# Origin of electrical noise near charge neutrality in dual gated graphene
device

Aaryan Mehra  Roshan Jesus Mathew  Chandan Kumar<EMAIL_ADDRESS>Centre
for Nano Science and Engineering, Indian Institute of Science,
Bangalore-560012, India

###### Abstract

This letter investigates low frequency $1/f$ noise in hBN encapsulated
graphene device in a dual gated geometry. The noise study is performed as a
function of top gate carrier density ($n_{TG}$) at different back gate
densities ($n_{BG}$). The noise at low $n_{BG}$ is found to be independent of
top gate carrier density. With increasing $n_{BG}$, noise value increases and
a noise peak is observed near charge inhomogeneity of the device. Further
increase in $n_{BG}$ leads to decrease in noise magnitude. The shape of the
noise is found to be closely related to charge inhomogeneity region of the
device. Moreover, the noise and conductivity data near charge neutrality shows
clear evidence of noise emanating from combination of charge number and
mobility fluctuation.

††preprint: AIP/123-QED

Graphene, a single sheet of carbon has emerged as one of the most promising
candidate for the future device applications which can supersede the silicon
technology. However, graphene devices are very sensitive to disorder
fluctuations which leads to fluctuation in channel current[1, 2]. Thus,
quantifying the effect of the disorder becomes crucial for using the graphene
based devices for practical applications. The low frequency $1/f$ noise
technique has been used extensively to study the effect of various disorders
like roughness and trap states in the substrate[3], short and long range
disorder[4], metal contact to graphene[5], charge inhomogeneity[6], grain
boundaries[7] etc. The $1/f$ noise not only affects the low frequency
$<100kHz$ but its possible up-conversion also affects the phase noise at
higher frequency. Thus, understanding the noise spectrum in graphene devices
becomes cardinal for technological applications.

However, even after a more than a decade of extensive research, there is no
consensus on the origin and mechanism of $1/f$ noise in graphene devices[1, 8,
9, 10]. The noise study in graphene has been performed on both silicon[11, 12]
and hBN substrate [13, 8, 9, 10]. The study reveals that the graphene devices
on hBN substrate shows much smaller noise as compared to the graphene on a
silicon substrate. The noise reduction in hBN encapsulated devices are
associated to the better screening of graphene from the trap states in silicon
substrate and to very smooth surface of hBN which is free of dangling
bonds[13, 9, 8, 10]. However, noise in hBN encapsulated graphene devices show
different shapes and magnitude as a function of carrier density[13, 9, 8, 10].
Moreover, the noise magnitude varies from device to device and also depends on
the type of encapsulating hBN - commercial hBN or NIMS hBN[10]. Furthermore,
it is reported that noise magnitude in hBN encapsulated graphene can be
further reduced by adding an additional graphite gate[10]. Thus, there is an
increasing evidence that carrier number fluctuation is the major noise source
in graphene devices both on silicon and hBN encapsulated devices. However,
there are also evidences of noise stemming from mobility fluctuation as well.
The claim is made on the basis of electron-beam irradiated graphene[14, 11] on
silicon substrate and from non-monotonous variation of noise with magnetic
field in hBN encapsulated[15] and suspended[16] graphene device.

In this letter, we present a schematic study of low frequency $1/f$ noise in
hBN encapsulated graphene device in a dual gated geometry. We have performed
noise study in a small region of graphene which is underneath the top gate.
The noise measurements are performed as a function of $n_{TG}$ at different
$n_{BG}$. We find that noise is small and almost independent of $n_{TG}$ at
small values of $n_{BG}$. With increasing $n_{BG}$, noise starts to increase
and obtains a maximum value near $n_{BG}=1.5\times 10^{12}cm^{-2}$. With the
further increase in $n_{BG}$, noise value decreases and becomes independent of
$n_{BG}$. Moreover, our measurements shows that noise amplitude is minimum at
the charge neutrality point (CNP) and it increases on both side with electron
and hole doping with its peak appearing close to the charge inhomogeneity of
the device. Our results shows that noise in hBN encapsulated graphene can be
tuned by nearly two orders of magnitude. Furthermore, our analysis of noise
and channel conductivity provides a clear evidence of noise emanating from
combination of charge number and mobility fluctuation.

Figure 1: (color online) (a) Schematic of the measured hBN encapsulated
graphene device along with resistance measurement scheme. The device is
fabricated using dry transfer technique[17]. The graphene is encapsulated
between the top and bottom hBN with thickness $\sim$14 nm and $\sim$25 nm,
respectively. The back gate controls the carrier density throughout the device
while the top gate controls the carrier density only beneath the top gate
region. (b) 2D color map of conductivity as a function of top gate and back
gate carrier density. The horizontal blue stripe shows the Dirac point
throughout the device while the diagonal stripe shows the Dirac point from a
small section of graphene underneath the top gate region. The vertical and
horizontal cut line from panel (b) is presented in (c) and (d), respectively.
As can be seen, the conductivity is minimum at the Dirac point and increases
with electron or hole doping. All the measurements are performed at 77 K.

The hBN encapsulated graphene device is fabricated by following standard dry
transfer technique[17]. In brief, a glass slide is prepared with Poly methyl
methacrylate (PMMA) layer and graphene is exfoliated on it. hBN is exfoliated
on a freshly cleaved highly p doped silicon wafer. The glass slide containing
PMMA and graphene is loaded in a micro-manipulator and graphene is transferred
on hBN. This is followed by cleaning the stack of hBN/graphene in acetone
followed by IPA cleaning. The standard electron beam lithography is used to
pattern contacts on the graphene heterostructure, followed by thermal
evaporation of 5 nm chromium and 70 nm gold at the base pressure of 3e-7 mbar.
The device is annealed in vacuum for 3 hours at 400 ∘ C. To define the top
gate, a thin top hBN is transferred on the hBN/graphene stack. This is
followed by final step where electron beam lithography is used to pattern the
contacts, followed by metal deposition.

Figure 1 (a) presents the schematic of our device along with resistance
measurement scheme. The optical image of the device is shown in Supplementary
material (SM) Fig. S1. All the measurements are performed at 77 K. The
graphene is encapsulated between the top and bottom hBN. The carrier density
in the device is tuned by the combination of top gate and back gate voltages.
The thin top hBN and 300 nm thick $SiO_{2}$ act as the top and back gate
dielectric, respectively. The back gate controls the density throughout the
graphene channel while the top gate tunes the carrier density in a small
portion of graphene, underneath the top gate. The total channel length and
width is $8\mu m$ and $2\mu m$, respectively. The distance between voltage
probes are $4\mu m$ and the top gate width is $2\mu m$. Figure 1 (b) shows the
four probe conductivity data as a function of top gate and back gate carrier
density. A horizontal and diagonal stripe can be seen in the figure where the
conductivity is minimum. The horizontal stripe corresponds to Dirac point
throughout the graphene channel which is independent of top gate carrier
density while the diagonal stripe represents the Dirac cone in a small portion
of graphene under the top gate region. Doping of carrier density (either
electron or hole) leads to an increase in conductivity. This is shown
explicitly in figure (c) and (d) by taking horizontal and vertical cut lines
from figure 1(b), denoted by dashed arrow. We estimate the top hBN thickness
to be $\sim 14nm$.

Figure 2: (color online) (a) Normalized noise spectral density in hBN
encapsulated graphene device at few representative $n_{eff}$ at
$n_{BG}=1.5\times 10^{12}cm^{-2}$. The dashed line represents a pure $1/f$
line with $\beta=1$. (b-c) Noise amplitude (A) as a function of effective
carrier density $n_{eff}=n_{TG}-n_{Dirac}$ underneath the top gate at
different $n_{BG}$. At low $n_{BG}$ the noise amplitude is almost independent
of $n_{eff}$. With increasing $n_{BG}$, noise increases and maximum noise is
observed at $n_{BG}=1.5\times 10^{12}cm^{-2}$. The further increase in
$n_{BG}$, leads to reduction in noise amplitude.

The low frequency noise is a versatile tool to study the effect of metal
contact, charge fluctuation, screening etc. on the device conductivity which
can not be accessed by the traditional conductivity measurement methods[1, 2].
It is characterized by normalized power spectral density which is inversely
proportional to the frequency, $S_{V}/V^{2}\varpropto f^{-\beta}$, with
$\beta=1$. In most of the semiconducting devices the noise power spectral
density follows an empirical relation[1, 2]:

$S_{V}/V^{2}=\alpha_{H}/Nf^{\beta}.$ (1)

where $S_{V}/V^{2}$ is the normalized voltage noise density, $f$ is the
frequency, $V$ is the bias voltage, $N$ is the total number of charge carriers
and $\alpha_{H}$ is the Hooge’s parameter.

Figure 2 (a) shows the normalized voltage noise density ($S_{V}/V^{2}$) at
fixed $n_{BG}$ for a few representative $n_{eff}$. The measurement setup is
discussed in SM. A clear $1/f$ noise behaviour with $\beta\thickapprox 1$ can
be seen from the figure. The noise value is quantified by calculating the
noise amplitude (A) which is defined as[1]:

$A=\frac{1}{N}\sum_{n=1}^{N}f_{n}S_{Vn}/V_{n}^{2}$ (2)

Here $S_{Vn}/V_{n}^{2}$ is the normalized noise spectral density at $n$
different frequencies $f_{n}$. The other method to quantify noise amplitude is
to measure normalized noise spectral density at a fixed frequency, however
this method is more prone to errors[18].
Fig. 2(b) and Fig. 2(c) shows the noise amplitude as a function of effective
carrier density ($n_{eff}=n_{TG}-n_{Dirac}$) beneath the top gate for
different values of $n_{BG}$ with $n_{BG}>0$. At low $n_{BG}$ (close to back
gate Dirac point) the noise is almost independent of effective carrier
concentration. As the back gate density is increased, the noise amplitude
starts increasing. The noise amplitude shows a clear "M" shape with effective
density at $n_{BG}=1.3\times 10^{12}cm^{-2}$. With the further increase in
$n_{BG}$, noise amplitude increases, and obtains a maximum peak value of
$\approx 1.1\times 10^{-7}$ at $n_{BG}=1.5\times 10^{12}cm^{-2}$. Further
increase in $n_{BG}$ leads to an over all decrease in noise amplitude. Our
measurements show similar trend for negative values of $n_{BG}$ (SM Fig. S4).
To elucidate the noise behaviour with back gate carrier concentration, we plot
noise amplitude with $n_{BG}$ in Fig. 3. It is obtained by finding maximum
noise amplitude value for different $n_{BG}$ from Fig. 2(b), 2(c) and SM Fig.
S4.

The noise in graphene show various shapes and noise magnitudes[3, 1, 9, 15,
13, 8, 10, 16, 19, 20, 21, 22, 23, 24, 25], they have been explained
considering the charge number fluctuation or carrier mobility fluctuation
model or the combination of both. According to McWhorter charge number
fluctuation model the $1/f$ noise originates from charge trapping and
detrapping events occurring between the channel and the trap states in gate
oxides, which are located at different distances from the channel. The noise
spectral density is written as[26]

$S_{I}/I^{2}=\frac{\lambda kTN_{t}}{fAVn^{2}}$ (3)

Here $\lambda$ is the tunnelling constant, $N_{t}$ is the trap concentration
near Fermi energy, $A$ is the gate area and $n$ is the carrier concentration
in the device.

Figure 3: (color online) Noise amplitude (A) and $l_{0}^{2}$ on right and left
y axis, respectively as a function of back gate carrier density at fixed
$n_{eff}\approx 1\times 10^{11}cm^{-2}$. The noise amplitude is obtained by
finding the maximum noise amplitude value for different $n_{BG}$ from Fig. 2
(b), 2(c) and SM Fig. S4. The mean free path is obtained from $\sigma$ Vs
$n_{BG}$ plot (SM Fig. S2), obtained by taking vertical cut line at $n_{TG}=0$
from Fig. 1 (b). The noise is found to increase with increasing mean free path
till a critical point beyond which noise decreases.

It is important to note that our device has two different regions with
different carrier concentration - One underneath the top gate, where the
density is controlled by the combination of back gate and top gate and other
where density is determined only by the back gate. Although our measurements
are performed as a function of top gate carrier density, but since the voltage
probes are present in the back gate region, the noise in our device can
originate from a combination of back gate and top gate regions. Since,
$n_{eff}$ is kept constant in top gate region the McWhorter charge number
fluctuation models predicts decrease in noise with increasing $n_{BG}$, which
is in stark contrast with our experimental results which show non-monotonous
density dependence (Fig. 3).

Figure 4: (color online) (a) $(d\sigma/dn_{eff})^{2}$ as a function of
effective carrier density in the top gate region. The dip and peaks in
$(d\sigma/dn_{eff})^{2}$ corresponds to Dirac point and charge inhomogeneity
region underneath the top gate region. The estimated $\delta n\sim 1\times
10^{11}cm^{2}$ is found to be independent of $n_{BG}$. The shaded region
represents the $\delta_{n}$ region. (b) Conductivity as a function of
effective carrier density in the top gate region. The $\delta n$ region is
represented by the shaded grey region. An increase in conductivity is observed
even in the $\delta n$ region. (c) Conductivity as a function of $n_{BG}$ at
few representative points in $\delta n$ region, obtained by taking a vertical
cut line from Fig. 4 (b). A clear increase in conductivity is observed with
$n_{BG}$ even in $\delta n$ region.

Recent years have seen increasing evidence of mobility fluctuation in graphene
devices[14, 1, 11, 15, 16]. The mobility fluctuation model predicts that noise
originates from the superposition of multiple events which changes the the
scattering cross section $\sigma$. The power spectral density is given by[1,
15, 16]

$S_{I}/I^{2}\propto\frac{N_{t}^{\mu}}{V}\frac{\tau\zeta(1-\zeta)}{1+(\omega\tau)^{2}}l_{0}^{2}(\sigma_{2}-\sigma_{1})^{2}$
(4)

where $N_{t}^{\mu}$ is the concentration of centres contributing to mobility
fluctuation, $\tau$ is the characteristic time, $\zeta(1-\zeta)$ is the
probability of with cross section $\sigma_{1}(1-\sigma)$, $V$ is the volume of
the sample and $l_{0}^{2}$ is the mean free path. The model predicts that
noise is directly proportional to $N_{t}^{\mu}$ and $l_{0}^{2}$. The mean free
path ($l_{0}$) is calculated from $\sigma$ Vs $n_{BG}$ curve, obtained from
Fig. 1 (a) by taking vertical cut at $n_{TG}=0$ (SM). The resultant
$l_{0}^{2}$ is plotted in the left axis of Fig. 3. We find that noise
amplitude, $A$ and $l_{0}^{2}$ increases with increasing $n_{BG}$. which is
consistent with the mobility fluctuation model. However, beyond a critical
$l_{0}^{2}$, noise starts to decrease. This can be due to decrease in
$N_{t}^{\mu}$ which compensate the effect of increasing $l_{0}^{2}$. Similar
results were also reported by Cultrera et al.[11].
The enhanced mean free path through out the device should also increase the
conductivity in the top gated region. For this we plot
$(d\sigma/dn_{eff})^{2}$ for different $n_{BG}$ in Fig. 4 (a). As can be seen
from the figure, the charge inhomogeneity region is almost independent of
$n_{BG}$, for $n_{BG}>0.8\times 10^{12}cm^{-2}$. We estimate $\delta n\approx
1\times 10^{11}cm^{-2}$, this is highlighted by grey stripe. In fig. 4 (b), we
plot conductivity as a function of $n_{eff}$ for different $n_{BG}$ close to
Dirac point. We find an increase in conductivity even in $\delta n$ region. To
highlight the same, we plot conductivity in $\delta n$ region as a function of
$n_{BG}$ in Fig. 4 (c). It is obtained by taking vertical line traces from
Fig. 4 (b) in $\delta n$ region. A clear increase in conductivity is found in
$\delta n$ region. However, by definition the conductivity in $\delta n$
region should not change. The increase in conductivity in $\delta n$ region
underneath the top gate can arise from an overall increase in device mean free
path (Fig. 3), which is consistent with the mobility fluctuation model.

Figure 5: (color online) (a-c) Noise amplitude (A) and
$(d\sigma/dn_{eff})^{2}$ on right and left y axis, respectively as a function
of effective carrier density at few representative $n_{BG}$. The dip in
$(d\sigma/dn_{eff})^{2}$ corresponds to Dirac point underneath the top gate
region and the peaks corresponds to charge inhomogeneity region, $\delta n$.
We estimate $\delta n\sim 1\times 10^{11}cm^{-2}$. The dip observed in noise
corresponds to the Dirac point while the peaks in noise is near the $\delta
n$.

To understand the shape and position of noise in Fig. 2 (b-c), we plot noise
amplitude (A) and $(d\sigma/dn_{eff})^{2}$ as a function of $n_{eff}$ for
different set of back gate densities in Fig. 5 on right and left axes,
respectively. The dip in $(d\sigma/dn_{eff})^{2}$ corresponds to Dirac point
and the peaks corresponds to the charge inhomogeneity ($\delta n$) of the
graphene channel underneath the top gate We find that the noise minimum
coincides with the Dirac point while the noise peak values are close to charge
inhomogeneity point of the graphene channel underneath the top gate (Fig. 5).
Similar "M" shape noise behaviour was also reported in hBN encapsulated
graphene as a function of back gate carrier density[27]. The charge
inhomogeneity model of Xu et al.[28] can qualitatively explain the position of
noise maximum in Fig. 5 (a) and 5 (c). In the charge inhomogeneity region the
noise amplitude can be written as combination of noise originating from the
electron and hole charge puddles i.e
$A\sim\alpha_{H}/(n_{e}D_{e})+\alpha_{H}/(n_{h}D_{h})$; here $n$ and $D$
represents the density and puddle size, and subscript represents electron-
hole. Thus, electron doping in the channel will not only increase the electron
density but will also increase the electron puddle size and decrease the hole
puddle size. This leads to less noise due to electron charge carriers and
large noise due to minority charge carriers leading to increase in noise in
charge inhomogeneity region. With the further increase in carrier density, the
minority (hole) puddle size starts shrinking and can no longer contribute to
the noise. Thus, beyond $\delta n$ region, main contribution to noise comes
from majority (electron) carriers and hence noise decreases following Hooges
Empirical relation, $A\sim\alpha/N_{e}$. The disappearance of "M" shape noise
in Fig. 5 (b) can be due to spatial variation of charge inhomogeneity or Fermi
energy fluctuation[29].

In conclusion, our measurements show that various different noise shapes and
noise magnitude reported till date in hBN encapsulated devices can be realized
in a single graphene device by performing noise measurement in dual gated
geometry. Our results show that at low back gate density, noise is small and
is almost independent of effective carrier density. As the back gate density
is increased, the noise magnitude increases by almost two orders of magnitude.
With the further increase in back gate density, the noise magnitude decreases.
From our analysis we trace the origin of noise emanating from combination of
charge inhomogeneity and mobility fluctuation mechanism.

Supplementary Material
See the supplementary material for optical image of the device, mean free path
calculation, $1/f$ noise measurement technique, noise contribution from top
gated and non-top-gated region

###### Acknowledgements.

The authors gratefully acknowledge National Nano Fabrication Centre (NNFC) and
Micro and Nano Characterization Facility (MNCF) at Center for Nano Science and
Engineering (CeNSE), IISc for help and support in carrying out this work. The
authors acknowledge funding support for CeNSE facilities from the Ministry of
Human Resource Development (MHRD), Ministry of Electronics and Information
Technology (MeitY), and Department of Science and Technology (DST). The
authors thank Anindya Das for insightful discussion. C.K thanks IISc start up
grant and QuRP seed fund grant for supporting this work.

Data Availability: The data that support the findings of this study are
available from the corresponding author upon reasonable request.

References:

## References

  * Balandin [2013] A. A. Balandin, “Low-frequency 1/f noise in graphene devices,” Nature nanotechnology 8, 549–555 (2013).
  * Islam, Shamim, and Ghosh [2022] S. Islam, S. Shamim, and A. Ghosh, “Benchmarking noise and dephasing in emerging electrical materials for quantum technologies,” Advanced Materials , 2109671 (2022).
  * Kaverzin  _et al._ [2012] A. Kaverzin, A. S. Mayorov, A. Shytov, and D. Horsell, “Impurities as a source of 1/f noise in graphene,” Physical Review B 85, 075435 (2012).
  * Zhang, Mendez, and Du [2011] Y. Zhang, E. E. Mendez, and X. Du, “Mobility-dependent low-frequency noise in graphene field-effect transistors,” ACS nano 5, 8124–8130 (2011).
  * Karnatak  _et al._ [2016] P. Karnatak, T. P. Sai, S. Goswami, S. Ghatak, S. Kaushal, and A. Ghosh, “Current crowding mediated large contact noise in graphene field-effect transistors,” Nature communications 7, 13703 (2016).
  * Heller  _et al._ [2010] I. Heller, S. Chatoor, J. Maannik, M. A. Zevenbergen, J. B. Oostinga, A. F. Morpurgo, C. Dekker, and S. G. Lemay, “Charge noise in graphene transistors,” Nano letters 10, 1563–1567 (2010).
  * Kochat  _et al._ [2016] V. Kochat, C. S. Tiwary, T. Biswas, G. Ramalingam, K. Hsieh, K. Chattopadhyay, S. Raghavan, M. Jain, and A. Ghosh, “Magnitude and origin of electrical noise at individual grain boundaries in graphene,” Nano Letters 16, 562–567 (2016).
  * Kayyalha and Chen [2015] M. Kayyalha and Y. P. Chen, “Observation of reduced 1/f noise in graphene field effect transistors on boron nitride substrates,” Applied Physics Letters 107, 113101 (2015).
  * Kumar  _et al._ [2016] C. Kumar, M. Kuiri, J. Jung, T. Das, and A. Das, “Tunability of 1/f noise at multiple dirac cones in hbn encapsulated graphene devices,” Nano letters 16, 1042–1049 (2016).
  * Kakkar  _et al._ [2020] S. Kakkar, P. Karnatak, M. A. Aamir, K. Watanabe, T. Taniguchi, and A. Ghosh, “Optimal architecture for ultralow noise graphene transistors at room temperature,” Nanoscale 12, 17762–17768 (2020).
  * Cultrera  _et al._ [2018] A. Cultrera, L. Callegaro, M. Marzano, M. Ortolano, and G. Amato, “Role of plasma-induced defects in the generation of 1/f noise in graphene,” Applied Physics Letters 112, 093504 (2018).
  * Wu  _et al._ [2018] T. Wu, A. Alharbi, T. Taniguchi, K. Watanabe, and D. Shahrjerdi, “Low-frequency noise in irradiated graphene fets,” Applied Physics Letters 113, 193502 (2018).
  * [13] M. A. Stolyarov, G. Liu, S. L. Rumyantsev, M. Shur, and A. A. Balandin, “Suppression of 1/f noise in near-ballistic h-bn-graphene-h-bn heterostructure field-effect transistors,” Applied Physics Letters 107.
  * Zahid Hossain  _et al._ [2013] M. Zahid Hossain, S. Rumyantsev, M. S. Shur, and A. A. Balandin, “Reduction of 1/f noise in graphene after electron-beam irradiation,” Applied Physics Letters 102, 153512 (2013).
  * Rehman  _et al._ [2022] A. Rehman, J. A. D. Notario, J. S. Sanchez, Y. M. Meziani, G. Cywiński, W. Knap, A. A. Balandin, M. Levinshtein, and S. Rumyantsev, “Nature of the 1/f noise in graphene—direct evidence for the mobility fluctuation mechanism,” Nanoscale 14, 7242–7249 (2022).
  * Kamada  _et al._ [2021] M. Kamada, W. Zeng, A. Laitinen, J. Sarkar, S.-S. Yeh, K. Tappura, H. Seppä, and P. Hakonen, “Suppression of $1/f$ noise in graphene due to non-scalar mobility fluctuations induced by impurity motion,” arXiv preprint arXiv:2112.11933 (2021).
  * Zomer  _et al._ [2011] P. Zomer, S. Dash, N. Tombros, and B. Van Wees, “A transfer technique for high mobility graphene devices on commercially available hexagonal boron nitride,” Applied Physics Letters 99, 232104 (2011).
  * Lin and Avouris [2008] Y.-M. Lin and P. Avouris, “Strong suppression of electrical noise in bilayer graphene nanodevices,” Nano letters 8, 2119–2125 (2008).
  * Aamir and Ghosh [2019] M. A. Aamir and A. Ghosh, “Electrical noise inside the band gap of bilayer graphene,” 2D Materials 6, 025018 (2019).
  * Tian  _et al._ [2020] M. Tian, Q. Hu, C. Gu, X. Xiong, Z. Zhang, X. Li, and Y. Wu, “Tunable 1/f noise in cvd bernal-stacked bilayer graphene transistors,” ACS applied materials & interfaces 12, 17686–17690 (2020).
  * Liu  _et al._ [2013] G. Liu, S. Rumyantsev, M. S. Shur, and A. A. Balandin, “Origin of 1/f noise in graphene multilayers: Surface vs. volume,” Applied Physics Letters 102, 093111 (2013).
  * Mavredakis  _et al._ [2018] N. Mavredakis, R. G. Cortadella, A. B. Calia, J. A. Garrido, and D. Jiménez, “Understanding the bias dependence of low frequency noise in single layer graphene fets,” Nanoscale 10, 14947–14956 (2018).
  * Peng  _et al._ [2017] S. Peng, Z. Jin, D. Zhang, J. Shi, D. Mao, S. Wang, and G. Yu, “Carrier-number-fluctuation induced ultralow 1/f noise level in top-gated graphene field effect transistor,” ACS applied materials & interfaces 9, 6661–6665 (2017).
  * Kumar  _et al._ [2015] M. Kumar, A. Laitinen, D. Cox, and P. J. Hakonen, “Ultra low 1/f noise in suspended bilayer graphene,” Applied Physics Letters 106, 263505 (2015).
  * Pal, Bol, and Ghosh [2010] A. N. Pal, A. A. Bol, and A. Ghosh, “Large low-frequency resistance noise in chemical vapor deposited graphene,” Applied Physics Letters 97, 133504 (2010).
  * Dmitriev, Levinshtein, and Rumyantsev [2009] A. Dmitriev, M. Levinshtein, and S. Rumyantsev, “On the hooge relation in semiconductors and metals,” Journal of Applied Physics 106, 024514 (2009).
  * Kumar and Das [2021] C. Kumar and A. Das, “Effect of boron nitride defects and charge inhomogeneity on 1/f noise in encapsulated graphene,” Applied Physics Letters 119, 223106 (2021).
  * Xu  _et al._ [2010] G. Xu, C. M. Torres Jr, Y. Zhang, F. Liu, E. B. Song, M. Wang, Y. Zhou, C. Zeng, and K. L. Wang, “Effect of spatial charge inhomogeneity on 1/f noise behavior in graphene,” Nano letters 10, 3312–3317 (2010).
  * Pellegrini [2013] B. Pellegrini, “1/f noise in graphene,” The European Physical Journal B 86, 1–12 (2013).


††institutetext: Department of Physics, College of Science, Swansea
University, Singleton Park, Swansea, SA2 8PP, UK.

#  A Model of Gravitational Leptogenesis

Graham M. Shore<EMAIL_ADDRESS>
(today)

###### Abstract

Gravitational leptogenesis is an elegant way of explaining the matter-
antimatter asymmetry in the universe. This paper is a review of the recently
proposed mechanism of radiatively-induced gravitational leptogenesis (RIGL),
in which loop effects in QFT in curved spacetime automatically generate an
asymmetry between leptons and antileptons in thermal quasi-equilibrium in the
early universe. The mechanism is illustrated in a simple see-saw BSM model of
neutrinos, where the lepton-number violating interactions required by the
Sakharov conditions are mediated by right-handed neutrinos with Majorana
masses of $O(10^{10})\,{\rm GeV}$. The Boltzmann equations are extended to
include new, loop-induced gravitational effects and solved to describe the
evolution of the lepton number asymmetry in the early universe. With natural
choices of neutrino parameters, the RIGL mechanism is able to generate the
observed baryon-to-photon ratio in the universe today.

##  1 Radiatively-Induced Gravitational Leptogenesis – overview

The origin of the matter-antimatter asymmetry of the universe is one of the
most important outstanding issues in cosmology. In this review,111This paper
is an edited arXiv version of a contribution to the Festschrift celebrating
the physics career of Peter Suranyi. based on the recent paper
McDonald:2020ghc , we describe a proposal in which a lepton number asymmetry
is generated dynamically in a minimal extension of the standard model by
gravitational effects in the expanding universe.

The fundamental mechanism, radiatively-induced gravitational leptogenesis, was
introduced in the series of papers McDonald:2014yfg ; McDonald:2015ooa ;
McDonald:2015iwt ; McDonald:2016ehm ; McDonald:2020ghc . The key idea is that
in a minimal extension of the standard model to include heavy right-handed
neutrinos $\nu_{R}^{\,\alpha}$ ($\alpha=1,2,3$), an asymmetry in the
propagation of the light neutrinos $\nu_{L}^{\,i}$ ($i=e,\mu,\tau$) and
antineutrinos arises due to gravitational tidal effects in loop diagrams in
which the heavy neutrinos appear as virtual intermediate states. This induces
an effective gravitational interaction in which the rate of change of the
Ricci scalar plays the role of a chemical potential for lepton number
Davoudiasl:2004gf . In the high temperature environment of the early universe,
this generates a lepton number asymmetry in quasi-equilibrium. Later, as the
universe expands and cools, this lepton asymmetry is transformed into the
observed baryon asymmetry through the well-known sphaleron mechanism
Khlebnikov:1988sr .

These loop effects violate the strong equivalence principle (SEP) of classical
general relativity, and are most usefully expressed in terms of a weak-
curvature effective action which includes SEP-violating direct couplings of
the curvature tensor to the light lepton fields. The curvature couplings
involve inverse powers of the heavy neutrino masses $M_{\alpha}$, since these
set the effective length scale of the virtual loops on which the tidal
gravitational forces act.

The necessary conditions for a successful model of baryogenesis, or
leptogenesis, were set out long ago in the famous Sakharov conditions
Sakharov:1967dj , according to which the fundamental theory should exhibit (i)
baryon or lepton violation, (ii) C and CP violation, and (iii) the mechanism
must involve non-equilibrium dynamics.

The fundamental BSM theory we consider here is the familiar extension of the
standard model with sterile right-handed neutrinos (i.e. coupling only to the
Higgs field and the light leptons through Yukawa interactions, and neutral
under the gauge interactions of the SM). We include Majorana mass terms of
$O(10^{10})\,{\rm GeV}$ and above for the right-handed neutrinos, which allows
the observed light neutrino masses to be generated by the see-saw mechanism.
The addition of the right-handed neutrinos is the only BSM ingredient - our
leptogenesis mechanism then follows automatically and necessarily from the
gravitational dynamics at one and two-loop order in this model.

The BSM Lagrangian is therefore

$S\,=\,\int d^{4}x\sqrt{-g}\biggl{[}{\cal L}_{\rm
SM}+\biggl{(}\frac{1}{2}i\,\overline{\nu_{R}}\,\gamma.\overleftrightarrow{D}\nu_{R}-\frac{1}{2}\overline{\nu_{R}^{\,\,c}}\,M\,\nu_{R}-\overline{\ell_{L}}\,\lambda\,\phi\,\nu_{R}+{\rm
h.c.}\biggr{)}\biggr{]}\ .$ (1)

Here, $M_{\alpha\beta}$ is the Majorana mass matrix, which we take to be
diagonal. The fields $\ell_{L}^{i}$ ($i=e,\mu,\tau$) are the SM lepton
doublets and $\phi$ is the Higgs field.222The notation is
$\phi_{r}=\epsilon_{rs}\tilde{\phi}_{s}^{*}$ where $\tilde{\phi}$ is the usual
Higgs doublet giving mass to the lower fields in the $SU(2)$ lepton doublets.
The Yukawa couplings $\lambda_{i\alpha}$ are complex and are responsible for
CP violation in this model. These flavour indices are suppressed here and in
(2), (3) below.

The inclusion of Majorana masses for the sterile neutrinos allows the lepton
number violating scattering processes
$\nu_{L}\,H\leftrightarrow\nu_{L}^{\,c}\,H$ and
$\nu_{L}\,\nu_{L}\leftrightarrow H\,H$, where $H$ is the physical Higgs boson,
as required by the first Sakharov condition. As is well-known, the model also
exhibits a non-gravitational mechanism for generating a lepton asymmetry –
thermal leptogenesis– through the out-of-equilibrium lepton number violating
decays of the heavy neutrinos $\nu_{R}\rightarrow\nu_{L}H$ Fukugita:1986hr .
See, for example, ref. Buchmuller:2004nz  for a review.

In the radiatively-induced gravitational leptogenesis (RIGL) mechanism, the
third Sakharov condition is replaced by the time-dependence of the background
gravitational field. Tidal effects, arising first at two-loop order in this
model, generate the following CP odd effective interaction of the lepton
(neutrino) number current $J^{\mu}=\overline{\nu_{L}}\gamma^{\mu}\nu_{L}$ with
the rate of change of the Ricci scalar,

$S_{b}\,=\,\int d^{4}x\sqrt{-g}\,\,b\,\partial_{\mu}R\,J^{\mu}\ ,$ (2)

where the coupling $b$ is $O(\lambda^{4}/M^{2})$. Evidently, a time-dependence
of the Ricci scalar reduces this interaction to $b\,\dot{R}\,Q$, where $Q$ is
the lepton number charge, which is equivalent to adding a chemical potential
$\mu=b\,\dot{R}\,$.333Note that in the original gravitational baryogenesis
(leptogenesis) proposal of Davoudiasl et al. Davoudiasl:2004gf , the
interaction (2) with $J^{\mu}$ interpreted as the baryon (lepton) number
current is postulated essentially a priori, with an arbitrary mass-dependent
coupling. Here, we show how it is necessarily generated in the effective
action of the simple and phenomenologically motivated BSM model with heavy
Majorana masses for the right-handed neutrinos. In an earlier proposal of
‘spontaneous baryogenesis’ Cohen:1987vi , an interaction of the form (2) but
with the Ricci scalar replaced by a time-dependent background scalar field was
proposed.  At finite temperature, this will induce a non-vanishing lepton
number density $n_{L}^{eq}=\tfrac{1}{3}b\,\dot{R}\,T^{2}$ in quasi-
equilibrium.

Further, CP even, interactions are generated already at one-loop level and the
complete effective action for the light neutrinos is

$\displaystyle S_{eff}=\int
d^{4}x\sqrt{-g}\biggl{[}\frac{1}{2}i\,\overline{\nu_{L}}\gamma.\overleftrightarrow{D}\,\nu_{L}$
$\displaystyle\,+\,(a-\tfrac{1}{2}d)\,R_{\mu\nu}\,i\,\overline{\nu_{L}}\gamma^{\mu}\overleftrightarrow{D}^{\nu}\,\nu_{L}\,+\,b\,\partial_{\mu}R\,\,\overline{\nu_{L}}\gamma^{\mu}\,\nu_{L}$
$\displaystyle~{}~{}~{}\,+c\,R\,i\,\overline{\nu_{L}}\gamma.\overleftrightarrow{D}\,\nu_{L}\,-\,d\,i\left(D_{\mu}\overline{\nu_{L}}\right)\gamma.\overleftrightarrow{D}\,D^{\mu}\nu_{L}\biggr{]}\
.$ (3)

Here, $a_{ij},c_{ij},d_{ij}$ are coefficients of $O(\lambda^{2}/M^{2})$, and
we have suppressed the corresponding $i,j$ flavour indices in the fermion
bilinears for simplicity of notation. The effective action is valid to lowest
order in $\mathfrak{R}/M^{2}$, where $\mathfrak{R}$ denotes a typical
curvature component, and also in the ‘low-energy’ regime
$E\sqrt{\mathfrak{R}}/M^{2}\ll 1$ (see ref. McDonald:2020ghc  for a careful
discussion, including the justification for using the effective Lagrangian at
temperatures in excess of $M$).

These CP even interactions modify the conservation law of the lepton number
current, giving

$D_{\mu}J^{\mu}\,=\,-2a\,R_{\mu\nu}D^{\mu}J^{\nu}\,-\,2\hat{b}\,\partial_{\mu}RJ^{\mu}\
,$ (4)

where $\hat{b}=\tfrac{1}{2}a+c+\tfrac{1}{4}d$. In a FRW universe, this implies
the following equation for the time evolution of the lepton number density
$n_{L}$,

$\frac{dn_{L}}{dt}\,+\,3Hn_{L}\,+\,2a\left(-3R^{0}{}_{0}+R^{i}{}_{i}\right)Hn_{L}\,+\,2\hat{b}\,\dot{R}\,n_{L}\,=\,0\
,$ (5)

where $H$ is the Hubble parameter. This shows that the lepton number density
$n_{L}$ evolves non-trivially in a gravitational background due to quantum
loop effects. Whether this effect tends to amplify or suppress the magnitude
of $n_{L}$ depends on the signs and relative magnitudes of the coefficients,
especially $a$, which are not arbitrary in RIGL but are determined by the
fundamental BSM theory.

Together with the gravitationally-induced equilibrium lepton asymmetry
$n_{L}^{eq}$, these two effects combine to generate a novel dynamical
evolution McDonald:2020ghc  of the lepton number density in the expanding
universe. To describe this quantitatively in the high temperature environment
of the early FRW universe, we need the corresponding Boltzmann equations.
Neglecting for the moment the contribution to the asymmetry from decays of the
heavy $\nu_{R}$ neutrinos, the gravitationally modified Boltzmann equation is

$\frac{dN_{L}}{dz}\,=\,-\,W\,\bigl{(}N_{L}\,-\,N_{L}^{eq}\bigr{)}\,-\,\mathcal{W}\,N_{L}\
.$ (6)

Here, $N_{L}(z)=n_{L}/n_{\gamma}$ is the ratio of the number density of light
leptons to photons, and we have replaced the time variable with $z=M_{1}/T$,
where $T$ is temperature and $M_{1}$ is the mass of the dominant heavy
neutrino.

The new gravitational terms in (6) are the equilibrium number asymmetry
$N_{L}^{eq}(z)$ generated by the interaction (2) and the evolution term
$\mathcal{W}(z)$ arising from the CP even interactions in (3), which reflects
the curvature-induced evolution terms in (5). $W(z)=\Gamma/zH$ is the usual
factor determined by the interaction rate $\Gamma(z)$ for the lepton number
violating interactions. Without the new gravitational terms, it acts as a
‘washout’ factor – here, it has the different role of driving the lepton
number asymmetry towards its non-vanishing equilibrium value $N_{L}^{eq}(z)$.

Figure 1: Illustration of the key stages in the evolution of the lepton
asymmetry $N_{L}(z)$ with temperature, given by the Boltzmann equation (6).
The temperature dependence of the normalised rate factor $W(z)$ for the
lepton-violating reactions, which drive the asymmetry towards its equilibrium
value $N_{L}^{eq}(z)$, is $W(z)\sim 1/z^{2}$, while both the gravitationally-
induced terms $N_{L}^{eq}(z)$ and $\mathcal{W}(z)$ fall off as $1/z^{5}$.

Putting all this together, the entire evolution predicted by the RIGL
Boltzmann equation is shown in Fig. 1. Essentially we find four stages. First,
there is a very early high temperature phase (region (a)) in which the new
evolution term $\mathcal{W}(z)$ keeps $N_{L}$ below equilibrium, followed by a
phase where $W(z)>\mathcal{W}(z)$ and $N_{L}$ is driven to its
gravitationally-induced equilibrium value $N_{L}^{eq}(z)$ (region (b)). Next
is the vital decoupling transition (region (c)) when $W(z)$ becomes too weak
to hold $N_{L}$ to $N_{L}^{eq}(z)$ (i.e. the $L$-violating reactions are too
slow compared to the Hubble expansion to maintain equilibrium) and it
decouples leaving a constant, frozen asymmetry. Finally, there is a late dip
in $N_{L}$ as $W(z)$ develops a resonance peak around $T\sim M_{1}$,
temporarily pushing $N_{L}$ lower towards the rapidly decreasing
$N_{L}^{eq}(z)$ (region (d)).

The transitions between these stages depend on the dynamical balance between
the temperature and curvature dependent rates $W(z)$ and $\mathcal{W}(z)$, and
$N_{L}^{eq}(z)$, as the universe expands and cools. In particular, the sign of
$\mathcal{W}(z)$ is key to whether the new evolution term amplifies or
suppresses the lepton asymmetry at early times. In the BSM model described
here, the sign is such as to suppress the asymmetry, as shown in Fig. 1.

In the remainder of this paper, we describe this scenario in more detail and
describe the quantitative predictions for the baryon asymmetry of the universe
in this model.

##  2 Matter-Antimatter Asymmetry from Gravitational Interactions

In this section, we explain in principle how gravitational interactions can
induce an asymmetry in the dynamics of matter and antimatter, and outline the
methods used to obtain the gravitational effective action (3) from the
fundamental BSM theory (1).

It is useful to begin by rewriting the action (1) explicitly in terms of the
fields $\nu_{L}$ and $\nu_{L}^{\,c}$ corresponding to the light neutrinos and
their antiparticles444The charge conjugate field is defined as
$\nu_{L}^{\,c}\equiv(\nu_{L})^{c}=-i\gamma^{2}\nu_{L}^{\,*}$ and is right-
handed. With a Higgs VEV $v$, the see-saw mechanism gives rise to three light
Majorana neutrinos with mass matrix
$(m_{\nu})_{ij}~{}=~{}\sum_{\alpha}\lambda_{i\alpha}\,\frac{1}{M_{\alpha}}\,\lambda_{\alpha
j}^{T}\,v^{2}\ ,$ along with the heavy sterile Majorana neutrinos with masses
$M_{\alpha}$. This is diagonalised by the PMNS matrix $U$ such that
$U^{T}m_{\nu}U=m_{diag}$, with the corresponding relation between the mass and
flavour eigenstates. 555The coupling to gravity is through the connection
alone, which is the requirement for the classical Lagrangian to satisfy the
strong equivalence principle. The covariant derivative acting on spinors is
$D_{\mu}=\partial_{\mu}-\tfrac{i}{4}\omega_{\mu ab}\sigma^{ab}$, where
$\omega_{\mu ab}$ is the spin connection and
$\sigma^{ab}=\tfrac{i}{2}\left[\gamma^{a},\gamma^{b}\right]$.

$\displaystyle S\,=\,\int d^{4}x\sqrt{-g}$ $\displaystyle\biggl{[}{\cal
L}_{\rm
SM}+\frac{1}{4}\left(i\,\overline{\nu_{R}}\,\gamma.\overleftrightarrow{D}\,\nu_{R}+i\,\overline{\nu_{R}^{\,\,c}}\,\gamma.\overleftrightarrow{D}\,\nu_{R}^{\,\,c}\right)-\frac{1}{2}\Bigl{(}\overline{\nu_{R}^{\,\,c}}\,M\,\nu_{R}+\overline{\nu_{R}}\,M\,\nu_{R}^{\,\,c}\Bigr{)}$
$\displaystyle-\frac{1}{2}\left(\overline{\ell_{L}}\,\lambda\,\phi\,\nu_{R}+\overline{\nu_{R}}\,\lambda^{\dagger}\phi^{\dagger}\,\ell_{L}+\overline{\ell_{L}^{\,\,c}}\,\lambda^{*}\phi^{*}\,\nu_{R}^{\,\,c}+\overline{\nu_{R}^{\,\,c}}\,\lambda^{T}\phi^{T}\,\ell_{L}^{\,\,c}\right)\biggr{]}\
.$ (7)

$\nu_{L}$$H$$\nu_{L}^{\,c}$$H$$S_{\alpha}^{\times}$

$\nu_{L}$$H$$\nu_{L}^{\,c}$$H$$S_{\alpha}^{\times}$

$\nu_{L}$$H$$H$$\nu_{L}$$S_{\alpha}^{\times}$

$\nu_{L}$$H$$H$$\nu_{L}$$S_{\alpha}^{\times}$

Figure 2: Feynman diagrams for the $\Delta L=2$ lepton number violating
reactions $\nu_{L}~{}H\leftrightarrow\nu_{L}^{\,c}~{}H$ and
$\nu_{L}~{}\nu_{L}\leftrightarrow H~{}H$, mediated by the ‘charge violating’
$\langle\,\nu_{R}~{}~{}\overline{\nu_{R}^{\,\,c}}\,\rangle$ propagator
$S_{\alpha}^{\times}$.

The Majorana mass for the right-handed neutrinos allows the occurrence of both
leptonic “charge-conserving” and “charge-violating” propagators:

$\displaystyle\langle\,\nu_{R}~{}~{}\overline{\nu_{R}}\,\rangle~{}$
$\displaystyle=~{}\langle\,\nu_{R}^{\,\,c}~{}~{}\overline{\nu_{R}^{\,\,c}}\,\rangle~{}=~{}S(x,y)\
,$ (8)
$\displaystyle\langle\,\nu_{R}~{}~{}\overline{\nu_{R}^{\,\,c}}\,\rangle~{}$
$\displaystyle=~{}\langle\,\nu_{R}^{\,\,c}~{}~{}\overline{\nu_{R}}\,\rangle~{}=~{}S^{\times}(x,y)\
.$ (9)

In flat spacetime, these are

$S_{\alpha}(p)~{}=~{}\frac{i\,\gamma.p}{p^{2}-M_{\alpha}^{2}}\
,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}S_{\alpha}^{\times}(p)~{}=~{}\frac{i\,M_{\alpha}}{p^{2}-M_{\alpha}^{2}}\
.$ (10)

The existence of the charge-violating $S^{\times}(x,y)$ propagator plays a key
role in generating the lepton asymmetry. First, it allows the $\Delta L=2$
scattering reactions $\nu_{L}~{}H\leftrightarrow\nu_{L}^{\,\,c}~{}H$ and
$\nu_{L}~{}\nu_{L}\leftrightarrow H~{}H$, illustrated in Fig. 2. These
diagrams depend on the Yukawa coupling factor through
$\lambda\,S^{\times}\lambda^{T}\,=\,\sum_{\alpha}\lambda_{i\alpha}S_{\alpha}^{\times}\lambda_{\alpha
j}^{T}$. This is the source of the lepton number violation required by the
first Sakharov condition.

Next, to implement the RIGL mechanism, we need to show that the propagation of
leptons and antileptons is different in a gravitational field. Specifically,
we find that at two-loop level, the self-energies $\Sigma$ and $\Sigma^{c}$
for the leptons and antileptons differ when translation invariance no longer
holds, leading to distinct dispersion relations.

$\ell_{L}$$\overline{\ell_{L}}$$\nu_{R}$$\overline{\nu_{R}}$$\ell_{L}$$\overline{\ell_{L}}$$S_{\alpha}$$\phi$$\phi^{\dagger}$
Figure 3:  One-loop self-energy diagram for the light $\nu_{L}$ neutrinos with
an intermediate charge-conserving $\nu_{R}$ propagator $S$. The fields at the
vertices are shown explicitly for comparison with the Lagrangian (7).

At one loop, there is a single self-energy diagram for the light neutrinos
$\nu_{L}$ (Fig. 3), which involves the charge-conserving right-handed neutrino
propagator $S_{\alpha}$. (For neutrino self-energies in the standard model,
see refs. Ohkuwa:1980jx ; McDonald:2014yfg ) There is no corresponding one-
loop diagram with the charge-violating propagator $S_{\alpha}^{\times}$. The
self-energy is therefore,

$\Sigma_{ij}(x,y)~{}=~{}\sum_{\alpha}\lambda_{i\alpha}\lambda_{aj}^{\dagger}\,G(x,y)\,S_{\alpha}(x,y)\
.$ (11)

A similar diagram gives the self-energy for the antineutrinos. Since we are
interested in the violation of total lepton number, we trace over the light
lepton flavours, and find

$\displaystyle{\rm tr}\left(\Sigma_{ij}(x,y)-\Sigma_{ij}^{c}(x,y)\right)~{}$
$\displaystyle=~{}\sum_{\alpha}\left(\lambda^{\dagger}\lambda-\lambda^{T}\lambda^{*}\right)_{\alpha\alpha}\,G(x,y)\,S_{\alpha}(x,y)$
$\displaystyle=~{}2i\,\sum_{\alpha}{\rm
Im}\left(\lambda^{\dagger}\lambda\right)_{\alpha\alpha}\,G(x,y)\,S_{\alpha}(x,y)~{}=~{}0\
.$ (12)

These diagrams therefore imply that the gravitational influence on the
propagation of the light neutrinos and antineutrinos is identical. In the
effective Lagrangian (3), they correspond to the CP even terms. However, they
do not contribute to the CP odd term which is responsible for generating the
lepton-antilepton asymmetry. For this, we require the two-loop self-energies,
which have the $O(\lambda^{4})$ dependence necessary to exhibit CP violation.

$\ell_{L}$$\overline{\ell_{L}}$$\nu_{R}$$\overline{\nu_{R}}$$\ell_{L}$$\overline{\ell_{L}}$$\nu_{R}$$\overline{\nu_{R}}$$\ell_{L}$$\overline{\ell_{L}}$$S_{\alpha}$$S_{\beta}$$\phi$$\phi^{\dagger}$$\phi$$\phi^{\dagger}$

$\ell_{L}$$\overline{\ell_{L}}$$\nu_{R}$$\overline{\nu_{R}^{\,c}}$$\ell_{L}^{\,c}$$\overline{\ell_{L}^{\,c}}$$\nu_{R}^{\,c}$$\overline{\nu_{R}}$$\ell_{L}$$\overline{\ell_{L}}$$S_{\alpha}^{\times}$$S_{\beta}^{\times}$$\phi$$\phi^{T}$$\phi^{*}$$\phi^{\dagger}$

$\ell_{L}$$\overline{\ell_{L}}$$\nu_{R}$$\overline{\nu_{R}^{\,c}}$$\ell_{L}^{\,c}$$\overline{\ell_{L}^{\,c}}$$\nu_{R}^{\,c}$$\overline{\nu_{R}}$$\ell_{L}$$\overline{\ell_{L}}$$S_{\alpha}^{\times}$$S_{\beta}^{\times}$$\phi$$\phi^{T}$$\phi^{*}$$\phi^{\dagger}$

Figure 4: Two-loop self-energy diagrams for the light $\nu_{L}$ neutrinos
giving rise to a lepton-antilepton asymmetry in curved spacetime.

At two loops, we find the three self-energy diagrams illustrated in Fig. 4.
The corresponding self-energies are:

$\displaystyle\Sigma_{ij}^{(1)}(x,y)~{}=~{}\int d^{4}z\sqrt{-g}$
$\displaystyle\int
d^{4}z^{\prime}\sqrt{-g}\,\sum_{\alpha,\beta,k}\left(\lambda_{i\alpha}\lambda_{\alpha
k}^{\dagger}\lambda_{k\beta}\lambda_{\beta j}^{\dagger}\right)$
$\displaystyle\times~{}G(x,y)\,G(z,z^{\prime})\,S_{\alpha}(x,z)\,\Delta(z,z^{\prime})\,S_{\beta}(z^{\prime},y)\
,$ (13)

from the ‘nested’ diagram with two $S$ propagators, and

$\displaystyle\Sigma_{ij}^{(2)}(x,y)~{}=~{}\int d^{4}z\sqrt{-g}$
$\displaystyle\int
d^{4}z^{\prime}\sqrt{-g}\,\sum_{\alpha,\beta,k}\left(\lambda_{i\alpha}\lambda_{\alpha
k}^{T}\lambda_{k\beta}^{*}\lambda_{\beta j}^{\dagger}\right)$
$\displaystyle\times~{}G(x,y)\,G(z,z^{\prime})\,S_{\alpha}^{\times}(x,z)\,\Delta(z,z^{\prime})\,S_{\beta}^{\times}(z^{\prime},y)\
,$ (14)

and

$\displaystyle\Sigma_{ij}^{(3)}(x,y)~{}=~{}\int d^{4}z\sqrt{-g}$
$\displaystyle\int
d^{4}z^{\prime}\sqrt{-g}\,\sum_{\alpha,\beta,k}\left(\lambda_{i\alpha}\lambda_{\alpha
k}^{T}\lambda_{k\beta}^{*}\lambda_{\beta j}^{\dagger}\right)$
$\displaystyle\times~{}G(x,z^{\prime})\,G(z,y)\,S_{\alpha}^{\times}(x,z)\,\Delta(z,z^{\prime})\,S_{\beta}^{\times}(z^{\prime},y)\
,$ (15)

for the ‘nested’ and ‘overlapping’ diagrams with two $S^{\times}$ propagators.
Note that there is no overlapping-type diagram with two $S$ propagators.

Tracing over the light flavours as before, we find the following structure of
the Yukawa couplings for the three diagrams:

${\rm
tr}\left(\Sigma_{ij}^{(1)}-\Sigma_{ij}^{(1)c}\right)~{}=~{}2i\sum_{\alpha,\beta}\,{\rm
Im}\left[(\lambda^{\dagger}\lambda)_{\beta\alpha}\,(\lambda^{\dagger}\lambda)_{\alpha\beta}\right]\,I_{\alpha\beta}^{(1)}~{}=~{}0\
,$ (16)

whereas

$\displaystyle{\rm tr}\left(\Sigma_{ij}^{(2)}-\Sigma_{ij}^{(2)c}\right)~{}$
$\displaystyle=~{}2i\sum_{\alpha,\beta}\,{\rm
Im}\left[(\lambda^{\dagger}\lambda)_{\beta\alpha}\,(\lambda^{T}\lambda^{*})_{\alpha\beta}\right]\,I_{\alpha\beta}^{(2)}$
$\displaystyle=~{}2i\sum_{\alpha,\beta}\,{\rm
Im}\left[(\lambda^{\dagger}\lambda)_{\beta\alpha}\,(\lambda^{\dagger}\lambda)_{\beta\alpha}\right]\,I_{\alpha\beta}^{(2)}\
,$ (17)

and

${\rm
tr}\left(\Sigma_{ij}^{(3)}-\Sigma_{ij}^{(3)c}\right)~{}=~{}2i\sum_{\alpha,\beta}\,{\rm
Im}\left[(\lambda^{\dagger}\lambda)_{\beta\alpha}\,(\lambda^{\dagger}\lambda)_{\beta\alpha}\right]\,I_{\alpha\beta}^{(3)}\
.$ (18)

have the same dependence on the $\lambda_{i\alpha}$. Note that ${\rm
Im}\left[(\lambda^{\dagger}\lambda)_{\beta\alpha}(\lambda^{\dagger}\lambda)_{\beta\alpha}\right]$
is antisymmetric in $\alpha,\beta$, so only the antisymmetric part of the
dynamical factors $I_{[\alpha\beta]}^{(2)}$ and $I_{[\alpha\beta]}^{(3)}$
contributes to the lepton-antilepton asymmetry.

Now, we can prove in general that CPT invariance666In QFT in curved spacetime,
we regard the discrete symmetries C, P and T as defined with respect to the
local Minkowski spacetime that exists, due to its pseudo-Riemannian nature, at
each point in spacetime. (See, for example, ref. McDonald:2014yfg  for a
detailed discussion.) This is also true of the quantum fields themselves, as
evident in the definition of spinor fields as representations of the SL(2,C)
symmetry of the local Minkowski spacetime. With this understanding, CPT
symmetry and its standard consequences in QFT for the properties of particles
and antiparticles, the spin-statistics theorem, etc. holds independently of
the curvature of the background spacetime, which is not required to be P or T
symmetric. This may be contrasted with an interesting recent proposal for a
“CPT symmetric” cosmology Boyle:2018tzc , in which the gravitational
background itself – an extension to $t<0$ of FRW spacetime – is taken to be
time-reversal symmetric about a single privileged point, interpreted as a
bounce or creation event. In our interpretation, this “CPT” would be viewed as
an environmental symmetry of the background fields (in this case
gravitational), rather than the fundamental CPT symmetry of the QFT itself.
Similar remarks apply to the use of “CPT” in the ‘spontaneous’ and
gravitational baryogenesis papers in refs. Davoudiasl:2004gf ; Cohen:1987vi
and Poincaré invariance together imply that the propagation of particles and
antiparticles is identical McDonald:2015iwt . To see how this is realised in
this context, note that in flat spacetime translation invariance implies that
the propagators are functions only of the difference in coordinates, i.e.
$\Delta(x,y)\rightarrow\Delta(x-y)$, etc. But given this, we can readily show
that the factors $I_{\alpha\beta}^{(2)}$ and $I_{\alpha\beta}^{(3)}$ are
symmetric in $\alpha,\beta$. For example, for diagram (2), translation
invariance implies

$\displaystyle I_{\alpha\beta}^{(2)}(x,y)~{}$ $\displaystyle=~{}\int
d^{4}z\int
d^{4}z^{\prime}\,G(x-y)\,G(z-z^{\prime})\,S_{\alpha}^{\times}(x-z)\,\Delta(z-z^{\prime})\,S_{\beta}^{\times}(z^{\prime}-y)$
$\displaystyle=~{}\int d^{4}u\int
d^{4}u^{\prime}\,G(x-y)\,G(u-u^{\prime})\,S_{\beta}^{\times}(x-u)\,\Delta(u-u^{\prime})\,S_{\alpha}^{\times}(u^{\prime}-y)$
$\displaystyle=~{}I_{\beta\alpha}^{(2)}(x,y)\ ,$ (19)

under the change of dummy variables $u=x+y-z^{\prime}$ and $u^{\prime}=x+y-z$.

This is no longer necessarily true in curved spacetime, and indeed we find by
explicit calculation that, provided the Majorana masses of the sterile
neutrinos are non-degenerate, the diagrams with the charge-violating
propagators $S_{\alpha}^{\times}$ have non-vanishing antisymmetric factors
$I_{[\alpha\beta]}^{(2)}$ and $I_{[\alpha\beta]}^{(3)}$. We therefore find a
difference in the self-energies of the light leptons and antileptons given by

${\rm
tr}\left(\Sigma_{ij}-\Sigma_{ij}^{c}\right)~{}=~{}2i\sum_{\alpha,\beta}{\rm
Im}\,[(\lambda^{\dagger}\lambda)_{\beta\alpha}(\lambda^{\dagger}\lambda)_{\beta\alpha}]\,\left(I_{[\alpha\beta]}^{(2)}+I_{[\alpha\beta]}^{(3)}\right)\
.$ (20)

This is the fundamental observation allowing the generation of a lepton-
antilepton asymmetry in curved spacetime through two-loop contributions to the
light neutrino propagators in this CP-violating BSM model.

The next step is to determine the coefficients of the effective Lagrangian (3)
by explicit evaluation of these diagrams in curved spacetime. Since we only
need $L_{eff}$ in the weak curvature regime $\mathfrak{R}/M_{\alpha}^{2}\ll
1$, the quickest way is to expand the metric around Minkowski spacetime as
$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ in (3) and evaluate the corrections to
the self-energies to $O(h)$. We then match the coefficients of the
corresponding expansions with explicit calculations of the Feynman diagrams
above in flat spacetime but with a single graviton insertion in the
propagators or vertices. Especially at two-loops, these calculations are
highly non-trivial, so here we simply refer to the original papers
McDonald:2014yfg ; McDonald:2015ooa ; McDonald:2015iwt ; McDonald:2016ehm ;
McDonald:2020ghc  and quote the results:

$\bigl{(}a_{ij},~{}c_{ij},~{}d_{ij}\bigr{)}~{}=~{}\frac{1}{12}\frac{1}{(4\pi)^{2}}\,\sum_{\alpha}\lambda_{\alpha
j}^{\dagger}\lambda_{i\alpha}\,\frac{1}{M_{\alpha}^{2}}\,\Bigl{(}-\frac{4}{3},~{}\frac{3}{4},\,-3\Bigr{)}\,\
.$ (21)

and

$b_{ij}~{}=~{}\frac{1}{9}\frac{1}{(4\pi)^{4}}\,\sum_{\alpha,\beta,k}\,\lambda_{\beta
j}^{\dagger}\lambda_{i\alpha}\lambda_{\beta
k}^{\dagger}\lambda_{k\alpha}\,\frac{1}{M_{\alpha}M_{\beta}}\,I_{[\alpha\beta]}\
,$ (22)

where

$I_{[\alpha\beta]}~{}\sim~{}\left(\frac{M_{\beta}}{M_{\alpha}}\right)^{2p}\,\log\left(\frac{M_{\beta}}{M_{\alpha}}\right)\
,$ (23)

up to an O(1) numerical factor.

The question of whether the hierarchy parameter $p$ is 0 or 1 was left
unresolved in ref. McDonald:2015iwt . The most natural result, which holds in
all the diagrams we calculated to a conclusion, is $p=0$. This would accord
with the expected decoupling of heavy mass intermediate states in the Feynman
diagram. Nevertheless, in calculating the arbitary-momentum, two-loop triangle
diagram involved – which appears to be at the limit of known techniques – we
found contributions with $p=1$ which would require a remarkable cancellation
if they were to be absent in $I_{[\alpha\beta]}$. For this reason, we retain
the possibility that $p$ could be 1 in what follows since, as we see in the
cosmological scenarios, the presence of a large sterile mass hierachy
dependence could significantly enhance the ultimate prediction for the baryon-
to-photon ratio in this BSM model.

##  3 Gravity-Extended Boltzmann Equations in FRW Spacetime

At this point, we have established that loop effects in the BSM model in a
gravitational field may be encoded in the effective Lagrangian (3), with
dynamically generated couplings $a_{ij},\ldots,d_{ij}$. The next step is to
use this effective action to describe the gravitational generalisation of the
Boltzmann equations which determine the evolution of the lepton number density
in a finite-temperature, expanding universe.

Using the standard Noether procedure with the effective action, we find that
the lepton number current
$J^{\mu}=\sum_{i}\overline{\nu_{L}^{\,i}}\gamma^{\mu}\nu_{L}^{\,i}$ for the
light neutrinos satisfies the broken conservation equation,

$\displaystyle(1+2cR)D_{\mu}J^{\mu}\,+\,2aR_{\mu\nu}D^{\mu}J^{\nu}\,$
$\displaystyle+\,\Bigl{(}a+2c+\frac{1}{2}d\Bigr{)}\partial_{\mu}RJ^{\mu}$
$\displaystyle+\,2d\left(\overline{\nu_{L}}D^{2}\gamma.D\nu_{L}\,+\,\overline{\nu_{L}}\gamma.\overleftarrow{D}\overleftarrow{D^{2}}\nu_{L}\right)~{}~{}\sim~{}0\
,$ (24)

with the same suppression of flavour indices $i,j$ in the couplings, currents
and bilinears employed in (3). Since $D_{\mu}J^{\mu}\sim O(\lambda^{2})$, the
pre-factor $2cR$ of the $D_{\mu}J^{\mu}$ term must be omitted to consistent
perturbative order so, up to terms vanishing by the equation of motion, we
find simply,777From this point, we disregard any light flavour dependence for
simplicity and take (25) as an equation with all the $J^{\mu}$ being flavour
singlet currents and with coefficients $a\sim{\rm tr}\,a_{ij}$, etc. depending
on the Yukawa couplings through $(\lambda^{\dagger}\lambda)_{\alpha\alpha}$.
888See also ref. Antunes:2019phe  for related ideas.

$\displaystyle
D_{\mu}J^{\mu}\,+\,2a\,R_{\mu\nu}D^{\mu}J^{\nu}\,+\,2\hat{b}\,\partial_{\mu}RJ^{\mu}~{}\sim~{}0\
,$ (25)

defining $\hat{b}=\tfrac{1}{2}a+c+\tfrac{1}{4}d$. Note that (25) involves only
the CP even terms – the CP odd term in the effective action with coupling
$b_{ij}$ does not contribute to the current conservation equation.

For applications in cosmology, we need to evaluate this in a FRW spacetime. We
consider the spatially flat FRW metric,

$ds^{2}\,=\,dt^{2}-a(t)^{2}\,\delta_{ij}dx^{i}dx^{j}\ ,$ (26)

with non-vanishing Christoffel symbols $\Gamma^{0}_{ij}\,=\,-H\delta_{ij}$,
$\Gamma^{i}_{j0}\,=\,H\delta^{i}{}_{j}$ and Ricci tensor components
$R_{00}\,=\,4\pi G\rho(1+3w)$, $R_{0i}\,=\,0$, $R_{ij}\,=\,-4\pi
G\rho(1-w)\delta_{ij}$. The Ricci scalar is $R\,=\,-8\pi G\rho(1-3w)$ and
$\dot{R}\,=\,8\pi G\rho\,3H(1-3w)(1+w)$. Here, $\rho$ is the energy density
and the equation of state is $p=w\rho$. The Hubble constant $H=\dot{a}/a$ is
related to $\rho$ through the Friedmann equation $3H^{2}=\rho/M_{p}^{2}$,
where $M_{p}$ is the reduced Planck mass, $8\pi G=1/M_{p}^{2}$.

Evaluating (25) in this background, and identifying the current component
$J^{0}$ as lepton number density $n_{L}=n_{\nu}-n_{\nu^{c}}$, gives the
following equation for the time evolution of $n_{L}$,

$(1+2aR_{00})\frac{dn_{L}}{dt}\,+\,3Hn_{L}\,+\,2aR^{i}{}_{i}Hn_{L}\,+\,2\hat{b}\,\dot{R}\,n_{L}\,=\,0\
.$ (27)

Writing this to consistent $O(\lambda^{2})$ perturbative order, we find the
form quoted in (5) and substituting for the curvatures gives finally,

$\frac{dn_{L}}{dt}\,+\,3Hn_{L}~{}=~{}3H\frac{\rho}{M_{p}^{2}}(1+w)\bigl{[}2a-2\hat{b}(1-3w)\bigr{]}\,n_{L}\
.$ (28)

To develop this into the full Boltzmann equation, we also need to take account
of the lepton number violating interactions. This is where the non-vanishing
equilibrium density $n_{L}^{eq}$ induced by the CP odd term in the effective
Lagrangian enters. The $\Delta L=2$ reactions are shown in Fig. 2, and the
model also allows contributions from the ‘inverse decays’
$\nu_{L}H\rightarrow\nu_{R}$ at finite temperature. (We neglect $\Delta L=1$
scattering reactions involving other SM particles here for simplicity
Buchmuller:2004nz .) Collecting these effects into a single rate factor
$\Gamma=\Gamma_{ID}+2\Gamma_{\Delta L=2}$, a standard kinetic theory analysis
shows that (28) should be extended to McDonald:2020ghc

$\frac{dn_{L}}{dt}\,+\,3Hn_{L}~{}=~{}3H\frac{\rho}{M_{p}^{2}}(1+w)\bigl{[}2a-2\hat{b}(1-3w)\bigr{]}\,n_{L}~{}+~{}\Gamma\left(n_{L}-n_{L}^{eq}\right)\
,$ (29)

with $n_{L}^{eq}=\tfrac{1}{3}b\,\dot{R}\,T^{2}$.

Finally, this model – which after all is a conventional BSM theory for thermal
leptogenesis – also includes the standard mechanism for generating a lepton
asymmetry through the out-of-equilibrium decays of the heavy sterile
neutrinos, $\nu_{R}^{\,\alpha}\rightarrow\nu_{L}^{\,i}H$ and
$\nu_{R}^{\,\alpha}\rightarrow\nu_{L}^{\,ic}H$, shown in Fig. 5. These display
an asymmetry dependent on the CP-violating combination of Yukawa couplings
Buchmuller:2003gz ,

$\varepsilon_{\alpha}\,\simeq\,\frac{3}{16\pi}\sum_{\beta\neq\alpha}\frac{{\rm
Im}((\lambda^{\dagger}\lambda)_{\alpha\beta}^{2})}{(\lambda^{\dagger}\lambda)_{\alpha\alpha}}\,\frac{M_{\alpha}}{M_{\beta}}\
.$ (30)

arising from the interference of the tree and one-loop diagrams for the decays
in Fig. 5. Note that the charge-violating propagators $S_{\alpha}^{\times}$
are crucial in establishing the lepton-antilepton asymmetry in these diagrams.

$\nu_{R}$$H$$\nu_{L}$$S_{\alpha}$

$\nu_{R}$$H$$\nu_{L}$$S_{\alpha}^{\times}$$S_{\beta}^{\times}$

$\nu_{R}$$H$$\nu_{L}$$S_{\alpha}^{\times}$$S_{\beta}^{\times}$

Figure 5: Diagrams for the decay
$\nu_{R}^{\,\alpha}\rightarrow\nu_{L}^{\,i}\,H$ which contribute to the decay
rate asymmetry factor $\varepsilon_{\alpha}$. At $O(\lambda^{4})$ the relevant
contribution to the $\Gamma(\nu_{R}\rightarrow\nu_{L}\,H)$ decay rate arises
from the interference of the tree and one-loop diagrams shown. Similar
diagrams, where the $S_{\alpha}$ and $S_{\alpha}^{\times}$ type $\nu_{R}$
propagators are switched compared to the figure, give the decay rate for
$\nu_{R}^{\,\alpha}\rightarrow\nu_{L}^{\,i\,c}\,H$.

It is usual in discussing leptogenesis to express the evolution in terms of
temperature rather than time, and as in section 1 we introduce the variable
$z=M_{1}/T$ and express the Boltzmann equation in terms of the lepton-to-
photon ratio $N_{L}(z)=n_{L}/n_{\gamma}$.999Since $n_{\gamma}\sim T^{3}$ and
$T\sim 1/a$, we have
$\frac{dN_{L}}{dt}~{}=~{}\frac{1}{n_{\gamma}}\left(\frac{dn_{L}}{dt}\,+\,3Hn_{L}\right)\
,$ and the change of variables gives $d/dz=(1/Hz)d/dt$. So in terms of these
variables, the final coupled Boltzmann equations for $N_{L}$ and $N_{\nu_{R}}$
read,

$\displaystyle\frac{dN_{\nu_{R}}}{dz}\,$
$\displaystyle=\,-D\left(N_{\nu_{R}}\,-\,N_{\nu_{R}}^{eq}\right)$ (31)
$\displaystyle\frac{dN_{L}}{dz}\,$
$\displaystyle=\,-D\,\varepsilon_{1}\left(N_{\nu_{R}}\,-\,N_{\nu_{R}}^{eq}\right)\,-\,W\left(N_{L}\,-\,N_{L}^{eq}\right)\,-\,\mathcal{W}\,N_{L}\
,$ (32)

where recall that $W(z)=\Gamma/zH$, where $\Gamma(z)$ is the finite-
temperature rate factor appearing in (29) and we similarly define
$D(z)=\Gamma(\nu_{R}\rightarrow\nu_{L}\,H)/zH$.

The gravitational terms $N_{L}^{eq}(z)$ and $\mathcal{W}(z)$ in (32) can be
read off in terms of the radiatively-generated couplings from (29). We find,

$\mathcal{W}(z)~{}=~{}-3(1+w)\left[2a-2\hat{b}(1-3w)\right]\,\frac{\rho}{M_{p}^{2}}\,\frac{1}{z}\
,$ (33)

and

$N_{L}^{eq}(z)~{}=~{}(1+w)(1-3w)\,b\,M_{1}^{2}\,\frac{H}{n_{\gamma}}\,\frac{\rho}{M_{p}^{2}}\,\frac{1}{z^{2}}\
.$ (34)

Note here that for a radiation-dominated FRW universe, $w\simeq 1/3$, with the
deviation from this value arising purely from the beta functions
characterising the energy-momentum trace anomaly, $T^{\mu}{}_{\mu}\neq 0$.
With standard model fields, this gives $(1-3w)\simeq 0.1$. This quantum
deviation from the classical conformal symmetry value is essential to
realising a non-vanishing $N_{L}^{eq}(z)$ in (34). Moreover, in the radiation-
dominated case, $H\sim 1/z^{2}$ while $\rho\sim 1/z^{4}$, so both the
gravitational terms $N_{L}^{eq}(z)$ and $\mathcal{W}(z)$ fall sharply as
$1/z^{5}$ as the universe cools.

From (33) we see that whether the radiative curvature corrections encoded in
$\mathcal{W}(z)$ act to amplify or reduce the lepton number density as the
universe evolves depends on the sign of the combination
$\big{[}2a-2\hat{b}(1-3w)\bigr{]}$. Since $(1-3w)$ is small, this essentially
depends just on the sign of the coupling $a$; negative $a$ tends towards
damping (washout) of the lepton number with time, while a positive $a$ would
imply an amplification. In the BSM model studied here we found in (21) that
$a$ is negative. This was assumed in the picture sketched in Fig. 1, where it
controls the early, high-temperature evolution in region (a).

##  4 Gravitational Leptogenesis in the Early Universe

In this final section, we solve these coupled Boltzmann equations in the BSM
model with physical neutrino parameters in realistic cosmological settings and
discuss the viability of the gravitational leptogenesis mechanism for
generating the observed baryon asymmetry in the universe today.

First we need expressions for the conventional finite-temperature rate factors
$W(z)$ and $D(z)$ in (31), (32). The necessary results are quoted in refs.
McDonald:2016ehm ; McDonald:2020ghc  and are discussed in detail in, for
example, the review Buchmuller:2004nz . With standard definitions101010The
definitions of the neutrino parameters and notation used here are explained in
refs.McDonald:2020ghc ; Buchmuller:2004nz . From the see-saw mechanism we have
the sum of the light neutrino masses,
$\bar{m}^{2}~{}=~{}v^{4}\sum_{\alpha,\beta}\frac{1}{M_{\alpha}M_{\beta}}{\rm
Re}(\lambda^{\dagger}\lambda)_{\alpha\beta}^{2}~{}\simeq~{}\Delta
m_{sol}^{2}+\Delta m_{atm}^{2}\ ,$ where the ‘solar’ and ‘atmospheric’ masses
are $\Delta m_{sol}^{2}=7.53\times 10^{-5}\,{\rm eV}^{2}$ and $\Delta
m_{atm}^{2}=2.44\times 10^{-3}\,{\rm eV}^{2}$. A useful mass scale is set by
$m_{*}=8\pi(\sigma/3)^{1/2}\,v^{2}/M_{p}=1.08\times 10^{-3}\,{\rm eV}$, where
$v=174\,{\rm GeV}$ is the electroweak scale, $M_{p}=2.4\times 10^{18}\,{\rm
GeV}$ is the reduced Planck mass, where $8\pi G=1/M_{p}^{2}$, and
$\sigma=\pi^{2}g_{*}/30$ is the constant appearing in the radiation energy
density $\rho=\sigma T^{4}$, where $g_{*}$ is the effective number of
relativistic degrees of freedom at temperature $T\sim M_{1}$. A key parameter
is the combination
$K\,=\,\frac{v^{2}}{M_{1}m_{*}}\,(\lambda^{\dagger}\lambda)_{11}\ .$ for
parameters related to the light neutrino masses, we have the following
expressions for the lepton number violating rate factor $W(z)$. For
temperatures well above $T\simeq M_{1}$,

$W(z\ll
1)\,\simeq\,\frac{12}{\pi^{2}}\,\frac{m_{*}M_{1}}{v^{2}}\,\left(\frac{\bar{m}^{2}}{m_{*}^{2}}\,+\,K^{2}\right)\,\frac{1}{z^{2}}\
,$ (35)

with the same formula without the $K^{2}$ term giving $W(z\gg 1)$. Near
$z\simeq 1$, $W(z)$ shows a resonance enhancement from the intermediate
$\nu_{R}^{\,1}$ state, as is evident from Fig. 2. This is illustrated in Fig.
6. The key features are the asymptotic behaviour $W(z)\sim 1/z^{2}$, which is
a much weaker $z$-dependence than the $1/z^{5}$ of the gravitational terms
$\mathcal{W}(z)$ and $N_{L}^{eq}(z)$, and the resonance around $z=1$.

Figure 6: This shows the dependence on $z=M_{1}/T$ of the coefficient $W(z)$
in the Boltzmann equation for $K=1$ and $K=10$, with $M_{1}=5\times
10^{10}\,{\rm GeV}$ and the neutrino parameters in the text. The resonance
peak around $T\simeq M_{1}$ increases with $K$.

Next, the sterile neutrino decay rate factor is given in terms of the Yukawa
couplings and Bessel functions by

$D(z)\,=\,K\,z\,\frac{K_{1}(z)}{K_{2}(z)}\ ,$ (36)

and has an equilibrium value at finite temperature,
$N_{\nu_{R}}^{eq}(z)=\tfrac{3}{8}\,z^{2}\,K_{2}(z)$.

For convenience, we also quote here the formulae for the gravitational terms
in a radiation-dominated universe with $w\simeq 0.3$ and with the radiatively-
induced couplings $a,\ldots d$ found in (21), (22).

These are,

$\mathcal{W}(z)\,\simeq\,0.075\,\sigma^{3/2}\,K\left(\frac{M_{1}}{M_{p}}\right)^{3}\frac{1}{z^{5}}\
,$ (37)

and

$N_{L}^{eq}(z)\,\simeq\,0.034\,\sigma^{3/2}\,\left(\frac{M_{1}}{M_{p}}\right)^{3}\,\sum_{\beta\neq
1}\frac{{\rm
Im}(\lambda^{\dagger}\lambda)_{1\beta}^{2}}{(4\pi)^{4}}\,\left(\frac{M_{\beta}}{M_{1}}\right)^{2p-1}\,\log\left(\frac{M_{\beta}}{M_{1}}\right)\,\frac{1}{z^{5}}\
.$ (38)

We have now collected all the ingredients to solve the Boltzmann equations and
determine the evolution of the lepton number asymmetry $N_{L}(z)$ with
temperature in this BSM model, assuming a radiation-dominated background
spacetime.

First, recall that in leptogenesis models, we assume that at much lower
temperatures around the electroweak scale, the lepton number asymmetry
generated in the early universe is converted through sphaleron processes into
a baryon asymmetry. Defining $\eta=n_{B}/n_{\gamma}$ as the baryon-to-photon
ratio, the sphaleron conversion in this model gives $\eta\simeq 0.02\,|N_{L}|$
Buchmuller:2002zs . Since the observed value is $\eta\simeq 6\times 10^{-10}$,
we see that successful leptogenesis requires a final value of $N_{L}(z)$ for
$z\gg 1$ of $|N_{L}|\simeq 10^{-8}$.

Figure 7: The left-hand diagram shows the evolution of the sterile neutrino
density starting from an initial condition with $N_{\nu_{R}}(z\ll 1)=0$. The
right-hand figure shows the corresponding absolute value of the lepton
asymmetry induced by the out-of-equilibrium decays of $\nu_{R}^{\,1}$. The
cusp in the plot indicates that the asymmetry $N_{L}(z)$ changes sign as the
sterile neutrino density $N_{\nu_{R}}(z)$ switches from under to just over its
equilibrium value. The parameters here are $M_{1}=5\times 10^{10}\,{\rm GeV}$
and $K=5$, with $\varepsilon_{1}=10^{-6}$ .

Now consider the Boltzmann equation (31) for $N_{\nu_{R}}(z)$. Fig. 7 shows
$N_{\nu_{R}}(z)$ rising from an initial condition $N_{\nu_{R}}(z\ll 1)=0$ at
early times, then slightly overshooting its equilibrium value around $z\simeq
1$ before rapidly being driven back to equilibrium. Ignoring all other effects
in the Boltzmann equation (32) for $N_{L}(z)$, we see that during this out-of-
equilibrium phase, the lepton number violating decays
$\nu_{R}\rightarrow\nu_{L}\,H$ generate the lepton asymmetry shown in the
right-hand plot in Fig. 7. This is of course the familiar mechanism of thermal
leptogenesis through out-of-equilibrium decays, satisfying the third Sakharov
condition. Note that the size of the induced asymmetry depends on the decay
rate $D(z)$ and, crucially, on the CP violating parameter $\varepsilon_{1}$
which is itself sensitive to the phases of the Yukawa couplings.

Now consider the evolution of the lepton number asymmetry including the
gravitational effects encoded in the full Boltzmann equation (32). This is
shown, in the case of a hierarchy enhancement $p=1$, in Fig. 8.

Figure 8: Dynamical evolution of the lepton number asymmetry $N_{L}(z)$ in the
case of a hierarchy enhancement, $p=1$, with parameters chosen so that the
gravitational leptogenesis mechanism dominates over the out-of-equilibrium
$\nu_{R}^{\,1}$ decays. Here, $M_{1}=5\times 10^{10}\,{\rm GeV}$, $K=1$, with
$M_{3}=10^{16}\,{\rm GeV}$, ${\rm
Im}(\lambda^{\dagger}\lambda)_{13}^{2}/(4\pi)^{2}=5\times 10^{-4}$ and
$\varepsilon_{1}=10^{-10}$.

From an initial condition $N_{L}(z\ll 1)\simeq 0$ at ultra-high temperatures,
$N_{L}$ is initially driven towards its equilibrium value $N_{L}^{eq}(z)\sim
1/z^{5}$ by the rate factor $W(z)\sim 1/z^{2}$ for the $\Delta L\neq 0$
reactions. However, at these early times, the gravitational ‘washout’ factor
$\mathcal{W}(z)\sim 1/z^{5}$ dominates over $W(z)$ and delays the approach to
equilibrium (labelled as region (a) in Fig. 1). $N_{L}$ then follows its
equilibrium trajectory (region (b)) until at some lower temperature ($z\simeq
10^{-3}$ with the parameters shown), the $\Delta L\neq 0$ reaction rate falls
below the Hubble expansion rate, i.e. $W(z)\simeq 1$, and can no longer
maintain $N_{L}$ in equilibrium. It then decouples and is frozen out at a
constant value (region (c)) as the universe continues to expand and cool. Much
later, when $W(z)$ grows around the resonance at $z\simeq 1$, it is strong
enough to temporarily force $N_{L}$ back towards the equilibrium value, which
is falling away extremely rapidly, resulting in a late dip (region (d)) before
settling to its final asymptotic value $|N_{L}|\simeq 10^{-8}$.

The ultimate value of the residual asymmetry $|N_{L}|$ for temperatures below
the Majorana mass scale, $z\gg 1$, is therefore the result of a competition
between the gravitational leptogenesis mechanism and thermal leptogenesis.
Fig. 8 shows the case where $\varepsilon_{1}$ is small enough and the
gravitational leptogenesis mechanism dominates.

Figure 9: Dynamical evolution of the lepton number asymmetry $N_{L}(z)$ in the
case of no hierarchy enhancement, $p=0$. Here, the final lepton asymmetry is
determined by the out-of-equilibrium $\nu_{R}^{\,1}$ decays, with the
gravitational mechanism producing a steep rise in the asymmetry at earlier
times for temperatures $z\lesssim 0.01$. Here, $M_{1}=10^{10}\,{\rm GeV}$,
$K=5$, with $M_{2}=10^{12}\,{\rm GeV}$, ${\rm
Im}(\lambda^{\dagger}\lambda)_{12}^{2}/(4\pi)^{2}=5\times 10^{-4}$ and
$\varepsilon_{1}=10^{-7}$.

In contrast, Fig. 9, where we have assumed no hierarchy enhancement, $p=0$,
and chosen a larger CP violating factor $\varepsilon_{1}$, shows a scenario in
which the thermal leptogenesis mechanism determines the final value of the
asymmetry $|N_{L}|$. Even in this case, however, the gravitational effects
describe the early-time evolution of $N_{L}(z)$ and predict a large lepton
number asymmetry in the early universe.

We therefore see from these quantitative examples that radiatively-induced
gravitational leptogenesis provides a mechanism which can generate the
observed baryon-to-photon ratio in the universe today.

As we have shown, while the RIGL mechanism always determines the evolution of
the lepton asymmetry in the early universe above $T\simeq M_{1}$, whether the
gravitational or sterile neutrino decay mechanisms determine the final value
of $|N_{L}|$ depends on the masses and Yukawa couplings in the BSM model. A
comprehensive investigation of the parameter space of this model,
incorporating flavour effects and varying the sterile neutrino mass spectrum,
has been given in an interesting recent paper Samanta:2020tcl . In particular,
it is shown that exploiting flavour effects, the RIGL mechanism is naturally
able to achieve successful leptogenesis for ranges of the sterile neutrino
masses which are not compatible with standard thermal leptogenesis. This paper
also analyses the implications of the model parameters favouring gravitational
leptogenesis for low-energy neutrino phenomenology.

Figure 10: Dynamical evolution of the gravitationally-induced lepton number
asymmetry $N_{L}(z)$ for effective equation of state parameters $w=0.3$ and
$0.5$, with $z_{*}=0.1$. The corresponding equilibrium values $N_{L}^{eq}$ are
shown as the dashed lines. Decoupling from $N_{L}^{eq}$ occurs earlier with
higher values of $w$, resulting in a larger final asymmetry. The parameters
here are $M_{1}=10^{10}\,{\rm GeV}$, $K=5$, with $M_{2}=10^{12}\,{\rm GeV}$,
${\rm Im}(\lambda^{\dagger}\lambda)^{2}_{12}/(4\pi)^{2}=5\times 10^{-4}$ and
no hierarchy enhancement, $p=0$.

Finally, we describe briefly the implementation of the RIGL mechanism in an
alternative cosmological scenario Davoudiasl:2004gf ; McDonald:2016ehm ;
McDonald:2020ghc . This picture is especially economical in the sense of
minimising the introduction of new fields and interactions, relying on
gravitational effects to drive the dynamical evolution of the early universe.
We therefore suppose that in place of the usual post-inflationary reheating
phase, the relativistic particles giving rise to the entropy of the universe
arise through gravitational particle creation at the transition from the de
Sitter inflationary vacuum to the FRW vacuum at the end of inflation
Ford:1986sy ; Peebles:1998qn . This removes the need for an unknown direct
coupling of the inflaton to the standard model fields to induce reheating.
Instead, immediately after inflation the energy density is dominated by the
inflaton component with equation of state $w>1/3$, which gradually dilutes
relative to the thermalised relativistic particles as the universe expands,
eventually leading, beyond a crossover temperature $z_{*}$, to a conventional
radiation-dominated FRW universe. The stiff equation of state $w>1/3$ is
achieved if the potential at the end of inflation is sufficiently steep
Turner:1983he , the limiting value $w=1$ in which the kinetic energy dominates
being known as ‘kination’. If the temperature during this phase is greater
than the lightest sterile neutrino mass, $T>M_{1}$, then gravitational
leptogenesis occurs during this era.

It is relatively straightforward to trace the dependence on $w$ through the
various terms in the Boltzmann equations and investigate their solutions. In
particuar, the power dependence of the gravitational terms becomes
$\mathcal{W}(z)\sim 1/z^{5-(1-3w)}$ while $N_{L}^{eq}(z)\sim
1/z^{5-\tfrac{3}{2}(1-3w)}$, so for higher $w$ the fall-off of $N_{L}^{eq}$ is
steeper, but decoupling occurs earlier giving a higher final value for
$|N_{L}|$. In Fig. 10 we show the dynamical evolution of $N_{L}(z)$ in an
illustrative scenario with no hierarchy enhancement but with $w=0.5$, which
gives the required asymptotic asymmetry $|N_{L}|\simeq 10^{-8}$, compared to
the radiation-dominated case $w\simeq 0.3$ with the same parameters.

Clearly there are many avenues still to explore in developing RIGL as a
potential explanation of the matter-antimatter asymmetry of the universe, both
in its implementation in a variety of BSM models and in different cosmological
scenarios. More generally, this work once again illustrates the importance of
loop effects in QFT in curved spacetime in determining the physics of the
early universe.

Acknowledgements

I am grateful to Jamie McDonald for the collaboration on which this review is
based. This work is supported in part by the STFC theoretical particle physics
grant ST/P0005X/1.

## References

  * (1) J. I. McDonald and G. M. Shore, JHEP 1502 (2015) 076, [arXiv:1411.3669 [hep-th]].
  * (2) J. I. McDonald and G. M. Shore, Phys. Lett. B 751 (2015) 469, [arXiv:1508.04119 [hep-ph]].
  * (3) J. I. McDonald and G. M. Shore, JHEP 1604 (2016) 030, [arXiv:1512.02238 [hep-ph]].
  * (4) J. I. McDonald and G. M. Shore, Phys. Lett. B 766 (2017) 162, [arXiv:1604.08213 [hep-ph]].
  * (5) J. I. McDonald and G. M. Shore, JHEP 10 (2020), 025, [arXiv:2006.09425 [hep-ph]].
  * (6) H. Davoudiasl, R. Kitano, G. D. Kribs, H. Murayama and P. J. Steinhardt, Phys. Rev. Lett. 93 (2004) 201301, [arXiv:hep-ph/0403019 [hep-ph]].
  * (7) S. Khlebnikov and M. Shaposhnikov, Nucl. Phys. B 308 (1988) 885.
  * (8) A. Sakharov, Sov. Phys. Usp. 34 (1991) no.5, 392.
  * (9) M. Fukugita and T. Yanagida, Phys. Lett. B 174 (1986) 45.
  * (10) W. Buchmuller, P. Di Bari and M. Plumacher, Annals Phys. 315 (2005) 305, [arXiv:hep-ph/0401240 [hep-ph]].
  * (11) A. G. Cohen and D. B. Kaplan, Phys. Lett. B 199 (1987), 251.
  * (12) Y. Ohkuwa, Prog. Theor. Phys. 65 (1981) 1058.
  * (13) L. Boyle, K. Finn and N. Turok, Phys. Rev. Lett. 121 (2018) 251301, [arXiv:1803.08928 [hep-ph]].
  * (14) V. Antunes, I. Bediaga and M. Novello, JCAP 10 (2019) 076, [arXiv:1909.03034 [gr-qc]]
  * (15) W. Buchmuller, P. Di Bari and M. Plumacher, Nucl. Phys. B 665 (2003) 445, [arXiv:hep-ph/0302092 [hep-ph]].
  * (16) W. Buchmuller, “Baryo- and leptogenesis (brief summary),” ICTP Lect. Notes Ser. 14 (2003) 41.
  * (17) R. Samanta and S. Datta, JHEP 12 (2020), 067, [arXiv:2007.11725 [hep-ph]].
  * (18) L. Ford, Phys. Rev. D 35 (1987) 2955.
  * (19) P. Peebles and A. Vilenkin, Phys. Rev. D 59 (1999) 063505, [arXiv:astro-ph/9810509 [astro-ph]].
  * (20) M. S. Turner, Phys. Rev. D 28 (1983) 1243.