# Corrections to the apparent value of the cosmological constant due to local
inhomogeneities

Antonio Enea Romano1,2,5<EMAIL_ADDRESS>Pisin Chen1,2,3,4
<EMAIL_ADDRESS>1Department of Physics, National Taiwan University,
Taipei 10617, Taiwan, R.O.C.
2Leung Center for Cosmology and Particle Astrophysics, National Taiwan
University, Taipei 10617, Taiwan, R.O.C.
3Graduate Institute of Astrophysics, National Taiwan University, Taipei 10617,
Taiwan, R.O.C.
4Kavli Institute for Particle Astrophysics and Cosmology, SLAC National
Accelerator Laboratory, Menlo Park, CA 94025, U.S.A.
5Instituto de Fisica, Universidad de Antioquia, A.A.1226, Medellin, Colombia

###### Abstract

Supernovae observations strongly support the presence of a cosmological
constant, but its value, which we will call apparent, is normally determined
assuming that the Universe can be accurately described by a homogeneous model.
Even in the presence of a cosmological constant we cannot exclude nevertheless
the presence of a small local inhomogeneity which could affect the apparent
value of the cosmological constant. Neglecting the presence of the
inhomogeneity can in fact introduce a systematic misinterpretation of
cosmological data, leading to the distinction between an apparent and true
value of the cosmological constant. We establish the theoretical framework to
calculate the corrections to the apparent value of the cosmological constant
by modeling the local inhomogeneity with a $\Lambda LTB$ solution. Our
assumption to be at the center of a spherically symmetric inhomogeneous matter
distribution correspond to effectively calculate the monopole contribution of
the large scale inhomogeneities surrounding us, which we expect to be the
dominant one, because of other observations supporting a high level of
isotropy of the Universe around us.

By performing a local Taylor expansion we analyze the number of independent
degrees of freedom which determine the local shape of the inhomogeneity, and
consider the issue of central smoothness, showing how the same correction can
correspond to different inhomogeneity profiles. Contrary to previous attempts
to fit data using large void models our approach is quite general. The
correction to the apparent value of the cosmological constant is in fact
present for local inhomogeneities of any size, and should always be taken
appropriately into account both theoretically and observationally.

##  I Introduction

High redshift luminosity distance measurements Perlmutter:1999np ;
Riess:1998cb ; Tonry:2003zg ; Knop:2003iy ; Barris:2003dq ; Riess:2004nr  and
the WMAP measurements WMAP2003 ; Spergel:2006hy  of cosmic microwave
background (CMB) interpreted in the context of standard FLRW cosmological
models have strongly disfavored a matter dominated universe, and strongly
supported a dominant dark energy component, giving rise to a positive
cosmological acceleration.

As an alternative to dark energy, it has been proposed Nambu:2005zn ;
Kai:2006ws  that we may be at the center of an inhomogeneous isotropic
universe without cosmological constant described by a Lemaitre-Tolman-Bondi
(LTB) solution of Einstein’s field equations, where spatial averaging over one
expanding and one contracting region is producing a positive averaged
acceleration $a_{D}$, but it has been shown how spatial averaging can give
rise to averaged quantities which are not observable Romano:2006yc . Another
more general approach to map luminosity distance as a function of redshift
$D_{L}(z)$ to LTB models has been recently proposed Chung:2006xh ; Yoo:2008su
, showing that an inversion method can be applied successfully to reproduce
the observed $D_{L}(z)$. Interesting analysis of observational data in
inhomogeneous models without dark energy and of other theoretically related
problems is given for example in Alexander:2007xx ; Alnes:2005rw ;
GarciaBellido:2008nz ; GarciaBellido:2008gd ; GarciaBellido:2008yq ;
February:2009pv ; Uzan:2008qp ; Quartin:2009xr ; Quercellini:2009ni ;
Clarkson:2007bc ; Ishibashi:2005sj ; Clifton:2008hv ; Celerier:2009sv ;
Romano:2007zz ; Romano:2009qx ; Romano:2009ej ; Romano:2009mr ;
Mustapha:1998jb

Here in this paper we will adopt a different approach. We will consider a
Universe with a cosmological constant and some local large scale inhomogeneity
modeled by a $\Lambda LTB$ solution Romano:2010nc . For simplicity we will
also assume that we are located at its center. In this regard this can be
considered a first attempt to model local large scale inhomogeneities in the
presence of the cosmological constant or, more in general, dark energy. Given
the spherical symmetry of the LTB solution and the assumption to be located at
the center our calculation can be interpreted as the monopole contribution of
the large inhomogeneities which surround us. Since we know from other
observations such as CMB radiation that the Universe appears to be highly
isotropic, we can safely assume that the monopole contribution we calculate
should also be the dominant one, making our results even more relevant. After
calculating the null radial geodesics for a central observer we then compute
the luminosity distance and compare it to that of $\Lambda CDM$ model, finding
the relation between the two different cosmological constants appearing in the
two models, where we call apparent the one in the $\Lambda CDM$ and true the
one in $\Lambda LTB$. Our calculations show that the corrections to
$\Omega^{app}_{\Lambda}$, which is the value of the cosmological constant
obtained from analyzing supernovae data assuming homogeneity, can be important
and should be taken into account.

##  II LTB solution with a cosmological constant

The LTB solution can be written as Lemaitre:1933qe ; Tolman:1934za ;
Bondi:1947av  as

$\displaystyle
ds^{2}=-dt^{2}+\frac{\left(R,_{r}\right)^{2}dr^{2}}{1+2\,E(r)}+R^{2}d\Omega^{2}\,,$
(1)

where $R$ is a function of the time coordinate $t$ and the radial coordinate
$r$, $E(r)$ is an arbitrary function of $r$, and $R_{,r}=\partial_{r}R(t,r)$.
The Einstein equations with dust and a cosmological constant give

$\displaystyle\left({\frac{\dot{R}}{R}}\right)^{2}$ $\displaystyle=$
$\displaystyle\frac{2E(r)}{R^{2}}+\frac{2M(r)}{R^{3}}+\frac{\Lambda}{3}\,,$
(2) $\displaystyle\rho(t,r)$ $\displaystyle=$
$\displaystyle\frac{2M,_{r}}{R^{2}R,_{r}}\,,$ (3)

with $M(r)$ being an arbitrary function of $r$, $\dot{R}=\partial_{t}R(t,r)$
and $c=8\pi G=1$ is assumed throughout the paper. Since Eq. (2) contains
partial derivatives respect to time only, its general solution can be obtained
from the FLRW equivalent solution by making every constant in the latter one
an arbitrary function of $r$.

The general analytical solution for a FLRW model with dust and cosmological
constant was obtained by Edwards Dilwyn  in terms of elliptic functions. By an
appropriate choice of variables and coordinates, we may extend it to the LTB
case thanks to the spherical symmetry of both LTB and FLRW models, and to the
fact that dust follows geodesics without being affected by adjacent regions.
An anaytical solution can be found by introducing a new coordinate
$\eta=\eta(t,r)$ and a variable $a$ by

$\displaystyle\left(\frac{\partial\eta}{\partial
t}\right)_{r}=\frac{r}{R}\equiv\frac{1}{a}\,,$ (4)

and new functions by

$\displaystyle\rho_{0}(r)\equiv\frac{6M(r)}{r^{3}}\,,\quad
k(r)\equiv-\frac{2E(r)}{r^{2}}\,.$ (5)

Then Eq. (2) becomes

$\left(\frac{\partial
a}{\partial\eta}\right)^{2}=-k(r)a^{2}+\frac{\rho_{0}(r)}{3}a+\frac{\Lambda}{3}a^{4}\,,$
(6)

where $a$ is now regarded as a function of $\eta$ and $r$, $a=a(\eta,r)$. It
should be noted that the coordinate $\eta$, which is a generalization of the
conformal time in a homogeneous FLRW universe, has been only implicitly
defined by Eq. (4). The actual relation between $t$ and $\eta$ can be obtained
by integration once $a(\eta,r)$ is known:

$t(\eta,r)=\int_{0}^{\eta}{a(x,r)dx}+t_{b}(r)\,,$ (7)

which can be computed analytically, and involve elliptic integrals of the
third kindellint .

The function $t_{B}(r)$ plays the role of constant of integration, and is an
arbitrary function of $r$, sometime called bang function, since by
construction at time $t=t_{b}(r)$ we have $a(t_{b}(r),r)=0$, and correspond to
the fact that the big bang initial singularity can happen at different times
at different positions from the center in a LTB space. In the rest of this
paper we will assume homogeneous bang, i.e. we will set

$t_{b}(r)=0.$ (8)

Inspired by the construction of the solution for the FLRW case get:

$a(\eta,r)=\frac{\rho_{0}(r)}{3\phi\left(\frac{\eta}{2};g_{2}(r),g_{3}(r)\right)+k(r)}\,,$
(9)

where $\phi(x;g_{2},g_{3})$ is the Weierstrass elliptic function satisfying
the differential equation

$\left(\frac{d\phi}{dx}\right)^{2}=4\phi^{3}-g_{2}\phi-g_{3}\,,$ (10)

and

$\displaystyle\alpha=\rho_{0}(r)\,,\quad g_{2}=\frac{4}{3}k(r)^{2}\,,\quad
g_{3}=\frac{4}{27}\left(2k(r)^{3}-\Lambda\rho_{0}(r)^{2}\right)\,.$ (11)

In this paper we will choose the so called FLRW gauge, i.e. the coordinate
system in which $\rho_{0}(r)$ is constant.

##  III Geodesic equations and luminosity distance

We adopt the same method developed in Romano:2009xw  to solve the null
geodesic equation written in terms of the coordinates $(\eta,r)$. Instead of
integrating differential equations numerically, we perform a local expansion
of the solution around $z=0$ corresponding to the point $(t_{0},0)$, or
equivalently $(\eta_{0},0)$, where $t_{0}=t(\eta_{0},0)$. The change of
variables from $(t,r)$ to $(\eta,r)$ permits us to have r.h.s. of all
equations in a fully analytical form, in contrast to previous considerations
of this problem which require a numerical calculation of $R(t,r)$ from the
Einstein equation (2). Thus, this formulation is particularly suitable for
derivation of analytical results.

The luminosity distance for a central observer in the LTB space-time as a
function of the redshift $z$ is expressed as

$D_{L}(z)=(1+z)^{2}R\left(t(z),r(z)\right)=(1+z)^{2}r(z)a\left(\eta(z),r(z)\right)\,,$
(12)

where $\Bigl{(}t(z),r(z)\Bigr{)}$ or $\Bigl{(}(\eta(z),r(z)\Bigr{)}$ is the
solution of the radial geodesic equation as a function of $z$. The past-
directed radial null geodesics is given by

$\displaystyle\frac{dt}{dr}=-\frac{R_{,r}(t,r)}{\sqrt{1+2E(r)}}\,.$ (13)

In terms of $z$, Eq. (13) takes the form Celerier:1999hp :

$\displaystyle{dr\over dz}$ $\displaystyle=$
$\displaystyle{\sqrt{1+2E(r(z))}\over{(1+z){\dot{R}}_{,r}[r(z),t(z)]}}\,,$
$\displaystyle{dt\over dz}$ $\displaystyle=$
$\displaystyle-{R_{,r}[r(z),t(z)]\over{(1+z){\dot{R}}_{,r}[r(z),t(z)]}}\,.$
(14)

The inconvenience of using the $(t,r)$ coordinates is that there is no exact
analytical solution for $R(t,r)$. So the r.h.s. of Eqs. (14) cannot be
evaluated analytically, but we are required to find a numerical solution for
$R$ first Hellaby:2009vz , and then to integrate numerically the differential
equations, which is quite an inconvenient and cumbersome procedure, and cannot
be used to derive anaytical results.

It can be shown Romano:2009xw  that in the coordinates $(\eta,r)$ eqs. (14)
take the form:

$\displaystyle\frac{d\eta}{dz}$ $\displaystyle=$
$\displaystyle-\frac{\partial_{r}t(\eta,r)+F(\eta,r)}{(1+z)\partial_{\eta}F(\eta,r)}\equiv
p(\eta,r)\,,$ (15) $\displaystyle\frac{dr}{dz}$ $\displaystyle=$
$\displaystyle\frac{a(\eta,r)}{(1+z)\partial_{\eta}F(\eta,r)}\equiv
q(\eta,r)\,,$ (16)

where

$F(\eta,r)\equiv\frac{\
R_{,r}}{\sqrt{1+2E(r)}}=\frac{1}{\sqrt{1-k(r)r^{2}}}\left[\partial_{r}(a(\eta,r)r)-a^{-1}\partial_{\eta}(a(\eta,r)r)\,\partial_{r}t(\eta,r)\right]\,.$
(17)

It is important to observe that the functions $p,q,F$ have explicit analytical
forms, making it particularly useful do derive anaytical results.

##  IV Number of independent parameters and Taylor expansion accuracy

In order to find the relation between the apparent and true value of the
cosmological constant we need in to match the terms in the red-shift expansion
:

$\displaystyle D_{i}^{\Lambda CDM}=D^{{\Lambda LTB}}_{i}$ , (18)

Before proceeding in deriving this relation we need to understand clearly how
many independent parameters we can solve for at different order in the Taylor
expansion for $D_{L}(z)$. After defining the expansion of the function $k(r)$
in terms of the dimensionless function $K(r)$:

$k(r)=(a_{0}H_{0})^{2}K(r)=K_{0}+K_{1}r+K_{2}r^{2}+..\\\ $ (19)

we have

$\displaystyle D_{1}^{\Lambda LTB}$ $\displaystyle=$
$\displaystyle\frac{1}{H^{\Lambda LTB}_{0}}\,,$ (20) $\displaystyle D^{\Lambda
LTB}_{i}$ $\displaystyle=$ $\displaystyle
f_{i}(\Omega_{\Lambda},K_{0},K_{1},..,K_{i-1}),$ (21)

which implies that if we want to match the coefficient $D_{i}$ up to order n,
we will have a total of $n+2$ independent parameters to solve for :

$\\{H^{\Lambda}_{0},\Omega_{\Lambda},K_{0},K_{i},..,K_{i-1}\\}.$ (22)

The matching conditions will imply a constraint over the $n+2$ independent
parameters, but this will not be enough completely determine them, since two
of them will always be free. For a matter of computational convenience we will
choose $K_{0},K_{1}$ as free parameters and express all the other in terms of
them. For example from :

$D_{2}^{\Lambda CDM}=D^{{\Lambda LTB}}_{2},$ (23)

we can get

$\Omega^{app}_{\Lambda}(\Omega_{\Lambda},K_{0},K_{1}),$ (24)

from

$D_{3}^{\Lambda CDM}=D^{{\Lambda LTB}}_{3},$ (25)

we can get

$K_{2}(\Omega_{\Lambda}^{app},K_{0},K_{1}),$ (26)

and in general from

$D_{i}^{\Lambda CDM}=D^{{\Lambda LTB}}_{i},$ (27)

we can get

$K_{i-1}(\Omega_{\Lambda}^{app},K_{0},K_{1}).$ (28)

Since our purpose is to find the corrections to the apparent value of the
cosmological constant, the second order term $D_{2}$ is enough. Higher order
terms in the redshift expansion will provide $K_{2},K_{3},..K_{i-1}$ as
functions of $\\{\Omega_{\Lambda}^{app},K_{0},K_{1}\\}$, but will not change
the analytical relation between $\Omega^{app}_{\Lambda}$ and
$\Omega_{\Lambda}$ which can be derived from eq.(23). For this reason we will
only need the expansion up to second order for the luminosity distance. The
fact that we have more free parameters than constraints implies that the same
correction to the apparent value of the cosmological constant can correspond
to an infinite number of different inhomogeneity profiles.

The corrections we calculate are accurate within the limits of validity of the
Taylor expansion $D^{\Lambda CDM}_{Taylor}$. It turns out that in the flat
case we consider the error is quite large already at a redshift of about $0.2$
as shown in the figure. This implies that the corrections should also be valid
only within this low redshift range, since even if we are exactly matching the
coefficients, the Taylor expansion of the $\Lambda CDM$ best fit formula
itself is not very accurate. This could be overcome by implementing other
types of expansions or numerical methods, such as Padé for example, with
better convergence behavior, but we’ll leave this to a future work.

Figure 1: The percentual error $\Delta=100\frac{D^{\Lambda CDM}-D^{\Lambda
CDM}_{Taylor}}{D^{\Lambda CDM}}$ for a third order expansion is plotted as a
function of the redshift. As it can be seen the error is already quite large
at redshift 0.1. Higher order expansion does not improve the convergence.

##  V Central behavior

A function of the radial coordinate $f(r)$ is smooth at the center $r=0$ only
if all its odd derivatives vanish there. This can be shown easily by looking
at the partial derivatives of even order of this type for example:

$\displaystyle\partial^{2n}_{x}\partial^{2n}_{y}\partial^{2n}_{z}f(\sqrt{x^{2}+y^{2}+z^{2}})\,,$
(29)

where $\\{x,y,z\\}$ are the cartesian coordinates related to $r$ by
$r^{2}=x^{2}+y^{2}+z^{2}$. Quantities of the type above diverge at the center
if $\partial^{2m+1}_{r}f(r)\neq 0$ for $2m+1<2n$. If for example the first
derivative $f^{\prime}(0)$ is not zero, then the laplacian will diverge. This
implies that including linear terms expansions for $k(r)$ and $t_{b}(r)$ we
are considering models which are not smooth at the center. The general central
smoothness conditions are:

$\displaystyle k_{2m+1}$ $\displaystyle=$ $\displaystyle 0,$ (30)
$\displaystyle t_{b}^{2m+1}$ $\displaystyle=$ $\displaystyle 0\,,$ (31)
$\displaystyle 2m+1$ $\displaystyle<$ $\displaystyle i\,,$ (32)

which must be satisfied for all the relevant odd powers coefficients of the
central Taylor expansion. In our case this implies that if we only want to
consider centrally smooth inhomogeneities then we need to set to zero all the
odd derivatives of $K(r)$

$K_{2m+1}=0$ (33)

The consequence of this smoothness conditions is that the exact matching of
the Taylor expansion is possible only up to order five when we have five
constraints equations

$\displaystyle D_{i}^{\Lambda CDM}=D^{{\Lambda LTB}}_{i}$
$\displaystyle\quad,\quad 1\leq i\leq 5\,,$ (34)

and five free parameters

$\displaystyle{H_{0}^{\Lambda LTB},\Omega_{\Lambda},K_{0},K_{2},K_{4}}$ (35)

implying there is a unique solution. Going to higher order there will be more
equations than free parameters making the inversion problem impossible. This
means that the effects of a different value of the cosmological constant
cannot be mimicked by a smooth inhomogeneity, as far as the exact matching of
the Taylor expansion is concerned. From a data analysis point of view this
limitation could be easily circumvented, since these considerations are based
on matching the Taylor expansion of the best $\Lambda CDM$ fit, which is quite
different from fitting the actual data. Also it turns out that the Taylor
expansion $D^{\Lambda CDM}_{Taylor}(z)$ is more accurate at second order than
at any other order as shown in the figure, implying that exact matching beyond
second order is practically irrelevant from a data fitting point of view.
Under these considerations the inversion problem can be considered still
effectively undetermined since by matching up to second order we have two
equations and three parameters:

$\displaystyle{H_{0}^{\Lambda LTB},\Omega_{\Lambda},K_{0}}$ (36)

For completeness of the analysis we mention that after counting the number of
independent parameters we can easily conclude that the inversion problem
remain undetermined for the third order, and has a unique solution for the
fourth and fifth order as shown above.

##  VI Calculating the luminosity distance

In order to obtain the redshift expansion of the luminosity distance we need
to use the following:

$\displaystyle k(r)$ $\displaystyle=$
$\displaystyle(a_{0}H_{0})^{2}K(r)=K_{0}+K_{1}r+K_{2}r^{2}+..$ (37)
$\displaystyle t(\eta,r)$ $\displaystyle=$ $\displaystyle
b_{0}(\eta)+b_{1}(\eta)r+b_{2}(\eta)r^{2}+..$ (38)

It should be noted that linear terms will in fact lead to central divergences
of the laplacian in spherical coordinates, which correspond to a central spike
of the energy distribution Romano:2009ej ; Romano:2009mr , but an appropriate
local averaging of the solution can easily heal this behavior, and we include
them here because they give the leading order contribution. Since we are
interested in the effects due to the inhomogeneities we will neglect $k_{0}$
in the rest of the calculation because this corresponds to the homogeneous
component of the curvature function $k(r)$.

Following the same approach given in Romano:2010nc  , we can find a local
Taylor expansion in red-shift for the geodesics equations, and then calculate
the luminosity distance:

$\displaystyle D^{\Lambda LTB}_{L}(z)$ $\displaystyle=$
$\displaystyle(1+z)^{2}r(z)a^{\Lambda LTB}(\eta(z),r(z))=D^{\Lambda
LTB}_{1}z+D^{\Lambda LTB}_{2}z^{2}+D^{\Lambda LTB}_{3}z^{3}+..$ (39)
$\displaystyle D^{\Lambda LTB}_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{H_{0}},$ $\displaystyle D^{\Lambda LTB}_{2}$
$\displaystyle=$
$\displaystyle\frac{1}{{36H_{0}(\Omega_{\Lambda}^{true}-1)}}\bigg{[}54B_{1}(\Omega_{\Lambda}^{true}-1)^{2}+18B^{\prime}_{1}(\Omega_{\Lambda}^{true}-1)-18h_{0,r}(\Omega_{\Lambda}^{true})^{2}$
(40)
$\displaystyle+30h_{0,r}\Omega_{\Lambda}^{true}-12h_{0,r}+6K_{1}\Omega_{\Lambda}^{true}-10K_{1}+27(\Omega_{\Lambda}^{true})^{2}-18\Omega_{\Lambda}^{true}-9\bigg{]},$

where we have introduced the dimensionless quantities
$K_{0},K_{1},B_{1},B^{\prime}_{1},h_{0,r}$ according to

$\displaystyle H_{0}$ $\displaystyle=$
$\displaystyle\left(\frac{\partial_{t},a(t,r)}{a(t,r)}\right)^{2}\Biggr{|}_{t=t_{0},r=0}=\left(\frac{\partial_{\eta}a(\eta,r)}{a(\eta,r)^{2}}\right)^{2}\Biggr{|}_{\eta=\eta_{0},r=0},$
(41) $\displaystyle B_{1}(\eta)$ $\displaystyle=$ $\displaystyle
b_{1}(\eta)a_{0}^{-1},$ (42) $\displaystyle B_{1}$ $\displaystyle=$
$\displaystyle b_{1}(\eta_{0})a_{0}^{-1},$ (43) $\displaystyle B^{\prime}_{1}$
$\displaystyle=$ $\displaystyle\frac{\partial
B_{1}(\eta)}{\partial\eta}\Biggr{|}_{\eta=\eta_{0}}(a_{0}H_{0})^{-2},$ (44)
$\displaystyle h_{0,r}$ $\displaystyle=$
$\displaystyle\frac{1}{a_{0}H_{0}}\frac{\partial_{r}a(\eta,r)}{a(\eta,r)}\Biggr{|}_{\eta=\eta_{0},r=0},$
(45) $\displaystyle t_{0}$ $\displaystyle=$ $\displaystyle t(\eta_{0},0),$
(46)

and used the Einstein equation at the center $(\eta=\eta_{0},r=0)$

$\displaystyle 1$ $\displaystyle=$
$\displaystyle\Omega_{k}(0)+\Omega_{M}+\Omega_{\Lambda}=-K_{0}+\Omega_{M}+\Omega_{\Lambda},$
(47) $\displaystyle\Omega_{k}(r)$ $\displaystyle=$
$\displaystyle-\frac{k(r)}{H_{0}^{2}a_{0}^{2}},$ (48)
$\displaystyle\Omega_{M}$ $\displaystyle=$
$\displaystyle\frac{\rho_{0}}{3H_{0}^{2}a_{0}^{3}},$ (49)
$\displaystyle\Omega_{\Lambda}$ $\displaystyle=$
$\displaystyle\frac{\Lambda}{3H_{0}^{2}}.$ (50)

Because of our coordinate choice $\Omega_{M}$ is independent of $r$, and all
the radial dependence goes into $\Omega_{k}(r)$. Note that apart from the
central curvature term $K_{0}$, the inhomogeneity of the $LTB$ space is
expressed in $h_{0,r}$, which encodes the radial dependence of the scale
factor. Details of these rather cumbersome calculations are provided in a
separate companion paper, but it should be emphasized that in order to put the
formula for the luminosity distance in this form it is necessary to manipulate
appropriately the elliptic functions and then re-express everything in terms
of physically meaningful quantities such as $H_{0}$.

##  VII Calculating $D_{L}(z)$ for $\Lambda CDM$ models.

The metric of a $\Lambda CDM$ model is the FLRW metric, a special case of LTB
solution, where :

$\displaystyle\rho_{0}(r)$ $\displaystyle\propto$ $\displaystyle const,$ (51)
$\displaystyle k(r)$ $\displaystyle=$ $\displaystyle 0,$ (52) $\displaystyle
t_{b}(r)$ $\displaystyle=$ $\displaystyle 0,$ (53) $\displaystyle a(t,r)$
$\displaystyle=$ $\displaystyle a(t).$ (54)

We will calculate independently the expansion of the luminosity distance and
the redshift spherical shell mass for the case of a flat $\Lambda CDM$, to
clearly show the meaning of our notation, and in particular the distinction
between $\Omega_{\Lambda}^{app}$ and $\Omega_{\Lambda}^{true}$. We can also
use these formulas to check the results derive before, since in absence of
inhomogeneities they should coincide.

One of the Einstein equation can be expressed as:

$\displaystyle H^{\Lambda CDM}(z)$ $\displaystyle=$ $\displaystyle
H_{0}\sqrt{(1-\Omega_{\Lambda}^{app}){\left(\frac{a_{0}}{a}\right)}^{3}+\Omega_{\Lambda}^{app}}=H_{0}\sqrt{(1-\Omega_{\Lambda}^{app}){(1+z)}^{3}+\Omega_{\Lambda}^{app}}.$
(55)

We can then calculate the luminosity distance using the following relation,
which is only valid assuming flatness:

$\displaystyle D^{\Lambda
CDM}_{L}(z)=(1+z)\int^{z}_{0}{\frac{dz^{\prime}}{H^{\Lambda
CDM}(z^{\prime})}}=D^{\Lambda CDM}_{1}z+D^{\Lambda CDM}_{2}z^{2}+D^{\Lambda
CDM}_{3}z^{3}+...$ (56)

From which we can get:

$\displaystyle D^{\Lambda CDM}_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{H_{0}}\,,$ (57) $\displaystyle D^{\Lambda CDM}_{2}$
$\displaystyle=$ $\displaystyle\frac{3\Omega_{\Lambda}^{app}+1}{4H_{0}}\,.$
(58)

We can check the consistency between these formulae and the ones derived in
the case of LTB by setting:

$\displaystyle K_{1}=B_{1}=B^{\prime}_{1}=K_{0}=h_{0,r}=0\,,$ (59)

which corresponds to the case in which
$\Omega_{\Lambda}^{app}=\Omega_{\Lambda}^{true}$.

##  VIII Relation between apparent and true value of the cosmological constant

So far we have calculated the first two terms of the redshift expansion of the
luminosity distance for $\Lambda LTB$ and $\Lambda CDM$ model. Since we now
that the latter provides a good fitting for supernovae observations, we can
now look for the $\Lambda LTB$ models which give the same theoretical
prediction. From the above relations we can derive :

$\displaystyle H_{0}^{\Lambda LTB}$ $\displaystyle=$ $\displaystyle H^{\Lambda
CDM}_{0}\,,$ (61) $\displaystyle\Omega_{\Lambda}^{app}$ $\displaystyle=$
$\displaystyle\frac{1}{{27(\Omega_{\Lambda}^{true}-1)}}\Bigg{[}54B_{1}(\Omega_{\Lambda}^{true})^{2}-108B_{1}\Omega_{\Lambda}^{true}+54B_{1}+18B^{\prime}_{1}\Omega_{\Lambda}^{true}-18B^{\prime}_{1}$
(62)
$\displaystyle-18h_{0,r}(\Omega_{\Lambda}^{true})^{2}+30h_{0,r}\Omega_{\Lambda}^{true}-12h_{0,r}+6K_{1}\Omega_{\Lambda}^{true}-10K_{1}$
$\displaystyle+27\Omega_{\Lambda}^{true}(\Omega_{\Lambda}^{true}-1)\Bigg{]}\,,$
$\displaystyle\Omega_{\Lambda}^{true}$ $\displaystyle=$
$\displaystyle-\frac{1}{{6(6B_{1}-2h_{0,r}+3)}}\Bigg{[}\Bigg{(}(36B_{1}-6B^{\prime}_{1}-10h_{0,r}-2K_{1}+9\Omega_{\Lambda}^{app}+9)^{2}+$
(63)
$\displaystyle-4(6B_{1}-2h_{0,r}+3)(54B_{1}-18B^{\prime}_{1}-12h_{0,r}-10K_{1}+27\Omega_{\Lambda}^{app})\Bigg{)}^{1/2}-36B_{1}$
$\displaystyle+6B^{\prime}_{1}+10h_{0,r}+2K_{1}-9(\Omega_{\Lambda}^{app}-1)\Bigg{]}\,.$

We can also expand the above exact relations assuming that all the
inhomogeneities, can be treated perturbatively respect to the $\Lambda CDM$ ,
i.e. $\\{{K_{1},B_{1},B^{\prime}_{1}\\}}\propto\epsilon$, where $\epsilon$
stands for a small deviation from $FLRW$ solution :

$\displaystyle\Omega_{\Lambda}^{true}$ $\displaystyle=$
$\displaystyle\Omega_{\Lambda}^{app}-\frac{2}{{27(\Omega_{\Lambda}^{app}-1)}}(27B_{1}(\Omega_{\Lambda}^{app}-1)^{2}+9B^{\prime}_{1}(\Omega_{\Lambda}^{app}-1)-9h_{0,r}{(\Omega_{\Lambda}^{app})}^{2}+15h_{0,r}\Omega_{\Lambda}^{app}$
(64)
$\displaystyle-6h_{0,r}+3K_{1}\Omega_{\Lambda}^{app}-5K_{1})+O(\epsilon^{2})\,.$

As expected all these relations reduce to

$\displaystyle\Omega_{\Lambda}^{true}$ $\displaystyle=$
$\displaystyle\Omega_{\Lambda}^{app},$ (65)

in the limit in which there is no inhomogeneity, i.e. when
$K_{1}=B_{1}=B^{\prime}_{1}=h_{0,r}=0$.

##  IX Conclusions

We have derived for the first time the correction due to local large scale
inhomogeneities to the value of the apparent cosmological constant inferred
from low redshift supernovae observations. This analytical calculation shows
how the presence of a local inhomogeneity can affect the estimation of the
value of cosmological parameters, such as $\Omega_{\Lambda}$. This effects
should be properly taken into account both theoretically and observationally.
By performing a local Taylor expansion we analyzed the number of independent
degrees of freedom which determine the local shape of the inhomogeneity, and
consider the issue of central smoothness, showing how the same correction can
correspond to different inhomogeneity profiles. We will address in a future
work the estimation of the magnitude of this effect based on experimental
bounds which can be set on the size and shape of a local inhomogeneity and the
fitting of actual supernovae data. It is important to underline here that we
do not need a large void as normally assumed in previous studies of $LTB$
models in a cosmological context. Even a small inhomogeneity could in fact be
important.

In the future it will also be interesting to extend the same analysis to other
observables such as barionic acoustic oscillations (BAO) or the cosmic
microwave background radiation (CMBR), and we will report about this in
separate papers. Another direction in which the present work could be extended
is modeling the local inhomogeneity in a more general way, for example
considering not spherically symmetric solutions. From this point of view our
calculation could be considered the monopole contribution to the general
effect due to a local large scale inhomogeneity of arbitrary shape. Given the
high level of isotropy of the Universe shown by other observations such as the
CMB radiation, we can expect the monopole contribution we calculated to be the
dominant one.

While this should be considered only as the first step towards a full
inclusion of the effects of large scale inhomogeneities in the interpretation
of cosmological observations, it is important to emphasize that we have
introduced a general definition of the concept of apparent and true value of
cosmological parameters, and shown the general theoretical approach to
calculate the corrections to the apparent values obtained under the standard
assumption of homogeneity.

###### Acknowledgements.

We thank A. Starobinsky, M. Sasaki for useful comments and discussions. Chen
and Romano are supported by the Taiwan NSC under Project No.
NSC97-2112-M-002-026-MY3, by Taiwan’s National Center for Theoretical Sciences
(NCTS). Chen is also supported by the US Department of Energy under Contract
No. DE-AC03-76SF00515.

## References

  * (1) S. Perlmutter et al. [Supernova Cosmology Project Collaboration], “Measurements of Omega and Lambda from 42 High-Redshift Supernovae,” Astrophys. J. 517, 565 (1999) [arXiv:astro-ph/9812133].
  * (2) A. G. Riess et al. [Supernova Search Team Collaboration], Astron. J. 116, 1009 (1998) [arXiv:astro-ph/9805201].
  * (3) J. L. Tonry et al. [Supernova Search Team Collaboration], Astrophys. J. 594, 1 (2003) [arXiv:astro-ph/0305008].
  * (4) R. A. Knop et al. [The Supernova Cosmology Project Collaboration], Astrophys. J. 598, 102 (2003) [arXiv:astro-ph/0309368].
  * (5) B. J. Barris et al., Astrophys. J. 602, 571 (2004) [arXiv:astro-ph/0310843].
  * (6) A. G. Riess et al. [Supernova Search Team Collaboration], Astrophys. J. 607, 665 (2004) [arXiv:astro-ph/0402512].
  * (7) C. L. Bennett et al., Astrophys. J. Suppl. 148, 1 (2003) [arXiv:astro-ph/0302207];
  * (8) D. N. Spergel et al., arXiv:astro-ph/0603449.
  * (9) Y. Nambu and M. Tanimoto, arXiv:gr-qc/0507057.
  * (10) T. Kai, H. Kozaki, K. i. nakao, Y. Nambu and C. M. Yoo, Prog. Theor. Phys. 117, 229 (2007) [arXiv:gr-qc/0605120].
  * (11) A. E. Romano, Phys. Rev. D 75, 043509 (2007) [arXiv:astro-ph/0612002].
  * (12) D. J. H. Chung and A. E. Romano, Phys. Rev. D 74, 103507 (2006) [arXiv:astro-ph/0608403].
  * (13) C. M. Yoo, T. Kai and K. i. Nakao, Prog. Theor. Phys. 120, 937 (2008) [arXiv:0807.0932 [astro-ph]].
  * (14) S. Alexander, T. Biswas, A. Notari and D. Vaid, “Local Void vs Dark Energy: Confrontation with WMAP and Type Ia Supernovae,” arXiv:0712.0370 [astro-ph]. CITATION = ARXIV:0712.0370;
  * (15) H. Alnes, M. Amarzguioui and O. Gron, Phys. Rev. D 73, 083519 (2006) [arXiv:astro-ph/0512006].
  * (16) J. Garcia-Bellido and T. Haugboelle, JCAP 0804, 003 (2008) [arXiv:0802.1523 [astro-ph]].
  * (17) J. Garcia-Bellido and T. Haugboelle, JCAP 0809, 016 (2008) [arXiv:0807.1326 [astro-ph]].
  * (18) J. Garcia-Bellido and T. Haugboelle, JCAP 0909, 028 (2009) [arXiv:0810.4939 [astro-ph]].
  * (19) S. February, J. Larena, M. Smith and C. Clarkson, Mon. Not. Roy. Astron. Soc. 405, 2231 (2010) [arXiv:0909.1479 [astro-ph.CO]].
  * (20) J. P. Uzan, C. Clarkson and G. F. R. Ellis, Phys. Rev. Lett. 100, 191303 (2008) [arXiv:0801.0068 [astro-ph]].
  * (21) M. Quartin and L. Amendola, Phys. Rev. D 81, 043522 (2010) [arXiv:0909.4954 [astro-ph.CO]].
  * (22) C. Quercellini, P. Cabella, L. Amendola, M. Quartin and A. Balbi, Phys. Rev. D 80, 063527 (2009) [arXiv:0905.4853 [astro-ph.CO]].
  * (23) C. Clarkson, M. Cortes and B. A. Bassett, JCAP 0708, 011 (2007) [arXiv:astro-ph/0702670].
  * (24) A. Ishibashi and R. M. Wald, Class. Quant. Grav. 23, 235 (2006) [arXiv:gr-qc/0509108].
  * (25) T. Clifton, P. G. Ferreira and K. Land, Phys. Rev. Lett. 101, 131302 (2008) [arXiv:0807.1443 [astro-ph]].
  * (26) M. N. Celerier, K. Bolejko, A. Krasinski arXiv:0906.0905 [astro-ph.CO].
  * (27) A. E. Romano, Phys. Rev. D 76, 103525 (2007) [arXiv:astro-ph/0702229].
  * (28) A. E. Romano, JCAP 1001, 004 (2010) [arXiv:0911.2927 [astro-ph.CO]].
  * (29) A. E. Romano, JCAP 1005, 020 (2010) [arXiv:0912.2866 [astro-ph.CO]].
  * (30) A. E. Romano, Phys. Rev. D 82, 123528 (2010) [arXiv:0912.4108 [astro-ph.CO]].
  * (31) N. Mustapha, C. Hellaby and G. F. R. Ellis, Mon. Not. Roy. Astron. Soc. 292, 817 (1997) [arXiv:gr-qc/9808079].
  * (32) A. E. Romano, M. Sasaki and A. A. Starobinsky, arXiv:1006.4735 [astro-ph.CO].
  * (33) G. Lemaitre, Annales Soc. Sci. Brux. Ser. I Sci. Math. Astron. Phys. A 53, 51 (1933).
  * (34) R. C. Tolman, Proc. Nat. Acad. Sci. 20, 169 (1934).
  * (35) H. Bondi, Mon. Not. Roy. Astron. Soc. 107, 410 (1947).
  * (36) D. Edwards, Monthly Notices of the Royal Astronomical Society, 159, 51 (1972).
  * (37) Byrd, P.F. & Friedman, M,D,m1954. Handbook of Elliptic Integrals, Lange, Maxwell and Springer Ltd, London
  * (38) A. E. Romano and M. Sasaki, arXiv:0905.3342 [astro-ph.CO].
  * (39) M. N. Celerier, Astron. Astrophys. 353, 63 (2000) [arXiv:astro-ph/9907206].
  * (40) C. Hellaby, PoS ISFTG, 005 (2009) [arXiv:0910.0350 [gr-qc]].