11affiliationtext: Wellcome Centre for Human Neuroimaging, UCL Queen Square
Institute of Neurology,
University College London, London, UK22affiliationtext: Athinoula A. Martinos
Center for Biomedical Imaging,
Massachusetts General Hospital and Harvard Medical School, Boston,
USA33affiliationtext: MR Research Collaborations, Siemens Healthcare Limited,
Frimley, UK44affiliationtext: Centre de Résonance Magnétique des Systèmes
Biologiques, UMR5536,
CNRS/University Bordeaux, Bordeaux, France

# Correcting inter-scan motion artefacts
in quantitative $R_{1}$ mapping at 7T.

Yaël Balbastre  Ali Aghaeifar  Nadège Corbin  Mikael Brudfors  John Ashburner
Martina F. Callaghan111Corresponding author<EMAIL_ADDRESS>
###### Abstract

Purpose: Inter-scan motion is a substantial source of error in $R_{1}$
estimation, and can be expected to increase at 7T where $B_{1}$ fields are
more inhomogeneous. The established correction scheme does not translate to 7T
since it requires a body coil reference. Here we introduce two alternatives
that outperform the established method. Since they compute relative
sensitivities they do not require body coil images.
Theory: The proposed methods use coil-combined magnitude images to obtain the
relative coil sensitivities. The first method efficiently computes the
relative sensitivities via a simple ratio; the second by fitting a more
sophisticated generative model.
Methods: $R_{1}$ maps were computed using the variable flip angle (VFA)
approach. Multiple datasets were acquired at 3T and 7T, with and without
motion between the acquisition of the VFA volumes. $R_{1}$ maps were
constructed without correction, with the proposed corrections, and (at 3T)
with the previously established correction scheme.
Results: At 3T, the proposed methods outperform the baseline method. Inter-
scan motion artefacts were also reduced at 7T. However, reproducibility only
converged on that of the no motion condition if position-specific transmit
field effects were also incorporated.
Conclusion: The proposed methods simplify inter-scan motion correction of
$R_{1}$ maps and are applicable at both 3T and 7T, where a body coil is
typically not available. The open-source code for all methods is made publicly
available.

Keywords— qMRI, inter-scan motion, sensitivity, generative modelling, $R_{1}$,
7T

††This work was supported by the MRC and Spinal Research Charity through the
ERA-NET Neuron joint call (MR/R000050/1). The Wellcome Centre for Human
Neuroimaging is supported by core funding from the Wellcome [203147/Z/16/Z].

##  1 Introduction

Quantitative MRI, and the push towards _in vivo_ histology, aims to extract
tissue-specific parameters from a series of weighted volumes [1]. For example,
the longitudinal relaxation rate, $R_{1}$, which is sensitive to important
biological features, such as myelin and iron content, can be quantified with
the variable flip angle (VFA) approach, e.g. Deoni et al. [2], Helms et al.
[3]. A common assumption when computing quantitative metrics is that certain
multiplicative factors, such as the signal intensity modulation imposed by the
receiver coil’s net sensitivity profile, are constant across the weighted
volumes. However, this is invalid if motion occurs between the volume
acquisitions. In the case of neuroimaging, rigid body co-registration can be
used to realign the brain but will not correct for the differential coil
sensitivity modulation, which in $R_{1}$ maps computed with the VFA approach
can lead to mean absolute error approaching 20% [4].

A correction scheme has previously been proposed by Papp et al. [4] and
validated for $R_{1}$ mapping at 3T. The position-specific net receive
sensitivity is estimated from two rapid low-resolution magnitude images,
received on the body and array coils respectively prior to each VFA
acquisition. The more homogeneous profile of the body coil is used as a
reference to compute the net receiver sensitivity, which is then removed from
the VFA acquisitions. This approach effectively assumes that the body coil’s
modulation is consistent across volumes instead of that of the array coil.
This in itself is a potential limitation, as is the general unavailability at
body coils at higher field strengths.

Here we propose an alternative whereby we estimate the _relative_ sensitivity
_between_ volumes. This approach does not fully remove the receiver’s
sensitivity modulation but does remove the bias that differential modulation
introduces in quantitative metrics. Only the calibration images obtained with
the array coil are required, i.e. less data than the originally proposed
method [4]. To validate the approach, we focus on $R_{1}$ maps computed with
the multi-parameter mapping (MPM) protocol [5]. We first compare performance
with the established method of Papp _et al._ at 3T [4] and then demonstrate a
reduction of inter-scan motion artefacts at 7T under a range of different
motion conditions. We further demonstrate that, unlike at 3T, the transmit
field $B_{1}^{+}$ also exhibits substantial position-specific variability at
7T. As a result, the most precise $R_{1}$ estimates were obtained by
accounting for both position-specific $B_{1}^{+}$ and $B_{1}^{-}$ effects.

##  2 Methods

###  2.1 Theory

$R_{1}$ mapping can be achieved by acquiring spoiled gradient echo volumes
with at least two different flip angles [2, 3]. At a given spatial location,
$r$, the image intensity, $I$, for a given nominal flip angle $\alpha$ is:

$I_{k}(r)=s_{k}(r)\rho(r)\frac{1-\exp(-\text{TR}_{k}R_{1}(r))}{1-\cos(f_{\mathrm{T}_{k}}(r)\alpha_{k})\exp(-\text{TR}_{k}R_{1}(r))}~{},$
(1)

where $s$ is the receive sensitivity, $\rho$ is the proton density,
$f_{\mathrm{T}}$ is the transmit field, $R_{1}$ is the longitudinal relaxation
rate, TR is the repetition time, and $k$ indexes the VFA acquisition. Co-
registration allows for inter-scan motion by realigning anatomical structure
across acquisitions. Under the small flip angle approximation [3], with two
nominal flip angles ($k$ = {1,2}), $R_{1}$ can be computed as follows:

$\displaystyle
R_{1}(r)=\frac{1}{2}\frac{\frac{s_{2}(r)I_{2}(r)f_{\mathrm{T}_{2}}(r)\alpha_{2}}{\text{TR}_{2}}-\frac{s_{1}(r)I_{1}(r)f_{\mathrm{T}_{1}}(r)\alpha_{1}}{\text{TR}_{1}}}{\frac{s_{1}(r)I_{1}(r)}{f_{\mathrm{T}_{1}}(r)\alpha_{1}}-\frac{s_{2}(r)I_{2}(r)}{f_{\mathrm{T}_{2}}(r)\alpha_{2}}}~{},$
(2)

Typically, it is assumed that $s_{1}(r)=s_{2}(r)$ and the sensitivities
simplify out. However, this assumption is invalid if inter-scan motion has
occurred leading to substantial bias in $R_{1}$ [4]. This can be avoided by
accounting for the _relative_ sensitivity across positions:
$\Delta_{1,2}(r)=s_{1}(r)/s_{2}(r)$. Substitution for $s_{1}(r)$ in equation
(2) gives:

$\displaystyle
R_{1}(r)=\frac{1}{2}\frac{\frac{I_{2}(r)f_{\mathrm{T}_{2}}(r)\alpha_{2}}{\text{TR}_{2}}-\frac{\Delta_{1,2}(r)I_{1}(r)f_{\mathrm{T}_{1}}(r)\alpha_{1}}{\text{TR}_{1}}}{\frac{\Delta_{1,2}(r)I_{1}(r)}{f_{\mathrm{T}_{1}}(r)\alpha_{1}}-\frac{I_{2}(r)}{f_{\mathrm{T}_{2}}(r)\alpha_{2}}}~{},$
(3)

The method of Papp et al. [4] did not include the relative sensitivity but
referenced to an additional calibration image acquired on the body coil,
assuming that the body coil modulation was position-independent.

It is commonly assumed that the transmit field is sufficiently smooth as to be
considered position-independent, i.e.
$f_{\mathrm{T}_{1}}(r)=f_{\mathrm{T}_{2}}(r)$, such that:

$\displaystyle
R_{1}(r)=\frac{f_{\mathrm{T}}(r)^{2}}{2}\frac{\frac{I_{2}(r)\alpha_{2}}{\text{TR}_{2}}-\frac{\Delta_{1,2}(r)I_{1}(r)\alpha_{1}}{\text{TR}_{1}}}{\frac{\Delta_{1,2}(r)I_{1}(r)}{\alpha_{1}}-\frac{I_{2}(r)}{\alpha_{2}}}~{},$
(4)

However, in this work we show that this assumption does not hold at 7T, and
that incorporating position-specific transmit field estimates maximises the
precision of $R_{1}$.

Figure 1: 7T Example. The acquired calibration images, $x_{k}$ have different
orientation due to participant movement between acquisitions. Co-registration
can align the images spatially, but does not correct for their differential
sensitivity field modulation, visible via their ratio, $x_{1}$/$x_{2}$ =
$\Delta_{1,2}$. Alternatively, the joint log-likelihood of a generative
forward model that embeds the spatial transformation from a mean image, $r$,
to native space can be maximised to determine the mean image and modulating
sensitivities, $s_{k}$, that best explain the acquired images $x_{k}$. The
generative modelling approach produces a similar relative modulation
($s_{1}$/$s_{2}$) but allows the sensitivity modulations to be explicitly
estimated, and for the corrected images to have the minimal modulation of the
mean image.

Ratio approach The calibration data used to correct inter-scan motion
artefacts comprised rapid low resolution, coil-combined magnitude images
acquired immediately prior to each high resolution VFA acquisition. These
images,
$\left\\{\mathbf{\boldsymbol{x}}_{k}\in\mathbb{R}_{+}^{N}\right\\}_{j=1}^{K}$,
assumed to have been rigidly co-registered and resliced to a grid of $N$
voxels, can be written as the product of a common image
$\mathbf{\boldsymbol{r}}\in\mathbb{R}_{+}^{N}$ and a net sensitivity field
$\left\\{\mathbf{\boldsymbol{s}}_{k}\in\mathbb{R}_{+}^{N}\right\\}_{k=1}^{K}$.
The relative sensitivity,
$\mathbf{\boldsymbol{\Delta}}_{k,\text{ref}}\in\mathbb{R}_{+}^{N}$ can be
computed with respect to one of the calibration acquisitions, used as a
reference:

$\mathbf{\boldsymbol{\Delta}}_{k,\text{ref}}=\frac{\mathbf{\boldsymbol{x}}_{k}}{\mathbf{\boldsymbol{x}}_{\text{ref}}}=\frac{\mathbf{\boldsymbol{r}}\mathbf{\boldsymbol{s}}_{k}}{\mathbf{\boldsymbol{r}}\mathbf{\boldsymbol{s}}_{\text{ref}}}=\frac{\mathbf{\boldsymbol{s}}_{k}}{\mathbf{\boldsymbol{s}}_{\text{ref}}}$
(5)

Dividing each VFA acquisition by its relative sensitivity
$\mathbf{\boldsymbol{\Delta}}_{k,\text{ref}}$ results in a common modulation,
$\mathbf{\boldsymbol{s}}_{\text{ref}}$, which, although less homogeneous than
the body coil used by Papp _et al._ , more faithfully restores the validity of
assuming common modulation when computing $R_{1}$. This ratio approach risks
noise amplification, particularly in regions of low signal-to-noise ratio
(SNR). This is combatted by isotropically smoothing
$\mathbf{\boldsymbol{x}}_{k}$ and $\mathbf{\boldsymbol{x}}_{\text{ref}}$
before taking their ratio.

Generative approach A potentially more robust alternative is to cast the
computation of the relative coil sensitivities, and a common image modulated
by them, as an inference problem in a probabilistic generative model of
$\mathbf{\boldsymbol{x}}_{k}$ that incorporates noise and can also embed
knowledge about the spatial smoothness of the sensitivities. Full details are
in Appendix A.

###  2.2 Experiment

Participants One participant (F, 31 years) was scanned at 3T (MAGNETOM Prisma,
Siemens, Erlangen, Germany) using a body coil for transmission and either the
body coil or a 32-channel head array coil for reception. Three additional
participants (2F, 1M; 32 - 41 years) were scanned at 7T (MAGNETOM Terra,
Siemens, Erlangen, Germany) using an 8-channel transmit, 32-channel receive
head array coil (Nova Medical, Wilmington, MA, USA) in TrueForm mode. All data
were acquired with approval from the UCL research ethics committee.

MPM Datasets MPM data were acquired using a multi-echo spoiled gradient echo
sequence with flip angles of 6∘ (PD-weighted, ”PDw”) and 26∘ (T1-weighted,
”T1w”), a TR of 19.5 ms and an RF spoiling increment of 117∘ with a total
dephasing gradient moment per TR of 6$\pi$. Eight echoes were acquired with TE
ranging from 2.56 ms to 15.02 ms in steps of 1.78 ms using a bandwidth of 651
Hz/pixel. Data were acquired with 1mm isotropic resolution over a field of
view of 160 mm right-left and 192 mm in the anterior-posterior and superior-
inferior directions. Elliptical sampling and partial Fourier, with factor 6/8
in each phase-encoded direction, were used to accelerate the acquisition,
leading to a scan time of 5 minutes per volume. A $B_{1}^{+}$ map was
estimated by acquiring a series of spin and stimulated echoes using previously
described 3T and 7T protocols [6, 7].

For inter-scan motion correction, additional single echo acquisitions were
acquired prior to each VFA acquisition with a flip angle of 6∘, TE = 2.4 ms,
TR = 6.5 ms, a bandwidth of 488 Hz/pixel and no acceleration schemes. At 3T,
these data were acquired, receiving sequentially on the array and body coils,
with 8 mm isotropic resolution leading to a scan time of 6 s per volume. To
capture the greater spatial variation in the net sensitivity field at 7T, the
resolution was increased to 4 mm isotropic leading to a scan times of 18 s per
volume, but acquired only on the array coil due to the absence of a body coil.

Motion Conditions Two MPM datasets were acquired to define baseline
reproducibility. Participants were then instructed to move to a new, arbitrary
position within the confines of the coil. A localiser was acquired and the
field of view repositioned as necessary to ensure appropriate brain coverage
in the new position. A third MPM data set was then acquired.

$R_{1}$ Analysis $R_{1}$ maps were computed using the hMRI toolbox [8], which
uses the small flip angle approximation [3] and corrects for imperfect
spoiling [9], which always used the $B_{1}^{+}$ map acquired in the space of
the PDw acquisition. Maps were computed with and without inter-scan motion
using all possible PDw and T1w combinations. To ease comparisons, all maps
were constructed in the space of the first PDw volume. Rigid transformations
between all volumes and the first PDw volume were estimated using SPM12
(Wellcome Centre for Human Neuroimaging) having first corrected for intensity
non-uniformity and skull-stripped the images. $R_{1}$ maps were computed with
and without the proposed inter-scan motion correction schemes. At 3T, $R_{1}$
maps were also computed with the method of Papp et al. [4]. An isotropic
kernel of 12mm full-width-at-half-maximum – the default in the hMRI toolbox –
was used to smooth the calibration images prior to computing the relative
(proposed) and absolute (Papp _et al._) sensitivities. The generative
modelling approaches used $\lambda_{\text{array}}=10^{3}$,
$\lambda_{\text{body}}=10^{8}$, and 15 iterations.

$B_{1}^{+}$ per contrast In each case, two different corrections for transmit
field inhomogeneity were performed: the first assumed the transmit field was
identical across head positions, and used that of the PD-weighted acquisition;
the second used position-specific $B_{1}^{+}$ maps. In the latter case
effective flip angle maps were computed for each volume and used in equation
(3).

Error metric $R_{1}$ maps with no inter-scan motion, and no corrections, were
averaged to produce a ‘ground-truth’ map, $\hat{R}_{1}$. This set of $R_{1}$
maps was also segmented to create a mask selecting those voxels with a mean
probability of being in WM, GM or CSF greater than 50%. For participant 3, the
cerebellum was excluded, using the SUIT toolbox [10] in SPM, as a result of
$B_{1}^{+}$ mapping failure caused by excessively large off-resonance. For the
$N$ voxels within the resulting participant-specific mask, the mean absolute
error, MAE, for each $R_{1}$ map was computed with respect to the ‘ground-
truth’ map, $\hat{R}_{1}$ as:

$\text{MAE}=\frac{1}{N}\sum_{n=1}^{N}\frac{\left|\hat{R}_{1}(n)-R_{1}(n)\right|}{\hat{R}_{1}(n)}~{}.$
(6)

These errors are reported as percentages.

##  3 Results

Exemplar images, relative sensitivities, and results from the generative
modelling are shown in figure 1. The $R_{1}$ and error maps obtained at 3T and
7T are shown in figures 2 and 3 respectively. The means and standard
deviations of the MAE are reported in Table 3.3. The differential impact of
correcting for transmit and receive field effects is illustrated in figure 4.
This shows $R_{1}$ and error maps without motion, and with motion having
implemented (i) no correction, (ii) correction only for receive field effects,
$B_{1}^{-}$, (iii) only for transmit field effects, $B_{1}^{+}$, or (iv) for
both effects in combination.

Figure 2: Results at 3T. The first row shows example $R_{1}$ maps constructed
with each method. The second row shows (normalised) error maps with respect to
the ground-truth map $\hat{R}_{1}$. The third row shows histograms (filled
area) of $R_{1}$ within the GM (green) and WM (purple), and their log-Normal
fit (solid line); these histograms display probability distributions and
therefore integrate to 1.

###  3.1 3T Validation

When the T1w and PDw volumes were acquired in the same position, the MAE
captured the test-retest variability, which was approximately 3% at 3T and
4-5% at 7T. In the absence of overt motion, correcting for the differential
sensitivity modulation did not substantially change the MAE. In the presence
of overt motion, the MAE rose to 10%. It was reduced to 4.7% by the method of
Papp _et al._ and to less than 4.4% by the proposed correction schemes, with
or without incorporating the body coil in the generative modelling approach.
The histograms in figure 2 confirm that the method did not introduce any bias
to the $R_{1}$ estimates.

###  3.2 Extension to 7T

At 7T, the range of motion varied across participants. The net (root-mean-
square across axes) translation ranged from 1.6 to 7.9 mm, while the net
rotation ranged from 2.7 to 11.2 degrees. The overall amplitude of motion
dictated the increase in MAE, which reached a maximum of 13.4% under the
tested conditions (c.f. motion summaries in figure 3 and MAE in table 3.3).
The proposed correction scheme reduced the MAE (5-8%), though not to the level
of no overt motion.

The variability of the transmit field, $B_{1}^{+}$, across head positions was
found to be much higher than at 3T. Incorporating position-specific
$B_{1}^{+}$ maps reduced the MAE (5-12%) even without correcting for the
differential receive sensitivity modulation.

The greatest reductions in MAE were obtained by correcting for position-
specific transmit _and_ receive fields, reaching 4-7%, converging on the level
obtained in the absence of overt motion (i.e. 4-5%).

###  3.3 Comparison of Methods

Overall, the ratio and generative modelling approaches to estimating the
relative sensitivities performed similarly. The generative modelling approach
performed marginally better at 3T (0.1% lower MAE), while the simpler ratio
approach performed better at 7T (up to 0.5% lower MAE).

Table 1: MAE (mean $\pm$ s.d. across repeats, in %) with respect to the
average reference $\hat{R}_{1}$.

rcc — cccccc Dataset Motion No correction Ratio Generative Generative (array +
body) Papp _et al._
3T #1 No 3.0 $\pm$ 0.1 3.0 $\pm$ 0.1 3.0 $\pm$ 0.1 3.1 $\pm$ 0.2 3.7 $\pm$ 0.2
Yes 10.1 $\pm$ 0.8 4.4 $\pm$ 0.4 4.3 $\pm$ 0.3 4.3 $\pm$ 0.1 4.7 $\pm$ 0.3
Dataset Motion No correction Ratio Generative B${}_{1}^{+}$ per contrast Ratio
& B${}_{1}^{+}$ per contrast Generative & B${}_{1}^{+}$ per contrast
#1 No 4.0 $\pm$ 0.3 4.1 $\pm$ 0.4 4.1 $\pm$ 0.3 4.1 $\pm$ 0.2 4.1 $\pm$ 0.4
4.1 $\pm$ 0.3
Yes 5.8 $\pm$ 0.7 4.8 $\pm$ 0.3 4.9 $\pm$ 0.4 4.9 $\pm$ 0.3 3.8 $\pm$ 0.3 3.9
$\pm$ 0.3
7T #2 No 4.8 $\pm$ 0.2 4.9 $\pm$ 0.3 4.9 $\pm$ 0.3 4.8 $\pm$ 0.2 5.0 $\pm$ 0.2
5.5 $\pm$ 0.1
Yes 8.4 $\pm$ 0.4 6.1 $\pm$ 0.3 6.2 $\pm$ 0.3 8.0 $\pm$ 0.4 5.3 $\pm$ 0.1 5.5
$\pm$ 0.1

#3 No 5.1 $\pm$0.2 5.0 $\pm$ 0.2 5.0 $\pm$0.2 5.0 $\pm$0.2 5.0 $\pm$0.3 5.0
$\pm$0.2
Yes 13.4 $\pm$ 1.9 7.9 $\pm$ 1.1 8.3 $\pm$ 1.2 11.8 $\pm$ 0.9 6.3 $\pm$ 0.3
6.8 $\pm$ 0.3

Figure 3: Results at 7T. The first column shows the ground-truth map
$\hat{R}_{1}$ for the three participants sorted based on the magnitude of
inter-scan motion. Uncorrected and corrected $R_{1}$ maps are shown in the
middle with the corresponding (normalised) error maps with respect to the
ground-truth on the right. Correction is only applied for net receive
sensitivity modulation, $B_{1}^{-}$ and not for $B_{1}^{+}$. Rows 1 to 3 of
the figure correspond to datasets 1, 2 and 3 as reported in Table 3.3.

Figure 4: Combining $B_{1}^{-}$ and $B_{1}^{+}$ correction at 7T, for
participant 3 (last row in figure 3). The first row shows an example $R_{1}$
map without and with inter-scan motion, before and after net receive
sensitivity correction, and employing a separate $B_{1}^{+}$ map for each
contrast in the case of inter-scan motion. The second row shows (normalised)
error maps with respect to the ground-truth map $\hat{R}_{1}$. In this
example, inter-scan motion correction was performed with the generative
modelling approach.

##  4 Discussion

We have introduced methods for correcting inter-scan motion artefacts in
quantitative MRI that do not rely on the availability or spatial homogeneity
of a body coil. The approaches are based on estimating the _relative_
sensitivity modulation across positions, and successfully reduced error in
$R_{1}$ maps at both 3T and 7T.

At 3T, the proposed approaches outperformed a previously established
correction method [4]. This can be attributed to the fact that the method of
Papp _et al._ assumes that the reference modulation of the body coil is
independent of position, whereas the proposed methods do not. Instead they
specifically account for the _relative_ sensitivity across positions thereby
restoring consistent modulations.

In the motion conditions tested here, both proposed approaches (ratio with
Gaussian smoothing, or generative modelling) produced comparable improvements
in $R_{1}$ reproducibility in the presence of inter-scan motion. Equally
importantly, when there was no overt motion neither method decreased
reproducibility, which was at a level in keeping with previous reports for
similar resolution MPM data [11, 5].

The simpler ratio method defines one calibration image as the reference
(denominator in equation (5)), and may therefore be vulnerable to low SNR. The
alternative generative modelling approach inherently adapts to variable SNR by
estimating the position-specific net sensitivity modulation relative to a
common image, which is their barycentre mean. This common image dictates the
final modulation of _all_ the corrected volumes. The generative model can also
easily incorporate any additional data, e.g. body coil images as done at 3T,
which flattens the final modulation. Furthermore, rigid registration could be
interleaved with model fitting [12] to reach a better global optimum. Finally,
the generative model could naturally be integrated with any fitting approach
that defines a joint probability over all acquired data, such as Balbastre et
al. [13] in the context of MPM.

The impact of movement on the effective transmit field has previously been
investigated in the context of specific absorption rate management [14, 15,
16, 17]. An important additional finding of the present work is the impact
this can have on $R_{1}$ estimates at 7T, which was negligible at 3T as
demonstrated previously [4]. Although $R_{1}$ errors were more substantially
reduced by correcting for receive field effects, they could only be reduced to
the approximate level of no overt motion by additionally accounting for the
positional-dependence of the transmit field. However, acquiring a $B_{1}^{+}$
map at multiple positions comes at a cost of increased scan time.

##  5 Conclusions

Inter-scan motion causes serially acquired weighted volumes to be
differentially modulated by position-specific coil sensitivities leading to
substantial errors when they are combined to compute quantitative metrics. We
have demonstrated the efficacy of two methods at reducing these artefacts in
the context of $R_{1}$ mapping. The proposed methods do not require a body
coil making them ideally suited for use at 7T, and can be extended to the
computation of other quantitative metrics, such as magnetisation transfer
saturation [18], that similarly assume constant modulation across multiple
weighted acquisitions.

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##  Appendix A Generative Model of Sensitivity-Modulated Images

The proposed generative model is similar to that of Ashburner and Ridgway [12]
(section 2.3), without non-linear deformations. The magnitude images of the
calibration dataset, $\mathbf{\boldsymbol{x}}_{k}$, can be written as the
voxel-wise product of a mean image $\mathbf{\boldsymbol{r}}$ and the net
sensitivity field $\mathbf{\boldsymbol{s}}_{k}$ plus additive noise,
approximated as Gaussian with variance $\sigma_{k}^{2}$, which is assumed to
be uncorrelated across $\mathbf{\boldsymbol{x}}_{k}$. The net sensitivity
fields can be written as diagonal matrices
$\mathbf{\boldsymbol{S}}_{k}=\operatorname{diag}\left(\mathbf{\boldsymbol{s}}_{k}\right)$,
such that the corresponding conditional probability is:

$p\left(\mathbf{\boldsymbol{x}}_{k}\mid\mathbf{\boldsymbol{s}}_{k},\mathbf{\boldsymbol{r}}\right)=\mathcal{N}\left(\mathbf{\boldsymbol{x}}_{k}~{}\middle|~{}\mathbf{\boldsymbol{S}}_{k}\mathbf{\boldsymbol{r}},\sigma_{k}^{2}\mathbf{\boldsymbol{I}}\right)~{}.$
(7)

If the images have been co-registered but not resliced, a mean space can be
defined by computing the barycentre of all aligned orientation matrices [12].
The linear operation of resampling an image from mean to acquired space can be
encoded by the matrix $\mathbf{\boldsymbol{A}}_{k}$ such that the conditional
probability becomes:

$p\left(\mathbf{\boldsymbol{x}}_{k}\mid\mathbf{\boldsymbol{s}}_{k},\mathbf{\boldsymbol{r}}\right)=\mathcal{N}\left(\mathbf{\boldsymbol{x}}_{k}~{}\middle|~{}\mathbf{\boldsymbol{A}}_{k}\mathbf{\boldsymbol{S}}_{k}\mathbf{\boldsymbol{r}},\sigma_{k}^{2}\mathbf{\boldsymbol{I}}\right)~{}.$
(8)

While this is the approach taken in practice, the following derivation is
restricted to the resliced case for clarity.

The magnitude of each sensitivity field is unknown since the intensity scaling
depends on many parameters. However, the sensitivities are known to vary
smoothly in space, which is captured by a probability distribution that
penalises the field’s bending energy [19], which integrates the squared
curvature of the sensitivity field, making it invariant to intensity scaling.
The net sensitivities are encoded by their logs, (_i.e._ ,
$\mathbf{\boldsymbol{s}}_{k}=\exp\mathbf{\boldsymbol{z}}_{k}$), such that
invariance under shifts in log-space implies invariance under scales in
exponentiated space, which is also incorporated into the prior. In a discrete
setting, computing the bending energy reduces to the quadratic term
$\mathbf{\boldsymbol{z}}^{\mathrm{T}}\mathbf{\boldsymbol{L}}\mathbf{\boldsymbol{z}}$;
The prior distribution over sensitivities is therefore defined as a Normal
distribution over their logs:

$p\left(\mathbf{\boldsymbol{z}}_{k}\right)=\mathcal{N}\left(\mathbf{\boldsymbol{z}}_{k}~{}\middle|~{}\mathbf{\boldsymbol{0}},\left(\lambda_{k}\mathbf{\boldsymbol{L}}\right)^{-1}\right)~{},$
(9)

where $\lambda_{k}$ is an image-specific regularisation factor.

The joint model likelihood, or its negative log, is obtained by combining the
likelihood in (7) and the prior in (9): $\mathcal{L}=-\ln
p\left(\left\\{\mathbf{\boldsymbol{x}}_{k},\mathbf{\boldsymbol{z}}_{k}\right\\}_{k=1}^{K}~{}\middle|~{}\mathbf{\boldsymbol{r}}\right)$.
This is minimised with respect to
$\left\\{\mathbf{\boldsymbol{z}}_{k}\right\\}_{k=1}^{K}$ and
$\mathbf{\boldsymbol{r}}$. Neglecting terms that do not depend on these
variables, yields the objective function:

$\mathcal{L}=\sum_{k=1}^{K}\left\\{\frac{1}{2\sigma_{k}^{2}}\left(\mathbf{\boldsymbol{x}}_{k}-\mathbf{\boldsymbol{S}}_{k}\mathbf{\boldsymbol{r}}\right)^{\mathrm{T}}\left(\mathbf{\boldsymbol{x}}_{k}-\mathbf{\boldsymbol{S}}_{k}\mathbf{\boldsymbol{r}}\right)+\frac{\lambda_{k}}{2}\mathbf{\boldsymbol{z}}_{k}^{\mathrm{T}}\mathbf{\boldsymbol{L}}\mathbf{\boldsymbol{z}}_{k}\right\\}+\text{const}~{}.$
(10)

Differentiating with respect to the mean image, $\mathbf{\boldsymbol{r}}$,
while keeping the sensitivities fixed gives a voxel-wise ($n$) closed-form
update:

$r_{n}\leftarrow\frac{\sum_{k=1}^{K}s_{kn}x_{kn}/\sigma_{k}^{2}}{\sum_{k=1}^{K}s_{kn}^{2}/\sigma_{k}^{2}}~{}.$
(11)

The log-sensitivities have no closed-form solution necessitating an iterative
method. The objective function is not everywhere convex, but the likelihood
term resembles that of a nonlinear least-squares problem, which can be solved
using Gauss-Newton optimisation. Gauss-Newton is a modification of Newton-
Raphson that uses Fisher’s method of scoring, which amounts to replacing the
Hessian at any point with its value at the optimum. With
$\mathbf{\boldsymbol{R}}=\operatorname{diag}\left(\mathbf{\boldsymbol{r}}\right)$
and
$\mathbf{\boldsymbol{X}}_{k}=\operatorname{diag}\left(\mathbf{\boldsymbol{x}}_{k}\right)$,
the gradient $\mathbf{\boldsymbol{g}}_{k}$ and Hessian
$\mathbf{\boldsymbol{H}}_{k}$ used in the Newton-Raphson iteration are:

$\displaystyle\mathbf{\boldsymbol{g}}_{k}$
$\displaystyle{}=\frac{1}{\sigma_{k}^{2}}\mathbf{\boldsymbol{S}}_{k}\mathbf{\boldsymbol{R}}\left(\mathbf{\boldsymbol{R}}\mathbf{\boldsymbol{s}}_{k}-\mathbf{\boldsymbol{x}}_{k}\right)+\lambda_{k}\mathbf{\boldsymbol{L}}\mathbf{\boldsymbol{z}}_{k}~{},$
(12) $\displaystyle\mathbf{\boldsymbol{H}}_{k}$
$\displaystyle{}=\frac{1}{\sigma_{k}^{2}}\mathbf{\boldsymbol{S}}_{k}\mathbf{\boldsymbol{R}}^{2}\mathbf{\boldsymbol{S}}_{k}+\underbrace{\frac{1}{\sigma_{k}^{2}}\mathbf{\boldsymbol{S}}_{k}\mathbf{\boldsymbol{R}}\left(\mathbf{\boldsymbol{R}}\mathbf{\boldsymbol{S}}_{k}-\mathbf{\boldsymbol{X}}_{k}\right)}_{\text{Fisher's
scoring}~{}\Rightarrow~{}\mathbf{\boldsymbol{0}}}+\lambda_{k}\mathbf{\boldsymbol{L}}~{}.$
(13)

This ensures that the Hessian used in the Newton-Raphson iteration is positive
definite, but does not ensure that the iteration monotonically improves the
objective function. We therefore replace the Gauss-Newton Hessian with the
more robust preconditioner:

$\displaystyle\mathbf{\boldsymbol{P}}_{k}$
$\displaystyle{}=\frac{1}{\sigma_{k}^{2}}\mathbf{\boldsymbol{S}}_{k}\mathbf{\boldsymbol{R}}^{2}\mathbf{\boldsymbol{S}}_{k}+\frac{1}{\sigma_{k}^{2}}\mathbf{\boldsymbol{S}}_{k}\mathbf{\boldsymbol{R}}\left|\mathbf{\boldsymbol{R}}\mathbf{\boldsymbol{S}}_{k}-\mathbf{\boldsymbol{X}}_{k}\right|+\lambda_{k}\mathbf{\boldsymbol{L}}~{},$
(14) $\displaystyle\mathbf{\boldsymbol{z}}_{k}$
$\displaystyle{}\leftarrow\mathbf{\boldsymbol{z}}_{k}-\mathbf{\boldsymbol{P}}_{k}^{-1}\mathbf{\boldsymbol{g}}_{k}~{},$
(15)

which has been shown to yield monotonic convergence [13]. The inversion in
equation (15) is performed with a full multi-grid solver that leverages the
sparsity and structure of the preconditioner [19].

Finally, a global scaling field
$\bar{\mathbf{\boldsymbol{s}}}=\exp\bar{\mathbf{\boldsymbol{z}}}$, applied to
both the sensitivities ($s_{cn}\leftarrow s_{cn}/\bar{s}_{n}$) and mean image
($r_{n}\leftarrow\bar{s}_{n}r_{n}$), ensures that the product $s_{cn}r_{n}$ is
unchanged. Keeping only terms of the objective function that depend on this
scaling field gives:

$\mathcal{L}\overset{c}{=}\sum_{c=1}^{C}\frac{\lambda_{k}}{2}\left(\mathbf{\boldsymbol{z}}_{k}-\bar{\mathbf{\boldsymbol{z}}}\right)^{\mathrm{T}}\mathbf{\boldsymbol{L}}\left(\mathbf{\boldsymbol{z}}_{k}-\bar{\mathbf{\boldsymbol{z}}}\right)~{}.$
(16)

By differentiating, the optimal scaling field is:

$\bar{\mathbf{\boldsymbol{z}}}=\frac{\sum_{c=1}^{C}\lambda_{k}\mathbf{\boldsymbol{z}}_{k}}{\sum_{c=1}^{C}\lambda_{k}}~{}.$
(17)

Therefore, at the optimum, the (weighted) mean log-sensitivity field must be
zero, and the mean image is a barycentre of the calibration images. To
accelerate convergence, this condition is enforced after each global
iteration.

In general, the variability of the sensitivity modulation smoothness across
images is unknown. Therefore all $\lambda_{k}$ are given the same value
$\lambda_{\text{array}}$. However, images acquired with the body coil will
have a flatter sensitivity field modulation, which can be incorporated by
setting $\lambda_{\text{body}}\gg\lambda_{\text{array}}$.

Code availability: The code used to fit the generative model is available at
https://github.com/balbasty/multi-bias. A modified version of the hMRI toolbox
that integrates this approach and enables $B_{1}^{+}$ correction on a per-
contrast basis is available at https://github.com/balbasty/hMRI-toolbox. The
ratio approach can be performed natively with the hMRI toolbox:
https://github.com/hMRI-group/hMRI-toolbox.


# Incorporating the Cosmological Constant in a Modified Uncertainty Principle

S. Ahmadi<EMAIL_ADDRESS>Department of Physics, Ayatollah Amoli
Branch, Islamic Azad University, Amol, Iran  E. Yusofi
e.yusofi@ipm.ir(Corresponding author) Department of Physics, Ayatollah Amoli
Branch, Islamic Azad University, Amol, Iran  School of Astronomy, Institute
for Research in Fundamental Sciences(IPM), P. O. Box 19395-5531,Tehran, Iran
Innovation and Management Research Center, Ayatollah Amoli Branch, Islamic
Azad University, Amol, Mazandaran, Iran  M. A. Ramzanpour
<EMAIL_ADDRESS>Department of Physics, Ayatollah Amoli Branch, Islamic
Azad University, Amol, Iran  Innovation and Management Research Center,
Ayatollah Amoli Branch, Islamic Azad University, Amol, Mazandaran, Iran

###### Abstract

The existence of a tiny but non-zero cosmological constant seems to be a
fundamental challenge for physics. This study examines the cosmological
constant problem and modified uncertainty principle within a unified framework
inspired by a void-dominated cosmology. We model voids/halos as spherical
bubbles/drops for simplification and analysis. Our heuristic calculations show
significant variations in surface energy values from the largest to smallest
scales, resulting in a substantial disparity (approximately $122$ orders of
magnitude) in the values of the cosmological constant. Our method suggests
that the difference in the values of the cosmological constant is inherent and
should be considered natural. As a main outcome of this research, we propose a
new form of extended uncertainty principle that incorporates cosmological
constant.

Keywords: Cosmological Constant; Surface Energy; Cosmic Void; Uncertainty
Principle

PACS: 98.80.Bp; 95.36.+x; 98.80.Es

##  I Introduction and Motivation

Some of the key challenges in modern cosmology include dark matter, dark
energy, the cosmological constant problem, and Hubble tension Silk (2017);
Peracaula (2022); Abdalla  _et al._ (2022); Vagnozzi (2023). These significant
physical hurdles highlight the inability of the standard model of cosmology to
accurately address many of these issues Perivolaropoulos and Skara (2021);
Abdalla  _et al._ (2022). Dark energy, which drives the accelerating expansion
of the universe at large scales Amendola and Tsujikawa (2015), has been
explored as potentially originating from the amplification of quantum
fluctuations of a light field during inflation Belgacem and Prokopec (2022).
Additionally, dark matter, as an invisible substance with gravitational
influence, plays a crucial role in the interconnection and balance of
galaxies, clusters, and superclusters. Both dark matter and dark energy have
been thoroughly confirmed through various direct and indirect observational
methods. However, despite decades of research, no plausible physical source
has been identified for these major contributions to the universe’s matter and
energy Zwicky (1937); Peebles and Ratra (2003). By the way, some models do not
find it necessary to postulate dark matter and dark energy Sanders and McGaugh
(2002); Verlinde (2011); Joyce  _et al._ (2016).

The cosmological constant problem is rooted in the fact that the observed
acceleration is not easily explained by the known matter and energy content of
the universe. The most commonly accepted explanation for the acceleration is
the existence of dark energy as a cosmological constant, a hypothetical form
of energy that permeates all of space and exerts a negative pressure that
drives the acceleration. However, the predicted value of the dark energy
density from quantum field theory is far larger than the observed value,
leading to a discrepancy of more than $120$ orders of magnitude, making it one
of the most significant discrepancies between theory and observation in all of
physics. This discrepancy is known as the fine tuning problem, and its
resolution remains one of the most significant challenges in modern cosmology
Weinberg (1989); Carroll (2001); Martin (2012); Amendola and Tsujikawa (2015).

In this study, we propose a bubble-drop model for the universe, which is
populated by two types of spherical objects: underdense ”cosmic voids” and
overdense ”matter halos”. The assumption is that cosmic voids (bubbles) are
encircled by matter halos (drops) with a shell-like distribution on their
borderEinasto  _et al._ (2016). Considering the shells as the ideal separating
surface between the under-density voids and the over-density clusters, we can
calculate the resulting surface tension (energy) through dimensional
calculation (see Table 1 in Yusofi  _et al._ (2022)). Based on this ideal
assumption, we can take into account the mass density and surface energy of
these spherical objects. We then calculate the surface energy for the possible
largest and smallest objects in our modeled universe to find a potential
explanation for the cosmological constant problem. In addition, we will offer
a possible relation between surface energy, cosmological constant and surface
gravity to find a plausible source for the acceleration of the universe at the
large scales of the universe. Since the proposed model covers spherical
objects from the largest to the smallest possible radii (lengths), as a
potential result of this study, we will present a modified uncertainty
principle (MUP), which may include a variable cosmological constant.

The MUP with minimum length (GUP) Ong (2018); Heydarzadeh  _et al._ (2022);
Bosso (2017) and with maximum length (EUP) Perivolaropoulos (2017) is a
theoretical concept that extends the traditional Heisenberg uncertainty
principle to incorporate the effects of quantum gravity Nozari and Pedram
(2010); Nozari and Etemadi (2012). In this modified version, the uncertainty
in the measurement of certain pairs of physical properties, such as position
and momentum, is constrained by both a minimum and maximum length scale. This
suggests that there are fundamental limits to the precision with which these
properties can be simultaneously known, and it implies a fundamental
granularity to space-time at the quantum level.

Considering that the average diameter of cosmic voids is typically $\simeq
100{~{}\rm Mpc}$, we explore whether a cosmic void, as a cosmic object, can
effectively represent the local and global behavior of the universe. It is
worth noting that given the web-like structure of the universe and its
hierarchical formationVan de Weygaert and Platen (2011); Sheth and van de
Weygaert (2004), present cosmic voids/clusters may be much larger than
$100{~{}\rm Mpc}$ Rudnick  _et al._ (2007). For instance, the Boötes and KBC
voids are the largest known spaces, encompassing a significant portion of the
visible universe’s diameter Einasto  _et al._ (2016); Cai  _et al._ (2022);
Nadathur  _et al._ (2014).
In Section II, we explore the coexistence and interplay of cosmic voids and
matter halos within the context of void-dominated cosmology. Moving to Section
III, we employ heuristic calculations to derive mass density, surface tension,
and cosmological constant for various cosmic objects, comparing these values
to the order of magnitude of the entire universe. Furthermore, we will discuss
the possible relationship between surface gravity and surface energy to
explain the present repulsive acceleration on the large scale of the universe.
In Section IV, we will have a brief discussion on the cosmological constant
problem within this framework, drawing upon the concept of surface energy
associated with cosmic voids. As a potential outcome of this study, we propose
a new kind of extended uncertainty principle that incorporates a scale-
dependent cosmological constant in In Section V. In the final section, we will
summarize the results of this study.

##  II Cosmic voids in Void-dominated Cosmology

In the cosmic web’s structure, cosmic voids constitute the main volume of
space, and the interconnected void shells serve as the scaffolding for this
expansive structure, encompassing both normal and dark matter. Despite cosmic
voids being significantly larger than local scales and smaller than cosmic
scales, we will demonstrate that they can be effectively studied as a
fundamental component within the cellular structure of the cosmic web.

The conventional models of cosmology overlook the statics, dynamics, and
evolution of supervoids, which form the dominant volume of local and global
scales within the cosmos. Supervoids are not only non-empty but also possess
energy density and undergo evolution. Due to their substantial size, they are
more prone to merging and are therefore strong candidates for influencing
cosmic scales Van de Weygaert and Platen (2011); Higuchi and Inoue (2018);
Yusofi  _et al._ (2022); Pandey (2019); Pandey and Das (2019). Consequently,
it seems plausible that these large inhomogeneities play a role in the
dynamics of the universe and have an impact on the evolution of cosmic
parameters Nan and Yamamoto (2022); Vigneron and Buchert (2019); Buchert
(2018); Srivastava and Dutta (2007).

The present cosmos contains a network of cosmic voids where multiple
superclusters and small to large galactic objects are merging. While much of
the mass in the Universe is concentrated in virialized structures, the
majority of the volume is taken up by large underdense voids.Voids dominate
the galaxy and matter distributions on a megaparsec (or larger) scale Sheth
and van de Weygaert (2004). In a void-based explanation of structure
formation, matter is compressed between expanding voids, and sheets and
filaments emerge at the void walls’ intersections. In the large scale
overview, the universe is void-dominant, while in smaller over-dense regions,
it consists of clusters, filaments, and nodes of matter in a matter-dominant
state. Research has shown that voids are counterparts of massive clusters, and
their coexistence and continuous integration with superclusters, as well as
the merging of supervoids, contribute significantly to the universe’s
structure formation. The increase in the size of cosmic voids after merging
exerts an effective repulsive force on the galaxies situated on their shell
Van de Weygaert and Platen (2011); Yusofi  _et al._ (2022). Under these
conditions, it can be assumed that the cosmic fluid at a large-scale overview
consists of merging and expanding vast voids, leading to a universe
increasingly dominated over time by larger cosmic voids and consequently an
accelerating expansion universe.

In a void-dominated universe Yusofi  _et al._ (2022); Correa (2021); Chan  _et
al._ (2019); Redmount (1988), voids (bubbles) constitute the main distribution
of the two-phase cosmic webYusofi (2018). By equating the energy density of
cosmic voids with the vacuum energy density, it has been shown that the value
estimated for the cosmological constant is very close to that reported by
Planck’s observations and has the same order of magnitude Yusofi  _et al._
(2022). Our hypothesis about the effect of supervoids on the large-scale
dynamics of the universe and cosmology in Yusofi  _et al._ (2022); Mohammadi
_et al._ (2023) could be consistent with the backreaction effects of density
inhomogeneities in cosmology Buchert  _et al._ (2015) and the configurational
entropy of the cosmic web and its evolution Pandey (2019); Pandey and Das
(2019).

The void-dominated cosmology Yusofi  _et al._ (2022) challenges the
traditional notion of matter distribution in the universe. Instead of
considering matter as filling the space between cosmic voids, we now view our
web-like universe as consisting of almost inter-empty inter-connected bubbles.
This means that matter, both visible (light) and invisible (dark), is
primarily concentrated on the borders of these cosmic voids, forming sheet-
like structures. This proposed change has significant implications for our
understanding of the cosmos. Firstly, it suggests that the matter in the
universe is not randomly and irregularly distributed in space but rather
concentrated in specific regions. Secondly, this alternative viewpoint
provides a potential explanation for the observed large-scale structures in
the universe, such as galaxy clusters and superclusters. These structures
could arise from the sheet-like arrangements of matter along the borders of
cosmic voids. Furthermore, this perspective may shed light on the nature of
dark matter and dark energyMohammadi  _et al._ (2023); Yusofi  _et al._
(2022).

In this study, over-dense objects, such as galaxies, clusters, their
superclusters and the dark matter around them, are modeled as ”drops” inside
the spherical-shaped ”matter halos”Navarro  _et al._ (1996), while the small
and large voids within the spherical cosmic voids. are modeled as ”bubbles”
Sheth and Tormen (1999); Yusofi  _et al._ (2022); Yusofi and Mohsenzadeh
(2010). Imagine that we were able to populate our Universe with these spheres.
In a two-phase mixture of drops and bubbles, not only do drops absorb drops,
but bubbles also absorb other bubbles and push droplets from the center to
their boundary (the direction of the red arrows in Fig. LABEL:Fig:I on the
left is from the center to the border of the bubble). The merger of bubbles
increases their size and decreases their density. Physical simulations of
redshifts with various displacements indicate that local scales become denser
over time, while the density of large scales decreases. Van de Weygaert and
Platen (2011).

Cosmic voids refer to vast, less-dense regions in the universe where there is
very little matter concentrated on the border of them, while matter halos are
dense regions where matter, such as galaxies and dark matter, is concentrated
in the center of them. The phrase ”cosmic voids versus matter halos”
essentially contrasts these two types of structures in the universe,
highlighting the difference between the empty voids and the dense,
concentrated matter halos.

##  III Mass Density and Surface Energy of Cosmic Voids

The discovery of the cell structure of the Universe revealed surprising empty
regions: voids. For years, the under-dense regions of the cosmic web were
unexplored due to a lack of information and the inability of surveys to
measure large portions of the sky with sufficient depth. This is now changing.
The era of large-scale surveys is providing us with an incredible amount of
data to work with, along with the possibility of extracting information using
statistical averages of quantities in the sky. These features are essential
for transforming voids into a potentially powerful tool for constraining
cosmology Pisani  _et al._ (2015).

In this section, we want to show whether a single cosmic void can be a good
representation of local and global scales or not? For this purpose, we first
calculate the mass density of an ideal spherical supervoid and show that its
magnitude is about one-tenth of the average density of the whole universe.
Then, considering the surface energy of a cosmic void.

###  III.1 Mass Density of Cosmic Voids

Cosmic voids are, on average, spherical, making the application of the Alcock-
Paczyński test not only feasible but also with reduced systematics Lavaux and
Wandelt (2012a); Hamaus  _et al._ (2020). Assume a perfectly empty spherical
void with a total mass accumulated on the shell. The mass density of it can be
calculated from the simple relation below,

$\rho_{\rm i}=\frac{3{M_{\rm{i}}}}{4\pi{\bar{r}_{\rm{v}}^{3}}}.$ (1)

Here $M_{\rm{i}}$ is the mass of the supercluster enclosed the cosmic void and
$\bar{r}_{\rm{v}}$ is the average (effective) radius of a cosmic void. Taking
into account the values of Table 1. for the mass and radius of the Laniakia
supercluster, we obtain

$\rho_{3}=1.70\times 10^{-27}{\rm{kg.m^{-3}}}.$ (2)

is very close to the universe’s average critical mass density of the universe
Cheng (2010a)i.e.

$\rho_{0,c}=1.88\times 10^{-26}{\rm{kg.m^{-3}}}.$ (3)

and is about one order smaller than that. This density deficit appears to be
due to the assumption of an empty bubble, while in a more real model inside
the cosmic voids, there are some galactic gas and sub-voids with a lower
density than the shell in the form of stacked voids Lavaux and Wandelt
(2012a).

###  III.2 Surface Energy of Cosmic Voids

The pressure difference between the inside and outside of a bubble is obtained
from the following equation,

$\Delta{P}=\frac{2\sigma}{\bar{r}_{v}}.$ (4)

Here, $\sigma$ represents the surface tension for bubble (drop). The relation
(4) may related to the Darmois-Israel theory is a way of describing the
boundary conditions for a thin shell of matter in general relativity. It tells
us how the geometry and the stress-energy tensor of the shell are related to
the metrics of the two regions that the shell separates Darmois (1927). The
basic idea of Darmois-Israel theory is to integrate the Einstein field
equations across the shell, which is assumed to be a singular hypersurface
with a Dirac delta distribution of matter. This leads to the following
equation:

$[K_{ij}]-[K]h_{ij}=-8\pi S_{ij}$ (5)

where $[K_{ij}]$ and $[K]$ are the jumps of the extrinsic curvature and its
trace across the shell, $h_{ij}$ is the induced metric on the shell, and
$S_{ij}$ is the stress-energy tensor of the shell. It is similar to equation
(4), which connects the pressure difference inside the bubble and outside it
to the surface energy of the shell containing the distribution of matter. The
equation (5) is known as the Darmois-Israel junction condition, and it
determines how the shell affects the spacetime curvature Chu and Tan (2022).
Further investigation of this matter will be addressed in the following
researches.

To calculate the surface tension of a single void/cluster, we can use the
following heuristic method Yusofi  _et al._ (2019, 2022),

$\sigma_{\rm i}\equiv\frac{\rm Energy}{\rm Area}={\frac{M_{\rm i}c^{2}}{\pi
R_{\rm i}^{2}}}.$ (6)

The surface energy (energy per unit area) of some cosmic objects are listed in
tables 1 and 2, respectively.

Upon examining table 1, it becomes evident that larger cosmic bodies exhibit
lesser surface energy as a result of their diminishing density. Conversely,
table 2 demonstrates that larger objects possess higher surface energy due to
their increased density. This suggests that cosmic bodies are situated within
void-dominated regions, while smaller astronomical objects are located within
matter-dominated regions such as the galactic (and dark matter) halos.
Consequently, in cosmological investigations, it is imperative to regard the
universe as a void-dominant region, as this approach aligns more logically
with the available evidence.

Since most of the calculations in this paper are dimensional and approximate,
slight changes in the formulas and their coefficients do not have a
significant impact on our results. It should also be noted that in the more
actual model, the voids are highly non-spherical, and our results are at best
a rough order-of-magnitude estimate.

Cosmic simulations, such as large-scale numerical simulations used in
cosmology, do indeed show the expansion of cosmic voids over time van de
Weygaert (2014).

Table 1: Surface tension (energy) $\sigma_{\rm i}$ and cosmological constant $\Lambda_{\rm i}$, for the clustered cosmic objects including superclusters (Sc), clusters (C), and galaxies i. Clustered |  $M_{\rm i}$ |  $R_{\rm i}$ |  $\sigma_{\rm i}$ |  $\Lambda_{\rm i}$
---|---|---|---|---
Objects | ($10^{47}{\rm kg}$) | ($10^{24}{\rm m}$) | ($10^{15}{\rm J.m^{-2}}$) | ($10^{-52}{\rm m^{-2}}$)
1\. Corona Sc |  $0.20$ |  $1.50$ |  $0.25$ |  $0.6645$
2\. Virgo Sc |  $0.03$ |  $0.50$ |  $0.34$ |  $0.8970$
3\. Laniakea Sc |  $1.00$ |  $2.40$ |  $0.50$ |  $1.2979$
4\. Caelum Sc |  $4.00$ |  $4.30$ |  $0.62$ |  $1.6172$
5\. Virgo C |  $0.002$ |  $0.045$ |  $2.83$ |  $7.3833$
6\. Coma C |  $0.012$ |  $0.10$ |  $3.44$ |  $8.9707$
7\. Galaxies G |  $0.0004$ |  $0.016$ |  $4.48$ |  $11.6806$
8\. Milky Way |  $0.00002$ |  $0.00087$ |  $75.74$ |  $1975.3200$
9\. Andromeda |  $0.00003$ |  $0.001$ |  $85.98$ |  $2242.6800$
10\. UGC02885 |  $0.000004$ |  $0.00036$ |  $88.46$ |  $2307.2800$
Table 2: The surface tension $\sigma_{\rm i}$, and cosmological constant $\Lambda_{\rm i}$ for a bubbly cosmos in the possible smallest and the largest scales. Scales |  $M_{\rm i}({\rm kg})$ |  $R_{\rm i}({\rm m})$ |  $\sigma_{\rm i}({\rm j.m^{-2}})$ | $\qquad\Lambda_{\rm i}({\rm m^{-2}})$
---|---|---|---|---
Minimum (Planck) |  $2.18\times 10^{-8}$ |  $1.62$$\times 10^{-35}$ |  $2.38\times 10^{78}$ |  $6.08\times 10^{70}$
Maximum (Caelum Sc) |  $4.00\times 10^{47}$ |  $4.30$$\times 10^{24}$ |  $0.62\times 10^{15}$ |  $1.62\times 10^{-52}$

##  IV From Surface Energy to Cosmological Constant

Based on the analysis and heuristic calculations conducted in this article so
far, maybe the isotropy in the background temperature of the universe (CMB) is
due to the homogeneity distribution of such cosmic voids across the cosmos.
With this point of view, our universe has a cellular structure, and each cell
of it is a cosmic void with almost the same size, the same mass density, and
the same surface energy. Also, considering the impactful idea of stacked voids
has addressed many obstacles in utilizing voids in cosmology. By identifying
voids within a particular redshift range and stacking them according to their
size, it is now possible to determine their average shape. In a universe that
is homogeneous and isotropic, voids display an average spherical shape without
any preferred directions Lavaux and Wandelt (2012b).

Based on this rationale, our proposed void-dominated model satisfies the
requirements of homogeneity and isotropy, allowing us to apply Friedman’s
equations for further investigation. In the standard $\Lambda$CDM model of
cosmology, the Friedman equation is written as follows Cheng (2010b);
Srivastava (2008),

$H^{2}={\frac{8\pi{G}}{3}}\rho_{\rm
m}+\frac{\Lambda{c^{2}}}{3}={\frac{8\pi{G}}{3}}(\rho_{\rm
m}+\rho_{\rm\Lambda}).$ (7)

In the standard $\Lambda$CDM model, the energy density is mainly divided into
two parts: $\Omega_{\rm m}$ and $\Omega_{\Lambda}(=1-\Omega_{\rm m})$. We want
to attribute the contribution of the cosmological constant (whose source is
still unknown) to the energy density and pressure of cosmic voids. For the
current state of the universe, in addition to the agreed-upon matter
contribution, there is also an important contribution attributed to the
cosmological constant. However, from our perspective, it is the energy density
and expanding pressure of the cosmic voids that affect the large-scale
behavior and dynamics of the universe, rather than the cosmological constant.
From our point of view, the cosmic fluid (web) is a two-phase fluid consisting
of two parts: the over-dense matter halos denoted by $\Omega_{\rm m}$ and the
under-dense cosmic voids denoted by $\Omega_{\rm v}$ (instead of
$\Omega_{\Lambda}$) Libeskind  _et al._ (2018); Yusofi  _et al._ (2022). The
difference here is that the source of $\Omega_{\rm v}$, the cosmic void, is
completely obvious, compared with the cosmological constant,
$\Omega_{\Lambda}$, which is unknown (dark). _The proposed model views cosmic
voids as the largest inhomogeneities in the universe, distributed almost
homogeneously throughout. There is a uniform distribution of cosmic voids,
with most of the matter halos situated on their borders._ As a result, we can
modify the Friedmann equation as follows

$H^{2}={\frac{8\pi{G}}{3}}(\rho_{\rm m}+\rho_{\rm v}).$ (8)

By equating relations (7) and (8), we can obtain

$\Lambda=\frac{8\pi{G}\rho_{\rm v}}{c^{2}}.$ (9)

With consideration $P_{\rm v}=w\rho_{\rm v}c^{2}$, we have

$\Lambda=\frac{8\pi{G}{P_{\rm v}}}{w{c^{4}}}.$ (10)

In this relation, the $P_{\rm v}\simeq\Delta P$ is difference pressure between
the inside and outside of the void in void-dominated cosmic web (web-like
cosmic fluid 111It is necessary to pay attention to the fact that most of the
energy density of the bubbles is concentrated on their border, while the
energy of drops is located in their volume and concentrated in their central
points.). If we consider the cosmic web as consisting of interconnected ideal
spherical bubbles, we can use the following formula for the internal pressure
of each void Butt  _et al._ (2003); Reichl (2016),

${P_{\rm v}}=\frac{2\sigma_{\rm i}}{\bar{r}_{\rm v}},$ (11)

Therefore from (10), we can write Yusofi  _et al._ (2022),

$\Lambda_{\rm i}=\frac{8\pi{G}}{{w{c^{4}}}}\frac{2\sigma_{\rm
i}}{\bar{r}_{\rm{v}}}.$ (12)

By inputting the required values, we achieve the findings in Table 1.

###  IV.1 Variable Cosmological Constant from Smallest to Largest Scales

In void-dominated cosmology, the cosmological constant has variable values on
different scales Xu  _et al._ (2010); Carroll (2001); Peebles and Ratra
(1988). Since the sizes of voids are diverse, we must obtain the largest
values for surface tension and cosmological constant in the smallest scales
and vice versa. To illustrate this more precisely, we consider the initial
web-like structure as a collection of very small Planck-sized voids connected
to each other in the foam-shape. By utilizing relation (12), we calculate the
surface tension and cosmological constant ($\Lambda_{\rm{Planck}}\sim
10^{+70}$) for a hypothetical single bubble with the Planck radius in Table 2.
The cosmological constant in the latest observational data is reported as
Aghanim  _et al._ (2020),

$\Lambda_{\rm{obs}}=1.1056\times 10^{-52}{\rm{m^{-2}}}.$ (13)

In our model, by using (12) for the largest object such as Laniakia
supercluster (please refer the row 3 in Table 1), the cosmological constant of
the model is estimated as Yusofi  _et al._ (2022),

$\Lambda_{3}=1.2979\times 10^{-52}{\rm{m^{-2}}}.$ (14)

In Table 1, we observe that the cosmological constant and mass density for
each cosmic void are equivalent to the values for the entire universe and are
very close to it. Thus, given the values obtained for the $\rho_{\rm i}$,
$\sigma_{\rm i}$ and $\Lambda_{\rm i}$, it seems that a cosmic void can be a
good indicator of the global behavior of the accelerating universe. Therefore,
the surface tension of expanding cosmic voids may be regarded as a possible
source to produce accelerating universe Yusofi  _et al._ (2022).

Our model can now address the discrepancy in cosmological constant values at
the Planck scale with the smallest radius (minimum possible length) and the
present universe (see Table 2), which related to the superclusters/voids with
the largest radius (maximum possible length) and is of the following order of
magnitude,

$\frac{\Lambda_{\rm Planck}}{\Lambda_{\rm Obs}}=\frac{\Lambda_{\rm
max}}{\Lambda_{\rm min}}\sim 10^{+122}.$ (15)

Here, $\Lambda_{\rm max}=\Lambda_{\rm UV}$ could be seen as a high-energy
extension of the traditional cosmological constant, resulting in $\Lambda_{\rm
min}=\Lambda_{\rm IR}$ in the low-energy range Shalyt-Margolin (2008); Jejjala
and Minic (2007); Padmanabhan (2006). In an intriguing outcome, Tables 1 and 2
along with finding (15) may validate the conjugate connection between the
cosmological constant and the volume of the four-dimensional space-time $V$,
i.e. $\Delta\Lambda~{}\Delta V\sim\hbar$ as discussed in Jejjala and Minic
(2007).

##  V Modified Uncertainty Principle with Cosmological Constant

The cosmological constant is a concept that has its origins in gravity, as a
modification to Einstein’s equations of general relativity, and in cosmology,
as a crucial component in understanding the dynamics and evolution of the
universe. The existence of a small, non-zero cosmological constant remains a
major puzzle in fundamental physics Bishop  _et al._ (2022); Lake  _et al._
(2023). At first glance, arguments from quantum field theory would suggest a
cosmological constant that is potentially $10^{120}$ times greater than the
one observed. It is widely held that a comprehensive theory of quantum gravity
could reconcile this substantial disparity between theory and observation
Bishop  _et al._ (2022). According to the discussions and results of the
previous sections, in this section, we examine the potential of MUP models,
which are models of quantum gravity motivated by phenomenology, in addressing
the cosmological constant problem.

In the realm of gravity, the application of quantum mechanics imposes a
restriction on measurements through a minimum length. This length, is
intricately linked to Planck’s length by (GUP). Similarly, (EUP) tackles the
limiting factor on the minimum momentum in (anti) de Sitter spacetime. This
restriction is characterized by the parameter $\alpha$, which symbolizes the
radius of the Hubble horizon. The modified uncertainty principle (MUP) expands
upon the commutation relation between position and momentum operators
Perivolaropoulos (2017),

$[X,P]=\frac{\hbar}{2}(1+\alpha X^{2}+\beta P^{2}).$ (16)

There are multiple ways to determine the operator representation related to
this the commutation relation . The position and momentum operators satisfying
this commutation can be defined by utilizing the operators $x$ and $p$, which
follow the conventional commutation relation $[x,p]=i\hbar$. As an
illustration, one may consider the following,

$X=x,\;\;\;\mbox{and}\;\;P=p(1+\alpha x^{2}+\beta p^{2}).$ (17)

Based on this definition, the more general form of the MUP includes explicit
maximum and minimum values for both position and momentum uncertainties and
can be written as follows,

$\displaystyle\Delta X\Delta P\geq\frac{\hbar}{2}\frac{1}{1-\beta\Delta
P^{2}}\frac{1}{1-\alpha\Delta X^{2}}~{}.$ (18)

Here $\Delta X$ and $\Delta P$ represent position and momentum uncertainties,
while $\alpha$ and $\beta$ are the parameters.

Motivated by the existence of the Planck length and the largest super
cluster/voids as the scales which set natural minimum and maximum possible
lengths, we now turn our attention to relation (18) by considering the
following relationship among $\Delta X$, $\alpha$ and $\beta$ (please refer to
section III of Perivolaropoulos (2017)),

$\Delta X\propto\frac{1}{\sqrt{\alpha}}\propto\sqrt{\beta}~{}.$

In the proposed bubbly model of the universe (as per Tables 1 and 2), the size
of spherical objects can vary from the smallest to the largest radius. As a
result, the maximum position uncertainty can be expressed as,

$\displaystyle l_{max}\equiv\Delta
X_{max}\propto\frac{1}{\sqrt{\alpha_{min}}}\propto\sqrt{\beta_{max}}~{},$ (19)

and the minimum position uncertainty can be expressed as,

$\displaystyle l_{min}\equiv\Delta
X_{min}\propto\frac{1}{\sqrt{\alpha_{max}}}\propto\sqrt{\beta_{min}}~{}.$ (20)

By combining equations (19) and (20), we can obtain,

$\frac{\alpha_{\rm max}}{\alpha_{\rm min}}\sim\left(\frac{l_{\rm min}}{l_{\rm
max}}\right)^{-2}\sim\left(\frac{10^{-35}}{10^{+26}}\right)^{-2}=10^{+122}.$
(21)

Based on the Table 2 and the above discussions, especially the relations (15)
and (21), it can be concluded that the parameter $\alpha$ in (21) is
equivalent to the cosmological constant $\Lambda$. Dimensional analysis, as
suggested in previous works Bolen and Cavaglia (2005); Bambi and Urban (2008),
implies that the primary correction’s structure is proportional to
$\Lambda\sim{l_{\Lambda}}^{-2}\sim(\frac{c}{H_{0}})^{-2}$ Padmanabhan (2006).
Considering the merger of voids and their expansion, voids can evolve into
genuine super-Hubble bubbles Sheth and van de Weygaert (2004), which can have
a radius in the size of the horizon i.e. $l_{\rm horizon}$. Therefore, we can
express $l_{\rm horizon}\equiv
l_{\Lambda}=\sqrt{\frac{3}{\Lambda}}\propto\frac{1}{\sqrt{\Lambda}}$ as the
(anti-) de Sitter horizon Luminet (2013); Bambi and Urban (2008). As a final
result of this discussion, we can propose a new extension of uncertainty
principle including the cosmological constant as follows:

$\displaystyle\Delta X\Delta
P\geq\frac{\hbar}{2}\frac{1}{1-\beta_{0}\Lambda^{-1}\Delta
P^{2}}\frac{1}{1-\alpha_{0}\Lambda\Delta X^{2}}~{}.$ (22)

Here $\alpha\equiv\alpha_{0}\Lambda$, and $\beta\equiv\beta_{0}\Lambda^{-1}$.
In the final relation (22), $\Lambda$ have approximate values, so we require
non-zero $\alpha_{0}$ and $\beta_{0}$ as the dimensionless coefficients. Our
new extension of the uncertainty principle (22) is directly linked to the non-
zero cosmological constant, becoming an essential element for a more
comprehensive understanding of quantum spacetime in both small and large
scales of the cosmos. The constant $\alpha$ in our proposed MUP is not
meaningless but rather is proportional to the energy of the vacuum (quasi-
empty or void) space. In our view, the cosmological constant varies at
different scales as $\Lambda\sim R^{-2}$ (this point has been mentioned in
Luminet (2013) by G. Lemaitre in 1927), with the highest value of energy-
momentum ($\Delta P_{\rm max}$) at the smallest scale ($\Delta X_{\rm min}$)
and vice versa. But neither $\Delta X$ nor $\Delta P$ can be assigned a zero
value because the cosmological constant never disappears at any scale.

Although we didn’t resolve the cosmological constant issue or introduce a new
version of the uncertainty principle, we have determined that presence of a
non-zero cosmological constant is essential at all levels, and its value is
contingent on the selected regime. In our view, the significant contrast
between the highest and lowest values of the cosmological constant is not a
concern. Our proposal suggests that the difference in the values of
cosmological constant is inherent and should be considered natural. In other
hand, the presence of cosmological constant in the corrected relation of
Heisenberg’s uncertainty principle may be a sign of the unification of quantum
gravity and quantum cosmology in a single framework Rovelli (2004);
Chataignier  _et al._ (2023). It is essential to acknowledge that the findings
presented in this paper are approximate and heuristic. Nevertheless, given the
extremely small or large orders of magnitude of the values under
investigation, these approximations closely reflect reality.

##  VI Conclusions

We have examined supervoids as interconnected spherical bubbles within a void-
dominated cosmic fluid, with the total mass of a supervoid situated on the
shell formed by disk-shaped superclusters. Through heuristic calculations of
the mass density and cosmological constant of a single supervoid, we have
demonstrated that the obtained order of magnitude aligns interestingly with
their corresponding values for the entire observable universe.

The web-like cosmic fluid comprises two coexisting dynamical parts: galaxies,
clusters, superclusters, and nodes on one hand, and small voids and supervoids
on the other. In void-dominated cosmology, voids are dominant and their
existence and evolution play a crucial role in the accelerating expansion of
the cosmos. Our findings indicate that a cosmic void can serve as an excellent
representative for studying both local and global scales. For each cosmic
void, we derived a specific mass density and cosmological constant that, in
terms of order of magnitude, mirror those of the entire universe on any scale.
Additionally, we have demonstrated that the largest discrepancy between the
observable cosmological constant and the early foam-like cosmos (approximately
$10^{+122}$) can be logically explained within the void-based model of the
universe. We have been concluded that the presence of a cosmological constant
is necessary at any scales of cosmos, with its value depending on the chosen
regime. The substantial difference between the highest and lowest value of the
cosmological constant is not an issue, and according to our proposal, this
difference is natural. Furthermore, by introducing an extended version of the
uncertainty principle, we have demonstrated that a non-zero cosmological
constant is essential for establishing the minimum and maximum length in
physics. From our perspective, this non-zero cosmological constant could be
attributed to the existence of energy in near-empty space at any scale.

While our analysis is less rigorous and largely heuristic, it provides a clear
view of problems in gravity and cosmology,by keeping the mathematics to the
bare minimum. An intriguing consequence of this study is that a cosmic void,
as a cell within the cellular structure of the cosmic web scaffolding, can
effectively describe the behavior of the universe on any scale. We anticipate
that by more thoroughly investigating the behavior of a single cosmic void as
a cosmos laboratory, both theoretically and observationally, we can provide
potential solutions to the significant challenges of physical cosmology on
local and global scales, and these are the topics we will address in future
projects.

## Declaration of Competing Interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

## Acknowledgements

EY would like to acknowledge A. Talebian and H. Sheikhahmadi for their
constructive discussions and for his help in drawing the tables. This work has
been supported by the Islamic Azad University, Ayatollah Amoli Branch, Amol,
Iran.

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