A flat FRW model with dynamical $\Lambda$ as function of matter and geometry
Anirudh Pradhan1, De Avik 2, Tee How Loo3, Dinesh Chandra Maurya4

1Department of Mathematics, Institute of Applied Sciences & Humanities, GLA
University,

Mathura -281 406, Uttar Pradesh, India

2Department of Mathematical and Actuarial Sciences, Universiti Tunku Abdul
Rahman, Jalan Sungai Long, 43000 heras, Malaysia

3Institute of Mathematical Sciences, University of Malaya, 50603 Kuala Lumpur,
Malaysia

4Department of Mathematics, Faculty of Engineering & Technology, IASE (Deemed
to be University), Sardarshahar-331 403, Rajsthan, India

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###### Abstract

We revisit the evolution of the scale factor in a flat FRW spacetime with a
new generalized decay rule for the dynamic $\Lambda$-term under modified
theories of gravity. It analyses certain cosmological parameters and examines
their behaviours in this generalized setting which includes several decay laws
in the literature. We have also obtained observational constraints on various
model parameters and estimated the present values of cosmological parameters
$\\{\Omega_{m_{0}}$, $\Omega_{\Lambda_{0}}$, $q_{0},t_{0}$, $\omega_{0}\\}$
and have discussed with various observational results. Finite time past and
future singularities in this model are also discussed.

Keywords FLRW universe, Modified gravity, Dynamical cosmological constant,
Observational parameters
PACS No.: 98.80.Jk; 98.80.-k; 04.50.-h

##  1 Introduction

The idea of the cosmological constant $\Lambda$ was first introduced by
Einstein in 1917 to obtain a static universe because of Einstein’s field
equations in general relativity were showing a non-static universe i.e. the
universe has either contracting or expanding nature and at that time
observational studies supported an static universe due to presence limited
observational data sets. Therefore, he revised his original field equations
proposed in 1915 to remove this feature of size-changing. But the discovery of
Hubble in 1929 suggested that our universe is in expanding nature which was
already obtained by the Friedmann-Lamatire-Rabortson-Walker in 1922 in their
solution of Einstein’s field equations using the same metric and this finding
was once removed the newly added cosmological term $\Lambda$.

After the discovery of accelerating universe in 1998, the cosmological term
came back in its place but at this time it did not prevent the expanding
nature of universe, but in order to explain why it did not continues expanding
nature with a constant rate. Although it should be possible for the known
amount of matter present in the universe to exert adequate gravitational force
to slow down the expansion, initiated with Big Bang. However, modern studies
of standard candles showed that instead of decreasing, the rate at which it is
expanding actually increases. The new generation of cosmologists proposes the
existence of an unknown, exotic, homogeneous energy density with high negative
pressure referred to as dark energy (DE) to explain these observational
findings. And the idea of cosmological term $\Lambda$ is the simplest
contributing factor of this dark energy.

We still face the critical question of whether $\Lambda$ is always a
fundamental constant or a mild dynamical variable constant after almost two
decades. It turns out that the selection of $\Lambda$ as dark energy faces
some serious issues in terms of (a) the “old” cosmological constant problem
(or fine-tuning problem) and (b) the cosmic coincidence problem, if we
consider it as a constant rather than a specific dynamical variable. Moreover,
the overall fit to the cosmological observables SNIa+BAO+H(z)+LSS+BBN+CMB has
also been shown to support the class of unique dynamic models of the so-called
fundamental constant $\Lambda$ as well. Dynamic model of $\Lambda$ may
ultimately be necessary not only as a new epitome to improve the EFE, but also
phenomenologically to unwind a number of tensions between the concordance
model and the observation that is indicating towards DE. There are innumerable
$\Lambda(t)$ laws available in the literature although all of which might not
be viable under theoretical and observational ground and they might not follow
covariant action, but they are still interesting to study from
phenomenological ground. For example, Carvalho et al. [1] and Waga [2] first
considered the case $\Lambda\propto(\frac{\dot{a}}{a})^{2}$ as a
generalization of Chen and Wu’s result advocating the possibility that the
(effective) cosmological constant $\Lambda$ varies in time as $a^{-2}$. The
case $\Lambda\propto\rho$ was first studied by Viswakarma [3] and the case
$\Lambda\propto\frac{\ddot{a}}{a}$ by Arbab [4]. Ray et al. [5] and Viswakarma
[6] showed that if we consider these three cases separately, the free
parameters satisfy certain relations. Recently, Pan [7] investigated dynamical
$\Lambda$ as a function of several form of polynomials of $H$ and $\dot{H}$,
discussing the finite time future singularities for each case.

Although several studies being done on these particular phenomenological
models in the past, with more powerful observational analysis of the present
days, most of these are detected to be suffering from certain theoretical and
observational limitations. The $\Lambda\propto H^{2}$ and
$\Lambda\propto\dot{H}$ cases are largely overruled now since the
corresponding linear growth of cosmic perturbations is strongly disfavored by
the observational data [8], [9]. If $\Lambda\propto\rho$ alone, one can
trivially absorb it to the constant and hence there will be no dark energy
scenario in the cosmological evolution. This is indeed very clear for a
perfect fluid with equation of state $p=\omega\rho$. The introduction of
$\Lambda\propto\rho$ effectively modifies the proportionality constant
$\omega$ and the model becomes equivalent to one with merely a new fluid only.
We tried to overcome this issue by introducing a new decay law which combines
both matter and geometry into the dynamical cosmological term. Moreover, we
can show that a motivation towards the present decay law of $\Lambda$ can be
drawn from the f(R)-gravity or equivalently from Brans-Dicke scalar-tensor
theory derived from a covariant action.

In cosmology, the fundamental variables are either constructed from a
spacetime metric directly (geometrical), or depend upon properties of physical
fields. Naturally the physical variables are model-dependent, whereas the
geometrical variables are more universal. The most popular geometric variables
are the Hubble constant $H$ and the deceleration parameter $q$ whose values
depend on the first and second time derivatives of the scale factor,
respectively. We use these two parameters extensively to resolve the mystery
of the inflationary universe. However, a more sensitive discrimination of the
expansion rate and hence dark energy can be performed by considering a third
time derivative in the general form for the scale factor of the Universe

$a(t)=a(t_{0})+\dot{a}|_{0}(t-t_{0})+\frac{\ddot{a}|_{0}}{2}(t-t_{0})^{2}+\frac{\dddot{a}|_{0}}{3}(t-t_{0})^{2}+\dots$

This led Sahni et al. [10] and Alam et al. [11] to introduce the statefinder
parameters, a dimensionless pair $\\{r,s\\}$ from the scale factor and its
third order time derivatives in concordance cosmology. In [12] this result is
generalized to show that the hierarchy, $A_{n}=\frac{a^{(n)}}{aH^{n}}$, can
take into account the deceleration parameter, the statefinder parameters, the
snap parameter, the lerk parameter etc and further it can be expressed in
terms of the deceleration parameter or the matter density parameter.

In observational cosmology, the beginning of the 21st century saw yet another
significant milestone. BOOMERanG in the year 2000 collaboration (Balloon
Measurements of Millimetric Extragalactic Radiation and Geophysics), a
research on cosmic microwave history (CMB) a cosmos of flat spatial geometry
([13], [14]) was recorded using balloon-borne instruments. The MAXIMA
(Millimeter Anisotropy Experiment Imaging Array) collaboration [15, 16], DASI
(Degree Angular Scale Interferometer) [17], CBI (Cosmic Background Imager)
[18] and WMAP (Wilkinson Microwave Anisotropy Probe) [19] have also reported a
similar result. These observations were an significant landmark in modern
cosmology; assuming astronomy’s value of $\Omega M\sim 0.3,$ the data directly
pointed to a positive cosmological constant with a contribution of energy
density of the order of $\Omega\Lambda\sim 0.7.$ The observations from the
supernova probes and the Hubble Space Telescope were ideally matched. The
precise values of these parameters $\Omega_{M}$ and $\Omega_{\Lambda}$ as
obtained by Sievers et al. [18] and Spergel et al. [20] are $[0.34\pm
0.12,0.67^{+0.10}_{-0.13}]$ and
$[0.249^{+0.024}_{-0.031},0.719^{+0.021}_{-0.029}]$, respectively. More
recently, the Lyman-$\alpha$ forest measurement of the baryon acoustic
oscillations (BAO) by the Baryon Oscillation Spectroscopic Survey preferred an
even smaller value of the matter density $\Omega_{M}$ than that obtained by
cosmic microwave background data [21]. We show that our model perfectly adapts
to these observational data.

The present paper is organized in the following format: a brief review of
literature is given in section-1, section-2 contains field equations with
cosmological term $\Lambda$ and some specific cosmological solutions like
scale factor $a(t)$, Hubble function $H(t)$, energy density parameters
$\Omega_{m}$ and $\Omega_{\Lambda}$. An observational constraints on various
model parameters using available observational data sets like $H(z)$, union
2.1 compilation of SNe Ia data sets and Joint Light Curve Analysis (JLA) etc.
by applying $R^{2}$-test formula are obtained and discussed with various
observational results in section-3. Some finite time singularities of the
current model are discussed in the section-4 and in section-5 we have
discussed some more cosmological parameters like statefinder with deceleration
parameters. Finally conclusions are given in section-6.

##  2 Field equations with $\Lambda(t)$

We consider an action [22]

$S=\frac{1}{16\pi G}\int f(R)\sqrt{-g}d^{4}x+\int L_{m}\sqrt{-g}d^{4}x,$

where $f(R)$ is an arbitrary function of the Ricci scalar $R$, $L_{m}$ is the
matter Lagrangian density, and we define the stress-energy tensor of matter as
$T_{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}L_{m})}{\delta
g^{\mu\nu}}$.

Assuming that the Lagrangian density of matter $Lm$ depends only on the
components of the metric tensor $g{\mu\nu}$ and not on its components, we
obtain derivatives,

$T_{\mu\nu}=g_{\mu\nu}L_{m}-2\frac{\partial L_{m}}{\partial g^{\mu\nu}}.$

By varying the action $S$ of the gravitational field with respect to the
metric tensor components $g^{\mu\nu}$ and using the least action principle we
obtain the field equation

$f_{R}(R)R_{\mu\nu}-\frac{1}{2}f(R)g_{\mu\nu}+(g_{\mu\nu}\Box-\nabla_{\mu}\nabla_{\nu})f_{R}(R)=8\pi
G\,T_{\mu\nu},$ (1)

where $\Box=\nabla^{2}=\nabla_{\mu}\nabla^{\mu}$ represents the d’Alembertian
operator and $f_{R}=\frac{\partial f(R)}{\partial R}$. EFE can be reawakened
by putting $f(R)=R$. The contracted equation is given by

$Rf_{R}(R)-2f(R)+3\Box f_{R}(R)=8\pi GT,$ (2)

which transforms (1) to

$R_{\mu\nu}-\frac{R}{2}g_{\mu\nu}+\frac{1}{6f_{R}(R)}[2kT+f(R)+Rf_{R}(R)]g_{\mu\nu}=8\pi
G\,T^{\text{eff}}_{\mu\nu},$ (3)

$T^{\text{eff}}_{\mu\nu}=\frac{1}{f_{R}(R)}[T_{\mu\nu}+\frac{1}{8\pi\,G}\nabla_{\mu}\nabla_{\nu}f_{R}(R)]$.

The Friedmann-Robertson-Walker (FRW) metric is given by the line element

$ds^{2}=-dt^{2}+a^{2}(t)\left[\frac{dr^{2}}{1-kr^{2}}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})\right],$ (4)

where the curvature parameter $k=-1,0,1$ for open, flat and closed models of
the universe, respectively and $a(t)$ is the scale factor shaping the
universe. Since current observations from CMB detectors such as BOOMERanG,
MAXIMA, DASI, CBI and WMAP confirm a spatially flat universe, $k=0$. Hence the
Ricci scalar becomes $R=6((\frac{\dot{a}}{a})^{2}+\frac{\ddot{a}}{a})$.
Comparing the equation (3) with the Einstein’s field equations with
cosmological constant

$R_{\mu\nu}-\frac{R}{2}g_{\mu\nu}+\Lambda g_{\mu\nu}=8\pi GT_{\mu\nu},$ (5)

we conclude that the dynamic cosmological term $\Lambda(t)$ is possibly a
function of $\frac{\ddot{a}}{a},\,(\frac{\dot{a}}{a})^{2}$ and $\rho$ and we
consider the simplest model as a linear combination

$\Lambda=l\frac{\ddot{a}}{a}+\lambda\left(\frac{\dot{a}}{a}\right)^{2}+4\pi
G\eta\rho,$ (6)

where $l$, $\lambda$ and $\delta$ are arbitrary constants.

As usual, the Friedmann and Raychaudhuri equations for flat FRW line element
are given by

$3\frac{\ddot{a}}{a}+4\pi G(1+3w)\rho=\Lambda,$ (7)

$\frac{\ddot{a}}{a}+2\left(\frac{\dot{a}}{a}\right)^{2}-4\pi
G(1-w)\rho=\Lambda,$ (8)

where $w=\frac{p}{\rho}$.

From Eqs. (7) and (8) we obtain

$\left(\frac{\dot{a}}{a}\right)^{2}-\frac{8\pi G\rho}{3}=\frac{\Lambda}{3}.$
(9)

Using Eq. (6), in terms of Hubble parameter $H=\frac{\dot{a}}{a}$, Eq. (9)
transforms into

$3H^{2}-8\pi G\rho=l\dot{H}+(l+\lambda)H^{2}+4\pi G\eta\rho.$ (10)

Therefore,

$4\pi G\rho=-\frac{l}{2+\eta}\dot{H}+\frac{3-l-\lambda}{2+\eta}H^{2}$ (11)

and

$\Lambda=\frac{2l}{2+\eta}\dot{H}+\frac{2l+2\lambda+3\eta}{2+\eta}H^{2}.$ (12)

From Eq. (8) we obtain

$\frac{\dot{H}}{H^{2}}=-\frac{(1+w)(3-l-\lambda)}{(2+\eta)-l(1+w)},$ (13)

which gives us

$H=\frac{H_{0}[(2+\eta)-l(1+w)]}{H_{0}(1+w)(3-l-\lambda)(t-t_{0})+(2+\eta)-l(1+w)}.$
(14)

Integrating again, we obtain

$\frac{a(t)}{a_{0}}=\left[1+\frac{H_{0}(1+w)(3-l-\lambda)}{(2+\eta)-l(1+w)}(t-t_{0})\right]^{\frac{(2+\eta)-l(1+w)}{(1+w)(3-l-\lambda)}}.$
(15)

Therefore, the vacuum energy density parameter
($\Omega_{\Lambda}=\frac{\Lambda}{3H^{2}}$) and the cosmic matter density
parameter ($\Omega_{m}=\frac{8\pi G\rho}{3H^{2}}$) are calculated as

$3\Omega_{\Lambda}=\frac{2l+2\lambda+3\eta-3l(1+w)}{(2+\eta)-l(1+w)}$ (16)

and

$3\Omega_{m}=2\frac{3-l-\lambda}{(2+\eta)-l(1+w)},$ (17)

respectively, which confirms

$\Omega_{m}+\Omega_{\Lambda}=1.$

##  3 Observational constraints

The Hubble parameter $H$ is one of the key parameter in observational
cosmology as well as in theoretical cosmology and it measures expansion rate
of the universe. In our paper, we have obtained the Hubble function using Eqs.
$(14)$ & $(15)$ as

$H(t)=H_{0}\left(\frac{a_{0}}{a(t)}\right)^{\frac{(1+\omega)(3-l-\lambda)}{2+\eta-l(1+\omega)}}$
(18)

Several observational data sets are available in terms of redshift $z$ and
hence, to obtain constraints on various model parameters we have to convert
the above relationship in terms of the redshift. Therefore, using the relation
$\frac{a_{0}}{a(t)}=1+z$ between scale factor $a(t)$ and redshift $z$, we
obtain:

$H(z)=H_{0}(1+z)^{\frac{(1+\omega)(3-l-\lambda)}{2+\eta-l(1+\omega)}}$ (19)

We can also obtain the expression for luminosity distance $D_{L}$ and apparent
magnitude $m(z)$ in the purpose of observational constraints (Union 2.1
compilation of SNe Ia data sets and Joint light curve analysis data sets)
respectively as

$D_{L}=\frac{c}{H_{0}}\frac{2+\eta-l(1+\omega)}{2+\eta+(\lambda-3)(1+\omega)}(1+z)^{\frac{4+2\eta+(\lambda-l-3)(1+\omega)}{2+\eta-l(1+\omega)}}$
(20)

and

$m(z)=16.08+5\times
log10\left(\frac{1}{0.026}\frac{2+\eta-l(1+\omega)}{2+\eta+(\lambda-3)(1+\omega)}(1+z)^{\frac{4+2\eta+(\lambda-l-3)(1+\omega)}{2+\eta-l(1+\omega)}}\right)$
(21)

Now, we consider the $29$ Hubble data sets $H(z)$ [23]-[29], $580$ data sets
of apparent magnitude $m(z)$ from union 2.1 compilation of SNe Ia data sets
[30] and $51$ data sets of $m(z)$ from Joint Light Curve Analysis (JLA) data
sets [31] and using the $R^{2}$-test formula, we obtain the best fit values of
various model parameters $l$, $\lambda$, $\eta$, $\omega$ and $H_{0}$ for the
best fit curve of Hubble function $H(z)$ and apparent magnitude $m(z)$. The
case $R^{2}=1$ means the curve of the Hubble function is exact match with
observational curve and using this technique of fitting we have obtained the
constraints on model parameters for maximum $R^{2}$ values. The best fit
values of $l$, $\lambda$, $\eta$, $\omega$ and $H_{0}$ for the various data
sets are mentioned in below Table-1 & 2 with their maximum $R^{2}$ value and
root mean square error (RMSE).

Parameters(For $\omega=0$) | $H(z)$ | SNe Ia | JLA
---|---|---|---
$l$ | $1.435$ | $0.1707$ | $0.2171$
$\lambda$ | $0.8113$ | $0.2855$ | $0.7471$
$\eta$ | $0.1044$ | $1.119$ | $0.8394$
$H_{0}$ | $59.22$ | $-$ | $-$
$R^{2}$ | $0.9115$ | $0.9938$ | $0.9214$
$RMSE$ | $13.2672$ | $0.2486$ | $0.2782$
Table 1: Case-I dusty universe: The best fit values for the model parameters $l$, $\lambda$, $\eta$ and $H{0}$ are used in the best fit curve of the Hubble function $H(z)$ and the apparent magnitude $m(z)$ with different observational data sets ($H(z)$, JLA, SNe Ia) with confidence level of constraints of $95\%$.  Parameters | $H(z)$ | SNe Ia | JLA
---|---|---|---
$\omega$ | $0.1256$ | $0.1156$ | $0.1218$
$l$ | $1.764$ | $0.1881$ | $0.1137$
$\lambda$ | $0.7351$ | $0.8632$ | $0.9196$
$\eta$ | $0.4916$ | $0.7292$ | $0.9696$
$H_{0}$ | $59.83$ | $-$ | $-$
$R^{2}$ | $0.9114$ | $0.9938$ | $0.9214$
$RMSE$ | $13.5632$ | $0.2488$ | $0.2812$
Table 2: Case-II $0<\omega<\frac{1}{3}$: The best fit values for the model parameters $l$, $\lambda$, $\eta$ and $H{0}$ are used for the best fit curve for the Hubble function $H(z)$ and the apparent magnitude $m(z)$ for different observational data sets ($H(z)$, JLA, SNe Ia) for confidence level of boundaries of $95\%$.  Parameters (for $\omega=0$) | $H(z)$ | SNe Ia | JLA
---|---|---|---
$\Omega_{m}$ | $0.7506224480$ | $0.5752015285$ | $0.5175609198$
$\Omega_{\Lambda}$ | $0.2493775520$ | $0.4247984713$ | $0.4824390803$
$q$ | $0.125933672$ | $-0.1371977072$ | $-0.2236586203$
$t_{0}$ | $14.56595745$ | $18.74417427$ | $20.62684904$
$r$ | $0.1576522515$ | $-0.09955128548$ | $-0.1236122635$
$s$ | $0.7506224480$ | $0.5752015286$ | $0.5175609198$
Table 3: Case-I dusty universe: The present values of various parameters obtained for different data sets. Parameters | $H(z)$ | SNe Ia | JLA
---|---|---|---
$\omega$ | $0.1256$ | $0.1156$ | $0.1218$
$\Omega_{m}$ | $0.6598930470$ | $0.5156609542$ | $0.4613334442$
$\Omega_{\Lambda}$ | $0.3401069530$ | $0.4843390457$ | $0.5386665557$
$q$ | $-0.3115722925$ | $-0.2331273858$ | $-0.3113554396$
$t_{0}$ | $14.54821996$ | $18.53672071$ | $20.41654688$
$r$ | $-0.1174177056$ | $-0.1244306298$ | $-0.1174710201$
$s$ | $0.4589518050$ | $0.5112484095$ | $0.4590963736$
Table 4: Case-II of $0<\omega<\frac{1}{3}$: The present values of various
parameters obtained for different data sets.

Using the above estimated values of various parameters (as mentioned in
Table-1 & 2) we have calculated the values of matter energy density parameter
$\Omega_{m}$, dark energy density parameter $\Omega_{\Lambda}$, deceleration
parameter $q$, age of the present universe and statefinder parameters $r$, $s$
for two types of universes one is dusty universe and second is the case of
$0<\omega<\frac{1}{3}$ which are mentioned in Table-3 & 4 given below.

Table-3 & 4 shows the present values of various cosmological parameters
$\\{\Omega_{m_{0}},\Omega_{\Lambda_{0}},q_{0},t_{0},r_{0},s_{0}\\}$ and we see
that for the deceleration in expansion of the universe dark energy parameter
must be $\Omega_{\Lambda_{0}}>0.2493775520$ and matter energy density
parameter $\Omega_{m_{0}}<0.7506224480$. In our model we have calculated the
present value of deceleration parameter $q_{0}$ lies between
$-0.3115722925\leq q_{0}\leq 0.125933672$. From these results we concludes
that our present universe is in accelerating phase.

a. b.

Figure 1: The plot of Hubble function $H(z)$ versus redshift $z$ for the best
fit values of model parameters $l$, $\lambda$, $\eta$ and $H_{0}$ in both
cases of $\omega$.

Figures $1a$ & $1b$ represent the best fit curve of the Hubble function $H(z)$
for the best fit values of the model parameters as well as the best fit values
of the model parameters where the curvature parameter $k=-1,0,1$ for the open,
flat and closed models displayed in Tables-$1$ & $2$ for the three data sets
as shown in the figures, respectively. Figures $2a$ & $2b$ depict the
behaviour of cosmological constant $\Lambda(z)$ over redshift $z$ and we see
that it is an increasing function of $z$ which is consistent with
observational findings. Figures $3a$ & $3b$ show the age of the universe
defined as $(t_{0}-t)=\int_{0}^{z}\frac{dz}{(1+z)H(z)}$ for both cases of
$\omega$ and we have calculated the age of the present universe $t_{0}$ as
$14.54821996\leq t_{0}\leq 20.62684904$ Gyrs that is consistent with the
observational results $t_{0}=13.6$ Gyrs.

a. b.

Figure 2: The plot of Cosmological constant $\Lambda(z)$ versus redshift $z$
for the best fit values of model parameters $l$, $\lambda$, $\eta$ and $H_{0}$
in both cases of $\omega$.

a. b.

Figure 3: The plot of Cosmic time $(t_{0}-t)H_{0}$ versus redshift $z$ for the
best fit values of model parameters $l$, $\lambda$, $\eta$ and $H_{0}$ in both
cases of $\omega$.

From Table-3 & 4 we see that the value of total energy density parameter
$0.9999999998\leq\Omega\leq 1$. These findings are consistent with the
observational constraints obtained by MAXIMA-1 flight and COBE-DMR experiments
[32], $\Omega=1.05\pm 0.08$from CBI-DMR observations [18] and $\Omega=1.05\pm
0.08$on the total energy density parameter $\Omega=1.00^{+0.25}{-0.30}$.
$\Omega=1.01\pm 0.03$calculated from the first acoustic peak in the CMB
fluctuation angular power spectrum [33].

Moreover, we can obtain ranges of $l_{0},\lambda_{0}$ and $\eta_{0}$ (the
values of $l,\lambda,\eta$ in the present epoch) in the dust case ($w=0$) from
the above equations. The first flight of the MAXIMA combined with COBE-DMR
resulted in $0.25<\Omega_{m_{0}}<0.50$ and $0.45<\Omega_{\Lambda_{0}}<0.75$.
Observations of SNeIa combined with the total energy density constraints from
CMB and combined gravitational lens and steller dynamical analysis lead to
$\Omega_{m_{0}}\sim 0.3$ and $\Omega_{\Lambda_{0}}\sim 0.7$. Sievers et al.
and Sperger et al. obtained the values of
$[\Omega_{m_{0}},\Omega_{\Lambda_{0}}]=[0.34\pm 0.12,0.67^{+0.10}_{-0.13}]$
and $[0.249^{+0.024}_{-0.031},0.719^{+0.021}_{-0.029}]$ respectively [18, 20].
Considering the varied observational dataset, many researchers prefer a range
of $0.295\leq\Omega_{m_{0}}\leq 0.365$ [5].

For dust case, from (17) we obtain
$3\Omega_{m}=2\frac{3-l-\lambda}{2+\eta-l}$. If we consider the case
$\lambda=0=\eta$, we obtain $3.7937\leq l_{0}\leq 4.2099$. We note that using
our model we are able to obtain the range of $l$ smaller than that of Arbab’s
[34] which was $(3,4.5)$ and Ray’s [5] which was $[3.417,4.674]$. On a similar
fashion we can find ranges of $1.905\leq\lambda_{0}\leq 2.115$ and
$3.479\leq\eta_{0}\leq 4.780$. So we see that using our model, we can
generalize the previous results and also obtain the respective range of the
parameters.

##  4 Finite time singularities in the current model

We investigate equation (15) for possible finite time past or future cosmic
singularities present in the current model. We consider distinct cases
depending on the sign of $(2+\eta)-l(1+w)$.

Case 1: $\frac{2+\eta}{l}>1+w$.

If $(1+w)(3-l-\lambda)>0$, then there is a finite past singularity at
$t_{p}=t_{0}-\frac{2+\eta-l(1+w)}{H_{0}((1+w)(3-l-\lambda))}<t_{0}$ whereas if
$(1+w)(3-l-\lambda)<0$, then the scale factor diverges at some finite future
time $t_{f}=t_{0}-\frac{2+\eta-l(1+w)}{H_{0}((1+w)(3-l-\lambda))}>t_{0}$.

Case 2: $\frac{2+\eta}{l}<1+w$.

If $(1+w)(3-l-\lambda)<0$, then there is a finite past singularity at
$t_{p}=t_{0}-\frac{2+\eta-l(1+w)}{H_{0}((1+w)(3-l-\lambda))}<t_{0}$ whereas if
$(1+w)(3-l-\lambda)>0$, then the scale factor diverges at some finite future
time $t_{f}=t_{0}-\frac{2+\eta-l(1+w)}{H_{0}((1+w)(3-l-\lambda))}>t_{0}$. From
this discussion it is clear that we can control the free parameters
$l,\lambda,\eta$ to avoid any future singularity in the system.

##  5 Some cosmological parameters

The deceleration parameter
$q=-\frac{a\ddot{a}}{\dot{a}^{2}}=-\frac{\dot{H}}{H^{2}}-1$ in cosmology is a
dimensionless measure of the cosmic acceleration of the expansion of space.
The statefinder parameters $(r,s)$ is defined originally by

$r=\frac{\dddot{a}}{aH^{3}},\qquad s=\frac{r-1}{3(q-1/2)}.$

We may link the $(r,s)$ parameters to the Hubble parameters $H$ and the
deceleration parameter $q$ for our present model as

$r=2\left(\frac{\dot{H}}{H^{2}}\right)^{2}+3\frac{\dot{H}}{H^{2}}+1=q(2q+1)$
(22)

$s=-\frac{2\dot{H}}{3H^{2}}=\frac{2}{3}(q+1).$ (23)

We also obtain a relation between $r$ and $s$ as

$r=\frac{9}{2}s(s-1)+1.$ (24)

In general, the statefinder hierarchy $A_{n}=\frac{a^{(n)}}{aH^{n}}$ for our
present model can be calculated as

$\displaystyle A_{2k+1}=q(2q+1)(3q+2)\dots(2kq+2k-1),$ $\displaystyle
A_{2k}=-q(2q+1)(3q+2)\dots((2k-1)q+2k-2).$ (25)

In this regard, it is to be mentioned that these handy relations between the
various cosmological parameters as obtained in this section are valid for any
time independent $w$ and with any of the decay laws considered in [1, 2, 3, 4]
or any of their linear combinations.

For a $\Lambda$CDM model the statefinder pair ($r,s$) have the value ($1,0$).
In our present model we obtain this if and only if $\frac{\dot{H}}{H^{2}}=0$
which means the scale factor $a(t)$ increases exponentially with time. It also
implies that $q=-1$, which is true if $\alpha+\beta=3$, but this condition
also provide $\Lambda$ is truly a constant.

Dust Case:

Below we obtain some important results for the particular case $w=0$.

First and foremost, we have

$q=\frac{3-l-\lambda}{2+\eta-l}-1.$ (26)

Hence for an accelerating universe ($q<0$) we obtain a relation
$\lambda+\eta>1$.

We notice that for dust case, the standard model formula $\Omega_{m}=2q$ is
now replaced by $\Omega_{m}=\frac{2}{3}q+\frac{2}{3}$, same as the result of
Arbab [4]. However, both models give $q=1/2$ for a critical density
$\rho_{c}=\frac{3H^{2}}{8\pi G}$. We also obtain $s=\Omega_{m}$.

Furthermore, using equations (13) and (16) we obtain that the deceleration
parameter $q$ and the vacuum density parameter $\Omega_{\Lambda}$ are
connected by the relation

$3\Omega_{\Lambda}=1-2q+2w\frac{3-l-\lambda}{(2+\eta)-l(1+w)},$ (27)

and so for dust case, an accelerating universe $(q<0)$ requires
$\Omega_{\Lambda}>1/3$, which perfectly fits the modern observational data as
the present accepted value of $\Omega_{\Lambda}$ is 0.7, much larger than 1/3.
Hence our model fits an accelerating universe.

##  6 Conclusion

We have considered a spatially homogeneous and isotropic spacetime in the
presence of a dynamic cosmological term $\Lambda(t)$ satisfying
$\Lambda=l(\frac{\dot{a}}{a})^{2}+\lambda\frac{\ddot{a}}{a}+4\pi G\eta\rho$
motivated by modification of the Einstein-Hilbert action. This generalizes
several results of the previous models in flat FRW spacetime and also enables
us to find the ranges for $l,\lambda,\eta$ from the observational data apart
from the interrelation we obtain between these parameters for accelerating
universe ($q<0$). We explore the recently defined statefinder hierarchy for
this generalized model of ours and prove some effective results for any time
independent $w$. Furthermore, we show that our model fits in perfectly with
the modern observational datasets.

We have found that the scale factor $a(t)$ vanishes at
$t=t_{0}-\frac{\\{(2+\eta)-l(1+w)\\}}{H_{0}(1+w)(3-l-\lambda)}$ while for this
value of time the Hubble function $H$ and the cosmological term $\Lambda$ get
infinitely large values. We have found that with increasing time or decreasing
redshift the scale factor increases but the $H$ and $\Lambda$ decreases to a
finite small values in late time universe. These behaviours of cosmological
parameters reveal that our universe model starts with a finite time big-bang
singularity stage and goes on expanding till late time. We have also noticed
that the total energy density parameter $\Omega\leq 1$ which indicates towards
the closed and flat geometry of the universe and it was compatible with
several observational results. We have also found a constraint on dark energy
density parameter $\Omega_{\Lambda_{0}}>\frac{1}{4}$ to provide an
acceleration in expansion of the universe which is compatible with the result
$\Omega_{\Lambda_{0}}>\frac{1}{3}$. We have also calculated the age of the
present universe in the range $14<t_{0}<21$ Gyrs.

## Acknowledgments

De Avik and Tee How Loo are supported by the grant FRGS/1/2019/STG06/UM/02/6.
A. Pradhan would like to express his appreciation to the Inter-University
Centre for Astronomy & Astrophysics (IUCAA), India for supporting under the
visiting associateship.

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# Strong coupling to circularly polarized photons: towards field-induced
enantioselectivity

Rosario R. Riso,1 Enrico Ronca ${}^{2^{\ast}}$ and Henrik
Koch${}^{1,3^{\ast}}$
1Department of Chemistry, Norwegian University of Science and Technology,
7491 Trondheim, Norway
2Department of Chemistry, Biology and Biotechnology, University of Perugia,
Via Elce di Sotto, 8, 06123, Perugia, Italy
3Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy

∗ E-mail<EMAIL_ADDRESS>and<EMAIL_ADDRESS>
> The development of new methodologies for the selective synthesis of
> individual enantiomers is still one of the major challenges in synthetic
> chemistry. Many biomolecules, and therefore many pharmaceutical compounds,
> are indeed chiral. While the use of chiral reactants or catalysts has led to
> significant progress in the field of asymmetric synthesis, a systematic
> approach applicable to general reactions has still not been proposed. In
> this work, we show that strong coupling to circularly polarized fields
> represents a promising alternative for reaching highly selective asymmetric
> synthesis in a non-invasive fashion. We demonstrate that the field induces
> stereoselectivity in the early stages of the chemical reaction, by selecting
> an energetically preferred direction of approach for the reagents.

## Introduction

Asymmetric synthesis is a formidable challenge in chemistry, as demonstrated
by the 2021 Nobel prize in chemistry to List and MacMillan (?, ?, ?, ?, ?).
Most biologically relevant molecules belong to a special class of systems that
cannot be superimposed with their own mirror image. These molecules, known as
chiral molecules, exist in two possible configurations called enantiomers,
which have identical energy levels and share most physical properties. As a
result, achieving a selective synthesis of only one of the two enantiomers is
a complicated task and reactions often produce a mixture of enantiomers
instead of an enantiomerically pure product. This is a major problem in the
pharmaceutical industry since most often only one enantiomer has the desired
healing behavior, while the other may be ineffective or even harmful (?).
Living organisms have developed remarkable techniques to produce only one of
the two configurations through the use of enzymes and catalysts (?, ?).
Therefore, the most successful research lines in asymmetric synthesis have so
far mostly focused in this direction (?, ?). However, catalysts are often
reaction specific and a systematic approach is yet to be developed (?). Strong
coupling between molecules and vacuum photons fields has recently emerged as a
promising technique to engineer molecular properties in a non-invasive way (?,
?, ?, ?, ?). When light and matter interact strongly, for example inside
optical cavities, they form new hybrid electron-photon states with unique
properties, the polaritons (?, ?, ?). Intriguing applications of polaritonic
chemistry include modifying molecular absorption and fluorescence (?, ?, ?),
altering photochemical processes (?) and even speeding up or slowing down
chemical reactions (?, ?, ?, ?). In a previous work, we demonstrated that
circularly polarized fields can induce energy differences between enantiomers,
even in the ground state. This result stems from the fact that the two
enantiomers exhibit differential absorption with respect to left‐ and right-
circularly polarised light (LHCP and RHCP, respectively), a phenomenon known
as circular dichroism. Notably, these energy differences can be exploited to
create enantioselective signatures in the rotational spectra of the two mirror
images (?).
In this work, we investigate whether the enantiomeric energy differences in
chiral cavities can induce enantioselectivity in reactions that are normally
non-selective. Indeed, while the effect on the formed enantiomer is intriguing
in itself, an even more captivating prospect is whether the field can bias
non-chiral reagents towards favoring the formation of one configuration over
the other (see Fig. 1a). In particular, we suggest that appropriate
engineering of the field can favor a desired approach direction of the
reagents, leading to the preferential formation of only one of the two
possible products (?, ?, ?). Furthermore, since strong coupling to quantized
fields induces long-range correlation between molecules, we expect that the
field footprint becomes increasingly relevant at large distances where the
electronic effects become smaller and smaller. Our studies suggest that
harnessing the power of quantum fields may hold the key to achieving high
levels of selectivity in asymmetric synthesis. Specifically, we confirm that
field discrimination has an increasingly larger role at long distances while
at short ranges the Coulomb interaction dominates.

## Results and discussion

Figure 1: Chiral cavities and their discriminating power. a, Pictorial
representation of a benzaldehyde reacting with ethanol inside an LHCP (red on
the left) and RHCP (blue on the right) chiral cavity. Due to the field effects
the reaction product has a different chirality. b, Example of mirrors
reflecting only RHCP and LHCP light without changing the field circular
polarization, as reported by Baranov and co-workers (?, ?). c, Energy
difference between two enantiomers of 1-ethoxy-1,2-diphenylethanol in a LHCP
cavity.

Strong coupling between molecules and circularly polarized fields can be
achieved using chiral cavities, devices that confine only one circular
polarization of the field within a reduced volume, known as the quantization
volume (?, ?, ?, ?). Experimental realization of such devices has been
reported by Gautier et al. (?) as well as by Baranov and co-workers (?, ?),
see Fig.1b for a pictorial representation. Exploiting the unique properties of
the circularly polarized fields, chiral cavities effectively break the energy
degeneracy between the two mirror images of a chiral molecule. To illustrate
this phenomenon, we report in Fig. 1c the field-induced energy difference
between the two enantiomers of a chiral molecule,
1-ethoxy-1,2-diphenylethanol, placed inside a LHCP chiral cavity. The field
frequency is set to $\omega$ = 2.7 eV and the light-matter coupling is
$\lambda$ = 0.05 atomic units (a.u.). Detailed information regarding these
parameters and their physical meaning can be found in the Methods. In line
with Ref. (?), we refer to the enantiomeric energy differences induced by the
cavity as the field discriminating power. For the 1-ethoxy-1,2-diphenylethanol
molecule in Fig.1c, the ground state field discriminating power is on the
order of 40 $\mu$eV, consistent with the findings reported previously in the
literature (?, ?). Even though this energy range can be experimentally
detected using modern instruments (e.g. nuclear magnetic resonance operates in
a similar energy range), it is important to note that the chiral effects are
several orders of magnitude smaller than both room temperature thermal energy
and molecular binding energies, respectively on the order of 25 meV and
hundreds meV. Although the field-induced discrimination increases for larger
chiral systems and stronger light-matter coupling strengths, the magnitude of
the discriminating power remains within the $\mu$eV range per molecule for
real systems. The effects can however be enhanced up to the chemically
relevant range (meV) when increasing the number of chiral molecules inside the
cavity.

### Benzaldehyde-ethanol reaction

Figure 2: Field effects on the short-range benzaldehyde-ethanol reaction. a,
Reaction mechanism and stereoselectivity for the benzaldehyde-ethanol
reaction. b, Potential energy surface for the reaction, the ethanol is
approaching parallel to the cavity wave vector $\mathbf{k}$. c, Cavity
discriminating power, computed by subtracting the potential energy surfaces
for right and left approaches, see panel a. We notice that the sign of the
field discrimination power remains constant along the full pathway.

In Refs. (?, ?), we show that since the cavity field introduces a spacial
anisotropy, the molecular energy is highly dependent on the system
orientation. As a consequence, during a reactive event every reaction pathway
is either stabilized or destabilized by the field (?, ?). This observation
suggests that strong coupling to circularly polarized fields has the potential
to influence the stereoselectivity of chemical reactions by favoring specific
approach directions. To investigate this idea, we focus on the field effects
on the non-enantioselective reaction between benzaldehyde and ethanol leading
to the formation of a hemiacetal (see Fig.2a). The approach direction of the
reagent plays a crucial role in determining the chirality of the final
product. Specifically, the orientation of the alcohol group relative to the
aromatic ring is the key variable. A crossing of this plane results in a
chirality inversion of the product. In Fig.2 we illustrate the cavity effect
on two different approach directions of the alcohol group: from left and right
with respect to the aromatic ring plane. In particular, the benzaldehyde and
the ethanol approach each other parallel with respect to the field wave vector
$\mathbf{k}$. The chiral cavity utilized in these calculations is a RHCP
cavity with an optical frequency of $\omega=2.7$ eV and a coupling strength of
$\lambda=0.05$ a.u. . In Fig.2b, we show the potential energy surfaces (PES)
of the reaction while in Fig.2c we report the energy difference between the
two approach directions, (left minus right), as a function of the distance. We
point out that in the non-cavity case, the energy difference in Fig.2c would
be zero. The chiral field can instead discriminate between the two approach
directions leading to a net difference in the PES profile. These field-induced
energy differences between reactive pathways leading to two different
enantiomers is exactly the effect this paper focuses on. The sign of the field
stabilization is constant along the reactive path, indicating that the cavity
consistently favors the right approach of the alcohol in Fig.2a, favouring a R
enantiomer product. However, the electronic effects overwhelmingly dominate
compared to the contributions from the cavity, which remain in the $\mu$eV
range, and only a negligible fraction of the reactive pathways will be
redirected due to the effects discussed in Fig.2. Yet we notice that the
cavity stabilization of the approach creating a R enantiomer increases as the
distance grows. This is a crucial observation because while the electronic
effects diminish quite rapidly with distance, strong coupling to quantized
fields introduces long-range correlation between molecules. In an ideal gas
phase experiment, where the reagents are approaching from a large distance,
the discriminating power of chiral fields might therefore become the
dominating effect. To exemplify the persistence of the cavity discriminating
power for large separations, we illustrate the field effects for the
benzaldehyde-ethanol reaction when the reagents are 200 Å apart. Specifically,
Fig.3 displays the angular dispersion of the energy for a RHCP and a LHCP
cavity, with the field-induced effects computed as the difference between the
red and the blue curves.

Figure 3: Angular dispersion of the energy for the hemiacetal reaction with
the reagents placed 200 Å apart. The sign change in the chiral discriminating
power shows the inversion of the preferred chirality between LHCP and RHCP
cavities.

The discriminating power of the cavity (obtained subtracting the LHCP results
with the RHCP results) exhibits a prominet sign change around the $90^{\circ}$
mark, i.e. where the system’s chirality is inverted. Specifically, RHCP fields
stabilize the R enantiomer ($\theta$ less than $90^{\circ}$) more than the S
enantiomer ($\theta$ larger than $90^{\circ}$), while the opposite behavior is
observed in the LHCP cavity. This result validates the intuitive picture that
when the system inverts its chirality, the field stabilization also changes
sign. Although the cavity-induced discrimination remains in the $\mu$eV range,
it is crucial to consider that in Fig.3 the Coulomb interactions between the
reagents are negligible and the field effects can accumulate over the whole
approach path. The shape of the angular dispersion is predominantly determined
by dipole effects, which are, in agreement with the theoretical description,
larger than the enantiomeric discrimination. However, at these large
distances, the enantioselective field effects are comparable to the dipole
effects and the dispersion curves for the rotation in the RHCP and LHCP
cavities are therefore qualitatively different.

Figure 4: Field-induced stabilization of the R (blue) or S (red) enantiomers
as a function of the relative orientation between the two reagents.

In the long distance range, it is critical to account for orientational
effects as the molecules are free to reorient with respect to each other. To
determine whether the orientational effects will cancel the long-range
contributions of the cavity discriminating power, we present in Fig.4 the
field-induced energy differences between the top and bottom reactions for all
possible approach directions of the alcohol in a RHCP cavity, as obtained by
varying the spherical angles $\theta$ and $\phi$. The plot in Fig.4 clearly
presents peaks and valleys, with the deepest minimum (in blue) corresponding
to the path where the R enantiomer formation is maximally favoured, and the
highest maximum (in red) indicating the approach direction that most
significantly favours the S enantiomer. Furthermore, we notice that the
minimum is more pronounced than the maximum. When a reaction is carried out
inside a chiral cavity, like in Fig.4, every configuration of the potential
energy surface is populated following the Boltzmann distribution

$P(\theta,\phi)=\frac{\textrm{exp}\left(-E(\theta,\phi)/k_{b}T\right)}{\sum_{\theta,\phi}\textrm{exp}\left(-E(\theta,\phi)/k_{b}T\right)},$
(1)

where $P(\theta,\phi)$ is the probability of finding the molecule in a certain
configuration, $T$ is the reaction temperature and $k_{b}$ is the Boltzmann
constant.

Figure 5: Field-induced enantioselectivity as a function of the distance and
of the reaction temperature. We notice that the enantioselectivity always
displays the same sign, becoming more pronounced for larger distances. On the
right we show how the Coulomb contributions decay much faster than the field
effects.

In Fig.5 we therefore compare the probability of an approach from above (blue
hemisphere) with an approach from below (red hemisphere) for the benzaldehyde-
ethanol reaction at different distances. To check the robustness of the
results over the reaction parameters, we consider the reactive approach at
different temperatures: 298 K (room temperature), 273 K ($0\;^{\circ}$C), 183
K (lowest temperature ever measured on earth) and 77 K (nitrogen
condensation). The results in Fig.5 demonstrate that the field-induced effects
persist even after rotational averaging. The same enantiomer is favored at all
distances and for all temperatures confirming that the cavity discriminating
power is robust. Furthermore, the field discrimination is more pronounced at
larger distances. This is a crucial observation because, as the Coulomb
interaction becomes negligible the chirality effects dominate the interaction
between the fragments. Specifically, on the right side of Fig.5 we plot the
magnitude of the Coulomb and chiral interactions for the approach at
$\theta=40^{\circ}$ and $\phi=20^{\circ}$, see Methods section. We observe
that the field-induced discrimination is significantly larger than the Coulomb
contribution before 50 Å. As expected, the cavity effect on the
stereoselectivity increases as the temperature decreases because the lower
energy approach direction is more populated at lower temperatures. The field-
induced enantioselectivity reported in Fig.5 may be small, with only a few
fractions of a percentage gained even at 77 K. However, it is important to
note that the effect is not zero even at very large distances, meaning that
accumulation of the chirality effects needs to be taken into account. We
firmly believe that with careful engineering of the field and reaction
conditions, a significant impact on the chirality of the final product can be
achieved.

### Origin of the field-induced discrimination

In Ref. (?), we demonstrate that the field-induced discrimination of a chiral
cavity is directly linked to the optical activity of the molecule. This
observation substantiates the claim that field-induced energy discrimination
arises due to the molecular circular dichroism. This is a remarkable result
since it connects to the idea of homochirality, where the field promotes the
enantiomer with the same chirality (?). A more detailed investigation of the
homochirality hypothesis in chiral cavities will be the subject of a future
work as our findings indicate that correlation between optical rotatory power
and field stabilization is not straightforward (only the positive pole of the
optical activity determines the effect). Even so, given a certain reaction, it
is possible to choose the field polarization based on the desired enantiomer
we wish to produce. The field discriminating effects can also be attributed to
a different mechanism, well established in the chemistry community. When
dealing with racemic mixtures containing both enantiomers, one effective
approach for their separation involves the reaction of the mixture with an
enantiomerically pure system. Despite still being isomers, the two reaction
products are indeed no longer mirror images of one another; they are instead
diastereoisomers. Diastereoisomers have different physical and chemical
properties and therefore techniques such as distillation, crystallization,
chromatography, or extraction can be employed to separate them (?). Nowadays
this procedure is not used as much because the reactive steps significantly
reduce the yield of the separation process. The field discriminating power in
chiral cavities parallels the diastereomeric mechanism described above.
Specifically, while the two enantiomers of a chiral molecule have the same
energy outside chiral cavities, upon interaction with a chirally pure entity,
the circularly polarized field in our framework, two different
diastereoisomers are formed. Inside a chiral cavity, no symmetry operation can
interconvert one enantiomer into the other (?). While drawing an analogy
between light-matter interaction and chemical bonding might appear bold, in
the strong coupling regime the field effects often mirror molecular
interactions, see the solvent caging effect reported by Li et al (?, ?).

## Discussion

Regardless of the conceptual framework chosen to rationalize the
discriminating power of the field, our work demonstrates that strong coupling
between molecules and tightly confined circularly polarized fields has the
potential to affect reactivity, eventually leading to enantioselective
synthesis. Despite being small compared to the electronic effects when the
reagents are close, the cavity discriminating power becomes increasingly
relevant at large separations. The field does indeed create long-range
correlation between the fragments that can reorient at long distances.
Overall, this research opens the way to new and unexplored paths toward
asymmetrical synthesis, with potential implications across multiple areas,
from drug design to material science. Considering the recent fast advances
made from the experimental side in the fabrication of chiral cavities (?, ?,
?, ?), we believe that the proposed methodology will, in a near future, find
an experimental validation. Future efforts will be devoted to studying the
field-induced discrimination for excited states with a particular focus on
collective effects (?, ?, ?). Additionally, the study of strong coupling to
circularly polarized fields in the vibrational energy (?) range will be the
subject of further investigations.

##  1 Acknowledgements

R. R. R and H. K. acknowledge funding from the Research Council of Norway
through FRINATEK Project No. 275506. This work has received funding from the
European Research Council (ERC) under the European Union’s Horizon 2020
Research and Innovation Programme (Grant Agreement No. 101020016). E. R.
acknowledges funding from the European Research Council (ERC) under the
European Union’s Horizon Europe Research and Innovation Programme (Grant No.
ERC-StG-2021-101040197—QED-Spin).

##  2 Author contributions

R.R.R., E.R. and H.K. developed the theory. E.R. conceived the project. R.R.R.
implemented the computational methodology in the eT program. All the authors
proposed the investigated applications. R.R.R. obtained the data. R.R.R.
prepared a first draft of the paper. R.R.R. and E.R. worked at the graphical
part of the manuscript. H.K. and E.R. oversaw the project. All the authors
discussed the results and edited the manuscript.

##  3 Competing interests

The authors declare no competing interests.

##  4 Data availability statement

The geometries used for the reported calculations can be found on the Zenodo
open repository (?). The eT program (?) is open source and instructions on how
to install can be found on the Zenodo link. Additional data on a SN1 reaction,
on the cavity effects on Van Der Waals complexes and on the theoretical
methodology used to compute the results of this paper can be found in the
Supplementary Materials.

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See pages - of Supplementary_materials.pdf