DSolve[{D[R[t],t]^2==H0^2 (OmegaM/R[t]+OmegaL R[t]^2),R[0]==0},R[t],t]
FullSimplify[%[[4]],{H0>0,OmegaL>0,OmegaM>0,t>0}]
                    OmegaM 1/3      3 H0 Sqrt[OmegaL] t 2/3
 Out[3]= {R[t] ->  (------)    Sinh[-------------------]   }
                    OmegaL                   2
R[t_]= R[t] /. % 

* set OL=.694 OM=.306
* let H0=67.9e3/(1e6*pc)
* ? 1/H0/(1e9*year)
    14.40078028934001    !Billion years

* ? 3/2*H0*sqrt(OL)
    .2749731085385721E-17
* ? 1/res/(1e9*year)
    11.52431260173826    !Billion years
* ? (Om/OL)^(1/3)
    .7611214918829410    

r[t_]=.7611 Sinh[t/11.52]^(2/3)

FindRoot[r[t]==1,{t,14}]                                                

Out[20]= {t -> 13.8022}

In[21]:= r[.00038]                                                              

Out[21]= 0.000782662

In[22]:= 2.735/%                                                                

Out[22]= 3494.48
In[23]:= 1/r[.00038] -1                                                         

Out[23]= 1276.69

Remark: T & z are a bit high, because t=.00038 is not 380,000 years after BB
Recall that when light density dominates (at about 50,000 years), R(t) becomes
steeper (t^(1/2) vs t^(2.3)) and so the BB actually happened after t=0, thus
"380,000 years after BB" is t>380,000 years

Better values would be z=1100, T=3000 K
In our model r=1/(1101) happens at t=0.00048

* let rho=OM*(3*H0^2)/(8*pi*Gn)
* ? rho
    .2650024392872864E-26  !(kg/m^3)
* ? rho/mp
    1.584353520686038    !(H atoms/m^3)