necessary, as the singular support condition could be understood just as a
usual support condition over $\mathbb{A}^{2}/\mathbb{G}_{m}.$ However, if we
had picked the opposite sign for our stability parameter, then 4.5.6 would be
replaced with a “purely microlocal” — that is, disjoint from the zero-section
— localization. The easiest way to compute the result of this localization on
4.5.5 would be to apply Koszul duality to replace the $(-1)$-shifted normal
bundles with $2$-shifted conormal bundles, and then apply the fact that Koszul
duality exchanges singular-support conditions (and microlocalizations) with
usual support conditions (and usual localizations).

##  5\. Proof of Theorems A & B

We now prove our first main theorem, an equivalence between the (Gale dual)
pair of 2-categories $\mathcal{O}$ we have defined:

###### Proof of A.

Our starting point is the equivalence of 2-categories 0.2.3 proved as
[GMH]*Theorem G, namely

(5.0.1)
$\mathsf{PervCat}_{\mathbb{L}_{G}}(\mathbb{C}^{n}/G)\simeq\mathsf{CohCat}_{\mathbb{L}_{F^{\vee}}}(\mathbb{C}^{n}/F^{\vee}),$

which sends the projective $\mathcal{P}^{\alpha}_{G}$ in the left-hand side to
the
$\mathsf{Coh}(X_{F^{\vee}}\times_{\mathbb{C}^{n}/F^{\vee}}X_{F^{\vee}})$-module
category
$\mathcal{S}_{F^{\vee}}^{\alpha}:=\mathsf{Coh}(X_{F^{\vee}}\times_{\mathbb{C}^{n}/F^{\vee}}X_{F^{\vee}}^{\alpha}).$
Observe that for $I\subset I,$ the subcategory of
$\mathsf{CohCat}_{\mathbb{L}_{F^{\vee}}}(\mathbb{C}^{n}/F^{\vee})$ generated
by the $\mathcal{S}_{F^{\vee}}^{\alpha}$ for $\alpha\in I$ is
$\mathsf{CohCat}_{\mathbb{L}_{F^{\vee}}^{I}}(\mathbb{C}^{n}/F^{\vee}),$ where
we write $\mathbb{L}_{F^{\vee}}^{I}=\bigcup_{\alpha\in
I}\mathbb{L}_{F^{\vee}}^{\alpha}$ for the union of components of
$\mathbb{L}_{F^{\vee}}$ corresponding to $\alpha\in I.$

Now recall that Lemma 3.13 gives a description of
$\mu\mathsf{PervCat}(\mathbb{L}_{G}(t,m))$ as the subquotient

(5.0.2)
$\mu\mathsf{PervCat}(\mathbb{L}_{G}(t,m))\simeq\frac{\langle\mathcal{P}_{G}^{\alpha}\mid\text{$\alpha$
$t$-semistable}\rangle}{\langle\mathcal{P}_{G}^{\beta}\mid\text{$\beta$
$m$-unbounded}\rangle}$

of the left-hand side of 5.0.1.

By the above observations, 5.0.2 is equivalent to the quotient 2-category

(5.0.3)
$\mathsf{CohCat}_{\mathbb{L}^{t\text{-bdd}}}(\mathbb{C}^{n}/F^{\vee})/\mathsf{CohCat}_{\mathbb{L}^{t\text{-bdd},m\text{-unstab}}}(\mathbb{C}^{n}/F^{\vee}),$

and by Corollary 4.31, the quotient 5.0.3 is equivalent to the 2-category
$\mu\mathsf{CohCat}(\mathbb{L}_{F^{\vee}}(m,t)).$ By construction, this
equivalence has identified $\mathcal{P}^{\alpha}_{G}$ with
$\mathcal{S}^{\alpha}_{F^{\vee}}.$ ∎

We now proceed to the proof of B. This theorem is really two theorems, with
equivalences 0.4.1 and 0.4.2 identifying the decategorifications of the A- and
B-side 2-categories, respectively. We will deal with each of these separately,
beginning with the B-side.

######  Notation 5.1.

From now on, $\Bbbk$ is assumed to be a field of characteristic 0. For a
$\Bbbk$-algebra or $\Bbbk$-linear category, we write
$(-)_{\mathbb{Z}/2}:=(-)\otimes_{\Bbbk}\Bbbk((u))$ for the 2-periodization,
where $u$ is a variable of degree 2.

Our main tool will be an identification of the periodic cyclic homology of
coherent-sheaf categories of stacks. From [Feigin-Tsygan], when $X$ is an
affine scheme there is an identification ${\sf HP}({\sf Perf}(X))\simeq{\sf
C}^{*}_{{\sf dR}}(X)_{\mathbb{Z}/2}$ between the periodic cyclic homology of
${\sf Perf}(X)$ and the 2-periodized infinitesimal cohomology of $X$. In
[Preygel-loopspaces], this result was generalized to the case where we replace
${\sf Perf}$ with $\mathsf{Coh}$ (allowing $X$ more generally to be a quasi-
compact and separated algebraic space): in this case the statement remains
true if we replace de Rham cochains ${\sf C}^{*}_{{\sf dR}}(X)_{\mathbb{Z}/2}$
with Borel-Moore chains ${\sf C}_{*}^{{\sf BM},{\sf dR}}(X)_{\mathbb{Z}/2}.$

Generalizing from algebraic spaces to stacks presents additional difficulties.
This case was studied in [Chen-Koszul], which gave a periodic-cyclic version
of the Atiyah-Segal completion theorem.

######  Notation 5.2.

Let $G$ be a reductive group. If $A$ is an algebra linear over the 2-periodic
representation ring ${\sf HP}({\sf Perf}(BG))=\mathcal{R}(G)_{\mathbb{Z}/2}$
of $G,$ we write $A_{\hat{e}}$ for the completion of $A$ at the augmentation
ideal.

######  Theorem 5.3 ([Chen-localization]*Theorem 4.3.2).

Let $G$ be a reductive group acting on a smooth quasi-projective variety $X.$
Then ${\sf HP}({\sf Perf}(X/G))$ is linear over ${\sf HP}({\sf Perf}(BG)),$
and there is an equivalence

(5.0.4) ${\sf HP}({\sf Perf}(X/G))_{\hat{e}}\simeq{\sf C}^{*}_{{\sf
dR}}(X/G)_{\mathbb{Z}/2}$

between the completion of ${\sf HP}({\sf Perf}(X/G))$ and 2-periodic de Rham
cochains on $X/G.$

For a smooth variety, the categories of coherent sheaves and perfect complexes
are equivalent. But we need a slight enhancement of this calculation beyond
the smooth case.

######  Corollary 5.4.

Let $X=\bigsqcup X^{\alpha}\to\mathbb{C}^{n}$ as in Definition 1.14, equipped
with its action of $G\subset(\mathbb{G}_{m})^{n}$ as usual, and write
$\mathfrak{X}$ for the fiber product
$\mathfrak{X}:=X\times_{\mathbb{C}^{n}}X$. Then there is an equivalence

${\sf HP}(\mathsf{Coh}(\mathfrak{X}/G))_{\hat{e}}\simeq{\sf C}_{*}^{{\sf
BM},{\sf dR}}(\mathfrak{X}/G)_{\mathbb{Z}/2}$

between the completion of ${\sf HP}(\mathsf{Coh}(\mathfrak{X}/G))$ and the
2-periodic Borel-Moore chains on $\mathfrak{X}.$

###### Proof.

First we observe that the statement is true when $G=D=(\mathbb{G}_{m})^{n},$
since by combining Koszul duality with a shearing equivalence, we have

$\mathsf{Coh}(\mathbb{A}^{n}[-1]/D)\simeq{\sf
Perf}(\mathbb{A}^{n}[2]/D)\simeq{\sf Perf}(\mathbb{A}^{n}/D),$

so we are reduced to the case when $\mathfrak{X}$ is smooth. From here, we can
reduce to the case of general $G$ by deequivariantization. ∎

The only other fact we will need about periodic cylic homology is the
following:

######  Lemma 5.5.

${\sf HP}(-)$ is a localizing invariant: it takes exact sequences of stable
categories to fiber sequences.111111We follow the convention of [Land-Tamme-
pullbacks], which differs from that of [BGT] in not requiring localizing
invariants to commute with filtered colimits. In the conventions of [BGT],
Hochschild and cyclic homology are localizing but periodic cyclic homology is
not.

###### Proof.

For Hochschild homology this is proved in [Keller-cyclic] or [Blumberg-
Mandell-localization]*Theorem 7.1. The result for ${\sf HP}$ follows, as ${\sf
HP}$ is obtained from ${\sf HH}$ by the exact functor $(-)^{tS^{1}}.$ ∎

###### Proof of B (B-side).

The canonical generator $\mathcal{S}$ of
$\mu\mathsf{CohCat}(\mathbb{L}_{F^{\vee}}(m,t))$ has endomorphism category
$\mathcal{A}_{F^{\vee}}^{t\text{-bdd}}/\mathcal{J}$, where
$\mathcal{J}\hookrightarrow\mathcal{A}_{F^{\vee}}^{t\text{-bdd}}$ is the ideal
generated by $\mathcal{A}_{F^{\vee}}^{t\text{-bdd},m\text{-unstab}}$ (and the
monoidal categories $\mathcal{A}_{F^{\vee}}$ are as in 4.26). We therefore
conclude that ${\sf HP}(\mathsf{end}(\mathcal{S}))$ is equivalent to the
quotient of ${\sf HP}(\mathcal{A}_{F^{\vee}}^{t\text{-bdd}})$ by the ideal
${\sf HP}(\mathcal{J})$ generated by ${\sf
HP}(\mathcal{A}_{F^{\vee}}^{t\text{-bdd},m\text{-unstab}})$. Observe that the
augmentation ideal for ${\sf HP}(BF^{\vee})$ is contained inside of ${\sf
HP}(\mathcal{J}),$ so that completion at the augmentation ideal does not
affect the resulting quotient algebra. By Corollary 5.4, the completion of
this algebra at the augmentation ideal is equivalent to the quotient of the
algebra ${\sf C}_{*}^{{\sf BM},{\sf
dR}}(X^{t\text{-bdd}}_{F^{\vee}}\times_{\mathbb{A}^{n}/F^{\vee}}X^{t\text{-bdd}}_{F^{\vee}})_{\mathbb{Z}/2}$
by the ideal generated by ${\sf C}_{*}^{{\sf BM},{\sf
dR}}(X^{t\text{-bdd},m\text{-unstab}}_{F^{\vee}}\times_{\mathbb{A}^{n}/F^{\vee}}X^{t\text{-bdd},m\text{-unstab}}_{F^{\vee}})_{\mathbb{Z}/2}.$
This agrees with the description of de Rham category $\mathcal{O}$ from
Theorem 2.6, up to 2-periodization. ∎

On the A-side, we begin with the non-microlocal computation.

######  Lemma 5.6.

Let $\mathcal{P}=\bigoplus\mathcal{P}^{\alpha}$ be the sum of generating
objects of $\mathsf{PervCat}_{\mathbb{L}}(\mathbb{C}^{n}),$ and similarly
$P=\bigoplus P^{\alpha}$ the sum of the projectives in
$\mathsf{Perv}_{\mathbb{L}}(\mathbb{C}^{n}).$ Then there is an equivalence

(5.0.5) ${\sf
HP}(\mathsf{end}_{\mathsf{Perv}_{\mathbb{L}}(\mathbb{C}^{n})}(\mathcal{P}))\simeq\mathsf{end}_{\mathsf{Perv}_{\mathbb{L}}(\mathbb{C})}(P)_{\mathbb{Z}/2}.$

Moreover, this equivalence intertwines the $G$-actions.

###### Proof.

It is sufficient to prove the case $n=1,$ since the general case is recovered
from this one as a tensor product. Let $\mathcal{A}=\mathsf{end}(\mathcal{P})$
and $A=\mathsf{end}(P).$ From the spectral decomposition

(5.0.6) $\mathcal{A}\simeq\mathsf{Coh}^{\mathbb{G}_{m}}\left((\mathbb{A}\sqcup
0)\times_{\mathbb{A}}(\mathbb{A}\sqcup
0)\right)\simeq\mathsf{Coh}^{\mathbb{G}_{m}}\left(\begin{array}[]{ll}\mathbb{A}^{1}&0\\\
0&\mathrm{N}^{*}_{0}\mathbb{A}^{1}\end{array}\right),$

we can see that there is an equivalence ${\sf HP}(\mathcal{A})\simeq{\sf
HH}(\mathcal{A})\otimes_{\Bbbk}\Bbbk((u))$ (for which see for instance [Chen-
localization]*Example 1.0.5), so it is sufficient to describe an isomorphism
$A\simeq{\sf HH}(\mathcal{A}).$

The map $A\to{\sf HH}(\mathcal{A})$ may be defined in the obvious way: observe
the algebra $A$ is generated by the vertex idempotents $e_{\Phi},e_{\Psi}$ and
the maps $u,v$ with $1-uv,1-vu$ invertible, and define the map $A\to{\sf
HH}(\mathcal{A})$ by sending the vertex idempotents to the classes of the
identity maps $[{\sf id}_{\Phi}],[{\sf id}_{\Psi}],$ and the maps $u,v$ to the
classes of the generating maps between $\Phi$ and $\Psi.$ The defining exact
triangles for the twist and cotwist ensure that $1-uv,1-vu$ map to these
invertible elements, so that the map is well-defined.

To see that this map is an equivalence, observe that the Hochschild homology
of the matrix category 5.0.6 may be written as

$\left(\begin{array}[]{ll}\Bbbk[T^{\pm}]&\Bbbk[T^{\pm}]\\\
\Bbbk[T^{\pm}]&\Bbbk[T^{\pm}]\end{array}\right).$

The explicit description of $P_{\Phi},P_{\Psi}$ given in Example 2.13 shows
that $A$ has an identical expression, and it is easy to see that the obvious
bases of $A$ and ${\sf HH}(\mathcal{A})$ are sent to each other by this map.

For the statement about $G$-actions, it is necessary to show that the
universal twist map $\mathcal{T}:\Bbbk[\pi_{1}G]\to
Z(\mathsf{PervCat}_{\mathbb{L}}(\mathbb{C}))$ induces on ${\sf
HP}(\mathcal{A})\simeq A_{\mathbb{Z}/2}$ the monodromy automorphism. This
follows immediately from the formula for the universal twist, and that our map
$A\to{\sf HH}(\mathcal{A})$ was defined so that the monodromy automorphisms
were sent to the classes of the twist and cotwist. ∎

######  Corollary 5.7.

There is an equivalence

(5.0.7) ${\sf
HP}(\mathsf{end}_{\mathsf{PervCat}_{\mathbb{L}_{G}}(\mathbb{C}^{n}/G)}(\mathcal{P}))\simeq\mathsf{end}_{\mathsf{Perv}_{\mathbb{L}_{G}}(\mathbb{C}^{n}/G)}(P)_{\mathbb{Z}/2}$

###### Proof.

Each of the algebras in 5.0.7 is obtained from the corresponding algebra in
5.0.5 by trivializing the respective action of $\Bbbk[\pi_{1}G],$ and we saw
that the equivalence 5.0.5 intertwined these actions. ∎

###### Proof of B (A-side).

The theorem now follows immediately from Corollary 5.7 by matching the
subquotient presentations of Corollary 2.22 and Lemma 3.13 and applying the
fact that ${\sf HP}$ is a localizing invariant. ∎

## References

(5.0.8)
$\mu\mathsf{PervCat}(\mathbb{L}_{G}(t,m))\simeq\frac{\mathsf{PervCat}_{\mathbb{L}_{G}}(\mathbb{C}^{n}/G)}{\langle\text{$t$-unstable
simples, $m$-unbounded projectives}\rangle}.$

Let $\mathcal{P}_{\alpha}$ denote the projective object in
$\mathsf{PervCat}_{\mathbb{L}_{G}}(\mathbb{C}^{n}/G)$ corresponding to
Lagrangian component $\mathbb{L}_{G}^{\alpha}.$ Write
$\mathcal{P}^{t-ss}:=\bigoplus_{t-ss}\mathcal{P}_{\alpha}$ (resp.
$\mathcal{P}^{t-ss,m\text{-unbdd}}:=\bigoplus_{t-ss,m\text{-unbdd}}\mathcal{P}_{\alpha}$)
for the sum over $t$-semistable (resp. $t$-semistable, $m$-unbounded) sign
vectors $\alpha.$ Need to fix what follows. By Lemma 3.13, we can rewrite
5.0.8 as the quotient

$\mu\mathsf{PervCat}(\mathbb{L}_{G}(t,m))\simeq\frac{\langle\mathcal{P}^{t-ss}\rangle}{\langle\mathcal{P}^{t-ss,m\text{-unbdd}}\rangle}.$

We can therefore write $\mu\mathsf{PervCat}(\mathbb{L}_{G}(t,m))\simeq{\sf
Mod}_{\mathcal{M}}({\sf St}),$ where $\mathcal{M}$ is the monoidal category
obtained as the quotient of $\mathsf{end}(\mathcal{P}^{t-ss})$ by the ideal
generated by $\mathsf{end}(\mathcal{P}^{t-ss,m\text{-unbdd}}).$ After applying
$\mathsf{tr}^{tS^{1}},$ we therefore recover the quotient description of
$\mu\mathsf{Perv}(\mathbb{L}_{G}(t,m))$ given in LABEL:eq:bcato-subquotient.

###  5.1. Notes from Justin

${\ignorespaces X\times_{Y}X}$ |  | ${\ignorespaces X}$ |
---|---|---|---
${\ignorespaces X}$ |  | ${\ignorespaces Y}$ |
$\scriptstyle{\ignorespaces p_{\hat{2}}}$$\scriptstyle{\ignorespaces
p_{\hat{1}}}$$\scriptstyle{\ignorespaces f}$$\scriptstyle{\ignorespaces f}$

The statement of the first lemma is

$\mathbf{T}_{X\times_{Y}X}\simeq\lim\left(\leavevmode\hbox to106.75pt{\vbox
to53.6pt{\pgfpicture\makeatletter\hbox{\hskip 53.37682pt\lower-26.84843pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
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}{{}}\ignorespaces\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
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}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\ignorespaces\hskip
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}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{\ignorespaces}}}}\ignorespaces
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
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}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}\ignorespaces{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-11.79999pt}{-24.2486pt}\pgfsys@lineto{11.40002pt}{-24.2486pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{\ignorespaces}}}}{{}{{}}{}{}{{}}{{{\ignorespaces}}{{{\ignorespaces}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{11.6pt}{-24.2486pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{\ignorespaces}}}}\ignorespaces
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\right),$

where the vertical maps are the pullbacks along $p$ of the canonical map
$df:\mathbf{T}_{X}\to f^{*}\mathbf{T}_{Y},$ and the horizontal map is the
isomorphism given by $\gamma.$ The dual statement is that
$\mathbf{L}_{X\times_{Y}X}$ is the colimit of the diagram with vertical arrows
reversed. The takeaway is that there are two maps
$\mathbf{L}_{X\times_{Y}X}[-1]\to p_{\hat{i}}^{*}f^{*}\mathbf{L}_{Y},$
homotopic by $\gamma.$ So we can fix one of them: call it

$\iota_{1}:\mathbf{L}_{X\times_{Y}X}[-1]\to
p_{\hat{2}}^{*}f^{*}\mathbf{L}_{Y}$

Now, given a singular support condition $\Lambda\subset\mathbf{L}_{Y},$ we can
pull it back to $\mathsf{Sing}(X\times_{Y}X)$ via $\iota_{1}$ (followed by

$\mathbf{L}_{Y}\leftarrow p_{\hat{i}}^{*}f^{*}\mathbf{L}_{Y},$

which is a map of total spaces of vector bundles).

######  Proposition 5.8.

The smallest ideal (at some point we may need to be more careful about what
“ideal” means: should be idempotent-complete, closed under small colimits) in
$\mathsf{Coh}(X\times_{Y}X)$ containing
$E_{\alpha}:=\Delta_{*}\mathcal{O}_{X_{\alpha}}$ is
$\mathsf{Coh}_{\Lambda_{\alpha}}(X\times_{Y}X),$ where $\Lambda_{\alpha}=??$

Easier (almost tautological) is the following

######  Lemma 5.9.

The smallest ideal in $\mathsf{Coh}(X\times_{Y}X)$ containing $E_{\alpha}$ is

(5.1.1)
$\mathsf{Coh}(X\times_{Y}X_{\alpha})\otimes_{\mathsf{Coh}(X_{\alpha}\times_{Y}X_{\alpha})}\mathsf{Coh}(X_{\alpha}\times_{Y}X)$

And [BZCHN, prop. 3.30] implies that 5.1.1 is equivalent to

$\Lambda_{\alpha}:=\mathsf{Coh}_{\mathsf{Sing}(X\times_{Y}X_{\alpha})\star\mathsf{Sing}(X_{\alpha}\times_{Y}X)}(X\times_{Y}X).$

We would like to describe this without convolution. (Starting here, everything
is in the appendix.) The claim is that $\Lambda_{\alpha}$ is equivalent to the
image in $\mathsf{Sing}(X\times_{Y}X)$ of the fiber product
$\mathsf{Sing}(X\times_{Y}X)\times_{\mathrm{T}^{*}Y}\mathrm{T}^{*}_{X_{\alpha}}Y.$
To understand this, we chase through the big diagram. The first key bit is to
understand why

${\ignorespaces X\times_{Y}X_{\alpha}\times_{Y}X}$ |  | ${\ignorespaces X\times_{Y}X}$ |
---|---|---|---
${\ignorespaces X_{\alpha}}$ |  | ${\ignorespaces Y.}$ |

is a pullback diagram. Then you pull back $\mathrm{T}^{*}Y\to Y$ along all the
vertices in this diagram. And you also pull back
$\mathsf{Sing}(X\times_{Y}X).$ The final step in the proof will happen in
$X\times_{Y}X_{\alpha}\times_{Y}X.$ We have a pullback diagram of vector
bundles

${\ignorespaces p_{13}^{*}\mathbf{L}_{X\times_{Y}X}[-1]}$ |  | ${\ignorespaces\mathbf{L}_{X\times_{Y}X_{\alpha}\times_{Y}X}[-1]}$ |
---|---|---|---
${\ignorespaces C_{\alpha}}$ |  | ${\ignorespaces p_{12}^{*}\mathbf{L}_{X\times_{Y}X_{\alpha}}[-1]\oplus p_{23}^{*}\mathbf{L}_{X_{\alpha}\times_{Y}X},}$ |

And there is a map from $C_{\alpha}$ to the upper-left corner, which has a
canonical map to $\mathsf{Sing}(X\times_{Y}X).$ The image of $C_{\alpha}$ is
the convolution
$\mathsf{Sing}(X\times_{Y}X_{\alpha})\star\mathsf{Sing}(X_{\alpha}\times_{Y}X).$
(This is what the definition of convolution of singular supports is.)

************************************************

Now assume we know the theorem that the smallest ideal containing $E_{\alpha}$
is $\mathsf{Coh}_{\Lambda_{\alpha}}(X\times_{Y}X).$ The remaining question is:
what is the smallest ideal $\mathcal{I}$ containing
$\\{E_{\alpha}\mid\alpha\in m\text{-unstab}\\}$? Clearly
$\mathcal{I}\supset\mathsf{Coh}_{\Lambda_{\alpha}}(X\times_{Y}X)$ for each
$\alpha.$ The claim is that
$\mathcal{I}=\mathsf{Coh}_{\bigcup_{\alpha}\Lambda_{\alpha}}(X\times_{Y}X).$

##  6\. Justin

A correspondence

${\ignorespaces X}$ |  | ${\ignorespaces C}$ |  | ${\ignorespaces Y}$ |
---|---|---|---|---|---
$\scriptstyle{\ignorespaces f}$$\scriptstyle{\ignorespaces g}$

with $g$ proper gives rise to a functor

$g_{*}f^{*}:\mathsf{Coh}(X)\to\mathsf{Coh}(Y).$

Using the fiber negation symplectomorphism

$\overline{\mathrm{T}^{*}[-1]X}\cong\mathrm{T}^{*}[-1]X$

one gets a Lagrangian correspondence

${\ignorespaces\mathrm{T}^{*}[-1]X}$ |  | ${\ignorespaces\mathrm{T}^{*}_{C}[-1](X\times Y)}$ |  | ${\ignorespaces\mathrm{T}^{*}[-1]Y.}$ |
---|---|---|---|---|---

That control the behavior of singular support under $g_{*}f^{*}$. Moreover
composition of correspondence goes to composition of Lagrangian
correspondences In particular we have

${\ignorespaces X}$ |  | ${\ignorespaces X}$ |  | ${\ignorespaces Y}$ |  |  | ${\ignorespaces\mathrm{T}^{*}[-1]X}$ |  | ${\ignorespaces g^{*}(\mathrm{T}^{*}[-1]Y)}$ |  | ${\ignorespaces\mathrm{T}^{*}[-1]Y}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---
${\ignorespaces X}$ |  | ${\ignorespaces Y}$ |  | ${\ignorespaces Y}$ |  |  | ${\ignorespaces\mathrm{T}^{*}[-1]X}$ |  | ${\ignorespaces f^{*}(\mathrm{T}^{*}[-1]X)}$ |  | ${\ignorespaces\mathrm{T}^{*}[-1]Y}$ |
$\scriptstyle{\ignorespaces
g}$$\scriptstyle{\ignorespaces(dg)^{*}}$$\scriptstyle{\ignorespaces
f}$$\scriptstyle{\ignorespaces(df)^{*}}$

. Let $f_{i}:X_{i}\to Y$ for $i\in\\{1,2,3\\}$ be proper maps of smooth
stacks. Consider the strictly cartesian cube

${\ignorespaces X_{1,3}}$ |  |  |  | ${\ignorespaces X_{1}}$ |  |  |
---|---|---|---|---|---|---|---
|  | ${\ignorespaces X_{1,2,3}}$ |  |  |  | ${\ignorespaces X_{1,2}}$ |
${\ignorespaces X_{3}}$ |  |  |  | ${\ignorespaces Y}$ |  |  |
|  | ${\ignorespaces X_{2,3}}$ |  |  |  | ${\ignorespaces X_{2}}$ |
$\scriptstyle{\ignorespaces p_{1}}$$\scriptstyle{\ignorespaces
p_{2}}$$\scriptstyle{\ignorespaces f_{1}}$$\scriptstyle{\ignorespaces
p_{1,3}}$$\scriptstyle{\ignorespaces p_{1,2}}$$\scriptstyle{\ignorespaces
p_{1}}$$\scriptstyle{\ignorespaces p_{2}}$$\scriptstyle{\ignorespaces
f_{3}}$$\scriptstyle{\ignorespaces p_{2,3}}$$\scriptstyle{\ignorespaces
p_{3}}$$\scriptstyle{\ignorespaces p_{2}}$$\scriptstyle{\ignorespaces f_{2}}$

Using the correspondence

|  | ${\ignorespaces X_{1,2,3}}$ |  |  |
---|---|---|---|---|---
${\ignorespaces X_{1,2}\times X_{2,3}}$ |  |  |  | ${\ignorespaces X_{1,3}}$ |
$\scriptstyle{\ignorespaces(p_{1,2}\,,\,p_{2,3})}$$\scriptstyle{\ignorespaces
p_{1,3}}$

we can define convolution of coherent sheaves

$\star=(p_{1,3})_{*}(p_{1,2},p_{2,3})^{!}:\mathsf{Coh}(X_{1,2})\otimes\mathsf{Coh}(X_{2,3})\to\mathsf{Coh}(X_{1,3}).$

The behavior of convolution on singular supports can be described using the
Lagrangian correspondences

|  | ${\ignorespaces p_{1,2}^{*}(\mathrm{T}^{*}[-1]X_{1,2})\oplus p_{1,3}^{*}(\mathrm{T}^{*}[-1]X_{2,3})}$ |  |  |
---|---|---|---|---|---
${\ignorespaces\mathrm{T}^{*}[-1]X_{1,2}\times\mathrm{T}^{*}[-1]X_{2,3}}$ |  |  |  | ${\ignorespaces\mathrm{T}^{*}[-1]X_{1,2,3}}$ |
$\scriptstyle{\ignorespaces[(p_{1,2})^{*},(p_{2,3})^{*}]}$

and

|  | ${\ignorespaces p_{1,3}^{*}(\mathrm{T}^{*}[-1]X_{1,3})}$ |  |  |
---|---|---|---|---|---
${\ignorespaces\mathrm{T}^{*}[-1]X_{1,2,3}}$ |  |  |  | ${\ignorespaces\mathrm{T}^{*}[-1]X_{1,3}.}$ |
$\scriptstyle{\ignorespaces(p_{1,3})^{*}}$

######  Proposition 6.1.

Using the fiber negation symplectomorphism

$\overline{\mathrm{T}^{*}(X_{1,2}\times
X_{2,3})}\cong\mathrm{T}^{*}(X_{1,2}\times X_{2,3})$

one can identify the composition of these two Lagrangian correspondences with
the conormal

${\ignorespaces\mathrm{T}^{*}_{X_{1,2,3}}[-1](X_{1,2}\times X_{2,3}\times X_{1,3}).}$ |
---|---

Let $\mathrm{N}^{*}_{i}=\mathrm{T}^{*}_{X_{i}}Y$ be the conormal of
$f_{i}:X\to Y$. One has a strictly cartesian cube

${\ignorespaces\mathrm{N}^{*}_{1,3}}$ |  |  |  | ${\ignorespaces\mathrm{N}^{*}_{1}}$ |  |  |
---|---|---|---|---|---|---|---
|  | ${\ignorespaces\mathrm{N}^{*}_{1,2,3}}$ |  |  |  | ${\ignorespaces\mathrm{N}^{*}_{1,2}}$ |
${\ignorespaces L_{3}}$ |  |  |  | ${\ignorespaces\mathrm{T}^{*}Y}$ |  |  |
|  | ${\ignorespaces\mathrm{N}^{*}_{2,3}}$ |  |  |  | ${\ignorespaces\mathrm{N}^{*}_{2}}$ |
$\scriptstyle{\ignorespaces q_{1}}$$\scriptstyle{\ignorespaces
q_{2}}$$\scriptstyle{\ignorespaces g_{1}}$$\scriptstyle{\ignorespaces
q_{1,3}}$$\scriptstyle{\ignorespaces q_{1,2}}$$\scriptstyle{\ignorespaces
q_{1}}$$\scriptstyle{\ignorespaces q_{2}}$$\scriptstyle{\ignorespaces
g_{3}}$$\scriptstyle{\ignorespaces q_{2,3}}$$\scriptstyle{\ignorespaces
q_{3}}$$\scriptstyle{\ignorespaces q_{2}}$$\scriptstyle{\ignorespaces g_{2}}$

######  Observation 6.2.

For $i=1,3$ we have iterated pullback squares

${\ignorespaces\mathrm{N}^{*}_{1,2,3}}$ |  | ${\ignorespaces\mathrm{N}^{*}_{i,2}}$ |  | ${\ignorespaces\mathrm{N}^{*}_{2}}$ |
---|---|---|---|---|---
${\ignorespaces\mathrm{N}^{*}_{1,3}}$ |  | ${\ignorespaces\mathrm{N}^{*}_{i}}$ |  | ${\ignorespaces\mathrm{T}^{*}Y.}$ |
$\scriptstyle{\ignorespaces q_{1,3}}$$\scriptstyle{\ignorespaces
q_{i,2}}$$\scriptstyle{\ignorespaces q_{i}}$$\scriptstyle{\ignorespaces
q_{2}}$$\scriptstyle{\ignorespaces g_{2}}$$\scriptstyle{\ignorespaces
q_{i}}$$\scriptstyle{\ignorespaces g_{i}}$

In particular, if the image of $\mathrm{N}^{*}_{1}$ or $\mathrm{N}^{*}_{3}$ is
contained in the image of $\mathrm{N}^{*}_{2}$ then $q_{1,3}$ is surjective.

######  Theorem 6.3.

The correspondence

|  | ${\ignorespaces\mathrm{N}^{*}_{1,2,3}}$ |  |  |
---|---|---|---|---|---
${\ignorespaces\mathrm{N}^{*}_{1,2}\times\mathrm{N}^{*}_{2,3}}$ |  |  |  | ${\ignorespaces\mathrm{N}^{*}_{1,3}}$ |
$\scriptstyle{\ignorespaces(q_{1,2},q_{2,3})}$$\scriptstyle{\ignorespaces
q_{1,3}}$

can be equipped with a Lagrangian structure.

######  Lemma 6.4.

There is isomorphism
$\alpha_{i,j}:\mathrm{T}^{*}[-1]X_{i,j}\cong\mathrm{N}^{*}_{i,j}$ of
(-1)-shifted symplectic stacks.

###### Proof.

There is a cocartesian diagram

${\ignorespaces\mathbb{L}_{Y}}$ |  | ${\ignorespaces\mathbb{L}_{X_{j}}}$ |  | ${\ignorespaces 0}$ |
---|---|---|---|---|---
${\ignorespaces\mathbb{L}_{X_{i}}}$ |  | ${\ignorespaces\mathbb{L}_{X_{i,j}}}$ |  | ${\ignorespaces\mathbb{L}_{f_{i}}}$ |
${\ignorespaces 0}$ |  | ${\ignorespaces\mathbb{L}_{f_{j}}}$ |  | ${\ignorespaces\mathbb{L}_{Y}[1]}$ |

of sheaves on $X_{i,j}$. Here the upper left cartesian square is the standard
formula for the cotangent complex of a pullback, the left and upper cartesian
rectangles define the relative cotangent complexes, and the outer cartesian
square is the definition of the suspension. The desired map $\alpha_{i,j}$
comes from noticing that $\mathrm{N}^{*}_{i}$ and $\mathrm{N}^{*}_{j}$ are the
total spaces of $\mathbb{L}_{f_{i}}[-1]$ and $\mathbb{L}_{f_{j}}[-1]$
respectively. The fact that this is a symplectomorphism is Remark ? in
Calaque. ∎

######  Proposition 6.5.

There is an isomorphism

${\ignorespaces\mathrm{T}^{*}[-1]X_{1,2}\times\mathrm{T}^{*}[-1]X_{2,3}}$ |  | ${\ignorespaces\mathrm{T}^{*}_{X_{1,2,3}}[-1](X_{1,2}\times X_{2,3}\times X_{1,3})}$ |  | ${\ignorespaces\mathrm{T}^{*}[-1]X_{1,3}}$ |
---|---|---|---|---|---
${\ignorespaces\mathrm{N}^{*}_{1,2}\times\mathrm{N}^{*}_{2,3}}$ |  | ${\ignorespaces\mathrm{N}^{*}_{1,2,3}}$ |  | ${\ignorespaces\mathrm{N}^{*}_{1,3}}$ |
$\scriptstyle{\ignorespaces\alpha_{1,2}\times\alpha_{2,3}}$$\scriptstyle{\ignorespaces\alpha_{1,2,3}}$$\scriptstyle{\ignorespaces\alpha_{1,3}}$

of Lagrangian correspondences.

There is a strongly cocartesian diagram

${\ignorespaces\mathbb{L}_{X_{2}}}$ |  |  |  | ${\ignorespaces\mathbb{L}_{X_{2,3}}}$ |  |  |
---|---|---|---|---|---|---|---
|  | ${\ignorespaces\mathbb{L}_{Y}}$ |  |  |  | ${\ignorespaces\mathbb{L}_{X_{3}}}$ |
${\ignorespaces\mathbb{L}_{X_{1,2}}}$ |  |  |  | ${\ignorespaces\mathbb{L}_{X_{1,2,3}}}$ |  |  |
|  | ${\ignorespaces\mathbb{L}_{X_{1}}}$ |  |  |  | ${\ignorespaces\mathbb{L}_{X_{1,3}}}$ |

of sheaves on $X_{1,2,3}$. Using the fact that the total cofiber of any face
of the cube must vanish we find that

$\displaystyle{\sf
cofib}(\mathbb{L}_{X_{1,2}}\to\mathbb{L}_{X_{1,2,3}})\cong{\sf
cofib}(\mathbb{L}_{X_{3}}\to\mathbb{L}_{X_{2,3}})\cong{\sf
cofib}(\mathbb{L}_{Y}\to\mathbb{L}_{X_{2}})=\mathbb{L}_{f_{2}}.$

The $(-1)$-shifted conormal bundle
$\mathrm{T}^{*}_{X_{1,2,3}}[-1](X_{1,2}\times X_{1,3}\times X_{2,3})$ is the
total space of the complex

$\displaystyle{\sf
fib}(\mathbb{L}_{X_{1,2}}\oplus\mathbb{L}_{X_{1,3}}\oplus\mathbb{L}_{X_{2,3}}\to\mathbb{L}_{X_{1,2,3}})[-1]$
$\displaystyle\cong{\sf
cofib}(\mathbb{L}_{X_{1,2}}\oplus\mathbb{L}_{X_{1,3}}\oplus\mathbb{L}_{X_{2,3}}\to\mathbb{L}_{X_{1,2,3}})[-2]$
$\displaystyle\cong{\sf
cofib}(\mathbb{L}_{X_{1,2}}\oplus\mathbb{L}_{X_{2,3}}\to{\sf
cofib}(\mathbb{L}_{X_{1,2}}\to\mathbb{L}_{X_{1,2,3}}))[-2]$
$\displaystyle\cong{\sf
cofib}(\mathbb{L}_{X_{1,2}}\oplus\mathbb{L}_{X_{2,3}}\to\mathbb{L}_{f_{2}})[-2]$
$\displaystyle\cong{\sf
fib}(\mathbb{L}_{X_{1,2}}\oplus\mathbb{L}_{X_{2,3}}\to\mathbb{L}_{f_{2}})[-1].$

The total space of the last complex is

$\mathrm{N}^{*}_{1,2}\times_{\mathrm{N}^{*}_{2}}\mathrm{N}^{*}_{2,3}\cong\mathrm{N}^{*}_{1,2,3}$

which completes the theorem.

##  7\. Category $\mathcal{O}$ stuff

######  Definition 7.1.

The Betti category $\mathcal{O}$ is the abelian category

$\mathcal{O}^{{\sf
Bet}}(\mathfrak{M}_{G}(t),\mathbb{L}_{G}(t,m)):=\mu\mathsf{Sh}_{\mathbb{L}_{G}(t,m)}(\mathbb{L}_{G}(t,m))^{\heartsuit}$

of microlocal perverse sheaves [perverse-microsheaves] on the Lagrangian
$\mathbb{L}_{G}(t,m),$ with stable polarization induced from the fiber
polarization of $\mathrm{T}^{*}(\mathbb{C}^{n}/G)$ by [NS-higgs]*Corollary
4.5.

######  Proposition 7.2.

Betti category $\mathcal{O}$ admits a presentation as the quotient

$\mu\mathsf{Sh}_{\mathbb{L}_{G}(t,m)}(\mathbb{L}_{G}(t,m))^{\heartsuit}\simeq\mathsf{Sh}_{\mathbb{L}_{G}^{m\text{-bdd}}}(\mathbb{C}^{n}/G)^{\heartsuit}/\mathsf{Sh}_{\mathbb{L}_{G}^{m\text{-bdd},t\text{-unstab}}}(\mathbb{C}^{n}/G)^{\heartsuit}$

of the category of perverse sheaves on $\mathbb{C}^{n}/G$ with microsupport in
$\mathbb{L}_{G}^{m\text{-bdd}}$ by the subcategory whose microsupport is
contained in the $t$-unstable locus.

###### Proof.

The open inclusion
$\mathbb{L}_{G}(t,m)\hookrightarrow\mathbb{L}_{G}^{m\text{-bdd}},$ with closed
complement $\mathbb{L}_{G}^{m\text{-bdd},t\text{-unstab}},$ defines a
microlocal restriction functor

(7.0.1)
$\mathsf{Sh}_{\mathbb{L}_{G}^{m\text{-bdd}}}(\mathbb{C}^{n}/G)^{\heartsuit}\simeq\mu\mathsf{Sh}(\mathbb{L}_{G}^{m\text{-bdd}})^{\heartsuit}\to\mu\mathsf{Sh}(\mathbb{L}_{G}(t,m))^{\heartsuit}$

whose kernel is the subcategory
$\mathsf{Sh}_{\mathbb{L}_{G}^{m\text{-bdd},t\text{-unstab}}}(\mathbb{C}^{n}/G)^{\heartsuit}$
of sheaves supported on the unstable locus. We need to prove that the functor
7.0.1 is essentially surjective. The microlocal Riemann-Hilbert correspondence
of [Waschkies-RH] gives an embedding

(7.0.2) $\mathcal{O}^{{\sf
dR}}(\mathfrak{M}(t),\mathbb{L}_{G}(t,m))\hookrightarrow\mu\mathsf{Sh}_{\mathbb{L}_{G}(t,m)}(\mathbb{L}_{G}(t,m))^{\heartsuit},$

whose image is generated by the simple microlocal perverse sheaves supported
on the smooth components of $\mathbb{L}_{G}(t,m).$ The description of
$\mathcal{O}^{{\sf dR}}(\mathfrak{M}(t),\mathbb{L}_{G}(t,m))$ from [BLPW10]
shows that each simple object $L_{\alpha}$ in $\mathcal{O}^{{\sf dR}}$ has a
projective cover; under the embedding 7.0.2, these map to the projective
objects corepresenting the microstalk functors at smooth points of the
skeleton $\mathbb{L}_{G}(t,m).$ These objects generate $\mathcal{O}^{{\sf
Bet}}(\mathfrak{M}_{G}(t),\mathbb{L}_{G}(t,m)),$ so we deduce that the simple
objects are also a set of generators. As the simple objects in
$\mu\mathsf{Sh}(\mathbb{L}_{G}(t,m))$ are the images of simple objects under
the microlocal restriction functor 7.0.1, we conclude that this functor is
essentially surjective. ∎


# Cosmic evolution and thermal stability of Barrow holographic dark energy in
nonflat Friedmann-Robertson-Walker Universe

G. G. Luciano<EMAIL_ADDRESS>Applied Physics Section of Environmental
Science Department, Universitat de Lleida, Av. Jaume II, 69, 25001 Lleida,
Catalonia, Spain  INFN, Sezione di Napoli, Gruppo collegato di Salerno, Via
Giovanni Paolo II, 132 I-84084 Fisciano (SA), Italy

###### Abstract

We study the cosmological evolution of a nonflat Friedmann-Robertson-Walker
Universe filled by pressureless dark matter and Barrow Holographic Dark Energy
(BHDE). The latter is a dark energy model based on the holographic principle
with Barrow entropy instead of the standard Bekenstein-Hawking one. By
assuming the apparent horizon of the Universe as IR cutoff, we explore both
the cases where a mutual interaction between the dark components of the cosmos
is absent/present. We analyze the behavior of various model parameters, such
as the BHDE density parameter, the equation of state parameter, the
deceleration parameter, the jerk parameter and the square of sound speed. We
also comment on the observational consistency of our predictions, showing that
the interacting model turns out to be favored by recent experimental
constraints from Planck+WP+BAO, SNIa+CMB+LSS and Union2 SNIa joint analysis
over the non-interacting one. The thermal stability of our framework is
finally discussed by demanding the positivity of the heat capacities and
compressibilities of the cosmic fluid.

##  I Introduction

Despite the recent progress at both theoretical and experimental level, there
are still several open problems in modern Cosmology Open . Among these, the
issue of the observed accelerating expansion of the Universe has been getting
greater attention in the last decades Supern ; Supernbis ; Supernter ;
Supernquar ; Supernquin , providing one of the most exciting challenges to the
Standard Model of particle physics and Cosmology. A widely accepted paradigm
to explain this phenomenon is the existence of a mysterious form of energy
with negative pressure - the Dark Energy (DE) - which would dominate the
energy-matter budget of the present Universe.

An interesting scenario for the quantitative description of DE is Holographic
Dark Energy (HDE) Cohen:1998zx ; Horava:2000tb ; Thomas:2002pq ; Li:2004rb ;
Hsu:2004ri ; Huang:2004ai ; Wang:2005ph ; Setare:2006sv ; Granda ;
Sheykhi:2011cn ; Bamba:2012cp ; Ghaffari:2014pxa ; Wang:2016och ;
Moradpour:2020dfm , which proves to be in good agreement with observations
Zhang ; Li ; Zhang2 ; Lu . A crucial ingredient of this framework is the
application of the holographic principle at cosmological scales. According to
this recipe, the total entropy content of the Universe should be proportional
to the area of its two-dimensional boundary, similarly to the Bekenstein-
Hawking entropy-area law for black holes. Recently, however, it has been
argued that quantum-gravitational effects may induce nontrivial fractal
features on the black hole structure, leading to a deformed horizon entropy in
the form BarrowBH

$S_{\Delta}\,\propto\,A^{1+\Delta/2}\,,$ (1)

where $A$ is the area enclosed by the horizon surface. Corrections to the
standard entropy-area law are parameterized by the exponent $0\leq\Delta\leq
1$, where $\Delta=0$ recovers the Bekenstein-Hawking limit, while $\Delta=1$
corresponds to the maximal entropy deformation. It is worth noting that upper
bounds on $\Delta$ have been derived in different contexts in
Anagnostopoulos:2020ctz ; Leon:2021wyx ; Barrow:2020kug ; Jusufi:2021fek ;
Dabrowski:2020atl ; Saridakis:2020cqq ; Luciano:2022pzg ; Vagnozzi:2022moj .

Based on the gravity-thermodynamic conjecture GTC , Eq. (1) is commonly
adopted also in Cosmology BarSar ; Sheykhi:2021fwh ; Adhikary:2021xym ;
Nojiri:2021jxf ; LucLiu ; GhafLuc  to derive modified Friedmann equations from
the first law of thermodynamics on the apparent horizon of a Friedmann-
Robertson-Walker (FRW) Universe. The ensuing framework is referred to as
Barrow Holographic Dark Energy (BHDE) and exhibits a richer phenomenology
compared to HDE, allowing us to predict effects beyond the domain of the
Standard Model of Cosmology for certain values of $\Delta$ (see, for instance,
Luciano:2022pzg ).

In line with the above studies, the properties of some DE models have been
largely investigated in Radicella ; Mimoso ; MoradInt ; Santos ; Barboza ;
Bhandari  at thermodynamic level. Specifically, in Barboza  the limits imposed
by thermodynamics on a DE fluid have been analyzed by considering the heat
capacities and compressibilities. On the other hand, in Bhandari  the thermal
stability criteria have been explored for interacting DE. A similar analysis
has been carried out in Abdollah  for the case of Tsallis Holographic Dark
Energy Tavayef:2018xwx ; Saridakis:2018unr ; Nojiri:2019skr ; DAgostino ;
LucGine , which is a generalization of HDE based on the use of nonextensive
Tsallis entropy for the horizon degrees of freedom of Universe Cirto . All
these studies suggest the existence of an intimate relation between the cosmic
expansion of the Universe, the thermodynamic stability conditions and DE
candidates, which deserves careful attention.

Starting from the above premises, in this work we examine the cosmic evolution
of a nonflat FRW Universe filled by pressureless dark matter and BHDE. By
considering the apparent horizon as IR cutoff and assuming a mutual
interaction between the dark sectors of the cosmos, we analyze the behavior of
various model parameters, such as the BHDE density parameter, the equation of
state parameter, the deceleration parameter, the jerk parameter and the square
of sound speed. We then discuss the thermal stability of our model by
demanding the positivity of the heat capacities and compressibilities of the
cosmic fluid.

Some comments are in order here to motivate our study: first, we emphasize
that in the absence of a fully consistent theory of quantum gravity, BHDE
represents a first, naive approach to accommodate quantum-gravity corrections
within the HDE framework based on simple physical principles BarSar .
Furthermore, the usage of the apparent horizon as IR cutoff is substantiated
by the fact that, unlike the event horizon, this boundary is consistent with
the generalized second law of thermodynamics She1 ; She2 . Last but not least,
although it is commonly believed that inflation washed out curvature
inhomogeneities in the early Universe, observational evidences suggest that
there is still room for a small but non-negligible spatial curvature Chernin ;
Ellis ; PLB ; Divale . This is why we consider a nonflat FRW Universe as
background for our analysis.

The remainder of the work is organized as follows: in the next section we
briefly review the theoretical framework of BHDE. The evolution of the
characteristic model parameters is studied in Sec. III for both the
noninteracting and interacting cases. In Sec. IV we discuss the thermal
stability of BHDE, while conclusions and outlook are finally summarized in
Sec. V. Throughout the work, we use natural units $\hbar=c=k_{B}=G=1$.

##  II Barrow Holographic Dark Energy: a review

Let us start by briefly reviewing the main features of BHDE. It is well-known
that the standard HDE is given by the inequality Wang:2016och

$\rho_{D}L^{4}\leq S\,,$ (2)

with the further assumption $S\propto A\propto L^{2}$, where $\rho_{D}$
denotes the dark energy density, while $L$ and $S$ are the horizon length and
entropy, respectively.

On the other hand, the usage of Barrow entropy (1) leads to BarSar

$\rho_{D}=CL^{\Delta-2}\,,$ (3)

where $C$ is an unknown parameter with dimensions $[L]^{-2-\Delta}$. Notice
that, for $\Delta=0$, the above relation reduces to the usual HDE, provided
that $C=3c^{2}m_{p}^{2}$, where $m_{p}$ is the reduced Planck mass and $c$ the
model parameter.

Now, in a nonflat FRW Universe containing pressureless dark matter and BHDE,
the first Friedmann equation takes the form

$H^{2}+\frac{k}{a^{2}}=\frac{8\pi}{3}\left(\rho_{M}+\rho_{D}\right),$ (4)

where $a$ is the scale factor, $H\equiv\dot{a}/a$ is the Hubble parameter and
the dot denotes derivative with respect to the cosmic time $t$.

We also assume that the apparent horizon of radius

$\tilde{r}_{A}=\frac{1}{\sqrt{H^{2}+k/a^{2}}}$ (5)

acts as IR cutoff111There is no universal consensus on the choice of the IR
cutoff. Following Tavayef:2018xwx ; IRCUT ; IRCUTbis , here we resort to Eq.
(5). However, other possible choices are the particle horizon, the future
event horizon, the GO cutoff GO  or combination thereof. Given the degree of
arbitrariness in the selection of the most reliable description of dark
energy, we leave the analysis of BHDE with different IR cutoffs for future
work. IRCUT ; IRCUTbis , so that Eq. (3) can be cast as

$\rho_{D}=C\tilde{r}_{A}^{\Delta-2}\,.$ (6)

The spatial curvature $k$ takes value $+1,0$ or $-1$, depending on whether the
shape of the Universe is a closed $3$-sphere, flat space or an open
$3$-hyperboloid, respectively. Consistently with recent observations Chernin ;
Ellis ; PLB ; Divale , henceforth we consider a closed Universe with a small
positive curvature.

By introducing the fractional energy densities

$\Omega_{D}=\frac{8\pi\rho_{D}}{3H^{2}}\,,$ (7a)
$\Omega_{M}=\frac{8\pi\rho_{M}}{3H^{2}}\,,$ (7b)
$\Omega_{k}=\frac{k}{H^{2}a^{2}}\,,$ (7c)

for BHDE, DM and curvature terms, respectively, Eq. (4) can be rearranged as

$1=\Omega_{D}+\Omega_{M}-\Omega_{k}\,,$ (8)

which can be further manipulated with the definition of the parameter
$r=\rho_{M}/\rho_{D}=\Omega_{M}/\Omega_{D}$ to give

$r=-1+\frac{1}{\Omega_{D}}\left(1+\Omega_{k}\right).$ (9)

If now assume the existence of a mutual interaction $Q$ between the dark
sectors of the cosmos, the conservation equations for BHDE and DM can be
coupled as

$\displaystyle\dot{\rho}_{D}+3H\rho_{D}\left(1+\omega_{D}\right)=-Q\,,$ (10)
$\displaystyle\dot{\rho}_{M}+3H\rho_{M}=Q\,,$ (11)

where $\omega_{D}\equiv p_{D}/\rho_{D}$ is the equation of state (EoS)
parameter of BHDE and $p_{D}$ its pressure.

Following He , we consider the rate $Q$ of energy exchange between BHDE and DM
in the form

$Q=3b^{2}H\left(\rho_{D}+\rho_{M}\right),$ (12)

where $b^{2}$ is a dimensionless constant that quantifies the coupling between
the dark sectors. Hence, we have $Q>0$, which implies that a net flux of
energy flows from BHDE to DM. This is consistent with the second law of
thermodynamics and Le Chatelier-Braun principle Q2 .

##  III Cosmological evolution of Barrow Holographic Dark Energy in nonflat
Universe

Let us now examine the time evolution of the above model. To this aim, we
insert the time derivative of Friedmann equation (4) into Eq. (10) and combine
the resulting expression with Eqs. (8) and (11). After some algebra, we get

$\frac{\dot{H}}{H^{2}}=\Omega_{k}-\frac{3\Omega_{D}}{2}\left(1+r+\omega_{D}\right).$
(13)

From this equation, we can straightforwardly derive the deceleration parameter
as

$\displaystyle q$ $\displaystyle=$
$\displaystyle-\frac{\ddot{a}}{aH^{2}}=-1-\frac{\dot{H}}{H^{2}}$ (14)
$\displaystyle=$
$\displaystyle-1-\Omega_{k}+\frac{3\Omega_{D}}{2}\left(1+r+\omega_{D}\right).$

It is easy to see that positive values of $q$ indicate a decelerated expansion
of the Universe ($\ddot{a}<0$), while negative values correspond to an
accelerated phase ($\ddot{a}>0$).

In a similar fashion, by combining the time derivative of Eq. (6) with (5) and
(13), we obtain

$\dot{\rho}_{D}=\left(2-\Delta\right)\rho_{D}\,H\left[\Omega_{k}-\frac{3\Omega_{D}}{2}\left(1+r+\omega_{D}\right)\right],$
(15)

while the use of Eqs. (7) and (13) leads to

$\displaystyle\Omega_{D}^{\prime}$ $\displaystyle=$
$\displaystyle-\Omega_{D}\left[\frac{1}{2}\left(2-\Delta\right)\left(3+\Omega_{k}+3\Omega_{D}\,\omega_{D}\right)\right.$
(16)
$\displaystyle+\,2\Omega_{k}-3\Omega_{D}\left(1+r+\omega_{D}\right)\bigg{]}.$

Here we have used the standard notation
$\Omega^{\prime}_{D}\equiv\frac{d\Omega_{D}}{d(\log a)}$.

A relevant quantity to establish whether the model is classically stable
against small perturbations is the square of sound speed, which is defined by

$v_{s}^{2}=\frac{dp_{D}}{d\rho_{D}}=\omega_{D}+\dot{\omega}_{D}\frac{\rho_{D}}{\dot{\rho}_{D}}\,.$
(17)

We remark that classical stability requirement is satisfied, provided that
$v_{s}^{2}>0$. Indeed, for a density perturbation, positive values of the
squared sound speed correspond to a regular propagation mode. Vice versa,
negative values imply that the perturbation equation becomes an irregular wave
equation Pert1 ; Pert2 . As a consequence, with a density perturbation, the
negative squared speed indicates an escalating mode. In other terms, when the
density perturbation increases, the pressure decreases, favoring the
development of an instability Pert1 ; Pert2 .

In what follows we study the evolution trajectories of the model parameters
introduced above. We analyze the noninteracting ($b^{2}=0$) and interacting
($b^{2}\neq 0$) cases separately, discussing their consistency with
observations.

###  III.1 Noninteracting model

In the case where there is no interaction between the dark sectors of the
cosmos, insertion of Eq. (15) into (10) allows us to infer the following
expression for the BHDE EoS parameter

$\omega_{D}=-\frac{3+\left(\frac{\Delta}{2}-1\right)\left(\Omega_{k}+3\right)}{3\left[1+\left(\frac{\Delta}{2}-1\right)\Omega_{D}\right]}\,.$
(18)

In turn, from Eq. (16) we get

$\Omega_{D}^{\prime}=\frac{\Omega_{D}\,\Delta}{2}\,\frac{3+\Omega_{k}-3\Omega_{D}}{1+\left(\frac{\Delta}{2}-1\right)\Omega_{D}}\,.$
(19)

Figure 1: The evolution of $\Omega_{D}$ versus $z$ for different values of
$\Delta$. We have set $\Omega_{k}=0.01$ and $\Omega_{D}^{0}=0.73$ as initial
condition. Figure 2: The evolution of $\Omega_{M}$ versus $z$ for different
values of $\Delta$. We have set $\Omega_{k}=0.01$ and $\Omega_{D}^{0}=0.73$ as
initial condition. Figure 3: The evolution of $\omega_{D}$ versus $z$ for
different values of $\Delta$. We have set $\Omega_{k}=0.01$ and
$\Omega_{D}^{0}=0.73$ as initial condition. Figure 4: The evolution of $q$
versus $z$ for different values of $\Delta$. We have set $\Omega_{k}=0.01$ and
$\Omega_{D}^{0}=0.73$ as initial condition.

To determine the behavior of BHDE, we solve this equation numerically. The
evolution of $\Omega_{D}$ and $\Omega_{M}$ versus the redshift
$z=\left(1-a\right)/a$ is plotted in Fig. 1 and Fig. 2 for different values of
$\Delta$, respectively. From Fig. 1 it is evident that $\Omega_{D}$ increases
monotonically going from early times to the far future. This indicates the
evolution of the present model from an initial dark matter dominated phase
toward a completely dark energy dominated epoch. The complementary behavior is
exhibited by $\Omega_{M}$ (see Fig. 2).

The dynamics of EoS parameter is shown in Fig. 3. As we can see, $\omega_{D}$
lies in the quintessence regime ($-1<\omega_{D}<-1/3$) at present (i.e. $z=0$)
and tends to the cosmological constant behavior $\omega_{D}=-1$ in the far
future ($z\rightarrow-1$), regardless of $\Delta$.222Notice that any
comparison with the results of Adhikary:2021xym  is prevented by the fact that
a different IR cutoff is used in that case.  Specifically, for the current
value of the EoS parameter the present model predicts
$\omega_{D}^{0}\in[-0.72,-0.47]$ for the considered values of $\Delta$. Notice
that this range is consistent with neither the recent observational
constraints obtained from Planck+WP+Union 2.1 ($-1.26<\omega_{0}<-0.92$) and
Planck+WP+BAO ($-1.38<\omega_{0}<-0.89$) data, nor with WMAP+eCMB+BAO+H0
($-1.162<\omega_{0}<-0.983$) measurements Planck , thus requiring some proper
amendment to this model.

For completeness, let us analyze the evolution of the deceleration parameter.
By plugging Eqs. (18) and (19) into (14), this takes the form

$q=\frac{1+\Omega_{k}-\left(1+\Delta\right)\Omega_{D}}{2+\left(\Delta-2\right)\Omega_{D}}\,.$
(20)

The behavior of $q$ is plotted in Fig. 4 for different values of $\Delta$. As
explained above, our model predicts the successive sequence of an early matter
dominated era with a decelerated expansion ($q>0$), followed by a late time DE
dominated epoch with an accelerated phase ($q<0$), In particular, the present
framework is capable of explaining the observed accelerated phase of the
Universe today ($z=0$), in contrast to the standard Holographic Dark Energy
model. However, quantitatively speaking the redshift $z_{t}$ at which the
transition from the decelerated to accelerated Universe occurs, i.e.
$q(z_{t})=0$, lies within the interval $z_{t}\in[0.02,0.46]$ for the
considered values of the model parameters. This range does not overlap with
the observational constraint obtained via SNIa+BAO/CMB ($z_{t}=0.72\pm 0.05$)
MamonEPJC2 , SNIa+CMB ($z_{t}=0.57\pm 0.07$) Alam  and SNIa+CMB+LSS joint
analysis ($z_{t}=0.61$) MamonMPA . Also, the value of the deceleration
parameter for the current epoch is estimated as $q_{0}\in[-0.26,-0.01]$, to be
compared with the experimental outcome $q_{0}=-0.64\pm 0.22$ from Union2 SNIa
data WuHu . These results provide further evidence of the observational
inconsistency of the non-interacting model.

Figure 5: The evolution of $v_{s}^{2}$ versus $z$ for different values of
$\Delta$. We have set $\Omega_{k}=0.01$ and $\Omega_{D}^{0}=0.73$ as initial
condition.

From Eq. (17), we can finally derive the square of sound speed as

$v_{s}^{2}=\frac{\left[\frac{3\Delta}{2}+\left(\frac{\Delta}{2}-1\right)\Omega_{k}\right]\left(\Omega_{D}-1\right)}{3\left[1-\left(1-\frac{\Delta}{2}\right)\Omega_{D}\right]^{2}}\,.$
(21)

This is plotted as a function of the redshift $z$ in Fig. 5. Interestingly
enough, we see that $v_{s}^{2}$ takes negative values at early and present
times, while it tends to positive values in the far future. In other terms,
the present model evolves towards a classically stable configuration. A
similar result has been exhibited in hybrid  for the case of BHDE with hybrid
expansion law.

Figure 6 finally shows the behavior of the jerk parameter, which is given by
the third derivative of the scale factor respect to the cosmic time, i.e.
Rapetti

$j\,=\,\frac{1}{aH^{3}}\frac{d^{3}a}{dt^{3}}=q\left(2q+1\right)+\left(1+z\right)\frac{dq}{dz}\,.$
(22)

This quantity allows us to quantify departures from $\Lambda$CDM model, which
is characterized by $j=1$. From Fig. 6, we find that $j>0$ for any redshift
and approaches unity in the far future (i.e. $z\rightarrow-1$), where our
model fits with $\Lambda$CDM. Also, we have $j_{0}\in[0.36,0.62]$ in the
present epoch for the considered values of model parameters. Notice that
deviations of $j_{0}$ from the $\Lambda$CDM value need attention as the real
cause behind the cosmic acceleration is still unknown Bambabis .

Figure 6: The evolution of $j$ versus $z$ for different values of $\Delta$. We
have set $\Omega_{k}=0.01$ and $\Omega_{D}^{0}=0.73$ as initial condition.

###  III.2 Interacting model

We now extend the above analysis to the case where the interaction (12)
between the dark sectors of the cosmos is taken into account. By repeating the
same computations as above, we derive the following generalized expression for
the EoS parameter

$\omega_{D}=-\frac{3+\left(\frac{\Delta}{2}-1\right)\left(\Omega_{k}+3\right)+3b^{2}\left(1+r\right)}{3\left[1+\left(\frac{\Delta}{2}-1\right)\Omega_{D}\right]}\,.$
(23)

Similarly, we get

$\displaystyle\Omega_{D}^{\prime}$ $\displaystyle=$
$\displaystyle\frac{\Omega_{D}\,\Delta}{2}\,\frac{3+\Omega_{k}-3\Omega_{D}-3b^{2}\left(1+\Omega_{k}\right)}{1+\left(\frac{\Delta}{2}-1\right)\Omega_{D}}\,,$
(24) $\displaystyle q$ $\displaystyle=$
$\displaystyle\frac{1+\Omega_{k}-\left(1+\Delta\right)\Omega_{D}-3b^{2}\left(1+\Omega_{k}\right)}{2+\left(\Delta-2\right)\Omega_{D}}\,,$
(25) $\displaystyle v_{s}^{2}$ $\displaystyle=$
$\displaystyle\frac{\left[\frac{3\Delta}{2}+\left(\frac{\Delta}{2}-1\right)\Omega_{k}\right]\left(\Omega_{D}-1\right)}{3\left[1-\left(1-\frac{\Delta}{2}\right)\Omega_{D}\right]^{2}}$
(26)
$\displaystyle+\frac{b^{2}\left(1+\Omega_{k}\right)\left[1+\frac{\Delta}{2}+\frac{2}{\left(\Delta-2\right)\Omega_{D}}\right]}{\left[1-\left(1-\frac{\Delta}{2}\right)\Omega_{D}\right]^{2}}\,,$

for the BHDE density, the deceleration parameter and the squared sound speed,
respectively. It is straightforward to check that these relations correctly
reduce to Eqs. (18), (19), (20) and (21) for $b^{2}=0$.

Fig. 7 shows the evolution of $\Omega_{D}$ versus $z$ for different values of
$b^{2}$ (upper panel) and $\Delta$ (lower panel). On the other hand, in Fig. 8
we plot the behavior of $\Omega_{M}$. We can see that both these two densities
exhibit the same profile as for the noninteracting case (see Fig. 1 and 2 for
comparison). Remarkable differences can instead be highlighted from the
analysis of the EoS parameter in Fig. 9. Indeed, in the present study the
phantom line $\omega_{D}=-1$ is crossed in the far future. In particular, the
higher the coupling, the more evident such a behavior (see upper panel of Fig.
9). This feature is peculiar to this model and has no correspondence in the
noninteracting description (see Fig. 3). To check the observational
consistency of the present model, let us compute the current value of the EoS
parameter. From the lower panel of Fig. 9, we find
$\omega_{D}^{0}\in[-0.96,-0.74]$, which is in agreement with constraints from
Planck+WP+BAO and Planck+WP+Union 2.1 measurements (see the discussion above
Eq. (20)).

Figure 7: The evolution of $\Omega_{D}$ versus $z$ for different values of
$b^{2}$ (upper panel) and $\Delta$ (lower panel). We have set
$\Omega_{k}=0.01$ and $\Omega_{D}^{0}=0.73$ as initial condition. For the
upper plot, we have considered $\Delta=0.4$, while for the lower one
$b^{2}=0.01$.

Figure 8: The evolution of $\Omega_{M}$ versus $z$ for different values of
$b^{2}$ (upper panel) and $\Delta$ (lower panel). We have set
$\Omega_{k}=0.01$ and $\Omega_{D}^{0}=0.73$ as initial condition. For the
upper plot, we have considered $\Delta=0.4$, while for the lower one
$b^{2}=0.01$.

Figure 9: The evolution of $\omega_{D}$ versus $z$ for different values of
$b^{2}$ (upper panel) and $\Delta$ (lower panel). We have set
$\Omega_{k}=0.01$ and $\Omega_{D}^{0}=0.73$ as initial condition. For the
upper plot, we have considered $\Delta=0.4$, while for the lower one
$b^{2}=0.01$.

In Fig. 10 we show the evolution of the deceleration parameter $q$. We see
that the interacting BHDE model still predicts a smooth transition of the
Universe from an initial decelerated phase to a late-time accelerated
expansion. However, the advantage of this framework over the non-interacting
one is that the transition redshift is now estimated as $z_{t}\in[0.30,0.64]$,
which is in good agreement with SNIa+CMB+LSS joint analysis. Also, the
deceleration parameter for the current epoch is found to lie in the range
$q_{0}\in[-0.33,-0.03]$ (see the lower panel in Fig. 10), the lower bound of
which is closer to the experimental value exhibited in WuHu . Thus, the
present model turns out to be favored by observations.

Figure 10: The evolution of $q$ versus $z$ for different values of $b^{2}$
(upper panel) and $\Delta$ (lower panel). We have set $\Omega_{k}=0.01$ and
$\Omega_{D}^{0}=0.73$ as initial condition. For the upper plot, we have
considered $\Delta=0.4$, while for the lower one $b^{2}=0.01$.

The behavior of the squared sound speed is plotted in Fig. 11. By comparison
with Fig. 5, we see that $v_{s}^{2}$ is now always negative, which implies
that interacting BHDE is unstable at classical level against small
perturbations.

Figure 11: The evolution of $v_{s}^{2}$ versus $z$ for different values of
$b^{2}$ (upper panel) and $\Delta$ (lower panel). We have set
$\Omega_{k}=0.01$ and $\Omega_{D}^{0}=0.73$ as initial condition. For the
upper plot, we have considered $\Delta=0.4$, while for the lower one
$b^{2}=0.01$.

Figure 12 shows the jerk parameter (22) versus the redshift. As in the absence
of interactions, we find that $j>0$ for any redshift, $j_{0}\in[0.35,0.64]$
(see the lower panel in Fig. 12) and $j\rightarrow 1$ in the far future (i.e.
$z\rightarrow-1$), thus reproducing $\Lambda$CDM in this limit.

Figure 12: The evolution of $j$ versus $z$ for different values of $b^{2}$
(upper panel) and $\Delta$ (lower panel). We have set $\Omega_{k}=0.01$ and
$\Omega_{D}^{0}=0.73$ as initial condition. For the upper plot, we have
considered $\Delta=0.4$, while for the lower one $b^{2}=0.01$.

To further investigate the experimental relevance of the present model and
constrain the free parameters $\Delta$, $b$ and $H_{0}$, let us consider the
evolution of $H(z)$ from Eqs. (14) and (25) for fixed $\Omega_{k}=0.01$ and
$\Omega_{D}^{0}=0.73$, and compare it with the data points obtained from the
latest compilation of 57 Hubble’s parameter measurements in the range
$0.07\leq z\leq 2.36$. Such points have been derived via Differential Age (31
points), BAO and other methods (the remaining 26 points) and are listed in
Table 1 (see Koussour  and references therein for more details on the data
set).

Following Koussour , we use the statistical $R^{2}$-test to find the best fit
values of model parameters. This is defined by

$R^{2}\,=\,1-\frac{\sum_{i=1}^{57}\left[(H_{i})_{ob}-(H_{i})_{th}\right]^{2}}{\sum_{i=1}^{57}\left[(H_{i})_{ob}-(H_{i})_{mean}\right]^{2}}\,,$
(27)

where $(H_{i})_{ob}$ and $(H_{i})_{th}$ are the observed and predicted values
of Hubble’s parameter, respectively. Minimization of deviation of $R^{2}$ from
unity gives the best-fit values $b=0.12$ and $\Delta=0.08$ at $95\%$
confidence level. We notice that the obtained estimation of Barrow parameter
fits with the result $\Delta=0.094^{+0.093}_{-0.101}$ derived in
Anagnostopoulos:2020ctz  and is very close to the constraint
$\Delta=0.075^{+0.001}_{-0.002}$ of Mathew , but it lies outside the intervals
$0.005\leq\Delta\leq 0.008$ Luciano:2022pzg  and $\Delta\lesssim 1.4\times
10^{-4}$ Barrow:2020kug  found via baryogenesis and Big Bang Nucleosynthesis
measurements, respectively. This discrepancy might be understood by assuming a
HDE description of the Universe with a running Barrow entropy, as discussed in
Dige .

From the fit in Fig. (13), we can also infer the current value of Hubble’s
parameter predicted by the present model as $H_{0}=(63.3\pm
4.3)\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$. This is consistent with the recent
observation from Planck Collaboration $H_{0}=\left(67.27\pm
0.60\right)\mathrm{km\,s^{-1}\,Mpc^{-1}}$ Planck , but still deviates from the
value $H_{0}=\left(74.03\pm 1.42\right)\mathrm{km\,s^{-1}\,Mpc^{-1}}$ based on
the analysis of 2019 SH0ES collaboration SH0ES . More discussion on this point
can be found in the concluding section.

$z$ |  $H(z)\,$ |  $\sigma_{H}$ |  $z\,$ |  $H(z)$ |  $\sigma_{H}$
---|---|---|---|---|---
0.070 |  69.0 |  19.6 |  0.4783 | 80 |  99
0.90 | 69 | 12 | 0.480 | 97 | 62
0.120 | 68.6 | 26.2 | 0.593 | 104 | 13
0.170 | 83 | 8 | 0.6797 | 92 | 8
0.1791 | 75 | 4 | 0.7812 | 105 | 12
0.1993 | 75 | 5 | 0.8754 | 125 | 17
0.200 | 72.9 | 29.6 | 0.880 | 90 | 40
0.270 | 77 | 14 | 0.900 | 117 | 23
0.280 | 88.8 | 36.6 | 1.037 | 154 | 20
0.3519 | 83 | 14 | 1.300 | 168 | 17
0.3802 | 83.0 | 13.5 | 1.363 | 160.0 | 33.6
0.400 | 95 | 17 | 1.430 | 177 | 18
0.4004 | 77.0 | 10.2 | 1.530 | 140 | 14
0.4247 | 87.1 | 11.2 | 1.750 | 202 | 40
0.4497 | 92.8 | 12.9 | 1.965 | 186.5 | 50.4
0.470 | 89 | 34 |  |  |

$z$ |  $H(z)\,$ |  $\sigma_{H}$ |  $z\,$ |  $H(z)$ |  $\sigma_{H}$
---|---|---|---|---|---
0.24 |  79.69 |  2.99 |  0.52 |  94.35 |  2.64
0.30 |  81.70 | 6.22 | 0.56 | 93.34 | 2.30
0.31 | 78.18 | 4.74 | 0.57 | 87.6 | 7.8
0.34 | 83.80 | 3.66 | 0.57 | 96.8 | 3.4
0.35 | 82.7 | 9.1 | 0.59 | 98.48 | 3.18
0.36 | 79.94 | 3.38 | 0.60 | 87.9 | 6.1
0.38 | 81.5 | 1.9 | 0.61 | 97.3 | 2.1
0.40 | 82.04 | 2.03 | 0.64 | 98.82 | 2.98
0.43 | 86.45 | 3.97 | 0.73 | 97.3 | 7.0
0.44 | 82.6 | 7.8 | 2.30 | 224.0 | 8.6
0.44 | 84.81 | 1.83 | 2.33 | 224 | 8
0.48 | 87.90 | 2.03 | 2.34 | 222.0 | 8.5
0.51 | 90.4 | 1.9 | 2.36 | 226.0 | 9.3

Table 1: 57 experimental points of $H(z)$: the first 31 have been obtained
from the method of Differential Age, while the last 26 from BAO and other
approaches ($H$ is expressed in $\mathrm{km\,s^{-1}\,Mpc^{-1}}$ and
$\sigma_{H}$ represents the uncertainty for each data point). Figure 13: Best
fit curve of Hubble’s parameter $H$ versus redshift $z$. Dots represent
observed values, while the red curve is the theoretical fit.

##  IV Thermal stability of Barrow Holographic Dark Energy

Let us now study our model from the thermodynamic point of view. Toward this
end, we follow the analysis of Abdollah ; Bhandari . In particular, we are
going to investigate whether the thermal stability criteria are satisfied.

We start by defining the internal energy of the cosmic fluid (dark energy +
dark matter) in the Universe as

$E=V\left(\rho_{D}+\rho_{M}\right)\equiv V\rho_{T}\,,$ (28)

where $V$ and $\rho_{T}$ are the volume and total energy density of the cosmic
system, respectively. We can then write the thermodynamic relation Callen

$TS=E+p_{D}V\,,$ (29)

where $T$ denotes the temperature of the cosmic system and we have implicitly
taken into account that DM is pressureless (i.e. $p_{M}=0$).

Inverting Eq. (29) respect to $E$ and using Eq. (28), we get

$E=\frac{S\,T}{1+\omega_{T}}\,,$ (30)

where $\omega_{T}\equiv p_{D}/\rho_{T}$ is the total EoS parameter of the
cosmic fluid. The behavior of $\omega_{T}$ versus $z$ is plotted in Fig. 14
and Figs. 16-17 for noninteracting and interacting BHDE models, respectively.
It can be seen that the evolution trajectories of this parameter are not
considerably affected by either the curvature $\Omega_{k}$ or the interaction
term $b^{2}$.

Figure 14: The evolution of $\omega_{T}$ versus $z$ for noninteracting BHDE
model and different values of $\Delta$. We have set $\Omega_{D}^{0}=0.73$ as
initial condition and $\Omega_{k}=0.01$ in the upper panel, while
$\Omega_{k}=0$ in the lower one.

Figure 15: Thermal stability condition (TSC) for noninteracting BHDE. We have
set $\Omega_{D}^{0}=0.73$ as initial condition and $\Omega_{k}=0.01$ in the
upper panel, while $\Omega_{k}=0$ in the lower one.

Figure 16: The evolution of $\omega_{T}$ versus $z$ for interacting BHDE model
and different values of $b^{2}$. We have set $\Delta=0.4$ and
$\Omega_{D}^{0}=0.73$ as initial condition. In the upper panel we have
considered $\Omega_{k}=0.01$, while in the lower one $\Omega_{k}=0$.

Figure 17: The evolution of $\omega_{T}$ versus $z$ for interacting BHDE model
and different values of $\Delta$. We have set $b^{2}=0.01$ and
$\Omega_{D}^{0}=0.73$ as initial condition. In the upper panel we have
considered $\Omega_{k}=0.01$, while in the lower one $\Omega_{k}=0$.

Figure 18: Thermal stability condition (TSC) for interacting BHDE and
different values of $b^{2}$. We have set $\Delta=0.4$ and
$\Omega_{D}^{0}=0.73$ as initial condition. In the upper panel we have
considered $\Omega_{k}=0.01$, while in the lower one $\Omega_{k}=0$.

Let us now consider a reversible adiabatic expansion ($dS=0$). By using the
first law of thermodyanmic combined with Eq. (30), we are led to Bhandari

$E=E_{0}\left(\frac{1+\omega_{0}}{1+\omega_{T}}\right)\frac{T}{T_{0}}\,,$ (31)

where $E_{0}$ and $T_{0}$ are the present values of the cosmic fluid energy
and temperature, respectively, while $\omega_{0}$ is the present value of the
total EoS parameter. Hence, comparison of Eqs. (30) and (31) gives

$S=\frac{\left(1+\omega_{0}\right)E_{0}}{T_{0}}=const.$ (32)

Additionally, using the heat capacities and the total energy-momentum
conservation law, we obtain Barboza ; Bhandari

$\displaystyle C_{v}$ $\displaystyle=$ $\displaystyle\left(\frac{\partial
E}{\partial T}\right)_{V}=\frac{C_{p}\,d\log V}{\left(1+\omega_{T}\right)d\log
V-d\log|\omega_{T}|}\,,$ (33) $\displaystyle C_{p}$ $\displaystyle=$
$\displaystyle\left(\frac{\partial h}{\partial
T}\right)_{p}=\frac{\left(1+\omega_{T}\right)E}{T}=\frac{\left(1+\omega_{0}\right)E_{0}}{T_{0}}=S\,,$
(34)

where we have resorted to Eq. (31) in the third step of the second relation.
Here, $C_{v}$ and $C_{p}$ are the heat capacities at constant volume and
pressure, respectively, and

$h=E+p\,V\,,$ (35)

is the enthalpy of the cosmic fluid.

By combining Eqs. (33) and (34), one can show that

$C_{p}=\left(1+\omega_{T}-\frac{d\log|\omega_{T}|}{d\log V}\right)C_{v}\,.$
(36)

Let us also introduce the definitions of thermal expansivity, isothermal
compressibility and adiabatic compressibility. They are given by

$\displaystyle\alpha$ $\displaystyle=$
$\displaystyle\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_{p}\,,$
(37) $\displaystyle\kappa_{T}$ $\displaystyle=$
$\displaystyle-\frac{1}{V}\left(\frac{\partial V}{\partial p}\right)_{T}\,,$
(38) $\displaystyle\kappa_{S}$ $\displaystyle=$
$\displaystyle-\frac{1}{V}\left(\frac{\partial V}{\partial p}\right)_{S}\,,$
(39)

respectively. By use of Eqs. (33)-(37), we then have Abdollah

$\alpha=\frac{C_{v}}{V\rho_{T}\,\omega_{T}}\left(\omega_{T}-\frac{d\log|\omega_{T}|}{d\log
V}\right).$ (40)

Moreover, the following relations between the heat capacities $\kappa_{T}$ and
$\kappa_{S}$ hold

$\displaystyle\kappa_{T}=\frac{\alpha
V}{C_{p}}\,,\qquad\kappa_{S}=\frac{\alpha VC_{v}}{C_{p}^{2}}\,,$ (41)
$\displaystyle\hskip
38.41121pt\frac{\kappa_{T}}{\kappa_{S}}=\frac{C_{p}}{C_{v}}\,.$ (42)

We now analyze the thermal stability of the present system assuming that work
is done due to the volume variation of the thermal system only. For stable
equilibrium, it has been shown that the second order variation of the internal
energy

$\delta^{2}E=\delta T\,\delta S-\delta p\,\delta V\,,$ (43)

must satisfy the condition $\delta^{2}E\geq 0$ Callen .

Notice that, by using Eqs. (33), (34), (38) and (39), we can equivalently cast
Eq. (43) as

$\delta^{2}E=V\,\kappa_{S}\,\delta p^{2}+\frac{T}{C_{p}}\delta S^{2}\,,$ (44)

or

$\delta^{2}E=\frac{1}{V\hskip 0.85358pt\kappa_{T}}\,\delta
V^{2}+\frac{C_{v}}{T}\,\delta T^{2}\,.$ (45)

Thus, for the stability of the system, the heat capacities and
compressibilities should obey

$C_{p},C_{v},\kappa_{S},\kappa_{T}\geq 0\,,$ (46)

which, combined with Eqs. (41) and (42), leads to Abdollah

$C_{p}\geq C_{v}\,,\qquad\kappa_{T}\geq\kappa_{S}\,.$ (47)

From Eq. (34), it is straightforward to see that $C_{p}$ is constant and
positive-definite. Moreover, Eqs. (36) and (40) imply that the constraints
(47) and, thus, (46) are met only if the total EoS parameter $\omega_{T}$
obeys the Thermal Stability Condition (TSC)

$\omega_{T}-\frac{d\log|\omega_{T}|}{d\log V}\geq 0\,.$ (48)

In Fig. 15 and Fig. 18 the TSC has been plotted for the noninteracting and
interacting cases, respectively. One can see that such criterion is not
satisfied, indicating that the cosmic fluid is not thermodynamically stable in
BHDE model.

##  V Conclusions and Outlook

We have analyzed the cosmic evolution of a nonflat FRW Universe filled by
pressureless dark matter and BHDE. By considering the apparent horizon as IR
cutoff, we have examined the evolutionary trajectories of various model
parameters, such as the BHDE density parameter, the EoS parameter, the
deceleration parameter, the jerk parameter and the squared sound speed. In the
absence of interaction, we have found that $\omega_{D}$ never crosses the
phantom line. Furthermore, the model successfully predicts the sequence of an
early matter dominated era with a decelerated expansion, followed by a late
time DE dominated epoch with an accelerated phase. Nevertheless, the values of
the current EoS parameter, transition redshift and deceleration parameter are
inconsistent with recent results in the literature (see Table 2).

On the other hand, assuming a mutual interaction between the dark sectors of
the cosmos, we have obtained that BHDE enters the phantom regime in the
future, with such a behavior being driven by the coupling $b^{2}$. The model
still predicts the usual thermal history of the Universe and is consistent
with observations for what concerns the values of the EoS parameter,
transition redshift and deceleration parameter in the current epoch.

|  Non-interacting model |  Interacting model |  Observational value
---|---|---|---
$\omega_{0}$ |  $[-0.72,-0.47]$ |  $[-0.96,-0.74]$ |  $[-1.38,-0.89]$ (Planck+WP+BAO)
$z_{t}$ |  $[0.02,0.46]$ |  $[0.30,0.64]$ |  $0.61$ (SNIa+CMB+LSS)
$q_{0}$ |  $[-0.26,-0.01]$ |  $[-0.33,-0.03]$ |  $[-0.86,-0.42]$ (Union2 SNIa)
$j_{0}$ |  $[0.36,0.62]$ |  $[0.35,0.64]$ |  $j=1$ ($\Lambda$CDM)
$H_{0}$ |  $-$ |  $(63.3\pm 4.3)\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$ |  $\left(67.27\pm 0.60\right)\mathrm{km\,s^{-1}\,Mpc^{-1}}$ (Planck)
Table 2: Theoretical and observational values of EoS parameter, transition
redshift, deceleration parameter, jerk parameter and Hubble parameter in the
current epoch (for the jerk parameter we have considered the $\Lambda$CDM
prediction as reference value).

Furthermore, for the interacting model we have studied the evolution of $H(z)$
and compared it with the data points obtained from the latest compilation of
57 Hubble’s parameter measurements from Differential Age, BAO and other
approaches. By using the $R^{2}$-test, we have found the best fit of $H(z)$
for the model parameters $b=0.12$ and $\Delta=0.08$ (with $95\%$ confidence
bounds). Also, the current value of Hubble parameter turns out to be
$H_{0}=(63.3\pm 4.3)\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$, which is consistent with
the recent constraint from Planck Collaboration Planck , but is in tension
with the observation from 2019 SH0ES Collaboration SH0ES . Along this line, it
could be interesting to explore whether BHDE (or, more general, HDE based on
deformed entropies) can play some role in solving the $H_{0}$ tension
H0Tension . However, a detailed investigation of this issue would require the
additional incorporation of CMB data and the development of a joint analysis.
This aspect goes beyond the scope of our analysis and will be addressed in
future works.

We have finally studied the thermal stability of our model by considering the
heat capacities and compressibilities. We have shown that both the
noninteracting and interacting frameworks suffer from the satisfaction of the
Thermal Stability Condition (48). This result is in line with the achievement
of Abdollah  for the case of Tsallis Holographic Dark Energy and, more
general, with the outcome of Barboza , where it has been found that DE fluids
with a time-dependent EoS parameter are in conflict with the physical
constraints imposed by thermodynamics.

Further aspects are still to be considered: first, one could investigate how
the present study interfaces with the results of Mamon:2018flx , where it has
been shown that some DE models, such as Chevallier-Polarski-Linder model,
Generalized Chaplygin Gas and Modified Chaplygin Gas are thermodynamically
stable for certain values of the model parameters. A potential path to explore
is to see whether thermal stability for BHDE is achieved by considering
different IR cutoffs and/or generalized interactions between the dark sectors
of the cosmos. Of course, it is possible that even after such modifications
the TSC is violated, indicating that BHDE may not be the answer to the
question of the unknown DE.

Additionally, since our framework provides a preliminary attempt to study
quantum gravity effects on the cosmic history of the Universe, it is worth
analyzing to what extent it is consistent with predictions of more fundamental
candidate theories of quantum gravity. Finally, we aim to examine in more
detail the connection between BHDE and other models proposed to explain the
origin of dark energy Yoo , and possibly fix more stringent constraints on the
deformation parameter $\Delta$. These lines of research are under active
investigation and are left for future projects.

###### Acknowledgements.

The author acknowledges Manos Saridakis for useful discussions and the
anonymous Referee for insightful comments that helped to improve the quality
of the manuscript. He is also grateful to the Spanish “Ministerio de
Universidades” for the awarded Maria Zambrano fellowship and funding received
from the European Union - NextGenerationEU. He finally acknowledges
participation in the COST Association Action CA18108 “Quantum Gravity
Phenomenology in the Multimessenger Approach”.

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