# Cosmological complexity of the modified dispersion relation

Tao Li<EMAIL_ADDRESS>Lei-Hua Liu<EMAIL_ADDRESS>Department of Physics, College of Physics, Mechanical and Electrical
Engineering, Jishou University, Jishou 416000, China

###### Abstract

Complexity will be more and more essential in high-energy physics. It is
naturally extended into the very early universe. Considering the universe as a
quantum chaotic system, the curvature perturbation of the scalar field is
identified with the two-mode squeezed state. By solving the
Schr$\ddot{o}$dinger equation, one can obtain the numerical solutions of the
angle parameter and squeezing parameter. The solution of the squeezing
parameter mainly determines the evolution of complexity. Our numeric indicates
that the complexity of the modified dispersion relation will have a non-linear
pattern after the horizon exits. Meanwhile, its corresponding Lyapunov index
is also larger compared with the standard case. During the inflationary
period, the complexity will irregularly oscillate and its scrambling time is
also shorter compared with the standard case. Since the modified dispersion
relation can be dubbed as the consequences of various frameworks of quantum
gravity, it could be applicable to these frameworks. Finally, one can expect
the framework of quantum gravity will lead to the fruitful evolution of
complexity, which guides us in distinguishing various inflationary models.

##  I Introduction

The finding of anti-de Sitter/conformal field theory (AdS/CFT) has opened a
totally new window and direction for exploring the nature of gravity and
strong-coupling system in the condensed field Maldacena:1997re . In light of
this, spacetime can naturally emerge from quantum entanglement
VanRaamsdonk:2010pw . Moreover, one can even suspect that ER (Einstein-Rosen
bridge) = EPR (Einstein–Prodolsky–Rosen pair) which relates two distinctive
physical concepts Maldacena:2013xja . Further development can be found in
Refs. Swingle:2014uza ; Laanemets:2022rmn ; Bianchi:2012ev ; Lashkari:2013koa
; Balasubramanian:2013lsa . Thus, quantum entanglement will be more and more
essential in high-energy physics. However, the researcher has discovered that
the boundary of quantum field theory (QFT) reaches thermal equilibrium within
a short temporal interval, and the evolution of wormhole will take a much
longer time compared with QFT Stanford:2014jda . In order to solve this issue,
the so-called quantum complexity is introduced to describe the evolution of
the corresponding wormhole Hartman:2013qma ; Liu:2013iza . In order to relate
this concept to the holographic principle, the CA conjecture was proposed by
Brown:2015bva , in which the complexity is equaled to the computation of
action evaluated on a bulk subregion called Wheeler–DeWitt patch. Afterward,
there are many profound results were produced Faulkner:2013ica ; Cai:2016xho ;
Carmi:2017jqz ; Carmi:2016wjl ; Reynolds:2016rvl ; Chapman:2018dem ;
Chapman:2018lsv ; Reynolds:2017lwq .

With the development of complexity, two main methods for investigating the so-
called circuit complexity are: (a). Nielsen et al. utilize a geometrical
method within the phase space of quantum gates Nielsen2006AGA ;
Nielsen2006QuantumCA ; Dowling2008TheGO . (b). One can also use the ”Fubini-
Study” distance to investigate the circuit complexity Chapman:2017rqy . In
light of these ones, the complexity can be explicitly obtained by using the
wave function Jefferson:2017sdb ; Bhattacharyya:2018bbv ; Guo:2018kzl  and
covariance matrix Chapman:2018hou ; Hackl:2018ptj ; Alves:2018qfv ;
Camargo:2018eof ; Ali:2018aon ; Khan:2018rzm . Even though the definition of
complexity is still preliminary, Ref. Jefferson:2017sdb  already tries to give
it in quantum field theory Jefferson:2017sdb . Based on it, the Hawking
radiation can be interpreted by circuit complexity Zhao:2019nxk ;
Brown:2019rox . Futhermore, spacetime can be explained by this complexity
Chandra:2021kdv .

In the geometric method Nielsen2006AGA , the most convenient quantum system is
the inverted Harmonic system. For the single inflationary theory, the Hamilton
of curvature perturbation is similar to the inverted Harmonic system in
momentum space since the sign of kinetic term and potential is the opposite.
Thus, one can implement the technology of quantum circuit complexity to
investigate various cosmological periods. Ref. Bhargava:2020fhl  has
investigated the complexity that is the most essential after the bounce
period, in which they also estimated the scrambling time. More generic
cosmological models were studied that show the complexity grows fastest in
matter-dominated era Lehners:2020pem . By the consideration of the
backreaction of background expansion Bhattacharyya:2020rpy , they found that
inflation has the most simple pattern of complexity that will linearly grow
after inflation. Ref. Wang:2022mix  investigated the cosmological complexity
of the thermal state in the expansion space. In our previous work Liu:2021nzx
, we have shown that the formation of primordial blackhole will lead to the
different evolution of complexity in inflation, but it is similar after
inflation. The complexity is also related to an integral part of the web of
diagnostics for quantum chaos Roberts:2016hpo ; Balasubramanian:2019wgd ;
Ali:2019zcj ; Yang:2019iav ; Bhattacharyya:2019txx ; Barbon:2019wsy , which
provides the Lyapunov index that could test the Kerr blackhole in various
modified gravitational models Deich:2023oox .

Based on Ref. Liu:2021nzx , it has shown that the various patterns of
complexity in inflation compared with Bhattacharyya:2020rpy ; Bhargava:2020fhl
; Lehners:2020pem , in which they show the complexity in the simple
inflationary models. Another point is that our study Liu:2021nzx  of
complexity is only different in inflation. For the late universe, the trend of
evolution is similar. In order to capture the various information of
complexity in different eras, we will utilize the modified dispersion relation
via Cai:2009hc  since it will impact the power spectrum in various periods due
to the modified dispersion relation. From another perspective, the modified
dispersion relation can be dubbed as the consequence of quantum gravity
Armendariz-Picon:2003jjq ; Armendariz-Picon:2006vgx ; Magueijo:2008sx ;
Piao:2006ja ; Martin:2000xs ; Arkani-Hamed:2003pdi ; Bojowald:2006zb ;
Jacobson:2000gw ; Cai:2007gs . It has so many phenomenological implications
Cai:2009in ; Li:2009rt ; Cai:2009zp ; Cai:2012yf ; Cai:2018tuh ; Zheng:2017qfs
; Chen:2017dhi ; Bianco:2016yib ; Pan:2015tza , $\it i.e.$ the string
cosmology, the DBI inflation, the cosmology of loop gravity, $\it e.t.c$.
Thus, the investigation of the cosmological complexity of the modified
relation is applicable to many theoretical frameworks.

The structure of this paper is organized as follows. In section II, we will
review the so-called dispersion relation according to Cai:2009hc . In section
III, we will treat the cosmological perturbation as the two-mode squeezed
state to evaluate the evolution of angle parameter $\phi_{k}$ and squeezing
parameter $r_{k}$. Section IV will investigate the evolution of complexity for
the modified dispersion relation. In section V, we will give our conclusions
and outlooks.

##  II The modified dispersion relation

In this section, we will work in the Friedman–Lemaitre––Robertson Walker
background metric ,

$ds^{2}=a(\eta)^{2}(-d\eta^{2}+d\vec{x}^{2}),$ (1)

where $\vec{x}=(x,y,z)$ (denoting the three-dimensional spatial part) is the
spatial vector and $a(\eta)$ is the scale factor in conformal time. Since the
curvature perturbation is inside the Hubble radius due to the inflationary
period, then the wavelength will become larger, meanwhile, the universe
undergoes the expansion all the time, and this inhomogeneity will re-enter the
Hubble radius again. This physical process can be characterized by which the
wavelength $k/a$ is longer than $1/H$. Thus, the conformal time is more
convenient compared with physical time. Under metric (1), one can define the
perturbation of scalar field like
$\phi(x_{\mu})=\phi_{0}(\eta)+\delta\phi(x_{\mu})$, the corresponding metric
can be read as follows,

$ds^{2}=a(\eta)^{2}\left(-(1+\psi(\eta,x)d\eta^{2}+(1-\psi(\eta,x)d\vec{x}^{2}\right),$
(2)

where $\psi(\eta,x)$ is the perturbation of metric. Once obtaining metric (1),
(2) and the curvature perturbation of scalar field, the perturbated action can
be written by

$S=\frac{1}{2}\int
dtd^{3}xa^{3}\frac{\dot{\phi}}{H^{2}}\left[\mathcal{\dot{R}}-\frac{1}{a}(\partial_{i}\mathcal{R})^{2}\right],$
(3)

where $H=\frac{\dot{a}}{a}$, $\mathcal{R}=\psi+\frac{{H}}{\phi_{0}}\delta\phi$
, $z=\sqrt{2\epsilon}a$,
$\epsilon=-\frac{\dot{H}}{H}=1-\frac{{H}^{\prime}}{H}$. Action (3) can be
transferred into a canonical form in terms of the Mukhanov variable
$v=z\mathcal{R}$,

$S=\frac{1}{2}\int d\eta d^{3}x\left[{v}^{\prime
2}-(\partial_{i}v)^{2}+\frac{{z}^{\prime}}{z}{v}^{2}-2\frac{{z}^{\prime}}{z}{v}^{\prime}v\right],$
(4)

where $\prime$ means the derivative with respect to the conformal time. If we
approximate $\epsilon$ to be a constant, then one can get
$\frac{{z}^{\prime}}{z}=\frac{{a}^{\prime}}{a}$,

$S=\frac{1}{2}\int d\eta d^{3}x\left[{v}^{\prime 2}-(f(k_{\rm
ph})\partial_{i}v)^{2}+\frac{{a}^{\prime}}{a}{v}^{2}-2\frac{{a}^{\prime}}{a}{v}^{\prime}v\right],$
(5)

where $f(k_{\rm ph})$ comes the definititon via Ref. Cai:2009hc ,

$\begin{cases}&\text{ if }(k_{\rm ph})>M;f=(\frac{k_{\rm ph}}{M})^{\alpha}\\\
&\text{ if }(k_{\rm ph})\leq M;f=1.\end{cases}$ (6)

One can see that $f=1$ will nicely recover the standard dispersion relation,
which is also equivalent to the sound speed is one as shown in Ref.
Liu:2021nzx . Furthermore, one could see that $f(k_{\rm ph})$ plays a role of
non-trivial sound speed $c_{s}$ but the origin is different. As discussed in
introduction I, the high energy scale (denoted by $M$) will lead to the
modified dispersion relation, such as the trans-Planckian physics and Lorentz
violating effects, $\it etc$. However, the formula is different in various
theoretical frameworks. As discussed in Refs. Bianco:2016yib ; Lu:2009he , the
modified dispersion relation is all proportional to $k_{\rm ph}^{\alpha}$ (the
notation for describing the modified dispersion is different) where $\alpha$
is non-zero in many gravitational models. That is the reason we adopt the
formula $\nu=kf(k_{\rm ph})$ for the dispersion relation. We should emphasize
this formula is applicable in many gravitational models since the modified
dispersion relation is of the structure $k_{\rm ph}^{\alpha}$.

Action (5) is our starting point for investigating its evolution of circuit
complexity. Varying action with respect to $v$ and then transforming into the
momentum space, its equation of motion (EOM) of $v_{k}$ ($k$ denotes the
momentum space) can be derived by

${v_{k}}^{\prime\prime}+(\nu^{2}-\frac{{a}^{\prime\prime}}{a})v_{k}=0.$ (7)

$f=1$ corresponds to the EOM of single field inflationary model of curvature
perturbation.

According to the condition (6), one can clearly see that the dispersion
relation can be classified into two regions, one is the so-called high energy
scale dubbed as the ultraviolet (UV) regime when $k_{\rm ph}>M$. On the
contrary, there is also an infrared (IR) regime as $k_{\rm Ph}<M$. It is
naturally considered $M$ as the criterion for assessing the energy scale.
Being armed with this logic, the main feature of the dispersion relation can
described by the value of $\alpha$, which corresponds to the UV regime or the
IR regime. Here, we summarized the values in light of Cai:2009hc .

In this kind of relation of the modified dispersion relation, its main feature
is characterized by the value of $\alpha$.

$(a)$. The standard dispersion relation corresponds to $\alpha=0$ is also the
standard inflationary model Guth:1980zm , which belongs to the IR regime

$(b)$. As $0<\alpha<2$, it also belongs to the UV regime.

$(c)$. As $\alpha=2$, it is a special kind of Horv$\check{a}$-Lifshitz
cosmology Kiritsis:2009sh ; Calcagni:2009ar .

$(d)$. As $\alpha>2$, it also belongs to the UV regime.

According to the above understanding, it explicitly shows that theory will be
in the UV regime as $\alpha>0$, which is consistent with our previous
discussion since the modified dispersion relation comes via the quantum
gravitational effects corresponding to $\alpha$ is nonzero. From another
aspect, we only assume that $\alpha>0$, since the power spectrum will be
conflicted with observation according to $P_{\delta\phi}\propto k^{\alpha}$ as
$\alpha<0$. Our discussion only focuses on the power spectrum in light of Ref.
Cai:2009hc . Therefore, the value of $\alpha$ is independent of specific
cosmological models. Furthermore, we will adopt some specific values of
$\alpha$ for the following investigations. For intuitively understanding its
various cases of $f(k_{\rm ph})$, we give the plot of $f^{2}$ since it plays
the role of non-trivial sound speed.

Figure 1: It shows that the modified dispersion relation varies with $k/a$. We
have set $M=20$ and set $\alpha=0$ (corresponding to the standard dispersion
relation), $\alpha=0.05$, $\alpha=1.5$, $\alpha=2$ (Horv$\check{a}$-Lifshitz
gravity) and $\alpha=3$.

Fig. 1 clearly indicates the various cases of $f^{2}$ varying with comoving
momentum. Since it is a simple power-law function in terms of $\alpha$, in
which we give five cases: $\alpha=0$, $\alpha=0.05$, $\alpha=1.5$, $\alpha=2$
and $\alpha=3$ that corresponds to our previous illustrations. Here, there is
only one thing needs to be noticed that $\alpha_{2}=1.5$, which we expect that
its corresponding complexity could show the deviation compared with the case
of the standard dispersion relation.

##  III The squeezed quantum states for cosmological perturbations

The complexity describes the unstable and chaotic features of a statistical
system. The most simple system is the inverted harmonic oscillator, where its
corresponding Hamilton is denoted by $H=\frac{1}{2}p^{2}-\frac{1}{2}kx^{2}$
with $k$ is the frequency and $p$ is the momentum. Thereafter, one can follow
the standard procedure of canonical quantization to investigate its
corresponding complexity. Noticing that the sign between the kinetic term and
the potential term is opposite, in which the Hamilton of curvature
perturbation (can be canonical quantization) has the same situation in
momentum space. Thus, it is naturally implemented the technology of quantum
information to investigate the complexity of curvature perturbation. Ref.
Albrecht:1992kf  has shown that the curvature perturbation can be transited
via the squeezed state, and the wave function of the quantum inverted harmonic
oscillator is the Gaussian distribution. Thus, the squeezed state will be
implemented into the investigation of complexity for the curvature
perturbation of inflation.

According to EOM (7), one can construct its corresponding action as follows

$S=\int d\eta L=\frac{1}{2}\int d\eta d^{3}x\left[{v}^{\prime
2}-f^{2}(\partial_{i}v)^{2}+\frac{{a}^{\prime}}{a}{v}^{2}-2\frac{{a}^{\prime}}{a}{v}^{\prime}v\right],$
(8)

where we do not transform it into the momentum space and $f$ is (6). Once
obtaining the action, one can define its canonical momentum,

$\pi(\eta,\vec{x})=\frac{\delta
L}{\delta{v}^{\prime}(\eta,\vec{x})}={v}^{\prime}-\frac{{a}^{\prime}}{a}v.$
(9)

And the Hamiltonian $H=\int d^{3}x(\pi v^{\prime}-\mathcal{L})$, thus we could
obtain

$H=\frac{1}{2}\int
d^{3}x\left[{\pi}^{2}+f^{2}(\partial_{i}v)^{2}+\frac{{a}^{\prime}}{a}(\pi
v+v\pi)\right].$ (10)

Following the standard procedure of quantum field theory, one promote the
variable $v$ and $\pi$ as Fourier modes,

$\hat{v}(\eta,\vec{x})=\int\frac{d^{3}k}{(2\pi)^{3/2}}\sqrt{\frac{1}{2k}}(\hat{c}_{-\vec{k}}^{\dagger}v_{\vec{k}}^{\ast}(\eta)e^{-i\vec{k\cdot}\vec{x}}+{c_{\vec{k}}}v_{\vec{k}}e^{i\vec{k\cdot}\vec{x}}),$
(11)
$\hat{\pi}(\eta,\vec{x})=i\int\frac{d^{3}k}{(2\pi)^{3/2}}\sqrt{\frac{p}{2}}(\hat{c}_{-\vec{k}}^{\dagger}u_{\vec{k}}^{\ast}(\eta)e^{-i\vec{k\cdot}\vec{x}}-\hat{c}_{\vec{k}}u_{\vec{k}}e^{i\vec{k\cdot}\vec{x}}),$
(12)

where $\hat{c}_{-\vec{k}}^{\dagger}$ and $\hat{c}_{\vec{k}}$ represent the
creation and annihilation operators, respectively. And then we choose an
appropriate normalization condition for mode functions $u_{k}(\eta)$,
$v_{k}(\eta)$, and we can get the following Hamiltonian

$\begin{split}\hat{H}=\int{d^{3}k}\hat{H}_{k}=&\int{d^{3}k}[\frac{k}{2}(f_{s}^{2}+1)\hat{c}_{-\vec{k}}^{\dagger}\hat{c}_{-\vec{k}}+\frac{k}{2}(f_{s}^{2}+1)\hat{c}_{\vec{k}}\hat{c}_{\vec{k}}^{\dagger}\\\
&+(\frac{k}{2}(f_{s}^{2}-1)+i\frac{a{}^{\prime}}{a})\hat{c}_{\vec{k}}^{\dagger}\hat{c}_{-\vec{k}}^{\dagger}\\\
&+(\frac{k}{2}(f_{s}^{2}-1)-i\frac{a{}^{\prime}}{a})\hat{c}_{\vec{k}}\hat{c}_{-\vec{k}}].\end{split}$
(13)

If $f^{2}=1$, it will nicely recover the Hamilton of the standard dispersion
relation, which is the same as sound speed is one as shown in Liu:2021nzx .
Observing that Fourier mode contains $\hat{c}_{-\vec{k}}$ and
$\hat{c}_{\vec{k}^{\dagger}}$, which the unitary operator should be of two
modes. The wave function is Gaussian distribution, one usually implements the
unitary evolution operator acting on this Gaussian-type wave function whose
form can be parameterized in the factorized form Albrecht:1992kf ;
Grishchuk:1990bj  as follows,

$\hat{\mathcal{U}}_{\vec{k}}(\eta,\eta_{0})=\hat{\mathcal{S}}_{\vec{k}}(r_{k},\phi_{k})\hat{\mathcal{R}}_{\vec{k}}(\theta_{k}).$
(14)

In the above equation, $\hat{\mathcal{R}}_{\vec{k}}$ is the two-mode rotation
operator, which could be written in terms of the rotation angle
$\theta_{k}(\eta)$

$\hat{\mathcal{R}}_{\vec{k}}(\theta_{k})=\exp[-i\theta_{k(\eta)}(\hat{c}_{\vec{k}}\hat{c}_{\vec{k}}^{\dagger}+\hat{c}_{-\vec{k}}^{\dagger}\hat{c}_{-\vec{k}})].$
(15)

Meanwhile, $\hat{\mathcal{S}}_{\vec{k}}$ is the two-mode squeeze operator
written in terms of the squeezing parameter $r_{k}(\eta)$ and the squeezing
angle $\phi_{k}(\eta)$, respectively.

$\hat{\mathcal{S}}_{\vec{k}}(r_{k},\phi_{k})=\exp[r_{k}(\eta)(e^{-2i\phi_{k}(\eta)}\hat{c}_{\vec{k}}\hat{c}_{-\vec{k}}-e^{2i\phi_{k}(\eta)}\hat{c}^{\dagger}_{-\vec{k}}\hat{c}^{\dagger}_{\vec{k}})].$
(16)

when $\eqref{rotation operator}$ acts on the initial value of vacuum, it only
generates the irrelevant phase factor which will not impact the evolution of
wave functions, thus it can be neglected. When using squeezed operator (16) to
act on the vacuum $\left|0;0\right\rangle_{\vec{k},-\vec{k}}$, one can obtain
the two-mode squeezed state as follows,

$\left|\psi\right\rangle_{\rm
sq}=\sum_{n=0}^{\infty}(-1)^{n}e^{2in\phi_{k}}\tanh^{n}r_{k}\left|n;n\right\rangle_{\vec{k},-\vec{k}},$
(17)

where $\left|n;n\right\rangle_{\vec{k},-\vec{k}}$ represents the two-mode
excited state, which has the following relationship with the two-mode vacuum
state $\left|0;0\right\rangle_{\vec{k},-\vec{k}}$

$\left|n;n\right\rangle_{\vec{k},-\vec{k}}=\frac{1}{n!}(\hat{c}^{\dagger}_{\vec{k}})^{n}(\hat{c}^{\dagger}_{-\vec{k}})^{n}\left|0;0\right\rangle_{\vec{k},-\vec{k}}.$
(18)

Combine Eqs. (13) (17) with Schr$\ddot{o}$dinger equation,

$i\frac{d}{d\eta}\left|\psi\right\rangle_{\rm
sq}=\hat{H}_{k}\left|\psi\right\rangle_{\rm sq},$ (19)

when we use the Hamilton operator and $i\frac{d}{d\eta}$ acting on the wave
function of squeezed state, the real part and imaginary part would generate
the following equations for $\phi_{k}(\eta)$ and $r_{k}(\eta)$,

$-\frac{dr_{k}}{d\eta}=-\frac{k}{2}(f^{2}-1)\sin(2\phi_{k})+\frac{a{}^{\prime}}{a}\cos(2\phi_{k}),$
(20)
$\begin{split}\frac{d\phi_{k}}{d\eta}&=-\frac{k}{2}(f^{2}+1)+\frac{k}{2}(f^{2}-1)\cos(2\phi_{k})\coth(2r_{k})\\\
&+\frac{a{}^{\prime}}{a}\sin(2\phi_{k})\coth(2r_{k}),\end{split}$ (21)

where Eq. (20) comes via the imaginary part of Schr$\ddot{o}$dinger equation
(19) and Eq. (21) is the real part of Schr$\ddot{o}$dinger equation (19). To
be honest, these two equations are very difficult to solve even for the
numerical simulations, thus we change the variable $\eta$ as $\log_{10}a$, and
then Eqs. (20), (21) will become as follows,

$\begin{split}\frac{10^{y}H_{0}}{\rm
ln10}\frac{dr}{dy}=\frac{k}{2}[(\frac{k_{\rm
ph}}{M})^{2\alpha}-1]\sin(2\phi_{k})-aH_{0}\cos(2\phi_{k}),\end{split}$ (22)
$\begin{split}\frac{10^{y}H_{0}}{\rm
ln10}\frac{d\phi_{k}}{dy}=&-\frac{k}{2}[(\frac{k_{\rm
ph}}{M})^{2\alpha}+1]+aH_{0}\sin(2\phi_{k})\coth(2r_{k})\\\
&+\frac{k}{2}[(\frac{k_{\rm
ph}}{M})^{2\alpha}-1]\cos(2\phi_{k})\coth(2r_{k}),\end{split}$ (23)

where we have defined $y=\log_{10}{a}$. With these two equations, one
numerically simulate $\phi_{k}$ and $r_{k}$ in Figs. (2,3).

Figure 2: The numerical solutions of $r_{k}(\eta)$ in terms of $\log_{10}{a}$
with $\alpha=0$, $\alpha=0.05$, $\alpha=1.5$, $\alpha=2$, and $\alpha=3$. Our
plots adopt $H_{0}=1$.

Figure 3: The numerical solutions of $\phi_{k}(\eta)$ in terms of
$\log_{10}{a}$ with $\alpha=0$, $\alpha=0.05$, $\alpha=1.5$, $\alpha=2$, and
$\alpha=3$. Our plots adopt $H_{0}=1$

.

Fig. 2, it shows that the evolution of $r_{k}$. As $\alpha=0$, it will recover
into the standard dispersion relation, also equal to the sound speed $c_{s}=1$
as shown in Liu:2021nzx ; Bhargava:2020fhl . In Liu:2021nzx , it shows that
the non-trivial sound speed will lead, where the deviation cannot be large
corresponding to $\alpha<1$, to the fast oscillation and lagging the linear
growth. In Fig. 2, it clearly indicates the same trend Liu:2021nzx  when
$\alpha<1$ which the $f$ plays a role of $c_{s}$. The difference comes via the
case of $\alpha>1$, in which it shows that there is the damping of the
oscillation and the damping will be more significant as enhancing the value of
$\alpha$. Another difference is that the time scale for reaching the linear
growth after the horizon exit will be shortened as $\alpha>1$, which these
cases all belong to the UV regime. Physically speaking, the string theory,
trans-Planckian physics, and Lorentz-violating physics will lead to this
damping behavior of $r_{k}$, where it also includes the
Horv$\check{a}$-Lifshitz cosmology Kiritsis:2009sh ; Calcagni:2009ar . In the
later investigation of complexity, we will show that the complexity will be
mainly influenced by $r_{k}$. As for $\phi_{k}$ in Fig. 3, the trend of
various cases is the same where it will grow until reaching a constant. The
only difference is that the constant will be larger as enhancing the value of
$\alpha$ which will not significantly impact the complexity. Being armed with
these two parameters, one can investigate the evolution of the modified
dispersion relation according to Figs. (2, 3). Here, we only consider the tree
level of the quantum perturbations where we have ignored the contribution from
the backreaction of perturbations. However, it may play a significant role
from the inflationary scale to the energy scale below self-reproduction
Mukhanov:1996ak .

##  IV The complexity of modified dispersion relation

Nielson’s geometric method will be implemented for investigating the evolution
of complexity Nielsen2006AGA ; Nielsen2006QuantumCA ; Dowling2008TheGO . For a
given reference state, one can define it as $\left|\psi^{R}\right\rangle$ at
$\tau=0$. At $\tau=1$, the corresponding target state will be connected with
the reference state via the unitary operator as follows,

$\left|\psi^{T}\right\rangle_{\tau=1}=U(\tau=1)\left|\psi^{R}\right\rangle_{\tau=0},$
(24)

where $\tau$ could parameterize the Hilbert space in quantum mechanics.
Following QFT, this unitary operator will be constructed from a path-ordered
exponential of a Hamiltonian operator

$U(\tau)=\overleftarrow{\mathcal{P}}exp\left(-i\int_{0}^{\tau}dsH(s)\right),$
(25)

where $\overleftarrow{\mathcal{P}}$ is the path ordering from right to left.
In this framework, Hamiltonian operator will be formed in terms of a basis of
Hermitian operators $M_{I}$ base, which are the generators for elementary
logic gates

$H(s)=Y(s)^{I}M_{I},$ (26)

where $Y(s)^{I}$ is identified with the control function which determines
which gate will be switched on of switched off. Meanwhile, it also satisfies
with the Schr$\ddot{o}$dinger equation

$\frac{dU}{ds}=-iY(s)^{I}M_{I}U(s),$ (27)

In order to define the complexity, one should introduce the function to
associated with circuit complexity

$C(U)=\int_{0}^{1}\mathcal{F}(U,\dot{U})d\tau.$ (28)

Then, one can obtain the complexity by minimizing the cost function (28).
Afterword, one can find shortest (geodesic) line between the reference state
and target state. First, we will pay attention to the quadratic cost function,

$\mathcal{F}(U,Y)=\sqrt{\sum_{l}(Y^{I})^{2}}.$ (29)

The target wave function is the two-mode squeezed state (17), then one can
transform it into the momentum space and we could obtain this,

$\begin{split}\Psi_{\rm
sq}(q_{\vec{{k}}},q_{-\vec{k}})&=\sum_{n=0}^{\infty}(-1)^{n}\frac{\tanh^{n}r_{k}}{\cosh^{n}r_{k}}\left\langle
q_{\vec{k}};q_{-\vec{k}}|n;n\right\rangle_{\vec{k},-\vec{k}}\\\
&=\frac{\exp[A(r_{k},\phi_{k})\cdot(q^{2}_{k}+q^{2}_{-k})-B(r_{k},\phi_{k})q_{\vec{{k}}}q_{-\vec{k}}]}{\cosh
r_{k}\sqrt{\pi}\sqrt{1-e^{-4i\phi_{k}}}\tanh^{2}r_{k}},\end{split}$ (30)

where $A(r_{k},\phi_{k})$ and $B(r_{k},\phi_{k})$ are

$A(r_{k},\phi_{k})=\frac{k}{2}\left(\frac{e^{-4i\phi_{k}}\tanh^{2}r_{k}+1}{e^{-4i\phi_{k}}\tanh^{2}r_{k}-1}\right),$
(31)
$B(r_{k},\phi_{k})=\frac{k}{2}\left(\frac{e^{-2i\phi_{k}}\tanh^{2}r_{k}}{e^{-4i\phi_{k}}\tanh^{2}r_{k}-1}\right).$
(32)

In vector spaces of $(q_{\vec{{k}}},q_{-\vec{k}})$, Eq. (30) can be written in
terms of diagonal matrix form after some rotation,

$\begin{split}&\Psi_{\rm
sq}(q_{\vec{{k}}},q_{-\vec{k}})=\frac{\exp[-\frac{1}{2}\tilde{M}^{ab}q_{a}q_{b}]}{{\cosh
r_{k}\sqrt{\pi}\sqrt{1-e^{-4i\phi_{k}}}\tanh^{2}r_{k}}},\\\
&\tilde{M}=\begin{pmatrix}\Omega_{\vec{{k}^{\prime}}}&0\\\
0&\Omega_{{-\vec{{k}^{\prime}}}}\end{pmatrix}=\begin{pmatrix}-2A+B&0\\\
0&-2A-B\end{pmatrix},\end{split}$ (33)

Naturally, one can consider the squeezed vacuum state as the reference state.
Then using the same standard procedure to denote it as

$\begin{split}\Psi_{00}(q_{\vec{{k}}},q_{-\vec{k}})&=\left\langle
q_{\vec{k}};q_{-\vec{k}}|0;0\right\rangle_{\vec{k},-\vec{k}}\\\
&=\frac{\exp[-\frac{1}{2}(\omega_{\vec{k}}q^{2}_{\vec{k}}+\omega_{-\vec{k}}q^{2}_{-\vec{k}})]}{\pi^{1/2}}\\\
&=\frac{\exp[-\frac{1}{2}\tilde{M}^{ab}q_{a}q_{b}]}{\pi^{1/2}}\end{split}$
(34)

where

$\tilde{M}=\begin{pmatrix}\Omega_{\vec{{k}^{\prime}}}&0\\\
0&\Omega_{{-\vec{{k}^{\prime}}}}\end{pmatrix}.$ (35)

Note that Eq. (34) is the Gaussian distribution that agreed with our previous
discussion, and meanwhile the wave function (33) is also of the Gaussian
distribution. Thus, the unitary operator (25) will not change the structure of
wave functions. According to (24), one can relate the reference states (34)
(33) to their corresponding target state via (25),

$\Psi_{\tau}(q_{\vec{{k}}},q_{-\vec{k}})=\tilde{U}(\tau)\Psi_{00}(q_{\vec{{k}}},q_{-\vec{k}})\tilde{U}^{\dagger}(\tau),$
(36)

$\Psi_{\tau=0}(q_{\vec{{k}}},q_{-\vec{k}})=\Psi_{00}(q_{\vec{{k}}},q_{-\vec{k}}),$
(37) $\Psi_{\tau=1}(q_{\vec{{k}}},q_{-\vec{k}})=\Psi_{\rm
sq}(q_{\vec{{k}}},q_{-\vec{k}}),$ (38)

where $U(\tau)$ is a $GL(2,C)$ unitary matrix that gives the geodesic line of
parameter space between the reference state and target state. Following
Jefferson:2017sdb , $U(\tau)$ will take the form as follows

$\tilde{U}(\tau)=\exp[\sum_{k=1}^{2}Y^{k}(\tau)M^{\rm diag}_{k}],$ (39)

where ${M^{\rm diag}_{k}}$ denotes two generator of $GL(2,C)$ that defines as

$M^{\rm diag}_{1}=\begin{pmatrix}1&0\\\ 0&0\end{pmatrix},M^{\rm
diag}_{2}=\begin{pmatrix}0&0\\\ 0&1\end{pmatrix}.$ (40)

As we discussed, $U(\tau)$ could parametrize the geodesic line in this group
manifold. Thus, the off-diagonal elements are zero since it will generate the
nontrivial curvature of this group manifold. As for ${Y_{I}(\tau)}$, it could
be constructed from Ali:2018fcz  as follows,

$Y_{I}(\tau)=Y_{I}(\tau=1)\cdot\tau+Y_{I}(\tau=0).$ (41)

From the boundary conditions (37) and (38), it could obtain that

$\rm Im(Y^{1,2})|_{\tau=0}=\rm Re(Y^{I})|_{\tau=0}=0,$ (42) $\rm
Im(Y^{1,2})|_{\tau=1}=\frac{1}{2}\ln\frac{|\Omega_{\vec{k},-\vec{k}}|}{\omega_{\vec{k},-\vec{k}}},$
(43) $\rm Re(Y^{1,2})|_{\tau=1}=\frac{1}{2}\arctan\frac{\rm
Im(\Omega_{\vec{k},-\vec{k}})}{Re(\omega_{\vec{k},-\vec{k}})}.$ (44)

Being armed with these conditions, we can write the complexity as the geodesic
line in the parametric manifold as follows,

$C(\tilde{U})=\int_{0}^{1}d\tau\sqrt{G_{IJ}\dot{Y}^{I}(\tau)\dot{Y}^{I}(\tau)^{*}},$
(45)

where $G_{ij}$ is the induced metric of the group manifold not for the
spacetime. In Ref. Jefferson:2017sdb , it shows that $G_{IJ}$ could have an
arbitrary structure corresponding to various structures of the group manifold.
As we mentioned, our group structure is $GL(2,C)$ whose induced metric is
flat. Therefore, one can substitute Eq. (41) into Eq. (45) for obtaining its
corresponding complexity,

$\begin{split}C(k)&=\frac{1}{2}[(\ln\frac{|\Omega_{\vec{\rm
k}}|}{\omega_{\vec{\rm k}}})^{2}+(\arctan\frac{\rm{Im}(\Omega_{\vec{k}})}{\rm
Re(\omega_{\vec{k}})})^{2}\\\ &+(\ln\frac{|\Omega_{-{\vec{\rm
k}}}|}{\omega_{-{\vec{\rm k}}}})^{2}+(\arctan\frac{\rm
Im(\Omega_{-{\vec{k}}})}{\rm Re(\omega_{-{\vec{k}}})})^{2}],\end{split}$ (46)

where the information of $\phi_{k}$ and $r_{k}$ are including in the
coefficients (31) and (32). Then, one can implement the numeric as shown in
Figs. (2,3) to investigate the evolution of complexity.

Figure 4: The numerical solutions of $\phi_{k}(\eta)$ in terms of
$\log_{10}{a}$ with $\alpha=0$, $\alpha=0.05$, $\alpha=1.5$, $\alpha=2$, and
$\alpha=3$. Our plots adopt $H_{0}=1$. The scrambling time is identified with
the complexity begins to increase and the Lyapunov index can be dubbed as the
complexity becomes the linear growth Ali:2019zcj .

Fig. 4 shows the evolution of complexity varying with scale $\ln_{10}(a)$. The
most essential physical quantities are scrambling time and the Lyapunov index.
In light of Fig. 4, the scrambling time can be defined as the complexity
begins to increase and the Lyapunov index can be dubbed as the slope of
complexity when complexity becomes linear as shown in Ali:2019zcj . In Fig. 4,
we can see that the Lyapunov index with $\alpha>1$ is larger compared with
$\alpha<1$, which physically means that the framework of string theory, trans-
Planckian physics and Lorentz violating physics will lead to the more chaos
comparing with the standard dispersion relation. Our new finding is that the
complex will manifest the nonlinear growth as $\alpha>1$ compared with
$\alpha<1$ not included in Liu:2021nzx ; Bhattacharyya:2020rpy ;
Bhargava:2020fhl ; Lehners:2020pem . Thus, we conjecture that if the inflation
comes via the string theory, trans-Planckian physics, and Lorentz violating
physics, the complexity would show the non-linear evolution after the horizon
exit. As for the scrambling time, the cases of $\alpha>1$ will be shorter
compared with $\alpha<1$ including the standard dispersion relation. Finally,
the oscillation of complexity will occur around the inflationary period, in
which one could see that there is no exact set of rules for their damping
behaviors as $\alpha>1$ which may be influenced by the evolution of $r_{k}$ as
shown in Fig. 2. To sum up, the complexity of $\alpha>1$ will show more
chaotic features compared with $\alpha<1$.

##  V Summary and discussion

The modified dispersion relation (6) can be considered as the consequences of
quantum gravity, including the string cosmology, DBI inflation, cosmology of
loop gravity $\it e.t.c.$ As we discussed in Sec. II, the modified dispersion
relation is applicable in many frameworks. From the perspective of quantum
information, the dispersion relation could show the various patterns of the
evolution of complexity. Based on the above two points, we implement the two-
mode squeezed state to investigate the complexity. First and foremost, we
write down the Hamilton in terms of creation and annihilation operators (13)
in light of the inverted harmonic oscillator. Afterward, we numerically solve
(19) so as to obtain the numeric of $\phi_{k}$ and $r_{k}$ as shown in Figs.
(2,3). Then, we could use them to show how the complexity varies with scale as
shown in Fig. 4.

The main results of this paper can be summarized as follows: $(a)$. Fig. 2
clearly indicates that there will be damping behavior as $\alpha>1$, which
manifests the effects of the modified dispersion relation. And the oscillation
is shorter compared with $\alpha<1$. All of these cases will grow linearly
after the horizon exit. As for $\alpha<1$ including the standard dispersion
relation, it only shows the oscillation before the horizon exit. $(b)$. Fig. 3
indicates that the varying trend of $\phi_{k}$ is the same, where it is
growing first and then it approaches a constant value. The only difference is
that the maximal value of $\alpha$ will be enhanced by improving $\alpha$.
$(c)$. Our new findings are mainly included in Fig. 4. First, it will present
the nonlinear evolution after horizon exit in $\alpha>1$, meanwhile this case
will also tell the irregular damping oscillations comparing with $\alpha<1$.
The slope of complexity when it grows linearly can be identified with the
Lyapunov index, which describes the chaotic features of a statistical system.
Thus, Fig. 4 clearly indicates the Lyapunov index is larger compared with the
standard dispersion relation. Another important quantity describing the
chaotic system is the scrambling time, which can dubbed as the complexity
begins to increase. Obviously, the scrambling time of $\alpha>1$ is shorter
compared with $\alpha<1$. Thus, we conjecture that various frameworks of
quantum gravity will lead to a more fruitful evolution of the complexity.

Our work is based on the single field inflation and the gravitational part
only contains the Hilbert-Einstein action, namely $R$ (Ricci scalar). Thus, it
naturally extends our methods into $f(R)$ gravitational models Liu:2018hno ;
Liu:2018htf  and the multi-field inflationary models Liu:2020zzv ; Liu:2019xhn
; Zhang:2022bde ; Liu:2021rgq . The key place is that we need to develop the
technology for obtaining their quantum Hamilton in terms of creation and
annihilation operators. The k-essence and D-brane naturally contain the
modified dispersion relation (equaled to the non-trivial sound speed in some
sense) Li:2021kfq ; Li:2021sro , thus our analysis could also apply to them.
Further, we can implement the modified dispersion relation to the Krylov
Complexity Adhikari:2022whf ; Adhikari:2022oxr .

## Acknowledgements

LH and TL are funded by NSFC grant NO. 12165009 and Hunan Natural Science
Foundation NO. 2023JJ30487.

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