of the following form:

μνρσ

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× 8x2 (1 − x)2 (1 − 2x)2

(di v )

70

× T r (γμ S 0 ( p + k)γρ S 1 ( p ))



ξ

Planck mass), where T a is the temperature at which the asymmetry is generated. This paper is organised as follows; in Section 2 we describe the calculation of the graviton polarisation tensor in the presence of a lepton asymmetric Cν B, and consider the associated effective action. Section 3 illustrates the birefringent behaviour of gravitational waves in such a background, while in Section 4 we derive constraints on the Cν B lepton asymmetry through the induced gravitational instabilities, before concluding in Section 5.

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To evaluate the divergent loop integral in (6) we employ the dimensional regularization method (d = 4 −  ,  → 0) and utilise the relations given in Appendix A. We hence obtain:

μνρσ =

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(2p + k)ν (2p + k)σ

+ (μ ↔ ν ) + (ρ ↔ σ ) + (μ ↔ ν , ρ ↔ σ ) .

17/3 ,

(2π )4



Fig. 1. Parity violating contribution to the fermion propagator.

12 13

d4 p

μνρσ = −

10

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73



9

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μν α =− k εμρα 0 m2 ηνσ  2π 2 + (μ ↔ ν ) + (ρ ↔ σ ) + (μ ↔ ν , ρ ↔ σ ) , 1

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(9)

107 108

where γ is Euler’s constant. A straightforward inspection reveals that this divergent term does not satisfy the gravitational Ward (di v ) identity, kν μνρσ = 0, and hence violates the gauge invariance of the effective gravitational action. This has also been observed previously in a somewhat related calculation in Ref. [12]. The origin of this violation is rooted in the method of dimensional regularization, which violates Local Lorentz invariance explicitly through the extrapolation to non-integer spacetime dimensions d = 4 −  . Therefore, following the standard lore, we introduce non-invariant counter-terms to renormalise away this unphysical divergent term. The polarisation tensor then takes the following simple form:

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μνρσ = μν εμρα 0kα [kν kσ − k2 ηνσ ]C (k2 )

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+ (μ ↔ ν ) + (ρ ↔ σ ) + (μ ↔ ν , ρ ↔ σ ) ,

(10)

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where

125

C (k2 ) =

p / −m (5)

1 192π 2

×



−

k2

m2

126

16π 2 (k2 )3/2

−



4m2

− k2 tan−1

127



√ √

k2

4m2

− k2



128

.

(11)

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This further reduces to:

2 3 4

C (k2 ) =

1 k − 1920 , if k2 /m2  1 π 2 m2

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μν d4 xεμρα 0 hμν ∂ α (2hρσ ηνσ − ∂ν ∂σ hρσ ) 192π 2  μν = d4 x K 0 , (13) 48π 2

Sef f = −

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which contains the same number of derivatives as the standard graviton kinetic term in the weak field approximation. In fact, K 0 is the linearised 0th component of the four dimensional Chern– Simons topological current: σ ∂ ρ − 2 σ ρ λ ) . K β = ε β αμν (αρ μ νσ 3 αρ μλ νσ

(14)

Therefore, the presence of an asymmetry in the Cν B replicates Chern–Simons modified gravity:



24 25

2



11

14

(12)

if k /m 1 2

We wish to investigate the second of these two possible limiting cases, k2 /m2ν 1. In this limit we obtain the following contribution to the graviton action,

9 10

present neutrino chemical potential. For this lepton asymmetric Cν B,

2

1 , 192π 2

SC S =



d4 x (∂μ θ) K μ =

d4 x θ(∗ R R ) ,

(15)

26 27

where ∂μ θ =

28 29

32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

Ai j exp[−i (φ(η) a(η)

parametrise the gravitational waves as: h i j = − k · x)], which can be decomposed into the two circularly polarised states: e iRj and e iLj . These two possible circularly polarised states

× L √1 (e + − ie × ), which are defined as: e iRj = √1 (e + i j + ie i j ) and e i j = ij ij

2 2 j R ,L satisfy ni i jk ekl = i λ R , L (el ) R , L , where λ R , L = ±1. The phase tor λ R , L leads to exponential suppression or enhancement of

ε

facthe left and right circular polarisations of the propagating gravitational waves, the magnitude of which we shall now calculate. From the equations of motion for the action S = S E H + S e f f [13] we obtain:

51

R ,L

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R ,L 2

=

i λ R , L |k|

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2

a2

λ R , L κ θ,η



a2

(θ,ηη − 2Hθ,η )(φ,Rη, L − i H) .

(16)

We will first solve the above equation, assuming propagation in a the matter dominated epoch a(η) = a0 η2 = 1+0 z . The accumulated phase over the length of propagation, to first order in θ , is given by,

59 60

2

(i φ,ηη + (φ,η ) + H,η + H − |k| ) 1 −

52

288π 2 M 2p

R ,L φmat = i λ R , L |k| H 0

1 

1 4

η

θ,ηη −

1

η

 θ,η

dη

η

4

hR hL

∝e

R ,L −2|φmat |

ing identification θ,η =

a(η0 ) a(η)

2

μ0 , where 48π 2 M 2p

(18)

(19)

.

fore, the accumulated phase difference for z ∼ 30 sources is too small to be observable by any conceivable gravitational wave detector. The more interesting scenario to consider is the propagation of gravitational waves from sources in the very early in the universe. At early times the chemical potential would have been larger and the longer accumulated propagation time. Conceivably, any early universe sources could provide constraints, if the different polarisations are measurable. Therefore, we now consider gravitational waves produced at very early times, during the radiation dominated epoch after reheating. The accumulated phase now reads: R ,L φrad

= i λ R ,L

1 

|k| r,0 H 02

η

1 2

θ,ηη −

 R ,L φrad  − i λ R , L ξν

(17)

μ0 = a(η)μν is the

|k|



1

η

1 GeV

Ts 1 TeV

67

 θ,η

69 70

Taking into account the current bounds  on the Cν B asymmetry | k| parameter, ξ , we find |i φ R , L |  10−87 1 GeV , for z ∼ 30. There-

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dη

η

2

,

(20)

90 91 92 93 94 95

4

96

,

(21)

97 98

where we have redefined the redshift in terms of the temperature at which the gravitational waves are produced, T s , or when the asymmetry is generated, whichever is lowest. From Eq. (21), it can be seen that if an asymmetry in the Cν B is present at early times, which equilibrium sphalerons transitions may assure, then it is possible to get significant birefringent behaviour in the propagation of gravitational waves from primordial sources. This is dependent on the momenta |k| of the gravitational waves and size of the asymmetry.

99 100 101 102 103 104 105 106 107 108

4. Induced ghost-like modes and vacuum decay

109 110

Another interesting consequence of the induced Chern–Simons term in Eq. (13) is that short-scale gravitational fluctuations exhibit negative energy modes, which lead to a rapid decay of a vacuum state e.g., into negative energy graviton and photons [14]. Since the graviton energy is not bounded from below, the phase space for this process is formally infinite [15,16], and as such will develop very rapidly. We investigate the production of two photons and a negative energy graviton via this process, to obtain constraints on the neutrino asymmetry at early times. The relevant effective interaction is of the form:

S int ∼

1



m∗  1 m∗

(22)

mcan = M p

112 113 114 115 116 117 118 119 120 122 123

1 μα F ν , d4 x hcan F μν F μν − hcan α μν F 2



111

121

μν d4 xhcan μν T

124 125

can where the canonically normalised graviton field is: hcan μν = mcan hμν , with

In the case considered in this manuscript, we make the follow-



1 GeV

66 68

(1 + z)4 .

Hence the ratio of the amplitudes of each polarisation is given by:

= .

|k|



where r,0 ∼ 9.2 · After solving the integral we find:

The Chern–Simons term in Eq. (13) is found to induce a birefringence effect on the propagation of gravitational waves. The planned gravitational wave detectors, such as eLISA, DECIGO and BBO, can potentially measure the polarization of gravitational waves and hence this birefringence effect. With this in mind, we consider the propagation of gravitational waves in a lepton asymmetric background over cosmological distances. To this end we



μν H 0

1



10−5 is the radiation density parameter today.

3. Gravitational wave propagation in asymmetric Cν B

49 50

R ,L φmat = −i

μν . 48π 2

30 31

3

126 127 128

1 + λ R ,L

|k| amC S

,

(23)

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a3 dt

2 3 4

M 2p

mC S (t ) =

μν

5 6

M 2p

=

ξT

=

a(t ) M 2p

μ0

(24)

.

4.1. Photon energy spectrum from induced vacuum decay

7 8 9 10 11 12 13 14

To obtain a finite result for the decay rate we need to constrain the phase space. In the absence of a fundamental physical reason for such a truncation, we, following [15,16], simply cut-off the three momenta at |k|max = . In the analysis that follows, we consider decays into this mode as it will have the largest contribution to the energy density of the generated photons. In addition, we take the reasonable approximation:

15 16 17

 mcan 

|k|μν a

18 19 20 21 23 24 25 26 27

(25)

,

and consider the dynamics of our scenario after reheating and prior to BBN, when the universe is radiation dominated and evolves as follows,

√

22

a(t ) = a0 t =



1/2

2r,0 H 0 t ,

(26)

where r,0 ∼ 9.2 · 10−5 is the radiation density parameter today. The time at which this ghost term is no longer present will be defined as t ∗ and can be found in terms of the scale factor:



30 31 32 33 34 35 36 37 38 39 40 41 42

1



45 46 47 48

 a(t ) 

51

90r,0

54 55 56 57 58 59 60 61 62 63 64 65

  1 4

H0 M p

g∗π 2



or a(t ∗ ) 

M 2p

,

T∗ =

T

90r,0 H 0 M p g∗π 2

440

√

ξν

GeV



ξν2 Mp



|k| −1

 (30)

,

where n(k, t ) is the number of photons per unit logarithmic wave number |k| and  is the total decay width, which we take to approximately be:

∼

6 2 mcan

=

a(t ) 6

|k|μν

=

a(t )2 5

|k|n∗ (|k|) ∼

a(t ∗ )2 a 1/2

μ0

(27)

(28)

Mp

(29)

.

Given that the maximum reheating temperature is ∼ 1015 GeV, Eq. (29) implies we can constrain the production temperature of

5r,0 H 0

69 70 71 73 75 76 77 78 79 80 82 83 84

universe since the end of photon production to today,

a(t ∗ ) a0

3

=

85 86 87

a(t ∗ ) , we obtain:

88

5

a(t ∗ ) a 1/2 5r,0 H 0

89

(33)

.

90 91 92

Therefore, the energy density for a given momenta k is:



ξ T∗

M 3p

10T a2

H0

4

∼ |k|n0 (|k|) ∼

5





93

11

94

.

Mp

(34)

95 96

We can obtain a bound on the energy density of the produced photons, through the observation that the universe is not radiation dominated today:

97 98 99 100

dE

 M 2p H 02 .

(35)

101

This means we get the following constraint on ξν , assuming the asymmetry is generated above the characteristic temperature T ∗ , when requiring consistency with observation:

103

d3 xd ln |k|

ξν  2 · 10−41



4/3 

Ta 1015

 T ∗  1023 GeV

Mp

GeV



for which it is assumed T a 

440 √ ξν



1015 GeV



ην  10−41



Ta 1015 GeV

Mp

ην  0.033 where

Ta

Mp



108

112

.

(37)

113 114

17/3



2

106

111

17/6

115 116 117 118 119 120 121

(38)

. 

122 123

Mp

, and hence vacuum ξν decay does not occur, then we get the following constraint on ην ,

2000 GeV

105

110

. Equivalently,

440 If we instead assume that T a  √ GeV



104

109 Mp

Mp

4/3 

102

107

(36)

,

GeV

−2/3 

Ta

17/3

Thus we arrive at the conclusion that, unless  M p , the resulting photon energy density from the induced vacuum decay can hardly be accommodated with observation. Substituting the constraint in Eq. (36) into that for the asymmetry stored in the Cν B as a function of ξν , in Eq. (2), we find the following bound:

M

neutrino asymmetries to be in the range ξν  2 · 10−25 p , with smaller ξ ’s not generating ghost like modes after reheating. We also assume here that ξ is approximately constant, and hence is the same parameter currently constrained by BBN measurements in the calculation of the lepton asymmetry stored in the Cν B. Next we compute the spectrum of photons generated by the induced vacuum decay, and subsequently the energy density, which can be constrained by experiment. It is given by:

68

81

(32)

. 

dE

67

74

Taking into account the dilution factor due to the expansion of the

|k|n0 (|k|) ∼

66

72

(31)

.

Since the above decay rate is much faster than the expansion rate of the universe, we may safely assume that the decay happens instantaneously. Therefore, we fix the scale factor in Eq. (31) at time ta , when the asymmetric background is first produced. We then integrate Eq. (30) between the time ta and when the ghost terms are no longer present t ∗ :

d3 xd ln |k|

,

 14  3

52 53

M 2p

ξν T ∗

where g ∗ = 106.75. Equating Eq. (27) and (28) to find the temperature at which this effect ends we find:

49 50

μ0

where T ∗ is the temperature at which the ghost terms stop contributing. This fixes the time at which the ghost modes no longer exist, and decay of the vacuum ceases. We can reinterpret this as a temperature, so that it is possible to associate this with the maximal reheating temperature, and also ensure it does not have adverse implications on BBN. If we assume that the asymmetry is present/produced during the reheating epoch, and prior to BBN, the scale factor has a T1 dependence, if ignoring the decoupling of radiation degrees of freedom. The scale factor takes the following form:

43 44

⇒ a(t ∗ ) 

a(t ∗ )mC S (t ∗ )

(a3n(k, t )) = δ

3

28 29



1 d

where mC S is the analogous Chern–Simons mass scale:

124 125 126 127

(39)

ην  0.033 is the current upper limit from BBN constraints.

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5. Conclusions

Dimensional regularisation of loop integrals:



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

In this paper we have argued that the relic neutrino background with non-zero lepton number exhibits gravitational instabilities. Fundamentally, these instabilities are related to the gravity-lepton number mixed quantum anomaly. Indeed, in the relevant limit of vanishing neutrino masses, we have explicitly calculated the parity odd part of the graviton polarization tensor in the presence of a lepton asymmetric Cν B, which replicates the gravitational Chern– Simons term in the effective action. The induced Chern–Simons term leads to birefringent behaviour leading to an enhancement/suppression of the gravitational wave amplitudes depending on the polarisation. While this effect is negligibly small for local sources, we demonstrate that it could be sizeable for gravitational waves produced in very early universe, e.g. during a first-order phase transition. In addition to the above, we have also argued that short-scale gravitational fluctuations in the presence of an asymmetric Cν B exhibit negative energy modes, which lead to a rapid decay of a vacuum state e.g., into negative energy graviton and photons. Since the graviton energy is not bounded from below, the phase space for this process is formally infinite, that is the instability is expected to develop very rapidly. From the constraint on the radiation energy density today, we have obtained an interesting bound on the neutrino degeneracy parameter in Eq. (4). We believe that the findings reported in this paper will prove to be useful for further understanding the properties of the Cν B and putting constraints on particle physics models with a lepton asymmetry.

30 31

Acknowledgements

32 33 34 35 36 37 38

We would like to thank Rohana Wijewardhana for useful discussions. This work was supported by the Australian Research Council. Appendix A. Further details of calculations

39

Useful equations:

40 41 42 43

T r (γμ γα γρ γβ γ 5 ) = −4i εμαρ β Taking

46 47 48 49 50 51 52

(40)

 → 0:

44 45

(1 +  )| →0  1,

( )| →0 

1



1

−γ,

( − 1)| →0  − + γ − 1

(41)

ημν ημν  4 − 2

(42)





4 π λ2 M2

5



| →0  1 +  ln



4 π λ2 M2



(43)

i

 i

 i

i

1

(2π ) N ( p 2 − m2 )2 N

d p

1

(2π ) N ( p 2 − m2 )3 N

d p

pμ pν

(2π ) N ( p 2 − m2 )2 

i

dN p

dN p

pμ pν

) N ( p 2 − m2 )3  (2π N

=− = =



1 16π 2



1 32π 2 1



=−

64π 2

67

( )

M2

2 2 

4π λ

(1 +  )

M2

M2

2 2 

4π λ

32π 2 1

4π 2 λ

66

2 

M2



4 π 2 λ2 M2

(44) (45)

71 72 73

M 2 ( − 1) g μν

74 75

(46)



76 77

( ) g μν

(47)

78 79 80

(2π ) N ( p 2 − m2 )3  2 2  1 4π λ = M 2 ( − 1)( g μν g ρσ + g μσ g νρ 2 2 + g μρ g νσ )

69 70

d p pμ pν pρ pσ

128π

68

81 82 83

M

(48)

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