In our first case under examination and based on Planck observations, ${\Omega }_{k,0}>0$ which corresponds to a slight open Universe in Equation (23), where we extract the first equation of interest for the transverse comoving distance

Alternatively, in order to simplify its mathematical expression, we can write that

where we introduced the main integral of the transverse comoving distance $I_{tc}$, which we know was previously part of the comoving distance, given by

As noticed, we do not neglect the contribution given by the omega density of radiation and by the curvature term, both commonly ignored in modern computational methods due to their small values. This topic will actually define the next approach in the next paragraph when we discuss the other cosmological case. Focusing on our current study case, therefore, by developing the binomials with different powers in the square root at the denominator, Equation (26) results in

$I_{tc}={\displaystyle {\int }_0^z\frac{\text{d}z}{\sqrt{{\Omega }_{r,0}{z}^4+4{\Omega }_{r,0}{z}^3+6{\Omega }_{r,0}{z}^2+4{\Omega }_{r,0}z+{\Omega }_{r,0}+{\Omega }_{m,0}{z}^3+3{\Omega }_{m,0}{z}^2+3{\Omega }_{m,0}z+{\Omega }_{m,0}+{\Omega }_{k,0}{z}^2+2{\Omega }_{k,0}z+{\Omega }_{k,0}+{\Omega }_{\Lambda ,0}}}}$ (28)

Moreover, due to Equation (21), the denominator of Equation (29) changes into

$I_{tc}={\displaystyle {\int }_0^z\frac{\text{d}z}{\sqrt{{\Omega }_{r,0}{z}^4+\left(4{\Omega }_{r,0}+{\Omega }_{m,0}\right)z^3+\left(6{\Omega }_{r,0}+3{\Omega }_{m,0}+{\Omega }_{k,0}\right)z^2+\left(4{\Omega }_{r,0}+3{\Omega }_{m,0}+2{\Omega }_{k,0}\right)z+1}}}.$ (30)

Inserting the values of the omega density parameters of Equations (12), (13), (14) and (15), it yields

$I_{tc}={\displaystyle {\int }_0^z\frac{\text{d}z}{\sqrt{9.173\times {10}^{-5}\times z^4+\left(4\times 9.173\times {10}^{-5}+0.315\right)z^3+\left(6\times 9.173\times {10}^{-5}+3\times 0.315+0.0007\right)z^2+\left(4\times 9.173\times {10}^{-5}+3\times 0.315+2\times 0.0007\right)z+1}}}$ (31)

or rather

We can identify a quartic polynomial in the square root which admits the following roots (determined by different very reliable computational tools available online such as Wolfram Mathematica)

where i is the imaginary number and ${\bar{r}}_3$ is the complex conjugated of $r_4$. Therefore, the main integral of Equation (32) takes now the form

The order of the roots in the parenthesis is not casual but follows the rules of the incomplete elliptic integral of the first order containing two complex roots and one of them complex conjugated [11] , integral 260.00, in which

Thus, the integral of Equation (34) containing the roots of Equation (33) can be written as

$I_{tc}={\displaystyle {\int }_0^z\frac{\text{d}z}{\sqrt{\left[z-\left(-2.297\right)\right]\left[z-\left(-3430.986\right)\right]\left[z-\left(-0.353-1.121i\right)\right]\left[z-\left(-0.353+1.121i\right)\right]}}}.$ (36)

We refer to that specific integral found in the scientific literature for which the solution is provided by the following expression

where $g_{*}$ is a constant associated with further coefficients as function of the polynomial roots whereas the incomplete elliptic integral of the first kind in the interval $\left[0,z\right]$ is

and it assumes exactly this kind of expression as the main solution of the elliptic integral in the literature considers only the following integral limits $\left[r_1,z\right]$. However, our integral extremes lay in-between values. For this reason, we have to subtract the integral contribution in the interval $\left[r_1,0\right]$. Accordingly, Equation (37) becomes

We have to calculate the solution in our desired interval by knowing that

$g_{*}=\frac{1}{\sqrt{AB}},$ (40)

in which the coefficients A and B are computed as follows

$A=\sqrt{{\left[r_1-\frac{r_3+{\bar{r}}_3}{2}\right]}^2-\left[\frac{{\left(r_3-{\bar{r}}_3\right)}^2}{4}\right]},$ (41)

$A=\sqrt{{\left[-2.297-\frac{-0.353-1.121i-0.353+1.121i}{2}\right]}^2-\left[\frac{{\left(-0.353-1.121i-\left(-0.353+1.121i\right)\right)}^2}{4}\right]}$ (42)

which leads to

$A=2.244,$ (43)

and

$B=\sqrt{{\left[r_2-\frac{r_3+{\bar{r}}_3}{2}\right]}^2-\left[\frac{{\left(r_3-{\bar{r}}_3\right)}^2}{4}\right]},$ (44)

$B=\sqrt{{\left[-3430.986-\frac{-0.353-1.121i-0.353+1.121i}{2}\right]}^2-\left[\frac{{\left(-0.353-1.121i-\left(-0.353+1.121i\right)\right)}^2}{4}\right]}$ (45)

which is ultimately

$B=\mathrm{3430.634.}$ (46)

Therefore, Equation 40 becomes

$g_{*}=\frac{1}{\sqrt{2.244\times 3430.634}}=\mathrm{0.017.}$ (47)

The general formulation of the incomplete elliptical integral of the first order, underlining the upper limit $l_{up}$, is

${F\left(\phi ,k_{*}\right)|}_{\left[r_1,l_{up}\right]}={\displaystyle {\int }_0^{{\phi |}_{\left[r_1,l_{up}\right]}}\frac{\text{d}\theta }{\sqrt{1-k_{*}^2{\sin }^2\theta }}},$ (48)

where the Jacobi’s amplitude is given by

${\phi |}_{\left[r_1,l_{up}\right]}=\arccos \left[\frac{\left(A-B\right)l_{up}+r_{1}B-r_{2}A}{\left(A+B\right)l_{up}-r_{1}B-r_{2}A}\right],$ (49)

whereas the elliptic modulus is

$k_{*}=\sqrt{\frac{{\left(A+B\right)}^2-{\left(r_1-r_2\right)}^2}{4AB}}.$ (50)

We can start from the calculation of the latter, as $k_{*}$ has the same value for both intervals in the elliptic integrals ${F\left(\phi ,k_{*}\right)|}_{\left[r_1,z\right]}$ and ${F\left(\phi ,k_{*}\right)|}_{\left[r_1,0\right]}$. Therefore,

$k_{*}=\sqrt{\frac{{\left(2.244+3430.634\right)}^2-{\left[-2.297-\left(-3430.986\right)\right]}^2}{4\times 2.244\times 3430.634}}=\mathrm{0.966.}$ (51)

It is a valid result as a condition to verify is

$-1<k_{*}<1.$ (52)

Based on the logic that we previously discussed in Equation (38), we can start from ${F\left(\phi ,k_{*}\right)|}_{\left[r_1,z\right]}$ so that $l_{up}=z$. Based on the scientific literature, we have to first evaluate

${\phi |}_{\left[r_1,z\right]}=\arccos \left[\frac{\left(A-B\right)z+r_{1}B-r_{2}A}{\left(A+B\right)z-r_{1}B-r_{2}A}\right],$ (53)

${\phi |}_{\left[r_1,z\right]}=\arccos \left[\frac{\left(2.244-3430.634\right)z+\left(-2.297\times 3430.634\right)-\left(-3430.986\times 2.244\right)}{\left(2.244+3430.634\right)z-\left(-2.297\times 3430.634\right)-\left(-3430.986\times 2.244\right)}\right],$ (54)

which eventually leads to

${\phi |}_{\left[r_1,z\right]}=\arccos \left(\frac{-3428.39z-185.123}{3432.878z+15583.387}\right).$ (55)

This expression is exactly responsible for the request of a numerical method as integration of the analytical one. Therefore, the first incomplete elliptic integral of the first order of Equation (48) in the interval $\left[r_1,z\right]$ is given by

${F\left(\phi ,k_{*}\right)|}_{\left[r_1,z\right]}={\displaystyle {\int }_0^{{\phi |}_{\left[r_1,z\right]}}\frac{\text{d}\theta }{\sqrt{1-k_{*}^2{\sin }^2\theta }}},$ (56)

${F\left(\phi ,k_{*}\right)|}_{\left[r_1,z\right]}={\displaystyle {\int }_0^{\arccos \left(\frac{-3428.39z-185.123}{3432.878z+15583.387}\right)}\frac{\text{d}\theta }{\sqrt{1-{0.966}^2{\sin }^2\theta }}},$ (57)

${F\left(\phi ,k_{*}\right)|}_{\left[r_1,z\right]}={\displaystyle {\int }_0^{\arccos \left(\frac{-3428.39z-185.123}{3432.878z+15583.387}\right)}\frac{\text{d}\theta }{\sqrt{1-0.933{\sin }^2\theta }}}.$ (58)

Moreover, considering the remaining incomplete elliptic integral ${F\left(\phi ,k_{*}\right)|}_{\left[r_1,0\right]}$ with $l_{up}=0$, it yields

${\phi |}_{\left[r_1,0\right]}=\arccos \left[\frac{\left(A-B\right)0+r_{1}B-r_{2}A}{\left(A+B\right)0-r_{1}B-r_{2}A}\right],$ (59)

${\phi |}_{\left[r_1,0\right]}=\arccos \left[\frac{\left(-2.297\times 3430.634\right)-\left(-3430.986\times 2.244\right)}{-\left(-2.297\times 3430.634\right)-\left(-3430.986\times 2.244\right)}\right]=1.583\text{rad}.$ (60)

Therefore, the second incomplete elliptic integral of the first order of Equation (48) in the interval $\left[r_1,0\right]$ is provided by

${F\left(\phi ,k_{*}\right)|}_{\left[r_1,0\right]}={\displaystyle {\int }_0^{{\phi |}_{\left[r_1,0\right]}}\frac{\text{d}\theta }{\sqrt{1-k_{*}^2{\sin }^2\theta }}},$ (61)

${F\left(\phi ,k_{*}\right)|}_{\left[r_1,0\right]}={\displaystyle {\int }_0^{1.583}\frac{\text{d}\theta }{\sqrt{1-{0.966}^2{\sin }^2\theta }}}=\mathrm{2.8151.}$ (62)

If we step back to the expression main integral of the transverse comoving distance of Equation (25), in order to determine the value of the incomplete elliptic integral of the first order in the interval $\left[0,z\right]$ in Equation (39), we can write that

$I_{tc}=0.017\times \left[\left({\displaystyle {\int }_0^{\arccos \left(\frac{-3428.39z-185.123}{3432.878z+15583.387}\right)}\frac{\text{d}\theta }{\sqrt{1-0.933{\sin }^2\theta }}}\right)-2.8151\right].$ (63)

Therefore, Equation (25) expressed in meters, eventually becomes

$d_{tc}=\frac{299792458}{2.183\times {10}^{-18}}\frac{1}{\sqrt{0.0007}}\times \sinh \left\{\sqrt{0.0007}\times 0.017\times \left[\left({\displaystyle {\int }_0^{\arccos \left(\frac{-3428.39z-185.123}{3432.878z+15583.387}\right)}\frac{\text{d}\theta }{\sqrt{1-0.933{\sin }^2\theta }}}\right)-2.8151\right]\right\},$ (64)

$d_{tc}=5.19\times {10}^{27}\times \sinh \left\{4.497\times {10}^{-4}\times \left[\left({\displaystyle {\int }_0^{\arccos \left(\frac{-3428.39z-185.123}{3432.878z+15583.387}\right)}\frac{\text{d}\theta }{\sqrt{1-0.933{\sin }^2\theta }}}\right)-2.8151\right]\right\}.$ (65)

In order to evaluate the transverse comoving distance, we have to consider numerically different values of z in order to determine the integral.

Compared to the previous case, we consider in this scenario the following assumption

${\Omega }_{r,0}\cong 0,$ (66)

which is justified by the small observational value and it is basically the main hypothesis made by Carroll [13] . Similarly, ${\Omega }_{k,0}>0$ which corresponds to a slight open Universe. Basically, we are now dealing with one omega density parameter less but we are similarly involving the same mathematics and physics of an open Universe. Based on that, due to Equation (25) the integral of Equation (26) becomes

$I_{tc}={\displaystyle {\int }_0^z\frac{\text{d}z}{\sqrt{{\Omega }_{m,0}{\left(1+z\right)}^3+{\Omega }_{k,0}{\left(1+z\right)}^2+{\Omega }_{\Lambda ,0}}}}.$ (67)

As in this case Equation (21) has one parameter less, it yields

${\displaystyle \underset{j=1}{\overset{3}{\sum }}{\Omega }_{j,0}}={\Omega }_{m,0}+{\Omega }_{k,0}+{\Omega }_{\Lambda ,0}=1,$ (68)

from which we can write that

${\Omega }_{k,0}=1-{\Omega }_{\Lambda ,0}-{\Omega }_{m,0}.$ (69)

According to some in-between algebraic steps, we can exactly reach Carroll’s formula [13] as follows

$I_{tc}={\displaystyle {\int }_0^z\frac{\text{d}z}{\sqrt{{\Omega }_{m,0}{\left(1+z\right)}^3+\left(1-{\Omega }_{\Lambda ,0}-{\Omega }_{m,0}\right){\left(1+z\right)}^2+{\Omega }_{\Lambda ,0}}}},$ (70)

$I_{tc}={\displaystyle {\int }_0^z\frac{\text{d}z}{\sqrt{{\Omega }_{m,0}{\left(1+z\right)}^3+{\left(1+z\right)}^2\left(1-{\Omega }_{m,0}\right)-{\Omega }_{\Lambda ,0}\left[{\left(1+z\right)}^2-1\right]}}},$ (71)

$I_{tc}={\displaystyle {\int }_0^z\frac{\text{d}z}{\sqrt{{\left(1+z\right)}^2\left(1+{\Omega }_{m,0}z\right)-z\left(2+z\right){\Omega }_{\Lambda ,0}}}}.$ (72)

However, our aim is to continue with the algebraic steps in Equation (72) as we want to obtain a polynomial in the variable z with a certain grade in order to be able to discuss the corresponding incomplete elliptic integral. Therefore, substituting the values for ${\Omega }_{m,0}$ and ${\Omega }_{\Lambda ,0}$, respectively of Equation (12) and Equation (15), in Equation (72), it yields

$I_{tc}={\displaystyle {\int }_0^z\frac{\text{d}z}{\sqrt{\left(z^2+2z+1\right)\left(1+0.315z\right)-z\left(2+z\right)0.685}}},$ (73)

which leads to

$I_{tc}={\displaystyle {\int }_0^z\frac{\text{d}z}{\sqrt{0.315z^3+0.945z^2+0.945z+1}}}.$ (74)

This time, we recognize a cubic polynomial in the square root at the denominator which admits the following roots

$\{\begin{array}{l}{r}_1=-2.295\\ r_2=-0.352-1.122i\\ r_3={\bar{r}}_2=-0.352+1.122i\end{array}$ (75)

where i is the imaginary number and ${\bar{r}}_2$ is the complex conjugated of $r_2$. Therefore, the integral of Equation (74) takes now the form

$I_{tc}={\displaystyle {\int }_0^z\frac{\text{d}z}{\sqrt{\left(z-r_1\right)\left(z-r_2\right)\left(z-{\bar{r}}_2\right)}}},$ (76)

$I_{tc}={\displaystyle {\int }_0^z\frac{\text{d}z}{\sqrt{\left[z-\left(-2.295\right)\right]\left[z-\left(-0.352-1.122i\right)\right]\left[z-\left(-0.352+1.122i\right)\right]}}},$ (77)

which leads eventually to

$I_{tc}={\displaystyle {\int }_0^z\frac{\text{d}z}{\sqrt{\left[z-\left(-2.295\right)\right]\left\{{\left[z-\left(-0.352\right)\right]}^2+1.259\right\}}}}.$ (78)

Therefore, the expression of the transverse comoving distance of Equation (25) is now

$d_{tc}=\frac{c}{H_0}\frac{1}{\sqrt{{\Omega }_{k,0}}}\times \sinh \left[\sqrt{{\Omega }_{k,0}}{\displaystyle {\int }_0^z\frac{\text{d}z}{\sqrt{\left[z-\left(-2.295\right)\right]\left\{{\left[\left(z-(-0.352\right)\right]}^2+1.259\right\}}}}\right].$ (79)

Through Equation (78), we obtained exactly the formulation of the incomplete elliptic integral of the first kind [11] , this time corresponding in the literature to integral 239.00, where we can infer, according to the integral terminology, that its coefficients, which will be used for the calculation of the parameters, are the following

$b_1=-0.352,$ (80)

and

$a_1^2=\mathrm{1.259.}$ (81)

The integral admits the same type of solution of Equation (39) with the same reasoning concerning the interval calculations. However, due to the new incomplete elliptic integral of the first kind under examination, we have to calculate the solution in our desired interval by knowing that this time

$g_{*}=\frac{1}{\sqrt{A}},$ (82)

in which the coefficient A can be computed according to the literature as follows

$A=\sqrt{{\left[b_1-r_1\right]}^2+a_1^2},$ (83)

which leads to

$A=\sqrt{{\left[-0.352-\left(-2.295\right)\right]}^2+1.259}=\mathrm{2.244.}$ (84)

From this result, which is the same calculated in the non-flat cosmology case, we can calculate in Equation (82) that

$g_{*}=\frac{1}{\sqrt{2.244}}=\mathrm{0.667.}$ (85)

The incomplete elliptical integral of the first order, underlining the upper limit $l_{up}$, is given by Equation (48) where we know identify different intrinsic parameters of the integral such as the Jacobis amplitude given by

${\phi |}_{\left[r_1,l_{up}\right]}=\arccos \left[\frac{A+r_1-l_{up}}{A-r_1+l_{up}}\right],$ (86)

whereas the elliptic modulus is

$k_{*}=\sqrt{\frac{A+b_1-r_1}{2A}}.$ (87)

As previously done, we can start from the calculation of the latter, as $k_{*}$ has the same value for both intervals in the elliptic integrals ${F\left(\phi ,k_{*}\right)|}_{\left[r_1,z\right]}$ and ${F\left(\phi ,k_{*}\right)|}_{\left[r_1,0\right]}$. Therefore,

$k_{*}=\sqrt{\frac{2.244-0.352-\left(-2.295\right)}{2\times 2.244}}=\mathrm{0.966.}$ (88)

Despite we are dealing with different coefficients, also in this case we calculated the same elliptic modulus which verifies the condition Equation (52). Based on the logic that we previously discussed, we can start from ${F\left(\phi ,k_{*}\right)|}_{\left[r_1,z\right]}$ in Equation (86) so that $l_{up}=z$. It yields

${\phi |}_{\left[r_1,z\right]}=\arccos \left[\frac{A+r_1-z}{A-r_1+z}\right],$ (89)

${\phi |}_{\left[r_1,z\right]}=\arccos \left[\frac{2.244-2.295-z}{2.244-\left(-2.295\right)+z}\right],$ (90)

${\phi |}_{\left[r_1,z\right]}=\arccos \left(\frac{-0.051-z}{4.539+z}\right).$ (91)

Due to Equation (91), the first incomplete elliptic integral of the first kind in the interval $\left[r_1,z\right]$ of Equation (56) is given by

${F\left(\phi ,k_{*}\right)|}_{\left[r_1,z\right]}={\displaystyle {\int }_0^{\arccos \left(\frac{-0.051-z}{4.539+z}\right)}\frac{\text{d}\theta }{\sqrt{1-0.933{\sin }^2\theta }}}.$ (92)

Additionally, considering the remaining incomplete elliptic integral ${F\left(\phi ,k_{*}\right)|}_{\left[r_1,0\right]}$ from Equation (86) with $l_{up}=0$. It yields

${\phi |}_{\left[r_1,0\right]}=\arccos \left[\frac{A+r_1-0}{A-r_1+0}\right].$ (93)

It leads to

${\phi |}_{\left[r_1,0\right]}=\arccos \left[\frac{2.244-2.295}{2.244-\left(-2.295\right)}\right]=1.582\text{rad}.$ (94)

Therefore, the second incomplete elliptic integral of the first order in the interval $\left[r_1,0\right]$ of Equation (61) is provided by

${F\left(\phi ,k_{*}\right)|}_{\left[r_1,0\right]}={\displaystyle {\int }_0^{1.582}\frac{\text{d}\theta }{\sqrt{1-0.933{\sin }^2\theta }}}=\mathrm{2.811.}$ (95)

In order to determine the value of the incomplete elliptic integral of the first order in the interval $\left[0,z\right]$, we can write that Equation (39) becomes

$I_{tc}=0.667\times \left[\left({\displaystyle {\int }_0^{\arccos \left(\frac{-0.051-z}{4.539+z}\right)}\frac{\text{d}\theta }{\sqrt{1-0.933{\sin }^2\theta }}}\right)-2.811\right].$ (96)

If we step back to Equation (25), the expression main integral of the transverse comoving distance in meters becomes now

$d_{tc}=\frac{299792458}{2.183\times {10}^{-18}}\frac{1}{\sqrt{0.0007}}\times \sinh \left\{\sqrt{0.0007}\times 0.667\times \left[\left({\displaystyle {\int }_0^{\arccos \left(\frac{-0.051-z}{4.539+z}\right)}\frac{\text{d}\theta }{\sqrt{1-0.933{\sin }^2\theta }}}\right)-2.811\right]\right\},$ (97)

$d_{tc}=5.19\times {10}^{27}\times \sinh \left\{0.017\times \left[\left({\displaystyle {\int }_0^{\arccos \left(\frac{-0.051-z}{4.539+z}\right)}\frac{\text{d}\theta }{\sqrt{1-0.933{\sin }^2\theta }}}\right)-2.811\right]\right\}.$ (98)

Also in this case, it is necessary to add a numerical analysis to the analytical one, in order to evaluate the transverse comoving distance at each redshift.

A closed Universe implies ${\Omega }_{k,0}<0$ which can be obtained, for instance, subtracting the negative tolerance values from the omega curvature parameter in Equation (14) as follows

${\Omega }_{k,0}=0.0007-0.0019=-\mathrm{0.0012.}$ (99)

In this cosmological case, we extract the third equation from the set of Equation (23)

$d_{tc}=\frac{c}{H_0}\frac{1}{\sqrt{\left|{\Omega }_{k,0}\right|}}\times \sin \left[\sqrt{\left|{\Omega }_{k,0}\right|}\times I_{tc}\right],$ (100)

We are dealing with four omega density parameters which will surely ensure a quartic polynomial in the expression at the denominator of the integral of Equation (26). Accordingly, substituting the new obtained value of Equation (99) into previous Equation (26), it yields

$I_{tc}={\displaystyle {\int }_0^z\frac{\text{d}z}{\sqrt{9.173\times {10}^{-5}\times z^4+\left(4\times 9.173\times {10}^{-5}+0.315\right)z^3+\left(6\times 9.173\times {10}^{-5}+3\times 0.315-0.0012\right)z^2+\left(4\times 9.173\times {10}^{-5}+3\times 0.315-2\times 0.0012\right)z+1}}}$(101)

or rather

$I_{tc}={\displaystyle {\int }_0^z\frac{\text{d}z}{\sqrt{9.173\times {10}^{-5}{z}^4+0.315z^3+0.944z^2+0.943z+1}}}.$ (102)

The quartic polynomial in the square root admits the following roots

$\{\begin{array}{l}{r}_1=-2.297\\ r_2=-3430.99\\ r_3=-0.351-1.122i\\ r_4={\bar{r}}_3=-0.351+1.122i\end{array}$ (103)

where i is the imaginary number and ${\bar{r}}_3$ is the complex conjugated of $r_4$. Therefore, the main integral takes the same form of Equation (34). Moreover, we recognize once again the integral 260.00 [11] , where the condition of Equation (35) is also verified. Thus, we can explicit the main integral $I_{tc}$ of Equation (34) as

$I_{tc}={\displaystyle {\int }_0^z\frac{\text{d}z}{\sqrt{\left[z-\left(-2.297\right)\right]\left[z-\left(-3430.99\right)\right]\left[z-\left(-0.351-1.122i\right)\right]\left[z-\left(-0.351+1.122i\right)\right]}}}.$ (104)

The procedure is identical to the procedure undergone in a non-flat open Universe. However, the values in outcome are not identical. The solution of the main integral is given by Equation (37) in which the incomplete elliptic integral of the first kind in the interval $\left[0,z\right]$ is provided by Equation (38). The constant $g_{*}$ is calculated according to Equation (40). Its coefficients A and B have the same mathematical expressions, respectively, coming from Equation (41) and Equation (44). Going into detail with their calculation, we obtain that

$A=\sqrt{{\left[-2.297-\frac{-0.351-1.122i-0.351+1.122i}{2}\right]}^2-\left[\frac{{\left(-0.351-1.122i-\left(-0.351+1.122i\right)\right)}^2}{4}\right]}$ (105)

which leads to

$A=2.246,$ (106)

as well as

$B=\sqrt{{\left[-3430.99-\frac{-0.351-1.122i-0.351+1.122i}{2}\right]}^2-\left[\frac{{\left(-0.351-1.122i-\left(-0.351+1.122i\right)\right)}^2}{4}\right]}$ (107)

which ends up with

$B=\mathrm{3430.639.}$ (108)

Therefore, from Equation (40), we can calculate that

$g_{*}=\frac{1}{\sqrt{2.246\times 3430.639}}=\mathrm{0.0114.}$ (109)

The incomplete elliptical integral of the first order, underlining the upper limit $l_{up}$, has been already introduced in Equation (48) as well as the Jacobis amplitude in Equation (49) and the elliptic modulus in Equation (50). From the latter, we can calculate that

$k_{*}=\sqrt{\frac{{\left(2.246+3430.639\right)}^2-{\left[-2.297-\left(-3430.99\right)\right]}^2}{4\times 2.246\times 3430.639}}=\mathrm{0.966.}$ (110)

It verifies the condition of Equation (52) and starting with the same calculation logic, from the interval $\left[r_1,z\right]$, we can write from Equation (53) that

${\phi |}_{\left[r_1,z\right]}=\arccos \left[\frac{\left(2.246-3430.639\right)z+\left(-2.297\times 3430.639\right)-\left(-3430.99\times 2.246\right)}{\left(2.246+3430.639\right)z-\left(-2.297\times 3430.639\right)-\left(-3430.99\times 2.246\right)}\right]$ (111)

which leads to

${\phi |}_{\left[r_1,z\right]}=\arccos \left(\frac{-3428.393z-174.174}{3432.885z+15586.18}\right).$ (112)

Due to this, the first incomplete elliptic integral of the first order in the interval $\left[r_1,z\right]$ given by Equation (56), can be calculated as

${F\left(\phi ,k_{*}\right)|}_{\left[r_1,z\right]}={\displaystyle {\int }_0^{\arccos \left(\frac{-3428.393z-174.174}{3432.885z+15586.18}\right)}\frac{\text{d}\theta }{\sqrt{1-0.933{\sin }^2\theta }}}.$ (113)

Moreover, considering the remaining incomplete elliptic integral ${F\left(\phi ,k_{*}\right)|}_{\left[r_1,0\right]}$ with $l_{up}=0$ in Equation (59), it yields

${\phi |}_{\left[r_1,0\right]}=\arccos \left[\frac{\left(-2.297\times 3430.639\right)-\left(-3430.99\times 2.246\right)}{-\left(-2.297\times 3430.639\right)-\left(-3430.99\times 2.246\right)}\right]=1.582\text{rad}$ (114)

Therefore, the second incomplete elliptic integral of the first order in the interval $\left[r_1,0\right]$ in Equation (61) is provided by

${F\left(\phi ,k_{*}\right)|}_{\left[r_1,0\right]}={\displaystyle {\int }_0^{1.582}\frac{\text{d}\theta }{\sqrt{1-0.933{\sin }^2\theta }}}=\mathrm{2.811.}$ (115)

If we step back to the expression of the main integral of the transverse comoving distance of Equation (37), in order to determine the value of the incomplete elliptic integral of the first order in the interval $\left[0,z\right]$, we can write that

$I_{tc}=0.0114\times \left[\left({\displaystyle {\int }_0^{\arccos \left(\frac{-3428.393z-174.174}{3432.885z+15586.18}\right)}\frac{\text{d}\theta }{\sqrt{1-0.933{\sin }^2\theta }}}\right)-2.811\right].$ (116)

Therefore, Equation (100), expressed in meters, becomes

$d_{tc}=\frac{299792458}{2.183\times {10}^{-18}}\frac{1}{\sqrt{\left|-0.0012\right|}}\times \sin \left\{\sqrt{\left|-0.0012\right|}\times 0.0114\times \left[\left({\displaystyle {\int }_0^{\arccos \left(\frac{-3428.393z-174.174}{3432.885z+15586.18}\right)}\frac{\text{d}\theta }{\sqrt{1-0.933{\sin }^2\theta }}}\right)-2.811\right]\right\},$ (117)

$d_{tc}=3.96\times {10}^{27}\times \sin \left\{3.95\times {10}^{-4}\times \left[\left({\displaystyle {\int }_0^{\arccos \left(\frac{-3428.393z-174.174}{3432.885z+15586.18}\right)}\frac{\text{d}\theta }{\sqrt{1-0.933{\sin }^2\theta }}}\right)-2.811\right]\right\}.$ (118)

A numerical analysis is essential to complete the analytical calculation for the transverse comoving distance.

In this special study case, which is very common in the scientific literature, the comoving distance coincides with the transverse comoving distance in the second equation of the set Equation (23). Due to that,

$d_{tc}\equiv d_c=\frac{c}{H_0}{\displaystyle {\int }_0^z\frac{\text{d}z}{\sqrt{{\Omega }_{r,0}{\left(1+z\right)}^4+{\Omega }_{m,0}{\left(1+z\right)}^3+{\Omega }_{k,0}{\left(1+z\right)}^2+{\Omega }_{\Lambda ,0}}}}.$ (119)

However, precisely because we are dealing with a flat cosmology, we can neglect the following omega density parameters

${\Omega }_{k,0}={\Omega }_{r,0}\cong 0,$ (120)

and therefore, Equation (9) assumes the following expression

$d_{tc}=\frac{c}{H_0}{\displaystyle {\int }_0^z\frac{\text{d}z}{\sqrt{{\Omega }_{m,0}{\left(1+z\right)}^3+{\Omega }_{\Lambda ,0}}}}.$ (121)

Due to this, the previous relation of Equation (21) shows only the sum of two single contributions

${\displaystyle \underset{j=1}{\overset{2}{\sum }}{\Omega }_{j,0}}={\Omega }_{m,0}+{\Omega }_{\Lambda ,0}=1,$ (122)

from which, we can write that

${\Omega }_{\Lambda ,0}=1-{\Omega }_{m,0}.$ (123)

Thus, we can express the argument of the square root at the denominator as function only of the omega matter density parameter. Equation (121) reduces to

$d_{tc}=\frac{c}{H_0}{\displaystyle {\int }_0^z\frac{\text{d}z}{\sqrt{{\Omega }_{m,0}{\left(1+z\right)}^3+1-{\Omega }_{m,0}}}},$ (124)

$d_{tc}=\frac{c}{H_0}{\displaystyle {\int }_0^z\frac{\text{d}z}{\sqrt{{\Omega }_{m,0}\left[{\left(1+z\right)}^3-1\right]+1}}},$ (125)

$d_{tc}=\frac{c}{H_0}{\displaystyle {\int }_0^z\frac{\text{d}z}{\sqrt{{\Omega }_{m,0}\left[z^3+3z^2+3z\right]+1}}}.$ (126)

It is only function of the omega density parameter for matter, which has the value of Equation (12), leading to

$d_{tc}=\frac{c}{H_0}{\displaystyle {\int }_0^z\frac{\text{d}z}{\sqrt{0.315z^3+0.945z^2+0.945z+1}}}.$ (127)

We recognize exactly Equation (74) in a non-flat open cosmology. Because of that, we can copy all the coefficients calculated in the previous cosmological case. We consider valid the results of Equation (85) for $g_{\ast }$, Equation (84) for A, Equation (88) for $k_{\ast }$ and Equation (96) for the main integral $I_{tc}$. However, the final result provided by the transverse comoving distance is different from a non-flat open cosmology as there is no more the operator sinh in the formulation. Therefore, Equation ( 127 ), expressed in meters, becomes

$d_{tc}=\frac{299792458}{2.183\times {10}^{-18}}0.667\times \left\{\left[{\displaystyle {\int }_0^{\arccos \left(\frac{-0.051-z}{4.539+z}\right)}\frac{\text{d}\theta }{\sqrt{1-0.933{\sin }^2\theta }}}\right]-2.8112\right\},$ (128)

or rather

$d_{tc}=9.16\times {10}^{25}\times \left\{\left[{\displaystyle {\int }_0^{\arccos \left(\frac{-0.051-z}{4.539+z}\right)}\frac{\text{d}\theta }{\sqrt{1-0.933{\sin }^2\theta }}}\right]-2.8112\right\}.$ (129)

Also in this cosmological case, in order to evaluate the first incomplete elliptic integral of the first order in the parenthesis, we have to consider numerically different values of z in order to determine the integral. Doing this, we can calculate the transverse comoving distance at each redshift and, in turn, the luminosity distance. The plot of the transverse comoving distance and the luminosity distance will be shown in the next chapter.

Starting exactly from the transverse comoving distance, in order to evaluate the first incomplete elliptic integral of the first order in the parenthesis, we have to consider numerically different values of z in order to determine the integral. In this way, we can calculate a specific distance at each redshift and, in turn, also the luminosity distance. The plot of the transverse comoving distance is shown in Figure 1 .

As two curves overlap in the down part of the plot for small values of distance, we can focus on them by reducing the distance scale in the y axis, as shown in Figure 2 .

Once calculated the transverse comoving distance at each redshift, the luminosity distance of Equation (1) is represented by the plot in Figure 3 .

Also in this case, two curves overlap and a reduction of the distance scale on the y axis is require in order to distinguish their values. The related plot is shown in Figure 4 .

The angular diameter distance of Equation (2) is shown in Figure 5 .

Similar to previous reasoning, we reduce the distance scale and we obtain Figure 6 .

The plot of the angular size associated with Equation (3) is shown in Figure 7 . It decreases down to a minimum for than increasing again and it is one of the most important characteristics of standard cosmology.

In this specific case, even three curves overlap for small stances. Once reduced the scale, as shown in Figure 8 , we can clearly underline the difference between these three curves.

All calculations for the plots are resumed in the following Tables 1-4 . Each table represents a specific cosmological scenario analyzed in the previous chapters.